Handbook of Cognitive Mathematics 3031039440, 9783031039447

Table of contents :
Preface
Contents
About the Editor
Section Editors
Contributors
Introduction
Cognitive Mathematics
Section I: Mathematics and Cognition
Introduction
Magnani´s Contribution
Zalamea´s Contribution
West´s Contribution
Danesi´s Contribution
Conclusory Remarks
References
1 The Cognitive and Epistemic Value of Mathematics: Making the World Intelligible - The Role of Abduction, Diagrams, and Affor...
Introduction
Mathematics Is Knowledge
Mathematical Constructions as Cognitive Activities
Mathematics as Synthetic A Priori Knowledge
Beyond Metaphysics: Mathematics Generates ``Objective Knowledge´´
Beyond Kant: Seeing Mathematics as the Transformation of the A Priori Determination of Forms of Intuitions in Formal Intuitions
Mathematics Makes Up New ``Principles of Experience´´
The Cognitive Virtues of Mathematics
Mathematics and Ordinary Language
Mathematical Modeling in Science as an Antidote Against the Epistemological Pretensions of Big Data and Deep Learning
Mathematics, Abduction, and Models
Mathematics and Manipulative Abduction
Optical and Unveiling Diagrams in Mathematical Cognition
Mirroring und Unveiling Hidden Properties Through Optical Diagrams
Externalizing Internal Models to Represent and Discover Mathematical Entities: Resolving Conceptual Anomalies
Externalizing Diagrammatic Models to Unveil Imaginary Entities
Abducing First Principles Through Bodily Contact
Non-Euclidean Parallelism
Unveiling Diagrams in Lobachevsky´s Discovery as Affordances: Gateways to Imaginary Entities
One of the Main Mathematical Cognitive Virtues: Not Indicative of Intuition
Conclusion
References
2 Peirce and Grothendieck on Mathematical Cognition: A Merging of the Pragmaticist Maxim and Topos Theory
Introduction
Peirce´s Pragmaticist Maxim (PM)
Peirce´s Views on Mathematics
Grothendieck´s Topos Theory (TT)
Grothendieck´s Views on Mathematics
Merging Pragmaticism (PM) and Topos Theory (TT)
Mathematical Cognition within the Merging of the Four Theories (CT) - (TT) - (TSK) - (PM)
References
3 Linguistic and Visuospatial Chunking as Quantitative Constraints on Propositional Logic
Introduction
Chunking and Its Affordances
Unconscious Versus Conscious Chunking
Anticipatory Logic to Inform Chunking
Semiotic Influences
Icons and Indices as Chunking Devices
Application to Working Memory Genres
Further Advantages of Higher-Level Chunking
Concluding Remarks
References
4 Blending Theory and Mathematical Cognition
Introduction
Metaphor in Mathematics
Mathematics and Language
Blending Theory
Concluding Remarks
References
Section II: Ethnomathematics
References
5 Ethnomathematics and Cultural Identity to Promote Culturally Responsive Pedagogical Choices for Teachers in Early Childhood ...
Introduction
Ethnomathematics
Ethnomathematics and Culturally Responsive Education Makes Sense in Alaska
Cultural Identity and Mathematical Identity
Early Childhood and Elementary Teachers of Mathematics
Conclusion
References
6 Bundles of Ethnomathematical Expertise Residing with Handicrafts, Occupations, and Other Activities Across Cultures
Introduction
Ethnomathematics (and Mathematics)
(First) Definitions of Ethnomathematics
Definitions of Mathematics
About Methodology
(New) Definitions of Ethnomathematics
Mathematics Education and Cultural Context
Bundles of Ethnomathematical Expertise
Within Handicrafts
Braiding and Weaving Crafts
Basketry Craft
Wooden Sculptures
Within Occupations
Street Markets
Bus Workers
Masonry
Within Other Activities
Dance
Space Organization
Final Reflections
References
7 Emic, Etic, Dialogic, and Linguistic Perspectives on Ethnomodeling
Introduction
Ethnoscience as the Relation Between Humanity and Its Sociocultural Context
The Need for a More Culturally Bound Perspective on Mathematical Modeling
Cultural and Cognitive Features of Ethnomodeling
Ethnomodeling and the Cultural Aspects of Mathematics
Linkage Between Ethnomodeling and Ethnoscience
Cultural Components of Ethnomodels
Ethnomodeling of Landless Peoples´ Movement: Wood Cubing in Brazil
An Ethnomodel of Wood Cubing
The Dialogic (Emic-Etic) Approach in Ethnomodeling Research
An Ethnomodeling Perspective in the Mathematics Curriculum
Conclusions
References
8 Ethnomathematics in Education: The Need for Cultural Symmetry
Introduction
Ethnomathematics and Education
Cultural Symmetry: Blending Ethnomathematics into Mathematics Education for Indigenous Students
Cultural Symmetry Examples
Wharenui/Meeting House/Longhouse
Orientation in Space
Waka Migration
Show-and-Tell Software for Enhancing the Teaching of Māori Language, Māori Knowledge, and Mathematics
Conclusion
References
9 Ethnomathematics Affirmed Through Cognitive Mathematics and Academic Achievement: Quality Mathematics Teaching and Learning ...
Introduction
Ethnomathematics Defined
Connection Between Learning Theories and Ethnomathematics
Connection Between Cognitive Mathematics and Ethnomathematics
Connections Between Pedagogy and Ethnomathematics
Connections Between Culturally Responsive Teaching and Ethnomathematics
Connection Between Instruction and Ethnomathematics
Instruction Foundation
Recognize and Honor Students´ Cultural Experiences
Classroom Environment
Teaching and Learning
Establishing Cultural Experiences
Concrete Real-World Ethnomathematics Approaches
Metacognition
Math Groups
Differentiation
Assessment
Curriculum
Benefits of Ethnomathematics Teaching
Ethnomathematics Enhances Communication
Ethnomathematics Elevates Guided Inquiry
Ethnomathematics Emphasizes Pride in Cultural Identity
Ethnomathematics Esteems Cultural Knowledge
Ethnomathematics Empowers Engagement and Motivation
Ethnomathematics Enriches Academic Achievement
Ethnomathematics Obstacles Explained
Ethnomathematics Philosophy Elucidated to Teachers
Ethnomathematics Expanded in the Future
Conclusion
References
Section III: Cognitive Neuroscience of Mathematics
References
10 Developmental Brain Dynamics: From Quantity Processing to Arithmetic
General Introduction
The Foundation: Representations of Quantities and Numerical Order
The Representation of Numerical Quantities
The Approximate Number System (ANS)
The Object Tracking System
The Numerical Meaning of Symbols
Mapping Numerical Symbols onto Quantities?
Additional Challenges to the Mapping Account
Mapping Symbols to Symbols: The Case of Numerical Order
Arithmetic
Conclusions
References
11 Neurocognitive Foundations of Fraction Processing
Introduction
Key Findings from Behavioral Research on Fraction Processing
Eye-Tracking Research on Fraction Processing
Neuroscientific Research on Symbolic and Non-symbolic Fraction Processing
fMRI Studies on Fraction Magnitude Processing
EEG Studies on Fraction Magnitude Processing
fMRI Studies on Fraction Processing Not Specific to Magnitude
A Tentative Temporal Model of Fraction Processing
Conclusion
References
12 Individual Differences in Mathematical Abilities and Competencies
Introduction
From a Categorical to a Dimensional Approach
The Importance of Domain-General Cognitive Factors as Sources of Individual Differences
Working Memory and Mathematical Achievement
Executive Functions and Attentional Control in Mathematical Learning
Beyond the Purely Cognitive: Metacognition, Affect/Beliefs, and Motivation as Sources of Individual Differences
Metacognitive Abilities
Negative and Positive Attitudes
Motivation
From Cultural and Language Differences to Contextual Factors
Linguistic Factors
Contextual Factors: From Parental Support to Educational Systems
Conclusion
References
13 Mind, Brain, and Math Anxiety
Introduction
What Is Math Anxiety?
How Is Math Anxiety Identified?
Who Develops Math Anxiety?
Understanding Math Anxiety Through General Anxiety
Theoretical Background: Cognitive Interference Theory
Theoretical Background: Processing Efficiency Theory and Attentional Control Theory
Math Anxiety, Processing Efficiency Theory, and Attentional Control Theory
Neuroimaging, Math Anxiety, Processing Efficiency Theory, and Attentional Control Theory
Math Anxiety and Emotional Responses
Interventions and Emotion Regulation in Math Anxiety
Math Anxiety and Mathematical Cognition
Math Anxiety and Numerical Processing
Interventions and Math Competency in Math Anxiety
Conclusions
References
14 Neurocognitive Interventions to Foster Mathematical Learning
Introduction
Prevention
Prediction of Arithmetical Skills in School by Early Numerical Competencies
Does Training of Early Numerical Skills Prevent Later Math Problems?
Longitudinal Outcome of Early Prevention Programs
Developmental Dyscalculia
Behavioral Interventions to Foster Mathematical Learning
Behavioral Interventions for Math Learning
Effects of Numerical Interventions on the Brain
Brain Stimulation to Foster Mathematical Learning
Prospects
Conclusion
References
Section IV: Biological Approaches to Mathematics
Introduction
15 The Neurobiological Basis of Numerical Cognition: Decision-Making Processes as a New Line of Inquiry
Introduction
Domain-Specific Mechanisms Underlying Numerical Cognition
Numerical Processing Skills and Math Performance
Numerical Processing Skills and Mathematical Learning Disabilities
Domain-General Mechanisms Underlying Numerical Cognition
Working Memory and Numerical Cognition
Visual Form Perception and Numerical Cognition
Inhibition and Numerical Cognition
Emotion and Numerical Cognition: Math Anxiety
Interim Summary
Decision-Making and Numerical Cognition
Conclusion
References
16 Trusting Your Own Eyes: Visual Constructions, Proofs, and Fallacies in Mathematics
Introduction
The Faculty of Vision and Development of Mathematical Thinking
Vision and Imagery Development
Visualization and Conceptualization in Mathematics
Visual Fallacies
Visual Constructions
Geometrical Constructions Related to Art and Architecture
Geometrical Constructions as Solutions of an Interconnecting Problem
Gradual Geometrical Constructions in Dynamic Geometry Environments
Visual Reasoning
Visualization in Non-Euclidean Settings
Conclusion
References
17 Numerical Abilities in Nonhumans: The Perspective of Comparative Studies
Introduction
Historical Background of Animal Numerical Abilities
Proto-numerical and Numerical Discrimination
Arithmetic Abilities
Spatial Numerical Association
Nonhuman Animals Associate Numbers with Space
Ordinality
Number-Space Association in Magnitude-Estimation Tasks
Conclusions
References
18 Executive Dysfunction Among Children with ADHD: Contributions to Deficits in Mathematics
Introduction
What Is ADHD?
ADHD Diagnostic Criteria and Subtypes
Functional Impairments Associated with ADHD
Findings from Structural and Functional Imaging Studies
ADHD as a Neurodevelopmental Disorder
Executive Functioning in ADHD and Relations to Math Achievement
Working Memory
Behavioral Inhibition
Set-Shifting
Interventions for ADHD-Related Executive Function and Math Deficits
Behavioral Interventions
Psychostimulant Medication
Cognitive Training Programs
Direct Math Instruction
Conclusion
References
Section V: Mathematics and the Arts
Introduction
19 The Challenge of Formal Logics and Metaphysical Systems to Semiotics
Introduction
Magic Formulas
Formal Logic
From Lévi-Strauss to the Paris School
Deontic Logic
Rudolf Carnap and Vienna Circle
Principles of Formalization: Peirce
Cybernetics
Some Words About Digitalization
Formalization in Music and Its Existential Analysis
Conclusion: Advantages and Disadvantages
References
20 Cultural Symmetry: From Group Theory to Semiotics
Introduction
Symmetry and Cognition: From Harmony to Invariance
Group Theory, Finite Designs, and Plane Patterns
Cultural Symmetry and Plane Pattern Analysis
Symmetries of Culture: Establishing the Field in Group Theory and Mathematics
Symmetry Comes of Age: Opening the Field to Metaphor and Cognition
Embedded Symmetries: Expanding the Field to Semiotics and Cross-Disciplinary Perspectives
Semiotic Expansions, Embodied Oppositions, and the Anatomical Planes
Chiastic Cognition and the In-Between: From Alteroception to Ritual
Conclusion
References
21 Fractals, Narrative, and Cognition
Introduction
What Are Fractals?
What Is the Monomyth?
On the System of Functions and the Procession of Archetypes
Monomyth as Universal Structure
Self-Similarity and Narrative
Propp´s Recursive Morphology
Palumbo, Monomyth, and Chaos Theory
Bloom, Kabbalah, and the Fractality of the Trace
Self-Similarity and Cognition
Re-Entry, Catalysis, and Emergence
Hofstadter´s Strange Loops
Recursion, Storytelling, and Thought
Zunshine´s Account of Theory of Mind and Fictional Consciousness
Long-Range Correlation and Stream of Consciousness
Conclusion: Strange Loops, Strange Attractors
References
22 Visage Mathematics: Semiotic Ideologies of Facial Measurement and Calculus
Introduction
Mathematics, Patterns, and Semiotics
Patterns, Calculus, and Measuring
The Human Body
The Human Face
Measuring Faces, Ranking Races
Measuring Devices
New Facial Angles
The Power of Facial Numbers
Conclusion
The Mismeasurement of Measures
References
Section VI: Learning and Teaching Mathematics
Conducive Mathematics Learning Environments
Conclusion
References
23 The Roles of Intelligence and Creativity for Learning Mathematics
Introduction
History, Definition, and Models of Intelligence
History, Definition, and Models of Creativity
Person
Process
Product
Press or Place
Models Integrating Intelligence and Creativity
The Special Case of Mathematical Creativity
Models of Mathematics Learning and Development
The Influence of Intelligence and Creativity for Learning Mathematics
Typically Developing Individuals
Individuals with Learning Difficulties in Mathematics
Mathematically Gifted Individuals
Fostering Intelligence and Creativity
References
24 Gestures in Mathematics Thinking and Learning
Introduction
Fundamental Background on Gesture Studies
Gestures as Characterized by Several Dimensions
Gesture-Speech Tight Relationship and Implications for Learning
The Role of Gestures in Thinking
The Gesture Revolution in Mathematics Education Research
Gestures as Semiotic Resources
Embodiment and Multimodality
Metaphorical Thinking in Mathematics
Blending Cognitive Mechanisms and Gestures
The Phenomenological Lens Behind Some Cognitive Mechanisms in Mathematics Problem-Solving
Gestures and Digital Technology
Gestures Beyond Mainstream Mathematics Education
Bi- and Multilingual Learners of Mathematics
Sensory-Diverse Students
Conclusions
Focusing on Gesture Beyond the Hegemony of Speech
Gestures in Their Self-Directed and Cognitive Function
Gestures as Resources Purposefully Used by Teachers
Gestures and Distance Teaching-Learning
Gestures and Augmented Reality
The Specificities of Nontypical Students´ Learning Processes
References
25 Teaching and Learning Authentic Mathematics: The Case of Proving
Introduction
Theoretical Perspective
Authentic Classroom Mathematical Activity
Proving as a Mathematical Activity and a Classroom Activity
Classroom Proving as an Authentic Mathematical Activity
The Activity of Proving in Authentic Mathematics
Disciplinary Mathematics and University Mathematics Classrooms
School Mathematics Classrooms
An Episode from Lampert (1990)
An Episode from Herbst (2002a)
Comparison of the Two Episodes
The Role of Proving in Gaining Conviction in Authentic Mathematics
Proving as Convincing in Professional Mathematical Practice
An Episode from Ball and Bass (2008)
Description
Commentary
Students´ Standards of Conviction and Their Relation to Proof
The Role of the Interplay Between Proving and Refuting in Knowledge Growth in Authentic Mathematics
Lakatos-Style Mathematical Activity
An Episode from Komatsu (2017)
Description
Commentary
Mathematics Education Research Related to Lakatos-Style Mathematical Activity
Conclusion
References
26 Why Are Learning and Teaching Mathematics So Difficult?
Introduction
Part 1. The Nature of Mathematical Thinking
What Matters in Mathematical Thinking and Problem-Solving?
Mathematical Resources (Including Content, Processes, and Practices)
Problem-Solving Strategies
Metacognition: Monitoring and Self-Regulation
Belief Systems
Part 2. The Learning Environment
What Is ``Ambitious Instruction´´ or ``Teaching for Robust Understanding´´?
The Teaching for Robust Understanding (TRU) Framework
Part 3. The Cultural Surround
Barriers to Progress
Issues of Curriculum and Testing
Issues of Teacher Support
Discussion
Conclusion
References
Section VII: Mathematics Education and New Technologies
References
27 Computing in Mathematics Education: Past, Present, and Future
Introduction
Theoretical Framework
Language, Thought, Development, and Mediation in Vygotsky´s Perspective
The Role of Technologies in the Production of Mathematical Knowledge
Experimentation and Simulation with Technologies in Mathematics
The Reorganization of Thought in the Collective Human-with-Media
Computing in Education: The Logo Movement
Logo as a Landmark of the Insertion of Computing in Education and the Main Ideas Defended by Papert and His Collaborators
International Overview of the Implementation of Computing in Education
Research in Brazil on Computing in Mathematics Education
Robotics in Educational Contexts
Current Trends in Computing in Education
A Historical Path from the Perspectives of Computational Thinking
Characteristics of Computational Thinking
Research on Computational Thinking
Computational Thinking in Kindergarten and in Early Years (Elementary School)
Computational Thinking in Middle School and in High School
Conclusion
References
28 Computer Algebra Systems and Dynamic Geometry for Mathematical Thinking
Introduction
CAS and DGE Contributions to Mathematical Processes, Algebra, and Geometry Education
DGE and CAS in the Development of Mathematical Reasoning and Modeling Processes
Mathematical Modeling
Mathematical Reasoning
Ways of Conceptualizing CAS in Educational Settings
CAS and DGE in Algebra and Calculus Education
DGS in Geometry Education
Discussion and Concluding Remarks
References
29 Student Collaboration in Blending Digital Technology in the Learning of Mathematics
Introduction
Theoretical Background
Vygotsky´s Social Constructivist Learning Theory
Push and Pull in Education
Heutagogy - Self-Determined Learning
The Expanding Classroom
Blended (or Hybrid) Learning
Distance Learning - How Does It Differ?
Blended (Hybrid) Learning Distinctions
Issues, Features, and Examples of Collaboration in Blended Learning
Humans-with-Media
MOOCs
Communities of Practice and Learning Environments
Communities of Practice
Open Learning Networks
Learning Management Systems (LMS)
Personal Learning Environments (PLE)
Mash-ups and Mupples
Hyperpersonalization of Learning
Use of Social Media
What Has Changed?
How Might Social Media Best Be Used?
How Valuable Is Social Media for Learning Mathematics
The Role of Collaboration in Online Assessment
Conclusion
References
30 Multimodality, Systemic Functional-Multimodal Discourse Analysis and Production of Videos in Mathematics Education
Introduction
Video Production in Distance Education
Systemic Functional-Multimodal Discourse Analysis
Semiotic Resources in Digital Mathematical Discourse
Methodology and Procedures for Video Research
Analysis of the ``Civil Construction´´ and ``Practical Use of Analytical Geometry´´ Videos
Discussion
Conclusion
References
31 STEAM and Critical Making in Teacher Education
Introduction to STEAM
STEAM and Critical Making in Teacher Education
History of STEM Movements
The Rise of STEM/STEAM Education
Engineering Design Processes in Mathematics
STEAM Education
The ``M´´ in STEAM
The ``T´´ for Maker Education Pedagogies
Integrated STEAM for Teachers
Maker Pedagogy Frameworks
Critical Making in Mathematics Education
Constructionism and Low-Floor, High-Ceiling Learning Theories
Humans-with-Media
STEAM Teacher Education Models
Case 1. Making in Mathematics for Elementary Preservice Teachers
Case 2. Critical Making in a Preservice Teacher Education Program
Case 3. Music Production in Mathematics Teacher Education
STEAM Education Affordances
STEAM Cognition
STEAM Equity: The Critical Part
Concluding Remarks
References
Section VIII: Mathematics and Computer Science
32 Memory Consolidation: Neural Data Analysis and Mathematical Modeling
Introduction
Sleep and Memory
Neural Data Analysis
Reactivation with Coincident Neural Activity and Sequential Replay
Principal Component Reactivation Analysis of Motor Memory
Mathematical Modeling
Detailed Modeling
Hodgkin and Huxley Model
Compartmental Model
Abstract Modeling
McCulloch and Pitts Model
The Hopfield Model
Storage Capacity of the Hopfield Nets
Extension of the Hopfield Nets
Asymmetric Connections
Dilution
Sparse Patterns
Temporal Sequences
Boltzmann Machine
Speeding Up the Boltzmann Learning
Restricted Boltzmann Machine for Parallel Learning
Belief Net
Conclusion
References
Section IX: Mathematics and Linguistics
33 Mathematical Linguistics and Cognitive Complexity
Introduction
Mathematical Theories of Language and Cognition
Formal Language Theory and Cognitive Theories of Language
The Chomsky Hierarchy
Beyond the Chomsky Hierarchy: Subregular Languages
Formal Theories of Grammar Learning
Membership Problems
Enumeration and Universal Grammar
Grammar Identification in the Limit
Learning K-Strictly Local and K-Strictly Piecewise Languages
Cognitive Lessons from Learning Theory
Testing Formal Predictions with Artificial Grammar Learning
Conclusion
References
34 Quantifying Context With and Without Statistical Language Models
Introduction
Defining Context
Sources of Linguistic Data
Granularity and Tokenization
Discrete and Latent Definitions of Linguistic Context
Rule-Based Representations
Statistical n-gram Models of Context
Estimating Unseen Language
N-grams and Conditional Probability-Based Definitions of Context
Contexts as Vectors
Sparse Vector Representations
Dense Vector Representations
Sequence Encoders and Contextual Representations
Conclusion
References
35 Cognitive Models of Poetry Reading
Introduction
Unusual Syntactic Constructions and Cognitive Models
Shakespeare´s Sonnets and Literary Criticisms
A Historical Perspective on Inversion and Early Modern English
The Sonnets and Noncanonical Syntactic Constructions
Fronting and Inversion: A Thorough Study of Argument Focusing in the Sonnets
Enjambments and Semantic Constraints
Computing Complexity for Popularity
SPARSAR Reads and Recites
Conclusion
References
Works Consulted on Shakespeare´s Poetry, Style, and Grammar
36 Cognition and Computational Linguistic Creativity
Introduction
Computational Creativity Theories
Optimal Innovation: A Cognitive Hypothesis
Implementing Optimal Innovation: An Artificial Intelligence Challenge
Dealing with Creative Language and Images: The Subvertiser System
Algorithm
Heady-Lines: Generation of Catchy News Headylines
Architecture
Selecting Keywords
Selecting a Well-Known Expression
Selecting the Final Headline
Heady-Lines Evaluation
Results and Discussion
Mockingbird: Optimal Innovation and Songs Through Lyrics Parodies
Corpus
System Architecture
Key Concept Extraction and Expansion
Lyrics Modification
Mockingbird Evaluation
Results and Discussion
Conclusions
References
37 Understanding Dialogue for Human Communication
Introduction
Characteristics of Human Dialogue
Turns, Utterances, and Dialogue Coherence
From Speech Acts to Dialogue Acts
Principle of Cooperation
Dialogue and Grounding
Cognitive Features of Dialogue
Types of Dialogue
Dialogue Systems
Early Dialogue Systems
Initiative in User-System Interaction
Subdialogues
Dialogue and Domain Knowledge
Dialogue Collections
Human-Human Dialogue Collection
Human-Machine Dialogue Collection
Task-Oriented Dialogue Systems
Natural Language Understanding
Dialogue Management
Dialogue State Tracking (DST)
Dialogue Policy
Natural Language Generation
Challenges and Future Directions
Portability
Robustness
Persona-Based and Empathic Dialogue Systems
Ethical Issues
Conclusion
References
Section X: Mathematics Cognition, Semiotics, and Hermeneutic Theories
38 Peirce on Abduction and Diagrams in Mathematical Reasoning
Introduction
Peirce´s Tiffany Watch
Abductive Reasoning
Mathematics as the Epitome of Abduction
Diagrams and Abductive Reasoning
Diagramming the Parallel Postulate
Peirce and Non-Euclidean Geometry
Peirce and Nonclassical Logic
Moving Pictures of Thought
Representation and Semeiotic
Diagrammatic Logic
Inquiry and Ingenuity
Modeling and Analysis
Practical and Ethical Reasoning
Guessing Right
Conclusion
References
39 Pragmaticism as a Philosophy of Cognitive Mathematics
Introduction
Pragmatism and Pragmaticism
What (Cognitive) Mathematics Should Not Be
Pragmaticism as a Logical Study of Mental and Cognitive Phenomena
Synechism: The Motivation of Pragmaticism
The Experiential Content of Mathematics
Pragmaticism Not Falling Prey to the ``Ten misconceptions of actual mathematics´´
The Reproducible Properties of Mathematics
The Origins of Three Kinds of Mathematical Experiences
The Experiential Content of Geometry, Analysis, and Algebra
The Framing of Mathematical Hypotheses
Fallibility and Error in Mathematical Knowledge
Definitions, Mental Models, and Forms of Relations
Pragmaticism as the Theory of Real Definitions
Pragmaticism and Mental Models
Pragmaticism Not a Psychological Theory
``Forms of Relations´´ and Pragmaticist Philosophy of Mathematics
Peirce´s Pragmaticism as a Philosophy of Cognitive Mathematics
Conclusions
Appendix. Which Philosophy of Mathematics Pragmaticism is Not
Pragmatism
Logicism
Axiomatic Program
Intuitionism
Platonism
Structuralism
Quasi-Empiricism
Coda
References
40 Diagrammatic Mathematics
Introduction
From Parentheses to Cap Forms, Topology, and More
Celtic Knots, Temperley-Lieb Algebra, Braids, Knots, and Categories
Conclusion
References
41 Peirce on Mathematical Reasoning and Discovery
Introduction
Abstraction and Generalization
Abstraction in Mathematics
Generalization in Mathematics in Logical Terms
Three Kinds of Reasoning in Mathematics: Abduction, Deduction, Induction
What Is Deduction?
Corollarial and Theorematic Deductions
Theorematic Proofs: Creativity and Invention in Mathematics
Probable Deductions
Images, Icons, and Metaphors
Ampliative Aspects of Mathematical Reasoning
Classification of Mathematical Inferences
Abduction in Mathematics
Induction in Mathematics
Practice of Mathematics and Diagrammatic Reasoning
Some Ramifications of Diagrammatic Reasoning to Cognitive Science
Mental Pictures, Diagrams, and Visual Bias
Diagrammatic Reasoning and the Dual-Process Theories
Philosophy of Mathematical Notations
Conclusions
References
42 Knowing by Drawing: Mathematics as Gesture
Introduction
Gesture: From Gesticulation to Mathematics
Diagrams, Charles S. Peirce, and Semiotics
The Existential Graphs as Gesture
Gesture: A New Definition
Advanced Studies: Zalamea´s Model
Education by Gesture
Conclusions
References
Index

Citation preview

Marcel Danesi Editor

Handbook of Cognitive Mathematics

Handbook of Cognitive Mathematics

Marcel Danesi Editor

Handbook of Cognitive Mathematics With 264 Figures and 34 Tables

Editor Marcel Danesi Anthropology, Victoria College University of Toronto Toronto, ON, Canada

ISBN 978-3-031-03944-7 ISBN 978-3-031-03945-4 (eBook) https://doi.org/10.1007/978-3-031-03945-4 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland.

Preface

Mathematics is both a creation and discovery of the human mind. Research has been showing that mathematical discoveries in general seem to be located in the same neural circuitry that sustains ordinary language and other symbol systems. Mathematics makes sense when it has meanings that fit our cognitive experiences of the world – experiences of quantity, space, motion, force, change, mass, shape, probability, self-regulating processes, and so on. The inspiration for new mathematics comes from these experiences. Studying the relation between cognition and mathematics comes under a general rubric – which can be called the cognitive science of mathematics, or cognitive mathematics for short. This handbook is intended as a reference volume for cognitive mathematics, which can be defined more concretely as the study of the relation of mathematics to other human faculties, from the arts to language. This is of interest to diverse disciplines, spanning education to computer modeling. Therefore, this handbook includes the background work in the field and sections dealing with the interconnections between mathematics and other faculties. The chapters are written by internationally renowned authors who are authorities in their fields. An overall discussion of the volume’s focus and areas covered is provided in the introduction to the handbook. In contemporary academia, the question of what is mathematics and how it is learned has been addressed through a variety of methods in a broad set of disciplines. Building on some of the developments in the different fields, the chapters in this volume, when considered cumulatively, discuss how mathematics involves a blend of imagination, abstraction, and notation-making processes – all of which have profound implications for the teaching and learning of mathematics. In his 2008 collection of studies on mathematical cognition (Mathematical Cognition, Charlotte: Information Age Publishing), James Royer pointed out that the study of mathematical cognition spanned a broad field of scientific, educational, and humanistic interests. The verity of this observation is being constantly evidenced in the studies published in diverse journals, which show the truly wide range of approaches to mathematics cognition, bringing together researchers from diverse fields to cast shreds of light on this truly fascinating phenomenon from different angles. The main implication that can be drawn from looking over works in the field is that mathematics cognition cannot be studied within the confines of a single discipline. The main focus seems to be discerning and explaining the neural basis of v

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Preface

mathematics. This handbook aims to cast a wide net to the study of mathematical cognition, bringing together scientists and humanists, so as to enlarge the purview. While the scientific study is fundamentally empirical, the addition of more humanistic disciplines to the mix allows the field to enter into more theoretical and speculative domains, thus perhaps both opening up a debate that is not based on empirical issues alone but on a more speculative ground – thus mirroring the origins of mathematics in philosophy. The study of mathematical cognition really took off after the publication of Lakoff and Núñez’s controversial 2000 book, Where Mathematics Comes From, which argued that mathematics is essentially no different from language or other symbol systems, since both share a basic modality – blending information from different parts of the brain to produce novel information. One simple verification of this is the fact that we use language to learn mathematics and that mathematics has many structural properties that are linguistic. The most salient manifestation of blending can be seen in metaphor, which undergirds how we think and learn. The question that Lakoff and Núñez asked was how metaphorical processes allow us to understand complex concepts such as infinity and limits. If metaphor is indeed at the core of mathematical reasoning, then it brings mathematics directly into the sphere of culture where it is shaped symbolically and textually. Whatever the truth, it is obvious that the study of mathematical cognition will benefit significantly by the participation of humanists and mathematicians in collaboration with the empirical scientists. In his groundbreaking 1962 study on the cognitive source of scientific theories, the American philosopher Max Black argued, before Lakoff and Núñez, that the genesis of theoretical notions and frameworks in the sciences and mathematics was not solely the result of scientists deducing them from empirical observations or experimental results, but also, and primarily, the result of scientists making inferences and connections between facts, other theories, and even everyday experience. Indirectly, Black laid the foundations for cognitive mathematics with his radical idea for the era in which it was written. In the ancient world, actually, the idea that mathematics was a part of a broader system of thought was a virtual given. The transition of mathematics from a practical counting, measuring, and generic problem-solving craft to a theoretical discipline is traced generally to the emergence of the method of proof. The chapters of this handbook span the interdisciplinary scope of cognitive mathematics, from the empirical to the educational and speculative, as well as examining aspects of mathematical methods and what they tell us about the nature of mathematical cognition. The objective here is twofold: to show how this line of inquiry can be enlarged profitably through an expanded pool of participating disciplines and to shed some new light on math cognition itself from within this pool. We hope that readers will find this handbook informative. We believe that mathematicians, cognitive scientists, educators of mathematics, philosophers of mathematics, and all the other kinds of scholars who are interested in the nature of mathematics will find something of interest in this volume. The implicit claim in all the studies is that in

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order to penetrate the phenomenon of mathematics, it is necessary to utilize methods and theoretical frameworks derived from a variety of disciplines. Toronto, Canada September 2022

Marcel Danesi

Contents

Volume 1 Section I Mathematics and Cognition . . . . . . . . . . . . . . . . . . . . . . . . Donna E. West 1

2

3

4

The Cognitive and Epistemic Value of Mathematics: Making the World Intelligible – The Role of Abduction, Diagrams, and Affordances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lorenzo Magnani

6

9

Peirce and Grothendieck on Mathematical Cognition: A Merging of the Pragmaticist Maxim and Topos Theory . . . . . . . . . . . . . . . . Fernando Zalamea

49

Linguistic and Visuospatial Chunking as Quantitative Constraints on Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Donna E. West

67

...............

89

Blending Theory and Mathematical Cognition Marcel Danesi

Section II Ethnomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Myrdene Anderson and Tod Shockey 5

1

Ethnomathematics and Cultural Identity to Promote Culturally Responsive Pedagogical Choices for Teachers in Early Childhood and Elementary Education . . . . . . . . . . . . . . . . . . . . . . Sandra Wildfeuer Bundles of Ethnomathematical Expertise Residing with Handicrafts, Occupations, and Other Activities Across Cultures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Veronica Albanese

111

115

129

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Contents

7

8

9

Emic, Etic, Dialogic, and Linguistic Perspectives on Ethnomodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Milton Rosa and Daniel Clark Orey

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Ethnomathematics in Education: The Need for Cultural Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tamsin Meaney, Tony Trinick, and Piata Allen

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Ethnomathematics Affirmed Through Cognitive Mathematics and Academic Achievement: Quality Mathematics Teaching and Learning Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jenni L. Harding

Section III Cognitive Neuroscience of Mathematics . . . . . . . . . . . . Roland H. Grabner 10

Developmental Brain Dynamics: From Quantity Processing to Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephan E. Vogel

11

Neurocognitive Foundations of Fraction Processing . . . . . . . . . . . . Silke M. Wortha, Andreas Obersteiner, and Thomas Dresler

12

Individual Differences in Mathematical Abilities and Competencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sara Caviola, Irene C. Mammarella, and Denes Szűcs

13

Mind, Brain, and Math Anxiety . . . . . . . . . . . . . . . . . . . . . . . . . . . Rachel Pizzie

14

Neurocognitive Interventions to Foster Mathematical Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Karin Kucian and Roi Cohen Kadosh

Section IV Biological Approaches to Mathematics . . . . . . . . . . . . . Dan Vilenchik 15

16

221

251

257 289

317 349

385

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The Neurobiological Basis of Numerical Cognition: Decision-Making Processes as a New Line of Inquiry . . . . . . . . . . Lital Daches Cohen and Orly Rubinsten

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Trusting Your Own Eyes: Visual Constructions, Proofs, and Fallacies in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Margo Kondratieva

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Contents

17

18

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Numerical Abilities in Nonhumans: The Perspective of Comparative Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rosa Rugani and Lucia Regolin Executive Dysfunction Among Children with ADHD: Contributions to Deficits in Mathematics . . . . . . . . . . . . . . . . . . . . Lauren M. Friedman, Gabrielle Fabrikant-Abzug, Sarah A. Orban, and Samuel J. Eckrich

Section V Mathematics and the Arts . . . . . . . . . . . . . . . . . . . . . . . . Stéphanie Walsh Matthews 19

The Challenge of Formal Logics and Metaphysical Systems to Semiotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eero Tarasti

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501

539

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Cultural Symmetry: From Group Theory to Semiotics . . . . . . . . . Jamin Pelkey

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21

Fractals, Narrative, and Cognition . . . . . . . . . . . . . . . . . . . . . . . . . Richard Rosenbaum

595

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Visage Mathematics: Semiotic Ideologies of Facial Measurement and Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Massimo Leone

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Volume 2 Section VI Learning and Teaching Mathematics Dragana Martinovic 23

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The Roles of Intelligence and Creativity for Learning Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michaela A. Meier and Roland H. Grabner

24

Gestures in Mathematics Thinking and Learning . . . . . . . . . . . . . Ornella Robutti, Cristina Sabena, Christina Krause, Carlotta Soldano, and Ferdinando Arzarello

25

Teaching and Learning Authentic Mathematics: The Case of Proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andreas J. Stylianides, Kotaro Komatsu, Keith Weber, and Gabriel J. Stylianides

26

Why Are Learning and Teaching Mathematics So Difficult? . . . . . Alan H. Schoenfeld

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647 685

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Contents

Section VII Mathematics Education and New Technologies . . . . . Marcelo C. Borba, Johann Engelbrecht, and Ricardo Scucuglia 27

28

29

30

31

Computing in Mathematics Education: Past, Present, and Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Gadanidis, S. L. Javaroni, S. C. Santos, and E. C. Silva

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Computer Algebra Systems and Dynamic Geometry for Mathematical Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jhony Alexander Villa-Ochoa and Liliana Suárez-Téllez

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Student Collaboration in Blending Digital Technology in the Learning of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Johann Engelbrecht and Greg Oates

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Multimodality, Systemic Functional-Multimodal Discourse Analysis and Production of Videos in Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marcelo C. Borba, Kay L. O’Halloran, and Liliane Xavier Neves STEAM and Critical Making in Teacher Education . . . . . . . . . . . Immaculate Kizito Namukasa, Janette Hughes, and Ricardo Scucuglia

Section VIII Mathematics and Computer Science . . . . . . . . . . . . . . Huaxiong Huang 32

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Memory Consolidation: Neural Data Analysis and Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Masami Tatsuno and Michael Eckert

Section IX Mathematics and Linguistics . . . . . . . . . . . . . . . . . . . . . Rodolfo Delmonte

909 939

971

973

1011

33

Mathematical Linguistics and Cognitive Complexity . . . . . . . . . . . 1015 Aniello De Santo and Jonathan Rawski

34

Quantifying Context With and Without Statistical Language Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053 Cassandra L. Jacobs

35

Cognitive Models of Poetry Reading Rodolfo Delmonte

36

Cognition and Computational Linguistic Creativity Lorenzo Gatti, Oliviero Stock, and Carlo Strapparava

37

Understanding Dialogue for Human Communication . . . . . . . . . . 1159 Bernardo Magnini and Samuel Louvan

. . . . . . . . . . . . . . . . . . . . . . . 1083 . . . . . . . . . . . 1121

Contents

Section X Mathematics Cognition, Semiotics, and Hermeneutic Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vitaly Kiryushenko

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38

Peirce on Abduction and Diagrams in Mathematical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209 Joseph W. Dauben, Gary J. Richmond, and Jon Alan Schmidt

39

Pragmaticism as a Philosophy of Cognitive Mathematics . . . . . . . 1243 Ahti-Veikko Pietarinen

40

Diagrammatic Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1281 Louis H. Kauffman

41

Peirce on Mathematical Reasoning and Discovery . . . . . . . . . . . . . 1313 Ahti-Veikko Pietarinen

42

Knowing by Drawing: Mathematics as Gesture . . . . . . . . . . . . . . . 1345 Giovanni Maddalena

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365

About the Editor

Marcel Danesi is Professor Emeritus of Anthropology at the University of Toronto. He has written extensively on the relation between mathematics and symbol systems, including how puzzles and problems in mathematics are part of a more general dialectic frame of mind for grasping the nature of reality. Among his works in the field are Language and Mathematics (2018) and Ahmes’ Legacy (2020). He also founded the CogSci Network at the Fields Institute for Research in Mathematical Sciences, which consists of internationally renowned researchers in the field of cognitive mathematics.

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Section Editors

Stéphanie Walsh Matthews Toronto Metropolitan University (formerly Ryerson) Toronto, Ontario, Canada

Marcelo C. Borba UNESP – São Paulo State University Rio Claro, São Paulo, Brazil

Dragana Martinovic Faculty of Education, University of Windsor Windsor, Ontario, Canada

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Section Editors

Huaxiong Huang York University Toronto, Ontario, Canada

Vitaly Kiryushenko York University Toronto, Ontario, Canada

Johann Engelbrecht Faculty of Education University of Pretoria Pretoria, Gauteng, South Africa

Roland H. Grabner Educational Neuroscience Institute of Psychology University of Graz Graz, Styria, Austria

Section Editors

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Myrdene Anderson Purdue University West Lafayette, Indiana, USA

Donna E. West State University of New York at Cortland Cortland, New York, USA

Tod Shockey The University of Toledo Toledo, Ohio, USA

Ricardo Scucuglia Sao Paulo State University (UNESP) Sao Jose do Rio Preto, Sao Paulo, Brazil

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Section Editors

Rodolfo Delmonte Computational Linguistics University of Venice Venezia, Italy

Dan Vilenchik School of Computer and Electrical Engineering Ben-Gurion University of the Negev Beersheba, Israel

Contributors

Veronica Albanese University of Granada, Melilla, Spain Piata Allen University of Auckland, Auckland, New Zealand Ferdinando Arzarello University of Turin, Torino, Italy Marcelo C. Borba State University of São Paulo, Rio Claro, Brazil Sara Caviola Department of Developmental Psychology, University of Padova, Padova, Italy School of Psychology, University of Leeds, Leeds, UK Lital Daches Cohen Edmond J. Safra Brain Research Center for the Study of Learning Disabilities, Department of Learning Disabilities, University of Haifa, Haifa, Israel Roi Cohen Kadosh Department of Experimental Psychology, University of Oxford, Oxford, UK Marcel Danesi Anthropology, Victoria College, University of Toronto, Toronto, ON, Canada Joseph W. Dauben Herbert H. Lehman College and the Graduate Center, City University of New York, New York, NY, USA Aniello De Santo Department of Linguistics, University of Utah, Salt Lake City, UT, USA Rodolfo Delmonte University of Venice, Venezia, Italy Thomas Dresler LEAD Graduate School & Research Network, University of Tübingen, Tübingen, Germany Department of Psychiatry and Psychotherapy, Tübingen Center for Mental Health, University of Tübingen, Tübingen, Germany Michael Eckert Department of Neuroscience, Canadian Centre for Behavioural Neuroscience, University of Lethbridge, Lethbridge, AB, Canada xxi

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Contributors

Samuel J. Eckrich Department of Pediatric Neuropsychology, Kennedy Krieger Institute; Johns Hopkins University School of Medicine, Baltimore, MD, USA Johann Engelbrecht Faculty of Education, University of Pretoria, Pretoria, South Africa Gabrielle Fabrikant-Abzug Department of Psychology, Arizona State University, Tempe, AZ, USA Lauren M. Friedman Department of Psychology, Arizona State University, Tempe, AZ, USA G. Gadanidis Western University, London, Canada Lorenzo Gatti University of Twente, Twente, The Netherlands Roland H. Grabner Educational Neuroscience, Institute of Psychology, University of Graz, Graz, Austria Jenni L. Harding University of Northern Colorado, Greeley, CO, USA Janette Hughes Ontario Tech University, Oshawa, ON, Canada Cassandra L. Jacobs Department of Psychology, University of Wisconsin, Madison, WI, USA S. L. Javaroni São Paulo State University (Unesp), Bauru, Brazil Louis H. Kauffman Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, USA Department of Mechanics and Mathematics, Novosibirsk State University, Novosibirsk, Russia Kotaro Komatsu University of Tsukuba, Tsukuba, Japan Margo Kondratieva Faculty of Education and the Department of Mathematics and Statistics, Memorial University of Newfoundland (MUN), St. John’s, NL, Canada Christina Krause University of Graz, Graz, Austria University of California, Berkeley, Berkeley, CA, USA University of Duisburg-Essen, Duisburg, Germany Karin Kucian Center for MR-Research, University Children’s Hospital Zurich, Zurich, Switzerland Massimo Leone University of Turin, Turin, Italy Shanghai University, Shanghai, China Bruno Kessler Foundation, Trento, Italy Cambridge University, Cambridge, UK Samuel Louvan Fondazione Bruno Kessler, University of Trento, Trento, Italy

Contributors

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Giovanni Maddalena University of Molise, Campobasso, Italy Lorenzo Magnani Department of Humanities, Philosophy Section and Computational Philosophy Laboratory, University of Pavia, Pavia, Italy Bernardo Magnini Fondazione Bruno Kessler, Trento, Italy Irene C. Mammarella Department of Developmental Psychology, University of Padova, Padova, Italy Tamsin Meaney Western Norway University of Applied Sciences, Bergen, Norway Michaela A. Meier Educational Neuroscience, Institute of Psychology, University of Graz, Graz, Austria Immaculate Kizito Namukasa Western University, London, ON, Canada Liliane Xavier Neves State University of Santa Cruz, Ilhéus, Brazil Kay L. O’Halloran University of Liverpool, Liverpool, UK Greg Oates School of Education, University of Tasmania, Launceston, TAS, Australia Andreas Obersteiner TUM School of Education, Technical University of Munich, Munich, Germany Sarah A. Orban Department of Psychology, University of Tampa, Tampa, FL, USA Daniel Clark Orey Departamento de Educação Matemática, Universidade Federal de Ouro Preto (UFOP), Ouro Preto, Minas Gerais, Brazil Jamin Pelkey Ryerson University, Toronto, Canada Ahti-Veikko Pietarinen Tallinn University of Technology, Tallinn, Estonia Research University Higher School of Economics, Moscow, Russia Rachel Pizzie Gallaudet University, Washington, DC, USA Jonathan Rawski Department of Linguistics & Language Development, San Jose State University, San Jose, CA, USA Lucia Regolin Department of General Psychology, University of Padova, Padova, Italy Gary J. Richmond Humanities Department, LaGuardia Community College, City University of New York, Long Island City, NY, USA Ornella Robutti University of Turin, Torino, Italy Milton Rosa Departamento de Educação Matemática, Universidade Federal de Ouro Preto (UFOP), Ouro Preto, Minas Gerais, Brazil

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Contributors

Richard Rosenbaum York/Ryerson Universities, Toronto, ON, Canada Orly Rubinsten Edmond J. Safra Brain Research Center for the Study of Learning Disabilities, Department of Learning Disabilities, University of Haifa, Haifa, Israel Rosa Rugani Department of General Psychology, University of Padova, Padova, Italy Cristina Sabena University of Turin, Torino, Italy S. C. Santos Federal University of Viçosa (UFV), Viçosa, Brazil Jon Alan Schmidt Independent Scholar, Olathe, KS, USA Alan H. Schoenfeld Graduate School of Education, University of California, Berkeley, CA, USA Ricardo Scucuglia Sao Paulo State University (UNESP), Sao Jose do Rio Preto, Sao Paulo, Brazil E. C. Silva Mathematic’s Education Graduate Program of São Paulo State University (Unesp), Rio Claro, Brazil Carlotta Soldano University of Turin, Torino, Italy Oliviero Stock FBK-irst, Trento, Italy Carlo Strapparava FBK-irst, Trento, Italy Andreas J. Stylianides Faculty of Education, University of Cambridge, Cambridge, UK Gabriel J. Stylianides University of Oxford, Oxford, UK Liliana Suárez-Téllez Instituto Politécnico Nacional, México City, Mexico Denes Szűcs Department of Psychology, University of Cambridge, Cambridge, UK Eero Tarasti University of Helsinki, Helsinki, Finland Masami Tatsuno Department of Neuroscience, Canadian Centre for Behavioural Neuroscience, University of Lethbridge, Lethbridge, AB, Canada Tony Trinick University of Auckland, Auckland, New Zealand Jhony Alexander Villa-Ochoa School of Education, Universidad de Antioquia, Medellín, Colombia Stephan E. Vogel Section of Educational Neuroscience, Institute of Psychology, University of Graz, Graz, Austria Keith Weber Rutgers University, New Brunswick, NJ, USA Donna E. West Modern Languages, State University of New York, Cortland, NY, USA

Contributors

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Sandra Wildfeuer University of Alaska Fairbanks, Fairbanks, AK, USA Silke M. Wortha Deparment of Neurology, University Medicine of Greifswald, Greifswald, Germany LEAD Graduate School & Research Network, University of Tübingen, Tübingen, Germany Fernando Zalamea Departamento de Matemáticas, Universidad Nacional de Colombia, Sede Bogotá, Colombia

Introduction

Cognitive Mathematics In the mid-1950s, psychologists started adopting insights and terms from the thenfledgling science of AI, seeking parallels between the functions of the human brain and those of the computer, borrowing terms such as “coding,” “storing,” “retrieving,” and “buffering” from computer science to apply to mental processes and functions. By the latter part of the 1960s, this approach became widespread. Ulrich Neisser (1967, p. 6) put it as follows: The task of the psychologist in trying to understand human cognition is analogous to that of a man trying to discover how a computer has been programmed. In particular, if the program seems to store and reuse information, he would like to know by what “routines” or “procedures” this is done. Given this purpose, he will not care much whether his particular computer stores information in magnetic cores or in thin films; he wants to understand the program, not the “hardware.” By the same token, it would not help the psychologist to know that memory is carried by RNA as opposed to some other medium. He wants to understand its utilization, not its incarnation.

Neisser realized, however, that the computer metaphor, if brought to an extreme, would actually lead psychology astray. So, only a few pages later he issued the following caveat (Neisser, 1967, p. 9): “Unlike men, artificially intelligent programs tend to be single-minded, undistractable, and unemotional; in my opinion, none does even remote justice to the complexity of mental processes.” By the end of the 1970s, psychologists had formed a partnership with AI – an alliance that eventually led to the emergence of a new discipline called cognitive science, whose aim was to study the human mind from different disciplinary angles, in addition to AI, computer science, psychology, linguistics, neuroscience, anthropology, biology, and philosophy. From the outset, two main schools within this new science surfaced. One was based directly on the notions and methods of AI researchers, portraying the mind as a kind of biological computing device operating separately from lived experience. As Howard Gardner (1985, p. 6) aptly put it, the guiding assumption of this “strong” version (as it was called) was that there exists “a level of analysis wholly separate from the biological or neurological, on the one hand, and the sociological or cultural, on the other,” and that “central to any understanding of the human mind is the xxvii

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Introduction

electronic computer.” The second version, called the “weak” version, aimed instead to study the mind as an interactive product of bodily, affective, and lived experiences. This version came eventually to be called the “embodied cognition” movement, based on the view that the brain organizes itself via the input it receives through the body and the emotions, making changes through specific anatomical and sensory-feeling systems (Damasio, 1994). The two strands are still operative, having developed sophisticated research methods on their own that are applied to all areas of cognitive science, from cognition to perception and emotions. An example of an approach that fell into the “strong version” category was so-called script theory, which proposed that cognition was guided by “internal scripts” that were very much like computer programs. For example, ordering a meal at a restaurant involves a sequenced series of activities, including a strategy for getting the waiter’s attention, a reaction to the waiter’s response, a strategy for ordering food to fit one’s particular tastes and financial capabilities, an optional strategy for commenting favorably or unfavorably on the quality of the food, and so on. Any radical departure from this script would result in a breakdown in both understanding and communication, or else lead to a chaotic, disorganized system for ordering food and likely an unsuccessful one. Scripts occur at different levels and in different ways, from solving mathematical problems to sensing meaning in the arts. Script theory was developed initially by computer scientist Roger Schank (1980, 1984, 1991), who saw cognitive scripts as unconscious knowledge structures, which manifest themselves in typical situations and can be modeled almost to precision in computer algorithms. As Gardner (1985, pp. 17–18) noted, such models of the mind were common at the start of cognitive science, reiterating Neisser’s early comment (above): The implications of these ideas were quickly seized upon by scientists interested in human thought, who realized that if they could describe with precision the behavior of thought processes of an organism, they might be able to design a computing machine that operated in identical fashion. It thus might be possible to test on the computer the plausibility of notions about how a human being actually functions, and perhaps even to construct machines about which one could confidently assert that they think just like human beings.

The contrary perspective in cognitive science (the embodied cognition view) was first articulated by philosopher Max Black (1962) who remarked that the idea of trying to discover how a computer has been programmed in order to extrapolate how the mind works was ultimately implausible because computers can never be intelligent in the same way as humans because the laws of nature will not allow it. In the early 1970s, Black’s caveat was given substance within biology and psychology by the notion of autopoiesis, put forth by Maturana and Varela (1973), who observed that organisms self-organize themselves according to the input they receive from the environment. Autopoietic systems are often contrasted to allopoietic systems, such as computer programs, defined as systems that produce something other than themselves. McGann (2000, p. 358) provides the following relevant characterization of the distinction:

Introduction

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An autopoietic system is a closed topological space that continuously generates and specifies its own organization through its operation as a system of production of its own components, and does this in an endless turnover of components. Autopoietic systems are thus distinguished from allopoietic systems, which are Cartesian and which have as the product of their functioning something different from themselves. Coding and markup appear allopoietic.

The embodied cognition movement was extended to the study of the mathematical mind in the year 2000, when Georg Lakoff and Rafael Núñez published their highly influential book, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. This was a key moment in embodied cognitive science, since it connected language to mathematics via metaphorical thinking, which led subsequently to a systematic study of the connection between numerical cognition and all other cognitive faculties, from language to aesthetics. This approach to math cognition has been designated recently as cognitive mathematics (e.g., Danesi, 2019; Costa et al., 2020). The distinguishing feature of this approach is its basis in a hermeneutic purview, thus extending the disciplinary amalgam of early cognitive science to embrace humanistic disciplines that might shed a meaningful light on the relation between mathematics and other faculties of mind. This Handbook is the first reference volume in this field, with chapters on ethnomathematics, math education, technology, linguistics, neuroscience, computer science, semiotics, psychology, biology, and the arts that are based (directly or indirectly) on a hermeneutic perspective. The hermeneutic approach has actually been an implicit one throughout the history of mathematics, even though it was never labeled as such (Danesi, 2020). Immanuel Kant (1798, p. 278), for example, discussed the nature of math cognition (MC) in a hermeneutic fashion, characterizing it as the process of “combining and comparing given concepts of magnitudes, which are clear and certain, with a view to establishing what can be inferred from them,” by examining the “visible signs” that mathematicians used to encode their particular form of knowledge. For example, a diagram of a triangle compared to that of a square will show how the cognitive differentiation is represented “visibly” – one consists of three intersecting lines, while the other has four parallel and equal sides that form a boundary. As trivial as this might seem, upon further consideration it suggests that mathematics is based on the perception of differential cues, thus indirectly projecting the study of MC into the domain of semiotics. Ferdinand de Saussure (1916), a modern-day founder of this discipline, called this perceptual process différence. So, in this framework, grasping cognitively what makes a triangle not a square (and vice versa) is an example of how perceptual différence works. Much of the empirical work on MC can, in fact, be explained in terms of the theory of différence, as the following few examples arguably show: 1. The ability to differentiate number cues may cross species at a rudimentary level, called numerosity (Dehaene, 1997, 2004). In one experiment (McComb et al., 1994), hidden speakers were used to transmit from 1 to 5 artificial male lion mating calls. As it turned out, if a lioness heard three calls, she would leave,

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presumably feeling outnumbered. If she was with four other lionesses, however, the five together would go and seek out the mating caller. It seems that lions possess the ability to distinguish number cues via their own innate sense of différence. 2. Starting with the work of Piaget (1952), studies have shown consistently that infants display a similar kind of rudimentary number sense, able to differentiate quantities as represented by objects and symbols. When presented with arrays of different numbers of dots on various surfaces or objects, infants respond in a patterned way: If they had become habituated to, say, 10 dots they would stop looking at novel displays involving that number of dots. But if presented with a different number, say a 5-dot display for the first time, they once again became intrigued, noting the perceptual différence as meaningful. 3. Neuroimaging studies have shown that the parietal lobe and the inferior parietal lobule are activated in subjects who are asked to carry out calculation tasks (Butterworth, 1999). The left parietal lobe controls fingers, explaining why we count instinctively on our fingers. But the same neural region is involved in hand and finger gesturing. Distinguishing between the functions of the two may well be a case of différence at a “meta” level – a level that processes the different functions of same signs. 4. Studies show that MC might overlap with spatial cognition – a topic treated extensively within math education research (e.g., Whiteley, 2012). This can be seen to involve a basic differential paradigm, quantity versus space, which can be seen, in turn, as a source of différence, again at a meta level. Various research projects have also shown, as Kant suggested, that MC is dependent on symbols, somewhat corroborating philosopher Ernst Cassirer’s (1944) observation that humans are “a symbolic species,” incapable of establishing knowledge without symbols, going on to suggest that systems of knowledge, such as mathematics, cohere into systems of symbols. In an in-depth study, Keith Devlin (2012) identified what he called the “symbol barrier” as the biggest obstacle to a mastery of mathematics. Ordinary people, Devlin asserted, can do practical mathematics (counting, measuring, comparing quantities, etc.). But they have more difficulty doing more complex mathematics without possessing the symbolism used to represent complicated ideas. As the mathematics becomes more complex and abstract, so too does the reliance on symbolism, which at the most abstract levels supersedes practical experience. Perhaps for this reason, Butterworth (1999) has argued that numerosity is located in the same areas of the brain that are responsible for symbolic activities. But, like Devlin, Butterworth suggests that this alone does not guarantee that knowledge of mathematics will emerge homogeneously in all individuals. Rather, the reason a person falters at mathematics is not because of a “wrong gene” or “engine part” in the left parietal lobe, but because particular individuals have not fully developed the number sense with which they were born, and the reason is due to environmental and personal psychological factors. Butterworth presents findings that neonates can add and subtract even a few weeks old and that people afflicted with Alzheimer’s have unexpected numerical abilities.

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But both lack the ability to control mathematical symbolism in any concrete way. The same kinds of visible signs, of which Kant spoke, may have universal validity, as a study on the presence of Euclidean ideas in a tribe (the Mundurucu) seems to have confirmed (Izard et al., 2011), given that the tribe had never been exposed to Euclidean geometry. However, contrastive research in ethnomathematics has abundantly shown that many of the mathematical concepts that are assumed to be universal turn out not to be so (Núñez et al., 1999). The jury is still out on this issue, which is being researched within all subfields of cognitive mathematics. In one relevant study, Lesh and Harel (2003) got students to develop their own models of a problem space, guided by instruction. Without the latter, the students had been incapable of coming up with them. Number sense may be innate, but many other mathematical concepts may have to be guided by input. Remarkably, pedagogical guidance can lead learners to a grasp of the concepts to different degrees, suggesting that the interplay between symbolism and innate tendencies is the crux to math education. A major area of concern in cognitive mathematics, since Lakoff and Johnson’s pivotal book, is the relation between mathematics and language. Neuroscientific studies, as discussed briefly above, have indicated that MC and language may arise from the same neural source; so too has the work in anthropology, which has suggested that the two faculties are united phylogenetically via four critical evolutionary events – bipedalism, a brain enlargement unparalleled among species, an extraordinary capacity for tool-making, and the advent of the tribe as the main form of human collective life (Cartmill et al., 1986). Bipedalism liberated the fingers to do several things – count and gesture. The former is the basis for numerosity, the latter for language. Both likely occurred in tandem – finger-use was used for counting and verbal gesturing at the same time. The history of number concepts and their corresponding words started when people indicated a number by pointing to a part of their body (even today, in indigenous languages of New Guinea, the word for six is “wrist”). These became the basis for abstract symbols, such as the Roman numerals (chosen for the ease with which they could be carved into wooden sticks), leading eventually to modern-day numerals and numeral systems. Although other species, including some non-primate ones, are capable of tool use, only in the human species did complete bipedalism free the hand sufficiently to allow it to become a supremely sensitive and precise manipulator and grasper, thus permitting proficient tool-making and tool use in the species. It also allowed humans to record their thoughts in various forms, such as art and pictography, leading to writing and symbolism. Shortly after becoming bipedal, and developing symbolism, the evidence suggests that the human species underwent rapid brain expansion. In the course of human evolution, the size of the brain has more than tripled. Modern humans have a braincase volume of between 1300 and 1500 cc. The brain has also developed three major structural components that undergird the unique mental capacities of the species – the large dome-shaped cerebrum, the smaller somewhat spherical cerebellum, and the brainstem. The size of the brain does not determine the degree of intelligence of the individual; this is determined instead by the number and type of functioning neurons and how they are structurally connected with one

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another. And since neuronal connections are conditioned by environmental input, the most likely hypothesis is that any form of intelligence, however it is defined, is a consequence of upbringing. Like most other species, humans have always lived in groups. Group life enhances survivability by providing a collective form of shelter. But at some point in their evolutionary history – probably around 100,000 years ago – bipedal hominids had become so adept at tool-making, communicating, and thinking in symbols that they became consciously aware of the advantages of a group life based on a common system of representational activities. By around 30,000–40,000 years ago, the archeological evidence suggests, in fact, that hominid groups became increasingly characterized by communal customs, language, symbolism, writing, and the transmission of technological knowledge to subsequent generations. The early tribal collectivities have left evidence that gesture (as inscribed on surfaces through pictography) and numerosity occurred in tandem. This suggests two things: (a) math and language do indeed have a common evolutionary origin and (b) mathematics, like language, is embedded as a social phenomenon (Hersh, 1997, 2014) Stanislas Dehaene, one of the leading researchers on MC, argues that the human brain does not work like a computer and that the physical world is not based on mathematics – rather, mathematics evolved to explain the physical world the way that the eye evolved to provide sight. Various case studies of brain-damaged patients support Dehaene’s basic ideas. Defects in grasping numbers (anarithmeia) have been shown to be associated with lesions in the left angular gyrus and with Gerstmann’s syndrome, which involves the inability to count with one’s fingers. Patients with acalculia (inability to calculate), who might read 14 as 4, have difficulty representing numbers with words. For example, they might have difficulty understanding the meaning of “hundred” in expressions such as “two hundred” and a “hundred thousand.” Acalculia is associated with Broca’s aphasia and, thus, with the left inferior frontal gyrus. But acalculia has also been found in patients suffering from Wernicke’s aphasia, which involves difficulties in saying, reading, and writing numbers. This is associated with the left posterior superior temporal gyrus. Patients with frontal acalculia have damage in the pre-frontal cortex. They have serious difficulties in carrying out arithmetical operations (particularly subtraction) and solving numerical problems. Dyscalculia (inability to calculate) is associated with the horizontal segment of the intraparietal sulcus, in both hemispheres. The list of relevant studies on such phenomena is extensive (e.g., Isaacs et al., 2001; Ardila & Rosselli, 2002; Dehaene et al., 2003; Butterworth et al., 2011). Overall, the hermeneutic approach within cognitive mathematics aims to unite the study of mathematics not only within the above scientific fields and enterprises, but also to activities such as music, dance, drawing, and other creative skills that are embedded in a “biological, psychological and cultural context,” as Rosch, Thomson, and Varela (1999, p. 5) so aptly observe. The term hermeneutics has actually been used with various designations, often as a catchphrase for any humanistic or cultural approach to mathematics. The term was introduced into philosophy by Aristotle in his Peri Hermeneias (c. 360 BCE), translated into Latin as De Interpretatione and

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later in English as On Interpretation. It is one of the first works to deal with the connection between language and logical thinking in a formal way. In the early medieval period, the term was used instead to designate the interpretation of scripture (Grondin, 1994, p. 21). In modernity, it resurfaced as a theory of understanding within philosophy with the writings of Friedrich Schleiermacher, Wilhelm Dilthey, Martin Heidegger, and Hans-Georg Gadamer among others (Seebohm, 2007; Zimmerman, 2015). Already in the 1980s and 1990s, math educators started using this term in reference to the interrelationships between mathematics, language, and symbolism in learning tasks (e.g., Varelas, 1989; Pimm, 1995; English, 1997). Douglas Hofstadter argued more generally that one cannot grasp the nature of mathematics without considering its connections to music and art in his two relevant books, Gödel, Escher, Bach: An Eternal Golden Braid (1979) and Metamagical Themas (1985). These made it obvious to many that isolating mathematics from other human faculties is ultimately futile. Although not named as hermeneutic, Hofstadter’s work laid the foundations, indirectly, for an ever-broadening hermeneutic approach to emerge in the investigation of math cognition generally and in the implications it bears for math education (see, e.g., Brown, 2001; Senechal, 2013; Gamwell, 2015; Emmer, 2016; Presmeg et al., 2018; Sriraman, 2020). It was in the late nineteenth century and the early twentieth that the term hermeneutics was first adopted to indicate a theory of interpretation, that is, of determining the meaning of something, from language to dreams (Freud, 1899; Heidegger, 2008). Within this new paradigm, Karl Popper (1972) included science, calling his approach “objective hermeneutics,” which led to a movement whose goal has since been to provide an interpretive framework for uniting all disciplines, scientific and humanistic, as Oevermann, Allert, Konau, and Krambeck remark (1987, pp. 436–437): Our approach has grown out of the empirical study of family interactions as well as reflection upon the procedures of interpretation employed in our research. For the time being we shall refer to it as objective hermeneutics in order to distinguish it clearly from traditional hermeneutic techniques and orientations. The general significance for sociological analysis of objective hermeneutics issues from the fact that, in the social sciences, interpretive methods constitute the fundamental procedures of measurement and of the generation of research data relevant to theory. From our perspective, the standard, nonhermeneutic methods of quantitative social research can only be justified because they permit a shortcut in generating data (and research “economy” comes about under specific conditions). Whereas the conventional methodological attitude in the social sciences justifies qualitative approaches as exploratory or preparatory activities, to be succeeded by standardized approaches and techniques as the actual scientific procedures (assuring precision, validity, and objectivity), we regard hermeneutic procedures as the basic method for gaining precise and valid knowledge in the social sciences. However, we do not simply reject alternative approaches dogmatically. They are in fact useful wherever the loss in precision and objectivity necessitated by the requirement of research economy can be condoned and tolerated in the light of prior hermeneutically elucidated research experiences.

Objective hermeneutics shares a large part of its theoretical territory with semiotics, the science of interpretation. Its goal in the area of math cognition is to explain

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connections between mathematical principles and ideas and other domains of human intellectual and aesthetic activity. Ultimately, the question that is of central interest to cognitive mathematics is whether or not it is a separate or interlinked faculty. One theory that has come forth to examine this question explicitly is so-called conceptual blending theory, a term first used by Fauconnier and Turner in 2002, implying that different areas of the brain are involved in all forms of cognition in a cooperative fashion, suggesting that the brain is a modular rather than locationist organ. The theory posits that knowledge emerges through linkage processes in neural pathways – that is, a conceptual blend is formed when the brain identifies two distinct inputs (concepts) in different neural regions as the same entity in a third neural region. The blend that results will contain more information than the two inputs. Blending is unconscious and that is why we are hardly ever aware of what we are doing when we think of, say, numbers. Consider a simple statement such as “7 is larger than 4.” In the Fauconnier-Turner paradigm, this is the result of a blend that amalgamates concepts of size and of numbers (Presmeg, 1997, 2005). In this case, the blend is a result of a metaphorical idea – namely, numbers are collections of objects of differing sizes (Lakoff & Núñez, 2000). The general mathematical concept of quantity involves at least two metaphorical blends. The first is more is up, less is down, which appears in common mathematical expressions such as “the height of those functions went up as the numerical value increased” and “the other functions sloped downwards as the numerical values decreased.” The other is linear scales are paths, which manifests itself in expressions such as “rational numbers are far more numerous than integers” and infinity is “something beyond any collection of finite sets.” As Lakoff (2012, p. 164) puts it: The metaphor maps the starting point of the path onto the bottom of the scale and maps distance traveled onto quantity in general. What is particularly interesting is that the logic of paths maps onto the logic of linear scales. Path inference: If you are going from A to C, and you are now at in intermediate point B, then you have been at all points between A and B and not at any points between B and C. Example: If you are going from San Francisco to N.Y. along route 80, and you are now at Chicago, then you have been to Denver but not to Pittsburgh. Linear scale inference: If you have exactly $50 in your bank account, then you have $40, $30, and so on, but not $60, $70, or any larger amount. The form of these inferences is the same. The path inference is a consequence of the cognitive topology of paths. It will be true of any path image-schema.

Research has been largely supportive of blending theory. Guhe et al. (2011), for instance, developed a computational model of how blending might be simulated – thus integrating both strands of cognitive science methodologically (the computational and the embodied versions). The researchers devised a system by which different conceptualizations of number can be blended together to form new ones via a recognition of common features, and a judicious combination of their features. The model of number the researchers used was based on Lakoff and Núñez’s grounding metaphors for arithmetic. The ideas were worked out using a so-called Heuristic-Driven Theory Projection (HDTP), a method that provides generalizations

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between domains, thus allowing for a mechanism of finding commonalities, transferring concepts from one domain to another, thus producing conceptual blends. Lakoff and Núñez (2000) had looked concretely at two main metaphorical blends, which they called grounding and linking. The former encode ideas that are grounded in experience. For example, addition develops from the experience of counting objects and then inserting them in a collection (a box or some other container). Linking metaphors, on the other hand, connect concepts within mathematics that may or may not be based on physical experiences, but on linked associations. Some examples are the number line, inequalities, and absolute value properties within an epsilon-delta proof of limit. Linking blends have been used to explain the emergence of negative numbers, which would otherwise seem to emerge in some disconnected fashion, as Alexander (2012, p. 28) elaborates: Using the natural numbers, we made a much bigger set, way too big in fact. So we judiciously collapsed the bigger set down. In this way, we collapse down to our original set of natural numbers, but we also picked up a whole new set of numbers, which we call the negative numbers, along with arithmetic operations, addition, multiplication, subtraction. And there is our payoff. With negative numbers, subtraction is always possible. This is but one example, but in it we can see a larger, and quite important, issue of cognition. The larger set of numbers, positive and negative, is a cognitive blend in mathematics. . .The numbers, now enlarged to include negative numbers, become an entity with its own identity. The collapse in notation reflects this. One quickly abandons the (minuend, subtrahend) formulation, so that rather than (6, 8) one uses -2. This is an essential feature of a cognitive blend; something new has emerged.

As Solomon Marcus (2012, p. 124) aptly remarked, this metaphorical mindset has always characterized the naming of ideas within mathematics, thus revealing its unconscious operation within MC: For a long time, metaphor was considered incompatible with the requirements of rigor and preciseness of mathematics. This happened because it was seen only as a rhetorical device such as “this girl is a flower.” However, the largest part of mathematical terminology is the result of some metaphorical processes, using transfers from ordinary language. Mathematical terms such as function, union, inclusion, border, frontier, distance, bounded, open, closed, imaginary number, rational/irrational number are only a few examples in this respect. Similar metaphorical processes take place in the artificial component of the mathematical sign system.

Using Russian psychologist Lev Vygotsky’s term (1961), such metaphors are “poetic,” namely, concepts that make connections among things “visible” inside the imagination (like poetry). They produce an “inner vision,” which abounds across the sciences – it is the reason why in physics sound waves are said to undulate through empty space like water waves ripple through a still pond, atoms leap from one quantum state to another, electrons to travel in circles around an atomic nucleus, and so on. Linking metaphors occur when gaps emerge in some system that require filling (Godino et al., 2011). Connectivity and linkage form the cognitive glue, so to speak,

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that gives mathematics its holistic structure. Linear and set algebras are a more general way of doing arithmetic – the connecting links are conceptual metaphors such as: arithmetic is motion along a path, sets are containers, recurrence is circular, and so on. Mathematics makes sense when it encodes concepts that fit our experiences of the world – experiences of quantity, space, motion, force, change, mass, shape, probability, self-regulating processes, and so on. The inspiration for new mathematics comes from these experiences as it does for new language. A classic example of blending is Gödel’s (1931) famous proof (Lakoff & Núñez, 2000). Gödel proved that within any formal logical system there are results that can be neither proved nor disproved, identifying the culprit statement in a set of statements that could be extracted by going through them in a diagonal fashion analogous to the one used by Georg Cantor in one of his own famous proofs – now called Gödel’s diagonal lemma. That produced a statement, S, like Cantor’s C, that does not exist in the set of statements. Lakoff and Núñez also pointed out that Cantor’s diagonal and one-to-one matching proofs are themselves mathematical metaphors – associations linking different domains in a specific way (one-to-one correspondences). Gödel’s proof revolves conceptually around the idea that a symbol in a statement system is the corresponding number in the Cantorian one-to-one matching system (whereby any two sets of symbols can be put into a one-to-one relation). As Lakoff and Núñez pointed out, this is an example of how the brain identifies two distinct entities in different neural regions as the same entity in a third neural region. The chapters in this handbook span a broad terrain of cognitive mathematics, ranging from technical subjects to more humanistic ones – a breadth that is the defining character of cognitive mathematics as a hermeneutic enterprise. Among the areas covered are the following: • • • • • • • •

Connections between mathematical modeling and AI research Associative processes in computational systems Historical contextualizations of mathematical ideas Connections between math cognition and symbolism, and between mathematical discovery and cultural processes Neuroscientific evidence that neural structures may (or may not) share the same areas as language and art The recruitment of everyday cognitive mechanisms that undergird imagination, abstraction, and notation-making processes and how these overlap among the various faculties and skills Determining which structures, if any, mathematics, language, and art might share Researching the notion of conceptual blending as a means to understand how mathematics emerges

In Greece, arithmetic, geometry, and grammar were considered to be intertwined branches of knowledge as “arts” of the mind. This interconnectedness was praised and practiced concretely in the Renaissance, when, for example, artists studied geometry and incorporated it into their works, thus shedding light on both art and

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mathematics. The separation of mathematics from the arts became a tendency after the Enlightenment. The hermeneutic approach entails, at the very least, a revisitation of ancient and Renaissance perspectives of mathematics. Needless to say, there are various interdisciplinary approaches today which aim to connect mathematics to expressive forms in philosophy, psychology, anthropology, and so on and so forth. The goal of cognitive mathematics is to integrate the diverse approaches into an overall understanding of mathematics, expanding the scientific paradigm of cognitive science to include the humanities – a goal that has been taken up also by such enterprises as the Journal of Humanistic Mathematics, which is part of a movement called Humanistic Mathematics. Building on some of the developments in these diverse fields, the chapters in this volume, when considered cumulatively, discuss how mathematics involves a blend of faculties that define all manifestations of human creativity and expressivity. A whole subset of chapters looks at the profound implications that the hermeneutic approach bears for the teaching and learning of mathematics. Mathematicians, cognitive scientists, educators of mathematics, philosophers of mathematics, and all the other kinds of scholars who are interested in the nature of mathematics will hopefully find something of interest in this volume. The implicit claim in all the studies is that in order to penetrate the phenomenon of mathematics it is necessary to utilize methods and theoretical frameworks derived from a variety of disciplines. Above all else, cognitive mathematics connects mathematics to social practices, historical forces, and expressive artifacts. It is relevant to note that this type of approach to mathematics was actually anticipated over half a century ago in Courant and Robins’ significant 1941 book What Is Mathematics? Their answer to the question they pose in the title is to simply illustrate what mathematics looks like and what it does to us, psychologically, socially, aesthetically, etc. Similarly, the only meaningful way to answer What is music? is to play it, sing it, or listen to it. A year before, in 1940, Kasner and Newman published another important popular book titled Mathematics and the Imagination. The authors also illustrated in that book what mathematics does to us intellectually and aesthetically, much like music and art. As Lynne Gamwell (2015) has cogently argued, mathematicians, poets, philosophers, artists, and others have been on a common quest since antiquity to understand the world they see before them. Their visions are complementary, not autonomous, ones. As literary critic John William Navin Sullivan (1925) so aptly put it, mathematics is perhaps itself best defined as an art: “The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.” Marcel Danesi

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References Alexander, J. (2012). On the cognitive and semiotic structure of mathematics. In M. Bockarova, M. Danesi, & R. Núñez (Eds.), Semiotic and cognitive science essays on the nature of mathematics (pp. 1–34). Lincom Europa. Ardila A., & Rosselli M. (2002). Acalculia and dyscalculia. Neuropsychology Review, 12, 179–231. Aristotle. (360 BCE). On interpretation. CreateSpace Independent Publishing Platform. Black, M. (1962). Models and metaphors. Cornell University Press. Brown, T. (2001). Mathematics education and language: Interpreting hermeneutics and post-structuralism. Kluwer. Butterworth, B. (1999). What counts: How every brain is hardwired for math. Free Press. Butterworth, B., Varma S., & Laurillard D. (2011). Dyscalculia: From brain to education. Science, 332, 1049–1053. Cartmill, M., Pilbeam, D., & Isaac, G. (1986). One hundred years of paleoanthropology. American Scientist, 74, 410–420. Costa, S., Martinovic, D., & Danesi, M. (2020). Mathematics (education) in the information age. Springer. Courant, R., & Robbins, H. (1941). What is mathematics? An elementary approach to ideas and methods. Oxford University Press. Damasio, A. R. (1994). Descartes’ error: Emotion, reason, and the human brain. G. P. Putnam’s Danesi, M. (Ed.). (2019). Interdisciplinary perspectives on math cognition. Springer. Danesi, M. (2020). Pi (π) in nature, art, and culture: Geometry as a hermeneutic science. Brill. Dehaene, S. (1997). The number sense: How the mind creates mathematics. Oxford University Press. Dehaene, S. (2004). Arithmetic and the brain. Current Opinion in Neurobiology, 14, 218–224. Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20, 487–506 Devlin, K. J. (2012). The symbol barrier to mathematics learning. In M. Bockarova, M. Danesi, & R. Núñez (Eds.), Semiotic and cognitive science essays on the nature of mathematics (pp. 54–60). Lincom Europa. Emmer, M. (Ed.). (2016). Mathematics and culture. Springer. English, L. D. (Ed.). (1997). Mathematical reasoning: Analogies, metaphors, and images. Lawrence Erlbaum Associates. Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. Basic Books. Freud, S. (1899). Die Traumdeutung. Franz Deuticke. Gamwell, L. (2015). Mathematics and art: A cultural history. Princeton University Press. Gardner, H. (1985). The mind’s new science: A history of the cognitive revolution. Basic Books.

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Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Teil I. Monatshefte für Mathematik und Physik, 38, 173–189. Godino, J. D., Font, V., Wilhelmi, M. R., & Lurduy, O. (2011). Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects. Educational Studies in Mathematics, 77, 247–265. Grondin, J. (1994). Introduction to philosophical hermeneutics. Yale University Press. Guhe, M., et al. (2011). A computational account of conceptual blending in basic mathematics. Cognitive Systems Research, 12, 249–265. Heidegger, M. (2008). Ontology: The hermeneutics of facticity. Indiana University Press. Hersh, R. (1997). What is mathematics really? Oxford University Press. Hersh, R. (2014). Experiencing mathematics. American Mathematical Society. Hofstadter, D. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books. Hofstadter, D. (1985). Metamagical Themas. Basic Books. Isaacs, E. B, Edmonds, C. J., Lucas, A., & Gadian, D. G. (2001). Calculation difficulties in children of very low birthweight: A neural correlate. Brain, 124, 1701–1707. Izard, V. Pica, P., Pelke, E. S., & Dehaene, S. (2011). Flexible intuitions of Euclidean geometry in an Amazonian Indigene Group. PNAS, 108, 9782–9787. Kant, I. (1781). Critique of pure reason. (Trans.: Kemp Smith, N.). St. Martin’s. Kasner, E., & Newman, J. R. (1940). Mathematics and the imagination. Simon and Schuster. Lakoff, G. (2012). The contemporary theory of metaphor. In M. Danesi & S. Maida–Nicol (Eds.), Foundational texts in linguistic anthropology (pp. 128–171). Canadian Scholars’ Press. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books. Marcus, S. (2012). Mathematics between semiosis and cognition. In M. Bockarova, M. Danesi, & R. Núñez (Eds.), Semiotic and cognitive science essays on the nature of mathematics (pp. 99–129). Lincom Europa. Maturana, H. R., & Varela, F. (1973). Autopoiesis and cognition: The realization of the living. Reidel. McComb, K., Packer, C., & Pusey, A. (1994). Roaring and numerical assessment in contests between groups of female lions, Panthera leo. Animal Behavior, 47, 379–387. McGann, J. (2000). Marking texts of many dimensions. In S. Schreibman, R. G. Siemens, & J. M. Unsworth (Eds.), A companion to digital humanities (pp. 358–376). Wiley. Navin Sullivan, J. W. (1925). Aspects of science. A. A. Knopf, Neisser, U. (1967). Cognitive psychology. Prentice-Hall.

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Núñez, R., Edwards, L. D., & Matos, F. J. (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics,39, 45–65. Oevermann, U. Allert, T., Konau, E., & Krambeck, J. (1987). Structures of meaning and objective hermeneutics. In V. Meja, D. Misgeld, & N. Stehr (Eds.), Modern German sociology: European Perspectives (pp. 436–447). Columbia University Press. Piaget, J. (1952). The child’s conception of number. Routledge and Kegan Paul. Pimm, D. (1995). Symbols and meanings in school mathematics. Routledge. Popper, K. (1972). Objective knowledge: An evolutionary approach. Oxford University Press. Presmeg, N. C. (1997). Reasoning with metaphors and metonymies in mathematics learning. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 267–280). Lawrence Erlbaum. Presmeg, N. C. (2005). Metaphor and metonymy in processes of semiosis in mathematics education. In J. Lenhard & F. Seeger (Eds.), Activity and sign (pp. 1050–1116). Springer. Presmeg, N., Radford, L., Roth, W.-M., & Kadunz, G. (Eds.). (2018). Signs of signification: Semiotics in mathematics education research. Springer. Rosch, E., Thompson, E., & Varela, F. (1991). The embodied mind: Cognitive science and human experience. MIT Press. Saussure, F. de (1916). Cours de linguistique générale. Payot. Schank, R. C. (1980). An artificial intelligence perspective of Chomsky’s view of language. The Behavioral and Brain Sciences, 3, 35–42. Schank, R. C. (1984). The cognitive computer. Addison-Wesley. Schank, R. C. (1991). The Connoisseur’s guide to the mind. Summit. Seebohm, T. M. (2007). Hermeneutics: Method and methodology. Springer. Senechal, M. (Ed.). (2013). Shaping space: Exploring Polyhedra in nature, art, and the geometrical imagination. Springer. Sriraman, B. (Ed.). (2020). Handbook of the mathematics of the arts and sciences. Springer. Varelas, M. (1989). Semiotic aspects of cognitive development: Illustrations from early mathematical cognition. Psychological Review, 100, 420–431. Vygotsky, L. S. (1961). Thought and language. MIT Press. Whiteley, W. (2012). Mathematical modeling as conceptual blending: Exploring an example within mathematics education. In M. Bockarova, M. Danesi, & R. Núñez (Eds.), Semiotic and cognitive science essays on the nature of mathematics (pp. 256–279). Lincom Europa. Zimmermann, J. (2015). Hermeneutics: A very short introduction. Oxford University Press.

Section I Mathematics and Cognition Donna E. West

Abstract

This section of the handbook is unique, in that it articulates the source for quantitative operations, both theoretically and pragmatically. Section contributors (Magnani, Zalamea, West, and Danesi) identify specific cognitive skills foundational to mathematics, namely: analogy, binding/chunking, classifying objects, and the like. These cognitive competencies provide vital intellectual and motivational building blocks to advance from the indeterminacy of objects and their comparisons to the determinant principles critical to inferencing abductively. Each contributor argues that the intellectual and affective competencies supplied by cognition facilitate several mathematical constructs (e.g., numerosity). In short, this section demonstrates how cognition affords us economy – to group individual entities into their functional characteristics, ultimately making practical sense of world knowledge. Keywords

Abduction · Neuro-networks · Affordances · Affect · Memory chunking · Metaphor

Introduction The chapters in this section validate how cognition establishes and undergirds structure-based phenomenon, particularly that of Mathematics. Topics range from unconscious inferences underlying action responses, namely, affordances, to the issue of feeling-based motivations which trigger workable hypotheses, enhancing memory units into episodic chunks. It moves to the claim that mathematical metaphors derive from spatial and temporal embodied experience. Each of these approaches demonstrates that aspects of the cognitive system influence mathematical structures. The contributors argue that the cognitive factors underpin the comprehension of how integers and geometric figures effect one another and that

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inferencing is a cognitive and a logical operation which integrates pragmatic as well as scientific reasoning skills. The reasoning competence animating the interplay between cognition and mathematics is abductive rationality – binding several propositions/arguments into a plausible, explanatory inference. For Magnani, inferences in the form of abduction constitute a cognitive mathematical process brought together by manipulating/reacting to affordances in the environment. These embodied responses to sets of environmental stimuli are driven by unconscious inferences – logical structures which Magnani refers to as manipulative abductions. These kinds of abduction demonstrate the involvement of neuronetworks, namely, mirror, unveiling diagrams. This embodied-neuro process becomes a primary catalyst to arrive at abductive inferences, plausible hypotheses informing the organism about how to orchestrate functional changes. Like Magnani, Zalamea demonstrates how feelings can direct the generation of new viable hypotheses. He refers to these determinative feelings as “corazón” (heart). He claims that “razón” (reason) is insufficient to supply the motivation for developing inferences which compel changes in theoretical orientations in Mathematics. Accordingly, Zalamea argues that Grothendieck’s Topos Theory transforms mathematics into a pragmatic science, such that intuitions guiding mathematics surface consequent to feeling and logic. This approach demonstrates the necessity of affect to resolve mathematical problems. West demonstrates how signs that are not symbolic in nature (icons, indices) can, by implication, display mathematical meanings. The cognitive operation of chunking units in working memory performs similar functions to scientific operations, in integrating existing meanings with new ones to formulate viable inferences. Because mathematical structures are informed by both intuitive meanings and acquired/explanatory ones, they necessarily rely upon bindings to experience insights foundational to alternative, plausible hypotheses. Danesi utilizes blending theory to show how logical and practical knowledge are inextricably bound, each facilitating the other. Danesi draws upon metaphor to illustrate binding from genre to unlike genres; he claims that the isomorphism of their origin (between primitive linguistic meanings and mathematical ones) validates the inferences which bind them, namely hypotheses regarding their similarities. Hence, meaning similarities between language and mathematical algorithms are accounted for by metaphoric paradigms. Danesi applies the metaphoric paradigm of similarity/analogy to everyday cognition and logical insight. His position is that both materialize from the need to infer rational reasons for unexpected experiences. Underlying all of the contributors’ claims regarding the interplay between cognition and Mathematics is Peirce’s notion of abductive logic. What truly enhances mathematical advancement is not chiefly the means to determine functional similarities; instead, Mathematics is enhanced by the perpetual drive to explain the nature of causal relations between happenings. Therefore, Peirce’s concept of insight (untaught, instinctual guesses explaining phenomenon) (1903: 5.181) constitutes the quintessential building block for both practical and theoretical understanding.

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Magnani’s Contribution Magnani’s notion of the interdependence between Mathematics and cognition is largely pragmatic. His position departs markedly from adherence of mathematics to mechanical rule-based logic/virtues. Magnani uses Lobachevsky’s model as a quintessential foundation for his own model-based reasoning paradigm, since both are discovery-based and are characterized as a “first non-Euclidean geometry approach.” As such, Magnani capitalizes on the diagrammatic and heuristic components of abductive models to demonstrate how mathematics is derivative from cognition. Magnani further illustrates the necessity of pragmatic operations to advance diagrammatic, hence mathematical meanings. Accordingly, the work of embodied cognition instructs mathematical algorithms, through manipulative abduction – a concept which Magnani coins. According to Magnani, manipulative abductions link structural hypotheses with extra-theoretical dimensions of geometrical cognition. For Magnani, abduction is a cognitive mathematical process, structure-based, brought together by responses to affordances in the spatiotemporal context, such that embodied components form the foundation for the generation of manipulative abductions, and illustrate the role played by so-called mirror, unveiling diagrams. Mirror and unveiling diagrams constitute indexical signs emanating from cognitive-epistemological reconstructions of non-Euclidean geometric discoveries. According to Magnani, “Manipulative abduction. . . is widespread in cognitive behaviors that aim at creating accounts of new communicable experiences; . . . [it] represents a kind of redistribution of the epistemic and cognitive effort to manage objects and information that cannot be immediately represented or found “internally” (Magnani, 2009, and this volume). Moreover, manipulative abduction for humans is analogous to “the construction of external models in a neural engineering laboratory or in mathematics, exploiting external diagrams, proofs, and computational artifacts.” Magnani analogizes the process of creative abduction to the operation of the construction of diagrams and mathematical proofs: “. . .creative abduction [is] formed by the application of heuristic procedures that involve all kinds of good and bad inferential actions.” Magnani then determines that it is only by means of the aforementioned heuristic procedures that “the acquisition of new truths is guaranteed.” Magnani’s incorporation of Gibson’s theory of affordances in physical and social contexts further supports the utility of his own reasoning paradigm, in view of the critical function of manipulation to generate alternative proposals incorporating more fitting contextually derived hypotheses for the purpose of, for example, determining how to avoid dangers in the face of threats. In the end, Magnani demonstrates an amplified function of manipulative abductions, namely, their means to integrate cognitions, as distributed cognitions. In this way, when manipulative abductions are applied to diverse genres of experience, they reach the status of creative abductions. In this way, Magnani demonstrates the value of manipulative abductions in generating creative ones – manipulative abductions become creative “when we are thinking through doing, not merely about doing” (cf. Magnani, 2009,

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and this volume). His rationale is that the embodied element of doing necessitates inquiry, particularly interrogative operations. Magnani additionally demonstrates that apprehension of affordances (what the environment permits/prohibits) while doing results in novel, diagrammatic representations whose meanings contain new pathways of thinking and acting.

Zalamea’s Contribution Zalamea shows how “mathematical cognition lies on the borders of art and science, profiting equally from compact esthetical intuitions, deep hypothetical visions, and lengthy rational deductions.” His use of “esthetical intuitions” clearly illustrates how the arts bolster the sciences, by providing a directed feeling for the creation of new viable hypotheses. Zalamea refers to this feeling as “corazón” (heart) as opposed to mere “razón” (reason). Zalamea’s integration of corazón and razon echoes C. S. Peirce’s contention, that reason without a feeling to support a new hypothesis is sterile, verbatim – resulting in mere mechanistic conduct/beliefs. In furtherance of this approach, Zalamea cites Grothendieck, namely, that the heart (the arts) facilitates the synthesis between the world of arithmetic and that of continuous magnitudes, so that individual integers benefit from their application in particular algorithms. Thus, Zalamea argues that Grothendieck’s Topos Theory transforms mathematics into a pragmatic science. This transformation permits cognition and mathematical meanings to “become welded together in a natural web of perspectives which enhances our grasp of the world.” In this way, Grothendieck explains how music synthesizes logic and affect, by contributing sequenced structures whose meanings are affectively driven. Zalamea utilizes this claim to emphasize that the operation of mathematics supersedes structural representations when its meaning-making is achieved by sculpting integer relations via emotive meanings. This affectively driven approach is the vehicle for transmitting explanatory elements unique to creative abductions. In short, Zalamea claims that the integration of feeling and logic in Mathematics is the source for insightful mathematical problem resolution.

West’s Contribution West’s contribution demonstrates how Peirce’s insight and foresight bring together the competencies upon which both mathematics and cognition rely. She mounts a compelling case that signs which are not symbolic in nature (icons, indices) can, by implication, mimic mathematical meanings. This operation surfaces through a kind of co-localization, in which icon and index are bound to new predicates; and previously conceived propositional subjects acquire novel meanings. This binding is accomplished by the work of informational indices – when iconic signs become associated with legends. West’s proposal echoes Stjernfelt’s characterization of Peirce’s Dicisign, whereby signs other than symbols acquire additional predicates; and the process of subject-predicate binding becomes the source of more amplified and more abstract meanings. West presents a convincing argument that binding

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viable predicates to icons is analogous to binding integers or expressions in mathematical frameworks, since both semiotic operations advance inferential reasoning. The cognitive operation of chunking units in working memory (depending largely upon iconic and/or indexical sign relations) performs similar functions to scientific operations, in that existing meanings are integrated with new ones – changing the habits and hypotheses which underlie them. This integration, in turn, alters the nature and purpose of the sign (as newly determined with expanded/restricted meanings), which uncovers the existence of previously undiscovered perceptual/functional similarities with other groupings. West articulates this claim as follows: “Analogies both unify smaller units into larger, more potent ones, and likewise conserve the number of meaning components permitted to be encoded, and stored in WM and eventually in LTM. Binding/chunking processes can be unconscious, or conscious; but, in either case, their composite meanings advantage propositional meanings by incorporating units (linguistic, sensory stimuli) into larger aggregates.” West likewise shows how chunking is not limited to human WM processing. Primitive forms of chunking such as numerosity illustrate how diverse species bind new meanings with existing signs, when they connect form (signs in the environment) with novel effects. West makes plain how this operation (numerosity) supports the emergence of number as an uncountable phenomenon (namely “more”ness). She presents evidence that in humans uncountable “moreness” is the most basic form of chunking ontogenetically. West advocates that mathematical structures are informed by both intuitive meanings and acquired/explanatory ones; hence they are open to bindings consequent to proposals of alternative, plausible hypotheses. Despite the oftenunconscious operation of chunking, it, nonetheless, is necessary to manufacture hypotheses which direct future habits of belief and action. Peirce’s pragmaticistic approach becomes particularly relevant to West’s discussion of WM binding given the need for unconscious groupings to provide a foundation for new hypotheses in that smaller perceptual and meaning units bundle into larger episodes (West, 2017, 2018). In this way, unconscious meanings figure in the process of binding subjects to different predicates: “When higher-level chunks bring together (analogize) elements of propositions and arguments, they acquire an episodic purpose.” Accordingly, West explains how the unconscious operation of chunking structural units (not unlike mathematical procedures) is necessary to exploit meaning paradigms. West convinces readers that binding smaller units into larger ones constitutes a mathematical operation inextricably dependent upon distinctive memory processes. The amplified but determinative meanings which larger memory bindings result in are indispensable to logical advances, namely, those which culminate in revisionary hypotheses.

Danesi’s Contribution Danesi’s unique contribution to this section is the utilization of Fauconnier and Turner’s “blending theory” to explain the relevance of cognition to mathematics. Blending theory demonstrates how the everyday mind draws upon and integrates “all our realities, from the social to the scientific.” In this way, blending theory shows

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how logical and practical knowledge are inextricably bound, and how each facilitates the other. Danesi further utilizes a metaphor-based pedagogy to argue the validity of this connection, drawing upon Lakoff and Nunez’ conceptual metaphor theory. He contends that such theory clarifies how cognitive knowledge of space and time underpin mathematical operations/concepts. Danesi supports this position by establishing the relevance of Lakoff and Nunez’s neuro-based metaphor, namely, that primitive linguistic meanings and mathematical ones draw upon the same origin and must possess similar meaning-structure units: “math cognition springs from the same neural processes that undergird language.” Danesi shows the pragmatic effects of this connection by explaining that every day cognition depends heavily upon logical insight gleaned from our need to infer rational reasons for unexpected phenomena. Danesi utilizes Peirce’s model of abduction to link mathematics as a science to the pragmaticism inherent in daily actions necessitating cognitive operation. For Danesi, the latter is instrumental in alighting upon viable rationale for what confronts us. Danesi implies that Peircean insight is the quintessential building block for both practical and theoretical understanding; hence it serves as a metaphor, or guiding light to advance derivative explanations across all disciplines. He returns to Fauconnier and Turner’s “blending theory” to frame other theories “[t]he notion that ideas come ‘like a flash’ coincides with the notion of blending considerably,” bringing Turner’s concept of “subjective creative inferences” into the orbit of Peirce’s abduction and hypothesis-generation (1903: 5.181). Although the LakoffNuñez model is critiqued as mere analogy, Danesi stops at the brink of declaring which model among many best accounts for blending theory. Danesi might have recognized the full advantage of Peirce’s abductive model, namely, that it supplies the capacity for blending propositions and arguments by way of its inferential property and production of new knowledge.

Conclusory Remarks This section of the handbook is unique, in that it articulates the source for quantitative operations, both theoretically and pragmatically. Section contributors identify specific cognitive skills foundational to mathematics, namely, analogy, binding/ chunking, classifying objects, and the like. In fact, they are more than ancillary to the delivery of propositional and argument-based logic, proving their necessity for the generation of inferential rationality. These cognitive competencies provide vital intellectual and motivational building blocks to advance from the indeterminacy of objects and their comparisons to the determinant principles critical to inferencing abductively. As such, cognition affords us economy – to group individual entities into their functional characteristics, ultimately making practical sense of world knowledge. Magnani’s incorporation of heuristics and manipulative abductions illustrates how organisms act upon their already conceived of functional object classifications – when they interact with, make novel discoveries about, and change underlying inferences. In this way, manipulative abductions manifest how earlier cognitions (bindings of object

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attributes) spur future conduct and propositions of revisionary hypotheses. Zalamea reminds us of the indispensability of emotional triggers to hasten inferential logic. West’s contribution demonstrates how the cognitive operation of binding memory chunks creates novel propositions/arguments, further enhancing the structure and content of inferences. Danesi convinces us of the relevance of primitive cognitive meanings (space and time) to the generation of metaphoric concepts within the realm of Mathematics. The four contributing authors emphasize the often-unrecognized interplay between psychological competencies and the onset of primary mathematical skills. Each contributor argues that the intellectual and affective competencies supplied by cognition facilitate several mathematical constructs. Absent their means to provide foundational meaning components for problem-solving, inferencing in logical genres would be cut off at the quick.

References Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. Basic. Gibson, J. J. (1979). The ecological approach to visual perception. Lawrence Erlbaum Associates. Grothendieck, A. (1958). The cohomology of abstract algebraic varieties. In Proceedings international congress of mathematicians (Edinburgh) (pp. 103–118). Cambridge University Press. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books. Magnani, L. (2009). Abductive cognition: The epistemological and eco-cognitive dimensions of hypothetical reasoning. Springer. Peirce, C. S. (i.1866–1913). The collected papers of Charles Sanders Peirce Vols. I–VI, ed. C. Hartshorne and P. Weiss (Cambridge, Massachusetts: Harvard University Press, 1931–1935); Vols. VII–VIII, ed. A. Burks (1958). Cited with the CP convention of volume and paragraph number CP X.yyy. Stjernfelt, F. (2014). Natural propositions: The actuality of Peirce’s doctrine of dicisigns. Docent Press. West, D. (2017). Virtual habit as episode-builder in the inferencing process. Cognitive Semiotics, 10(1), 55–75. West, D. (2018). Fashioning episodes through virtual habit: The efficacy of pre-lived experience. Studia Gilsoniana, 7(1), 81–99.

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The Cognitive and Epistemic Value of Mathematics: Making the World Intelligible – The Role of Abduction, Diagrams, and Affordances Lorenzo Magnani

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics Is Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Constructions as Cognitive Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics as Synthetic A Priori Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beyond Metaphysics: Mathematics Generates “Objective Knowledge” . . . . . . . . . . . . . . . . . . . . Beyond Kant: Seeing Mathematics as the Transformation of the A Priori Determination of Forms of Intuitions in Formal Intuitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics Makes Up New “Principles of Experience” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cognitive Virtues of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics and Ordinary Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Modeling in Science as an Antidote Against the Epistemological Pretensions of Big Data and Deep Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics, Abduction, and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics and Manipulative Abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical and Unveiling Diagrams in Mathematical Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mirroring und Unveiling Hidden Properties Through Optical Diagrams . . . . . . . . . . . . . . . . . . . Externalizing Internal Models to Represent and Discover Mathematical Entities: Resolving Conceptual Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Externalizing Diagrammatic Models to Unveil Imaginary Entities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abducing First Principles Through Bodily Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Euclidean Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unveiling Diagrams in Lobachevsky’s Discovery as Affordances: Gateways to Imaginary Entities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One of the Main Mathematical Cognitive Virtues: Not Indicative of Intuition . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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L. Magnani (*) Department of Humanities, Philosophy Section and Computational Philosophy Laboratory, University of Pavia, Pavia, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_42

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Abstract

When dealing with the relationship between mathematics and cognition, we face two main intellectual traditions. First of all the abundant studies about the role of mathematics in the human (and animal) development of cognitive abilities; second, the philosophical reflections upon the various ways provided by mathematics in generating specific kinds of “knowledge.” Among the various perspectives offered by the philosophical studies about the status of mathematics, I think that Immanuel Kant’s ideas represent a valuable and indispensable fil rouge able to furnish a conceptual instrument which can highlight how mathematics and cognition are strictly intertwined. I say that Kantian perspective constitutes a conceptual fil rouge because it is only through it that it is possible to synthetically understand the epistemological nature of the various approaches at play. Kant provides a philosophical anti-metaphysical framework for mathematics that constitutes a fundamental defense of its role in high-level cognitive activities and its capacity to make rational intelligibility of the world, avoiding old-fashioned ontological views: the empirical world becomes a world of mathematical relations. I contend that it is thanks to Kantian philosophy of mathematics that the door to the subsequent studies regarding the cognitive and epistemic value of mathematics is opened up. I will take advantage of this classical perspective to provide new insight into some of the main problems related to the issue: (1) the historicization/naturalization of mathematics, which shows that their cognitive mechanisms of discovery and application and their historical development are strictly interrelated; (2) the role of manipulative abduction, affordances, model-based and diagrammatic reasoning, and distributed cognition as ways for clarifying the cognitive aspects of mathematics in the context of discovery; (3) the emphasis on the cognitive virtues of mathematical modeling in science as an antidote against the recent exaggerated attention to the management of big data, as a way of reaching scientific results; and (4) the lack of a mathematical genuine cognitive schematic effort of creating scientific intelligibility, which often leads to mere surrogate “modeling,” unreasonably supposed to be scientific. Finally, taking advantage of the Lobachevskian discovery of the first non-Euclidean geometry I will exemplify the issue of the abductive, model-based, diagrammatic, heuristic, and the extra-theoretical dimension of geometrical cognition, by illustrating the role played by the so-called mirror and unveiling diagrams. Keywords

Mathematics · Cognition · Intelligibility · Kant · Mathematical discovery · Abduction · Models · Affordances · Model-based reasoning · Schematization · Axiomatics · Diagrammatic reasoning · Geometrical construction · Manipulative abduction · Mirror diagrams · Unveiling diagrams · Mental models · Internal and external representations

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Introduction When dealing with the relationship between mathematics and cognition, we face two main intellectual traditions. First of all, the abundant studies about the role of mathematics in the human (and animal) (Cf., for example, the classical study by Brannon (2019), concerning “what animals know about numbers”) development of cognitive abilities, more or less taking advantage of an interdisciplinary perspective and of recent research in the broad field of cognitive science. On this issue cf. for example, Campbell (2005): part 1 of this handbook is devoted to “Cognitive Representations for Numbers and Mathematics” and lists, among the others, the well-known leading authors Rafael Núñez and George Lakoff, who inaugurated novel ideas concerning the role of conceptual metaphor in the cognitive foundations of mathematics; part 2, on “Learning and Development of Numerical Skills” together with parts 3, on “Learning and Performance Disabilities in Math and Number Processing,” and 4 on “Calculation and Cognition” deal with basic and important problems that regard strict psychological research and the consequences for education and pedagogy. The last part 5, on “Neuropsychology of Number Processing and Calculation,” presents studies already belonging to the more extended and interdisciplinary field of cognitive science, also involving other renowned authors such as Stanislas Dehaene and Brian Butterworth and, again Núñez and Lakoff. Still devoted to psychology of mathematics is the old collection (Nesher & Kilpatrick, 1990), which deals with pioneering cognitive studies regarding arithmetics, geometry, and algebra in children learning and the role of the intertwining between language and mathematics in educational settings. Recent studies oriented by both classical and cognitive science multidisciplinary perspectives are collected in the rich Bockarova, Danesi, Martinovic, and Núñez (1990) and Danesi (2019). Second, the philosophical reflections upon the various ways provided by mathematics in generating specific kinds of “knowledge,” a perspective I will adopt in the present chapter. An interesting unconventional recent article (Karaali, 2019) usefully addresses the problem regarding the “mathematical ways of knowing,” stressing the attention to the intertwining between mathematics, rationality and imagination, universals and eclecticisms, certainty and ambiguity, and the applications of mathematical ways of knowing, presumed to affect both human identity and self-knowledge. Among the various perspectives offered by the philosophical studies about the status of mathematics, I think that Immanuel Kant’s ideas represent a valuable and indispensable fil rouge able to furnish a conceptual instrument which can highlight how mathematics and cognition are strictly intertwined. I say that Kantian perspective constitutes a conceptual fil rouge because it is only through it that it is possible to synthetically and appropriately understand the epistemological nature of the various approaches at play. Essentially, Kant provides a philosophical anti-metaphysical framework for mathematics that constitutes a fundamental defense of its role in high-level cognitive activities and its capacity to make rational intelligibility of the world, avoiding old-fashioned ontological views: the empirical world becomes a world of mathematical relations. It is an unsurpassed anti-metaphysical philosophical vision of the

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epistemological importance of mathematics and of its applicability in science that opens up the whole field not only of the cognitive value of mathematics but also of its “epistemic” centrality. The a priori determination of forms of intuition in formal intuitions, provided by mathematics, not only opens up the horizon of various axiomatics but also the horizon of a conceptual development which permits a schematization, in the sense of a mediated construction of concepts, of an indefinitely open series of regional categories as newly created “principles of experience” (e.g., the Newtonian schema of the physical concept of determinist evolution through the mathematical concept of the dynamic system). (The reader has to pay attention to this particular sense of the word “intuition,” adopted by Kant, which contrasts with the standard meaning, related to a psychological attitude.) Thanks to Kantian philosophy of mathematics, the door to the studies regarding the cognitive and epistemic value of mathematics is opened. I will take advantage of this classical perspective to provide new insight into some of the main problems related to the issue: 1. The generalization of Kantian “Aesthetics” and of “Logic” in favor of a historicization/naturalization of mathematics (which shows how the immanent “dialectics” of mathematical concepts, their cognitive mechanisms of discovery and application, and their historical development are strictly interrelated). 2. The role of manipulative abduction (Magnani, 2001a), affordances, model-based and diagrammatic reasoning, and distributed cognition (as ways of clarifying the cognitive aspects of mathematics in the context of discovery, taking into account both an eco-cognitive perspective and the role of external representations (Magnani, 2017)). (An interesting and rich article about the variety of numerical representations from the point of view of classical neuropsychological, experimental, and developmental studies is Fayol and Seron (2019).) 3. The emphasis on the cognitive virtues of mathematical modeling in science as an antidote against the recent exaggerated attention to the management of big data (currently and unreasonably presented as aiming at substituting human-centered scientific understanding, but leading to unsubstantial/spurious computerdiscovered correlations). 4. The lack of a mathematical genuine cognitive schematic work for creating scientific intelligibility (which often leads to mere surrogate “modeling,” supposed to be scientific). This epistemological situation, for example, affects (aspects of) psychology and (aspects of) economics, and other human sciences, areas which do not – or scarcely do – reach the most common received epistemological standards, such as predictive power.

Mathematics Is Knowledge Mathematical Constructions as Cognitive Activities In the “Transcendental Doctrine of Method,” Kant affirms: “Thus I construct a triangle by representing the object which corresponds to this concept either by imagination alone, in pure intuition, or in accordance therewith also on paper, in

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empirical intuition – in both cases completely a priori, without having borrowed the pattern from any experience” (Kant, 1787, A713-B741, p. 577). When I draw a triangle in a sheet of paper or in a blackboard – that is, in empirical intuition, in Kantian terms – or in the pure intuition, that is in what recent cognitive scientists would call “visual buffer” (Kosslyn & Koenig, 1992), I am constructing a mathematical concept taking advantage of a cognitive manipulation of the external environment. This last one is suitably made artifactual, exactly thanks to a diagram, as in the case above of the triangle, but, in general, also thanks to various kinds of imagined and written symbols, or to both diagrams and symbols. I have to stress that for Kant, the nature of mathematics is embedded in his philosophical concept of construction, which is the lifeblood of both mathematics as established knowledge and of its growth in terms of discoveries and novelties. Mathematical knowledge essentially presents a dynamical aspect. I also wholeheartedly agree with Kant, but also with Peirce’s following observation – surely indebted to his reading of Kant – written about a century before the new perspective on the dynamic of mathematical knowledge and on mathematical discovery offered by Lakatos’s work (Lakatos, 1976): (The famous 1976 Lakatos’ book Proofs and Refutations: The Logic of Mathematical Discovery opens up – following the epistemological Popperian tradition – the first authoritative perspective on the dynamics of mathematical knowledge and on the mathematical cognitive processes of discovery.) “It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in nature, and draws its conclusions apodictically, while in the other hand, it presents as rich and apparently unending a series of surprising discoveries as any observational science” [1866–1913 (1885), 3.363]. Let us continue to analyze Kant’s ideas regarding the cognitive status of mathematics. In the sentence above, he says that “in both cases,” that is both in a paper or in the internal mental visual buffer – “imagination” in Kant’s terms – the construction is occurring “completely a priori, without having borrowed the pattern from any experience”: these words express the Kantian conviction that mathematics is able to refer to the external world without learning anything from the external world. Mathematics is characterized by the so-called synthetic a priori judgments. Again, adopting the Kantian lexicon, the construction of a concept which Kant describes in the “Transcendental Doctrine of Method,” to be intended as the a priori “exposition” of the corresponding intuition, is isomorphic to the definition of schematism seen as a “procedure” through which imagination acts “in providing an image for a concept” (Kant, 1787, A140-B180, p. 182). The crucial Kantian concept of schematism refers to something capable to mediate between the cognitive agent – thanks to his productive imagination – and the things as they appear at the phenomenological level: we will soon see the role of schematization played by mathematics. The schema is therefore a rule, as Kant observed in A141-B180, but also a model, a procedure, or a method, always distinct from the image. It is activated by the productive imagination. The schema has an empirical nature, in the sense that it refers to things as they appear. Schematism is “[. . .] an art concealed in the depths of the human soul, whose real modes of activity nature is hardly likely ever to allow us to discover, and to have open to our gaze” (Kant, 1787, A141-B181, p. 183).

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An exemplification of the mechanism that characterizes schematism concerns geometrical cognition: a geometric schema is not only the index of the conditions of the “construction” but also the criterion of their identification. Once faced with a particular empirical intuition, the geometrical schema that selects the suitable concept (e.g., a triangle) is also able to recognize and identify the shape of the specific empirical intuition (because with it that concept is compatible, e.g., a triangular object). Thus it is possible to know a pure triangle, that is, imagine it as an object, but it is also possible to recognize an empirical object as triangular. In sum, we can conclude by affirming that schematism is the condition of possibility of constructions and that constructions substantiate the identification: that is, we can establish a world of objects that are mathematically recognized.

Mathematics as Synthetic A Priori Knowledge Kant further stresses the fact that in mathematics the determination of the concept “in conformity with” the conditions of intuition is necessary: this means that mathematical concepts always coherently refer to something regarding the external world, the environment as “intuition” (on the Kantian concept of intuition see above). These “determinations” are made possible by the activity of the schematism of imagination (to be intended not only as reproductive and associative, which are merely psychological properties, but also, as already said, “productive”). This explains how mathematics – as it is well-known – is considered by Kant as productive of synthetic a priori knowledge. For example, the geometrical concept of a straight line is related to the spatial conditions that in turn are expressed by those axioms (or postulates) – the Euclidean ones – describing the properties of the space in which certain constructions are appropriate. Kant clearly stresses that, in mathematics, the analytic (or discursive) method cannot produce new knowledge. Construction, on the contrary, allows us to “pass beyond” and, fundamentally, is responsible for the growth of mathematical knowledge: “For I must not restrict my attention to what I am actually thinking in my concept of a triangle (this is nothing more than the mere definition); I must pass beyond it to properties which are not contained in this concept, but yet belong to it” (Kant, 1787, A717-B746, p. 580). Here Kant implicitly refers to a kind of cognitive manipulation as the one which we can see in the diagrammatic demonstration illustrated in Fig. 1, taken from the field of elementary geometry. In this case a simple manipulation of the triangle in Fig. 1a gives rise to an external configuration – Fig. 1b – that carries relevant semiotic information about the internal angles of a triangle “anchoring” new and extended meanings. The representation is external – in Kantian terms – in the “empirical intuition,” for example, a sheet of paper or on a blackboard, and happens a priori. This example is explicitly quoted in a famous passage regarding mathematical reasoning in the “Transcendental Doctrine of Method,” the last part of the Critique of Pure Reason (cf. Magnani (2001b, p. 47)), in which the foundational role of constructions is emphasized:

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Fig. 1 Diagrammatic demonstration that the sum of the internal angles of any triangle is 180
. (a) Triangle. (b) Diagrammatic manipulation/construction

Suppose a philosopher be given the concept of a triangle and he be left to find out, in his own way, what relation the sum of its angles bears to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, and possessing three angles. However long he meditates on this concept, he will never produce anything new. He can analyse and clarify the concept of a straight line or of an angle or of the number three, but he can never arrive at any properties not already contained in these concepts. Now let the geometrician take up these questions. He at once begins by constructing a triangle. Since he knows that the sum of two right angles is exactly equal to the sum of all the adjacent angles which can be constructed from a single point on a straight line, he prolongs one side of his triangle and obtains two adjacent angles, which together are equal to two right angles. He then divides the external angle by drawing a line parallel to the opposite side of the triangle, and observes that he has thus obtained an external adjacent angle which is equal to an internal angle – and so on (it is Euclid’s Proposition XXXII, Book I of the Elements, cf. above Fig. 1.) In this fashion, through a chain of inferences guided throughout by intuition, he arrives at a fully evident and universally valid solution of the problem. (Kant, 1787, A716-B744, pp. 578–579)

We can depict the situation of the philosopher described by Kant at the beginning of the previous passage taking advantage of some ideas coming from catastrophe theory (further details are given in Magnani (2009, Ch. 8)). As a human being who is not able to produce anything new relating to the angles of the triangle, the philosopher experiences a feeling of frustration (just like Köhler’s monkey which cannot keep the banana out of reach). The negative affective experience “deforms” the organism’s regulatory structure by complicating it and the cognitive process stops altogether. The geometer instead “at once constructs the triangle,” that is, he makes an external representation of a triangle and acts on it with suitable manipulations. Thom, the creator of the mathematical theory of catastrophes, thinks that this action is triggered by a “sleeping phase” generated by possible previous frustrations which then change the cognitive status of the geometer’s available internal triangle and correct it (like the philosopher, he “has nothing but the concept of a figure enclosed by three straight lines, and possessing three angles,” but his action is triggered by a sleeping phase). Here the idea of the triangle is no longer the occasion for “meditation,” “analysis,” and “clarification” of the “concepts” at play, as in the case of the “philosopher.” Here the inner concept of triangle – symbolized as insufficient – is amplified and transformed thanks to the sleeping phase (which reminds us of Kantian imagination active through schematization, we have just quoted above) in a prosthetic triangle to be put outside, in some external support (i.e., Kantian

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“empirical intuition”). The instrument (here an external diagram) becomes the extension of an organ: What is strictly speaking the end [. . .] [in our case, to find the sum of the internal angles of a triangle] must be set aside in order to concentrate on the means of getting there. Thus the problem arises, a sort of vague notion altogether suggested by the state of privation. [. . .] As a science, heuristics does not exist. There is only one possible explanation: the affective trauma of privation leads to a folding of the regulation figure. But if it is to be stabilized, there must be some exterior form to hold on to. So this anchorage problem remains whole and the above considerations provide no answer as to why the folding is stabilized in certain animals or certain human beings whilst in others (the majority of cases, needless to say!) it fails. (Thom, 1988, pp. 63–64) (A full analysis of Köhler’s chimpanzee getting hold of a stick to knock a banana hanging out of reach in terms of the mathematical models of the perception and the capture catastrophes is given in Thom (1988, pp. 62–64). On the role of emotions, for example, frustration, in scientific discovery cf. (Thagard, 2002))

I have contended above, like Kant, that the geometrical concept of a straight line is related to the spatial conditions that in turn are substantiated by those axioms (or postulates) – the Euclidean ones – describing the properties of the space in which certain constructions are appropriate. Indeed, the Kantian Axioms of Intuition explain why we can apply geometry to experience. They subsume all appearances, as intuitions in space and time, under the concept of quantity, and is thus a principle of the “application of mathematics to experience” (Kant, 1783, § 24, p. 66). Mathematics can be applied to intuitions because they are extensive quantities: “This transcendental principle [as a principle of the possibility of axioms in general] of the mathematics of appearances greatly enlarges our a priori knowledge. For it alone can make pure mathematics, in its complete precision, applicable to objects of experience. Without this principle, such application would not be thus self-evident; and there has indeed been much confusion of thought in regard to it” (Kant, 1787, A165B206, p. 200). The fact that the sum of the internal angles of a triangle equals two right angles – see the simple example above – is external to the pure concept of triangle. Or, we can say in other words, the conditions of intuition are expressed by the axioms or postulates (exactly thought as “intuitive” principles) that “permit” the constructions (we know these constructions – in the case of Kant – as limited to a specific space, the Euclidean space). In sum we can conclude that in this Kantian perspective, the assertion about the synthetic a priori character of mathematics is equivalent to the assertion about its “axiomatic structure.”

Beyond Metaphysics: Mathematics Generates “Objective Knowledge” Mathematics is related to cognition because it provides “objective knowledge” that makes the world intelligible in a rational way. If Kant considered knowledge to be “objective” experience, which is the activity of experiencing objects, for this to be

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possible (it is the question of the “Analytic”), it is necessary that spatial concepts are activated, to mark the distinction between subjective and objective experience. As already stated in the previous subsection, the Axioms of Intuition, guaranteeing the applicability of geometry, function as concepts of this type; in fact, they permit the objects of experience to be specifically placed in space and time. It is necessary that the public space (and time) that embraces the objects we perceive is quite different. Space has to be (and it is) measurable (Brittan, 1978, p. 11). Moreover it is also guaranteed that we can apply geometry to this space, as an inter-subjective conceptual system point of view. In this sense mathematics is capable to produce the form of phenomena. In the meantime, this form becomes the form of an a priori field of potential experience. We could say it delineates a kind of “formal ontology,” objective. The term “formal ontology” was proposed by Edmund Husserl in the second edition of the Logical Investigations (1900–1901), to the aim of expressing the ontological counterpart of formal logic. As I will explain in the following sections, Husserl unfortunately reduces mathematics to axiomatics, and consequently the objective value of mathematics is basically lost. In the present chapter, formal ontology is intended, in a post Kantian perspective, as the fruit of the application of mathematics to the objects of experience, seen as emerged thanks to its special capacity of creating new levels of intelligibility. Kant exposes this problem by discussing the application of categories to objects of experience. Remember we have said that Euclidean geometry is applicable to experience due to the fact that the same geometric constructions can be “executed” in pure intuition as well as in empirical space, thanks to schematism. In Prolegomena, Kant very clearly expresses his opinion on this fundamental problem of the applicability of mathematics contending that his perspective goes “against all the chicaneries of a shallow metaphysics”: It will always remain a phenomenon in the history of philosophy, that there was a time, when even mathematicians, who were also philosophers, began to doubt, not indeed the correctness of their geometrical propositions in so far as they merely concern space, but the objective validity and application to nature of this concept itself and of all geometrical determinations of it. They were anxious whether a line in nature might not consist of physical points and true space in the object, of simple parts, although the space which the geometer thinks about can in no way consist of these. They did not recognise that it is this space in thought which itself makes possible physical space, i.e. the extension of matter; that it is not a quality of things in themselves but only a form of our faculty of sensible representation; [. . .] and that space [. . .], as the geometer thinks it, being precisely the form of sensible intuition which we find in ourselves a priori, and which contains the ground of the possibility of all outer appearances (as to their form), it must agree necessarily in the most precise way with the propositions of the geometer [. . .] In this and no other way can the geometer be secured as to the undoubted objective reality of his propositions against all the chicaneries of a shallow metaphysics, however strange this may seem to a metaphysics which does not go back to the sources of its concepts. (Kant, 1783, Prolegomena, 13, note 1, po. 44–45)

At the center of my theoretical perspective in this chapter is the Kantian antimetaphysical attitude that originally expresses a central defense of the role of

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mathematics in superior human cognitive activities, essentially its capacity to make rational intelligibility of the world.

Beyond Kant: Seeing Mathematics as the Transformation of the A Priori Determination of Forms of Intuitions in Formal Intuitions The form of intuition gives only a manifold, the formal intuition gives unity of representation: indeed Kant explains that first of all the forms of intuition described in the metaphysical exposition of “Aesthetics” are also (cf. transcendental exposition) formal intuitions, which have the statute of mathematically determined objects, and later, that this mathematical determination in some way actualizes the system of categories of pure intuition. For example, when considered to be the object of geometry, space is the basis of “formal intuitions,” that is, of intuitions that are directly determined from the mathematical point of view. That all intuitions (in this case, mathematical individuals) are, for us human beings, necessarily sensible (a result of the Aesthetics and not a corollary of the definition of “intuition”) explains how and in what sense the propositions of mathematics are evident and hence supplies an additional reason for saying that they are synthetic (Brittan, 1978, p. 57). The transformation of the phenomenon, from being aesthetic manifestation to an object of experience, is based on the possibility of mathematically representing the notional contents of the categories (Petitot, 1984, pp. 65–66). Beyond Kant we see modern mathematics in its cognitive capacity to elaborate structural mathematical concepts endowed with categorical contents, that is, concepts not immediately but mediately constructible. When speaking of categorical intuitions, Husserl had foreseen this possibility. Nevertheless, reducing, alas, as many others, mathematics to axiomatics, he could not conceive the objective value of mathematics if not in the form of pure formal ontology which uniformly subordinates regional ontologies (Petitot, 1984, p. 68).

Mathematics Makes Up New “Principles of Experience” The a priori determination of forms of intuition in formal intuitions (transcendental exposition) not only opens the horizon of various axiomatics but also the horizon of a conceptual development which permits schematization, in the sense of a mediated construction of concepts as principles of experience, of an indefinitely open series of regional categories. In this perspective, we face a generalization, in Kantian terms, of “Aesthetics” and of “Logic.” I have furnished a generalization of “Aesthetics” and of “Logic”: mathematics, which is inherently implicated in the construction of the objective experience, can be historicized, seeing when mathematical entities are created. An example of a new mathematical schematization is given by Thom’s theory of catastrophes (Thom, 1972). In this theory the concepts of transition of state, differentiation, and stratification have permitted us to define the eidetic unity of a new region, that of structural apperception (neither reductionist nor holistic) of

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critical and morphological phenomena (cf. Petitot (1984, 1985, 1999)). In this perspective, mathematical physics is the ontological transformer of the a priori structure of possible experience in objective knowledge of a material multiplicity: Thom’s theory of catastrophes is the ontological transformer of primitive “discontinuity” in objective knowledge of morphological multiplicity. The mathematical schematism (e.g., the Newtonian schema of the physical concept of determinist evolution through the mathematical concept of the dynamic system) is a cognitive opening for new empirical perspectives. For example, in the case of Newton, this schematization has permitted us to reunite two orders of phenomena, which had been considered to be “ontologically heterogeneous” since the times of Aristotle, in one eidetic unity of an apperception. The same can be said in the case of the schematization of morphological phenomena and so too the mathematical study of living organisms we have just quoted. We are beyond Kant’s framework, given the fact Kant thought that these phenomena escape a priori to scientific knowledge, to the point that they are thought of only in the Critique of Judgment.

The Cognitive Virtues of Mathematics Mathematics and Ordinary Language As it is well-known, in the passage from the philosophical considerations of the Tractatus to those of Philosophical Investigations, Wittgenstein seems to use some ideas about mathematics as the model for the mechanism of ordinary language. Mathematics and logic represent a formal and “calculating” type of knowledge that produces objectivity and creates scientific intelligibility for the world. But mathematics is also a language. In Wittgenstein, it is exactly the idea of calculation that suggests the philosophical concept of game, as a means for general ordering of “propositions.” On the other hand, mathematics would consist of internal regularities that create “games” and are the model for discovering the mechanisms of ordinary language. Mathematics itself becomes a language among languages and no longer belongs, as in Kant, to the pure forms of objectivity as determinations of pure multiplicity. On the contrary, the specific aspects of mathematics are nullified by the absolute priority of language games; both mathematics and ordinary language are “language games.” This commonplace interpretation discards the epistemological distinction between mathematics and ordinary language. A typical consequence is the tendency to eliminate the relevance of the problem of science and knowledge. As illustrated in the previous subsection, by relying on the Kant of the “Aesthetic” and of the “Analytic,” we can be immunized from this danger. When Kant speaks of geometry (and mathematics), he shows us that we are dealing with a knowledge that organizes the world by generating scientific intelligibility. This is the answer to a theoretical problem of scientific knowledge, of epistemology. The activity of ordinary language is different. This activity is

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explained by some functions of empirical schematism, as we have seen, where rules and schemata emerge from the empirical manifold. So to speak, mathematics is certainly a semiotic activity but it is endowed with exceptional characters, completely lacking in other disciplines. From a philosophical point of view, our post-Kantian philosophy of mathematics is beyond the Husserlian/Heideggerian perspective which interprets the Critique as a thought of transcendence of a finite Dasein, which on the one hand leads toward the Husserlian “egology” of the original temporality of consciousness and on the other toward the well-known Heideggerian existential analysis. I contend that these traditions too have eliminated the relevance of the problem of science and knowledge, legitimizing “our situation of Krisis” (Petitot, 1984, pp. 63–64 and 69–70).

Mathematical Modeling in Science as an Antidote Against the Epistemological Pretensions of Big Data and Deep Learning The emphasis on the virtues of mathematical modeling in science which we have seen in a Kantian perspective is also an antidote against the ambitions which originate from the management of big data, currently presented as aiming at substituting human-centered scientific understanding. In June 2008 C. Anderson, former editor-in-chief of Wired Magazine, wrote an article titled “The end of theory: the data deluge makes the scientific method obsolete” contending that “with enough data, the numbers speak for themselves,” science as we know it will be replaced by robust correlations in immense databases! Calude and Longo (2017) demonstrate, taking advantage of deep classical mathematical results from ergodic theory, Ramsey theory, and algorithmic information theory, how absurd is this contention and that instead very large databases present too many arbitrary – and seemingly spurious – correlations, which surely cannot be considered examples of pregnant scientific creative abduction (that is reasoning to novel relevant hypotheses), but just uninteresting generalizations, even if made thanks to sophisticated artifacts. The fundamental Greek practice of scientific observation, thinking, and debating on different theoretical interpretations of phenomena was enriched by the experimental method (since Galileo) and mathematics (since Descartes and Newton). Big data analytics cannot replace science based on the capacity of mathematics to build new regional categories/principles of experience; and, symmetrically, no theory can be so good to supplant the need for data and testing. Implicit or, better, explicit and revisable theorizing should accompany meaningful measurements of “evidences” and mathematical modeling, as well as reliable analyses of databases. The lack of a mathematical schematic effort of creating scientific intelligibility leads to surrogate “modeling” in human and social sciences. Following Cartwright an epistemologically “good” credible world has to be provided by models that are able to trigger hypotheses about the “causation of actual events,” that is, in cases in which “the fictional world of the model is one that could be real.” Cartwright’s (2009a) classical idea concerning capacities is very clear. For her, the function of a model is to demonstrate the reality of a capacity by isolating it – just as Galileo’s

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experiment demonstrates the constancy of the vertical component of the acceleration of a body acted on by gravity. Notice how Cartwright speaks of showing that C has the capacity to produce E and of deriving this conclusion from accepted principles. Of course in this activity of modeling, the schematic role of mathematics is fundamental. Sugden (2009, p. 20) prudently considers too strong Cartwright’s perspectives on models as tools for isolating the “capacities” of causal factors in the real world and provides other conceptual devices to save various aspects of epistemological – supposed to be weak – “sciences,” for example, some parts of biology, psychology, or economics, which never fulfill the target of revealing capacities. To save these sciences, he says that models can simply provide “conceptual explorations,” which ultimately contribute to the development of genuinely explanatory theories or credible counterfactual worlds which can trigger inductive (or “abductive”) inferences to explain the target systems. I think that it is virtuous to be prudent about strong methodological claims such as the ones advanced by Cartwright; but the epistemological problem remains open: in the cases of models as conceptual explorations, are they an excuse for providing ambitious but unjustified hypotheses, devoid of various good epistemological requisites? Adopting Cartwright’s rigid demarcation criterium clearly stated in the relatively recent “If no capacities then no credible worlds” (Cartwright, 2009a), it would seem that no more citizenship is allowed to some postmodern exaggeration in attributing the label “scientific” to various proliferating areas of academic production of knowledge, from (parts of) psychology to (parts of) economics, and so on, areas which do not – or scarcely – reach the most common received epistemological standards, for example, the predictivity of the phenomena that pertain to the explained systems. An example: research in psychology (Miller, 2010, p. 716) explores three contentions: “[. . .] that the dominant discourse in modern cognitive, affective, and clinical neuroscience assumes that we know how psychology/biology causation works when we do not; that there are serious intellectual, clinical, and policy costs to pretending we do know; and that crucial scientific and clinical progress will be stymied as long as we frame psychology, biology, and their relationship in currently dominant ways.” The last considerations implicitly resort to that emphasis, which characterizes the theoretical aim of this chapter, on the role of mathematics in building new “formal ontologies” thanks to its application to the experiential world. In this perspective mathematics makes us capable of recognizing its function in making intelligible regimes of causality and consequently the spurious character of certain epistemological views that too quickly concede to too many kinds of modeling the status of “rational knowledge.”

Mathematics, Abduction, and Models Manipulative abduction, which is widespread in cognitive behaviors that aim at creating accounts of new communicable experiences so that, for example, in the case of various kinds of scientific reasoning, the abductive process concerning the

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formation and evaluation of a hypothesis occurs by resorting to a basically extratheoretical and extra-sentential dimension: in this perspective manipulative abduction represents a kind of redistribution of the epistemic and cognitive effort to manage objects and information that cannot be immediately represented or found “internally” (Magnani, 2009, Ch. 1). An example of manipulative abduction is exactly the case of the human use of the construction of external models in a neural engineering laboratory or in mathematics, exploiting external diagrams, proofs, and computational artifacts (on “how technology has changed what it means to think mathematically” cf. (Devlin, 2019).) In these cases the external tools and representations are useful to make observations and “experiments” to transform one cognitive state into another to discover new properties of the target systems/theories. Manipulative abduction also refers to those more unplanned and unconscious actionbased cognitive processes which I have earlier characterized as forms of “thinking through doing” (Magnani, 2009, Ch. 1). Hence, manipulative abduction is a kind of abduction, usually model-based, that exploits external models endowed with delegated (and often implicit) cognitive roles and attributes. We have to add that, for example, model-based visual thinking is surely the kind of model-based cognition more extendedly studied in the epistemology of mathematics. An impressive and rich compendium is provided by Gianquinto (2020), which illustrates the relationships between visual thinking, formal and non-formal proofs, and their reliability, visual thinking in discovering strategies, and a priori and a posteriori roles of visual experience. Manipulative abduction happens when we are thinking through doing and not only, in a pragmatic sense, about doing (cf. Magnani (2009, Ch. 1)). An example of manipulative abduction can be seen in the case of elementary geometrical reasoning, which tales advantage of diagrams we can say that 1. The model (diagram) is external and the strategy that organizes the manipulations is unknown a priori. 2. The result achieved is new (if we, for instance, refer to the constructions of the first creators of geometry) and adds properties not contained before in the concept (that is the Kantian to “pass beyond” we have explained above in subsection “Mathematics as Synthetic A Priori Knowledge”). Of course in the case in which we are using diagrams to demonstrate already known theorems (for instance, in didactic settings), the strategy of manipulations is not necessarily unknown and the result is not new.

Mathematics and Manipulative Abduction I have just anticipated that a traditional and important example of model-based and manipulative abduction in mathematics is represented by the cognitive exploitation of diagrams. Let’s quote an interesting passage by Peirce about constructions. Peirce says that mathematical and geometrical reasoning “[. . .] consists in constructing a

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diagram according to a general precept, in observing certain relations between parts of that diagram not explicitly required by the precept, showing that these relations will hold for all such diagrams, and in formulating this conclusion in general terms. All valid necessary reasoning is in fact thus diagrammatic” (1866–1913 (1896), 1.54). Not dissimilarly Kant says, as we have already reported above in subsection “Mathematics as Synthetic A Priori Knowledge,” that in geometrical construction “[. . .] I must not restrict my attention to what I am actually thinking in my concept of a triangle (this is nothing more than the mere definition); I must pass beyond it to properties which are not contained in this concept, but yet belong to it” (Kant, 1787, A718-B746, p. 580). For Peirce, the whole mathematics consists in building diagrams that are “[. . .] (continuous in geometry and arrays of repeated signs/letters in algebra) according to general precepts and then [in] observing in the parts of these diagrams relations not explicitly required in the precepts” (1866–1913 (1896), 1.54). Peirce contends that this diagrammatic nature is not clear if we only consider syllogistic reasoning “which may be produced by a machine” but becomes extremely clear in the case of the “logic of relatives, where any premise whatever will yield an endless series of conclusions, and attention has to be directed to the particular kind of conclusion desired” (1866–1913/1985, pp. 11–23). In ordinary geometrical proofs available in textbooks, auxiliary constructions are present in terms of “conveniently chosen” figures and diagrams where strategic moves are important aspects of deduction. The system of reasoning exhibits a dual character: deductive and “hypothetical.” Also in other – for example, logical – deductive frameworks, there is room for strategic moves which play a fundamental role in the generation of proofs. These strategic moves correspond to particular forms of abductive reasoning. We know that the kind of reasoned inference that is involved in creative abduction goes beyond the mere relationship between premises and conclusions in valid deductions, where the truth of the premises “guarantees” the truth of the conclusions, and beyond the relationship that there is in probabilistic reasoning, which renders the conclusion just more or less probable. On the contrary, we have to see creative abduction as formed by the application of heuristic procedures that involve all kinds of good and bad inferential actions, and not only the mechanical application of rules. It is only by means of these heuristic procedures that the acquisition of new truths is guaranteed. Also Peirce’s mature view on creative abduction as a kind of inference stresses the strategic component of reasoning and its strict relationship with action: “It will be remarked that the result of both Practical and Scientific Retroduction [another name for abduction] is to recommend a course of action” [MS 637, 12, 1909] (Peirce, 1866–1913/1966). Many researchers in the field of philosophy, logic, and cognitive science have maintained that deductive reasoning also consists in the employment of logical rules in a heuristic manner, even maintaining the truth preserving character: the application of the rules is organized in a way that is able to recommend particular courses of actions instead of other ones. Moreover, very often the heuristic procedures of deductive reasoning are performed by means of model-based abductive steps

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where iconicity is central: to offer an example, the logical and mathematical proofs that are mainly composed of symbols, which do not involve diagrams, exhibit configurations that are characterized by a variety of iconic shapes. We have seen that the most common example of manipulative creative abduction is the usual experience people have when solving “problems” in elementary geometry in a model-based way, trying to devise proofs using diagrams and illustrations: of course the attribute of creativity we give to abduction in this case does not mean that it has never been performed before by anyone or that it is original in the history of some knowledge. (We have to say that model-based abductions – which, for example, exploit iconicity – also operate in deductive reasoning. On the role of strategies and heuristics in deductive proofs cf. Magnani (2009, Ch. 7).) As anticipated in section “Mathematics, Abduction, and Models” above, manipulative abduction is a kind of, usually model-based, abduction that takes advantage of external models endowed with delegated (and often implicit) cognitive roles and attributes. The concept of manipulative abduction – which also takes into account the external dimension of abductive reasoning in an eco-cognitive perspective – captures a large part of common and scientific thinking where the role of action and of external models (e.g., diagrams) and devices is central and where the features of this action are implicit and hard to be elicited. Action can provide otherwise unavailable information that enables the agent to solve problems by starting and by performing a suitable abductive process of generation and/or selection of hypotheses. Humans and other animals make a great use of perceptual reasoning and kinesthetic and motor abilities. We can catch a thrown ball, cross a busy street, read a musical score, go through a passage by imaging if we can contort our bodies to the way required, evaluate shape by touch, recognize that an obscurely seen face belongs to a friend of ours, etc. Usually the “computations” required to achieve these tasks are not accessible to a conscious description. Mathematical reasoning uses language explanations, but also non-linguistic notational devices and models. Geometrical constructions represent a classic example of this kind of extra-linguistic machinery we know as characterized in a model-based and manipulative – abductive – way. Certainly a considerable part of the complicated environment of a thinking mathematical agent is internal and consists of the proper “software” composed of the knowledge base and of the inferential expertise of that individual. Nevertheless, as I have already pointed out, any cognitive system consists of a “distributed cognition” among people and “external” technical artifacts (Hutchins, 1995; Zhang, 1997). In the case of the construction and examination of diagrams in geometry or, in general, mathematics, of written proofs, notes, and sketches (and, where appropriate, their computational counterparts), a specific sort of “experiments” is characterized as states, while the implied operators are the manipulations and observations that transform one state into another. The mathematical outcome is dependent upon practices and specific sensorimotor activities performed on an external entity, which acts as a dedicated external representational medium supporting the various operators at work. There is a kind of epistemic negotiation between the sensory framework of the mathematician and the external reality of diagrams, proofs, notes,

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and sketches. This process involves an external representation consisting of written symbols, figures, etc. that are manipulated “by hand.” The cognitive system is not merely the mind-brain of the person performing the mathematical task but the system consisting of the whole body (cognition is embodied) of the person plus the external physical representation. Tall presents a rich study on how the human brain makes sense of various sophisticated mathematical ideas by referring to neurophysiological knowledge together with observations of teachers and learners in the classroom (Tall, 2019). For example, in geometrical discovery the whole activity of cognition is located in the system consisting of a human with diagrams. A lot of recent mathematical research in the area of education has demonstrated that learners’ actions can affect how they think and vice versa, taking advantage of the role played by manipulations, gestures, and body movements. Edwards (2019) provides clear examples that show how the processes and ideas related to mathematical proofs are embodied phenomena, rather than something merely existing “in the head,” also indicating a conceptual continuity between mathematical proof and nonmathematical thinking and discourse. The mechanisms that underscore the role of embodied activities to explore how to harness the affordances of new technology to enhance mathematical thinking are, for example, analyzed (Tran et al., 2017). An external representation can modify the kind of computation that a human agent uses to reason about a problem: the Roman numeration system eliminates, by means of external signs, some of the hardest aspects of addition, whereas the Arabic system does the same in the case of the difficult computations in multiplication (Zhang, 1997). All external representations, if not too complex, can be transformed into internal representations by memorization. But this is not always necessary if the external representations are easily available. In turn, internal representations can be transformed into external ones by productive externalization “[. . .] if the benefit of using external representations can offset the cost associated with the externalization process” (Zhang, 1997, p. 181). Hence, contrary to the old view in cognitive science, not all cognitive processes happen in an internal model of the external environment. The information present in the external world can be directly picked out without the mediation of memory, deliberation, etc. Moreover, various different external devices can determine different internal ways of reasoning to solve the problems, as is wellknown. Even a simple arithmetic task can completely change in the presence of an external tool and representation. In Fig. 2 an ancient external tool for division is represented. In the following pages of this chapter, I will describe the so-called optical diagrams in mathematics, see below, section “Mirroring und Unveiling Hidden Properties Through Optical Diagrams,” and their role in removing obstacles and obscurities and in enhancing mathematical knowledge of critical situations (e.g., the problem of parallel lines, cf. below, section “Externalizing Internal Models to Represent and Discover Mathematical Entities: Resolving Conceptual Anomalies”). To summarize we can say mathematical diagrams play various roles in a typical abductive way; moreover, they are external representations which, in the cases I will present in the following sections, are devoted to providing abductive results. Two of them are central:

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Fig. 2 Galley division, sixteenth century, from an unpublished manuscript of a Venetian monk. The title of the work is Opus Artimetica D. Honorati veneti monachj coenobij S. Lauretij

• They provide an intuitive and mathematical explanation able to help us understand concepts difficult to grasp or that appear obscure and/or epistemologically unjustified. I will present some mirror diagrams which provided new puzzling mental representations of the concept of parallel lines. • They help abductively create new previously unknown concepts that are nonexplanatory, as, for example, illustrated in the case of the discovery of non-Euclidean geometry. (A full description of this interesting case of mathematical discovery is provided in Magnani (2009, Ch. 2).)

Optical and Unveiling Diagrams in Mathematical Cognition Mirroring und Unveiling Hidden Properties Through Optical Diagrams I have illustrated in the previous sections that in the whole history of geometry, many researchers used internal mental imagery and mental representations of diagrams but also self-generated diagrams (external) to facilitate thoughts (Otte & Panza, 1999). Indeed iconic geometrical constructions present situations that are curious and “at the limit.” Because of their iconicity, they are constitutively dynamic and artificial and offer various contingent ways of epistemic acting, like looking from different

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perspectives, comparing subsequent appearances, discarding, choosing, reordering, and evaluating. Moreover, they present the features typical of manipulative reasoning illustrated above, such as the simplification of the task and the capacity to get visual information otherwise unavailable. We have seen that manipulative abduction is a kind of abduction, usually modelbased and so intrinsically “iconic,” that exploits external models endowed with delegated (and often implicit) cognitive and semiotic roles and attributes. We can say that (1) the model (diagram) is external and the strategy that organizes the manipulations is unknown a priori; (2) the result achieved is new (if we, for instance, refer to the constructions of the first creators of geometry) and adds properties not contained before in the concept (the Kantian to “pass beyond” or “advance beyond” the given concept) (Kant, 1787, A154-B193/194, p. 192). (Other interesting applications of the concept of abduction in mathematical discovery and in the manipulation of symbols are illustrated in Heeffer (2007, 2008). On Cardano’s abductive discovery of negative numbers and negative solution to a linear problem cf. Heeffer (2007).) Hence, in the construction of mathematical concepts, many external representations are exploited, both in terms of diagrams and of symbols, but also propositions mixed with ordinary language and sketches, as we already said above. It is appropriate to refer in this chapter devoted to the intertwining between mathematics and cognition to special kinds of diagrams which play various iconic roles: an optical role, microscopes (that look at the infinitesimally small details), telescopes (that look at infinity), and windows (that look at a particular situation); a mirror role (to externalize rough mental models); and an unveiling role (to help to create new and interesting mathematical concepts, theories, and structures) (Cf. section “Externalizing Internal Models to Represent and Discover Mathematical Entities: Resolving Conceptual Anomalies”). I also describe these diagrams as those epistemic mediators able to perform various abductive tasks (discovery of new properties or new propositions/hypotheses, provision of suitable sequences of models able to convincingly verifying theorems, etc.). (Elsewhere I have presented some details concerning the role of optical diagrams in calculus (Dossena & Magnani, 2007; Magnani & Dossena, 2005).) An interesting epistemological situation is the one concerning the cognitive role played by some special epistemic mediators in the field of non-standard analysis, an “alternative calculus” invented by Abraham Robinson (1966), based on infinitesimal numbers in the spirit of Leibniz’s method. (Further details concerning Leibniz’s mathematics and philosophy of infinitesimals are illustrated in Mancosu (1996).) It is a kind of calculus that uses an extension of the real numbers system ℝ to the system ℝ* containing infinitesimals smaller in the absolute value than any positive real number. I maintain that in mathematics diagrams play various roles in a typical abductive way. Optical diagrams play the first of the two roles illustrated in the last part of the previous section: a fundamental explanatory (and didactic) role in removing obstacles and obscurities and in enhancing mathematical knowledge of critical situations. They facilitate new internal representations and new symbolic-propositional

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achievements. In the example from calculus, the extraordinary role of optical diagrams in the interplay standard/non-standard analysis is emphasized. In the case of our non-standard analysis examples, some diagrams (microscopes within microscopes) provide new mental representations of the concept of tangent line at the infinitesimally small regions. Hence, external representations which play an “optical” role can be used to provide us with a better understanding of many critical mathematical situations and, in some cases, to more easily discover (or rediscover) sophisticated properties. The role of an “optical microscope” that shows the behavior of a tangent line is illuminating. In standard analysis, the change dy in y along the tangent line is only an approximation of the change Δy in y along the curve. But through an optical microscope, which shows infinitesimal details, we can see that dy ¼ Δy and then the quotient Δy/Δx is the same of dy/dx when dx ¼ Δx is infinitesimal (see Fig. 3 and, for more details Magnani & Dossena (2005)). This removes some difficulties of the representation of the tangent line as limit of secants and introduces a more intuitive conceptualization: the tangent line “merges” with the curve in an infinitesimal neighborhood of the contact point. Only through a second more powerful optical microscope “within” the first (a kind of epistemic mediators called microscopes within microscopes) (again, see Fig. 3) we can see the difference between the tangent line and the curve. Under the first diagram, the curve looks like the graph of f 0 ðaÞx, i.e., a straight line with the same slope of its tangent line; under the second, the curve looks like (This is mathematically justified in Magnani and Dossena (2005)) f 0 ðaÞx 

1 00 f ðaÞ: 2

Δy Δy

dy

Δx

Fig. 3 An optical diagram shows an infinitesimal neighborhood of the graph of a real function

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This suggests nice new mental representations of the concept of tangent lines: through the optical lens, the tangent line can be seen as the curve, but through a more powerful optical lens, the graph of the function and the graph of the tangent are distinct, straight, and parallel lines. The fact that one line is either below or above the other depends on the sign of f00 (a), in accordance with the standard real theory: if f00 (x) is positive (or negative) in a neighborhood, then f is convex (or concave) here and the tangent line is below (or above) the graph of the function. Furthermore, this easily mirrors a sophisticated hidden property. Let f be a two times differentiable function and let a be a flex point of it. Then f00 (a) ¼ 0 and so the second microscope shows again the curve as the same straight line: this means that the curve is “very straight” in its flex point a. Of course, we already know this property – the curvature in a flex point of a differentiable two times function is null – which comes from standard analysis, but through optical diagrams, we can find it immediately and more easily (the standard concept of curvature is not immediate). To conclude, I have already noted that some diagrams could also play an unveiling role, providing new light on mathematical structures: it can be hypothesized that these diagrams can lead to further interesting creative results: we will see this case at work in the discovery of non-Euclidean geometry in section “Unveiling Diagrams in Lobachevsky’s Discovery as Affordances: Gateways to Imaginary Entities.” I stated that mathematical diagrams play various roles in a typical abductive way; we can further and finally emphasize that: • they are epistemic mediators able to perform various more or less creative abductive tasks in so far as • they are external representations which provide explanatory and non-explanatory abductive results

Externalizing Internal Models to Represent and Discover Mathematical Entities: Resolving Conceptual Anomalies The epistemologists that study natural sciences taught us the importance of empirical anomalies resulting from data that cannot currently be fully explained by a theory. They often derive from predictions that fail, which implies some element of incorrectness in the theory. In general terms, many theoretical constituents may be involved in accounting for a given domain item (anomaly) and hence they are potential points for modification. The detection of these points involves defining which theoretical constituents are employed in the explanation of the anomaly. Thus, the problem is to investigate all the relationships in the explanatory area. First and foremost, anomaly resolution involves the localization of the problem at hand within one or more constituents of the theory; it is then necessary to produce one or more new hypotheses to account for the anomaly, and, finally, these hypotheses need to be evaluated to establish which one best satisfies the criteria for theory

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justification. Hence, anomalies require a change in the theory. We know that empirical anomalies are not alone in generating impasses. The so-called conceptual problems represent a particular form of anomaly; mathematical cognition, and in general formal sciences, present various interesting cases. Resolving conceptual problems may involve satisfactorily answering questions about the status of theoretical entities: conceptual problems arise from the nature of the claims in the principles or in the hypotheses associated with the theory. Usually it is necessary to identify the conceptual problem that needs a resolution, for example, by delineating how it can concern the adequacy or the ambiguity of a theory, yet also its incompleteness (or lack of evidence). The discovery of non-Euclidean geometries presents an interesting case of visual/ spatial abductive reasoning. First of all it demonstrates a kind of visual/spatial abduction, as a strategy for anomaly resolution connected to a form of explanatory and productive visual thinking. Since ancient times, the fifth postulate has been held to be not evident. This “conceptual problem” has generated many difficulties about the reliability of the theory of parallels, consisting of the theorems that can be only derived with the help of the fifth postulate. The recognition of this anomaly was crucial to the development of the non-Euclidean revolution. Two thousand years of attempts to resolve the anomaly have produced many fallacious demonstrations of the fifth postulate: a typical attempt was that of trying to prove the fifth postulate from the others. Nevertheless, these attempts have also provided much theoretical speculation about the unicity of Euclidean geometry and about the status of its principles. Let us show how the anomaly is recognizable. A postulate that is equivalent to the fifth postulate states that for every line l and every point P that does not lie on l, there exists a unique line m through P that is parallel to l. If we consider its model-based (diagrammatic) counterpart (cf. Fig. 4), the postulate may seem “evident” to the reader, but this is because we have been conditioned to think in terms of Euclidean geometry. The definition above represents the most obvious level at which ancient Euclidean geometry was developed as a formal science – a level composed of symbols and propositions. Furthermore, when we consider the other fundamental level, where model-based aspects (diagrammatic) are at play, we can immediately detect a difference between this postulate and the other four if we regard the first principles of geometry as abstractions from experience that we can in turn represent by drawing figures on a blackboard or on a sheet of paper or on our “visual buffer” (Kosslyn & Koenig, 1992) in the mind. We have consequently a double passage from the sensorial experience to the abstraction (expressed by symbols and propositions) and from this abstraction to the experience (sensorial and/or mental). We immediately discover that the first two postulates are abstractions from our experiences drawing with a straightedge, the third postulate derives from our experiences drawing with a compass. The fourth postulate is less evident as an abstraction, nevertheless it derives from our measuring angles with a protractor (where the sum of supplementary angles is 180
, so that if supplementary angles are congruent, they must each measure 90
) (Greenberg, 1974, p. 17).

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Fig. 4 A diagram regarding Euclidean parallel lines

In the case of the fifth postulate, we are faced with the following serious problems: (1) we cannot verify empirically whether two lines meet, since we can draw only segments, not lines. Extending the segments further and further to find if they meet is not useful, and in fact we cannot continue indefinitely. We are forced to verify parallels indirectly, by using criteria other than the definition; (2) the same holds with regard to the representation in the “limited” visual buffer. The “experience” localizes a problem to solve, an ambiguity, only in the fifth case: in the first four cases, our “experience” verifies without difficulty the abstraction (propositional and symbolic) itself. In the fifth case the formed images (mental or not) are the images that are able to explain the “concept” expressed by the definition of the fifth postulate as problematic (an anomaly): we cannot draw or “imagine” the two lines at infinity, since we can draw and imagine only segments, not the lines themselves. The chosen visual/spatial image or imagery (in our case the concrete diagram depicted in Fig. 4, derived from the propositional and symbolic level of the definition) plays the role of an explanation of the anomaly previously envisaged in the definition itself. As stated above, the image demonstrates a kind of visual abduction, as a strategy for anomaly localization related to a form of explanatory visual/spatial thinking. Once the anomaly is detected, the way to anomaly resolution is opened up – in our case, this means that it becomes possible to discover non-Euclidean geometries. That Euclid himself did not fully trust the fifth postulate is revealed by the fact that he postponed using it in a proof for as long as possible – until the twenty-ninth proposition. As is well-known, Proclus tried to solve the anomaly by proving the fifth postulate from the other four. If we were able to prove the postulate in this way, it would become a theorem in a geometry which does not require that postulate (the future “absolute geometry”) and which would contain all of Euclid’s geometry. Without showing all the passages of Proclus’s argument (Greenberg, 1974, pp. 119–121), we only have to remember that the argument seemed correct because

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it was proved using a diagram. Yet we now know that we are not allowed to use that diagram to justify a step in a proof. Each step must be proved from stated axioms or from previously proven theorems. We may visualize parallel lines as railroad tracks, everywhere equidistant from each other, and the ties of the tracks as being perpendicular to both parallels. Yet this imagery is valid only in Euclidean geometry. In the absence of the parallel postulate, we can only consider two lines as “parallel” when, by the definition of “parallel,” they do not possess any points in common. It is not possible implicitly to assume that they are equidistant, nor can it be assumed that they have a common perpendicular. This is an example in which a selected abduced image is capable of compelling you to make a mistake, and in this way it was used as a means of evaluation in a proof: we have already stated that in this case it is not possible to use that image or imagery to justify a step in a proof because it is not possible to use that image or imagery that attributes to experience more than the experience itself can deliver. For over two thousand years, some of the greatest mathematicians tried to prove Euclid’s fifth postulate. For example, Saccheri’s strategy for anomaly resolution in the eighteenth century was to abduce two opposite hypotheses of the principle, that is, to negate the fifth postulate and derive, using new logical tools coming from non-geometrical sources of knowledge, all theorems from the two alternative hypotheses by trying to detect a contradiction. (On the “strategies” adopted in anomaly resolution cf. Darden (1991, pp. 272–275).) The aim was indeed that of demonstrating/explaining that the anomaly is simply apparent. We are faced with a kind of what is called “non-explanatory abduction.” New axioms are hypothesized and adopted in looking for outcomes which can possibly help in explaining how the fifth postulate is unique and so not anomalous. At a first sight, this case is similar to the case of non-explanatory abduction active in reverse mathematics, but the similarity is only structural (i.e., guessing “new axioms”). Indeed, non-explanatory modes of abduction are clearly exploited in the “reverse mathematics” pioneered by Harvey Friedman and his colleagues (2000), where propositions can be taken as axioms because they support the axiomatic proofs of target theorems. The target of reverse mathematics is to answer this fundamental question: What are the appropriate axioms for mathematics? The problem is to discover which are the appropriate axioms for proving particular theorems in central mathematical areas such as algebra, analysis, and topology (cf. Simpson (1999)). The idea of reverse mathematics originates with Russell’s notion of the regressive method in mathematics (Russell, 1973) and is also present in some remarks of Gödel (1944, 1990). (For more details about this, see Irvine (1989), who also compares Russell’s regressive method to Peirce’s abduction.) Gabbay and Woods (2005, p. 128) conclude, following Russell, that regressive abduction is both instrumental and non-explanatory and quote a Gödel’s passage, which confirms their statement: [. . .] even disregarding the intrinsic necessity of some new axiom, and even in case it has no intrinsic necessity at all, a probable decision about its truth is possible also in another way, namely inductively by studying its “success”. Success here means fruitfulness in

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consequences, in particular, “verifiable” consequences, i.e., consequences demonstrable without the new axioms, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make it possible to contract into one proof many different proofs. (Gödel, 1990, pp. 476–477)

In the case of reverse mathematics, axioms are hypothesized to account for already existing mathematical theories but do not aim at explanatory results; in the case of the parallel lines, the chance of finding a new axiom (a new postulate) concerns only one of the two following possibilities: (1) finding new axioms (postulates) to solve the anomaly by establishing that actually it is not an anomaly (Saccheri), (2) hypothesizing a new axiom (that is a new parallel postulate) able to dissolve the anomaly (so considering it a “true” anomaly) (this second case will lead to Lobachevsky’s discovery of non-Euclidean geometry). Gabbay and Woods (2005) contend that abduction is not intrinsically explanationist, like its description in terms of inference to the best explanation would suggest. Abduction can also be merely instrumental. In Chapter 2 of Magnani (2009), some examples of abductive reasoning that basically are non-explanatory and/or instrumentalist have been described. Gabbay and Woods’ distinction between explanatory, non-explanatory, and instrumental abduction is orthogonal to mine in terms of the theoretical and manipulative (including the subclasses of sentential and modelbased) and further allows us to explore fundamental features of abductive cognition. Hence, if we maintain that E explains E0 only if the first implies the second, certainly the reverse does not hold. This means that various cases of abduction are consequentialist but not explanationist (other cases are neither consequentialist nor explanationist). Let us come back to the important results provided by Saccheri. The contradiction in the elliptic case (“hypothesis of obtuse angle,” to use Saccheri’s term designing one of the two future elementary non-Euclidean geometries) was found, but the contradiction in the hyperbolic case (“hypothesis of the acute angle”) was not so easily discovered: having derived several conclusions that are now well-known propositions of non-Euclidean geometry, Saccheri was forced to resort to a metaphysical strategy for anomaly resolution, “Proposition XXXIII. The ‘hypothesis’ of acute angle [that is, the hyperbolic case] is absolutely false, because repugnant to the nature of the straight line [sic]” (Saccheri, 1920). (Lobachevsky’s discovery leads to a new geometry that will be called “hyperbolic.” Riemann’s one will be called “elliptic.”) But Saccheri chose to state this result with the help of the somewhat complicated imagery of infinitely distant points: two different straight lines cannot both meet another line perpendicularly at one point, if it is true that all right angles are equal (fourth postulate) and the two different straight lines cannot have a common segment. Saccheri did not ask himself whether everything that is true of ordinary points is necessarily true of an infinitely distant point. In Note II to proposition XXI, some “physico-geometrical” experiments to confirm the fifth postulate are also given, invalidated unfortunately by the same incorrect use of imagery that we have observed in Proclus’s case. In this way, the anomaly was resolved unsatisfactorily and Euclid was not freed of every fleck: although he did not

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recognize it, Saccheri had discovered many of the propositions of non-Euclidean geometry (Torretti, 1978, p. 48). In the following sections, I will illustrate the example of Lobachevsky’s discovery of non-Euclidean geometry where we can see the model-based abductive role played in a discovery process by new considerations concerning visual sense impressions and productive imagery representations.

Externalizing Diagrammatic Models to Unveil Imaginary Entities In the last passages of subsection “Mathematics and Manipulative Abduction,” I underlined the role of mathematical diagrams in abductively creating new previously unknown concepts that are non-explanatory: in the following, I will summarize the main cognitive and epistemological aspects of the discovery of Lobachevsky’s non-Euclidean geometry, which represents a clear example. (As I have already anticipated, a full description of this interesting case of mathematical discovery is provided in Magnani (2009, Ch. 2).) The process of discovery is characterized by various heuristic steps, as indicated in the following subsections.

Abducing First Principles Through Bodily Contact Lobachevsky was obliged first of all to rebuild the basic principles, and to this end, it was necessary to consider geometrical principles in a new way, as neither ideal nor a priori. New interrelations were created between two areas of knowledge: Euclidean geometry and the philosophical tradition of empiricism/sensualism. I have already said that for over two thousand years, some of the greatest mathematicians tried to prove Euclid’s fifth postulate. Geometers were not content to merely construct proofs in order to discover new theorems and thereby to try to resolve the anomaly (represented by its lack of evidence) without trying to reflect upon the status of the symbols of the principles underlying Euclidean geometry. Lobachevsky’s strategy for resolving the anomaly of the fifth postulate was 1. To manipulate the symbols 2. To rebuild the principles 3. To derive new proofs and provide a new mathematical apparatus Of course his analysis depended on some of the previous mathematical attempts to demonstrate the fifth postulate. The failure of the demonstrations – of the fifth postulate from the other four – present to the attention of Lobachevsky, led him to believe that the difficulties that had to be overcome were due to causes traceable at the level of the first principles of geometry. By using internal representations, Lobachevsky has to create new external visualizations and adjust them tweaking and manipulating (Trafton et al., 2005) the

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previous ones to generate appropriate spatial transformations (the so-called geometrical constructions). (I maintain that in general spatial transformations are represented by a visual component and a spatial component (Glasgow & Papadias, 1992).) In cognitive science, many kinds of spatial transformations have been studied, like mental rotation and other actions to improve and facilitate the understanding and simplification of the problem. It can be said that when a spatial transformation is performed on external visualizations, it is still generating or exploiting an internal representation. From this Lobachevskian perspective, the abductive attainment of the basic concepts of any science is in terms of senses: the basic concepts are always acquired through our sense impressions. Lobachevsky builds geometry upon the concepts of body and bodily contact, the latter being the only “property” common to all bodies that we ought to call geometrical. It is clear that in this inferential process, Lobachevsky performs a kind of model-based abduction, where the perceptual role of sense impressions and their experience with bodies and bodily contact is cardinal in the generation of new concepts. On the basis of these foundations, Lobachevsky develops the so-called absolute geometry, which is independent of the fifth postulate: “Instead of commencing geometry with the plane and the straight line as we do ordinarily, I have preferred to commence it with the sphere and the circle, whose definitions are not subject to the reproach of being incomplete, since they contain the generation of the magnitudes which they define” (Lobachevsky, 1929, p. 361). With the help of the explanatory abductive role played by the new sensualist considerations of the basic principles, by the empiricist view and by a very remarkable productive visual hypothesis, Lobachevsky had the possibility to proceed in discovering the new theorems. Following Lobachevsky’s discovery, the fifth postulate will no longer be considered in any way anomalous – we do not possess any proofs of the postulate, because this proof is simply impossible. Moreover, the new non-Euclidean hypothesis is reliable: indeed, to understand visual thinking, we have also to capture its status of guaranteeing the reliability of a hypothesis. (In order to prove the relative consistency of the new non-Euclidean geometries, we should also quote some very interesting visual and mathematical “models” proposed in the second half of the nineteenth century (i.e., the Beltrami-Klein and Poincarémodels), which involve new uses of visual images in theory assessment.) Together with the introduction of the new concept of parallelism, it is possible to derive new theorems of a new non-Euclidean geometrical system exempt from inconsistencies, just like the Euclidean system. There is no space here to provide all the details of Lobachevsky’s creative cognitive process. I would just like to indicate that the process is characterized by a continuous reference to diagrams. He continues to develop the absolute geometry defining the concept of plane and of straight line starting from the bodily perspective of the stereometric level (e.g., BB0 in the mirror diagram of Fig. 5) (Lobachevsky, 1829–1830, 1835–1838, §25). Further, rectilinear angles (which express arcs of circles) and dihedral angles (which express spherical lunes) are then considered and the solid angles too, as generic parts of spherical surfaces – and in particular the interesting spherical triangles. π means for Lobachevsky the length of a

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Fig. 5 The concept of straight line defined as the geometrical locus of the intersections of equal spheres described around two fixed points as centers (example of the use of a mirror diagram)

semicircumference but also the solid angle that corresponds to a semisphere (straight angle). The surface of the spherical triangles is always less than π and, if π, coincides with the semisphere. The theorems about the perpendicular straight lines and planes also belong to absolute geometry. At this point spherical geometry is always treated together with plane geometry, and thanks to this perspective, “infinite” can be perceived in “finite” constructions because the infinite is considered only as something potential that can be just mentally and artificially thought: “defined artificially by our understanding.” As I have already reported in Magnani (2009, p. 129), Lobachevsky states “Which part of the lines we would have to disregard is arbitrary” and adds “our senses are deficient,” and it is only by means of the “artifice” consisting of the continuum “enhancement of the instruments” that we can overcome these limitations (Lobachevsky, 1829–1830, 1835–1838, §38). Given this epistemological situation, it is easy to conclude saying that instruments are not just and only telescopes and laboratory tools but also diagrams.

Non-Euclidean Parallelism The basic unit is the manipulation of diagrams. Before the birth of the modern axiomatic method, geometers still and strongly had to exploit external diagrams, to enhance their thoughts. The new external diagram proposed by Lobachevsky (the diagram of the drawn parallel lines of Fig. 6) (Lobachevsky, 1840) is a kind of analogue both of the mental image we depicted in the mental visual buffer and of the symbolic-propositional level of the postulate definition. It no longer plays the explanatory role of showing an anomaly, as it was in the case of the diagram of Fig. 4 (and of other similar diagrams) during the previous centuries. I have already said I call this kind of external tool in geometrical reasoning mirror diagram. In general this diagram mirrors internal imagery and provides the possibility of

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Fig. 6 Non-Euclidean parallel lines

detecting anomalies, like it was in the case of the similar diagram of Fig. 4. The external representation of geometrical structures often activates direct perceptual operations (e.g., identify the parallels and search for the limits) to elicit consistency or inconsistency routines. Sometimes the mirror diagram biases are inconsistent with the task, and so they can make the task more difficult by misguiding actions away from the goal. If consistent, we have already said that they can make the task easier by instrumentally and non-explanatorily guiding actions toward the goal. In certain cases the mirror diagram biases are irrelevant, they should have no effects on the decision of abductive actions and play lower cognitive roles. In the case of the diagram of parallel lines of the similar Fig. 4, it was used in the history of geometry to make both consistent and inconsistent the fifth Euclidean postulate the new non-Euclidean perspective. I said that in some cases the mirror diagram plays a negative role and inhibits further creative abductive theoretical developments. As I have already indicated (p. 23), Proclus tried to solve the anomaly by proving the fifth postulate from the other four. If we were able to prove the postulate in this way, it would become a theorem in a geometry which does not require that postulate (the future “absolute geometry”) and which would contain all of Euclid’s geometry. We only have to remember that the argument seemed correct because it was proved using a diagram. In this case the mirror diagram biases were consistent with the task of justifying Euclidean geometry, and they made this task easier by guiding actions toward the goal, but they inhibited the discovery of non-Euclidean geometries (Greenberg, 1974, pp. 119–121; cf. also Magnani, 2001b, pp. 166–167). In sum, contrary to the diagram of Fig. 4, the diagram of Fig. 6 does not aim at explaining anything given, it is fruit of a non-explanatory and instrumental abduction, as I have intimated in section “Externalizing Internal Models to Represent and Discover Mathematical Entities: Resolving Conceptual Anomalies”: the new related

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principle/concept of parallelism offers the chance of further multimodal and distributed abductive steps (based on both visual and sentential aspects and on both internal and external representations) which are mainly non-explanatory. On the basis of the new concept of parallelism it will be possible to derive new theorems of a new non-Euclidean geometrical system exempt from inconsistencies just like the Euclidean system (cf. below section “Unveiling Diagrams in Lobachevsky’s Discovery as Affordances: Gateways to Imaginary Entities”). The diagram now favors the new definition of parallelism (Lobachevsky, 1840, Prop. 16), which introduces the non-Euclidean atmosphere of a new parallel postulate: “All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided in two classes – into cutting and notcutting. The boundary lines of the one and the other class of those lines will be called parallel to the given lines” (Lobachevsky, 1840, p. 13). Because diagrams can contemplate only finite parts of straight lines, it is easy to represent this new postulate in this mirror image: we cannot know what happens at the infinite neither in the internal representation (because of the limitations of visual buffer) nor in the external representation, “[. . .] in the uncertainty whether the perpendicular AE is the only line which does not meet DC, we will assume it may be possible that there are still other lines, for example AG, which do not cut DC, how far so ever they may be prolonged” (ibid.). So the mirror image in this case is seen as consistently supporting the new non-Euclidean perspective. The idea of constructing an external diagram of a non-Euclidean situation is considered normal and reasonable. The diagram of Fig. 6 is now exploited to “unveil” new fruitful consequences. A first analysis of the exploitation of what I call unveiling diagrams in the discovery of the notion of non-Euclidean parallelism is presented in the following section, related to the exploitation of diagrams, still related to a stereometric level.

Unveiling Diagrams in Lobachevsky’s Discovery as Affordances: Gateways to Imaginary Entities Lobachevsky’s target is to perform a geometrical abductive process able to create new and very abstract entities: the whole epistemic process is mediated by interesting manipulations of external unveiling diagrams. The first step toward the exploitation of what we have called unveiling diagrams is the use of the notion of non-Euclidean parallelism at the stereometric level, by establishing relationships between straight lines and planes and between planes: Proposition 27 (already proved by Lexell and Euler), “A three-sided solid angle equals the half sum of surface angles less a right-angle” (p. 24, Fig. 7). Proposition 28 (directly derived from Prop. 27): “If three planes cut each other in parallel lines, then the sum of the three surface angles equals two rights” (p. 28) (cf. Fig. 8). These achievements are absolutely important: it is established that for a certain geometrical configuration of the new geometry (the three planes cut each other in parallel lines that are parallel in the Lobachevskian sense), some properties of the ordinary geometry hold.

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Fig. 7 A three-sided solid angle equals the half sum of surface angles less a right angle

Fig. 8 If three planes cut each other in parallel lines, then the sum of the three surface angles equals two rights

The important notions of oricycle and orisphere (cf. Fig. 9) are now defined to search for a possible symbolic counterpart able to express a foreseen consistency (as a justification) of the non-Euclidean theory. This consistency is looked at from the point of view of a possible “analytic” solution that is in terms of verbal-symbolic (not diagrammatic) results (equations). The last constructions of the Lobachevskian abductive process give rise to two fundamental unveiling diagrams (cf. Figs. 10 and 12) that accompany the remaining proofs. They are more abstract and exploit “audacious” representations in the perspective of three-dimensional geometrical shapes. Inside the perspective representations (given by the fundamental unveiling diagram of a non-Euclidean structure, cf. Fig. 10), a Euclidean spherical triangle and the orisphere (and its boundary triangle where the Euclidean properties hold) are constructed. The directly perceivable information strongly guides the geometer’s selections of moves by eliciting what we can call the Euclidean-inside non-Euclidean “model matching strategy.” This maneuver also constitutes an important step in the affirmation of the modern “scientific” concept of model (Fig. 11). In the following I will adopt the cognitive/epistemological/psychological concept of affordance that requires a short description. As it is relatively well-known, affordance is what the environment offers the individual. James J. Gibson introduced

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Fig. 9 Oricycle: the curve lying in a plane for which all perpendiculars erected at the mid-points of chords are parallel to each other. The perpendicular DE erected upon the chord AC at its mid-point D will be parallel to the line AB, which is called the axis of the boundary line

Fig. 10 Unveiling diagram. Diagram that represents a stereometric non-Euclidean form built on a rectilinear right angled triangle ABC to which Theorem 28 can be applied (indeed the parallels AA0 , BB0 , CC0 , which lie on the three planes are parallels in non-Euclidean sense)

the term in his book (1966), and it was anticipated in many of his earlier articles. A clear definition is contained in his in formidable 1979 The Ecological Approach to Visual Perception: The affordances of the environment are what it offers the animal, what it provides or furnishes, either for good or ill. The verb to afford is found in the dictionary, the noun affordance is not. I have made it up. I mean by it something that refers to both the environment and the animal in a way that no existing term does. It implies the complementarity of the animal and the environment. (Gibson, 1979, p. 127)

Here I am referring to the capacities of mathematical externalizations, for example, diagrams or proofs, to afford explanations, discoveries, further ideas capable of improving research, etc. In the studies concerning the relationship between mathematics and cognition, the concept of affordance is usually exploited in many aspects

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of education (Hadjerrouit, 2019; Moyer-Packenham et al., 2016; Watson, 2003), for example, for connecting culture and mathematics (Madusise, 2020), but also when analyzing how affordances related to mathematical analogies improve teaching strategies (Vamvakoussi, 2019). The external representation in terms of the fundamental unveiling diagram illustrated in Fig. 10 activates new mathematical affordances as perceptual reorientation in the construction (that identifies possible further constructions); in the meantime, the consequent new generated internal representation of the external elements activates directly retrievable information (numerical values) that elicits the strategy of building further non-Euclidean structures together with their analytic counterpart (which are the non-Euclidean trigonometry equations). Finally, it is easy to identify in the proof the differences between perceptual and other cognitive operations and the differences between sequential – the various steps of the constructed unveiling diagram – and parallel perceptual operations. Similarly, it is easy to distinguish between the forms that are directly perceptually inspected and the

Fig. 11 Spherical triangle and rectilinear triangle Fig. 12 A final productive unveiling diagram

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elements that are mentally computed or computed in external symbolic configurations. To arrive at the second unveiling diagram, the old diagram (cf. Fig. 10) is further enhanced by a new construction, breaking the connection of the three principal planes along the line BB0 , and turning them out from each other so that they, together with all the lines lying in them, come to lie in one plane, where consequently the arcs p, q, and r will form single arc of a boundary line (oricycle). This goes through the point A and has AA0 as its axis, in such a manner that Fig. 12 on the one side will lie. The arcs q and p, side b of the triangle, which is perpendicular to AA0 at A, the axis CC0 going from the end of b parallel to AA0 and through C00 the union point of p and q, side a perpendicular to CC0 at point C, and from end-point of a the axis BB0 parallel to AA0 which goes through the end-point B00 of the arc p, etc. Finally, taking CC0 as axis, a new boundary line (an arc of oricycle) from point C to its intersection with the axis BB0 is constructed. What happens?

One of the Main Mathematical Cognitive Virtues: Not Indicative of Intuition In this case we see that the external representation completely abandons its spatial intuitive interest and/or its capacity to simulate internal spatial representations: it is not useful to represent it as an internal spatial model in order to enhance the problem solving activity. The diagram of Fig. 12 does not have to depict internal forms coherent from the intuitive spatial point of view; it is just devoted to suitably afford and so “unveil” the possibility of further calculations by directly activating perceptual information that, in conjunction with the non-spatial information and cognitive operations provided by internal representations in memory, determine subsequent problem-solving behaviors: in this perspective we can say that diagrams prompt perceptual models. This diagram does not have to prompt an internal “spatially” intuitively coherent model. Indeed perception often plays an autonomous and central role; it is not a peripheral device. In this case the end product of perception and motor operations coincides with the intermediate data highly analyzed, processed, and transformed that is prepared for high-level cognitive mechanisms in terms of further analytic achievements (equations). (In other problem-solving cases, the end product of perception – directly picked up – is the end affording product of the whole problem-solving process.) We have to note that of course it cannot be said that the external representation would work independently without the support of anything internal or mental. The mirror and unveiling diagrams have to be processed by perceptual mechanisms that are of course internal. And in this sense the end product of the perceptual mechanisms is also internal. But it is not an internal model of the external representation of the task: the internal representation is the knowledge and structure of the task in memory; and the external representation is the knowledge and structure of the task in the environment. The end product of perception is merely the situational information in working memory that usually only reflects a fraction (crucial) of the external

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representation (Zhang, 1997), that is, what affords the human mathematician. At this point, it is clear that the perceptual operations generated by the external representations “mediated” by the unveiling diagrams are central as mechanisms of the whole geometrical abductive and manipulative process; they are not less fundamental than the cognitive operations activated by internal representations, in terms of images and/or symbolic propositions. They constitute a superior example of complex and perfect coordination between perceptual, motor, and other inner cognitive operations. Let us conclude the survey on Lobachevsky’s route to an acceptable assessment of its non-Euclidean theory. By means of further symbolic/propositional designations taken from both internal representations followed from previous results and “externalized” calculations, the reasoning path is constrained to find a general “analytic” counterpart for (some aspects of) non-Euclidean geometry (we skip the exposition of this complicated passage – cf. Lobachevsky (1840)). Therefore we arrive at the equations sin Π ðcÞ ¼ sin Π ðaÞ sin Π ðbÞ sin Π ðβÞ ¼ cos Π ðαÞ sin Π ðaÞ Hence we obtain, by mutation of the letters sin Π ðαÞ ¼ cos Π ðβÞ sin Π ðbÞ cos Π ðbÞ ¼ cos Π ðcÞ cos Π ðαÞ cos Π ðaÞ ¼ cos Π ðcÞ cos Π ðβÞ that express the mutual dependence of the sides and the angles of a non-Euclidean triangle. In these equations of plane non-Euclidean geometry, we can pass over the equations for spherical triangles. If we designate in the right-angled spherical triangle (Fig. 11), the sides Π(c), Π(β), and Π(a), with the opposite angles Π(b) and Π(α0 ), by the letters a, b, c, A, and B, then the obtained equations take of the form of those which we know as the equations of spherical trigonometry for the rightangled triangle sin ðaÞ ¼ sin ðcÞ sin ðAÞ sin ðbÞ ¼ sin ðcÞ sin ðBÞ cos ðAÞ ¼ cos ðAÞ sin ðBÞ cos ðBÞ ¼ cos ðBÞ sin ðAÞ cos ðcÞ ¼ cos ðaÞ cos ðbÞ Lobachevsky assumes that the equations “[. . .] attain for themselves a sufficient foundation for considering the assumption of imaginary geometry as possible” (p. 44). The new geometry is considered to be exempt from possible inconsistencies together with the acknowledgment of the reassuring fact that it presents a very complex system full of surprisingly harmonious conclusions. A new contradiction

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which could have emerged and which would have forced us to reject the principles of the new geometry would have already been contained in the equations above. Of course this is not true from the point of view of modern deductive axiomatic systems. Furthermore a satisfactory model of non-Euclidean geometry has not yet been built (as Beltrami and Klein will do with the so-called Euclidean models of non-Euclidean geometry). (On the limitations of the Lobachevskian perspective cf. Torretti (1978) and Rosenfeld (1988).) As for now the argument rests on a formal agreement between two sets of equations, one of which is derived from the new non-Euclidean geometry. Moreover, the other equations do not pertain to Euclidean geometry; rather they are the equations of spherical trigonometry that do not depend on the fifth postulate (as maintained by Lobachevsky himself). Nevertheless, we can conclude that Lobachevsky is not far from the modern idea of what constitutes a scientific model. We can say that geometrical diagrammatic thinking represents the capacity to extend finite perceptual experiences to known (Euclidean) and infinite unknown (non-Euclidean) mathematical structures that appear consistent in themselves and that have quite different properties each other.

Conclusion In the relationship between mathematics and cognition, I adopted an interdisciplinary attitude guided by the clear philosophical aim of analyzing how mathematics generates specific kinds of “knowledge.” Immanuel Kant’s transcendental ideas have been evaluated as the best conceptual tools capable of showing how mathematics and cognition are strictly intertwined; and the main epistemological virtues of this interplay are outlined. In this perspective I illustrated (1) the anti-metaphysical defense of mathematical capacity to make rational intelligibility of the world; (2) how the empirical world becomes a world of mathematical relations; (3) the historicization/naturalization of mathematics, as related to the need to stress the cognitive processes of discovery and the role of manipulative abduction, modelbased reasoning, affordances, and distributed cognition (I contend that these issues represent synthetic and illuminating ways for illustrating the cognitive aspects of mathematics in the context of discovery); (4) the emphasis on the cognitive virtues of mathematical modeling in natural sciences as an epistemological remedy against the recent belief in certain presumed unreasonable “scientific” merits of the computational management of big data; and (5) how the absence of the “schematic” mathematical cognitive determination in creating scientific intelligibility often leads to mere surrogate “modeling,” unacceptably supposed to be scientific. Finally, the analysis of mirror and unveiling diagrams described in this chapter, taking advantage of the cognitive-epistemological reconstruction of the discovery of non-Euclidean geometry, furnishes an exemplification of the various concepts introduced before, such as abduction, affordance, heuristics, model-based and diagrammatic reasoning, departure from human intuition, and other mathematical virtues.

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Peirce and Grothendieck on Mathematical Cognition: A Merging of the Pragmaticist Maxim and Topos Theory Fernando Zalamea

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peirce’s Pragmaticist Maxim (PM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peirce’s Views on Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grothendieck’s Topos Theory (TT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grothendieck’s Views on Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merging Pragmaticism (PM) and Topos Theory (TT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Cognition within the Merging of the Four Theories (CT) – (TT) – (TSK) – (PM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Around the problematics of “mathematical cognition,” some Peircean and Grothendieckean tools are presented (Pragmaticist Maxim and Topos Theory), which help to assess the diverse perspectives and strata which enrich our understanding of mathematics. Keywords

Peirce · Grothendieck · Mathematics · Pragmatism · Category Theory · Cognition

Introduction Mathematical cognition lies on the borders of art and science, profiting equally from compact esthetical intuitions, deep hypothetical visions, and lengthy rational deductions. In Kantian terms, mathematics is situated between form and the formal, between sensibility and intelligibility. In Spanish, these dualities are expressed by F. Zalamea (*) Departamento de Matemáticas, Universidad Nacional de Colombia, Sede Bogotá, Colombia © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_43

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the use of the prefix “co,” capturing synthetically the back-and-forth between reason (“razón”) and heart (“corazón ¼ co/razón”). The study of those borders and dualities can be greatly enhanced, thanks to some powerful logical, topological, and methodological tools, coming from Peirce (the Pragmatic(ist) Maxim, 1870–1900) and from Grothendieck (Category Theory and Topos Theory, 1955–1990). In this text, we will review those tools, and use them in order to provide a rich canvas of alternating forces and cumulative strata in the assessment of “mathematical cognition.” Section “Introduction” explains Peirce’s pragmatic maxim (in actualized contexts) and its pragmaticist extension (in modal contexts). Section “Peirce’s Pragmaticist Maxim (PM)” describes some of Peirce’s views on mathematics. Section “Peirce’s Views on Mathematics” explores some paradigms of Category Theory and Grothendieck’s invention of Topos Theory. Section “Grothendieck’s Topos Theory (TT)” surveys some Grothendieckean reflections on mathematics. Section “Grothendieck’s Views on Mathematics” merges the Pragmaticist Maxim and Topos Theory. Finally, section “Merging Pragmaticism (PM) and Topos Theory (TT)” implements all these different perspectives, undergirded by the duality razón – co/razón, in order to appreciate better our understanding of “mathematical cognition.” Connections between different sections are illustrated in the following Hasse diagram: 6 5 1

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Peirce’s Pragmaticist Maxim (PM) The pragmatic maxim appears to have been formulated several times throughout the development of Peirce’s thought. The better known statement is from 1878, but more precise expressions appear in 1903 and 1905: Consider what effects which might conceivably have practical bearings we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object. (Peirce, 1931-1958, 5.402; “How to Make Our Ideas Clear”, 1878)

Pragmatism is the principle that every theoretical judgement expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood. (Peirce, 1931-1958, 5.18; “Harvard Lectures on Pragmatism”, 1903)

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The entire intellectual purport of any symbol consists in the total of all general modes of rational conduct which, conditionally upon all the possible different circumstances, would ensue upon the acceptance of the symbol. (Peirce, 1931-1958, 5.438; “Issues of Pragmaticism”, 1905)

The Pragmaticist Maxim (PM) (1903–1905, possible effects) is a modal extension of the pragmatic maxim (1878, actual effects) and signals that knowledge, seen as a semiotic-logical process, is preeminently contextual (versus absolute), relational (versus substantial), modal (versus determined), and synthetic (versus analytic). (PM) serves as a sophisticated sheaf of filters to decant reality. According to Peirce’s thought, we can only know through signs, and, according to the maxim, we can only know those signs through diverse correlations of its conceivable effects in interpretation contexts. The Pragmaticist Maxim “filters” the world by means of three complex webs which can “separate” the one into many and, conversely, can “integrate” the many into one, constituting a representational web, a relational web, and a modal web. Even though the twentieth century has clearly retrieved the importance of representations and has emphasized (e.g., since cubism) a privileged role for interpretations, both the relational and the modal webs seem to have been much less explored and understood (or made good use of) through the century. For Peirce, understanding of the use of a symbolic sign requires a consideration of all necessary reactions between the interpretations (sub-determinations) of the sign, encompassing all possible interpretative contexts. The pragmatic(ist) dimension emphasizes the correlation of all possible contexts: even if (PM) detects the fundamental importance of local interpretations, it also urges the reconstruction of global approaches – appropriate relational and modal gluing together all localities. A diagrammatic scheme of the Pragmaticist Maxim – which follows closely the 1903 and 1905 passages quoted above – can be the following. The shifting dynamic between differentiation and integration is one of the main strengths of the Pragmaticist Maxim, able to capture both postmodernist (local, differential, relative) forces and modernist (global, integral, universal) tensions. Going further beyond, (PM) can be fully mathematized using Category Theory and nonclassical logical systems (Arengas, 2019), yielding a vast array of local theorems in completely formalized contexts. In turn, the Pragmaticist Maxim is closely correlated to Peirce’s general phaneroscopy. Around the 1880s, Peirce had imagined (or discovered, according to our variable ontological commitment) a wonderful phenomenological tool (Peirce, 1981–, 5.300–301; “One, Two, Three: An Evolutionist Speculation,” 1886) which helps to unravel the multilayered geometry of the strata, obstructions, and transits of knowledge. Phaneroscopy, or the study of the phaneron, that is the complete collective spectrum present to the mind includes the doctrine of Peirce’s cenopythagorean categories (“ceno-” coming from the Greek kaíno, “fresh”), which observe the universal modes (or “tints”) occurring in phenomena. Peirce’s three categories are vague, general, and indeterminate and can be found simultaneously in every phenomenon. They are interlaced in several levels but can be prescised (distinguished, separated, detached) following recursive layers of interpretations,

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in progressively more and more determined contexts. A dialectic between the one and the many, the universal and the particular, the continuous and the discrete, the general and the concrete, and the integral and the differential is multilayered along a dense variety of theoretical and experimental fibers. Peirce’s Firstness detects the immediate, the spontaneous, whatever is independent of any conception or reference to anything else. Secondness is the category of facts, mutual oppositions, existence, actuality, material fight, action, and reaction in a given world. Thirdness proposes a mediation beyond clashes, a third place where the “one” and the “other” enter a dialogue, the category of sense, representation, and synthesis. As Peirce reckons (Peirce, 1981–, 5.300; “One, Two, Three: An Evolutionist Speculation,” 1886): By the Third, I understand the medium which has its being or peculiarity in connecting the more absolute first and second. The end is second, the means third. A fork in the road is third, it supposes three ways. (...) The first and second are hard, absolute, and discrete, like yes and no; the perfect third is plastic, relative, and continuous. Every process, and whatever is continuous, involves thirdness. (...) Action is second, but conduct third. Law as an active force is second, but order and legislation third. Sympathy, flesh and blood, that by which I feel my neighbor’s feelings, contains thirdness. Every kind of sign, representative, or deputy, everything which for any purpose stands instead of something else, whatever is helpful, or mediates between a man and his wish, is a Third.

Peirce’s vague categories are “characterized” by the following keywords: (1) immediacy, first impression, freshness, sensation, unary predicate, monad, chance, and possibility; (2) action-reaction, effect, resistance, binary relation, dyad, fact, and actuality; and (3) mediation, order, law, continuity, knowledge, ternary relation, triad, generality, and necessity. The three Peircean categories interweave recursively and produce a nested hierarchy of interpretative modulations (Zalamea, 2012). Dynamic cognition yields progressive precision through progressive prescision. Both surgery and gluing form part of a ubiquitous topology of comprehension. Intelligence grows with the definition of more and more contexts of interpretation, and the association of increasingly fine cenopythagorean tinctures inside each context. As we will see in section “Grothendieck’s Views on Mathematics” below, this topological flavor of the Pragmaticist Maxim will allow the central merging/gluing between Peirce’s and Grothendieck’s thoughts on mathematics.

Peirce’s Views on Mathematics Peirce’s categories permanently overlap in the phaneron. Phenomena are never isolated, because they are never wholly situated in some detached categorical realm. Nevertheless, some readings can emphasize determined categorical layers and can help to obtain important relative distinctions (the method shows, right away, that no absolute characterization is to be expected). Throughout his life, Peirce proposed more than 100 of such layered readings in reference to the classification of the sciences. In 1903, using his categories, Peirce came up with a lasting

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classification, designated by Beverley Kent (1987) as the “perennial” classification (see Fig. 2). The first recursive branching of the classification shows the places of mathematics and the continuum. Mathematics (1), ever-growing support of an ever-growing cathedral, emphasizes possibilia as Firstness: it studies the abstract relational realm without any actual or real constraints. In place (1.1) of the classification, the mathematical study of the immediately accessible is drawn: the study of finite collections. In place (1.2), the study of mathematical action-reactions on the finite realm is undertaken: colliding with the finite, infinite collections emerge. In place (1.3) mediation is realized: the general study of continuity emerges. The awesome richness of mathematics arises from its peculiar position in the panorama of knowledge: constructing its relational web with pure possibilities, it nevertheless reaches actuality (and even reality) by means of unsuspected applications, guaranteeing in each context its necessity. The fluid wanderings of mathematics – from the possible to the actual and necessary – are specific to the discipline. Peirce insisted on the hypothetical character of mathematics (possibility realm in the Pragmaticist Maxim) and its true environment (necessary context in the maxim): “Mathematics is the study of what is true of hypothetical states of things. That is its essence and definition” (4.233; 1902). This modal back-and-forth between possibility and necessity needs some fundamental abstraction and generality, also typical of mathematics: “Another characteristic of mathematical thought is the extraordinary use it makes of abstractions” (4.234); “mathematical thought (...) can have no success where it cannot generalize” (4.236). Peirce goes on to compare two definitions of mathematics, the one given by his father (“science which draws necessary conclusions,” 4.239) and his own (pendulum between possible hypotheses and necessary consequences) (4.238): It is difficult to decide between the two definitions of mathematics; the one by its method, that of drawing necessary conclusions; the other by its aim and subject matter, as the study of hypothetical states of things. The former makes or seems to make the deduction of the consequences of hypotheses the sole business of the mathematician as such. But it cannot be denied that immense genius has been exercised in the mere framing of such general hypotheses as the field of imaginary quantity and the allied idea of Riemann's surface, in imagining non-Euclidian measurement, ideal numbers, the perfect liquid.

The spatial, “ideal,” “perfect liquid” appearance of Riemann surfaces is of particular importance to us, as we will see in section “Grothendieck’s Views on Mathematics” below. In a sense, Peirce is looking at a sort of “geometric definition” of mathematics, beyond a merely deductive one, where the extra dimensions of the hypothetical, ideal realms provide the peculiar characteristics of mathematics. In fact, well beyond what will later be called the foundationalist programs for mathematics (logicism, formalism, intuitionism), Peirce does not look for a foundation for mathematics (based either on analytical logic or on synthetical intuition), but rather the inverse: he looks for an understanding of logic based on mathematics. This is reflected along the classification of the sciences (Fig. 2), where mathematics (1) becomes a soil for the development of logic (2.2.3). And this corresponds to

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our contemporary understanding of mathematical logic, whose main objectives (Proof Theory, Model Theory) study the (i) syntax and (ii) semantics of logical languages through (i) algebraic and (ii) topological tools. As often happens with Peirce, his ideas constitute a web of resistances against the trends of his epoch, which acquire their full sense only a century later. In particular, regarding mathematics, Peirce’s views offer fresh perspectives, beyond the usual “normalization” trends in the analytical philosophy of mathematics, where a search for number foundations hinders the possibility to look at independent, “real,” spatial mathematics (Corfield, 2003). As we will see now with Grothendieck, a true blend spacenumber (going even beyond Einstein’s “space-time”) is required to approach full mathematical cognition.

Grothendieck’s Topos Theory (TT) Grothendieck’s (1958) Edinburgh lecture offers his famous vision on a resolution of the Weil conjectures, through a cohomological blend of Galois extensions (number realm – schemes) and Riemann surfaces (spatial realm – toposes) (Zalamea, 2019, 2021). Afterward, his IHES decade (1960–1970) witnesses the gigantic construction of his Topos Theory (TT), which does not only serve as a key tool to solve the conjectures, but constitutes above all a far-reaching extension of the concept of space, with extraordinary applications beyond its bounded, technical emergence. In Grothendieck’s assessment: “The most fundamental seems to me the extension of general topology, in the spirit of sheaf theory (developed initially by Jean Leray), incorporated in the topos point of view. I introduced these toposes in 1958, to define an l-adic cohomology for algebraic varieties (more generally, for schemes), which in accord with a cohomological interpretation of the celebrated Weil conjectures. In fact, the traditional notion of topological space is not sufficient to treat the case of algebraic varieties over a field different from complex numbers, since the Zariski topology does not provide reasonable discrete cohomological invariants” (Grothendieck, 1972, pp. 3–4).1 Toposes blend the discrete and the continuous under the general framework of considering all sheaves over a category-theoretic notion of topology. First, in Fig. 3, we recall the concept of a sheaf (E, X, p), where E is an upper (“global”) topological space, X is a bottom (“local”) topological space, and p: E ! X is a projection from E to X well behaved (i.e., a “local homeomorphism,” meaning essentially that the upper space is constructed through small sections over the bottom space). A sheaf is thus conformed of a “folding” X and an “unfolding” E, where E can be seen as the disjoint union of the fibers of the sheaf, that is, the punctual inverses p1(x), where x  X. The sections of the sheaf are the neighborhood inverses p1(O), where Oopen  X. An understanding of the sheaf combines thus two complementary approaches: vertical (fibers) and horizontal (sections) (see Fig. 4). 1

Translation is the author’s, with editorial input (Ed.).

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The gluing (or, on the contrary, an eventual obstruction) of local sections into global ones is the main objective of the theory. Sheaves abound in all mathematical regions (geometrical, topological, differential, arithmetical, logical, etc.). Two paradigmatic examples are the sheaf of germs of holomorphic functions (following RiemannPoincaré-Cartan) and the structural sheaf of a ring (following Galois-DedekindGrothendieck). Generalizing and abstracting the topological concept of a sheaf, the construction of a topos follows three basic steps (1)–(3): • (1) An extended category-theoretic definition of a topology: over a category C, a Grothendieck topology is given by coverings (collections of maps) J(U) for each object U of the category, which are “well behaved” (an identity is a covering; covering of coverings is a covering; pulling back a covering is a covering). • (2) An extended category-theoretic definition of a sheaf: given a site (C,J) (i.e., a category C with a Grothendieck topology J), a sheaf (initially defined over a topological space) can now be described over a general site, through a universal ∃! (exists unique) definition. • (3) A consideration of all such sheaves in a category-theoretic environment (topos): Top(C,J) is by definition the category of all sheaves over the site (C,J), and the resulting topos reveals deep structural properties (“exactness”: limits, completeness, Cartesian closure, classifier subobject, etc.) which were not present in the original topological space. The mathematical “gesture” codified in Fig. 5 (see below) corresponds to the musical gesture of an orchestra conductor: over (1) a given score (base space), all the (2) instruments (sheaves over the base) develop the score and become unified under (3) the baton of the director (topos, or musical superstructure, in red). The elevation of the construction follows the path (1) one (site) ! (2) many (sheaves) ! (3) one (topos). In this third level, a profound mathematical structure emerges (transgressive, archetypal, META): a Grothendieck topos possesses all limits and co-limits (is complete and co-complete), possesses generator and co-generator, and is well-powered and co-well-powered. In the ascent from (1) the particular to (2) the differential and (3) the integral, toposes can be seen as sublimations of types, with a double connectivity between them: (2) types are injected in (3) toposes, and (3) toposes are projected into (2) types. The phenomenological (2) and the metaphysical (3) enter then in a rich back-and-forth between many levels of knowledge. Mathematical cognition is related to a precise phenomenological cognition. Depending on some additional structure on the sheaf fibers, the resulting categories of sheaves may tend to be more “numerical”/“algebraical” (if the fibers are, e.g., abelian groups or rings) or more “spatial”/“geometrical” (if the fibers are just sets). In Grothendieck’s words, what emerges is a “synthesis between two worlds, until then contiguous and tightly solidary, but nevertheless separated: the «arithmetical» world, in which reside the so called «spaces» without principle of continuity, and the world of continuous magnitudes, in which reside the «spaces» in the proper sense

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of the term (...) In the new vision, those worlds separated before, form only one” (Grothendieck, 1986, Preface, p. 30). Thus, beyond separation, thanks to the abstraction ( freeness, projectivity) provided by sheaves, a new mathematical smoothness governs the interrelations between geometry and arithmetic. Grothendieck’s revolutionary space-number shift is both simple and extremely deep. A double differential and integral process governs the situation: exploring multiplicity along all sheaves and threading unity along the exactness properties of the topos. From a semiotic and philosophical perspective, the general method (closely related to the Pragmaticist Maxim, as we will see in section “Grothendieck’s Views on Mathematics” below) is fascinating: to understand something, consider all possible points of view, and search for a structural connection between them. Only then will you be approaching true knowledge and, in particular, true mathematical cognition, since Truth requires a full multiplicity of perspectives. On the other hand, a systematic use of the liberty and multiplicity of Topos Theory in our ordinary life would have huge consequences on tolerance, ethics, politics, and social action, since we would be able to destroy the “Self” only in favor of the “Other” (map actions, representable functors, presheaves, sheaves). It is an example of how advanced mathematical operations may transform our everyday life, something expressed independently, for example, in the Solidarność movement (1980) or in the Black Lives Matter movement (2020). Nice toposes with many applications to culture and society are the Toposes of Sheaves over Kripke Models (TSK) (Zalamea, 2020). A Kripke model for (propositional) intuitionistic logic K can be understood as ramified (non-necessarily linear) time frame, with some coherence conditions: propositional information grows over time, a contradiction ⊥ never holds at any time, negation :α is defined as α ! ⊥, and satisfiability behaves classically for “or”/“and,” but acquires a new meaning for “implication,” related to its “future” (i.e., α ! β holds at a time t if and only if 8s  t; if α holds at time s, then β holds at time s). With a small calculation, this forces :: 6¼ id (in fact, one can show that ::α holds at a time t if and only if α holds densely in the future of t). Thus :: behaves as a nontrivial closure operator, inside the order topology on K. With this in mind, one can then imagine all sheaves over the topological space (K,), producing the topos of sheaves over the Kripke model K. In Fig. 6, one can see how a model (TSK) integrates the fundamental forces of mathematical thought (historicity, “unreasonable effectiveness” of mathematics applied to science, phenomenological multiplicity, metaphysical unity), but also fights against any dogmatic reductionism (i.e., all strata are independent). A nice analogical example of a (TSK) is our topos of existence: we take at the base the story of our life, and on each instant we situate the fiber of our beliefs at that instant. Our life gives rise to local sections, usually noncoherent, and we enter into the constant contradictions of our existence. With some perspective, we ask ourselves if our permanent agitation, in childhood, adolescence, mature life, or old age, has made any sense. We examine then if our local sections can be subsumed into a global property of the topos which would offer some sense of transcendence for our being. A positive or negative answer may plunge us in relative satisfaction or despair.

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Many reflection layers are operating in the (TSK) model, between lower and higher structures, extrinsic and intrinsic forces, types and archetypes. This continuous iteration may be seen as closely related to the many interpretative layers present in the Pragmaticist Maxim (PM) and to the continuous sem(e)iosis advocated by Peirce. Through (PM) we obtain a better understanding of signs. In the same vein, we will propose (section “Grothendieck’s Views on Mathematics”, below) that Topos Theory (TT) and, particularly, the Toposes of Sheaves over Kripke Models (TSK) can help us to obtain a better approach to cognizing mathematics.

Grothendieck’s Views on Mathematics Récoltes et semailles (“Harvests and Seeds”; Grothendieck, 1986) wanders meticulously (1500 pages) around mathematical thought and, particularly, around Grothendieck’s own creative paths in mathematics. The complexity of Récoltes et semailles reflects well the complexity of mathematical cognition: (1) an understanding of mathematical understanding, in its eternal (Kantian) fight between form (spaces, numbers, structures) and the formal (cohomologies, motives, derivators); (2) a reflection on the access modes to that understanding and, in particular, on a naive access to invention (freshness, smoothness, childish vision); (3) pondering the mathematical method and, in particular, an analysis of the needed constraints of perseverant work (tasks of the architect and the laborer); (4) a systematic study of the yin-yang slopes of creativity; (5) a conceptual biography of the author and a calibration of the social context where it is inserted; (6) a criticism of degenerative processes in the Western world (impressive anticipation of our present ecological crisis and sanitary crisis); and (7) a web construction of multi-temporal and multispatial stylistic strata, product of an open and inquisitive mind. Beyond Poincaré’s L’invention mathématique (Poincaré, 1908) – the other major twentieth-century reference around mathematical cognition – Grothendieck delves into a multivalent dialectic between the continuous and the discrete, magnitude and number, and geometry and arithmetic and explores axiomatically a back-and-forth between algebraic geometry (1955–1970: algebraic methods to understand space) and topological algebra (1980–1990: topological methods to understand number). The recognition of structural and formal archetypes (K-theory groups, toposes, motives, ncategories, derivators) and their projective distribution along many diverse types (Weil conjectures, standard conjectures, anabelian conjectures) offer new perspectives on the ways in which higher mathematical practice becomes intimately connected with mathematical cognition. Between the many insights offered by Grothendieck (1986), we emphasize here three main ideas (Zalamea, 2019). First, mathematical cognition deals with a backand-forth between discovery and invention, where the two swings of the pendulum – a discovery of mathematical structures and an invention of “languages” which reveal them– are fundamental and irreducible (Grothendieck, a fine musician, repeatedly insisted that he “heard” the structures “speaking” to him). Second, mathematical cognition often works through a rising sea, by “immersion, absorption, dissolution”

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(Grothendieck, 1986, Part 3, p. 553), thanks to Category Theory, where an object (“nut”) is immersed in an adequate category (“sea”) which explains both its “real” behavior and the “ideal” phantasmata which surround the object (Yoneda’s Lemma). Third, mathematical cognition can profit from an analogical use of mathematical tools (combinatorics, elementary geometry, group actions, Galois connections, homology, sheaves, Riemann surfaces) to understand the very realm of mathematics (Grothendieck, 1986, Les portes sur l’univers, Appendix to Récoltes et semailles, p. 1-127). In fact, many of the tasks advanced in the brilliant Appendix Les portes sur l’univers can be seen as gluings, inversions, stratifications, thresholds, and symmetry breaks: (i) variations, degrees, and intensities between warm (yin) and cold (yang) (16-17: numbers in this paragraph refer to pages in Les portes sur l’univers); (ii) inversions, through subgroup associations, between continent and content, between abstraction (yang) and concretion (yin) (19-20); (iii) dialectics multiplicity-unity (“I feel myself like a multiple in search of unity,” 23); (iv) diagrams (hexagons, icosahedrons, trees) to capture yin/yang tonalities (28-32); (v) dynamics between the ideal (yang) and the real (yin) (36-37); (vi) zigzag iterations and homologies between unity/mystery (yin) and order/simplicity (yang) (37-40); (vii) fruitful tension between discovery (yin) and invention (yang) (47-51); (viii) “accordion” between exterior (surface, light, yang) and interior (deepness, shadow, yin) (51-55); etc. In all these processes, “the spirit, hurled in the pursuit of the elusive flesh of things, goes like an Ahab after the White Whale” (66). The ever-growing Grothendieck search for mathematical archetypes (Grothendieck’s inequality, Ktheory group, classifier topos, absolute Galois group, universal homotopy, etc.) is reflected in the allusion to Moby-Dick, that major literary expression of the neverending quest for the metaphysical strata which govern our understanding of the world. Through Grothendieck, a major ontologico-mathematical inversion appears in contemporary mathematics. The emergence of deep archetypical constructs in the technical realms of Category Theory shows that, notwithstanding some naive illusions in analytical philosophy, “metaphysics” has never been dead. On the contrary, many connections with Leibniz’s analysis situs and monads; with Galois’ “métaphysique des équations”; with Riemann’s intuition of structural, complex-variable, unifying forces in mathematical physics; with Poincaré’s homotopical and homological invariants for topological spaces; with Gödel’s efforts to prove the existence of phantasmata (Cassou-Noguès, 2007) or the ontological existence of God (Gödel, 1970); and with Grothendieck’s thorough axiomatization of a full range of archetypes for the space-number connection show that a systematic quest for what lies “beyond,” what cannot be seen through our “blind eyes” (Tarkovsky, 1984), has always been one of the main forces which propels mathematical cognition. The fact that those systematic visions occur in advanced mathematics has been perhaps the basic obstruction which explains why the analytical philosophy of mathematics, reduced to considerations on elementary mathematics and set-theoretic reconstructions, has been naturally blind to aspects of mathematical cognition. Thanks to Grothendieck and to the enlargement of space-number obtained in Topos Theory, we

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are now situated in a richer framework that includes nonstandard, nonclassical, nonessentialist perspectives, which allow a larger and deeper comprehension of mathematical imagination and its stratified entanglement with our multiverse experience.

Merging Pragmaticism (PM) and Topos Theory (TT) Peirce’s Pragmaticist Maxim (PM), Category Theory (CT), Topos Theory (TT), and the Topos of Sheaves over a Kripke Model (TSK) share many features. Their fundamental core is to understand a sign, concept, or object, through its conceivable, contextual effects, either semiotic (PM), relational (CT), archetypical (TT), or dynamical (TSK). A multidimensional, multivalued, multi-stratified approach becomes mandatory, and our comprehension escapes reductive frameworks (either classical or analytical). A relativization trend marks all these perspectives, but they always possess a universal counterpart: integrating the semiotic differences through correlative interpretations (PM), capturing the back-and-forth between universal (∃!) definitions and concrete realizations (CT), projecting the archetypical exactness properties of the topos onto the different sheaf types (TT), synthesizing the dynamical development of time through the nonclassical logics encrypted into the classifier object (TSK). In short, a conceptual, abstract, differential and integral calculus governs the many layers of our understanding, through a new notion of universal relative (see Fig. 7). An apparent contradiction lies in the terminology “universal relative.” In fact, universalization has always been considered as an absolute, nonrelative process, but after Gödel’s Incompleteness Theorems (1931), it is well known that no absolute foundation for mathematics is possible and, a fortiori, no absolute foundation for knowledge is to be expected. Thus, the universalization idea must be relativized and cannot longer live in an impossible absolute. But the program of Category Theory consists precisely in obtaining universal, abstract, non-absolute definitions of the usual, concrete, mathematical structures. In this sense, relative universals do acquire a precise technical sense and can consistently be thought, thanks to the extremely precise axiomatics of the “categorical imperative” (CT) – (TT) – (TSK). On the other hand, Peirce’s maxim (PM) points to a general comprehension of particular signs, which takes into account both their differential, concrete representations and their integral, abstract correlations, gluings, and transfers (see Fig. 1). As Pavel Florensky recalls, the etymological analysis of “universal” comes from the merging “unum versus alia,” the One versus the Other (Florenskij, 1914, p. 146). In the same vein, beyond multiplicity, a path to unity can be imagined, following the merging (CT) – (TT) – (TSK) – (PM). Iterating the sheaf theoretic methods to this very problematic, one can also think of additional gluings, or blendings, along the conceptual line (*): (CT) – (TT) – (TSK) – (PM). In fact, a quadruple iteration – sheaves applied to sheaves applied to sheaves applied to sheaves – is at stake: (1) from a sheaf (S), we pass to (2) categories of sheaves (TT), to (3) categories of sheaves of sheaves (TSK), and to (4) sheaf theoretic expansions of the line (*), thanks to very general tools (CT, PM). In this tendency toward a geometric multiplication of our understanding, all the different

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strata provide interesting new paths: (1) a sheaf over (*) capturing (CT), (TT), (TSK), (CT), as four different fibers of knowledge, (2) a topos perspective multiplying those initial four-fibered sheaves through imaginary global sections connecting them, (3) a Kripke-topos approach exploring the dynamic development of the connections obtained, and (4) a global (CT, PM) reading integrating methods and meta-methods in a coherent framework. In practice, (1) if sheaves (S) capture local-global transits and obstructions, (S) applied to the merging (*) offers a good method to assess the advantages and drawbacks of the “relative universals” method; (2) if toposes (TT) synthesize typearchetype transits and obstructions, (TT) applied to (*) can calibrate the extent of successes and failures in the dialectics relativization-universalization; (3) if Kripketoposes (TSK) detect cognition dynamics around time, (TSK) applied to (*) reveals our crucial historical limitations; (4) if Category Theory (CT) and the Pragmaticist Maxim (PM) underline the back-and-forth between the differential and the integral, (CT) and (PM) applied to (*) strongly emphasize our need to ban any reductionist strategy in cognition. The results (1)–(4) offer, in particular, some basic multilayered levels required to express the complex richness of mathematical cognition (see section “Merging Pragmaticism (PM) and Topos Theory (TT)”).

Mathematical Cognition within the Merging of the Four Theories (CT) – (TT) – (TSK) – (PM) Kant distinguishes between the intelligible and the sensible through a dialectic between formal and form, where a functional drive helps to understand form through the formal. But the dialectic remains obscure, with all the deep forces of penumbrae beyond light (Zalamea, 2013). Already Pascal, with his famous calembour “Heart has its reasons of which reason knows nothing,” pointed to forms of intuition and sensibility that a purely formal treatment could never apprehend. In Spanish, Pascal’s limitation is well captured by an idiosyncrasy of the language: “razón” ¼ reason is contrasted with “corazón” ¼ heart, with a full duality inscribed in Spanish, thanks to the (category-theoretic) prefix “co.” The wonderful equation (**) corazón ¼ corazón is a unique characteristic of the Spanish-speaking realm and, better, of Hispanic America, extremely attentive to the merging and blending of opposite cultural and sociological currents (Zalamea, 2000). If we apply our vision (**) corazón ¼ co-razón, to the line of knowledge studied above (*) (CT) – (TT) – (TSK) – (PM), we obtain a fruitful inversion, or duality, of the perspectives in play. In fact, beyond reasons (CT), (TT), (TSK), (PM), their obverses display some rich penumbrae of the heart: (CT verso) thanks to Yoneda’s Lemma (which may well be understood as the “heart” of Category Theory; see Fig. 8 below), one observes the emergence of “ideal” phantasmata (presheaves) beyond “reality” (representable functors); (TT verso) thanks to the diverse logics that may be embedded in the classifier subobject, the duals of the Heyting algebras of subobjects (i.e., co-Heyting algebras) encapsulate the emergence of paraconsistent logics in the toposes, allowing local contradictions without destroying the system; (TSK verso)

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thanks to the multiple strata in the topos of dynamic sheaves, the double negation operator (in fact, a Lawvere’s elementary topology in the topos) discriminates actual (real) truth, from possible/dense (imaginary) truth; (PM verso) thanks to a diagram of the Pragmaticist Maxim on a sheet (see Fig. 1), one can imagine the verso of the drawing (following Peirce’s techniques in his existential graphs; see Zalamea (2012)), which inverts the pendulum differential-integral and suggests that many differential reasons (analytical types) can be obtained as projections of integral co-reasons (synthetic archetypes). Both (*) and (* verso) can now be projected into many features of mathematical cognition. First, L’invention mathématique (Poincaré, 1908), with its beautiful web

Fig. 1 Peirce’s Pragmaticist Maxim (PM)

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Fig. 2 Peirce’s triadic classification of the sciences

of reasons (proof, order, conscious work, voluntary efforts) and co-reasons (illumination, esthetic sense, harmony, unsuspected liaisons), which help to calibrate mathematical creativity, can be seen as profound psychological features of the human mind, capable both of exploring lucis et umbrae, to construct and deconstruct the positive and the negative, the classical and the nonclassical, invariants and variations, and form abductions and formal deductions. Second, Peirce’s views on mathematics (section “Peirce’s Pragmaticist Maxim (PM)” above), situating logic inside mathematics, contrary to foundationalist programs, express well the aerial dialectics (*) – (* verso), where many trends in Category Theory and Topos Theory explore the multifarious regions of mathematics, thinking in different universes for their development and looking for different mathematics in each arbitrary topos, well beyond the topos of sets: castles firmly travel in the air (counterpart to Murphey (1961, p. 407)), without any need to ground them. Third, Grothendieck’s views on mathematics (section “Grothendieck’s Topos Theory (TT)” above), emphasizing the

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Fig. 3 From analytic continuation in complex variables (Riemann) to a topological sheaf (Leray)

Fig. 4 A sheaf: spaces, projection, fibers, and sections

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Fig. 5 Basic gesture of a topos

Fig. 6 Topos of Sheaves over Kripke Models. Model (TSK)

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Fig. 7 Relative universals: a new life through (PM) and (CT)

Fig. 8 Yoneda’s Lemma: the heart of (CT). Lucis et umbrae: representable functors (hA) versus general presheaves (functors C ! Sets)

emergence of “relative universals,” make a systematic use of dualities, where the identity (* verso) ¼ co–(*) becomes just a particular situation of much general cognition “adjunctions.” The general abstract strategies (*)/co–(*) can be concretely detailed along some mathematical examples: • Around the infinite, possibly the most important concept in mathematics, the line (*) explains the initial structural role of the natural numbers (CT: via Lawvere’s NNO, natural numbers object) in apprehending infinities, while its dual (* verso) is used in size-independent proofs (TT: following Freyd’s use of Lawvere’s elementary topology ::, proving, e.g., the independence of the continuum hypothesis). • Around the crucial construction of ideal structures in mathematical cognition (Hilbert, 1925), the line (*) offers clues on the “ideality” of concepts (aura of an object in CT, archetypes in TT, general signs in PM), while (CT verso) explains the necessary appearance of ideal constructions, thanks to Yoneda’s Lemma

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(Zalamea 2012) and its fully general embedding of (incomplete) discreteness into (complete) continuity. • Around the basic space-number blending, the line (*) explores the multiplication of space (toposes), while its dual (* verso) captures the multiplication of number (schemes). • Around Grothendieck’s duality invention/discovery for mathematical cognition, the line (*) supports yang inventive architectures, while its dual (* verso) approaches better yin discovery patterns. In this way, the concrete and the abstract coalesce, the pragmatic view and the category-theoretic view complement each other, and material cognition and mathematical understanding become welded together in a natural web of perspectives which enhances our grasp of the world.

References Arengas, G. (2019). La máxima pragmática peirceana: modelos categóricos, dualización, aproximaciones algebraicas y modalizaciones lógicas. Ph. D. Thesis. Universidad Nacional de Colombia. Cassou-Noguès, P. (2007). Les démons de Gödel: logique et folie. Seuil. Corfield, D. (2003). Towards a philosophy of real mathematics. Cambridge University Press. Florenskij, P. (1914). Il significato dell'idealismo. Rusconi (1999). Gödel, K. (1970). Ontological Proof. In K. Gödel. Collected Works (vol. III) (p. 403). Oxford University Press (1995). Grothendieck, A. (1958). The cohomology of abstract algebraic varieties. In Proceedings International Congress of Mathematicians (Edinburgh) (p. 103-118). Cambridge University Press. Grothendieck, A. (1972). Esquisse thématique des principaux travaux mathématiques de A. Grothendieck. Technical Report. CNRS. Grothendieck, A. (1986). Récoltes et semailles. Unpublished Manuscript. Hilbert, D. (1925). On the infinite. In J. van Heijenoort (ed.), From Frege to Gödel. A Source Book in Mathematical Logic 1879-1931 (p. 367-392). Harvard University Press (1967). Kent, B. (1987). Charles S. Peirce. Logic and the Classification of Sciences. McGill – Queen’s University Press. Murphey, M. (1961). The Development of Peirce's Philosophy. Harvard University Press. Poincaré, H. (1908). L'invention mathématique. Institut Général Psychologique. Tarkovsky, A. (1984). Sculpting in time. University of Texas Press. Zalamea, F. (2000). Ariel y Arisbe. Evolución y evaluación del concepto de América Latina en el siglo XX. Andrés Bello - Tercer Mundo. Zalamea, F. (2012). Peirce's Logic of Continuity. Docent Press. (Extended translation of El continuo peirceano, Universidad Nacional de Colombia, 2001, and Los gráficos existenciales peirceanos, Universidad Nacional de Colombia, 2010). Zalamea, F. (2013). Antinomias de la creación. Las fuentes contradictorias de la invención en Valéry, Warburg, Florenski. Fondo de Cultura Económica. Zalamea, F. (2019). Grothendieck. Una guía a la obra matemática y filosófica. Universidad Nacional de Colombia. Zalamea, F. (2020). Modelos en haces para el pensamiento matemático. Universidad Nacional de Colombia (to appear). Zalamea, F. (2021). Grothendieck: A Short Guide to his Mathematical and Philosophical Work. In B. Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Springer (to appear).

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Linguistic and Visuospatial Chunking as Quantitative Constraints on Propositional Logic Donna E. West

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chunking and Its Affordances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unconscious Versus Conscious Chunking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anticipatory Logic to Inform Chunking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semiotic Influences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Icons and Indices as Chunking Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application to Working Memory Genres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Advantages of Higher-Level Chunking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This account explores the mathematical underpinnings of working memory (WM) processes. It assumes a semiotic approach, positing that constructing signs by analogy (with indexical and iconic values) is paramount in establishing and reforming units handled in limited (space, time) memory systems. Accordingly, analogies constitute mathematical tools, in that they increase meaning efficacy, while decreasing the processing load. Analogies both unify smaller units into larger, more potent ones, and likewise conserve the number of meaning components permitted to be encoded, and stored in WM and eventually in LTM. This chunking process can be unconscious, or conscious; but, in either case, it advantages propositional meanings by incorporating units (linguistic, sensory stimuli) into larger aggregates. Sacrificing the often very abstract meanings of smaller units does not result in meaning reduction; rather its embeddedness within meaning frames provides needed contextual amplification to enhance interpretive endeavors. As such, chunking is equivocal to an additive operation D. E. West (*) Modern Languages, State University of New York, Cortland, NY, USA e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_44

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in mathematics whereby new members of a class inform both the addition to the class, as well as the class’ identity. Armed with the means to classify, chunking provides quantitative and qualitative advantages, one of which is increased space and resources in WM to admit fewer items with more potent meaning associations. Keywords

Working Memory · Chunking · Icon · Index · Episodic Processing · Diagrammatic Reasoning

Introduction This account explores the mathematical underpinnings of working memory (WM) processes. It assumes a semiotic approach, positing that constructing signs by analogy (with indexical and iconic values) is paramount in establishing and reforming units handled in limited (space, time) memory systems. Accordingly, analogies constitute mathematical tools, in that they increase meaning efficacy, while decreasing the processing load. Analogies both unify smaller units into larger, more potent ones, and likewise conserve the number of meaning components permitted to be encoded, and stored in WM and eventually in LTM. This chunking process can be unconscious, or conscious; but, in either case, it advantages propositional meanings by incorporating units (linguistic, sensory stimuli) into larger aggregates. Sacrificing the often very abstract meanings of smaller units does not result in meaning reduction; rather its embeddedness within meaning frames provides needed contextual amplification to enhance interpretive endeavors. As such, chunking is equivocal to an additive operation in mathematics whereby new members of a class inform both the addition to the class, as well as the class’ identity. Armed with the means to classify, chunking provides quantitative and qualitative advantages, one of which is increased space and resources in WM to admit fewer items with more potent meaning associations. The competency to apprehend that unconscious chunking mechanisms scaffold WM processing, indicates that an a priori capacity underlies it. Discerning more meaning potency from fewer connected structures, and realizing the necessity of such strategy for extending propositional meanings, namely, numerosity, support the a priori assumption. The evidence in favor of the a priori (untaught/unlearned) capacity for this sense is predicated upon the following: its pre-existence to counting skills in human ontogeny (Butterworth, 1999: 101; Libertus & Brannon, 2010), its use across species (Butterworth, 1999: 139), and its universality across human cultures (Butterworth, 1999: 117–119). Numerosity constitutes a primitive form of chunking, in that, despite its global/undifferentiated knowledge (the expectation of additional affordances), it demonstrates a rudimentary analogy – items held together by some intrinsic motivational factor.

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Chunking and Its Affordances The propensity to chunk (commit to hierarchical representations) is formidable in the receipt and recovery of new information. As such, the chunking operation establishes boundaries, and hence has an algebraic and geometric function in logical genres; it determines quantities and qualities of instantiations. According to Baddeley (2007: 64) diagrammatic schemes of processing (although manufactured in all of his WM components: the phonological loop, the central executive, and the episodic buffer) are particularly apparent in the part of working memory referred to as the visuospatial sketchpad. These diagrammatic chunks (be they visuospatial, auditory, or otherwise) are characterized as a process by which smaller units are bound consequent to increasingly higher meaning affiliations. The chunking process is motivated by temporal and spatial limitations determined by each of the WM systems; accordingly, the chunks are arranged sequentially. Their temporal and spatial limitations (how many elements can fit in the defined space and for how long) actually advantage up-take of logical meanings. They serve a gatekeeping function – facilitating the generation of propositions. At the same time, their cognitive advantage is apparent – energizing increasingly larger meaning components in short-term, working, and long-term memory. Despite the limitations imposed by these constraints (and perhaps because of them), the process itself becomes indispensable; it provides the means to regulate information flow – affording more economical encoding, while taking advantage of episodic potential. This WM process allows improved storage for successful retrieval. According to Miller (1956), the capacity for processing information may be more determined by the number of chunks than by the number of discrete items (measured by digits in STM). Baddeley (2007) provides further evidence that chunking is exercised more prominently in LTM than in STM – his 2000 reform that the episodic buffer must exist to regulate logical chunking bears witness to this. Hence, the profound influence of chunking upon executive control and logic cannot be overstated (Baddeley, 2007: 145). As an executive process, interpreters utilize chunking to seek out affinities within informational strings, creating diagrammatic and analogous connections. Later it governs the unification of larger, more conceptual and more propositional meanings. Because these larger chunks emanate from more conscious effort, they obviously rely upon conscious intervention (Baddeley, 2007: 307), (“Good mnemonic strategies depend on their capacity to integrate previous unassociated material by conscious manipulation, for example by means of interacting visual imagery” (Baddeley, 2007: 307).) in that interpreters deliberately and intentionally seek out meaning to undergird structure clustering. Their power to infuse discrete pieces of structural uptake with potential goals and purposes accounts for their effectiveness to chunk at higher levels. This capacity to manage units of structural up-take is paramount – especially its power to draw out the purposes of constituents (e.g., participant roles) and hint at episodic connections among the constituents.

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Several memory systems are facilitated by chunking, namely, encoding, shortterm storage, forming episodes, and long-term storage. Chunks need not be linguistic units, e.g., phonetic aggregates; they can constitute image schemas, e.g., pictures/happenings framed in simple stories (Baddeley, 2007: 96–97; Mandler & Cánovas, 2014: 510). In fact, meaning assignments mediated by chunking do not contradict the psychologism which Peirce cautions against, because their operation establishes logical potential in designing subjects and thereafter predicates, hence initiating novel propositions. The cognitive readiness to receive and apply distinctive predicates to subjects; it supports the influence of logic (guarding against psychologism), in that readiness to chunk prepares the mind to actively attribute constituents to contributory ontological truths. This natural proclivity to identify affinities to explore truth supplies interpreters with the natural capacity to anticipate subject-predicate affiliations (see Baddeley’s, 2000, 2007 model), beginning with information up-take. The natural operation of seeking out affinities attunes the memory device to settle upon attributory meanings for inclusive storage and reliable retrieval. This process is similar to a priming device (cf. Schacter et al., 2004), in that it ensures more rapid synapses upon repeated appearance of a stimulus or set of stimuli. The natural expectation – seeking more discrete meaning packages for logical benefit – showcases how organisms rely substantially upon implicit meanings to construct higher level bundles. The natural drive to attach meaning (thereby reducing the quantity of units) during priming, is akin to an imprinting process – manufacturing episodic meaning structures from scraps of memory data. To form episodic memories, special vigilance is exercised to attend to and recognize analogies across neighboring components, be they visual, auditory, olfactory, or tactual. In priming, meaning-to-structure connections are automatic; as a consequence, more memory resources are available to focus on forming higher level meaning-to-structure chunks (consonant with episodically based constructs). These episodic bundles form the foundation for propositional content, in that smaller units which begin as structures absent discernable meaning are exploited consequent to automatic priming and resultant freeing of resources allocated to form non-automatic meaningful structures.

Unconscious Versus Conscious Chunking An episodic form of chunking scaffolds proposition/argument-building, in its propensity to turn unconscious meaning structures into more conscious and more logical ones. Nonetheless, even more unconscious chunking relies upon some meaning affiliations; otherwise there would be no affinity for the individual units to be subsumed into a bundle, however unconscious the operation might be. More unconsciously generated chunks are driven by some implicit meaning, as well as decisive quantity based procedural constraints, e.g., numerosity. Numerosity, as such, is described as a cognitive primitive (Coolidge & Overmann, 2012), (More specifically, they refer to it as “one of the feral cognitive bases for modern symbolic

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thinking” (Coolidge & Overmann, 2012: 204).) and as innate (Bagchi & Davis, 2016; Harvey et al., 2013). Assuming that numerosity creates constraints based on quantity, e.g., that a group of four is perceived to be more than a group of three, one can certainly argue that the emergence of numerosity demonstrates that as a species, humans (and arguably other species) are prone to chunk early in ontogeny. Because numerosity entails unlearned propensities toward perceiving holistically, it qualifies as chunking – binding neighboring objects despite incidental differences. This readiness to include more than a single item in contexts in which objects are proximate and perceptually similar is the essence of numerosity; it constitutes a first attempt at exploiting quantitative measures to begin learning classificatory attributes. This quantitative measurement supplies a holding place or marker to return to upon future instantiations of similar objects. Consequently, numerosity establishes a primacy effect, in that when remembered, future instantiations of similar items are processed more rapidly and effortlessly; and the pictorial shape of the three or four items form a composite, allowing rapid recognition of their context and properties, independent of awareness of sequence or amount. Priming is further consolidated with repetitions of the array, in view of the iterativity of color, shape, voice attributes, and the like. Numerosity has been noted in nonhuman species, e.g., squirrels, bees, lions, and chimps (Butterworth, 1999: 139–144). These species spontaneously estimate the amount of quantities, in that they differentiate two amounts, compare them, and determine the effect that the quantity might produce on them, be it advantageous or otherwise. With respect to the former, squirrels gravitate twice as often to tree branches which harbor more nuts. Gallistel (1990) notes that this preference demonstrates adaptive number capacity. This basic numerosity discernment is likewise operational in bees. Worker bees dance only after accessing a certain threshold of food (von Frisch, 1965/1967). It should be kept in mind that these findings, however, may have been influenced by a reward-based paradigm, namely, food, and may not reflect the extent of the bees’ number concept. To test this, honey bees were provided with a maximum or minimum of three dots; they preferred the array of three (to two or four), suggesting some counting-based awareness, namely, absolute numerosity, which is more advanced than relative numerosity (Bortot et al., 2019). Lionesses utilize relative numerosity competency when they attack other tribes – only when they perceive a threat (that the amount of the tribe is equal to or greater than their own). McComb et al. (1994), used the number of unique roars to measure the lions’ numerosity determinations; although the legitimacy of this measurement (number of unique roars is questionable (in that the experimenters determined uniqueness and overlap might have been a confounding factor), the findings do not abrogate altogether the possibility that the lions relied upon global number skills. Chimps likewise are considered to possess rudimentary number capacity. Alpha male chimps implemented a numerical system to communicate distinct concepts to their community, e.g., drumming once on two trees means proceed according to a certain path, whereas drumming twice means rest (Boesch, 1996). In short, all of these species demonstrate an association of distinct meanings with at least two different amounts of stimuli, hence demonstrating rudimentary numerosity skills; their responses

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toward particular groups of stimuli consequent to more/less parameters (not fewer because not necessarily countable) demonstrate preferences for particular amounts of items which are in their purview. The findings clearly show relative numerosity (only hinting at the possibility of absolute numerosity); more or less (not fewer) is discerned, absent the means to count individual stimuli (Davis, 1993). In human ontogeny, two kinds of numerosity are recognized: subitization and magnitude appreciation (Feigenson et al., 2004). The former is akin to relative numerosity exhibited in other species, and emerges earlier in ontogeny (Kaufman et al., 1949), given its global character. Subitization entails recognition and temporary storage of small numbers of items without apprehending their sequence. This early quantitative competency becomes operational prior to 6 months of age presumably because differences among items are not under scrutiny, and the skills of counting and arranging individual members have not yet emerged: “[it] is important to note that subitization is not synonymous with counting (i.e., counting requires ordinality or number sequencing)” (Coolidge & Overmann, 2012: 205). With competency of subitization, infants nearly instinctually ascribe meaning to single items; and individuated items may well be unprocessable at early ages. In fact, this innate competency demonstrates the first attempts to make intelligible arbitrary but discrete groups of items when attending to common properties among three or four objects (Feigenson et al., 2004). This apprehension of common properties illustrates early attempts to ascribe general quantitative meanings to objects (perhaps drawing upon proto-propositional logic); and shape similarities appear to be primary. More complex forms of numerosity are termed magnitude appreciation, “the ability to appreciate large but approximate numerical magnitudes” (Coolidge & Overmann, 2012: 205), since capturing clear interrelationships between larger groups of numbers defines its implementation (Feigenson et al., 2004). This numerosity skill is more advanced than subitization, in that comparisons between numbers is operational, whereas beforehand sequencing/ordinality is absent. Coolidge and Overmann likewise note that neither kind of numerosity is modality specific: “[i]nterestingly, both core systems appear to be robust across various sense modalities” (2012: 205). Curiously, Coolidge et al. attribute the early exhibition of numerosity to powers of intuition, and assume that more advanced cognitive skills which require classification of similar attributes, likewise arises from seemingly unexplained intuitions: “dual systems of numerosity provide an intuitive basis for analogies and metaphors” (Coolidge & Overmann, 2012: 209). Their presumption that analogies and metaphors have an “intuitive basis” is simple-minded; it ignores the influence of semiotic forces (per Peirce’s triadic system) to scaffold meaning upon perceptual aspects of ontology. For Peirce, the role of iconic signs to inform object meanings is hardly intuitional; iconic signs, rather than unexplained pre-existing cognitions, are the catalyst to capture analogous frames, even pertaining to objects of quite distinctive genres (in the case of metaphors). What Coolidge and Overmann fail to recognize is twofold: the semiotic (iconic, indexical) basis for numerosity, and how such lays a foundation for the development of propositions and arguments. The unlearned nature of numerosity makes obvious its reliance upon directional and diagrammatical systems, absent dependence upon symbolic

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representational skills (language). Numerosity’s unique dependence upon global/ undifferentiated object situatedness – drawing together items before the mind never before experienced – validates the early influence of nonsymbolic signs (as an analogous operation) to implicate foundational spatial and temporal relations. For Peirce, sudden inspirational hunches do not emerge from first cognitions (intuitions), but from instincts which imply togetherness (1868: 5.213). (Throughout this chapter, the term intuition will be taken as signifying a cognition not determined by a previous cognition of the same object, and therefore so determined by something out of the consciousness. Let me request the reader to note this. Intuition here will be nearly the same as “premiss not itself a conclusion”; the only difference being that premises and conclusions are judgments, whereas an intuition may, as far as its definition states, be any kind of cognition whatever. But just as a conclusion (good or bad) is determined in the mind of the reasoner by its premise, so cognitions not judgments may be determined by previous cognitions; and a cognition not so determined, and therefore determined directly by the transcendental object, is to be termed an intuition.) Coolidge and Overmann’s claim that numerosity skills are “automatic and preattentive” (2012: 209) falls short of illustrating that they stem from intuitions; rather their claim provides leverage for Peirce’s instinct model. After all, reliance upon unconscious processing does not presume that the source for the sudden hunch is a cognition never before conceived of. In characterizing analogies as outgrowths of intuitions, Coolidge et al. fail to account for the propensity to assign situational meanings to objects. The diagrammatic building-blocks which cognition uniquely provides (e.g., numerosity) are not explained by intuitions, but by emergent attempts to derive relatedness from the hereness and nowness in which objects are embedded. Coolidge et al.’s insistence that intuitions are responsible for perceiving basic analogies is misplaced; it ignores the mind’s need to access previously considered and stored memory chunks (situatedness of previous arrays). Furthermore, Coolidge et al.’s assumption that intuitions are responsible for generating analogies invalidates Peirce’s vehement rejection of intuitions altogether. In short, it is antithetical to the prospect of the emergence of all three forms of logic (abduction, induction, deduction), because it assumes some a priori competence of first cognitions – cognitions which do not require constructive mental devices to arrive at plausible hypotheses, etc. Instead, diagrammatic models (based upon mathematical foundations) convince us that cognitions emerge from structuring meanings, namely, chunking (consequent to meaning analogies). Be they unconsciously or consciously generated, chunks are inherently diagrammatic; they cohere diffuse items into a concept/proposition/argument; and each member/constituent contributes to the meaning boundaries. As such, what is included within chunks manages the container aspect (per Lakoff and Johnson’s (1980, 1999) schema – however subject to change the schema might be). (Danesi (2019: 141–153) likewise addresses this issue with regard to the concept of “gamifying” mathematics pedagogy with puzzles.) Nonetheless, Lakoff and Johnson’s proposal that spatial primitives alone establish and manage attentional matters fails to fully account for the handling of attentional matters, namely, the

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cognitive and logical need for living systems to allocate meaning to ontological features, hence to bind phenomena. This effort gives prominence to the efficacy of index and icon in interpreting spatial arrays by supplying the initiative to look at, listen to, and feel the properties which epitomize or call attention to what might cohere subunits into chunks – properties common to or reagents of the core meaning. Hence, the propensity to observe certain objects and arrays is guided, even at foundational stages, by a sign-meaning-object primitive, innervated by definite iconic operations (1898: MS 485). Notice of definite physical contexts (including resemblance to other entities in that context) and their boundaries cannot be explained by drawing upon spatial primitives alone, but must rely upon first-order semiotic devices which impel organisms to be especially vigilant in the face of ontology, such that they sense some intellectual meaning from assembling units into an inclusive chunk. The present approach proposes that Lakoff and Johnson’s model (1980, 1999) is remiss in assuming that forming container propositions from innate spatial schema is the most fundamental skill to make sense of the world. Instead, the profound influence of first-order semiotic chunks – the diagrammatic approach of processing analogies (whose meanings are underpinned by iconic and indexical information) is the cornerstone for examining the rightness of fit and boundedness critical to determine container schemas. Numerosity findings demonstrate the propensity of living systems to depend upon diagrammatic representations, because they are armed with the internal directive to ascribe recognizable identity to things in the universe. Accordingly, interpreters instinctually seek different kinds of and increasingly larger containments.

Anticipatory Logic to Inform Chunking Diagrammatic explanations for how it is that numerosity is a foundational quantitative competence, require analyses of how interpreters’ hunches about the meanings inherent in spatial arrays increases the number of constituents within chunks, while decreasing information load and up-take. The process is effective provided that interpreters ignore slight differences across members of the array/chunk – so that ultimately more meaning potency is gleaned from less/fewer components. In this way, chunks which are generated automatically ensure that other chunks can be admitted into the limited WM system. Priming chunks to be processed automatically makes space for additional items. WM resources can then incorporate additional items which require conscious deliberation. Reliance upon conscious, deliberative determinations of fit (rather than instinctual operations) ultimately requires realization of episodic features – making apparent the purpose for the object or its feature. Incorporating purpose into chunks produces higher-level chunks, in that the chunk’s meaning is defined by destinations, goals, and the like. For Peirce (1907: MS 318: 16–17), ascribing a purpose to ontological stimuli unequivocally

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validates that interpreters chunks are informed by triadic signification, ultimately improving not only encoding, but storage and retrieval, as well. In fact, the existence of purpose to drive spatial and temporal components introduces episodic value to chunks, increasing their relevance and guarding against forgetting. This semiotically-driven propensity to attach more analogous-based meanings to presentments (internal, external) establishes an information round-up strategy which invites both higher-order chunking and inferencing from raw data (including episodes conceived of in the mind). This affordance, in turn, checks earlier subjectpredicate assignments, sometimes renewing their validity, while in other cases, filtering out extraneous and faulty assignments (by annulling or defeating the predicative assumptions). This kind of defeasibility demonstrates how chunking constrains fitting new predicates to subjects, hence informing proposition-making, and the fit of ground-level hypothesis creation. As such, chunks possess the means to prioritize purposes of co-occurring entities. In fact, unifying objects based on purpose is a scaffolding device for hypothesis construction – for trying out subjects with different predicates. In turn, underutilized or implied predicative material rallies interpreters to take a first look as to the appropriateness of particular subjects to already established chunks, as well as the fit of subjects with certain predicates. This WM processing opportunity revitalizes proposition-making by revisiting the question of the chunk’s (the hypothesized subject-predicate affiliation’s) sustainability – the determination of which propositions still qualify as assertions after repeated exposure to similar objects. This kind of chunking requires hypothetical operations – projecting amplified subjects for propositions in the future, and predicting predicative extensions, e.g., metaphorical uses. When hypotheses are plausible, they express higher-level chunks, or laws whose propositions have some possibility of materializing. For Peirce, mathematical operations express this hypothetical thinking, when their hypotheses consist in real possibilities (see Cooke, 2011: 180). Wilson (2020: 365) elaborates on Peirce’s argument for how mathematics is hypothetical: Peirce is clear that there are real possibilities. However, in order to avoid attributing a hyperinflationary realism to Peirce, we must resist interpreting Peirce as claiming that all logical possibilities are real. (Here it becomes necessary to disambiguate our definitions of the word “real.” Far from being limited to the physical and empirical here and now, the Peircean definition of “real” extends to those things which are merely possible – e.g., because some of the technologies exist, flying cars are “real” under this definition.)

Peirce contends that apart from predicting the possible, under laws which can materialize, “the truth of the pure mathematical proposition is constituted by the impossibility of ever finding a case in which it fails” (1902: 5.567). In never failing to prove the law inaccurate, the potentiality for truth-value is guarded; and some plausibility of the hypothetical proposition/argument still stands. This claim (absence of proof to invalidate) permits mathematical laws to be real possibilities. In contrast, Wilson’s argument of a “hyperinflationary realism” results in an overly limited view of realism where mathematical cognitions are concerned (in excluding

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them altogether). Wilson’s use of “hyperinflationary” actually deflates the power of mathematics the possibility of substituting infinite integers, and to exist as a rather abstract representation. Moreover, the power of integers to represent broad values in algebraic or diagrammatical constructs would, in effect, not be considered among the possible. Nonetheless, Peirce’s concept of possibility was constructed to withstand expansive and recurrent meaning chains in which the real possibility of truth survives even in the absence of Logical Interpretants—since the object properties to which “A” can refer are not identifiable. Furthermore, the unknowability of the Immediate Objects associated with algebraic units, although militating in favor of impossibility, does allow for some semblance of possibility in the incessant search for the Immediate Object – the search keeps alive intermediate forms of the ultimate Final Interpretants. Individual instantiations, on the other hand, do possess Dynamical Objects when expressed in a particular string. Even in the absence of Logical Interpretants, algebraic constituents arguably still possess Energetic Interpretants. Although Energetic Interpretants derive from single instantiations only (see West, 2020), they may still provide sufficient potential meaning to allow algebraic instantiations to qualify as real possibilities. In short, despite their placeholder character, algebraic expressions and integers may still qualify as real possibilities, by virtue of predication upon Energetic Interpretants. Accordingly, single, idiosyncratic interpretive events which give rise to Energetic Interpretants, may preclude interpreters from drawing upon the combinatorial effects of events/conditions. This approach can overly pidgin-hole more general meanings inherent to Immediate Objects. Attentional, storage and retrieval Limitations within working memory (especially Baddeley’s episodic buffer) require converting Dynamical Objects present in Energetic Interpretants into Immediate Objects of Logical Interpretants – scaffolding the exercise of the hypothetical by considering competing propositions. The operation of larger, more abstractive chunks expedites comparisons of more than one hypothetical proposition, increasing simultaneous consideration of competing propositions. As such, interpreters can efficiently consider which has greater merit.

Semiotic Influences To handle and store information for predicative purposes, some foundational semiotic (triadic) operation must be in place to exploit binding – subject-based (meaning-objects) and predicative binding (objects’ contributory perceptual and functional features). This process generates higher-level structures, in which more propositional meanings prevail. The present model proposes that propositional logic based on analogic reasoning drives the chunking process from the outset, such that incessant search for underlying predicative meanings primes memory up-take; and interpreters unconsciously seek out interpretants even in the absence of linguistic representations. Hence, the search for purpose in primitive diagrammatic arrays clearly emerges as a primary catalyst for determining meaning chunks (see Pietarinen, 2006). It is the search for analogies which alone drives the unity of

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smaller chunks into larger ones. Chunking from analogous reasoning (however unconscious it may be) orients interpreters to seek out predicates for subjects, and funnels interpreters’ attention to the need to enhance memory flow via shrinking the number of informational chunks. In other words, because semiotic priming requires attaching meaning (especially purpose-based) to objects at foundational junctures in processing, it is responsible for the motivation to shrink the number of elements to be attended to within the working memory apparatus. This shrinkage has far-reaching consequences; it increases memory span (Baddeley, 2007: 186), which militates against clogging the memory system. Shrinking the number of units in WM regulates the amount of information that can fit into defined (limited) memory spaces by “making the many few.” Although “making the many few” is a quantitative process momentarily managing data receipt with fit advantages, qualitative measures steer its shrinkage. The incessant search for meaning to ensure semiotic efficacy demonstrates the integral influence of such qualitative measures. This process operates when several small components are folded into a single meaning structure, such that incidental meanings are encapsulated, having a bundling effect. Rather than having to attend to innumerable units with little discernible meaning, chunking affords attending to, and storing fewer (although larger) units; hence, meaning is exploited. Chunks can operate in any modality: visual, auditory, tactual, or olfactory; and it is obvious that the information bundling may derive from several distinct modalities. Conscious, as well as unconscious meaning-pairing operate in the flow of experience, requiring adaptations in the Gibsonian sense (1979: 88) of ambient perception. In short, chunking is responsible for the various and sundry layers of meaning which are often implicit when structures are combined. The episodic nature of chunks bears witness that logic operates even before more permanent storage in long-term memory. Logic is made obvious both in active connections between entities (agent affects X), and in more passive relations (resultative events). In either case, inferential logic surfaces when episodes are depicted/articulated as having novel states of affairs. In short, the propensity to chunk episodically constitutes convincing evidence for the existence of logical primitives, guiding future judgements after retrieval from storage in an effort to interpret subsequent intake. The logical primitives anticipate propositional meaning, making obvious their function to quicken assigning meaning to units. Groundlevel expectations for discovery of propositional meaning facilitate chunking, and hence make expedient (make automatic) lower-level chunks. This process guards against superfluity of seemingly irrelevant intake, while guarding against precipitous judgements. As such, effortlessly assigning and extracting meaning from form preempts novel predications, which, in turn, gives rise to expectations of predicative value. The effect wrought by these preemptive meanings provides evidence for the indispensability of semiotic operations from the outset. The reach of semiotic processes is so fundamental; it compels the application of triadic relations to the syntax of events. The presence of Thirdness invokes purposes for events, showcasing their episodic meanings. In this way, semiotic operations form the bedrock for logical thought.

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Icons and Indices as Chunking Devices The analogies which are inherent in icons (see Bellucci, 2018: 131), together with the attentional necessity for indexical signs to ascertain resemblances between different objects of the same and different subjects promotes building higher level chunks. Anticipating propositions hastens the operation of connecting what appears to initially be arbitrary elements in the memory stream. Rather than ascribing the power to assign meanings to physiological sources such as priming as a neurological device, the influence of foundational semiotic proclivities must be examined. Even in his late semiotic, Peirce is convinced that the search for icons is the mechanism by which analogies are discovered (see Bellucci, 2018:131), even at the most elementary levels. Icons are scaffolds to extract propositional meanings, as Peirce puts it, signs “. . .from which information may be derived” (c. 1902: 2.309). Here Peirce claims that a telling characteristic of icons is their potency to extract information, i.e., their facilitated means to disclose interpretants. Algebraic expressions constitute icons (with interpretants), in that they provide a roadmap for constructing informationally based meanings: “All icons, from mirror-images to algebraic formulae, are much alike, committing themselves to nothing at all, yet they are the source of all our information. They play in knowledge a part iconized by that played in evolution, according to the Darwinian theory, by fortuitous variations in reproduction” (c. 1902: MS 599: 42–43). Images in the mind and mathematical formulae qualify as icons because they supply “fortuitous variations in reproduction.” Peirce’s use of “fortuitous,” together with “variations,” describes the essence of how icons (analogies) function. Akin to sensors and dendrites, icons reproduce variations on previous representations, and play an active role in magnetizing future loose bundles of mental arrays/mathematical rules into a profoundly unique aggregate in which the components are so solidified that they deliver a new informational unit. Here, icons govern forces of intelligent affinity – scaffolding novel chunks of information and transferring such to expectant minds. Peirce holds icons responsible for generating future affirmative/negative variations on the original schema. The future variations can extend to the fringes of what is possible, hence Peirce’s use of “fortuitous.” In addition to manufacturing and disclosing novel chunks of information about its sign, the icon uncovers meanings pertaining to their objects: “For a great distinguishing property of the icon is that by the direct observation of it other truths concerning its object can be discovered than those which suffice to determine its construction” (1895: 2.278). In this passage, Peirce’s message to us regarding icons is two-fold: 1) they establish parameters for sign conformity (i.e., for the form of the newly constructed chunk), and 2) they reveal previously unconsidered “truths about the object.” Peirce elaborates on the standard to which, as signs, icons are held, namely, to ensure affinities within their representamen; otherwise, little incentive to manage previous meaning structures and to create future ones would be truncated. Second, Peirce ascribes to icons the delight of serving as intermediary with the ontological world, to indicate some previously unconceived of fact regarding its object – presumably the Dynamical Object. The fostering of new “truths regarding the object” provides icons with the added, practical responsibility to communicate

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such truths among interpreters. In short, the two-fold purpose of icons entails managing new propositions/arguments from what otherwise might be sterile past knowledge chunks to more informed chunks, as Peirce terms them, “ideas” (1895: 2.278). Whether directly or indirectly communicated, icons are uniquely responsible for transmitting ideas: “The only way of directly communicating an idea is by means of an icon; and every indirect method of communicating an idea must depend for its establishment upon the use of an icon” (1895: 2.278). The direct meanings which the icon reveals entail the nature of its sign and interpretant, while its capacity for indirect transfer entails revelation of “truths regarding its object.” Icons draw upon analogous features of meaning from affinities either between components of signs, or those of objects. In fact, icons qualify as the only signs which provide information (cf. Stjernfelt, 2011: 397). Whether in conveying information or ideas via icons, Peirce is explicit that the unique purpose of receiving information between interlocuters is to promote reasoning: it is “only by icons that we really reason” (c. 1893: 4.127). Misiewicz elaborates on just this issue – how iconic signs are the only signs from which information/ideas (signs sporting analogous meanings) can be drawn: “they begin the path of understanding by loading our predicates with some conceptual character, and generalizing those predicates requires diagrammatic – and so analogical – cognition for use in propositions” (2020: 304). As such, it is only by icons (either explicit or implicit) that distinct objects (the old and the new) are drawn together into bundles. In Peircean terms, anticipation of propositional logic to inform processing practices is foundational to iconic and subsumed indexical meanings, in that some icons are often rather difficult to decode, particularly when they qualify as hypoicons (see Stjernfelt, 2007: 277). This is so in part because the decoding relies upon inferential skills in which concerted consciousness is paramount. The meaning in such icons must be inferred, given its dependence upon “possibility involving a possibility” (c. c. 1895: 2.279; 1902: 2.311; and Sternfelt 2014: 208). In 1895, Peirce foreshadows the integral place of iconic signification as follows: “. . .a great distinguishing property of the icon is that by the direct observation of it other truths concerning its object can be discovered than those which suffice to determine its construction” (2.279). In other words, the construction itself does not determine its ultimate meanings, nor does it limit how its structure is processed. Here Peirce recognizes the power of anticipated iconic relations to inform the very processing of what might appear to be unrelated units at a perfunctory glance. In short, Peirce’s admission of the inherent informational nature of icons demonstrates a conviction that future instantiations of objects (stemming from analogy) unquestionably influence meaning assemblage at basic conscious processing levels (c. 1895: 2.341). (“. . .a mental construction, or diagram, of something possessing those characters, and the possession of those characters is kept in the foreground of consciousness.”) Stjernfelt further advances Peirce’s argument as follows: “. . .the decisive test for iconicity lies in whether it is possible to manipulate or develop the sign so that new information as to its object appears. Icons are thus signs with implicit information that may be made explicit” (2011: 397). The iconic acquires information when a

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predicate (already associated with another/other subject) is extended to a new subject. This process is motivated by analogy, in that it consciously/unconsciously recognizes some perceptual, action, or affective/rationale basis for the subjectpredicate affinity. The motion/action meaning affinities demonstrate the presence of indexical signs in making prominent primary truths underlying functions for objects (consisting of implicit and latent meanings). In fact, because all icons rely upon “involved indices” (1905: MS: 284: 43; West, 2016, 2018; Stjernfelt, 2014: 55), the influence of index in manufacturing iconic signs must be further explored in Peirce’s Dicisign (see Stjernfelt, 2014: 49–69; Bellucci, 2018: 207; West, 2018). In the Dicisign, the information underlying the analogy classifies the subjects of propositions, which is an indexical operation, given that indices are subjects of propositions, in that they existentially quantify variables (1903: EP2:209; 1903: EP2:168; 1905: MS: 289; and Stjernfelt, 2014: 59–60). Iconic operations are likewise express quantification on a universal plane; they accelerate sameness by extending the same predicates to different, but similar objects (see Stjernfelt, 2014: 60). Like iconic signs (Bellucci, 2018: 131), indexical signs promote diagrammatic reasoning, especially those which qualify as composite photographs (1903: EP2: 281; 2.317) in that they direct the intellectual eyeballs of the interpreter to examine objects (1908: 8.350) (“. . .B. Designatives (or Denotatives), or Indicatives, Denominatives, which like a Demonstrative pronoun, or a pointing finger, brutely direct the mental eyeballs of the interpreter to the object in question, which in this case cannot be given by independent reasoning.”) in novel contexts implying new meanings. In this way, indices call attention to objects’ location, and to their actual or purported movement. They likewise capture the interlocutors’ attentional stream (see West, 2013 chapter 2, West, 2016, 2018 for further explication of indexical functions). Beyond the resemblance communicated by the icon and the action/attentional operation of the index, a nugget of intellectual meaning is borne. The function of index in constructing and communicating analogous meanings should not be understated. Otte’s (2006: 13) argument and his 2015 (with de Barros) are in accord – that it is index which compels the implicit diagrammatical meanings epitomized in mathematical formulae. Otte (2006: 13) asserts the “notion of index becomes fundamentally important.” He adds that the nature of mathematics (particularly algebraic formulae) as “activity” accounts for its influence. The role of index in interpreting diagrams (as in algebraic expression) suggests that chunking diagrammatic units is more primary than is chunking language structures. Yet, the place of index in mathematics amplifies its contexts of influence. Were Peirce’s earlier characterizations of index as objects in the material world not expanded to linguistic and mental genres as a dicisign (1903 and 1906), the claim would be anomalous: “The indices occurring in pure mathematics refer to entities or objects that belong to a model, rather than to ‘the real world,’ that is, they indicate objects in constructed semantic universes” (Otte & de Barros, 2015: 760). With respect to WM processing, our propensity to conceive of these informational meanings (by way of analogies conceived of in icons and indices) scaffolds chunking in the episodic buffer. It does so by supplying propositional chunks of moving pictures (episodes depicting icons with involved indices). These moving

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pictures are diagrams, in that they depict to the mind of the beholder how objects perform resultative meanings, e.g., agent, patient, and beneficiary, implying goals/ destinations (c. 1905: 4.8). The chunks which interpreters form then contain purposive meanings – the potential consequences of objects, the operation of A upon B, for example. These higher-level episodic chunks drive interpreters to exploit meanings of simple propositions, converting them into arguments. These argument chunks are afterward employed to generate abductive hypotheses. In short, absent the natural gravitation toward analogous representations, meaning advancement (chunking at argumentative levels) would lack the mechanism to quickly and efficiently code information in limited storage systems.

Application to Working Memory Genres The role of working memory to arrive at and process analogous meanings is obvious, particularly in light of the organizational functions associated with the central executive and the episodic buffer. The latter has three primary functions: (1) to integrate information from the loop, the visuospatial sketchpad and LTM into a coherent whole; (2) to bind this information from diverse sources into meaningful chunks (static, dynamic), and (3) to temporarily store these larger chunks, ordinarily for fifteen seconds (Baddeley, 2007: 148). Static binding derives from two frequently experienced perceptual features (e.g., color, shape), while dynamic binding emanates from combining two or more novel characteristics (Hummel, 1999). The episodic buffer is the most responsible mechanism in WM to meaningfully chunk subjects with predicates, in view of its power to bind propositions, in such a way that purposes for their sequences are underscored, e.g., agent contributes to a surprising consequence. For example, the episodic buffer unites the subject and predicate of birds peep, but after a surprising consequence in which a parrot barked, the predicate of the proposition is altered to: birds bark. Tailoring arguments relies upon: which linguistic and visual units are admitted into the central executive and the episodic buffer, which become temporarily stored there, and ultimately how the encoded units become integrated with LTM units within more permanent storage systems (Baddeley, 2007: 203–205). Without these underlying meaning assignments permeated within WM, linking logic to representations (forming new concepts, new propositions, new arguments) is an unlikely prospect. In fact, whether inferences are instinctual or deliberative, they, nonetheless, depend upon contraction within WM; otherwise, the plethora of smaller meaning units would retain a grammatical and/or a parochial meaning, which would preclude meaning at higher cognitive/linguistic levels, e.g., semantic, syntactic, discourse. The preclusion is undoubtedly the result of ignoring indexical factors – contextual ones, such as spatial, temporal, and participant influences. These factors need first to be chunked into smaller units so that limited memory systems (in the phonological loop and the visuospatial sketchpad) are not overwhelmed consequent to attention to too many lower-level units, e.g., more than fifteen phonological, or syllabic components (see Erlam, 2009; West, 2012). Exercising rehearsal (sub-vocal or vocal) to

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remember lower-level chunks is paramount; otherwise the trace will fade (Baddeley, 2007: 49–51). But, to graduate to chunking at a higher level, articulatory rehearsal must be interrupted, unless the processing is so automatic that WM resources have sufficient means to call upon more conscious processing operations. In effect, the greater the number of inclusions (sub-chunks) within a chunk evidences that one has had to depend upon more conscious strategies to find meanings for binding – meanings which are innervated by purpose. This is the case because these meanings hold together more solidly, suggesting more complex associations between the subjects and predicates. As such, more explicit predicates in the form of perceptual judgements are, in turn, more discernable and extractible from more complex chunks. In this way, more extended and more compressed meanings within chunks is a necessary precursor to enhance predicative potency, which, in turn, intimates the purpose underlying the chunk. At this juncture, when chunk (sign) is associated with object, then with purpose (interpretant), some degree of awareness of representational relations is present, however abstract. Folding sub-chunks into higher level chunks requires more working memory resources, and hence increased need to consciously manage and expand memory span – the amount that one can retain in WM concurrently within a three second interval (see Baddeley, 2007: 181–184). When higher-level chunks bring together (analogize) elements of propositions and arguments, they acquire an episodic purpose. These episodic meaning chunks implicitly contain the purpose for the participants pivotal to the episode. Armed with episodic memory chunks, interpreters can encapsulate event elements into a pre-logical sequence (see West, 2014). Accordingly, chunks acquire a discourse function, making coherent inter-event connections; and the intra-sentential and inter-sentential cohesion critical for sound interpretation is likewise enhanced. In other words, hierarchizing chunks constitutes a semiotic device which affords the advantage of easy and more exacting storage and retrieval, consequent to event-based meanings. Moreover, absent the complex of the meaning component, the ability of memory span to give precedence to higher level meanings would be truncated. Accordingly, semiotically driven episodic chunking is indispensable; otherwise, lower-level chunks (those without clear predications) would govern – short-circuiting the effectiveness of the central executive to regulate chunks and shunt them to the LTM system (Baddeley, 2007: 124–145). Hence, cognitive control over these processes (even before arguments have emerged) relies upon the capacity to minimize (suppress) rehearsal/attention to lower-level chunks (see Baddeley, 2007: 91– 92, 50–51, 63–64; and West, 2012). It is obvious that analogizing is operating – folding more predicative meanings into chunks (higher-level), hence maximizing executive resources by prioritizing contextual episodic features over redundant inconsequential ones. Examples include verb lexemes (which entail semantic relations, namely, agentive and resultative) and their preferred status over smaller phoneme chunks (Labov & Waletzky, 1967: 13, 32). This higher-level chunking operation requires exertion of conscious control, such that phonological and syllabic units develop affinities to one another by repeated analogous associations, that they are largely automatic, freeing up attentional resources for representations which afford scrutiny of goal-related themes (Baddeley, 2007: 125–126).

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Further Advantages of Higher-Level Chunking It must be kept in mind that temporal and spatial limitations militate in favor of forming higher-level chunks in the episodic buffer and the central executive into LTM; otherwise, they are likely to be forgotten, shelved, or buried. Refreshing what one is about to say sub-vocally is an effective strategy to guard against forgetting (Baddeley, 2007: 80); such has the advantage of practice to determine the efficacy of the chunks as an aggregate. The motivation is always to move along units from Baddeley’s two slave systems, or points of percept entry (the phonological loop and the visuospatial sketchpad) into the working memory systems which combine and regulate meaning chunks, which increases memory span. Apart from encoding phonetic/phonological units, the central executive (by way of memory span) processes units from the visuospatial sketchpad. Items from the sketchpad destined for further integration in the episodic buffer ordinarily are visual and spatial in nature. These visuospatial chunks may take on greater significance, given the increased role of imagery in melding visual components into higher-level, situational chunks (see West, 2014). Because the sketchpad both integrates sensory units with other systems: the loop, and LTM (Baddeley, 2007: 64), it is most responsible for imagery-based memories. The sketchpad highlights shape (visually/tactually); and makes prominent spatial parameters (within, between) within the chunk. The chunks which emanate from the sketchpad do not merely define physical attributes, but individuate the outer boundaries of mental images – percepts which are internal. The latter image contributes to episode building within narratives by conjoining structural elements into a sign with obvious iconic and indexical meanings (see Stjernfelt, 2014: 59–60). The meaning caliber and effect of this sign within Baddeley’s visuospatial sketchpad is formidable; its nature as a Dicisign emphasizes resemblance along with spatiotemporal relevance; in so doing, it declares its meaning with a Legisign, as well as chunking that meaning in a consolidated image. At the same time that WM span imposes constraints (which at first glance might appear to be deleterious), its limited space and time forces consideration of the most relevant chunks within a larger meaning base. This operation offers the advantage of requiring the subversion of sound features to automatic status, while promoting increased meaning potency for each higher-level chunk by enlarging its units either in the phonological loop or in the visuospatial sketchpad. Ultimately, WM constraints have several beneficial functions: they force resolution of attentional competition; and they marshal the mind to apply episodic meanings to individual phonological elements or to features of diagrams (Dicisigns). Producing episodic meanings and extracting their meanings entail discourse related skills. Episodic meanings require support of Baddeley’s visuospatial sketchpad, made obvious by their deictic nature. Comprehending the exigencies exerted by participant role alterations, following changes in location and orientation, and switching temporal frame reference, constitute deictic competencies which facilitate a mutual and ultimate vision for episodes. These deictic (hence indexical) parameters affect episodic images, since they require vigilance in arranging units in

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WM given the need to instantaneously reinterpret/formulate different and perhaps opposing judgments – all instantaneously online. This incessant call to revisit temporary memory units challenges how and whether certain representations become concatenated with others to make obvious the episodic and explanatory construct of component aggregates. It is evident that the constraints and affordances of space and time encoding are crucial to sustain propositions/arguments, because they determine the core meaning elements which are actually kept and maintained in Baddeley’s buffer to associate terms with predicates. Because the number of units and their duration in working memory are finite (four for a three second period prior to being refreshed, Baddeley, 2007: 145; Cowan, 2000), and since propositions/arguments require particular space and resource load to sustain them, the structure of the working memory system must not be ignored. Viable claims regarding how hunches surface, and their character requires a deep seeded appreciation for the number of memory units in the proposition/argument, together with the amount of executive resources necessary to sustain their storage. The degree of memory resources to support simultaneous retrieval of long-term memory (LTM) elements and to make comparisons between propositions/arguments newly encoded items constitute still another compensatory issue meriting consideration. Abductive reasoning requires uniting a number of memory units, as well as ferreting out the relevance of LTM units (old information) of newly encoded information. This entire process not only draws upon encoding constraints of space and memory span, but also upon significant executive resources to coordinate WM units with those from LTM (shunted into the buffer). This is evidenced when previous LTM units (assumed to be relevant) surface concurrently with the newly encoded units (linguistic, visuospatial); and judgments as to the plausibility are called for. In the process of encoding a surprising consequence, consideration of which elements (phonetic, lexical, visual) need to be omitted/shelved/attended to is central to germinating hunches – obviously the viability of the hunch must be measured against previous LTM knowledge, so that decisions as to which has more merit can be determined. In fact, instinctual abductions are unlikely to require as many resources as do more deliberative abductions, since they ordinarily do not depend upon comparisons and adjustments with units from LTM. Accordingly, utilization of a well-anchored working memory model is in order (to reason abductively), especially one for which episodic units are given a central place. Baddeley (2000) and Baddeley (2007: 148–156) has updated his 1986 model to account for the role of the episodic buffer. The episodic buffer affords several functions: to control attention/focus, to divide/separate units, and to switch attentional frames. To orchestrate the latter, the buffer is additionally tasked with interfacing WM with LTM (Baddeley, 2007: 139–142). The buffer contributes to enhanced chunking in the following ways: it monitors encoding linguistic and visuospatial information from what Baddeley refers to as the “slave systems;” it temporarily stores this encoded information; and it integrates the former with relevant information from LTM. The two slave systems (the phonological loop and the visuospatial sketchpad) are the point of entry for perceptual information (be it diagrammatic or symbolic); and

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each has spatial and temporal limitations even more restrictive than those characteristic of the buffer. Baddeley’s phonological loop is responsible for encoding, brief storage, and recall of items which are deemed relevant enough to remain in the loop. Although phonological units can be chunked in the loop (representing an early stage in processing), the aggregating process is often rather automatic. As a consequence, strategies for recall are frequently unconscious (Baddeley, 2007: chapter 3). Some are barely conscious, as in subvocal rehearsal, which entails some underlying wish to recall such units and not allow them to evaporate. Baddeley describes the low-level conscious awareness/control necessary for rehearsal in the loop as follows: “One of the advantages of the loop is that rehearsal seems to proceed with minimal conscious control, allowing awareness to be utilized to maximize other aspects of processing” (316). Even in the buffer, conscious control of chunks is not always obvious, perhaps given the absence of rehearsal, and other memory devices which operate in the slave systems: “Hence, while I would be inclined to identify the episodic buffer with the representation of events that are currently in conscious awareness, much of the machinery that feeds the buffer is probably not typically itself open to conscious manipulation” (316). In the loop, rehearsal can be covert or overt. When it is covert (subvocal) it is often unconscious, such that phonological units are reviewed absent explicit refreshment (via articulation). When rehearsal is more overt and articulated, it is more likely to impose some degree of conscious awareness of the chunks which merit encoding and passage into the buffer and central executive. After meaning is assigned, when chunks are initially practiced subvocally only (in the loop or the sketchpad), the modality does not afford the sound or pictorial feedback which is natural especially to linguistic chunks. In the event that rehearsal is overt, phonological units are afforded a feedback loop (permitting practice of primary points in the narrative) – sound units are both produced and afterward are heard. The role of audition in this enterprise is substantial; it allows the producer to detect elements in the message that might deserve improvement. In either case (covert or overt rehearsal), recall is made more accurate via conscious or unconscious practice. Baddeley’s loop provides a very limited work space in which to encode perceptual information, prior to shunting it to the episodic buffer for further integration with other measures of the memory system (e.g., the visuospatial sketchpad and LTM). Holding on to phonological units creates a memory trace which is held in the loop, giving the item efficacy to be recalled and to advance to the buffer. While verbal information has greater potency to stay in the loop (resulting in a memory trace which can be beneficial when telling narratives), visual information can more easily be precluded from entry into the loop if subvocal rehearsal is blocked (Baddeley, 2007: 40). Without engaging in sub-vocal or vocal rehearsal, phonological units are especially subject to rapid decay (causing narrators to forget their premise), particularly when more than three seconds intervene between presentment of stimuli and repetition (Baddeley, 2007: 38–39). The upshot for enhancing memory is the likelihood that irrelevant units are unpracticed and become quickly forgotten/inaccessible, and of course do not pass beyond the loop into the buffer, while those which are practiced are strengthened. Factors which further limit encoding of phonological

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strings include the morphemic and semantic complexity which overlays the phonemic units, together with prosodic and distinct phonotactic patterns (Baddeley, 2007: 42). The latter two factors can interfere with accurate recall when stress/pitch is unexpected, or when the position of sounds with respect to one another conflicts with the conventional patterns of the particular language. Additionally, memory of the same sound across the same string is more likely to be forgotten (Baddeley, 2007: 49), perhaps as a consequence of lower salience and decreased attentional affordance. Accordingly, recognition of the same/analogous sound constitutes a more passive cognitive intervention to guard against forgetting, with the advantage of preserving iconic representations and their meanings. Nonetheless, memory of longer strings may require a more active strategy for remembering structural chunks coordinated by meaning; they need to be refreshed through rehearsal (rearticulating just a portion of the string). Determining the viability of meaning chunks and exercising smooth articulations within overt messages provide a scaffold to fill in any gaps. The ability to chunk a longer string bears witness to this more active (more conscious) memory strategy, because longer words (with greater numbers of morphemes) can, nevertheless, be successfully chunked. Moreover, strings containing more than a single word appear to rely upon selection of semantic and syntactic assignments to more consciously aggregate meanings into chunks. This process of folding less integral structures into wider and more concept-based and more predicate-based memory components has the benefit of ensuring the memory of what had been an arbitrary unit; but, clarifies the potency of semantic and conceptual meanings to bind initially arbitrary elements with concurrent event-based profiles. This active approach of building increasingly larger units of meaning demonstrates the advantages of Baddeley’s model; it demonstrates how otherwise arbitrary phonological and visual percepts are converted into semantic and syntactic units for ease of interpretation by other minds. This process shows how more linguistic units can be interpreted concurrently when compared with units whose meaning is disconnected and abstract. Baddeley’s (2007: 68) discussion of the capacity of only four chunks within the visuospatial sketchpad, as opposed to four and nine in the phonological loop suggests that the propositional knowledge which underlies more graphical items is closer iconically to processes inherent in event goals; hence fewer units need to be integrated into each visual aggregate – each chunk consisting in a greater number of features; whereas, phonological units have a less obvious reason to fit into a unitary semantic chunk (hence fewer sound units are included in each chunk).

Concluding Remarks This account bears witness to how chunking at higher levels of analogical meaning potential exploits episodic knowledge as action situatedness, e.g., agency and resultative meanings. The exploitation of diagrammatic sign implementation serves as a mathematical tool to uncover the anatomy of propositional logic. The present inquiry demonstrates how index and icon underlie the analogous processes inherent

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in mathematics which advance propositional logic. It proposes that the propensity to bind episodes demonstrates how indexes and icons introduce a code of chunked sequentiality, which arguably is the most primary measure of early mathematical thinking. In this way, episodic chunks code how A affects B, for example. The upshot is that more is less (more chunks produce fewer episodic meanings); and fewer is more (fewer chunks result in more meaning potential). The upshot is that meanings aggregated according to episodes embed smaller, isolated meaning units, making space for other, more potent meaningful chunks. When episodic thinking is hastened, inferring why the constituents of the episode are sequenced as they are further increases the potency of the chunk.

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Labov, W., & Waletzky, J. (1967). Narrative analysis. In J. Helm (Ed.), Essays on the verbal and visual arts (pp. 12–44). University of Washington Press. Lakoff, G., & Johnson, M. (1980). Metaphors we live by. University of Chicago Press. Lakoff, G., & Johnson, M. (1999). Philosophy in the flesh. Basic Books. Libertus, M., & Brannon, E. (2010). Stable individual differences in number discrimination in infancy. Developmental Science, 13(6), 900–906. Mandler, J. M., & Cánovas, C. P. (2014). On defining image schemas. Language and Cognition, 6, 1–23. McComb, K., Packer, C., & Pusey, A. (1994). Roaring and numerical assessment in contests between groups of female lions, Panthera leo. Animal Behaviour, 47, 379–387. Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63(2), 81–87. Misiewicz, R. (2020). Peirce on analogy. Transactions of the Charles S. Peirce Society, 56(3), 299–325. Otte, M. (2006). Mathematical epistemology from a Peircean semiotic point of view. Educational Studies in Mathematics, 61(1/2), 11–38. Otte, M., & de Barros, L. G. X. (2015). What is mathematics, really? Who wants to know? Bolema, Rio Claro, 29(52), 756–772. Peirce, C. S. (i. 1866–1913). (1931–1935). The collected papers of Charles Sanders Peirce (Vols. I– VI). In C. Hartshorne & P. Weiss (Eds.). Harvard University Press, Vols. VII–VIII ed. Arthur Burks (Same publisher, 1958). Cited as CP. Peirce, Charles S. (i. 1866–1913). (1967). Unpublished manuscripts are dated according to the Annotated catalogue of the papers of Charles S. Peirce. In R. Robin (Ed.). University of Massachusetts Press, and cited according to the convention of the Peirce Edition Project, using the numeral “0” as a place holder. Cited as MS. Peirce, Charles S. (i. 1866–1913). (1992–1998). The essential Peirce: Selected philosophical writings (Vol. 1). In N. Houser & C. Kloesel (Eds.), Vol. 2, Peirce edition project. University of Indiana Press. Cited as EP. Pietarinen, A.-V. (2006). Signs of logic: Peircean themes on the philosophy of language, games, and communication. Springer. Schacter, D. L., Dobbins, I. G., & Schnyer, D. M. (2004). Specificity of priming: A cognitive neuroscience perspective. Nature Reviews: Neuroscience, 5, 853–862. Stjernfelt, F. (2007). Diagrammatology: An Investigation on the Borderlines of Phenomenology, Ontology, and Semiotics. Springer. Stjernfelt, F. (2011). On operational and optimal iconicity in Peirce’s diagrammatology. Semiotica, 186(1/4), 395–419. Stjernfelt, F. (2014). Natural propositions: The actuality of Peirce’s doctrine of Dicisigns. Docent Press. von Frisch, K. (1965/1967). The dance language and orientation of bees. Harvard University Press. West, D. (2012). Elicited imitation to measure morphemic accuracy: Evidence from L2 Spanish. Language and Cognition, 4(3), 203–222. West, D. (2013). Deictic imaginings: Semiosis at work and at play. Springer. West, D. (2014). Perspective switching as event affordance: The ontogeny of abductive reasoning. Cognitive Semiotics, 7(2), 149–175. West, D. (2016). Indexical scaffolds to habit-formation. In D. West & M. Anderson (Eds.), Consensus on Peirce’s concept of habit: Before and beyond consciousness (pp. 215–240). Springer. West, D. (2018). The work of Peirce’s dicisign in representationalizing early deictic events. Semiotica. https://doi.org/10.1515/sem-2017-0042 West, D. (2020). Perfectivity in Peirce’s energetic interpretant. Cognitio, 21(1), 152–164. Wilson, A. (2020). Interpretation, realism, and truth: Is Peirce’s second grade of clearness independent of the third? Transactions of the Charles S. Peirce Society, 56(3), 349–373.

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Blending Theory and Mathematical Cognition Marcel Danesi

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Metaphor in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Mathematics and Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Blending Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Abstract

Blending theory emerged in the early part of the twenty-first century to model conceptualizations across faculties, from language to mathematics and art. It was applied for the first time to mathematics in a systematic way by Lakoff and Núñez in their 2000 book, Where mathematics Comes From, in which the two cognitive scientists show how mathematic ideas are forged in the same way as linguistic ones via conceptual metaphors and the image schemas that undergird them. This chapter looks at blending theory as a model of math cognition, and how it has evolved since Lakoff and Núñez’s book, comparing it to other approaches, including abduction theory as put forth by Charles Peirce. Keywords

Math cognition · Blending theory · Conceptual metaphors · Image schemas · Abduction · Innatism · Language

M. Danesi (*) Anthropology, Victoria College, University of Toronto, Toronto, ON, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_50

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Introduction The study of mathematical cognition became a full-fledged interdisciplinary field in the early 2000s, after the publication of George Lakoff and Rafael Núñez’ Where mathematics comes from (2000), which put forth a concrete, albeit controversial, proposal that math cognition springs from the same neural processes that undergird language. Subsequently known as blending theory (Fauconnier & Turner, 2002), relevant research on the theory has produced intriguing findings and insights that have shed light, putatively, on what happens in the brain as people do, use, learn, and discover mathematics (Danesi, 2016). Critiques and other approaches have emerged to challenge the theory (as for example, Sinclair & Schiralli, 2003; Voorhees, 2009), but it remains a viable one to this day, claiming in a nutshell that mathematics, like language, emerges and develops from everyday mechanisms of experience. The connecting link between the two is metaphor, a view that surfaces initially within linguistics in the late 1970s and developed by various linguists and psychologists throughout the 1980s and 1990s (Ortony, 1979; Honeck & Hoffman, 1980; Lakoff & Johnson, 1980; Lakoff, 1987; Lakoff & Johnson, 1999). This became a framework for studying math cognition after Lakoff and Núñez’ book, which laid the groundwork for an approach involving researchers from science, education, and the humanities (see, for instance, Berch et al., 2018; Danesi, 2019). In the Lakoff-Núñez approach, metaphor is not a mere rhetorical figure of speech, but rather a cognitive process that guides the flow of thought, manifesting itself in verbal and nonverbal ways. The historical record in mathematics appears to support their approach (Schlimm, 2013). A classic example is the origination of the concept of the number line (which was suggested by a metaphor) by John Wallis in his Treatise of algebra (1685). Wallis describes addition and subtraction as someone walking forward and backward on a linear path – hence the number line. This episode in mathematical history (of which there are many) illustrates in a nutshell what Lakoff and Núñez meant by the link between metaphor, diagrams, and mathematics – namely, that many (if not most) concepts in mathematics arise in this way. This view was prefigured by Immanuel Kant and Charles Peirce. Kant (1781: 278) defined mathematical thinking as the process of “combining and comparing given concepts of magnitudes, which are clear and certain, with a view to establishing what can be inferred from them.” He claimed further that this basic sense becomes explicit through the “visible signs” that we use to highlight the structural detail inherent in this type of knowledge – that is, through the actual diagrams used to do mathematics, from geometrical figures to number lines. Peirce’s (1882: W4: 391–399) notion of existential graphs extended the Kantian view, constituting a veritable diagrammatic theory of math cognition (as will be discussed below), largely corroborated by psychological studies which have shown that diagrams in mathematics are not mere devices for illustrating concepts, but reveal how these take shape in the mind (for example, Hammer & Shin, 1996; Kulpa, 2004; Cellucci, 2019).

Metaphor in Mathematics According to Lakoff and Núñez, metaphor undergirds not only the invention of mathematical concepts but also guides how children learn them, which is one of the practical areas to which their theory has been applied most fruitfully (for example,

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Presmeg, 2005; Yee, 2017). The viability of a “metaphor-based pedagogy” is evidenced in elementary school math classrooms every day, albeit not often recognized as such. It is inherent, for example, in the age-old pedagogical principle that children learn best by experiencing the meaning of new concepts through the senses and the body. Manipulatives, for instance, are used commonly to impart the concepts of quantity and numeration – a manipulative is any object, or set of objects, designed to get learners to grasp a concept such as larger versus smaller via manipulation, in terms of the object’s potential functions and meanings (see Magnani, 2001, 2009). By putting them into containers of varying sizes (larger versus smaller), and a numerical name applied to each container, corresponding to the quantity of objects within it, the concept of numeration is then presumed to emerge, as early experiments by Piaget (1952) suggested. This type of pedagogy is based on the association of numerical size to the experience of putting objects in containers of various sizes. In the Lakoff-Núñez framework, the container is called an image schema – a mental construct with spatial reasoning origins that is, thus, formed in the mind from observing and using containers, becoming the basis for building concepts of number, size, and quantity. Another common pedagogical technique which is often based on metaphorical reasoning is the use of puzzles to impart certain concepts creatively and imaginatively (Petkovic, 2009; Danesi, 2018). The following one, from the pen of Renaissance mathematician, Niccolò Tartaglia, is a case-in-point, used commonly in classrooms to teach fractions in a ludic way. It is based on the image schema of fractions as symbolizing the everyday experience of partitioning wholes into parts; but Tartaglia twists it in a clever way: A father dies, leaving 17 camels to be divided among his three sons, in the proportions 1/2, 1/3, 1/9. How can this be done?

Dividing up the camels in the manner decreed by the father would entail having to split up one of the camels, which would, of course, kill it. So, Tartaglia suggested “borrowing an extra camel,” for the sake of argument. With 18 camels, we arrive at a practical solution: one son was given 1/2 (of 18), or 9; another 1/3 (of 18), or 6; and the last one 1/9 (of 18), or 2. The 9 + 6 + 2 camels apportioned in this way, add up to the original seventeen. The extra camel could then be returned to its owner. Whatever the interpretation of this solution, in real terms, as a puzzle, it impresses on learners that the concept of fractions is derived from common partitioning experiences such as dividing an inheritance into (real) parts, and what this implies in mathematical terms. Virtually any mathematical concept can be taught in similar practical ways, including the notion of infinity, which derives from what Lakoff and Núñez call the basic metaphor of infinity (BMI). The BMI is based on the image schema of “adding one more” to any collection or sequence of things, as is done regularly every day, whenever we want, knowing that we can do this forever. This image schema guides the understanding and use of various notions and proofs, such as proof by induction. The latter implies that if some condition holds for the (n + 1)st case, given n, then it holds infinitely, because we can add the (n + 2)nd case, the (n + 3)rd case,

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and so on – one case at a time, until we decide to stop, which Lakoff and Núñez call completion. The BMI was behind key discoveries such as the one of infinite sets by Georg Cantor (1874), which itself harkened back to Galileo’s paradoxical observation in his book, Dialogue concerning two new sciences (1638/1914: 31–33), that the set of square integers can be compared, one-by-one, with all the positive integers, indicating the seemingly preposterous idea that there are as many square integers as there are numbers (even though the squares are themselves only a part of the set of integers): Integers

¼

Squares

¼

1 # 1 # 12

2 # 4 # 22

3 # 9 # 32

4 # 16 # 42

5 # 25 # 52

6 # 36 # 62

7 # 49 # 72

8 # 64 # 82

9 # 81 # 92

10 # 100 # 102

11 # 121 # 112

12 # 144 # 122

... ... ...

This shows that no matter how far we go down along the sequence, there will never be a gap. In 1872, Cantor showed that the same one-to-one correspondence schema can be used to prove that this pattern holds between the whole numbers and numbers raised to any power: Integers

¼

Powers

¼

1 # 1n

2 # 2n

3 # 3n

4 # 4n

5 # 5n

6 # 6n

7 # 7n

8 # 8n

9 # 9n

10 # 10n

11 # 11n

12 # 12n

... ...

Because the integers are called cardinal numbers, any set of numbers that can be put in a one-to-one correspondence with them are said to have the same cardinality. Cantor used this notion to investigate all kinds of infinite sets and, indeed, established a basic epistemology for set theory, allowing it to become a major approach in formal mathematics. It is relevant to note that this image schema has inspired various puzzles and pedagogical devices. One of these is a famous infinity paradox, formulated originally by mathematician David Hilbert in 1924. It was presented as a puzzle by George Gamow in his 1947 book, One, two, three. . .infinity. Here’s Gamow’s version: Let us imagine a hotel with a finite number of rooms, and assume that all the rooms are occupied. A new guest arrives and asks for a room. “Sorry,” says the proprietor, “but all the rooms are occupied.” Now let us imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel, too, comes a new guest and asks for a room. “But of course!” exclaims the proprietor, and he moves the person previously occupying room N1 into room N2, the person from room N2 into room N3, the person from room N3 into room N4, and so on. The new customer receives room N1, which became free as the result of these transpositions. Let us imagine now a hotel with an infinite number of rooms, all taken up, and an infinite number of new guests who come in and ask for rooms. “Certainly, gentlemen,” says the proprietor, “just wait a minute.” He moves the occupant of N1 into N2, the occupant of N2 into N4, and occupant of N3 into N6, and so on, and so on. Now, all odd-numbered rooms became free and the infinite of new guests can easily be accommodated in them. (Page 17)

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In an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is smaller than the total number of rooms. However, in Hilbert’s Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total number of rooms, because the sets have the same cardinality. As Godino et al. (2011: 250) have cogently argued, this image schema underlies the process of counting itself, where objects (stones, sticks, etc.) are put into a one-to-one correspondence to solve the problem of counting in an unlimited way: As we have freedom to invent symbols and objects as a means to express the cardinality of sets, that is to say, to respond to the question, how many are there?, the collection of possible numeral systems is unlimited. In principle, any limitless collection of objects, whatever its nature may be, could be used as a numeral system: diverse cultures have used sets of little stones, or parts of the human body, etc., as numeral systems to solve this problem.

As this citation suggests, the symbolism used to represent numerals does not affect the concept of infinity, since it is based on a “limitless collection of objects,” which has cross-cultural resonance. In effect, infinity is built into counting itself, which could go on literally ad infinitum. In a lecture at the Field’s Institute in 2011, Lakoff even explained how Kurt Gödel’s famous proof had adopted another image schema used by Cantor, called the diagonal method (see Danesi, 2011). While it is largely acknowledged that Gödel was influenced by Cantor’s method, Lakoff provided an explanation of its cognitive source. As is well known, Gödel proved, in a famous 1931 paper, that within any formal logical system, there are results that can be neither proved nor disproved and are thus undecidable. Gödel was apparently inspired and guided unconsciously by Cantor’s diagonal method which proves that the entire set of rational numbers have the same cardinality as integers:

In each row, the successive denominators (q) represent the infinite set of integers {1, 2, 3, 4, 5, 6, . . .}. The numerator (p) of all the numbers in the first row is 1, of all those in the second row 2, of all those in the third row 3, and so on. In this way, all

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numbers of the form p/q are covered in the above array. Cantor highlighted every fraction in which the numerator and the denominator have a common factor. If these fractions are deleted, then every rational number appears once and only once in the array. Now, Cantor went back to his initial image schema, setting up a one-to-one correspondence between the integers and the numbers in the array as follows: he let the cardinal number 1 correspond to 1/1 at the top left-hand corner of the array; 2 to the number below (2/1); following the arrow, he let 3 correspond to 1/2; following the arrow, he let 4 correspond to 1/3; and so on, ad infinitum. The path indicated by the arrows, therefore, allowed him to set up a one-to-one correspondence between the cardinal numbers and all the rational numbers: 1 # 1/1

2 # 2/1

3 # 1/2

4 # 1/3

5 # 3/1

6 # 4/1

7 # 3/2

8 # 2/3

9 # 1/4

10 # 1/5

11 # 5/1

12 # 6/1

13 # 5/2

... ...

It was this diagonal image schema that must have been in Gödel’s mind when he devised his own metaphorical proofs, which showed, essentially, that within a diagonal layout of symbols, there are some that do not fit. While this is a liberal reduction of Lakoff’s argument (see Danesi, 2011), the point is that the same kind of image schema used by Cantor, namely the BMI, influenced Gödel to make his own two famous proofs, which, Lakoff went on to argue, exemplify how blending works. The first proof showed that no consistent system of axioms whose theorems can be listed by some procedure is capable of proving all truths about the arithmetic of natural numbers; the second one is an extension of the first, showing that the system cannot demonstrate its own consistency. Both were guided by the metaphor of infinite correspondence within numerical arrays – hence, the BMI. As Rafael Núñez (2005: 1717) explained in an article he wrote several years after the publication of Where mathematics comes from, in such proofs the BMI can be adjusted to what he calls the Basic Mapping of Infinity, since this describes concretely what Cantor and Gödel actually did with their proofs: [Cantor’s] analysis is based on the Basic Metaphor of Infinity (BMI). The BMI is a human everyday conceptual mechanism, originally outside of mathematics, hypothesized to be responsible for the creation of all kinds of mathematical actual infinities, from points at infinity in projective geometry to infinite sets, to infinitesimal numbers, to least upper bounds Under this view “BMI” becomes the Basic Mapping of Infinity.

As Lakoff and Núñez claim throughout their 2000 book, metaphor is the cognitive mechanism that guides all kinds of proofs, even proofs of undecidability, such as the one by Gödel. The ancient Greeks grappled constantly with the fact that certain things could not be proved within their system of demonstration. Why, for example, was it seemingly impossible to trisect an angle with compass and ruler, given that bisection was such a simple procedure? For centuries afterward, mathematicians attempted trisection with compass and ruler, but always to no avail. The demonstration that it was impossible had to await the development and spread of Descartes’ method of converting every problem in geometry into a problem in algebra. The proof came in

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the nineteenth century after mathematicians had established that the equation which corresponds to trisection must be of degree 3 – that is, it must be an equation in which one of its variables is to the power of 3. A construction carried out with compass and ruler translates, on the other hand, into an equation to the second degree. Thus, trisection with compass and ruler is impossible. The formal proof was published by mathematician Pierre Laurent Wantzel in 1837, which was based on a blending of algebra and geometry, showing relationships among ideas and facts that were previously considered to be separate or unrelated but which, in effect, were related after all. Another example of how metaphor underlies the construction of specific proof methods is proof by contradiction, or reductio ad absurdum. This implies an image schema whereby some element does not fit into the same container, which can be called an exclusion metaphor, for the sake of argument. An example is Euclid’s famous proof that irrationals, such as √2, were different from rationals and, thus, could not be classified under the same rubric (that is, put into the same container of numbers). It is worth revisiting here for the sake of illustration. Euclid started by noting that the general form of a rational number is p/q (q 6¼ 0). So, if √2 could not be written in the form p/q, then we would have shown that it was not a rational, hence “exclusionary.” Euclid’s proof was truly remarkable since it started with the assumption that √2 was in the system (container) of rational numbers, showing that this would lead to a contradiction. Using a contemporary form of the proof, it proceeds like this: Squaring both sides of the equation: √2 ¼ p/q (assumption) (√2)2 ¼ (p/q)2 Therefore: 2 ¼ p2/q2 Multiply both sides by q2: 2q2 ¼ p2 Now, p2 is an even number because it equals 2q2, which has the form of an even number. So, let p ¼ 2n: 2q2 ¼ p2 Since p ¼ 2n: 2q2 ¼ (2n)2 ¼ 4n2: Therefore: 2q2 ¼ 4n2 This equation can be simplified by dividing both sides by 2: q2 ¼ 2n2

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This shows that q2 is an even number, and thus that q itself is an even number and can be written as 2m (to distinguish it from 2n): q ¼ 2m. Now, Euclid went right back to his original assumption – namely that √2 was a rational number: √2 ¼ p/q In this equation, he substituted what he had just proved (or its equivalent), namely, that p ¼ 2n and q ¼ 2m: √2 ¼ 2n/2m √2 ¼ n/m Now, the problem is that we find ourselves back to where we started. We have simply ended up replacing p/q with n/m. We could, clearly, continue on in this way, always coming up with a fraction with different numerators and denominators: √2 ¼ {p/q, n/m, a/b, x/y, . . .}. We have thus reached an impasse, caused by the assumption that √2 had the rational form p/q, but it obviously does not, because it produces an absurdity. More specifically, Euclid showed that one type of number (the irrationals) cannot be inserted into the category of another (the rationals) because its form is divergent. Interestingly, and revealingly, the proof itself is based on the BMI, since it shows that the same result is produced by symbol replacements ad infinitum. One of the early inventors of proof was the philosopher Thales around 600 BCE. But the one who developed the first methods of proof was Euclid, who began with axioms and postulates, from which he demonstrated 467 propositions of plane and solid geometry, using contradiction, induction, deduction, and other kinds of proof strategies. Euclid finished his proofs with QED, as it was later translated in Latin. The letters stand for Quod erat demonstrandum (“which was to be demonstrated”) – remaining the symbolic hallmark of what mathematical proofs are all about to this day. The claim here is that proof is convincing, no matter what form it takes, because of its metaphorical basis, thus linking it to image schemas of everyday life that come from reasoning about experience. A precursor to image schema theory is the notion of existential graph of Charles Peirce, mentioned above (1931–1956, vol. 2: 398–433, vol. 4: 347–584), which sees diagrams in logic and mathematics not simply as illustrative of information, but as forms showing how thinking about the information occurs in actu (Peirce, vol. 4: 6). Peirce called them, in fact, “a moving picture of thought” (1908: LI 381), because in their diagrammatic form, we can literally “see” a given argument or thought process. Related to this point is the following insight: “A verb is by its significance a mere dream, an imagination unattached to any particular occasion. It calls up in the mind an icon. A (relative) is just that, an icon, or an image without attachments to experience, without a local habitation and a name but with indications of the need of such attachments” (Peirce, 1887: 3.459). As Kiryushchenko (2012: 122) has aptly put it, for Peirce “graphic language allows us to experience a meaning visually as a set of transitional states, where the meaning is accessible in its entirety at any given ‘here and now’ during its transformation.” An existential graph is thus a pictorial representation of what goes on in the mind as it grapples with structural information (Stjernfelt, 2007; Roberts, 2009).

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A simple example of what an existential graph implies cognitively is the logical diagram that Peirce used to show the relation A > B (Roberts, 2009):

The diagram represents the relation between the two entities iconically – with A as the “greater” element on top and B the “smaller” one below. In terms of Lakoff and Núñez’ approach, this would indicate the presence of the metaphorical image schema, higher is greater and lower is smaller. This is reinforced by the straight lines, with the higher line and the lower line connected by a vertical line showing the metaphorical juxtaposition of the two entities (above and below). This diagram thus shows visually how quantity and orientation are linked metaphorically in the mind. The relation between existential graphs and algebraic ideas is described by Louis Kauffman (2001: 80) as follows: Peirce’s Existential Graphs are an economical way to write first order logic in diagrams on a plane, by using a combination of alphabetical symbols and circles and ovals. Existential graphs grow from these beginnings and become a well-formed two dimensional algebra. It is a calculus about the properties of the distinction made by any circle or oval in the plane, and by abduction it is about the properties of any distinction.

The connection between graphic representation and mathematical conceptualization comes out as well in the representation of sets. The diagrams devised by John Venn in the 1880s, for example, represent logical relations in terms of collections of objects, reflecting the container schema. Such diagrams actually started with Leonhard Euler (1768), who was the first to use intersecting circles and embedded circles to show relations among sets and their elements (Hammer & Shin, 1996). Image schemas underlie what Lakoff and Johnson (1980), Lakoff (1987), Johnson (1987) call conceptual metaphors, as distinct from, but related to, linguistic metaphors. For instance, metaphorical expressions such as “He is a snake,” “She is an eagle,” and so on are instantiations of the general concept, people are animals. They call this a conceptual metaphor, in which people is the target domain and animal the source domain. Conceptual metaphors are the mental formulas that underlie mathematical reasoning. Each one results from the operation of a specific image schema, such as the container or partitioning one, which is what Lakoff calls the Invariance Principle (Lakoff, 2012: 129): Metaphorical mappings preserve the cognitive topology (that is, the image-schema structure) of the source domain, in a way consistent with the inherent structure of the target domain. What the Invariance Principle does is guarantee that, for container schemas, interiors will be mapped onto interiors, exteriors onto exteriors, and boundaries onto boundaries; for pathschemas, sources will be mapped onto sources, goals onto goals, trajectories onto trajectories; and so on. . .As a consequence it will turn out that the image-schematic structure of the target domain cannot be violated: One cannot find cases where a source domain interior is

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Among the conceptual metaphors discussed by Lakoff and Núñez are: change is motion, sets are collections in containers, continuity is gapless, functions are sets of ordered pairs, geometric figures are objects in space, numbers are object collections, recurrence is circular, etc. Each of these underlies a specific mathematical conceptualization such as the calculus (change is motion), infinity (continuity is gapless), set theory (numbers are object collections), and so on. In this framework, metonymy (the part for the whole) is seen as the mechanism that allows for generalizations from particular instances to emerge. The ontological difference between metaphor and metonymy can be reduced to a simple paraphrase: metaphor amalgamates information, metonymy condenses it. So, metonymy is operative in how symbols arise to compress ideas; metaphor is operative in how different experiential inputs are amalgamated to produce the ideas. Both processes reflect blending in general, taking different inputs and putting them together in imageschematic ways. Previous work in the psychology of mathematics aimed to examine experimental conditions that led to the acquisition of such concepts, including how symbolism is acquired. This includes the work of Piaget (1952); but such work did not lead to a theoretical model of how these concepts are formed in connection with linguistic development, as did the one by Lakoff and Núñez. As mentioned, various critiques were aimed at this model from the start, some of which dealt with the actual mathematics used by the two cognitive scientists, which was seen as faulty. But one of the more pertinent ones for the present discussion concerned the use of the term metaphor. As Winter and Yoshimi (2020) have recently observed, Lakoff and Núñez simply assumed that mathematical concepts are constituted by metaphor, but the truth may well be that they are facilitated by it instead, as occurs in many other domains of human knowledge: We argue that the evidence collected in the embodied mathematics literature is inconclusive: It does not show that abstract mathematical thinking is constituted by metaphor; it may simply show that abstract thinking is facilitated by metaphor. Our arguments suggest that closer interaction between the philosophy and cognitive science of mathematics could yield a more precise, empirically informed account of what mathematics is and how we come to have knowledge of it.

The distinction is, on the surface, a seemingly important one. But, in effect, it is a moot one. The fact that metaphorical language cannot be avoided in describing mathematics is indirect evidence that the two are one and the same, as Solomon Marcus (2012: 124) has insightfully observed: For a long time, metaphor was considered incompatible with the requirements of rigor and preciseness of mathematics. This happened because it was seen only as a rhetorical device such as “this girl is a flower.” However, the largest part of mathematical terminology is the result of some metaphorical processes, using transfers from ordinary language. Mathematical terms such as function, union, inclusion, border, frontier, distance, bounded, open,

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closed, imaginary number, rational/irrational number are only a few examples in this respect. Similar metaphorical processes take place in the artificial component of the mathematical sign system.

Moreover, there is now neuroscientific research that has found that image schemas and the conceptual metaphors they underlie correspond to dynamic activation patterns shared across the regions in the sensorimotor cortex – the same regions from which linguistic expressions and metaphor originate (Rohrer, 2005; Bou et al., 2015). This suggests that metaphors are constructive mechanisms, not just facilitating ones. Whatever one’s views on the role of metaphor in mathematics, Lakoff and Núñez’ book continues to be provocative and beneficial to the overall study of math cognition. If nothing else, many practical pedagogical insights can be distilled from it to teach everyday mathematics, as discussed above. To demonstrate this, Lakoff and Núñez end their treatment by showing how their approach can be used to explicate Euler’s famous identity formula, relating five of the most significant numbers in mathematics: eiπ + 1 ¼ 0. In order to explain the inspiration for this equation, Lakoff and Núñez looked at the conceptual metaphors underlying analytic geometry and trigonometry, exponentials, imaginary numbers, and the imageschematic mechanisms that blend them into the identity formula. Specifically, the numbers in the identity formula correspond to conceptual meanings such as change, acceleration, recurrence, rotation, and self- regulation. Lakoff and Núñez (2000: 34) call this approach “mathematical idea analysis,” which aims to show how any notion, such as this equation can bring various systems of ideas together into an overall blending process.

Mathematics and Language While some mathematicians and cognitive scientists resisted the type of analysis put forth by Lakoff and Núñez, their notion that mathematical concepts emerge in human experience, expressed by metaphor, has not been seriously impugned. What Lakoff and Núñez showed, in essence, was how language and mathematics (and other faculties) are grounded on the same form of conceptualization, and that, overall, mathematical concepts, like linguistic ones, are acquired, rather than hardwired, raising the age-old question regarding whether mathematics is discovered or invented in a new theoretical light. As is well known, the discovery view is traced to Plato, who claimed that the human mind is equipped from birth to discover mathematical truths that are present in reality, simply giving them symbolic form – a view formalized millennia later by Gottlob Frege (1884). Lakoff and Núñez provide a serious anti-Platonist argument – namely, humans have derived mathematics from their practical experiences of the world and, thus, are able to use it to grasp the world according to needs. One of the main critiques has been that there exists a substantive literature showing that number sense – an intuitive understanding of numbers – may in fact

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be hardwired in the brain, and organized differently than language. In his 1999 book, What counts, neuroscientist Brian Butterworth looked at the evidence for an innate number sense, a year before Lakoff and Núñez’ book, coming to the conclusion that numbers (and math concepts generally) do not exist in the brain in the way that words do; they are part of a separate kind of intelligence with their own brain module, located in the left parietal lobe, which also controls the movement of fingers, explaining why we count instinctively on our fingers (Butterworth, 1999: 248). The nonlinguistic nature of math, according to Butterworth, is also evidenced by the fact that early cultures that had no symbols or words for numbers still managed to develop counting systems for practical purposes, that neonates can add and subtract even at a few weeks of age, and that people afflicted with Alzheimer’s have unexpected numerical abilities. A study by Izard et al. (2011) is particularly relevant. The researchers tested the comprehensibility of notions of Euclidean geometry in an indigenous Amazonian society, called the Mundurucu, which had never had any exposure to this type of geometry in its history. If they were able to comprehend them, then it could be claimed that the main ideas of Euclidean geometry are present a priori in all humans (such as points, lines, and surfaces), even in the absence of formal mathematical training. The subjects included Mundurucu adults and age-matched child control groups from the United States and France as well as younger American children without education in geometry. The responses of Mundurucu adults and children were analogous to those of mathematically educated adults and children, suggesting therefore that there is an intuitive understanding of essential properties of Euclidean geometry, regardless of age or culture. For instance, on a surface described to them as perfectly planar, the Mundurucus’ estimations of the internal angles of triangles added up to approximately 180 degrees, and they stated that there exists one single parallel line to any given line through a given point. These concepts were also present in the group of American child participants. Izard et al. concluded that, during childhood, humans naturally develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, no matter what language they speak. Indeed, the Mundurucu society did not have words for the Euclidean notions that corresponded to words in Western languages. On the other hand, the presence of geographically based terms are not needed to instantiate geometric ones. One may always find an analogy from the geographic to the geometric – a possibility that can never be precluded. Libertus et al. (2009) conducted an experiment that also seems to corroborate the innatist hypothesis. They presented 7-month-old infants with familiar and novel number concepts while electroencephalogram measures of their brain activity were recorded. The findings provided convergent evidence that the brains of infants can detect numerical novelty. Alpha-band and theta-band oscillations both differed for novel and familiar numerical values. The findings thus provide hard evidence that numerical discrimination in infancy is ratio dependent, suggesting the continuity of cognitive processes over development. These results are also consistent with the idea that networks in the frontal and parietal areas support ratio-dependent number

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discrimination in the first year of life, consistent with what has been reported in neuroimaging studies in adults and older children. In his book, The number sense: How the mind creates mathematics (1997), Stanislas Dehaene also argued that math ideas are processed separately from language, although they have many points in common conceptually. Dehaene brought forth experimental evidence which strongly suggests that animals such as rats, pigeons, raccoons, and chimpanzees can perform simple calculations. When a rat is trained to press a bar 8 or 16 times to receive a food reward, the number of bar presses will approximate a Gaussian distribution with peak around 8 or 16 bar presses. When rats are more hungry, their bar pressing behavior is more rapid; so by showing that the peak number of bar presses is the same for either well-fed or hungry rats, it is possible to disentangle time, rhythm, and number of bar presses. Additionally, researchers have set up hidden speakers in the African savannah to test natural (untrained) behavior in lions (McComb et al., 1994). The speakers play a number of lion calls, from 1 to 5. If a single lioness hears, for example, three calls from unknown lions, she will leave, while if she is with four of her sisters, they will go and explore, which might also be motivated by some social factor. This suggests that not only can lions tell when they are “outnumbered” but also that they can do this on the basis of signals from different sensory modalities, suggesting that numerosity is a multisensory concept. The difference between humans and nonhuman animals is that humans have a brain that produces symbols for things, which has allowed them to climb toward higher mathematics. These allow for reflection and understanding, not just instinctive responses. Using evidence from brain imaging techniques (PET and MRI), Dehaene pinpoints where in the brain numerical calculation and its symbolic forms take place. In sum, mathematics evolved to explain the physical world the way that the eye evolved to provide sight. Various case studies of brain-damaged patients have also come forth to support the separation of mathematics from language. Patients with acalculia (inability to calculate), who might read 14 as 4, have difficulty representing numbers with symbols, indirectly suggesting a link between the two. For example, they might have difficulty understanding the meaning of “hundred” in expressions such as “two hundred” and a “hundred thousand.” Acalculia is associated with Broca’s aphasia and, thus with the left inferior frontal gyrus (see, Gerstmann, 1940). But acalculia has also been found in patients suffering from Wernicke’s aphasia who also have difficulties saying, reading, and writing numbers – associated with the left posterior superior temporal gyrus. Patients with frontal acalculia have damage in the prefrontal cortex. They have serious difficulties in carrying out arithmetic operations (particularly subtraction), and solving numerical problems, and cannot match the instructions given in language to the math. Dyscalculia (difficulty in understanding calculations) is associated with the horizontal segment of the intraparietal sulcus, in both hemispheres (Butterworth, 2010). Studies on patients with these types of challenges indicate, overall, that they can continue to speak but not do math, thus suggesting that the two may be separate faculties (for example, Dehaene, 2004; Butterworth et al., 2011).

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In two key books, The math gene (2000) and The math instinct (2005), Keith Devlin suggests that there must be an innate capacity for mathematics, otherwise no one could do it. He raises two relevant questions: Why does it vary so widely, both among individuals in a specific culture and across cultures (Devlin, 2000: 4)? Why can we speak easily, but not do math so easily (in most cases) (Devlin, 2005: 2)? The answer to both, according to Devlin, is that the variation is not overly significant and that people can and do math as easily as they acquire language, but do not recognize that they are doing math when they do it. Our prehistoric ancestors’ brains were essentially the same as ours, so they must have had the same underlying abilities (Devlin, 2005: 34–39). But those brains could hardly have imagined how to multiply 15 by 36 or prove Fermat’s last theorem. To grasp abstractions, it is clear that mathematics and language were required in tandem. So, overall, it can be argued that there must be an evolutionary link between math and language. The question becomes: How does the link manifest itself? Lakoff and Núñez provided the first concrete answer to this question. Because image schemas are the mechanisms that create the linkage, the implication is that math and mental imagery are linked. The correspondence between the two is seen every time we solve a word problem in algebra. As George Pólya (1957: 174) aptly puts it: To set up equations means to express in mathematical symbols a condition stated in words; it is translation from ordinary language into the language of mathematical formulas. The difficulties which we may have in setting up equations are difficulties of translation.

There is now evidence that understanding what mathematics is, how it is learned, and how it varies is dependent on where it emerges and the forms it has taken in a specific culture, as has become apparent from the field of ethnomathematics – a field that has soared considerably since the publication of Lakoff and Núñez’ book. Within this field, studies have shown a link between language, culture, and math conceptualization, which need not concern us here (see, for example, D’Ambrosio, 1985; Ascher, 1991).

Blending Theory One of the first mentions of conceptual blending is found in Mark Turner’s book, The literary mind (1997: 93), in which he states that “Conceptual blending is a fundamental instrument of the everyday mind, used in our basic construal of all our realities, from the social to the scientific.” Blending theory was elaborated in an in-depth manner a few years later by Fauconnier and Turner (2002). Essentially, it provided a neural explanation of how conceptual metaphors work to produce mathematical concepts, whereby information inputs from different regions in the brain, which share a common experiential basis (for example, the container schema), are amalgamated to form a new concept, which is much greater than the sum of its inputs. Hence, new properties emerge from the blend, and new relationships are created that did not exist in the original inputs.

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The theory thus describes what happens when mathematical concepts are acquired or discovered. The question becomes: How is the blending process triggered? The prompt comes from subjective creative inferences (Turner, 2014), which recalls what Charles Peirce called abductions, or individual acts of insight. Peirce (1903: 5.181) described the process as follows: The abductive suggestion comes to us like a flash. It is an act of insight, although of extremely fallible insight. It is true that the different elements of the hypothesis were in our minds before; but it is the idea of putting together what we had never before dreamed of putting together which flashes the new suggestion before our contemplation.

The notion that ideas come “like a flash” coincides with the notion of blending considerably. It is clearly coincident with the assertion by blending theorists that new ideas are governed by image schemas (Peirce’s “flash”) that connect experiential events in the brain. But the notion of abduction also posits that the insight gained through a blend remains a possibility, rather than a completed blend, since it emanates from an observation or set of observations that the logical part of the process sees as potential. This is how theories come and go in science – they are abductions, rather than certainties. Alexander (2012: 28) uses blending theory to describe the invention of negative numbers, as a case-in-point for how the abductive process might unfold: Using the natural numbers, we made a much bigger set, way too big in fact. So we judiciously collapsed the bigger set down. In this way, we collapse down to our original set of natural numbers, but we also picked up a whole new set of numbers, which we call the negative numbers, along with arithmetic operations, addition, multiplication, subtraction. And there is our payoff. With negative numbers, subtraction is always possible. This is but one example, but in it we can see a larger, and quite important, issue of cognition. The larger set of numbers, positive and negative, is a cognitive blend in mathematics. . .The numbers, now enlarged to include negative numbers, become an entity with its own identity. The collapse in notation reflects this. One quickly abandons the (minuend, subtrahend) formulation, so that rather than (6, 8) one uses -2. This is an essential feature of a cognitive blend; something new has emerged.

Blending theory maintains, overall, that new concepts emerge through the recruitment of everyday experiential-cognitive mechanisms that set off fusional or “collapsing” processes (as Alexander calls them) in the brain. Mathematics makes sense when it encodes concepts that fit our experiences of the world – experiences of quantity, space, motion, force, change, mass, shape, probability, self-regulating processes, and so on. The inspiration for mathematics comes from these experiences as it does for new language and these lead to “collapses.” This notion is consistent with what René Thom (1975) called “catastrophes,” that is, discoveries that subvert or overturn existing knowledge. Thom assigns these catastrophes to the process of “semiogenesis,” defined as the emergence of “pregnant” forms within symbol systems themselves, that is, as forms that emerge by happenstance through contemplation and manipulation of the previous forms. As this goes on, every so often, a catastrophe occurs that leads to new insights, disrupting the previous system. The discovery of negative numbers is a catastrophe in Thom’s sense.

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A related idea, based on the foregoing discussion, is that abductions cannot be forced, or controlled. One never knows when and to whom the abduction will come. Consider a well-known anecdote that Henri Poincaré recounted in his book, Science and method (1908). Poincaré had been puzzling over an intractable mathematical problem, leaving it aside for a little while to embark on a geological expedition. As he was about to get onto a bus at one point, the crucial idea came to him in a flash of insight (an abduction). He claimed that without it, the solution would have remained buried somewhere in his mind, possibly forever. Now, as he elaborates, the idea was the result of linking (blending) Fuchsian functions with non-Euclidean geometry (Poincaré, 1908: 23): Just at this time I left Caen, where I then lived, to take part in a geologic excursion organized by the École des Mines. The circumstances of the journey made me forget my mathematical work; arrived at Coutances we boarded an omnibus for I don’t know what journey. At the moment when I put my foot on the step the idea came to me, without anything in my previous thoughts having prepared me for it; that the transformations I had made use of to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify this, I did not have time for it, since scarcely had I sat down in the bus than I resumed the conversation already begun, but I was entirely certain at once. On returning to Caen I verified the result at leisure to salve my conscience.

Poincaré’s anecdote is a relevant one, since it explicates how different domains are combined into a blend, via an abductive form of insight. The question of how to model such seemingly spontaneous thinking is a huge problem within cognitive science. For example, Guhe et al. (2011) developed a computational model of how blending might be simulated, devising a system by which different conceptualizations of number can be blended together to form new ones via the recognition of common features, and a particular combination of these features. The model is based on Lakoff and Núñez’ conceptual metaphors for arithmetic, such as the one based on the container image schema. The metaphors are amalgamated using a so-called heuristic-driven theory projection, which provides generalizations between domains, based on a mechanism that searches for commonalities and then transfers them from one domain to another, producing new conceptual blends. Blending is a continuous process. Reading a math theorem in a book might lead some individual mathematician to devise another one or to use it as part of some new idea, based on the individual’s experiences and background knowledge related to the theorem. When others take it on to develop it, the idea becomes a shared one. This implies that a blend, once completed, is available for subsequent or additional blending. The modus operandi of mathematicians is, in fact, to build upon ideas created by others – which is (of course) not unique to mathematics, but to many other disciplines. To quote Turner (2005): As long as mathematical conceptions are based in small stories at human scale, that is, fitting the kinds of scenes for which human cognition is evolved, mathematics can seem straightforward, even natural. The same is true of physics. If mathematics and physics stayed within these familiar story worlds, they might as disciplines have the cultural status of something like carpentry: very complicated and clever, and useful, too, but fitting human

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understanding. The problem comes when mathematical work runs up against structures that do not fit our basic stories. In that case, the way we think begins to fail to grasp the mathematical structures. The mathematician is someone who is trained to use conceptual blending to achieve new blends that bring what is not at human scale, not natural for human stories, back into human scale, so it can be grasped.

Turner’s notion of “story worlds” encapsulates how blends bring about change; they do so by changing narrative conceptualizations – that is, concepts that are organized in a fixed connected way. So, new blends are what allow the mind to break away from the embedded “stories.” Whatever one might think of blending theory, if nothing else, it has provided a framework for discussing the origination of mathematical concepts in relation to other faculties of mind. In a simple concept such as “7 is larger than 4,” we hardly realize that it involves the blending of a source domain based on concepts of size with the target domain of numbers. The conceptual metaphor that underlies this blend is numbers are collections of objects of differing sizes – with the target domain being numbers and the source domain sets of objects of different sizes: so, the greater the collection, the greater the number. Similarly, the more is up, less is down image schema, which appears in Peirce’s existential graph drawing above, underlies the representations of functions on the Cartesian plane. The linear scales are paths conceptual metaphor manifests itself in concepts such as rational numbers are far more numerous than integers, and infinity is way beyond any collection of finite sets – that is, the path (number line) is infinite as is the space between numbers. Lakoff (2012: 164) explains the path metaphor as follows: The metaphor maps the starting point of the path onto the bottom of the scale and maps distance traveled onto quantity in general. What is particularly interesting is that the logic of paths maps onto the logic of linear scales. Path inference: If you are going from A to C, and you are now at in intermediate point B, then you have been at all points between A and B and not at any points between B and C. Example: If you are going from San Francisco to NY along route 80, and you are now at Chicago, then you have been to Denver but not to Pittsburgh. Linear scale inference: If you have exactly $50 in your bank account, then you have $40, $30, and so on, but not $60, $70, or any larger amount. The form of these inferences is the same. The path inference is a consequence of the cognitive topology of paths. It will be true of any path image-schema.

By cognitive topology, Lakoff is referring to the common experience of paths and their structure. So, for example, walking on a path which can come to an end allows for assessing distance and scale, and this is the cognitive source of the path image schema. Blending theory has various precursors. For instance, it resonates with interaction theory, as developed by Richards (1936) and Black (1962), whereby a metaphor results from an attempt to establish a conceptual link between what is known (the vehicle source domain) and what needs to be known (the topic target domain). The interaction assumes that the two domains share in a common experiential ground, which is elicited in the metaphorical meaning. Soskice (1985) suggests, colorfully, that the two domains “animate” each other. Consider the concept of the number line again as a derivative of the path metaphor (originating from the image schema of

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walking, as Wallis remarked). It is a diagrammatic model of how we actually count and organize counting in a sequence from small to large to infinity. It both mirrors and then subsequently structures the actions we perform when we count. Now, the number line became a source for further mathematics, after Wallis, leading to more complex mathematical concepts and calculations. Without the number line, it is unlikely that such concepts as imaginary numbers would have developed in the first place. These new ideas likely emerged when a number line was drawn through the origin at right angles to the real number line in the Cartesian plane and used, simply, to represent the imaginary numbers. In effect, it extended the number line image schema to enfold the emergent complex number plane (Danesi, 2020: 56). The question becomes: Are these concepts, guided by image schemas, inventions or discoveries? The word invention derives from Inventio in Latin, where it meant both invention and discovery, indicating that the two are closely related. Discovery comes about through largely serendipitous (abductive) processes, whereas invention entails intentionality. For example, fire is discovered through the abduction of how the fire started, by rubbing sticks to start it, which is an invention. The general principles of arithmetic emanate from a similar process; they derive from the discovery that counting connects quantities, and numerals are inventions that instantiate this discovery. Naming the counting signs (numerals) allows us to turn these principles into ideas that can be manipulated intellectually and systematically. Analogously, the general principles of geometry derive from the experience and practice of measuring the size of fields, the angles in the corners of buildings, and so on. To carry out such measurements, diagrams were invented to represent them and names were assigned to the geometric figures employed. Around 2000 BCE, the Egyptians discovered that knotting and stretching a rope into sides of 3, 4, and 5 units in length produced a right triangle, with 5 the longest side (the hypotenuse). The Pythagoreans were aware of this discovery. Their goal was to show that it revealed a general structural pattern. Knotting any three stretches of rope according to this pattern – for example, 6, 8, and 10 units – will produce a right triangle because 62 + 82 ¼ 102 (36 + 64 ¼ 100). As the historian of science, Jacob Bronowski (1973: 168) has insightfully written, we hardly recognize today how important this demonstration was. It is a fruit of Inventio that leads to discoveries serendipitously. Invention is thus the precursor to discovery, as blending-abduction theory would have it. This faculty of mind has made it possible to reflect on stimuli and forms present in everyday experience, and to give them abstract form. It has endowed humans with the capacity to carry the world around in their heads, so to speak, and to transform the nonreflective consciousness that they share with the other animals – the physically bound type that reacts instinctively to urges and changes in the continuum of perceived events – into a reflective one, encoded in symbols of all kinds. As Bronowski, 1977: 24) has put it, this allows for projecting ideas into the future: The images play out for us events which are not present to our senses, and thereby guard the past and create the future—a future that does not yet exist, and may never come to exist in that form. By contrast, the lack of symbolic ideas, or their rudimentary poverty, cuts off an

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animal from the past and the future alike, and imprisons it in the present. Of all the distinctions between man and animal, the characteristic gift which makes us human is the power to work with symbolic images.

Concluding Remarks Whether or not blending theory can be “proven” in any empirical way is beside the point; it is cognitively plausible and highly interesting, and needs to be investigated if we are ever to come to an understating of what mathematical cognition is all about and how it is interconnected with other faculties. Today, there is a huge dataset of research findings and theories relating math cognition to math learning, and to how mathematics intersects with other neural faculties such as language and visual art. Math cognition is not easy to define psychologically, although we may have an intuitive sense of what it is. Neuroscientific research on it has brought a wave of experimental seriousness to the question of what mathematics is. But in the end, all research and theories are essentially descriptive of various processes involved in generating mathematics; they cannot really explain mathematics in its totality. Gödel made it obvious to mathematicians that mathematics was made by them, and that the exploration of “mathematical truth” would go on forever as long as humans were around. Mathematics lies within the minds of humans. In effect, mathematics is itself a “meta-theory” of reality, interpreting it in a particular way that reflects very closely how language does so. In the end, Lakoff and Johnson argued that mathematicians and linguists had a common goal – to study the common processes that unite mathematics and language. The greatest critique of the Lakoff-Núñez model is that it simply makes analogies between mathematics and metaphor, but it does not explain what mathematics is in any truly ontological way. But this belabors the raison d’être of any theoretical paradigm. In a significant book written for the general public by Courant and Robins in 1941, titled What is mathematics?, their answer to their own question is indirect – that is, they illustrate what mathematics looks like and what it does, allowing us to come to our own conclusions as to what mathematics is. And perhaps this is the only possible way to answer this question. The same can be said about music. The only way to answer What is music? is to play it, sing it, or listen to it. A year before, in 1940, Kasner and Newman published another important popular book titled Mathematics and the imagination. Again, by illustration the authors show how mathematics is tied to imaginative thought. We come away grasping intuitively that mathematics is both a system of thought and an art, allowing us to investigate reality. Lakoff and Núñez approached the topic of what is mathematics in a similar way – they illustrated how it is connected to metaphor, rather than provide any complex theoretical analysis of the connectivity. They made the claim that it arose from the same conceptual system that led to the origin of language, art, and other faculties. The connectivity is suggested by our symbolic artifacts, from notations and diagrams to proofs and theoretical organizations (such as set theory). Like language, no one aspect of mathematics can be taken in isolation. A theory of the

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mathematical mind has never really come forward. And the reason for this is that, as Courant and Robins and Kasner and Newman certainly knew, there can never really be one, given the multifarious vicissitudes that characterize discovery and invention in mathematics. The only thing that can be done is to guess and infer what is going on, since even a theory of the mathematical mind can only be an abduction.

References Alexander, J. (2012). On the cognitive and semiotic structure of mathematics. In M. Bockarova, M. Danesi, & R. Núñez (Eds.), Semiotic and cognitive science essays on the nature of mathematics (pp. 1–34). Lincom Europa. Ascher, M. (1991). Ethnomathematics: A multicultural view of mathematical ideas. Brooks/Cole. Berch, D. C., Geary, D. C., & Koepke, K. M. (Eds.). (2018). Language and culture in mathematical cognition. Academic. Black, M. (1962). Models and metaphors. Cornell University Press. Bou, F., Corneli, J., Gómez-Ramírez, D., Smaill, E., Maclean, A., & Pease, A. (2015). The role of blending in mathematical invention. In Proceedings of the sixth international conference on computational creativity (pp. 55–62). Association for Computational Creativity. Bronowski, J. (1973). The ascent of man. Little, Brown, and Co. Bronowski, J. (1977). A sense of the future. MIT Press. Butterworth, B. (1999). What counts: How every brain is hardwired for math. Free Press. Butterworth, B. (2010). Foundational numerical capacities and the origins of dyscalculia. Trends in Cognitive Sciences, 14, 534–541. Butterworth, B., Varma, S., & Laurillard, D. (2011). Dyscalculia: From brain to education. Science, 332, 1049–1053. Cantor, G. (1874). Über eine Eigneschaft des Inbegriffes aller reelen algebraischen Zahlen. Journal für die Reine und Angewandte Mathematik, 77, 258–262. Cellucci, C. (2019). Diagrams in mathematics. Foundations of Science, 24, 583–604. Courant, R., & Robbins, H. (1941). What is mathematics? An elementary approach to ideas and methods. Oxford University Press. D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5, 44–48. Danesi, M. (2011). George Lakoff on the cognitive and neural foundation of mathematics. Fields Notes, 11(3), 14–20. Danesi, M. (2016). Language and mathematics: An interdisciplinary approach. Mouton de Gruyter. Danesi, M. (2018). Ahmes’ Legacy: Puzzles and the mathematical mind. Springer. Danesi, M. (Ed.). (2019). Interdisciplinary perspectives on mathematical cognition. Springer. Danesi, M. (2020). Pythagoras’ legacy: Mathematics in ten great ideas. Oxford University Press. Dehaene, S. (1997). The number sense: How the mind creates mathematics. Oxford University Press. Dehaene, S. (2004). Arithmetic and the brain. Current Opinion in Neurobiology, 14, 218–224. Devlin, K. J. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. Basic. Devlin, K. J. (2005). The math instinct: Why you’re a mathematical genius (along with lobsters, birds, cats and dogs). Thunder’s Mouth Press. Euler, L. (1768). Lettres à une princesse d’Allemagne. l’Académie Imperiale des Sciences. Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. Basic.

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Frege, G. (1884). Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl. Verlag von Wilhelm Koebner. Galilei, G. (1638 [1914]). Dialogues concerning two new sciences. Macmillan. Gamow, G. (1947). One, two, three. . .infinity. Dover. Gerstmann, J. (1940). Syndrome of finger agnosia, disorientation for right and left, agraphia, acalculia. Archives of Neurology and Psychology, 44, 398–408. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Teil I. Monatshefte für Mathematik und Physik, 38, 173–189. Godino, J. D., Font, V., Wilhelmi, R., & Lurduy, O. (2011). Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects. Educational Studies in Mathematics, 77, 247–265. Guhe, M., et al. (2011). A computational account of conceptual blending in basic mathematics. Cognitive Systems Research, 12, 249–265. Hammer, E., & Shin, S. (1996). Euler and the role of visualization in logic. In J. Seligman & D. Westerståhl (Eds.), Logic, language and computation: Volume 1. CSLI Publications. Honeck, R. P., & Hoffman, R. R. (Eds.). (1980). Cognition and figurative language. Lawrence Erlbaum Associates. Izard, V., Pica, P., Pelke, E. S., & Dehaene, S. (2011). Flexible intuitions of Euclidean geometry in an Amazonian indigene group. PNAS, 108, 9782–9787. Kant, I. (2011 [1781]). Critique of pure reason (J. M. D. Meiklejohn, Trans.). CreateSpace Platform. Kasner, E., & Newman, J. R. (1940). Mathematics and the imagination. Simon and Schuster. Kauffman, L. K. (2001). The mathematics of Charles Sanders Peirce. Cybernetics & Human Knowing, 8, 79–110. Kiryushchenko, V. (2012). The visual and the virtual in theory, life and scientific practice: The case of Peirce’s quincuncial map projection. In M. Bockarova, M. Danesi, & R. Núñez (Eds.), Semiotic and cognitive science essays on the nature of mathematics (pp. 46–59). Lincom Europa. Kulpa, Z. (2004). On diagrammatic representation of mathematical knowledge. In A. Sperti, G. Bancerek, & A. Trybulec (Eds.), Mathematical knowledge management. Springer. Lakoff, G. (1987). Women, fire and dangerous things: What categories reveal about the mind. University of Chicago Press. Lakoff, G. (2012). The contemporary theory of metaphor. In M. Danesi & S. Maida-Nicol (Eds.), Foundational texts in linguistic anthropology (pp. 128–171). Canadian Scholars’ Press. Lakoff, G., & Johnson, M. (1980). Metaphors we live by. Chicago University Press. Lakoff, G., & Johnson, M. (1999). Philosophy in flesh: The embodied mind and its challenge to western thought. Basic. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books. Libertus, M. E., Pruitt, L. B., Woldorff, M. G., & Brannon, E. M. (2009). Induced alpha-band oscillations reflect ratio-dependent number discrimination in the infant brain. Journal of Cognitive Neuroscience, 21, 2398–2406. Magnani, L. (2001). Manipulative abduction. In Abduction, reason and science. Springer. https:// doi.org/10.1007/978-1-4419-8562-0_3 Magnani, L. (2009). Abductive cognition. Springer. Marcus, S. (2012). Mathematics between semiosis and cognition. In M. Bockarova, M. Danesi, & R. Núñez (Eds.), Semiotic and cognitive science essays on the nature of mathematics (pp. 98–182). Lincom Europa. McComb, K., Packer, C., & Pusey, A. (1994). Roaring and numerical assessment in contests between groups of female lions, Panthera leo. Animal Behavior, 47, 379–387. Núñez, R. (2005). Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals. Journal of Pragmatics, 37, 1717–1741. Ortony, A. (Ed.). (1979). Metaphor and thought. Cambridge University Press.

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Peirce, C. S. (1882 [1989]). On junctures and fractures in logic. In C. J. W. Kloesel (Ed.), Writings of Charles S. Peirce, vol. 4. Indiana University Press. Peirce, C. S. I. (1931–1958) Collected papers of Charles Sanders Peirce. 1862–1914. In C. Hartshorne, P. Weiss and A.W. Burks (Eds.), vols. 1–8. Harvard University Press. Petkovic, M. S. (2009). Famous puzzles of great mathematicians. American Mathematical Society. Piaget, J. (1952). The child’s conception of number. Routledge and Kegan Paul. Poincaré, H. (1908). Science and method. Dover. Presmeg, N. C. (2005). Metaphor and metonymy in processes of semiosis in mathematics education. In J. Lenhard & F. Seeger (Eds.), Activity and sign (pp. 105–116). Springer. Richards, I. A. (1936). The philosophy of rhetoric. Oxford University Press. Roberts, D. D. (2009). The existential graphs of Charles S. Peirce. Mouton. Rohrer, T. (2005). Image schemata in the brain. In B. Hampe & J. Grady (Eds.), Image schemas in cognitive linguistics (pp. 165–196). Mouton de Gruyter. Schlimm, D. (2013). Conceptual metaphors and mathematical practice: On cognitive studies of historical developments in mathematics. Topics, 5, 283–298. Sinclair, N., & Schiralli, M. (2003). A constructive response to ‘Where mathematics comes from’. Educational Studies in Mathematics, 52, 79–91. Soskice, J. M. (1985). Metaphor and religious language. Clarendon Press. Stjernfelt, F. (2007). Diagrammatology: An investigation on the borderlines of phenomenology, ontology, and semiotics. Springer. Thom, R. (1975). Structural stability and morphogenesis: An outline of a general theory of models. Benjamin. Turner, M. (1997). The literary mind. Oxford University Press. Turner, M. (2005). Mathematics and narrative. http://www.thalesandfriends.org/en/papers/pdf/ turnerpaper.pdf. Turner, M. (2014). The origin of ideas: Blending, creativity, and the human spark. Oxford University Press. Voorhees, B. (2009). Embodied mathematics: Comments on Lakoff and Núñez. Journal of Consciousness Studies, 11, 83–88. Winter, B., & Yoshimi, J. (2020). Metaphor and the philosophical implications of embodied mathematics. Frontiers in Psychology. https://doi.org/10.3389/fpsyg.2020.569487 Yee, S. P. (2017). Students’ and teachers’ conceptual metaphors for mathematical problem solving. School Science and Mathematics, 117, 146–157.

Section II Ethnomathematics Myrdene Anderson and Tod Shockey

Abstract

In 1985, D’Ambrosio wrote his seminal paper “Ethnomathematics and its place in the history and pedagogy of mathematics” (also cf. Pike, Language in relation to a unified theory of the structure of human behaviour. Mouton, 1967). Completely unaware that Fettweis had coined the word in the 1930s (Rohrer and Schubring, Learn Math 31(2):35–39, 2011), D’Ambrosio had initiated a program that has a global presence today. Scholars from around the globe are engaged in ethnomathematical research and this community has organized the International Study Group on Ethnomathematics (ISGEm) which hosts a global meeting every 4 years. The scholars represented in this section provide a valuable insight into a selection of ongoing ethnomathematical research. Keywords

Ethnomathematics · Culturally responsive · Identity · Ethnomodeling · Mãori medium education Fettweis (Rohrer & Schubring, 2011) is credited for coining “ethnomathematics” in his research of the 1930s. During the 1980s D’Ambrosio began an international conversation using the term “ethnomathematics,” unaware of Fettweis. In his seminal paper of 1985 D’Ambrosio defined ethnomathematics: We will call ethnomathematics the mathematics which is practised among identifiable cultural groups, such as national-tribal societies, labor groups, children of a certain age bracket, professional classes, and so on. Its identity depends largely on focuses of interest, on motivation, and on certain codes and jargons which do not belong to the realm of academic mathematics. We may go even further in this concept of ethnomathematics to include much of the mathematics which is currently practised by engineers, mainly calculus, which does not respond to the concept of rigor and formalism developed in academic courses of calculus. (p. 45)

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Since D’Ambrosio’s definition, scholars from around the world have contributed to the body of literature on ethnomathematics. Through the International Study Group for Ethnomathematics (ISGEm), the North American Study Group for Ethnomathematics (NASGEm) and their supported Journal of Mathematics and Culture, academics are sharing their research. In this section, it is our great pleasure to share the research of this group of scholars from around the globe. We hope the brief introduction to the included chapters stirs your curiosity as a reader to explore and appreciate the ethnomathematical scholarship from a few select societies. Veronica Albanese (University of Granada, Melilla, España) explores early work in ethnomathematics, with her current research to overcome early ‘simplistic conceptions’ of ethnomathematics. Albanese shows us “mathematical techniques and procedures belonging to diverse activities of different cultures and distant locations can become very similar.” Jenni Harding’s (University of Northern Colorado, Colorado, USA) research emphasizes the importance of the knowledge that children bring to our classrooms. She shares how students’ knowledge can influence the pedagogical practices in mathematics education, practices that “create an effective classroom environment.” Tamsin Meaney (Western Norway University of Applied Science, Bergen, Norway) with Tony Trinick and Piata Allen (University of Auckland, Auckland, New Zealand) uses the concept of Cultural Symmetry as a theoretical framework “as a way of supporting Indigenous students to see that their heritage included mathematical ideas.” Through this framework these researchers analyze distinctive cultural traditions of the Māori people. Milton Rosa and Daniel Orey (Universidade Federal de Ouro Preto, Ouro Preto, Minas Gerais. Brasil) bring to our attention their work in Ethnomodeling. “Bringing together ethnomathematics and mathematical modelling, we arrive at the concept of ethnomodelling.” It is through this conceptual lens that Rosa and Orey bring forth a different way for us to “interpret the world.” In the context of mathematics education in Alaska, Sandra Wildfeuer writes about the importance of “culturally responsive choices” in classrooms. She continues to remind us that mathematics “really looks like” what emerges in conversation between students and teachers, and by inference what transpires in autocommunication. Ethnomathematics scholarship is emerging from all corners of the world. While the bulk of the work continues to emerge from Brazil, we are seeing more and more work from other areas. Fifteen years ago, the North American Study Group on Ethnomathematics initiated the Journal of Mathematics and Culture. This open access journal provides an outlet for ethnomathematics scholars, oftentimes in their first language. The journal has a dedicated editorial board that considers submissions in Arabic, English, Italian, Norwegian, Portuguese, and Spanish. This growing body of scholars, primarily under the guise of the International Study Group on Ethnomathematics, comes together every 4 years to meet in a conference setting. In these meetings scholars gather to share their research from their locales. Since D’Ambrosio’s introduction in 1985, ethnomathematics is oftentimes on the agenda for international, national, and local mathematics education meetings. In 2000, ethnomathematics was a Topic Study Group at the International Congress on Mathematics

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Education and continues to be so today. We are becoming aware of more and more graduate study research using ethnomathematics as a framework. Nearly 40 years later, ethnomathematics is finding “its place in the history and pedagogy of mathematics.” We trust that readers will find these contributions as engaging as we do. It is with sincere gratitude we extend our thanks to this group of remarkable scholars, Tod Shockey and Myrdene Anderson.

References D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44–48. Pike, K. L. (1967). Language in relation to a unified theory of the structure of human behaviour. Mouton. Rohrer, A., & Schubring, G. (2011). Ethnomathematics in the 1930s – The contribution of Ewald Fettweis to the history of ethnomathematics. For the Learning of Mathematics, 31(2), 35–39.

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Ethnomathematics and Cultural Identity to Promote Culturally Responsive Pedagogical Choices for Teachers in Early Childhood and Elementary Education Sandra Wildfeuer

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics and Culturally Responsive Education Makes Sense in Alaska . . . . . . . . . . . . Cultural Identity and Mathematical Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Childhood and Elementary Teachers of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Teachers can include ethnomathematics to make culturally responsive choices that support the cultural and mathematical identity of students that will further contribute to equitable mathematics teaching and learning practices as well as student achievement. Building a shared vision of equitable mathematics achievement entails social, political, and cultural values, which means that everyone may not have the same idea in mind when talking about mathematics. From universal mathematical principles to political decisions regarding funding, and instructional and curricular decisions, mathematics achievement is discussed on many levels. Broad choices about mathematical progress are made through graduation requirements and for access to post-secondary education in workforce development or for university degrees. What mathematics looks like within those requirements is constructed by the mathematics educators and the mathematicians and scientists who explore deeply the meaning and application of mathematics. However, what mathematics really looks like has a lot to do with the types of experiences and discussions that occur between a teacher and student, in or outside of a classroom setting. Empowering current and future teachers to include ethnomathematics and culturally responsive choices can reduce the culture of math anxiety and increase S. Wildfeuer (*) University of Alaska Fairbanks, Fairbanks, AK, USA e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_1

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the culture of achievement and mathematical literacy. The history and potential of education in Alaska provides a rich example to explore these concepts. Keywords

Alaska native · Cultural identity · Culturally responsive · Ethnomathematics · Mathematics education · Pedagogy

Introduction Empowering current and future mathematics teachers to believe that they can understand and teach mathematics, and that they have the pedagogical skills to make culturally responsive choices that build on ethnomathematical principles supports the on-going discussion occurring in mathematics education that calls for equitable mathematics teaching and learning opportunities for all students (Gutiérrez, 2013; Jurdak et al., 2016; National Council of Supervisors of Mathematics (NCSM), 2020a, b). Access to and success at learning mathematics opens doors and pathways to opportunities and careers for students (Maltese & Tai, 2011; Stone et al., 2008). Hence, the teaching and learning of mathematics in school requires reflection about how mathematics is taught, what is considered mathematics learning, and who is participating. Access and equitable opportunities including culturally responsive teaching that recognizes the different cultural backgrounds of learners and strategies to promote student identity in mathematics are influencing the direction of teacher preparation and curricular choices (National Council of Teachers of Mathematics (NCTM), 2014; NCSM & TODOS, 2018). The discussion about what mathematics looks like and how it is different in school and out of school is shifting priorities and expectations. In short, mathematics education has social, cultural, and political dimensions. From my perspective as a mathematics educator and a teacher educator for over 25 years in Alaska, making the time and including ethnomathematics and culturally responsive learning is important and meaningful. Mathematics success in K-8 schooling sets a foundation influencing the opportunities that students have in high school and in post-secondary education. Mathematics education is public, in that political decisions impact funding and curriculum choices, and it is socially constructed through educational experiences and public discourse. It is also psychological and personal, and success or failure can influence one’s cultural and mathematical identity (Ashcraft, 2002; Ruge, 2018). Hence teachers need training in more than the math content, but also the pedagogy in how to support someone that is deepening their understanding of mathematics. It is important to support the whole person so more people see themselves as someone who belongs and can succeed in mathematics. Teachers need to be able to teach mathematics with confidence by using ethnomathematics examples and creating culturally responsive lessons to engage and motivate students. This chapter demonstrates how the movement is ripe for these changes to be implemented in Alaska.

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Ethnomathematics Ubiratan D’Ambrosio first used the word ethno mathematics in reference to the mathematics of Indigenous populations, as he was searching for ways to explain how different peoples develop systems of knowledge: “observing, comparing, classifying, evaluating, quantifying and measuring, counting, representing, and inferring” (Rosa et al., 2016). Ethnomathematics is more than the historical study of mathematics from a cultural perspective; it provides a theoretical framework for learning about the ways, the values, and the techniques developed to understand and make sense of the world (D’Ambrosio, 1985). It opens up the discussion about what mathematics is, how it is practiced, and who has access to it. In education, ethnomathematics provides a foundation for a culturally responsive pedagogy. In each different culture, we have to look into the ways, arts, and techniques that were developed to express understanding, to explain the natural and sociocultural environment, and the complementing ways of doing and knowing (D’Ambrosio, 1985). One of D’Ambrosio’s many contributions to mathematics education included exploring what mathematics and mathematics education looked like through different cultural lenses. What is absolute about mathematics, and what is influenced by cultural perspective? How does the universality of mathematical principles impact the discussion about mathematics education? He has surveyed these concepts and asked important questions about how culture influences and reinforces a view and perspective of what it means to comprehend mathematics and how experiences in mathematics impact opportunities. He argues that Ethnomathematics education can inform political and ethical choices regarding personal and national security, social and economic status, and care of natural and cultural resources (D’Ambrosio & D’Ambrosio, 2013). The growth of these ideas over the past 40 years has been a response to what had become an unacceptable status quo in mathematics education, wherein some students were successful, and many were not. The nationwide movement today toward equitable access and culturally responsive education shares its roots with the movement to recognize and include Alaska Native values and ways of knowing in Alaska Native education. In response to historical educational policies and “to provide a way for schools and communities to examine the extent to which they are attending to the educational and cultural wellbeing of the students in their care,” the Alaska State Legislature adopted the Alaska Standards for Culturally Responsive Schools in 1998 (Alaska Standards for Culturally Responsive Schools: Cultural Standards for: Students, Educators, Schools, Curriculum, Communities, 1998). The standards mirror the academic standards in Mathematics and Language Arts, and focus on five areas, including students, educators, curriculum, schools, and communities. The standards are not exclusive but provide a guide for what culturally responsive education can be, including making a connection between in school experiences and out-of-school living, and recognizing and validating use of Indigenous language and local knowledge as a source for curriculum, ways of knowing, and worldviews. In Alaska, Oscar Kawagley, a Yupiaq scholar and educator, described how mathematics and science were viewed from within his culture and how it differed from the emphasis of school

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mathematics. He described the historical consequences of the conflict between his Indigenous worldview and that of the Western worldview. He saw the disconnect between worldviews and how Native people would adjust their thinking to accommodate Western assumptions, and also saw the repercussions of this in an educational system. He provided a vision and framework of Native education that sought to include Native values and ways of teaching and learning in educational processes (Kawagley, 1993). A generation of Indigenous and non-Indigenous scholars have learned from and built upon the contributions he has made to Indigenous ways of knowing and Indigenous science (Lowan, 2012). A. Culturally responsive educators incorporate local ways of knowing and teaching in their work. B. Culturally responsive educators use the local environment and community resources on a regular basis to link what they are teaching to the everyday lives of the students. C. Culturally responsive educators participate in community events and activities in appropriate and supportive ways. D. Culturally responsive educators work closely with parents to achieve a high level of complementary educational expectations between home and school. E. Culturally responsive educators recognize the full educational potential of each student and provide the challenges necessary for them to achieve that potential. (Guide to Implementing the Alaska Cultural Standards for Educators, 2012)

Ethnomathematics and Culturally Responsive Education Makes Sense in Alaska According to the Bureau of Indian Affairs, there are 229 recognized sovereign Native entities, in what is now the state of Alaska (Bureau of Indian Affairs, 2017). The population of the state is estimated at about 730,000, with up to 80% of the population centered around urban areas that have populations greater than 2500. In Alaska, many towns and villages are accessible via the road system, but still many are only accessible by air or water transportation. In 2019, the state’s population was 65% White, 16% Alaska Native, with an additional 8% represented by two or more races (Department of Labor and Workforce Development, 2020). Outside of the two largest urban areas (Anchorage and Fairbanks), the population is 54% White, 31% Alaska Native, with 7% represented by two or more races (2020). Both the Northern region (the Nome census area, the North Slope, and the Northwest Arctic Borough) and the Southwest region (Aleutians, Bethel census area, and Dillingham) of the state of Alaska have populations that are nearly 70% Alaska Native, a much larger proportion of Indigenous people, than the entire state (Department of Labor and Workforce Development, 2020). In addition, there are at least 20 Indigenous languages that are spoken and being revitalized.

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In Alaska, there is a need to reflect on the role that cultural values play in the educational system (Barnhardt & Kawagley, 2010). For thousands of years, the Indigenous peoples of Alaska have lived and educated each successive generation on the life skills and values necessary for not only survival but their overall cultural values. Embedded in these skills are foundational principles of mathematics that include spatial and proportional reasoning (Lipka et al., 2015). Alaska Native communities and Elders share their knowledge and teach younger generations how to use local resources in ingenious ways, for practical purposes, and also for beauty. For example, geometric design is displayed in the art of basket-making and in the construction of clothing, and proportional reasoning is evident in ways of constructing a fish trap, and logic and experience in understanding how to navigate on a snowy tundra or at sea. Educational movements like No Child Left Behind and the adoption of the Alaska Math Standards, a version of the Common Core, have made an imprint on what mathematics education looks like in Alaska. School districts choose math curriculum with pacing guides and engage their teachers in professional development on how to implement it. A focus on standardized testing and measuring the success of the schools has put an emphasis on how mathematics is perceived and taught. This impacts the autonomy of the teacher and focuses lessons around the way they are interpreted and presented in the math curriculum. Students see math as a series of worksheets and something that occurs in the classroom. There is a need for students to develop a strong math foundation, and following the curriculum provides a structure needed by many teachers, especially those who themselves have struggled to learn and understand mathematics. Even though the state has cultural standards and mathematics standards, they are not always woven together and applied throughout the school curriculum. Grounding instruction in ethnomathematics that is relevant to the lives of the students can help them engage in and connect with the mathematics that they are learning. Mathematics does not just take place in a classroom. The mathematics traditionally taught in school today reflects the Western mathematical traditions that have evolved over several hundred years, these including assumptions of universal principles that are often stripped from any context. The textbook industry presents school mathematics in an ordered way, giving the learner, and often the teacher, the impression that they are to learn in that way. Mathematical performance and success are measured, and lack of success gives the impression that the learner is “less than” or not capable of learning. This can reinforce a lack of willingness to participate. The inclusive and humanizing mathematics education movement nationwide today needs to expand in Alaska. Native leaders created the cultural standards and the vision of how education can serve and not diminish Alaska Native student success and how it can help to build community. Teachers need to be able to teach with confidence and use ethnomathematics examples to engage and connect with students. This can become a self-replicating system that improves the educational pathway in mathematics, where culturally responsive teachers connect with students, and students see the value of mathematics to their own lives. Mathematics is not a barrier to academic success, but something that is relatable and doable.

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The responsibility of the educational system in Alaska has changed over the past century, as has the purpose of attending school and getting an education. Before and after Alaska became a state in 1959, a separate and unequal system existed for non-Native students, Native students, and those of two or more races (Gross et al., 1976). The legacy of establishing segregated schools, and sending Alaska Native children to boarding schools in Alaska and in the Lower 48, was driven by a push to develop a “civilized society.” In the 1960s, the Department of Education determined that the best way to give students separate but equal educational opportunities was to require rural students, mostly Native, to attend a boarding school, or to board in a home in an urban community. Boarding schools were expensive, and lacked the emotional support students needed. Villages lost teenagers, and the teenagers lost connections that made them a part of the community. Local elementary schools went to the sixth or eighth grade, and students who completed this far had to decide to stay home or leave home to further their education. This practice of separating Native youth from their home communities, which contributed to a loss of Native language and a loss of knowledge of place-based subsistence activities, continued until the Molly Hootch lawsuit in 1975. The Molly Hootch case in Alaska resulted in a ruling that the Department of Education would work with each local school district to provide a high school for any community with a minimum number of students (Gross et al., 1976). This shift meant that rural and Indigenous communities in Alaska did not need to send their youth away for an education. However, the boarding schools also provided opportunities that may not have been available to students locally, such as learning a musical instrument, and taking courses not offered at home. Boarding schools continue to offer students educational resources and opportunities for academic achievement. Lifelong friendships are made among boarding school students from places around Alaska that may not have occurred if the students were at home (Hirshberg & Sharp, 2005). In the 1950s and 1960s, non-Native educators worked within paradigms that assumed non-majority cultures to be deficient, and it was not until the 1980s that this mindset was broadly challenged (Hirshberg & Sharp, 2005). In the early 1970s educators and anthropologists at the University of Alaska Fairbanks investigated educational policies, including the teaching and learning styles of the teachers and their students. They identified differences in communication styles (Scollon & Scollon, 1979) and in the standard curriculum offered in textbooks that did not relate to the animals, people, and places in Alaska. This spurred discussion about the role of culture in education. Judith Kleinfeld sought to understand why some students were successful in boarding schools, and others were not. She studied the role of the teacher, and what the teacher did right (rather than focus on what was lacking), and developed what she coined the Warm Demander type of teacher (Kleinfeld, 1972). A Warm Demander has high expectations and creates opportunities to motivate and engage others in the learning process (Safir, 2019). Another educational leader, Dr. Ray Barnhardt believed that developing local and Native teachers who lived and worked in the communities could benefit students by giving them role models and educational experiences that were not separate from their

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culture. All of the reasons listed are still true today (Barnhardt & Cross-Cultural Education Development Program, 1977). In Alaska, the population of school age children is about 25% Alaska Native, while only about 5% of the school teachers and administrators in the state are Alaska Native (Hill & Hirshberg, 2013). This lack of representation and being able to see oneself as a teacher or researcher or scientist is one problem that needs to be addressed. Ethnomathematics as a program is important to the Indigenous populations but also the non-Native people of Alaska must recognize the contribution of Alaska Native people, and learn more about their state. Comparing graduation rates and drop-out rates from 2019–2021 report by the state of Alaska shows that high school students that identify as Alaska Native drop out at a higher proportion (near 5%) than their peers (2–3%) and a 4-year cohort graduation rate shows students that identify as Alaska Native or American Indian graduate less often (65–73%) than their peers (76–84%) (Alaska Department of Education and Early Development, 2022). The state of Alaska is not serving its own people (Lomawaima & McCarty, 2002). Because Alaska Native tribes are sovereign, they have the right to educate their own people. Currently, the proposed legislation is to form Tribal Compacts between tribes and the state, where the tribes will take responsibility for educating their own people. The proposal is to allow five schools to participate in the first phase of the project, with the first schools opening in Fall 2025 (Ebertz, 2022). As the First Alaskans Institute states, “Alaska Natives know what is best for Alaska Natives, and what is good for Alaska Natives is good for all Alaskans” (First Alaskans Institute, 2020). A brief overview of the history of public education in Alaska over the past century demonstrates how political and social policies impacted the role of education and how it is perceived. Ethnomathematics and examples from Alaska of culturally responsive decision making share strategies with current trends in mathematics education research and educational policy such as support of differentiation strategies for different types of learners and encouraging a growth, or mathematical mindset for the teaching and learning of mathematics (Boaler, 2015).

Cultural Identity and Mathematical Identity Historically the Western ideas of rational thought and Mathematics were viewed through the lens of the dominant culture. This included access to learning mathematics and also what mathematics looks like in and out of school and in work and in life. Mathematics has valued abstract rational thought, and was considered free from cultural bias. The assumptions and principles of mathematical thinking constructed over hundreds years of Western thought were considered absolute (Bishop, 1990). Access to learning mathematics was for those that had access to learning. In the last century, mathematics access has been considered more ubiquitous, but the social construction of mathematics, and what mathematics looks like in the school setting demonstrates that math learning can look different in different contexts. The discourse about what mathematics is and who it is for has been modified over time. The curricular and instructional choices that teachers make on a regular basis

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directly affect the students that are learning. Situating math education in a social context, a political context, a psychological context, in an educational context, in an equity context, expands the way and the type of discussion that can occur around the teaching and learning and usefulness of mathematics. This context enables the development of the six important dimensions of the ethnomathematics program: Cognitive, Conceptual, Educational, Epistemological, Historical, and Political (Rosa et al., 2016). Teaching ethnomathematics and responding culturally are strategies that are compatible with other movements in mathematics education. As written and described by Gutiérrez (2018), this is an opportunity to create windows and mirrors for students, mirrors so students see themselves in mathematics and windows to see into the world outside of their own (Goffney et al., 2018). Opening the window to create spaces and welcome everyone into the process is inclusive. It is our collective responsibility to cultivate the space to change and rethink who belongs in mathematics and what kind of mathematics we want for our future. Everyone has a cultural identity that includes family, geographic location, language, and history, and life experiences impact one’s worldview. Everyone also has a mathematics identity. Traditional classroom expectations tend to focus on mathematics as a set of topics to learn and problems to solve. Rehumanzing mathematics embraces the multiple dimensions of the ways we interact with mathematics, including considering who owns the mathematics, how one positions oneself in mathematics, what are the sociocultural factors, and what emotions and histories are present? Mathematics is abstract, but can also be concrete, like with hands-on construction and active exploration of mathematics concepts. As important for mathematics success is building a mathematical identity. Students need a strong self-efficacy and the confidence to use math to solve problems. By incorporating a growth mindset in a mathematics classroom, students engage in low floor-high ceiling math tasks and the social construction of mathematical concepts through opportunity that encourage sense-making, verification, and creativity (Boaler, 2015). National organizations propose shifting the focus of mathematics instruction toward equitable processes that engage students in sense-making and mathematical communication, while building a positive mathematics identity. The National Council of Teachers of Mathematics’ (NCTM) Catalyzing Change publications for early childhood through high school education argue several main points, and provide suggestions for how to get there (NCTM, 2020). First, the purpose of learning mathematics should be broadened to include that every student should develop deep mathematical understanding and also experience the joy and wonder of mathematics, which contributes to a positive mathematical identity. Also, equitable structures in mathematics instruction and learning may include use of data and detracking students and teachers, while encouraging the highest-quality mathematics education for each and every student (NCTM, 2020). The National Council of Supervisors of Mathematics’ (NCSM) Essential Actions: Framework for Mathematical Leadership series encourages mathematics teachers and researchers to make bold choices to make a change in the culture of how mathematics is taught, who is served, and what needs to be done to truly engage students in mathematical understanding and success (b; NCSM, 2020a). The foundation includes examining beliefs

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about mathematics education and a vision for how to improve the mathematical performance of all students. Equitable systems and structures mean that every student has access to meaningful and relevant mathematics teaching and learning, and that teachers and communities need to work together to build relationships and a shared vision (b, NCSM, 2020a). Position statements from NCSM and TODOS Mathematics for All call for supporting multi-lingual learners and including social justice in mathematics education, changing the culture of how teachers and schools approach curriculum development and student engagement by advocating for eliminating deficit views of mathematics learning; eradicating mathematics as a gatekeeper; and elevating the professional learning of mathematics teachers and leaders with a dual focus on mathematics and social justice (NCSM & TODOS, 2016, 2021). Ethnomathematics and culturally responsive education shift the focus to support all student learners. Students can build a strong mathematical identity when they see their race, ethnicity, language, and culture reflected in the classroom (Zavala, 2012). Native leader Bernice Joseph said in her keynote speech at the 2005 Alaska Federation of Natives Convention: My experiences are that most curricula are Western in nature. As a result, students do not see themselves represented in written materials, texts, movies, videos, or literature. From this, it is safe to say that students are learning that it was the Europeans that made history, discovered other lands, shaped the histories of science, the arts, and humanities; and made all the important contributions to the world. (Joseph, 2010, p. 122).

This statement highlights the need for school districts and teachers to embrace ethnomathematics and culturally responsive education to help develop strong and positive cultural and mathematical identities. The foundation for and the need for culturally responsive education has been here, and there are case-by-case examples of success. Yet, in Alaska, the educational system and common practices have often ended up putting the focus too narrowly on the curriculum, and less on empowering each teacher to make the best choices for their students.

Early Childhood and Elementary Teachers of Mathematics Early childhood and elementary teachers of mathematics work closely with their students, and play a huge role in determining what mathematics looks like within their own educational settings. Mathematical principles are considered absolute, but how do teachers address the socio-emotional aspects of learning mathematics? The psychological factors of stress about whether an individual understands the content, how quickly they can respond to questions, and their feeling of whether they are good enough or smart enough to succeed in mathematics are outcomes of how mathematics teaching and learning has been traditionally structured. The vision from professional organizations is that there needs to be a shift in priorities about what is labeled mathematics and what types of experiences and work products are valued.

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Teacher training also needs to address the reality that many of the teachers themselves have had negative and anxious experiences in learning mathematics (Ashcraft, 2002; Ruge, 2018). Their challenge is to grow into educator who themselves enjoy mathematics, and are confident in making choices to teach mathematics that will be different from how they had learned it (Malinsky et al., 2006). Teacher training today places a focus on learning the mathematical content through academic standards, while learning the pedagogy through classroom experiences and internships. Early childhood education is important because children are developing concepts of space, number, and worldview (Kagan & Roth, 2017). Math structures, questioning, and ways of thinking and reasoning can impact the math potential of young learners (English, 2016). Early childhood teachers need confidence to know what questions to ask in order to encourage student thinking. They also need to develop number sense, spatial awareness, probability, and reasoning skills. The more comfortable the teacher is with these concepts, the more the teacher will have the opportunity to think about how to relate what is going on in the classroom with mathematical ideas outside of the classroom and in the community. Responding culturally or introducing an ethnomathematics example takes planning, and has to be focused on the audience. Counting local objects, learning to count in the local language, and creating experiences for the students to measure and explore proportional reasoning are examples of this. Teachers also need to consider what mathematical strengths the students bring to school with them already. The benefits of learning mathematics early are for students to learn, to analyze, to reason, and to develop their concepts of number sense (English, 2016). Elementary educators are often tasked with teaching a full array of subjects, including, but not limited to: reading, writing, art, social studies, science, and math. Hence many elementary teachers are tasked to become experts in many content areas, as well as to develop pedagogical strategies in each discipline. As stated earlier, many school districts require that their teachers use an adopted mathematics curriculum, which will influence what math looks like in that setting (Louie, 2017). Several of the large publishers provide a plethora of materials for teachers to use, including pacing guides, worksheets, and online homework access. It is still up to the teacher how they organize instruction and how to implement instruction. Novice teachers can benefit from the resources, but in most cases, they will need to pick and choose what to focus on because there is too much to apply in one academic year. Some teachers follow the curriculum lessons, in the sequence they are presented. Others have collected their own teaching materials and math games. These teachers are being asked to embrace the local school district’s vision and to create a culture shift toward equity and inclusion. Detracking schools means eliminating the low, middle, and high tracking of students. The problem has been that students in the low group remain in the low group throughout their education, and hence lack opportunities to learn the mathematics that is required for post-secondary education (b; NCSM, 2020a). Teachers need strategies to develop equitable instruction, while maintaining adherence to academic standards and expectations.

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Elementary teachers need to include culturally responsive and real-world lessons that show the students that math is more than completing worksheets. Lessons that include aspects of the local culture, that share about life in Alaska, and that invite elders and the community into the classroom are ways that the teacher can make choices to support learning ethnomathematics. A challenge in preparing teachers to adopt ethnomathematics concepts in the classroom is that the teacher may not be an expert on the local knowledge or how to address social justice issues. Support of other teachers and the community can play a big role in filling in any gaps. Now is a time for the sociopolitical turn in mathematics education (Gutiérrez, 2013). We need effective teachers that represent all races and genders in the classroom for students to see themselves and to persist in learning mathematics and science (Ladson-Billings, 1995; Price, 2010). The use of hands-on activities and manipulatives to explore and understand mathematical concepts is not a weakness, but a strength (Battey et al., 2021). Solving abstract mathematics concepts without a context can be challenging for many learners of mathematics. Why not provide a context or other meaning to the actions of the students? It is time to transform mathematics instruction (Li et al., 2014). A discussion about mathematics education can be framed as educational, in terms of the teaching and learning decisions made within a classroom learning environment, and sociocultural, in terms of who has access to educational opportunities, and what those look like (Hand et al., 2012). The interplay of how mathematics is constructed in the classroom and the role that mathematics education plays in democratic access to society – in terms of academic achievement and career opportunities – are distinct yet related. This leads to different conversations about what is important and what is valued, and who has the power to make those decisions. Academic mathematics as compared with localized mathematical activities and practice includes length and complexity of historical development cultural contribution. We need to prepare culturally responsive teachers by rethinking the curriculum (Villegas & Lucas, 2002). There needs to be support for teachers of traditionally underrepresented students (Bonner, 2014) to make pedagogical choices to support those students. Teachers need to learn how to create equitable access to mathematical tasks while maintaining the cognitive demand of a mathematical task (Sararose et al., 2018). Non-Indigenous teachers need to learn to be culturally responsive and Indigenous teachers need to follow their culture as they position themselves in the community (Bonner, 2021).

Conclusion There is momentum to shift the culture of mathematics education to a more equitable and holistic experience, where individuals are valued and their contributions recognized. In Alaska, this shift to rehumanize mathematics instruction needs to occur to support student success and promote the well-being of both Indigenous and non-Indigenous people. One way to support this includes championing teachers to

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develop strategies to include ethnomathematics examples and culturally responsive teaching approaches. Mathematics success can help contribute to a healthy society and give individuals choices for careers after high school (Gravemeijer et al., 2017). Mathematics is needed to earn university degrees but also for other postsecondary work, like to be a successful plumber or nurse. Teachers need to support each student’s cultural identity while helping to build their mathematical identity. A student that can see themselves as someone who can do and achieve success in mathematics can also use math as a tool in daily life to solve simple problems and to be mathematically literate. In Alaska, it is time to take action to support the Indigenous students by recognizing and supporting their worldview and way of knowing. Early childhood and elementary teachers work directly with young children and can make choices toward equitable instruction and equal outcomes among their students. Every teacher does not have to be the expert of cultural knowledge but each needs to be able to respect it and facilitate instruction in a way that is meaningful to their students. The teacher can request guests and elders to the classroom to share their knowledge, and the community can support the teacher by sharing the responsibility. The future tribal compact in Alaska will engage Indigenous teachers and communities in new and exciting ways.

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Louie, N. L. (2017). The culture of exclusion in mathematics education and its persistence in equityoriented teaching. Journal for Research in Mathematics Education, 48(5), 488–519. https://doi. org/10.5951/jresematheduc.48.5.0488 Lowan, G. (2012). Expanding the conversation: Further explorations into indigenous environmental science education theory, research, and practice. Cultural Studies of Science Education, 7, 71–81. https://doi.org/10.1007/s11422-012-9379-1 Malinsky, M., Ross, A., Pannells, T., & McJunkin, M. (2006). Math anxiety in pre-service elementary school teachers. Education, 127(2), 274–279. https://www.questia.com/read/1G1158523257/math-anxiety-in-pre-service-elementary-school-teachers Maltese, A. V., & Tai, R. H. (2011). Pipeline persistence: Examining the association of educational experiences with earned degrees in STEM among U.S. students. Science Education, 95(5), 877–907. https://doi.org/10.1002/sce.20441 National Council of Supervisors of Mathematics (NCSM). (2020a). Closing the opportunity gap: A call for Detracking mathematics. Position Paper. https://www.mathedleadership.org/position-papers/ National Council of Supervisors of Mathematics (NCSM). (2020b). NCSM essential actions: Framework for leadership in mathematics education. Leadership in Mathematics Education. National Council of Supervisors of Mathematics (NCSM) & TODOS: Mathematics for All. (2016). Mathematics education through the lens of social justice: Acknowledgement, actions, and accountability. Joint Position Paper. https://www.mathedleadership.org/position-papers/ National Council of Supervisors of Mathematics (NCSM) & TODOS: Mathematics for All. (2021). Positioning multilingual learners for success in mathematics. Joint Position Paper. https://www. mathedleadership.org/position-papers/ National Council of Teachers of Mathematics (NCTM). (2020). Catalyzing change in early childhood and elementary mathematics: Initiating critical conversations. The National Council of Teachers of Mathematics. Price, J. (2010). The effect of instructor race and gender on student persistence in STEM fields. Economics of Education Review, 29(6), 901–910. https://doi.org/10.1016/j.econedurev.2010. 07.009 Rosa, M., D’Ambrosio, U., Orey, D. C., Shirley, L., Alangui, W. V., Palhares, P., & Gavarrete, M. E. (2016). Current and future perspectives of Ethnomathematics as a program. Springer. https:// doi.org/10.1007/978-3-319-30120-4 Ruge, J. (2018). On epistemological violence in mathematics education research – An exemplary study in the journal of mathematics teacher education. The Mathematics Enthusiast, 15(1), 320–344. https://scholarworks.umt.edu/tme/vol15/iss1/17 Safir, S. (2019). Becoming a warm demander. The Power of Instructional Leadership, 76(6), 64–69. http://www.ascd.org/publications/educational-leadership/mar19/vol76/num06/Becoming-aWarm-Demander.aspx Sararose, D. L., Hunt, J. H., & Lewis, K. E. (2018). Productive struggle for all: Differentiated instruction. Mathematics Teaching in the Middle School, 23(4), 194. https://doi.org/10.5951/ mathteacmiddscho.23.4.0194 Scollon, R., & Scollon, S. (1979). Bush consciousness and modernization, Ch 4. In Linguistic convergence: An ethnography of speaking at fort Chipewyan, Alberta (pp. 177–209). Academic Press. Stone, J. R., Alfeld, C., & Pearson, D. (2008). Rigor and relevance: Enhancing high school students’ math skills through career and technical education. American Educational Research Journal, 45(3), 767–795. https://doi.org/10.3102/0002831208317460 Villegas, A. M., & Lucas, T. (2002). Preparing culturally responsive teachers: Rethinking the curriculum. Journal of Teacher Education, 53(1), 20–32. https://doi.org/10.1177/ 0022487102053001003 Zavala, M. R. (2012). Race, language, and opportunities to learn: The mathematics identity negotiation of Latino/a youth (order no. 3521624). [doctoral dissertation from University of Washington]. ProQuest Dissertations & Theses Global. (1035319844). Retrieved from http:// uaf.idm.oclc.org/login?url¼https://www.proquest.com/dissertations-theses/race-languageopportunities-learn-mathematics/docview/1035319844/se-2?accountid¼14470

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Bundles of Ethnomathematical Expertise Residing with Handicrafts, Occupations, and Other Activities Across Cultures Veronica Albanese

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics (and Mathematics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (First) Definitions of Ethnomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . About Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (New) Definitions of Ethnomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematics Education and Cultural Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bundles of Ethnomathematical Expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Within Handicrafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Braiding and Weaving Crafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basketry Craft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wooden Sculptures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Within Occupations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Street Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bus Workers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Masonry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Within Other Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Space Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The ethnomathematics program is interested in the relations between mathematics and cultures and their implications for mathematical education. The program is rooted in an interdisciplinary set of theories that share a relativistic and constructivist view of the origin and development of knowledge. A brief overview of the history of the program allows us to tackle the tensions that arise in the first V. Albanese (*) University of Granada, Melilla, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_2

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researches and the solutions proposed by various researchers with regard to the definition of mathematics, ethnomathematics, and other methodological issues. This chapter presents different ethnomathematical experiences in the context of handicrafts, occupations, and other activities across cultures, with the aim of overcoming the simplistic conception of ethnomathematics as the study of the mathematics of indigenous populations. It will be highlighted how mathematical techniques and procedures belonging to diverse activities of different cultures and distant locations can become very similar. A brief reflection on the importance of the context in mathematics education allows us to link these ethnomathematical results with the contextualization of mathematical concepts and practices in daily activities. So the contributions to mathematical education made by the results of these experiences will also be emphasized. Keywords

Ethnomathematics · Handicraft · Culture · Mathematics education · Activity · Occupation · Contextualization

Introduction During the twentieth century, a great interest emerged in the relationship between mathematics and cultures or mathematics and everyday life, sometimes in relation to problems linked to mathematical education. Researchers in various areas coined different terms to indicate mathematics that is less formal than that studied in schools but equally or more useful in everyday life or in the development of specific activities in certain cultures: Mathematicians, and most of all mathematics educators who deal with indigenous peoples, started to look at these ways of conceptualizing mathematics with interest, giving them different names: sociomathematics, spontaneous mathematics, informal mathematics, oppressed mathematics, non-standardized mathematics, popular mathematics, mathematics of know-how, oral mathematics, implicit mathematics, non-professional mathematics, contextualized mathematics, folk mathematics, and indigenous mathematics. (Albanese et al. 2017, p. 308)

In 1985, Ubiratan D’Ambrosio began to use the term ethnomathematics and to theorize about it. Many researchers interested in the relationship between mathematics and culture (although not all, as it is the case of Alan Bishop (1991)) adopted the term. The ethnomathematics program, as defined by its founder Ubiratan D’Ambrosio (1985, 2006), insists that it is not just about research but also about influencing reality for effective change and is not an easy task due to the many facets of the program itself. A superficial and rudimentary approach associates ethnomathematics with the mathematics of indigenous populations as studied by anthropologists. But, since then, the program has evolved, and ethnomathematics is considered the mathematics not only of indigenous populations but also of any other particular cultural group,

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which includes occupational guilds, groups of professionals, or groups of people united by any ascription (age range, ethnicity, locality) or any achievement (skill, beliefs, any other way of life). This chapter demonstrates that ethnomathematics of crafts, professions, and other activities can be studied beyond indigenous populations. The first part of this chapter will review the definitions given of ethnomathematics, which will lead to an overview of its origins, the continuous evolution of the program, and its transdisciplinary nature. Some philosophical and cognitive questions will be briefly dealt with, which are essential to situate the research – to understand its scope and to deal with some tensions that have arisen in the literature – and which will guide us to relate the contributions that will be chosen. A possible relationship between ethnomathematics and its implications for mathematical education will also be addressed. In the second part of the chapter, various bundles of ethnomathematical experiences will be presented as paradigmatic examples of studies carried out in this framework with the intention of highlighting their cognitive contributions and their educational implications. Experiences that have been presented in Spanish- or Portuguese-language literature will be preferred, in order to provide the English-speaking reader with an overview of research in ethnomathematics in these languages.

Ethnomathematics (and Mathematics) In his first conceptualization, D’Ambrosio (1985) indicated that ethnomathematics is on the borderline between the history of mathematics and cultural anthropology. Later, he explicitly adds mathematical education and the science of cognition (D’Ambrosio 2006). Other ethnomathematics proposals include mathematical modeling (Rosa and Orey 2013), ethnology, ethnosciences, and ethnography (Rohrer and Schubring 2013), although influences can also be identified from the philosophy (and epistemology) of mathematics, politics of mathematics (Barton 1996), and sociology of mathematics (Bauchspies and Restivo 2001; Restivo 1994). It can be stated that ethnomathematics is a transdisciplinary program since it is nourished by the contributions and reflections of other areas as is clarified in the MEDIPSA model (Oliveras 1996) which identifies the roots of the program in a relativistic paradigm, a position also shared by Barton (1999). The MEDIPSA model (Oliveras 1996) – the acronym corresponding to mathematics, epistemology, didactics, investigation, psychology, sociology, and anthropology – integrates a multidisciplinary set of theories rooted in the same relativistic and contextualized conception of the nature of knowledge and its relationship with reality, based on epistemic, sociological, and anthropological questions, respectively, about the nature of knowledge; the root of the educational phenomenon; and, above all, the relativism of reality. Reality is not unique, it is socially constructed through diverse realities contextualized in different cultures. Human beings cannot be separated from their social structures; therefore, knowledge cannot be extracted from its sociocultural context because reality is known and understood according to the

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meaning that the cultural group attributes to it socially. Language itself and symbols are valid in relation to the internal interactions between elements of the group.

(First) Definitions of Ethnomathematics D0 Ambrosio’s first definition (1985) describes ethnomathematics as the mathematics practiced by identifiable cultural groups, such as “national-tribal societies, labor groups, children of a certain age brackets, professional classes, engineers” (p. 16), in contrast to the academic mathematics taught and learned at school. Further on, academic mathematics is included as a possible form of ethnomathematics, practiced by the cultural group identified by teachers and students in the schools and the academic community (that in our case happens to be in a familiar Western society). Bill Barton (1996) reflected on some inconsistencies of research in ethnomathematics in the first decade of its official existence: with respect to its epistemological and philosophical bases, mainly with respect to what is conceived as mathematics. Barton (1996) proposed a new definition: “Ethnomathematics is a research programme of the way, in which cultural groups understand, articulate and use the concepts and practices which we describe as mathematical, whether or not the cultural group has a concept of mathematics” (p. 214). The key question of Barton’s (1996) definition lies in the fact that it is necessary to determine what is considered to be mathematics, and in this case, the relativity of this question is made explicit because it depends on who the “we” in the definition is. This question is very delicate and shows the importance of rethinking what we understand by mathematics, in order to determine what is considered ethnomathematical. If we consider mathematics according to the criteria of the researcher who generally represents or is a bearer of the academic culture, we run the risk of not seeing or taking into account mathematical practices that are specific to the cultural group being studied. On the other hand, it is necessary to establish a certain limit to what is considered ethnomathematical, and at this early stage of his research life, Barton (1996) proposes that these limits are based on some consideration that belongs to the academic field. This dilemma of determining what is mathematical and ethnomathematical has been evident in ethnomathematical research that relies on ethnographic methodologies from the beginning. In fact, in Wendy Millroy’s (1991) paradox, it becomes clear that the (ethno)mathematics that she finds in her field work with the carpenters are not others – nor can they be others – but rather expressions of academic mathematics put into practice, precisely because it is the researcher with his or her academic gaze that recognizes them. Many researchers (Albertí 2007; Barton 1999, 2008; Bishop 1991; D’Ambrosio 1985, 2006; Dehaene 2011; Rosa and Orey 2012) have made proposals to address and solve this issue in various ways. Generally, these proposals are aimed at modifying the definition of ethnomathematics or mathematics and/or at developing methodological proposals to address this dilemma. The following is a brief mention of some of these proposals that should be taken into account when dealing with ethnomathematics research in handicrafts,

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occupations, and other activities, which is the object of this chapter, although some of them will be dealt with in more detail in other chapters of this book (Chap. 7, “Emic, Etic, Dialogic, and Linguistic Perspectives on Ethnomodeling,” about emic, etic, and dialogical perspectives and Chap. 8, “Ethnomathematics in Education: The Need for Cultural Symmetry,” about the need for cultural symmetry).

Definitions of Mathematics For the definition of ethnomathematics to make sense, D’Ambrosio (1985) himself indicates that we must broaden the interpretation of what we consider mathematical as it is understood in the academic and educational field. With regard to the proposals that redefine mathematics, there are those (Barton 1999, 2008; D’Ambrosio 1985; Dehaene 2011) that clearly indicate a duality between more formal mathematics associated with the school-academic environment and more intuitive mathematics associated with practical experiences. The historical origins of this duality go back to the times of ancient Greece, when Plato (D’Ambrosio 1985, p. 15) spoke of a distinction between scholarly mathematics aimed at a few people of the elite and practical mathematics aimed at working middle-class people for the sole purpose of managing business affairs. This idea has been maintained over the centuries; the following are more recent reflections from different areas of knowledge. Davis and Hersh (1981) take up this duality by describing two types of mathematical experience. Analogical mathematics is with easy, fast, and no or few mathematical symbols and within the reach of everyone; they give a relevant role to the experimental verification of mathematical facts through intuition, understanding, or the clinical eye. These are particularly useful for finding real-world solutions to everyday problems. On the contrary, analytical mathematics is difficult and complex, a specific preparation is needed to deal with it, and there is a predominance of symbolic notation; verification is based on formal demonstrations that must be accepted by the mathematical community. Analytical mathematics provides theoretical solutions to problems that live mainly in the academic field. Studies (Dehaene 2011; Saxe 1991) on the functioning of the brain seem to confirm that mental processes, when thinking about analogical or analytical solutions, are different. Cognitive neuroscience can identify the areas of the brain that are activated when a person performs a certain task. Stanislas Dehaene (2011) has shown that tasks involving complex arithmetic operations in which an exact result is requested – an analytical solution – activate areas of the brain related to language, memory, and mechanical procedures, whereas, if an approximate result is requested – which we can associate with analogue solutions – areas of the brain directly linked to quantification are activated. In his book The Language of Mathematics: Telling Mathematical Tales, Bill Barton (2008) proposes to refer to mathematics as it is conceived in the school and academic environment as “near-universal, conventional mathematics” or NUC mathematics (p. 10). Barton then addresses the aspects of a given culture that are considered

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mathematical; as an adjective, he proposes to use the phrase “(concerning) a system for dealing with quantitative, relational, or spatial aspects of human experience” or QRS system for short (Barton 2008, p. 10). In other words, Barton considers that when quantities are handled, when space is involved, and when relationships are established, the system of meanings constructed should be considered as mathematical (Barton 1999). Other authors have used different stratagems to differentiate in their writings this duality. For example, another option consists of indicating with the capital “M” and singular (this is especially effective in the Spanish language), the Mathematics of the academic school environment, and with the small “m” and plural, the mathematics of the cultural groups. Personally, I do not fully share these notations since with the capital “M” we are recognizing a higher hierarchy to which comes from the academic school environment with respect to what comes from other cultures. On the other hand, the strength of Barton’s proposal lies in the delimitation of some areas – quantity, relationships, and space – for the aspects of human experience that are considered mathematical, which is not evident in other proposals.

About Methodology Other proposals to address the above dilemma refer to methodological aspects and give indications on how to carry out ethnomathematical research. It seems appropriate to summarize a group of proposals that are related to the duality of the definitions of mathematics put forth by Barton (1996) that could be summarized in the following dichotomy: in ethnomathematical research, NUC mathematics can be recognized, and QRS systems can be searched for and discovered (Albanese et al. 2017). Bill Barton (1996) has conceptualized the methodological process in ethnomathematics by defining different activities that the researcher must carry out. These are descriptive activity, which is the first activity of the researcher who must describe the practices under study within the culture in which they are carried out, using a common language and respecting the conceptions of the cultural group; archeological activity, in which the researcher must reconstruct the mathematics hidden and frozen, implicit in the practice under study; mathematizing activity, which consists of translating the cultural material into mathematical language, leaving the context in which it has been found; and analytical activity, which studies why the practices under study are as they are, from a social historical perspective. In the descriptive activity, a search for a QRS system is evident, while in the mathematizing activity, a recognition of NUC mathematics is put into action. In the archeological activity and analytical activities, the researcher looks for relationships between QRS systems and NUC mathematics. Miquel Albertí (2007) develops a methodological procedure for ethnomathematical research on handicrafts. Albertí proposes to organize the collection of information around three objects: the work in progress, that is to say, the nonparticipating observation of the process of construction of the object by the craftsman; the finished work, that is to say, the observation of the object when it is finished; and the explained work, that is to say, the explanations that the craftsman provides

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regarding the techniques that he uses in the construction and in relation to the final form of the object. During observation of the work in process and the finished work, the researcher identifies mathematical projections, recognizing procedures in terms of NUC mathematics in the craftsman’s actions and practices. Albertí’s procedure compares the craftsman’s explanations and the researcher’s interpretation. If the craftsman actually explains his actions in the same way identified by the researcher, only in this case, then this is a situated interpretation of the craftsman’s practice, that is, a discovery of QRS systems. Milton Rosa and Daniel Orey (2012) indicate an already existing dichotomy in anthropological studies with respect to the actions of the researcher. The researcher can approach the analysis of the information assuming an emic perspective, that is to say, based on the categories and cultural schemes of the group being studied, trying to respect their vision of the world, which would imply looking for QRS systems. On the contrary, he could use the categories and schemes of his own academic culture, an etic perspective external to the cultural group, and then recognize the NUC mathematics. Furthermore, Rosa and Orey insist on the importance of assuming a dialogical perspective in which the two ways of proceeding complement each other (see for more details Chap. XX). It is interesting to note that the authors, in one way or another, promote the complementarity of both perspectives and the linking of QRS systems and NUC mathematics. Other proposals indicate which are the activities that generate mathematical knowledge, or rather QRS systems, common to the various cultures. In other words, it has been established which activities the ethnomathematical researcher should consider when searching for QRS systems. In this sense, Alan Bishop (1991, pp. 42–43) asks what are the universal activities, common to all cultures, that provoke the generation and development of a system of mathematical knowledge, warning that its definition is not as relevant as the idea of rethinking what is mathematical from the activities that generate such knowledge. With respect to the idea of quantity, Bishop (1991) identifies two activities, counting and measuring. The activity of counting is generated from everything that is discrete, while the activity of measuring is more associated with continuous phenomena. In relation to the idea of space, it is important to locate and design. The activity of locating involves the topographical and cartographic aspects of the environment, while the activity of designing refers to objects and generates the idea of form. With regard to the activities that are oriented toward the relationships between individuals, individual and society, and individual and environment, it considers the activities of playing and explaining. Playing is related to social rules and procedures to act and also to what the hypothetical behavior is. Explaining includes the aspects of investigating and conceptualizing the environment and sharing it with others. Ubiratan D’Ambrosio (2006, p. 39) also presents a list of mathematical ideas that are forms of thought present in all cultures: compare, classify, quantify, measure, explain, generalize, infer, and evaluate. Historically, these ideas are linked to the satisfaction of basic needs for survival (feeding oneself, then hunting and sowing, protecting oneself) and then to needs to transcend in order to understand and manipulate one’s environment (myths, songs and dances, social organization).

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Rik Pinxten (2015) makes an even more extensive list of complex mathematical activities, taking up some of Bishop’s, that can be used in mathematical education and that should take cultural aspects into account: locating and representing, related to ethnogeography and the organization of space and mapping the environment; measuring; designing, which is related to classifying forms; traditional building, especially in relation to the worldview that is related to its structure; music, through the division of time into regular intervals and its physical perception; dancing, being mathematics in motion, unites the conception of time and space; computer design as an expression of networks of relationships; storytelling, as a means of communicating the organization of social relations; and exchange and market activities, as their buying and selling or trading activities.

(New) Definitions of Ethnomathematics Due to the tensions that have been highlighted throughout the previous sections, Ubiratan D’Ambrosio (2006) proposes a new definition for ethnomathematics based on the etymology of the word. His definition maintains that the interest of the ethnomathematics program is centered on the creation and development of the instruments of observation and reflection, material and intellectual (the -tics), of understanding, knowing, explaining, and learning to know and to do (the mathema) as a response to the needs of surviving and transcending in different natural, social, and cultural environments (the ethno-). It should be noted that in D’Ambrosio’s (2006) definition, the relationship between mathematics from the academic school environment is lost a little, which would give more freedom to the ethnomathematical researcher in identifying QRS systems. On the other hand, this definition makes explicit the interest in the processes of creation, development, and also validation of the knowledge systems studied, a key turning point in the evolution of the ethnomathematics program (Albanese et al. 2017). Aldo Parra-Sánchez (2017) provides a new definition in which he proposes to overcome the abovementioned tensions. He criticizes an approach that states that ethnomathematics should be considered according to a model of intersection of the set(s) of mathematics and culture(s), as Barton (1996) conceptualized it, by interpreting what other authors have done up to that point. On the contrary, ParraSánchez proposes to identify the object of study of the ethnomathematics program with the relationships – the bundle of associations among the elements of the two sets defined by mathematics and culture: Following the intersection approach corresponding to Barton’s Venn diagrams, there is only one possible intersection between two sets. Each member of a set is examined with the principle of excluded third: It must belong or must not belong to the other set. Alternatively, in the second metaphor there are multiple possible relations between sets; a relation is defined as a bundle of associations among the elements of two sets. One element in a set can be associated with (i.e., translated as) another element in the other set, associated with more than one element, or even associated with no element. If a connection seems “unsatisfactory”, another relation is chosen, i.e., another bundle of associations is built. The

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relations are “customizable” while the intersection is not. It is noticeable that the change from intersections to relations is not a small one at all. Such change entails an entirely different role for an ethnomathematics researcher. In the intersection approach, the researcher behaves as a detective looking for, uncovering, and trying to prove facts based on evidence. Researcher pursues a factual truth (timeless and univocal), which requires proof. In contrast, within the relational approach, the ethnomathematician acts like an artist: creating, proposing, and performing interactions; researcher tries to make sense through translations of meanings. The truth that the researcher aims at is a poetical one (ephemeral and polysemic) that deserves to be experienced. (Parra-Sánchez 2017, p. 99)

Equally expressive is the image in Fig. 1, which represents the way to look for such relationships that must go through processes of communication and negotiation of meanings between the members of different cultures, the fruit of which is a hybrid knowledge that is particular, localized, and multiple and that does not belong to any of the starting cultures.

Mathematics Education and Cultural Context One of the current challenges of the ethnomathematics program concerns its influence on mathematics education. This section proposes a possibility related to the importance of the cultural context in the process of teaching and learning mathematics and therefore the need to address this issue in teacher training. Several researchers have shown that culture has an influence on how mathematics is conceived and made (Barton 1999, 2008; Bishop 1994; Nuñes-Carraher et al. 1985; Pinxten et al. 1983). Cognitive psychologists have also observed that learning mathematics is strongly related to context (Saxe 1991). Research in mathematics education with children and adolescents has concluded that mathematics is best learned when it is related to natural learning situations (Gasteiger 2012) and when the mathematization of the daily environment occurs (Alsina 2010) or is inspired in the cultural context close to the students (Canals 2013). International standards for mathematics education provide indications of this, for example, when they state that teachers should “build on children’s varying experiences, including their family, linguistic, and cultural backgrounds; their individual approaches to learning; and their informal knowledge” (NCTM – National Council of Teachers of Mathematics & NAEYC – National Association for the Education of Young Children 2002, p. 24). However, the training of mathematics teachers provides for the acquisition of pedagogical and mathematical knowledge (Ball et al. 2008; Shulman 1986) which must interact with each other. At the same time, these knowledge must be integrated with a competence for pedagogical-didactic action (Gasteiger 2012) so as to facilitate the development of mathematical competence in children. According to Gasteiger (2012), this competence in action allows the teacher to act flexibly and appropriately by interacting and responding to children’s behavior on the go. Teachers can identify and take advantage of mathematical learning opportunities that arise spontaneously in everyday and play situations which are inevitably related to children’s cultural context. Educators can detect mathematically

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Fig. 1 A multiplicity of intentioned interpretations of a practice becomes the practice. Illustration by Aldo Parra-Sanchez, inspired in Square Heads. (Source: Parra-Sánchez (2017, p. 103))

relevant aspects in the interaction with and between children and using them to ask questions and encourage mathematical reflection (Gasteiger 2012). It is about opening eyes, developing a kind of mathematical gaze toward the environment, and capturing the mathematics in the relationships between the children and the community around them and thus directing them toward those discoveries (Canals 2013). In order to achieve this, the pre-service or in-service teachers’ education should cover research literature on the mathematical thinking of children and actors in the environment. This education includes facilitating the analysis of cases in which

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mathematics is done and identifying practices that allow for the development of mathematics with the purpose of training this mathematical gaze, which Ginsburg et al. (2008) also support. In this context, the ethnomathematics program can play a decisive role in promoting the development of a mathematical perspective in teacher training and provide examples of activities that allow mathematical knowledge to be contextualized and put into practice.

Bundles of Ethnomathematical Expertise In this second part of the chapter, some ethnomathematical experiences are selected for elaboration. In order to facilitate the organization of these, I have differentiated between experiences that have been carried out with handicrafts, understanding these as works done by hand or with the help of nonautomated tools that involve a creative component and are linked to the cultural context. Other experiences have been carried out within professional occupations, which I understand as work that implies professional specialization, in terms of qualifications or training time, for those people who carry it out, and which tends to be less different depending on the geographical environment (although this is not always exactly the case). I conclude with some activities that cannot be considered either as handicrafts or as occupations, but which are determined by the cultural context in which they are carried out. It is reiterated that this classification, handicrafts or professional specialization, has a purely organizational purpose. It will be shown that for each craft, task, or activity, you can: 1. Study different aspects 2. Recognize NUC mathematics and/or search for and discover different ethnomathematics or QRS systems 3. Establish relationships between QRS systems that arise in various activities

Within Handicrafts For the study of handicrafts, the proposal of Albertí (2007) considers aspects related to the finished work, the work in process, and the explained work, wherein lies many potentialities. In this section, different investigations on ethnomathematics in handicrafts of various types are related.

Braiding and Weaving Crafts I will differentiate between braiding crafts and weaving crafts. Braiding is done by hand and in its product usually clearly predominates one dimension over the other, for example, the creation of ropes, braids, and bracelets. In the weaving crafts, a

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loom is used, and its product usually has two dimensions, for example, the weaving of carpets, blankets, and fabrics. For the braiding, the Leonel Vieira et al.’s (2008) studio has focused on the finished work and has looked for a model to describe mathematical aspects of the braid as a finished product. These braids are made in the town of Fafe, in the north of Portugal, with palm leaves. In particular, they have identified the angle that each palm leaf band forms with the sides of the braid and have recognized a pattern between the bands that are identified at the top of the braid as well as at the bottom; see Fig. 2 for the pattern that is repeated throughout the braid. The position in which the braid is analyzed in the images (horizontal) that accompany the work is relevant, as it does not respect the position in which the braid is in its construction (vertically), as it is seen by the artisan when making it. These observations are not accompanied by indications about the process of making the braid, that is, the work in progress, nor by the explanations of the craftsmen who do this work, so it can be said that this is a mathematical projection by the researcher who recognizes some elements of NUC mathematics in the object produced. Also, with regard to braiding, Aldo Parra-Sánchez’s (2003) study focused on the work in progress of braiding manillas (bracelets) made of fiber from the chambira plant in Macedonia, a town in the Brazilian Amazon. This process was translated into a computer language by means of a pseudo-code created ad hoc from the definition of the basic knots and the actions carried out by the artisan to create them. Parra-Sánchez conducted numerous interviews with the artisans, but his research does not specify the artisans’ possible contribution in the elaboration of the computational model. In my research of braiding crafts, I focus on the work in progress and the work explained. With regard to the braiding of sheep’s wool threads from Salta, in the north of Argentina (Albanese et al. 2014), a model inspired by the mathematical concept of graph theory has been identified that reflects the process of braiding the threads (Fig. 3). This model was developed by a craftsman who is also involved in teaching mathematics courses. In this case, it was the craftsman himself who used NUC mathematical tools to create a QRS system that allowed him to systematize his practical knowledge. Below is the description of the graph-inspired model:

Fig. 2 Patter of the braid identified in the top and in the bottom of the finished work. (Source: Vieira et al. (2008, p. 306))

1 2

3

2 3

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1 2

2 3

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Fig. 3 Graph modeling the braiding process of an eightthread braid. (Source: Albanese and Perales (2014b, p. 13))

141 a

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h

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g

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The structure graph is made up of 8 knots or vertices, as the threads used for braiding are 8. The eight knots are arranged two on each side of an imaginary square, and are named in a clockwise direction, starting from the first one on the top left as a, b, c, d, e, f, g, h. The sequence that models the process of realization is simple and its only step is exemplified by a graph made up of two circuits of four knots. The first circuit, clockwise, involves knots a, c, e, g, the second circuit is counter-clockwise and involves knots b, d, f, h. (Albanese et al. 2014, p. 13, own translation)

It should be noted that during the research process, a similar model inspired by graphs had been recorded (Owen 1995) to describe the process of making silk-thread braids in some Asian countries such as Japan, with the difference that the vertices of the graph are arranged on a circle instead of a square. Research has also been carried out on the mathematical thinking of the artisans who practice the handicraft of soguería – braiding of leather in Argentina (Albanese 2015; Albanese and Perales 2014b), with the aim of identifying ethnomodels – those very models that artisans use to communicate or teach their practices to their apprentices. In this case, it is not a question of a situated interpretation, but rather, a QRS system has been identified that is specific to the craftsmen’s guild which, although it uses some mathematical symbols, is characterized by its own rules and interpretations (Fig. 4, Table 1). The following is an explanation of this ethnomodel: It is based on the implicit-tacit convention that the external thread on the side of the working hand is the one that is braided. The working hand is indicated by a (I ¼ izquierda) for Left and (D ¼ derecha) for Right. An external thread realizes a pasada [movement] until it reaches its new position, then the external thread on the other side is moved [...]. The movement is always towards the center. When the braid is flat, it starts from the same side of the working thread (improperly described as from the front), while if the braid is round or square, it is passed behind and the threads passed by the movement start to be counted from the opposite side. In particular, if the braid is, for example, S2 B1, this means that the working threads pass on (S ¼ sobre) two threads, covering them from the view of the craftsman, and under (B ¼ bajo) one thread, hiding the working thread underneath, always counting from the external side to the center. (Albanese 2015, p. 502, own translation)

It should be stressed that one of the aspects of interest of ethnomathematics in the weaving crafts is the description of the weaving process itself, through a more or less shared model within the artisan community. In the description of this process,

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different mathematical aspects stand out, mainly related to the physical space of the objects (Pinxten et al. 1983), especially in the handling of relative positions between the object under construction and the craftsman who works it. In fact, except in the first case presented where the process of elaboration is not taken into account, the product of the craft is conceived in its verticality, following the process of braiding, either upward (in the case of the braids from Salta) or downward (in the case of the bracelets from Brazil and the soguería). It should also be noted that the models or ethnomodels identified involve the use of different types of language: verbal, iconic, symbolic, and sometimes mixed together. Numerical aspects are also recognized, which link the number of threads that allows the making of the braids with mathematical characteristics of the number, such as it being even or odd (in the soguería) or it being a factor of another number (four, for the Salta braids). In the weaving crafts, different angles can also be studied.

Fig. 4 Ethnomodels of braids of eight threads in the craft of soguería from craftsmen notes. (Source: Albanese (2015, p. 502)) Table 1 Ethnomodels of braids of three, five, and seven threads in the craft of soguería. (Source: Albanese (2015, p. 502)) Braid The only braid of three threads and the two equivalent way to model it The two possible braids of five threads The four possible braids of seven threads

Artisanal ethnomodel I D ~ S1 S1 ~ I D S2 S2 I D S3 S3 I D S1 B2 S1 B2

I B1 I S1 B1 I S2 B1 I S1 B1 S1

D B1 D S1 B1 D S2 B1 D S1 B1 S1

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María Luisa Oliveras (1996) analyzes various crafts of Andalusia (Spain) focusing on the ethnomathematics frozen in them. In the craft of carpet production, a weaving craft, she identifies a list of mathematical concepts, from her academic (etic) perspective, as she observes the production of the carpets, the work in process; that is to say, she carries out a mathematical projection, in Albertí’s (2007) terms. Oliveras (1996) first mentions the process of dyeing the threads which is done by mixing spoons of dyes of the primary colors (red, blue, and yellow), indicating that the artisans do not explain to her the proportions to obtain different colors and shades, but that these are coded with numbers. In the design of the carpet, Oliveras (1996) recognizes mathematical concepts used in practice, being generally geometric notions. Here, Oliveras shares the case of geometric transformations: [The craftsman] has to create carpet designs by adapting some basic decorative motifs to the measurements or shape requested, so he has to make changes of scale and other types of transformations. The most characteristic can be called topological, as it consists of changing the shape of the carpet, for example rectangular into another elliptical or circular one, maintaining the decorative motifs, but deforming them and adapting them to the new shape. (Oliveras 1996, p. 162, own translation)

Another example of the etic perspective of Oliveras is the use of a system of Cartesian coordinates: The performance (of the sketch artisan) is similar to that of the one who generates some Cartesian axes, placing in them the numerical codes of the color of the wool. The subsequent creation of this type of carpet is done by a single weaver, [...] she uses the coordinates of each point, decoding its color code. (Oliveras 1996, p. 163, own translation)

In specific moments of the analysis, an approach like the one that Albertí (2007) has defined as a situated interpretation is identified. The recognition of mathematical concepts (NUC mathematics) in the description of the craft work (work in progress) is followed by a reference to the explanations of the craftswomen themselves and how they handle or think of these concepts and their properties. Here is Oliveras’ case of symmetry: When weaving, they reinvent symmetry, repeating the number-color selections from the sides to the center for a certain number of knots. As in a mirror, both weavers must do the same until they meet in the center. Since they work perpendicular to the axis or central thread of the weft, all the properties of specular symmetry are fulfilled, and they know these properties. When asked, they say “so that if I make a mistake it is less noticeable because it is the same on both sides.” (Oliveras 1996, p. 162, own translation)

Basketry Craft A pioneer in the study of ethnomathematics is basketry of Mozambique by Paulus Gerdes (2003). In his research, he studies, among other objects, the sipatsi, wallets to keep money and documents. The sipatsi are constructed with two groups of flat

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strips of different colors that are crossed perpendicularly. Gerdes catalogued many ornamental patterns that are reproduced in the sipatsi, but he does not do it trying to achieve a symbolic model for the braiding patterns. Rather, he conforms to an iconic representation (Fig. 5) to which he assigns a code whose only relation to the pattern is the number of strips that serve to braid it. Leonel Vieira et al. (2008) study the creation of circular-base baskets in Vigo (Spain). At the base of the basket as a finished work, an axial symmetry is identified, determined by the number of wick sticks that constitute the radius of the circle, which depends on the size of the basket and how dense it must be according to its intended use. In addition, they identify a braiding pattern in the work in progress. “The base was woven with two osier sticks, alternately presenting an ABAB pattern” (Vieira et al. 2008, p. 298). They show that variations in the braiding pattern create different decorative motifs at the base of the baskets: The craftsman bypasses the situation using a simple strategy that consists of jumping two units simultaneously at the end of each lap, i.e. momentarily leaving the ABAB pattern, for each lap and only once using ABBA. In turn, at each lap, the craftsman goes forward to the point where it changes from ABAB to ABBA, resulting in a spiral which is perfectly visible. (Vieira et al. 2008, p. 300, own translation)

By analyzing the sides of the baskets, Vieira et al. (2008) continue to study how the braiding process produces the decorative motifs in the appearance of the finished work: To make the designs presented, the craftswoman has to abandon the ABAB pattern and apply an AABB pattern alternating with pink and beige and then beige and green. In the term of frieze, we are faced with a horizontal axis reflection and a vertical axis reflection, in which the initial motif is composed by a quadrangular shape. (Vieira et al. 2008, p. 303, own translation)

Their research does not explain whether this modeling of the braiding process has been provided by the craftsmen, which suggests that it is more of a mathematical projection by the researchers. It is also interesting to note that, although the

Fig. 5 Image of an ornamental pattern of the sipatsi. (Source: Gerdes (2003, p. 26))

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symbolization used is different, here, the same action as in the soguería is being modelled; actually, the pattern symbolizes passing over or under a certain number of threads that go in the other direction, precisely one in the case of ABAB and two in the case of AABB. Christian Fuentes-Leal (2011) researches basketry in Guacamayas, Colombia. His interest includes the finished work from which he rescues the geometric properties (mathematical NUC) that characterize his decorations: An important characteristic of this type of craft is the geometric patterns, where translations predominate. Concepts such as measurement and proportionality also arise, since they are present in the process of building the objects, where certain parameters must be met in order to make the desired design real. (Fuentes-Leal 2011, p. 65, own translation)

But he also investigates the work in progress and the explained work. Two interesting results of his research related to the manufacture of round plates or trays in which a spiral motif are often depicted. The first is a method used by craftsmen to split the length of a fiber into three equal parts (Fig. 6), which will then provide the circle partition once the fiber is closed. Based on the fact that the fiber they use is flexible and can be folded, they do the following: . . . As a result of the interviews held with the working group, a special characteristic was found. This type of handicraft is made with a long straw fibre, which the craftswomen divide into a certain length by twisting the fibre, then making marks with chalk or charcoal at the break points, then releasing the fibre back into its natural shape and obtaining the desired division. It is possible to think that this type of division of lengths is far from the Greek rationalist thought promoted by geometricians like Euclid. (Fuentes-Leal 2011, pp. 62–63)

The second result deals with the translation to be carried out in different cases (Fig. 7): In the construction of the spirals it is observed that in a sliding translation of a segment of length x a distance of x/2 predominates, this proportion varies depending on the spiral to be made, the greater the total length of the spiral, the greater the distance of the sliding translation will be. (Fuentes-Leal 2011, p. 58, own translation)

To carry out this translation, a craftswoman does not use length measurements, but measures the time she spends doing that part of the work (Fig. 7). The following can be considered a QRS system used by the basket craftswomen to carry out her work: [From an interview] I know that if I take half an hour to do one line, then I know that the other line I have to do for fifteen minutes straight and so on. The craftswoman uses time as a unit of measurement [of length], in this case that the sliding translation of x/2 would be used, since she mentions half the time between segment and segment, it is also very interesting to observe the craftswoman’s recursion where she relates these two contexts (time-space). (Fuentes-Leal 2011, pp. 59–60, own translation)

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Fig. 6 Method of dividing a segment into three equal parts. (Source: Fuentes-Leal (2011, p. 63))

a

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d e

45 minutes 1hours

Fig. 7 Procedimiento de traslación deslizante basado en el tiempo. (Source: Fuentes-Leal (2011, p. 60))

Wooden Sculptures In the research of María Luisa Oliveras (1996) on the taracea, an Andalusian craft (Spain) that covers surfaces of objects with wood mosaics through the composition of motifs previously composed by the most expert craftsmen, the recognition by the researcher of NUC mathematical concepts in the work of the craftsmen is again highlighted: This craftsman does not create the motifs, but he does make designs with them in the covering phase, using the compass and the ruler and producing irregular tessellations of the plane in which there are symmetries and twists, using Thales’ theorem and serials that become motifs by translation. [. . .] Using the stars at his disposal, he took one of the largest and placed it in the centre of the box, which he pointed out to the eye. He then checked the correct position of this centre with the compass. (Oliveras 1996, p. 154, own translation)

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This last paragraph shows the importance of procedures based on visual intuition (analogical mathematics) that comes from experience (by the eye) and is subsequently tested with other techniques. An approach that penetrates more into the thinking of the craftsman is that of Albertí (2007) who proposes to investigate the techniques used by the craftsmen of some Polynesian islands when decorating the roofs of their wooden houses. He goes so far as to describe, among other things, the Kira-Kira method for dividing a segment in equal parts as he observes it and then explains it to the craftsmen themselves (explained work), ensuring that it is a situated interpretation of the craft practice. This method consists of dividing a segment into equal parts first by the eye, as the taracea artisans do, and then checking for an error by bringing a wooden slat close to the parts. If an error is detected, this error is divided in two, and the procedure is carried out again, allowing the solution to be found by recursive approximations. It should be noted that this method is very efficient in the case of craftsmen because of the means available to them (and the lack of others) and because of the simplicity and operability of the method, which does not involve complex measurement operations and division calculations, as is usually the case with a task of dividing a segment in equal parts at school.

Within Occupations There are numerous studies, not only from ethnomathematics, on the mental calculation strategies that are used in different occupations. This section will detail some examples related to the activities of selling in street markets and in some professions linked to the bus transportation system. I conclude with a description of how bricklayers put geometric definitions and procedures into practice, sometimes detaching from the ways in which they are usually proposed in school classrooms.

Street Markets Various ethnomathematical researches in Portugal have focused on the mental calculation that is put into practice in the activity of gypsy street markets. Claudio Cadeia et al. (2008) proposed to an adult gypsy the question of how much they cost – 15 pieces at 8.10 euros each. He applies the distributive property of multiplication in relation to addition, moreover in two different ways, applying in one case the commutative property: When I placed the situation of 15 pieces at 8.10 euros, he decomposes it into 15 plus 5 and 8.10 euros into 8 euros plus 10 cents, then he applied the distributive property of the multiplication in relation to the addition (10 + 5) x (8 + 0.1) ¼ (10 x 8 + 5 x 8) + (10 x 0.1 + 5 x 0.1). Later, he applies the same reasoning and multiply 10 pieces for 8.10 euros by giving it eighty-one euro (10 + 5) x (8 + 0.1) ¼ (10 x 8 + 10 x 0.1) + (5 x 8 + 5 x 0.1). Finally, he answers that the total is one hundred and twenty euros and fifty cents. [...] Now 15 pieces at 8.10€- It is 80; 120. Giving 121.50€. 10 pieces, how much is it? It is 80€, isn’t it? No. It is 81€ [...] now, the middle is 40 and half. 121.5€. (Cadeia et al. 2008, p. 86, own translation)

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In the dialogue, another strategy used by the interviewee can also be perceived, the fact that, five being half of ten, he multiplies by ten and then calculate the half to know how much the multiplication by five gives. Claudio Cadeia et al. (2010) present another multiplication calculus made by a teenager to whom the problem of selling three pairs of shoes at 6.50€ each is proposed: He wanted to calculate three pairs of shoes at six euros and fifty cents. He multiplied three by fifty cents, getting one euro and fifty cents, then multiplied three by six euros, getting eighteen euros. He then added one euro and fifty cents to the eighteen, resulting in nineteen euro and fifty cents. He used the distributive property of the multiplication in relation to the addition 3  (0.50 + 6) ¼ (3  0.50) + (3 x 6) ¼ 1.50 + 18 ¼ 19.50 €. (Cadeia et al. 2010, p. 81, own translation)

The researchers then proposed to the same teenager the problem of selling eight pairs of shoes, always at the price of 6.50€: The same price was maintained but the number of pieces was changed. Thus the question of selling eight pairs of shoes was settled. He manifested once again that he never had anyone to buy him so many shoes. He did something curious because he did not multiply the eight by fifty cents but added two by two. So he said that two are one euro, four are two euros, six are three euros and eight are four euros. For the whole part he used the same strategy. He added the six euros eight times. Two is twelve euros, four is twenty-four euros. Then he went on to say six is thirty-six euros and eight is forty-eight euros. Finally he added the forty-eight euros with four euros, counting by the fingers. It seems from these accounts that [in this case] he has interpreted multiplication as a repeated addition. (Cadeia et al. 2010, p. 81, own translation)

In this regard, it is worth noting the difference between the strategies used and the conception of the operation carried out by the same person when calculations involve different numbers. Moreira and Pires (2012) show the calculation of a gypsy child who is asked to solve the problem: how much are five glasses at 15 euros each? The child shared the following calculation: At 15? I have to do 5 times 15, right? 15 and 15 makes 30, 30 and 30 . . .60 Then 60. . .70. . .75. (Moreira and Pires 2012, p. 130, own translation)

At the base of his reasoning is a conception of multiplication as a repeated sum, the application of the associative property – when grouping 15 with 15 and then 30 with 30 – and the application of the distributive property when separating the last 15 into 10 + 5.

Bus Workers Armando Aroca (2015) investigates the counting activity of the calibrators of the city of Cali, Colombia. The calibrators are positioned at certain points on the bus route and tells the driver how much time has passed since the previous bus on the same line. This

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information is very useful to the driver in making decisions on whether to slow down or speed up and thus maintain a service at regular intervals, an essential condition for the quality of the service and for a good profit for each driver working autonomously, since there is no centralization of the transport system in companies. The study highlights the importance of a gestural system for representing numbers. It also emphasizes the mental calculation strategies used by the calibrators very quickly. Aroca (2015) insists that subtractions are made as additions in the form of “how much I need to get to,” sometimes indicated as additions to complete, as shown in the examples below where the calculation of a time interval between an hour and 56 min and the following hour and 4 min is presented: I’ve got 60 stuck in my head... 60 is my number pattern. [...] I go for the smallest number, where it comes out the fastest.... We go from 1 to 60, for the matter of time,..., 4 for 60 and 4... 8. (Aroca 2015, p. 247, own translation)

Here, it is clear that the strategy of the calibrator is to complete the 60 min to reach the hour, making the calculation in the modular numerical system of the hours. The importance of the context is also stressed, since the choice of strategy depends on the numbers involved in the calculation, as stated by the calibrator itself: I propose [to another calibrator] these calculation, I asked him with times 47 and 06. He answered: “It gives 19. 10 to 57 and 9 to 06, it is not 10-10 (ten-ten), but 10-9, because 10-10 is to 07 then it goes over of 1 ¿can you see it?.” (Aroca 2015, p. 247–248, own translation)

Here, another strategy is presented that consists of rounding the calculation by counting from 10 to 10 and then correcting the excess by subtracting the unit to obtain the exact result. Nirmala Naresh (2015) studied the mental strategies and actions of bus drivers in Chennai, a city in South India. Here too, there is a strong conditioning of the context: the need to perform the calculations very quickly and the artifacts that are available to support the calculations condition the way in which the calculation is performed. One of the tasks that a bus driver must perform consists of calculating the cost of a ticket that differs according to the route of each passenger, that is, at which stop he/she gets on and off the bus (bus ride). The bus drivers have tables and specials rules to apply when calculating the price of a ticket. Once the cost of a ticket has been established, the driver must quickly get from the passenger the money that correspond to the number of tickets of in his familiar group: The implied computational task was: 20  (3  3.50). The conductor explained his mental strategy in this way: “I did 3 x 3 as 9. Three fifties are equal to 1.50. Adding it to 9, I got 10.50. Now I demanded another 50 paise since I gave back 10 rupees to the passenger.” (Naresh 2015, p. 1576)

Again, the use of the distributive property of multiplication with respect to addition is noted. It is also illustrated the ability to ask the passenger for correct change.

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Another task that a bus driver performs is the calculation of the profit at the end of the work day. This is based on the idea that different cost tickets are taken from different stubs and that the tickets on each stub are progressively numbered. In Fig. 8, a summary of the strategies identified by Naresh (2015) is presented. It is worth highlighting again the use of the distributive property, of the approximation to numbers that are easier to calculate and then correct (or compensate for) and the strategy of dividing by two when multiplied by 0.50. It is very interesting to note how most of these strategies, in street markets as well as occupations related to bus transportation, are quite different from the calculations based on the traditional algorithms used in most schools around the world. The techniques used are based on mental calculation strategies that allow to compensate and to apply different properties of the operations (the distributive property, the associative property, the conception of subtraction through a sum “how much I need to get to or to complete”), and they always present strong links to the context; to the artifacts at disposal; to the gestural, symbolic, or material representations; and to the particular cases of the numbers involved on each occasion.

Masonry The work of bricklayers is permeated by geometric knowledge put into practice. In her research, Gema Fioriti (Fioriti 2002; Fioriti and Gorgorió 2006) relates different techniques used by Argentine bricklayers when they carry out different tasks in their profession. From her results, it is worth highlighting how the analysis not only identifies concepts of NUC mathematics in the observation of practice but also shows procedures that are typical of this occupation and that do not always coincide with the procedures of the academic school environment. We could say that the masons’ QRS systems are based on the concepts of NUC mathematics, but they are put into practice with their own techniques that make these systems somewhat different. Some examples are described below. The construction of right angles is a very prominent task in the work of the bricklayers as it is needed for the construction of door and window frames, as well as it is needed for the perpendicularity of the walls with the floor and the ceiling. One of the techniques used by the masons is based on the use of Pythagorean triples (Fig. 9): Once the two legs of the right angle -0.60 meters and 0.8 meters- have been fixed, making the length of the hypotenuse 1 meter will ensure that the angle constructed is a right angle. The following interview excerpt relates this procedure “I’m going to tell you a trick we use that never fails. You take 80 here and 60 here, then you open or close the frame until it is one meter long; then the sides are squared [i.e. they form a right angle].” (Fioriti and Gorgorió 2006, p. 106, own translation)

It is clear that the direction of the implication of Pythagoras’ theorem used here is the opposite of that usually studied in school. In fact, the theorem studied in the school recites the following: If the triangle is a rectangle, then its sides verify the Pythagorean relationship, while here, the bricklayers use the opposite implication: if the sides of the triangle verify the Pythagorean relationship, then the triangle is a rectangle.

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Fig. 8 Mental strategies employed in completing the waybill. (Source: Naresh (2015, p. 1577))

The author herself shows how this procedure is used not only to construct right angles but also to verify that certain angles are right: [A bricklayer refers]: In order to get the structure straight, we take the measurement, that is 60 in one part and 80 in the other part, and there we measure. If it gives us a meter, it is because it is well square. . .. (Fioriti and Gorgorió 2006, p. 106, own translation)

Another technique used by bricklayers that puts geometric knowledge into practice is the construction of rectangles and squares. It is related that the bricklayers take care to build the sides parallel and of equal length; then, to verify that they are squared, they measure the diagonals and make sure that they are equal, as well as controlling the distances of the intersection of the diagonals to the vertices. If the latter are also equal, then the rectangle is well constructed (Fioriti 2002). This procedure is similar to the one described by Gerdes (1998) that is used in Mozambique to build the base of the houses that have rectangular shape. Again, it is observed that in NUC mathematics, the properties of the diagonals are presented as a consequence of the definition of the rectangle where the right angles play the protagonist role. While in practice, it is the properties of the diagonals (which are equal and intersect at the respective midpoints) that define the rectangle. In this respect, Fioriti and Gorgorió (2006) show how the bricklayers also use the characterization of the axis of a segment as the geometric location of the points equidistant from the ends of the segment, to draw a line perpendicular to a segment. But, once again, the direction of the implication is not the one that is usually

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Fig. 9 Modeling the process of construction of a right triangle from sides of determined length that verify a Pythagorean triple. (Source: Own elaboration)

presented at school; in fact, it is this: if a point is equidistant from the extremes, then the axis of the segment is traced over there point. Another technique that has aroused interest (Fioriti and Gorgorió 2006) is that used by workers to draw angles of less than 45 (Fig. 10): Draw a square of 45 cm side (the tangent of a 45 angle is 1) and then mark the number of centimeters corresponding to the angle measurement you need. The values obtained by dividing the measures of the sides are very close to those of the tangents in the angles of 10 , 20 , 30 , 40 . (Fioriti and Gorgorió 2006, p. 108, own translation)

According to the analysis and interpretation of Fioriti and Gorgorió (2006), this procedure concretizes and puts into use the concept of trigonometric tangent but, again, in the inverse direction of the implication with which it is initially defined in school, since usually in school tasks the calculation of the tangent is proposed, when the amplitude of the angle is given. On the contrary, here, the length of the tangent is used to construct an angle of a certain amplitude. Finally, it can be seen that in the construction of molds for bridge arches, the concept of circumference as the limit of a polygon with an infinite number of sides is implicit, given that first, the props are drawn and then their ends are joined to give them a round shape. Fig. 10 Drawing of the bricklayer accompanying the explanation of the construction of angles less than 45 . (Source: Fioriti & Gorgorió (2006, p. 108))

40°

30° 20° 10°

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Within Other Activities This section presents two activities that are related to different conceptions of space. Some similarities between conceptions that are presented in different environments and cultures are worth noting.

Dance As already mentioned, the activity of dancing involves mathematical thinking by unifying conceptions of time (and thus quantity) with space in movement (D’Ambrosio 2006; Pinxten 2015). Different aspects of dances can be analyzed, which link the concepts of time and space in different ways. For example, in the measurement of space, temporal elements such as the rhythms of music are involved, and these are understood in relation to the steps that the dancers perform in these times: [The research analyses] the use of non-conventional measurement systems and, in relation to this, how to measure space through the time needed to travel through it and time through the duration of an action. [...] The beat is born as a measure of time determined by the rhythm of the music; in most folk dances the beat is constant and does not change during the development of the same song. In choreographies, a step is performed in the time of one beat, so the beat is also used as a measure of space in relation to the dancer’s step. For example, it is said that the dancers of a couple have to be placed 4 beats apart, and this number 4 is not casual, but it depends on the structure of the music and the choreographies. (Albanese and Perales 2014a, p. 466, own translation)

With regard to the location in space, a topic that will also be addressed in the next section, the existence of a difference between professional dancers and amateur dancers is worth noting: [professional dancers] imagine and teach that the positions of the dancers are located at the midpoints of two opposite sides of an imaginary square, while those who learned according to a family tradition have as a reference the segment that joins the positions of the dancers of the couple. (Albanese and Perales 2014a, p. 466, own translation)

Geometric figures are another element that takes on special relevance in different dances. Some investigations have shown how the movement of the feet of the dancers in some dances draws geometric figures on the floor. In the case of the tango (Di Paola et al. 2008), different conics are recognized, such as the circumference in the case of the planeo and the rational circular quartz (Fig. 11) in the case of the lapis in the giro. Also in the Argentine folk dance of the Malambo, it is evident that the feet of the dancers act as a compass when drawing a circle (Albanese 2016). In other dances, it is interesting to study the geometric figures that draw the movements of the dancers in space during the choreographies. This has been the case

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Fig. 11 Figure left on a sand floor by the tango dancer while making a lapis in the giro. (Source: Di Paola et al. (2008, p. 160))

of another Argentine folk dance, the Chacarera. In one of its movement, the avanceretroceso (Fig. 12), quadrilaterals with congruent sides have been identified because they are performed with four steps by the dancers (Albanese and Perales 2014a). The analysis has generated an interesting discussion regarding the conception of rhombus: [a dancer] drew attention to the different lengths of the diagonals of the rhombus. This detail corresponds to the fact that in the dance, although there are lateral steps, the dominant direction that the dancer marks corresponds to the direction towards the couple. In fact, the same name of the figure of the advance-retroceso [forward-backward] movement underlines it. This conception of the difference between square and rhombus from the diagonals is very different from the school conception that usually insists on the difference of the angles in the rhombus with respect to the square that has all equal and right angles. (Gavarrete and Albanese 2015, p. 307, own translation)

Regarding the geometric figure of the circumference which, in the representation of the choreography, characterizes the steps that involve giros and rotations, the analysis shows that the dancers’ perception is that it is a regular polygon that tends not to have angles, while in the school, its definition is based on the equidistance of its points from a center (Albanese and Perales 2015). As has been presented in the practice of the bricklayers (Fioriti 2002; Gerdes 1998) with the rectangles, here too, the diagonals take on special relevance in the definition of the geometric figures, unlike what usually happens in school where the measurement of the angles is the protagonist of the definitions. The

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Fig. 12 Different representations of the dancers for the Chacarera’s movement of avanceretroceso. (Source: Gavarrete and Albanese (2015, p. 307))

similarity of the concept of the circumference as a polygon with an infinite number of sides can also be observed.

Space Organization The organization of the socio-geographical space (Pinxten et al. 1983) is expressed in the activity of localization as defined by Bishop (1991). In collaboration with a team from Costa Rica, I had the opportunity to reflect on how orientation and localization in the environment where we live are an activity that is permeated by the characteristics of the sociocultural context (Chavarría et al. 2017; Gavarrete and Albanese 2018). An analysis from an emic and etic perspective of ways of finding a direction in different countries or localities of the world has led to the identification of different mathematical conceptions. Here, I propose a recapitulation of these studies enriched by my own experiences. In Europe, at least this is the case in many cities in Italy and Spain (taking, respectively, Rome and Granada as examples), all the streets have a name, it could be of a historical character, of a key date in the history of the country, or of the main activity carried out in the street, be it commercial or recreational. Many of these streets, especially in the historical centers, tend to have winding routes and intersect with each other at all sorts of angles. The numbering of the buildings or houses is consecutive; sometimes, there are even numbers on one side of the street and odd numbers on the opposite side, but it is not uniformly established what the principle of numbering is, particularly what is considered to be the beginning of the street. Surely, the historical development of cities before the existence of urban plans (particularly in medieval times) has determined these labyrinthine structures. This way of organizing space is similar to the characterization of the movement that Barton (2008) indicates as Path Navigation. The navigators of the Pacific oceans identify kinds of paths in the sea and guide themselves along the islands and currents so as to know where they are on the path. In European cities, the streets represent the paths, and the numbering provides information as to how far along the street one is. In many cities in America, the urban plan of the city tends to be more regular, with a group of streets with orientation, for example, east to west, and another group of streets with orientation north to south – or other perpendicular orientations – which

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means that the streets form rather right angles to each other. The result is that the streets form blocks from the segmentation of a street by regular crossings with the streets of the other orientation. The streets can have names (this is the most common in Argentina) or also be identified by progressive numbers starting from the center, sometimes differentiating with the cardinal points (see the case of the city of Antigua in Guatemala) or using even and odd numbers to distinguish orientation (e.g., New York City in the United States). The numbering of the buildings or houses is determined by the distance of the house from the block in meters, plus a higher order number that identifies that block. For example, in Buenos Aires, the blocks are identified with second-order numbers, that is to say, by 100 in 100. Here generally, the center of the city is considered to be the origin of the numbering for the blocks and determines the direction to identify the corner (it is considered to be the one furthest from the center). To determine where a place is located, the closest corner is usually indicated by mentioning the names of the streets that make up the corner. Surely, these cities have been built in more recent times following urban planning plans for the land that have determined the regularity of their structure. It can be seen that, in the case of the American cities mentioned, the planning of the organization of space generates a reticulation of the city that allows the location of the places through a Cartesian conception (that is to say, based on a system of coordinates as it happens in the Cartesian plane). The city center usually determines the center of a system of coordinates that follows the orientations of the streets generally in two perpendicular orientations. In the case that the streets are numbered, this numbering determines a particular metric system in the axes, while in any case, another metric system is identified in the numbering of the blocks. In order to identify an area, the information related to what would be the equivalent of two coordinates is used, which can be the name of two streets or, alternatively, the name of a street and the numbering of the block in question. Some examples for the city of Buenos Aires are the following: “It is in Callao and Corrientes,” which means that the place of interest is located near the intersection of Callao Avenue and Corrientes Avenue, and “It is in Armenia at 1300,” which means that the place of interest is on Armenia Street in the 1300 block. To a city connoisseur, this would give the information that it is near the intersection of Armenia Street with Córdoba Avenue, since Córdoba Avenue cuts the numbering of the streets perpendicular to it in the 1000 block. In this case, this way of organizing space is similar to the characterization of movement that Barton (2008) indicates as Position Navigation. The navigators of Western culture use a coordinate system on the earth made up of meridians and parallels and identify a point in the sea with two coordinates provided by GPS (they are not Cartesian because the earth is round, but in the maps of each zone, they usually seem so). In many American cities, the streets themselves constitute a coordinate system that makes it possible to identify a point in the city by naming two perpendicular streets. A separate case is constituted in Costa Rica, where the so-called system of direcciones a la tica predominates. Here, again, urban plans tend to show a certain regularity. But the streets do not have names, or rather, people do not identify the streets by name, nor do the professionals who work with the location of houses or places, such as taxi drivers and postal personnel.

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Here, to provide the location of a house or a building, a reference point determined by a square, a public building (church, museum, etc.) or a trade (bank, supermarket, or neighborhood store) is provided, and then, the approximate meters are indicated toward a certain cardinal point that must be traveled to reach the place. Once again, at the base of this organization of space is a Cartesian conception but on a small and dynamic scale, as will now be explained. When the starting reference point is provided, it is identified as the center of a coordinate system determined by the orientations established by the cardinal points: north-south and east-west. A metric system based on distances in meters is applied to these axes: In the city of San José, Costa Rica, the old Porfirio Brenes School is located 150 meters west of La Dolorosa Church. [. . .] We insist that this form is adjusted to any rural environment, since the reference point from where the displacement is indicated is mobilized. (Gavarrete and Albanese 2018, p. 26, own translation)

It is very common for Costa Ricans to know how to provide different ways of locating the same place from different points of reference, since it is very important that the interlocutor knows this point of reference; therefore, they tend to test several. This is the reason why it is affirmed that this Cartesian conception is dynamic, since the reference system is translated according to need.

Final Reflections This chapter has revealed the existence of multiple situations that can enrich the study of mathematics through the observation of the practice of handicrafts, occupations, or other activities in society, regardless of indigenous contexts. A historical overview of the definitions of ethnomathematics has been developed, which has allowed us to discuss some of the tensions present in the program and some of its solutions that involve philosophical, cognitive, and methodological elements. All this has laid the foundations for the subsequent documentation of various ethnomathematical empirical practices. A brief reflection on the importance of the context in mathematics education allows us to link these ethnomathematical results with the contextualization of mathematical concepts and practices in daily activities, also demonstrating the importance that there may be in teacher education the training of a mathematical gaze that allows taking advantage of many occasions to do mathematics in the classroom provided by the environment. Throughout this overview of ethnomathematical research, different practices that would be enriching in formal mathematics education have been highlighted, for example, the diversity of mental calculation strategies put into practice by street markets’ sellers and by bus drivers, the properties of geometric figures used by bricklayers and dancers to deepen their understanding of geometries, or the way of conceptualizing space in cities (and on the sea) in different parts of the world. It should be noted how mathematical techniques and procedures belonging to diverse activities of different cultures and distant locations can become very similar.

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In the ethnomathematical experiences reported, the importance of going beyond the recognition of school-academic mathematics in cultural practices has been emphasized but also insisting on the need to penetrate the mathematical thinking of artisans and professionals with the purpose of searching and discovering their own QRS systems. In education, the acceptance of a diversity of modeling and mathematization practices of human activity, such as those shown for weaving crafts through symbol systems not common in NUC mathematics, would provide students with a broader view, democratically and socially more interesting than just about mathematical thinking. Acknowledgments The document has been produced as part of the project of the Spain Ministry of Education with reference: PID2019-105601GB-I00 / AEI / 10.13039/501100011033.

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Emic, Etic, Dialogic, and Linguistic Perspectives on Ethnomodeling Milton Rosa and Daniel Clark Orey

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnoscience as the Relation Between Humanity and Its Sociocultural Context . . . . . . . . . . . . . . The Need for a More Culturally Bound Perspective on Mathematical Modeling . . . . . . . . . . . . . Cultural and Cognitive Features of Ethnomodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomodeling and the Cultural Aspects of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linkage Between Ethnomodeling and Ethnoscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cultural Components of Ethnomodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomodeling of Landless Peoples’ Movement: Wood Cubing in Brazil . . . . . . . . . . . . . . . . An Ethnomodel of Wood Cubing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dialogic (Emic-Etic) Approach in Ethnomodeling Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Ethnomodeling Perspective in the Mathematics Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Ethnomodeling is considered the association of ethnomathematics and mathematical modeling and enables us to perceive different realities in relation to the nature of mathematical knowledge. It also provides insights into the many diverse forms of mathematics. In this context, ethnomodeling is defined as the study of mathematical phenomena that adds cultural components to the mathematical modeling processes, which are supported by the ethnoscience research field. The development of this connection has been conducted through the development of three cultural components used in the conduction of ethnomodeling investigations: emic, etic, and dialogic approaches. Ethnomodeling aims to work against colonialism in order to value and respect the sociocultural diversity found in the mathematics and scientific traditions of distinct cultural groups. Ethnomodeling, as an ethnoscientific approach, studies the connections between mathematics and M. Rosa (*) · D. C. Orey Departamento de Educação Matemática, Universidade Federal de Ouro Preto (UFOP), Ouro Preto, Minas Gerais, Brazil © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_3

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science in the direct relation with the social, economic, political, environmental, and cultural backgrounds. Since ethnomodeling seeks to promote the development of the understanding of differences through dialogue. We argue for the inclusion of ethnomodeling as a translation process for systems taken from the reality of the members of distinct cultural traditions. Ethnomodeling creates a firm foundation that allows for the integration of these three approaches in exploring mathematical knowledge developed by the members of distinct cultural groups. Thus, we argue that there is a significant dialogic interface between ethnomodeling and ethnoscience, which leads to important interdisciplinary reflections as a consequence of diverse and distinct ways of communicating, reading, and interpreting the world. Keywords

Cultural approaches · Dialogic approach · Ethnomathematics · Ethnomodeling · Ethnoscience · Mathematical modeling

Introduction Currently, there is an overwhelming bias against local orientations; this seems especially true in regard to the acknowledgment and presence of mathematics in many diverse contexts. By acknowledging the importance of local mathematical knowledge and experiences, ethnomodeling encourages connections, debates, and a sense of mindfulness of the real nature of mathematics as it relates to ongoing changes and the development of culture and society. It creates a democratic environment for the discussion of decolonization of mathematical thinking, teaching and learning, and its uses in the context of how members of distinct cultural groups perceive mathematics in their daily lives. By promoting cultural diversity developed in non-western perspectives, we are encouraged to become more mindful of diverse perspectives and worldviews held by others. This is one of the principal ways to concretely decolonize mathematical knowledge. For example, Battiste (2011) states that decolonization is the examination of our assumptions inherent in western knowledge, mathematics, and science in order to make it visible and dispel the assumption that local knowledge is primitive and in binary opposition to dominant scientific, western Eurocentric (modern) knowledge. The colonial strategy is related to the devaluation and disqualification of mathematical knowledge developed by the conquered, particularly in science, mathematics, and technology. Thus, mathematical ideas, procedures, and practices developed by members of distinct cultural groups have been disregarded in favor of powerful western advances. Part of this method is related to the predominant belief, especially in academia, that western mathematics is the privileged manifestation of the rationality of humanity; hence it is universal and culture-free of influences from impurities of sociocultural contexts (D’Ambrosio 2006).

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According to this assertion, it is necessary to develop techniques and strategies to counter Eurocentric discourse that positions western knowledge as the only, true, or superior science and local mathematical knowledge systems as inferior, or at best, exotic curiosities. As a part of a decolonized research paradigm, ethnomodeling recognizes that there are non-western forms of knowledge, which are in response to the ups and downs of survival and the transcendence among the members of distinct cultural groups who can be considered as insiders (emic, local) and outsiders (etic, global) in accordance to their position regarding a specific culture. The concepts of emic (insiders) and etic (outsiders) are taken from linguistic work and come from phonetic and phonemic. In this regard, all the possible sounds that people can make constitute the phonetics of the language. However, when people actually speak a particular language, they do not hear all possible sounds because not all of them make a difference. The sounds that are locally significant are the phonemics of that language. Emic, then, is about differences that make a difference from an insider’s point of view (Pike 1967). Historically, the emic and etic concepts were first introduced by Kenneth Pike who drew on analogies to linguistic terms phonemic and phonetic. By suggesting that just as in the study of a language’s sound system, Pike (1967) affirms that there were also two perspectives that could be applied in the study of a society or culture. Generally, emic refers to taking the viewpoint of the insiders, whereas etic means taking the outsiders’ viewpoint. This context enabled Berry (1969) to transfer Pike’s linguistic concepts to crosscultural psychology by applying the etic term to analyze human behavior from the perspective of people who focus on universals. An emic analysis of these behaviors focuses on unique cultural conducts and/or on the diverse and distinct ways in which daily activities are carried out in specific cultural settings. As well, it is possible to develop a similar analogy to ethnomodeling because emic approaches are developed when members of distinct cultural groups expand their own interpretation of their mathematical knowledge opposed to an outsider’s interpretation of this knowledge. Currently, the debate between emic and etic continues to be one of the most intriguing, indeed valuable questions in mathematics education research. It enables investigators to examine questions such as “are there mathematical patterns that are identifiable and similar across cultures?” and “is it better to focus on these patterns particularly arising from the culture under investigation?” This perspective enables us to identify three components: etic, emic, and dialogic approaches. These three components combine to assist us in investigating, studying, and discussing issues related to decolonization and culture, while assisting us to understand the mathematical ideas, procedures, and practices developed by and useful to members of distinct cultural groups. The etic (global, outsiders) approach deals with an outsider’s view of the beliefs, customs, and scientific and mathematical knowledge developed by the members of distinct cultures by describing similarities and differences among cultures through the use of accounts, descriptions, and analyses of mathematical ideas, procedures, and practices expressed in terms of the conceptual schemes and categories that are regarded as meaningful by the community of scientific observers (Lett 1996).

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Globalization emphasizes utilitarian and, more often than not, capitalist approaches of mathematics with the objective to reinforce the ongoing western bias and value system prevailing in mathematics. It is in danger of providing a critical filter for status in order to become a perpetuator of mistaken illusions of certainty and as an instrument of power that helped dominant elites to commercialize and globalize mathematical ideologies (Rosa and Orey 2019). For example, Pike (1967) argues that more influential trends in cross-cultural investigations privilege etic approaches based on outsiders’ accounts of other cultures. An etic description generates scientific theories about the causes of sociocultural differences and similarities. These constructs are associated with the structures and criteria developed by external observers as a framework for studying cultures. According to Sue and Sue (2003), this approach is known as culturally universal. An emic (local, insiders) approach concerns itself with an insider’s view on how they have come to develop mathematical ideas and procedures. It respects cultural practices, social understandings, customs, religion, gender, and beliefs by enabling members of distinct cultural groups to describe their own culture in its own terms. It seeks an understanding of daily phenomena through the eyes of members of a culture being studied in order to capture meanings of daily life activities. This approach represents the accounts, descriptions, and analyses expressed in terms of conceptual schemes and categories that are regarded as meaningful to the members of distinct cultural groups. Emic approach values and recognizes the contributions of local people to the development of scientific and mathematical knowledge because it has been validated within local contexts (Lett 1996). Local knowledge is characterized by integrated systems of cognition, beliefs, and practices. Emic approach creates a framework from which members of these groups are able to understand and interpret the world around them (Rosa and Orey 2019). In this regard, Lett (1996) emphasizes that emic approach matches shared perceptions that portray the features of a specific cultural group, which are in accordance with understandings deemed appropriate by the insiders’ culture. In this context, Sue and Sue (2003) affirm that this approach is known as culturally specific. The emic-etic (glocal, dialogic) approach represents a continuous interaction between the globalization (etic) and localization (emic) that offers a perspective in which both approaches develop elements of valuable perspectives related to the same phenomenon. For Rosa and Orey (2019), it is a blending, mixing, a give and take by all participants, and adaptation of the two approaches in which one component addresses; indeed, it involves the voices of the members of local cultures, systems of values, and daily practices. In this regard, D’Ambrosio (2006) states that the intense cultural dynamics caused by interactions between localization and globalization may produce innovative ways of thinking and reasoning and solve societal issues and problems concerning politic, economic, health, and environment. Similarly, Rosa and Orey (2016) affirm that this vivid encounter between cultures provokes the emergence of glocalized societies (Glocalization is the ability of a culture, when it encounters other

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cultures, to absorb influences that naturally fit into and can enrich that culture, to resist those things that are truly alien, and to compartmentalize those things that, while different, can nevertheless be enjoyed and celebrated as different (Rosa and Orey 2016)) in which members of distinct cultural groups develop active interactional processes that are in an ongoing negotiation between the local and the global mathematical, scientific, and technological knowledges in a dialogic manner through the development of cultural dynamism. In these societies, by focusing on emic (local) approach and then building on it to integrate etic (global) approach, it is possible to develop mathematical ideas, procedures, and practices that are rooted in local traditions and contexts, but also equipped with a global knowledge that creates a sort of localized globalization. This dialogic construct shows us the importance of emic knowledge that is related to the insiders’ perspective that provides insights into cultural nuances and complexities (Rosa and Orey 2017a). This context enables us to perceive ethnomodeling as an ethnoscientific tool that concerns itself with modes of scientific and mathematical thinking used and defined by the standards developed by members of distinct cultural groups themselves. There is a need to legitimize, systematize, formalize, and value local scientific and mathematical knowledge so that people can reach ideas, procedures, and practices developed locally and globally. In this regard, members of distinct cultural groups have developed scientific knowledge traditions in order to help them to comprehend the process of reading and interpreting their own world, and they enable these members to understand, comprehend, and explain phenomena they face in everyday life (Rosa and Orey 2017b). Mathematics knowledge is perceived as a creation by the members of distinct cultural groups (ethno) who develop their own jargons, codes, symbols, myths, and specific ways of reasoning, inferring, and modeling by using methods, procedures, strategies, and techniques (tics) developed to solve problems theses members face in their daily lives by applying local categories of analysis (mathema) (Rosa 2010). Therefore, it is not possible to conceive mathematics as a universal language because its principles, assumptions, and foundations are not always the same in the world (Rosa and Orey 2007). In this regard, the “choice among equivalent systems of representation can only be founded on considerations of simplicity, for no other consideration can adjudicate between equivalent systems that univocally designate reality” (Craig 1998, p. 540). This means that the processes of production of mathematical ideas, notions, procedures, and practices operate in the register of the interpretative singularities regarding the possibilities for the symbolic construction of the mathematical knowledge developed by the members of distinct cultural groups. Hence, cultural specificity may be better understood as the comprehension found in the background of contextualized methods independent of the subjectivity of the observers (Rosa and Orey 2019). In accordance to this context, the aim of this theoretical chapter is to share our understanding of ethnomodeling by discussing our concern for voices that have been silenced by colonialism and, at the same time, as we learn to adapt the three cultural

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approaches when developing investigations that seek to connect ethnomathematics and mathematical modeling. Our primary argument is that ethnomodeling has been shown to create a firm foundation based on ethnoscience that allows for the integration of emic, etic, and dialogic approaches in exploring both scientific and mathematical knowledge developed by the members of distinct cultural groups.

Ethnoscience as the Relation Between Humanity and Its Sociocultural Context Ethnoscience is rooted in scientific proposals elaborated at the end of the nineteenth century that sought to record and catalog a wide variety of plants and animals used by the members of distinct cultural groups. Early ethnoscience texts are probably a consequence of the strong relation between scientific knowledge and the fields of natural science (Barrau 1985). Ethnoscience is established as a multidisciplinary research field based on anthropological studies related to the studies of the role of scientific systems of knowledge and logics developed by members of different cultures (Clement 1998). It is important to state that ethnoscience is related to the relation between humanity and its environment, and it has been focused on the comprehension of the knowledge of others. Hence, ethnoscience is the study of scientific phenomena in direct relation to the social, cultural, political, economic, and environmental contexts of the members of distinct cultures (D’Ambrosio 2001). It also “may be taken to refer to the system of knowledge and cognition typical of a given culture” (Crump 1990, p. 160). Hence, D’Ambrosio (2006) argues that scientific and mathematical knowledges are responses to the environment and to cultural encounters, which are mutually interdependent. Thus, there is a symbiotic relation between these two knowledge fields because members of distinct cultural groups can be represented by their own mathematical and scientific classifications, which may become a form of ethnoscience for this group, mainly by the development of particular ways of classifying their material, natural, and social contexts. Ethnoscience proposed new anthropological approaches whereby cultures are perceived as more than a collection of artifacts and as a set of behavioral norms and associated knowledge systems (including the mathematics and sciences used by the members of a given group). This perspective considers knowledge as a set of skills, abilities, and competences spread throughout generations that aims to discover principles that govern the norms of organized cultures in order to determine if they are universal (Brown 1999). In this approach, ethnoscience seeks to focus on the ideational aspects of cultures and nature by representing an important rupture with the materialistic approaches. For example, D’Ambrosio (2006) has argued that the ethnoscience is a body of knowledge that establishes explanation systems and diverse ways of knowing and doing developed by members of diverse cultures who have accumulated it over generations in specific cultural environments.

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In this regard, Rosa and Orey (2017b) affirm that ethnoscience is understood as a collection of ways that humanity has developed for explaining, understanding, comprehending, problem-solving, and interacting with unique cultural and natural environments. The members of distinct cultural groups have learned to produce their own forms of scientific knowledge in order to help them to resolve problems and interact with phenomena that occur in their own context. Historically, humanity has been exposed to diverse and often very different forms of scientific knowledge, such as mathematical and which have been produced, accumulated, and diffused in different cultural contexts. This greater body of knowledge called ethnoscience is producing and documenting how humans interact within their own given contexts. Through studying the specific motivations that were often modified and altered with colonization or trade throughout history, we gain a more nuanced understanding of what it is to be human (D’Ambrosio 2001). It is important to state here that, over time, ethnoscience has evolved a deeper understanding and comprehension of science by developing a respect for cyclical relations that occur through dialogue in order to facilitate the establishment of symmetrical relations that permeate the encounters between these individuals.

The Need for a More Culturally Bound Perspective on Mathematical Modeling When researchers investigate the diversity of knowledge forms and traditions that are possessed by members of diverse cultural groups, they may be able to find distinctive mathematical ideas, procedures, and practices. For example, D’Ambrosio (2006) has affirmed that the description of these non-western mathematical systems that have been retained and further developed, by all players, is the major focus, indeed basis, of cultural anthropology. As well, we argue that an outsider’s understanding of cultural traits is in danger of misinterpretation and/or ignored because of bias that may overemphasize inessential features of cultures and create misconceptions in relation to the mathematical knowledge developed by its members. It is also in danger of delegating the others to curiosities that are considered primitive, exotic, or less powerful forms of mathematical knowledge. We emphasize that the term cultural traits are used for simple behavior patterns that are transmitted by the members of distinct cultural groups and to which they give recognition and meaning. These cultural traits are learned sociocultural systems that consist of patterns of traditions, beliefs, values, norms, meanings, and symbols that are passed on from one generation to the next and are shared to varying degrees by the interaction between these members (Ting-Toomey and Chung 2005). The challenge arising from this approach is related to how we determine and understand culturally bound mathematical ideas, procedures, and practices without allowing the cultural and academic background of researchers influence the cultural background of the members of a cultural group under study. We point out that this may happen when members of distinct cultural groups share the interpretation of

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their own culture (emic) opposed to an outsider’s interpretation (etic) in their own words. This is why it becomes necessary for researchers to be in tune with the cultural nuances of the phenomenon they are investigating by identifying different ways in which members of distinct cultural groups experience, interpret, understand, perceive, or conceptualize mathematical ideas, procedures, and practices from the perspectives of the members of that specific culture. Mathematics in academia is a western construct, which is a compilation of progressive discoveries and inventions from cultures originating in the Mediterranean basin, and through a history that forms a mosaic of cultural contributions, these cultures colonized and spread power and knowledge worldwide. It is important to recognize that in this process the contributions of other cultures and the importance of the dynamics of cultural encounters (D’Ambrosio 2006) have equal validity. It goes without saying, but still must be emphasized here, that western mathematics and sciences are invaluable and contribute to the search for solutions to specific problems. At the same time, a more local perspective helps in the development of mathematical ideas that are imbedded in cultural contexts. Thus, the identification of specific problems rather than mathematical content enables interactions between cultural perspectives. In order to understand how mathematics (tics) is created, it is necessary to comprehend problems (mathema) that precipitate it by considering its cultural contexts (ethnos) that drive them. In this perspective, ethnomodeling is the process of formulation of problems that grow from real situations, which form an image or sense of an idealized version of the mathema (D’Ambrosio 2001). Consequently, mathematics cannot be conceived as a universal language because its principles and foundations are not always the same everywhere around the world. Conversely, it is naive to state that members of distinct cultural groups do not share universal mathematical characteristics, or communalities. For example, Bishop (1994) stated that many of the everyday activities of these members involve a substantial amount of mathematical application. In this regard, there are six universal activities practiced by the members of any cultural group. These activities are counting, measuring, designing, locating, explaining, and playing, and they provide the fundamental facets used to probe traditional daily living practices. These universals are inseparably intertwined with other aspects of the daily life of the members of any cultural group. Through a study of these applications, it is possible to understand the wonder of the members of distinct cultural groups and their early experiences using mathematical knowledge. However, even though these activities may be universal, it is important to recognize that they are merely general to those members who share the same cultural characteristics and historical perspectives. However, it is equally naive to believe that mathematical concepts do not reflect the distinct cultural values and lifestyles of the members of any given cultural group. Therefore, a better approach to these opposing, yet complementary, views may be to understand the universality of mathematical ideas, notions, procedures, and practices, which are relevant to researchers, educators, and teachers. It is also

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necessary to state that these approaches take into consideration the relationship between cultural norms, values, attitudes, and the manifestation of mathematical knowledge in different educational fields. The notion that members of diverse cultures have developed different ways of knowing/doing mathematics in order to increase understanding and comprehension in their own cultural, social, political, economic, and natural contexts is often controversial within the mathematics community. Thus, it is important to show how non-western populations developed their own unique and distinct ways to mathematize their surroundings and realities (Rosa and Orey 2010). Mathematization is the process by which members of distinct cultural groups interpret their own surroundings by applying their own mathematical knowledge, most notably universal abilities that all cultures and peoples use, such as counting, locating, measuring, designing, playing, classifying, patterning, gaming, quantifying, explaining, reasoning, problem-solving, and modeling (Bishop 1988; D’Ambrosio 1985; Rosa and Orey 2007). All human beings have developed successful and specific mathematical activities that have allowed them to organize, analyze, comprehend, understand, and solve problems faced in their daily life and their unique historical/political contexts. These activities enable members of distinct cultures to apply unique procedures and techniques developed in diverse cultural contexts in order to schematize, formulate, and visualize problems in distinct ways, as well as to discover relations and regularities to translate real-world problems through mathematization. In this process, mathematical ideas, procedures, and practices developed by the members of distinct cultural groups are the results of experience that uses sophisticated schemes of observation, experimentation, visualization, and formulation of mental ethnomodels that help them to conceptualize patterns and create artifacts. Ethnomodels are considered as small units of information rooted in sociocultural contexts. They are also representations of reality that help members of distinct cultural groups to interpret, understand, and comprehend daily problems and phenomena in order to survive and transcend (Rosa and Orey 2017b). In other words, D’Ambrosio (2005) affirms that ethnomathematics deals with the concepts of reality and action as part of the advancement of schematizing, formulating, and visualizing processes, which are the bases of the development of different forms of knowledge developed in distinct contexts. This process of acquisition of knowledge develops this dialogic relation knowing/doing, which is propelled by consciousness and takes place in many dimensions. Rosa and Orey (2019) state that these actions for transcendence are always accompanied by the actions for survival, which have their effect on reality, creating new interpretations and uses of natural and artificial reality by modifying it through the elaboration of ethnomodels. In this context, ethnomodels are consistent representations of the knowledge socially constructed and shared by the members of distinct cultural groups. Thus, ethnomodels help to link the development of mathematical practices developed by members of different cultural groups with their cultural heritage (Rosa and Orey 2010). According to this context, in the ethnomodeling process, ethnomodels can be classified as emic, ethical, and dialogic.

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Emic ethnomodels are representations developed by the members of distinct cultural groups taken from their own reality as they are based on mathematical ideas, procedures, and practices rooted in their own cultural contexts, such as their own science, religion, clothing, ornaments, architecture, and lifestyles. Etic ethnomodels are elaborated according to the view of the external observers in relation to the systems taken from reality. In this regard, ethnomodelers use techniques to study mathematical practices developed by members of different cultural groups by using common definitions and metric categories. Dialogic ethnomodels are based on the shared understanding that complexity of mathematical phenomena is only verified in the context of cultural groups in which they are developed. In these ethnomodels, the emic approach seeks to understand a particular mathematical procedure based on the observation of the local internal dynamics, while the etic approach provides a cross-cultural understanding of these practices. These ethnomodels often depend on unique conceptions of space and time that are contextualized and culturally bound. This perspective has allowed us to justify the need for a culturally bound perspective on mathematical modeling process coupled with sources rooted on the theoretical basis of ethnomathematics through ethnomodeling, which is defined as the study of mathematical phenomena that adds cultural components to the mathematical modeling process.

Cultural and Cognitive Features of Ethnomodeling Similarly, Rosa and Orey (2013a) state that cultural components of mathematical knowledge range from coming to see mathematical practices as socially learned cultural traits, such as artifact, sociofacts, and mentifacts that are transmitted to the members of distinct cultural groups, to mathematical procedures viewed as a set of abstract symbolic systems with an internal logic that provides their mathematical structure. If the former is considered, then ethnomodeling is a process by which transmission takes place from one member to another and is central in order to elucidate the role of culture in the development of mathematical knowledge. We emphasize that the terms artifacts, mentifacts, and sociofacts are cultural traits introduced by biologist Julian Huxley (1887–1975) as the bases for a theory of culture. Sociofacts refer to objects that consist of interactions between members of cultural groups as well as describe interpersonal interactions and social structures. Mentifacts describe belief and behavioral systems, values, and ideas developed by these members. Artifacts are objects created by the members of distinct cultural groups that provide cultural clues and information about the culture of its creators and user (Huxley 1955). In this context, cultural traits are socially learned system of beliefs, values, traditions, symbols, and meanings that members of distinct cultural group acquire throughout history. Cultural traits identify these members because they are considered as deposits of knowledge, experiences, actions, cosmologies, attitudes, hierarchies,

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religion, notions of time, spatial and temporal relations, as well as concepts of universe and the diverse worldviews developed by members of distinct cultural groups from generation to generation (Samovar and Porter 2000). Cognitive aspects needed in this framework are related to decision-making processes by which members of distinct cultural groups either accept or reject ethnomodels as part of their own repertoire of local mathematical knowledge. If the latter is considered, then culture plays a far-reaching constructive role with respect to the development of mathematical practices in the ethnomodeling process that cannot be induced simply through its observation (Rosa and Orey 2013b). In this context, if mathematical knowledge consists of a set of abstract symbolic systems whose form is the consequence of an internal logic, then students may be able to learn specific instances of the usage of specific symbologies as well as to derive a cognatically based understanding of the internal logic of the mathematical symbology system developed by these members. This means that ethnomodeling also is concerned with the connection between cognition and culture. For example, D’Ambrosio (2005) states that “Cognitive abilities cannot be assessed outside their cultural contexts. Obviously, each individual has his/her own cognitive capacity. There are cognitive styles that must be recognized in different cultures, in an intercultural context, and also within the same culture, in an intracultural context” (p. 117). Thus, these mathematical cognitive processes are triggered through the development of “relations between individuals from a same culture (intracultural) and above all between individuals from different cultures (intercultural)” (p. 112). In this regard, Rosa and Orey (2015) argue that there are two ways in which we represent and make sense of mathematical phenomena present in our daily life through the elaboration of ethnomodels: (a) First, there is a level of cognition that members of distinct cultural groups share to varying degrees with the members of other cultures. This level may include cognitive modeling that these members may develop at a nonconscious level (mental ethnomodels) that serves to provide an internal organization of external mathematical phenomena that provide the basis upon which mathematical practices take place. (b) Second, there is a culturally constructed representation of external mathematical phenomena that provides an internal organization in which they arise through the formulation of abstract and conceptual structures that is not required to be consistent with the form and patterning prescribed by external observers. This context reveals that cultural constructs provide constructed realities. In this regard, D’Ambrosio (2005) states that it is necessary to recognize that cognitive and organizing practices are related to the historical, social, cultural, political, and environmental contexts in which these processes are developed and practiced by the members of distinct cultural groups.

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Ethnomodeling and the Cultural Aspects of Mathematics Investigators, philosophers, and anthropologists such as Ascher and Ascher (1997), D’Ambrosio (1985), Eglash et al. (2006), Gerdes (1991), Orey (2000), Urton (1997), and Zaslavsky (1973) have revealed in their investigations sophisticated mathematical ideas and procedures that include geometric principles in craft work, architectural concepts, and practices often unique in the activities developed by the members of local cultures. These mathematical practices are related to numeric relations found in measuring, calculations, games, divination, navigation, astronomy, and modeling, as well as in a wide variety of mathematical strategies and techniques used in the confection of cultural artifacts (Eglash et al. 2006). In this regard, ethnomodeling is a research area that responds to its surroundings, and it is culturally dependent because it is socially bounded. One of the goals of ethnomodeling is not to provide mathematical ideas, procedures, and practices developed in other cultures a western stamp of approval, but to value and recognize that they are, and always have been, just as valid in the overall human endowment of mathematics, sciences, and technologies (Rosa and Orey 2017a). Ethnomodeling privileges the organization and presentation of mathematical ideas and procedures developed by the members of distinct cultural groups in order to enable its communication and transmission through generations. The elaboration of ethnomodels that describes these systems is representations that help the members of these groups to understand and comprehend the world around them by using ethnomodels, which link their cultural heritage with the development of their mathematical ideas, procedures, and practices. In accordance to this context, Rosa and Orey (2017a) state that ethnomodeling is described as the intersection between cultural anthropology, ethnomathematics, and mathematical modeling (Fig. 1). In the ethnomodeling process, the intersection between mathematical modeling and ethnomathematics relates to the respect and the valorization of tacit knowledge (This knowledge is related to the ways in which members of distinct cultural groups appropriate mathematical knowledge relating them to their own experiences, beliefs, and cultural values) acquired by the members of distinct cultural groups and which enables us to access, translate, and assess problem situations faced daily as we elaborate ethnomodels in different contexts. From this perspective, Knijnik (1996) has stated that ethnomathematics is not considered merely as folklore, but as funds of knowledge (The funds of knowledge concept describes the historical accumulation of abilities, bodies of knowledge, assets, and cultural ways of interacting. Although these funds were demonstrated as culturally, socially, and cognitively complex, it was pointed out that educators were not using them as a resource to enhance their students’ academic progress. The results of the study conducted by Gonzalez, Moll, and Amanti (2005) provided rich examples of how to recognize particular funds of knowledge and apply them in school settings by elaborating curricular activities based on their lives) that must be rescued so that members of distinct cultural groups and their perspective knowledge are valued.

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Mathematical Modelling Validation Valuing and Respecting

Cultural Anthropology

Ethnomodelling

Dialogue

Ethnomathematics

Fig. 1 The intersection of three research fields described by ethnomodeling. (Source: Rosa and Orey (2017a, p. 36))

Local mathematical practices are interpreted and decoded in order to understand their internal coherence and their close connection with the practical world. Thus, ethnomodeling is a socioculturally bound construct that forms a basis for significant contributions of an ethnomathematical perspective in re-conceiving mathematics through innovative perspective for the modeling processes. When we look at how members of distinct cultural groups use their own (mathematical) knowledge and traditions to translate and solve problems faced in their own environments, local (emic) knowledge serves as an intersection between ethnomathematics and cultural anthropology. For example, Eglash et al. (2006) stated that cultural anthropology has always depended on acts of translation between emic and etic knowledge addressed to help the members of distinct cultural groups to understand specific mathematical practices developed in diverse contexts. In this context, Rosa and Orey (2017a) have affirmed that this translational process is conducted with the elaboration of emic, etic, and dialogic ethnomodels. Ethnomodeling research often applies the term translation to describe the process of modeling local systems into another mathematical knowledge systems, such as western academic mathematical representations (Rosa and Orey 2019). However, as with all translational processes, its success is always partial, and intentionality is one of the areas in which the process is particularly difficult to understand. Often local designs are merely analyzed from a western view such as the application of symmetry classifications from crystallography to local textile patterns (Eglash et al. 2006). It also attempts to establish relations between local mathematical conceptual framework and the mathematics embedded in the practices developed by the members of distinct cultural groups.

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This translational process is used to describe the development of modeling local (emic) cultural systems that may have western (etic) mathematical representations because the mathematical knowledge found in ethnomodeling arises from emic rather than etic origins. This means that ethnomodeling translates from local mathematical knowledge into the analogous knowledge forms found in distinct contexts. This is one way to approach ethnomodeling by applying emic and etic perspectives to express it and to represent it. In this case, this process is a matter of translation. Thus, ethnomodeling is used to help members of distinct cultural groups to translate mathematical ideas, procedures, and practices found in their own communities among diverse mathematical knowledge systems (Rosa and Orey 2017a). It is reasonable to expect that an ethnomathematical perspective applies modeling procedures to establish relations between local conceptual frameworks and the mathematical ideas embedded in global designs through translations (Eglash et al. 2006). For example, fourfold symmetry is a design theme used in many Native American cultures as an organizing principle for religion, society, and technology. It has emerged through native structures analogous to the Cartesian coordinate system that helps researchers to translate this mathematical practice among distinct cultural systems. However, it is important to emphasize that the epistemological basis of ethnomodeling is not restricted to methods of direct and/or literal translations of non-western mathematical practices into western traditions. This is because it is necessary to understand emic and etic approaches as a way to explain the validity of mathematical ideas, procedures, and practices from the insiders and/or outsiders’ points of view. In this case it is necessary to point out that western mathematics is not the only reference, but valid explications can also come from the insiders’ mathematical knowledge. For example, ethnomodeling consists of studies that highlight historic, cultural, and mathematical procedures, strategies, techniques, and practices like those found in First Nations peoples, Chinese, Hindu, and Islamic contexts. In this regard, the Chinese Chu Shih-Chieh triangle can be mapped onto Pascal’s triangle by a rotation of 90 degrees. According to Eglash et al. (2006), in some cases, the translation to western mathematics is direct and simple, such as with counting systems and calendars, while in other cases, the mathematical knowledge is embedded in complex processes such as iteration in beadwork and in Eulerian paths found in sand drawings.

Linkage Between Ethnomodeling and Ethnoscience Ethnomodeling is considered as a body of knowledge established as a system composed of explanations and different ways of doing (practices) and of knowing (theories) that characterize different cultures (Rosa and Orey 2017b). Among these systems, the growing body of knowledge derived from qualitative and quantitative practices that document how humans count, weigh, measure, draw, infer, classify,

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and model is important for the ongoing development of ethnomodeling. Because it focuses more on the knowledge produced by the others, the ethnoscience is a multidisciplinary field of study that investigates the many roles that knowledge systems and their construction of reality. Thus, the concept of the ethnoscience has influenced the development of ethnomodeling since it provided theoretical tools for the comprehension of interrelations between mathematical knowledge and the different cultural contexts, forms of cognition, and diverse social and spiritual practices. In this regard, ethnoscience and ethnomodeling possess a symbiotic relation because their implications take into account a variety of forces that have shaped scientific and mathematical modes of thought in the sense of looking into the generation, organization (both intellectual and social), and diffusion of knowledge (Rosa and Orey 2017b). Ethnomodeling deals with the concepts of reality and action, space and time, and the ways of comparing, classifying, explaining, generalizing, inferring, modeling, and, as part of every action, quantifying, measuring, and evaluating, which are the bases upon which all forms of knowledge, including the ethnoscience, are grounded (Rosa and Orey 2017a). Ethnomodeling relies on science and mathematics, which is a favorable argument to strengthen its theoretical basis. Indeed, the intense cultural dynamics caused by glocalization produces innovative mathematical and scientific thinking. Since ethnomodeling focuses on the study of mathematical ideas, notions, procedures, and practices produced and developed in the daily organization of the members of distinct cultural groups, it possesses characteristics that broaden the objectives of ethnoscience (Rosa and Orey 2017b). Implications of these two research fields are related to factors that have come to influence, shape, and model the scientific and mathematical thinking of humanity. Hence, ethnoscience and ethnomodeling can be considered research fields that are interdisciplinary because they interrelate the results from cognition, epistemology, history, and education (Rosa and Orey 2006). One objective of the theoretical character of science seeks to explain and understand the structures and modes of social and cultural life in its approximation to the theoretical and practical aspects using ethnoscience. The main objective of this approach recognizes and promotes different ways of knowing and doing that are developed by different cultures. Since ethnoscience is based on a scientific research paradigm that recognizes and values human cultural diversity, then it intersects with ethnomodeling because its studies are developed in relation to everyday practices developed by the members of distinct cultures. The articulation between ethnomodeling practices and ethnoscience allows for the investigation of objects and artifacts from the view of the members of distinct cultural groups. In this direction, studies related to these two research fields are also related to the anthropological, social, and cultural features developed by these members in their own sociocultural contexts. For example, one of the main objectives of ethnoscience is related to the promotion of a theoretical foundation capable of integrating different branches of the natural and social science with other scientific systems. Philosophically, these objectives serve as a link between these members in

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an attempt to clarify comprehension and mutual respect among the members of these groups. Corroborating with this perspective, ethnomodeling aims to stimulate broader reflections about the nature of mathematical knowledge in the scientific, cognitive, historical, social, and cultural spheres. This approach aims to understand the development of practical knowledge built over time by humanity (D’Ambrosio, 2006). Thus, there is a necessity for researchers, scientists, and ethnographers to catalog these knowledges by describing them in a way that members of distinct cultural groups understand, comprehend, and interpret them, yet they must be in agreement with the categories elaborated in accordance to the ethnoscience developed in each one of these cultures.

Cultural Components of Ethnomodels We argue that what is traditional in the sense of mathematical modeling does not always consider implications of many unique and diverse cultural aspects of human social systems. The cultural component in this process is critical because it accounts for and emphasizes the wide diversity of culture composed of a myriad of diverse and unique mathematical ideas, procedures, practices, and values that are incompatible with traditional one size fits all mathematical modeling process. It is important to recognize that mathematical knowledge, especially in regard to what is meant by a particular cultural component, varies widely. It ranges from viewing mathematical practices as socially learned and transmitted by members of distinct cultural groups to academic mathematical practices viewed as a set of abstract symbolic systems with an internal logic that provides them a defined structure. To further clarify this, we would like to revisit a particular and beloved ethnomodel.

Ethnomodeling of Landless Peoples’ Movement: Wood Cubing in Brazil The wood cubing method involves the calculation of the volume of a tree trunk; thus, cubing means to determine the volume of a given object by measuring it in cubic units. Performing calculations for wood cubing involves popular and scientific methods. In this context, Knijnik (2006) states that the wood cubing (cubagem da madeira) is a process associated with the sociocultural environment of the members of Landless Peoples’ Movement (Movimento dos Sem Terra – MST). Cubing wood is a traditional mathematical practice used by the members of this group to determine how many cubic meters of wood are needed in the construction of sheds, houses, and animal shelters. For example, Knijnik (1996) studied the elaboration of mathematical activities related to the determination of the volume of tree trunks with participants of this movement. It is important to state here that the emic knowledge related to the development of this method to determine the volume of a tree trunk was orally transmitted and

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shared by MST family members across generations. Thus, mathematical knowledge involved in these local methods is also related to productive activities that members of this cultural group performed in their daily routines. According to Knijnik (2006), cubing wood possesses the features and points of view of landless peasant culture. In this context, D’Ambrosio (2006) argued that the self-validation of these methods within agricultural communities and settlements results from the development of local agreements of signification that results from a long cumulative process of generation, intellectual and social organization, and diffusion of this knowledge.

An Ethnomodel of Wood Cubing The members of MST use their own practices to estimate the volume of wood in a tree trunk, which is called cubing. This practice was verbally diffused from generation to generation by the members of the group. The results of interviews conducted in the study done by Knijnik (2006) show that the members of this specific cultural group consider wood cubing as an important daily practice; this is because it consists of calculating how many cubic meters of wood there is in a tree trunk, in a forest, or in a truck load of lumber. For example, one of the MST members stated how he used a tree trunk found in the forest to explain the cubing process used to determine its volume by the following emic ethnomodel: To begin this process, I chose this point here in the middle of the log, because there it is thicker and here it is thinner [he was pointing out to the extremities of the tree trunk]. So, the point in the middle of the log gave us, more or less, its average. Now, I took this string and I turned it around this point. So, I folded it into four parts and then I measured it to see how many centimeters were there. There were 42 centimeters. Now, I took 42 and multiplied it by itself. Thus, 42 by 42 gave me 1764. Hence, I measured the length of the log, which is 1 meter and 50 centimeters. Now, I multiplied that length of the log by the number I had before, which is 1764. So, I multiplied 1764 by 1 and 50, which gave me 264600 cubic centimeters of wood. It is the same as doing side times side times length. (Knijnik 1996, p. 32–33)

Another group member explained his method of determining the volume of a tree trunk by stating that “The measurement process I know is almost the same, except that, I make the measure at the thin end of the tree trunk because at its thick end we will square the wood in the sawmill, and if you lose some wood, it will not disappear” (Knijnik 1996, p. 32–33). According to Murray (2012), these make use of jargons, which are considered as specific terminologies associated with a particular cultural group, field, or area of activity. In general, jargons are employed in distinct communicative contexts; therefore, they may not be well understood in other contexts related to distinct cultures, trades, professions, vernacular, and academic fields. Jargons are sometimes understood as a form of technical vocabularies that are distinguished from the official terminologies used in particular fields of activities (Polskaya 2011). Thus, jargons are considered as technical terminologies, linguistic features, and characteristic idioms of special activities or cultural groups.

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It is important to state that a main driving force in the creation of a particular jargon is its precision and efficiency of communication that enriches everyday vocabulary with significant contents and meanings (Wodak 1989) for the members of distinct cultural groups. In this regard, in the emic approach, information and observations are constructed to reflect as far as possible the target population’s own vocabulary, linguistic terms, scientific and mathematical knowledge, conceptual categories, language of expression, and cultural belief systems. This approach contrasts with the etic approach that refers to information collected in terms of the conceptual system, categories, and linguistic terminologies of the external observers. To collect emic data, it is usually necessary to use the local language or dialect and gather information in a very open-ended, nondirective way. For example, when we ask informants to group the food items in any groups they wish to or in any way that they happen to think of, referred to as the pile-sort technique, the resulting groups are emic categories. In accordance to this context, the results of the study conducted by Amorim et al. (2007) demonstrated that the cubing procedure used to calculate the volume of tree trunk is given by the following emic ethnomodel used by the members of this cultural group although it is presented in mathematical terms: (a) First, it is necessary to estimate the center point of the tree trunk, that is, the diameter is taken at half the length of the log.

(b) From this point, by using a string, the perimeter of the trunk (circumference) is determined.

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(c) Then, the string that is related to the perimeter that was previously determined is folded into four equal parts, which gives 2π r ¼ 4 sides or 2π r ¼ 4 s.

2πr ¼ 4s s¼

  2πr 4

s¼

  πr 2

(d) Then, the measure of the quarter of the string (circumference) is squared.

A¼

 2 π 2

(e) And the value of the quarter of the string (circumference) is multiplied by the height of the tree trunk in order to obtain the volume in cubic meters (m3) of the wood. The volume is calculated as if the log was a cylinder. In the etic ethnomodel below, the members of this cultural group approximate the truncated cone (tree trunk) to a cylinder. This approximation is given as perimeter by determining the average between the perimeters of the smallest and the largest bases of the tree trunk.

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The minor difference at the top of the tree trunk is compensated by the major difference at its bottom. By dividing the string into four parts and raising it to the square, the members of this distinct cultural group calculate the area of a square by transforming the circle into a square.

Although the perimeters are the same, the areas are different. Subsequently, the volume of a square prism is calculated by multiplying its area of the base by its height.

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The development of the dialogic ethnomodel shows that the volume calculated in this way is relatively accurate if the shape of the tree trunk approaches a cylinder. This context enables the members of this cultural group to develop a comparison between the volume of wood in the prism and in the truncated cone. This ethnomodel shows that this mathematical practice combines western mathematics (measurement and basic operations such as multiplication) and mathematical techniques developed locally. It also shows that there is no great gap between local (emic) procedures and western mathematics (etic) because they interact dialogically.

This method used to determine the volume of a tree trunk basically consists of two steps. In the first step, a tree trunk (essentially a cylinder) was identified through a mathematization process in which its circumference coincides with the middle part of the tree trunk. In the second step, a tree trunk (again a cylinder) was identified as a square prism whose side measurement is equal to a quarter of the perimeter of the cylinder base in this mathematization process. This method of cubing wood (cubagem) finds the volume of the trunk as the volume of a square prism whose side of the base was obtained by determining the fourth part of its circumference, which corresponds to the base of the cylinder, and was obtained through an ethnomodeling process, that is, as part of the elaboration of a dialogic ethnomodel of the tree trunk. In the dialogic approach of this particular mathematical practice, the emic observation sought to understand the mathematical practice of cubing wood from the perspective of the internal cultural dynamics of the members of this group and their relation with the environment in which they live. In the etic approach, we explain this mathematical practice through the understanding of more than one feature of this local knowledge. This particular type of mathematical knowledge developed by MST members consists of socially learned and transmitted mathematical practices, which are represented in the elaboration of ethnomodels taken from sociocultural systems. This process aims to translate procedures used in this mathematical practice for the understanding of those who have different cultural backgrounds, so that a comprehension and an explanation of this practice from the perspective of outsiders

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can be developed. Therefore, tacit procedures (emic knowledge) used in this particular mathematical practice have been shared to the members of MST through generations. Hence, D’Ambrosio (1985) stated that mathematical practices can be seen as socially learned and historically diffused from one generation to another between the members of groups.

The Dialogic (Emic-Etic) Approach in Ethnomodeling Research To recap, the relation between emics and etics is dynamic and neither is more significant than the other. An emic approach is developed when members of distinct cultural groups develop their own interpretation of their cultural group (emic) opposed to an outsider’s interpretation (etic) of this specific culture. We can develop a similar analogy to ethnomodeling because it is possible to state that the emic approach is about differences that make mathematical practices unique from an insider’s point of view. We argue that emic ethnomodels are grounded in what matters in the world of the members of distinct cultural groups in which that their mathematical reasoning is being modeled by investigating mathematical phenomena by means of their interrelationships and structures through the eyes of the people native to a specific cultural group. Etic ethnomodels represent how the modeler thinks the world works in the context of a person or group under study, through systems taken from etic modelers’ reality, while emic ethnomodels represent how people who live in such contexts think these systems work in their own reality. This approach plays an important role in ethnomodeling research, yet the emic approach should be also taken in consideration in this process because, in this perspective, the emic ethnomodels sharpen the question of what ethnomodels should include to serve cultural and practical goals in modeling investigations. Hence, ethnomodels can show how mathematical ideas and procedures are etic if they can be compared across cultures using common definitions and metrics, while the focus of the emic analysis of these aspects is emic if the mathematical concepts and practices are unique to a subset of cultures that are rooted on the diverse ways in which etic activities are carried out in a specific cultural setting. Usually, in these investigations, an emic analysis focuses on a single culture or an artifact and employs descriptive and qualitative methods to study mathematical ideas, procedures, and practices of interest. It also focuses on the study within a cultural group context in which the investigators attempt to develop research criteria relative to internal or logic characteristics of a given cultural system or context. A dialogic (emic-etic) approach includes the recognition of other epistemologies and the holistic nature of mathematical knowledge by combining ethnomathematics and mathematical modeling through ethnomodeling. In this context, Rosa and Orey (2019) have argued that “dialogic ethnomodels enable a translational process between emic and etic knowledge systems. In this cultural dynamism, these systems are used to describe, explain, understand, and comprehend knowledge generated, accumulated, transmitted, diffused, and internationalized by

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people from other cultures” (p. 16). Hence, an important goal of ethnomodeling investigations is the acknowledgment of the development of both emic and etic knowledge by the members of distinct cultural groups. Thus, we may invoke a notion of local vitality, which releases an unexpected and astonishing cultural power, reinforced by the advantage supplied by the continual full participation in the community simultaneously with the action in the glocal world in a cultural dynamism. It is important to emphasize that members of distinct cultural groups collaborate in providing information to assist in the elaboration of inventories of mathematical ideas, procedures, and practices occurring in a particular region. The mathematical practice of wood cubing shows an approximation between ethnomodeling and ethnoscience through integrating the use of the scientific method to research the knowledge found in diverse cultures and is an ancient concept used in various locations in the world. For example, Rosa and Orey (2017b) highlight that this connection can be strengthened by the development of knowledge gained and supported by cultural dynamics, which occurs when members of distinct cultural groups encounter, produce, generate, organize, disseminate, and institutionalize scientific and mathematical knowledge.

An Ethnomodeling Perspective in the Mathematics Curriculum Considering diverse educational fields of study, which approach to a mathematics curriculum must be applied in the schools? Should researchers, educators, and teachers be looked at the perspectives of culturally universal (etic, global) or culturally specific (emic, local)? These questions allow us to argue that some of these professionals believe in cultural universality, which focus on similarities and minimize cultural factors, and believe on cultural specificity that focus on cultural differences. Then, the question is whether it is necessary to understand cultural specificity that requires specific theoretical basis and concepts (emic) against the background of universal and generic theories and methods (etic). This means that these professionals must also take into account their own worldviews because if they become more mindful and self-aware of their own paradigms and values, then they can become more open to apply aspects of ethnomodeling in their pedagogical practices, which seeks the development of dialogic (cultural dynamism, glocal) perspective into the mathematics curriculum. This perspective may lead researchers, educators, and teachers to a clear decision between these two approaches. For example, Dossey (1992) argues that many of these professionals disagree in relation to the nature of mathematics by debating whether this subject is culturally bound (internalists) or culturally free (externalists). Internalists such as Bishop (1988) and D’Ambrosio (1985) believe that mathematics is a highly cultural product, which is developed as a result of various activities such as counting, locating, measuring, designing, playing, inferring, and modeling. Other mathematicians, such as Kline (1980), are externalists because they believe mathematics activity is

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a cultural-free activity. Thus, they do not believe in the connection between mathematics and culture. The results of the study conducted by Rosa (2010) show that a majority of educators and teachers possess an externalist view of mathematics, which means that they perceive mathematics as a cultural-free subject, while few of these professionals possess an internalist view of mathematics because they perceive mathematics as a cultural product. Therefore, one of the primary issues raised in mathematics education is concerned to the position of researchers, educators, and teachers in relation to the etic (culturally universal) and emic (culturally specific) approaches. For example, Rosa and Orey (2015) state that some of these professionals may operate from the etic position because they believe that mathematical ideas, concepts, procedures, and practices occur in the same way in every culture. Thus, they base their beliefs on western ideas in which the members of distinct cultural groups construct, develop, acquire, accumulate, and transmit the same kind of mathematical knowledge. In this regard, minimal modifications for the pedagogical practices of mathematics are required because these professionals consider mathematical knowledge universal and equally applicable across cultures. Therefore, if the assumption regarding the universal origin, process, and manifestation of mathematical knowledge is similar across cultures, then general guidelines and strategies for the pedagogical work would appear to be appropriate to apply in all cultural groups. According to Lonner and Berry (1986), from the nonuniversalistic viewpoint, distinctions can be made specifying a hypothetical construct as culture-specific and can be distinguished from the universal. The acquisition of mathematical knowledge is based on the applications of current mathematics curriculum (etics), which is assessed based on multiple instructional methodologies across various cultures. Researchers, educators, and teachers who take on an emic position believe that many factors such as cultural values, morals, and lifestyle come into play when mathematical ideas, notions, procedures, and practices are developed in regard to the cultural backgrounds of the members of distinct cultural groups. Since students come from different cultures, they have developed different ways of doing mathematics in order to understand and comprehend their own cultural, social, political, economic, and natural environments. These professionals understand that students may operate from an emic rather than an etic perspective (Rosa and Orey 2015). In this regard, Rosa (2010) argues that it is important that researchers, educators, and teachers acknowledge that lifestyles, cultural values, and different worldviews influence the development of students’ mathematical knowledge since its development arises from cultural contexts. This is one of the most important educational issues currently confronting these professionals because it is pointing out that worldwide current guidelines and standards for the development of mathematics curriculum and mathematical instruction are culturally bound. In an ethnomodeling perspective for the mathematics curriculum, educators can search for problems and phenomena taken from students’ or community’s reality in order to translate a deepened understanding of real-life situations through the

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elaboration of ethnomodels. This process enables students to take a position such as sociocultural, political, environmental, and economical in relation to the system under study. The main objective of this pedagogical action is to rehearse the established mathematical context that allows students to see the world as consisting of opportunities to employ mathematical knowledge that help them to make sense of any given situation through ethnomodeling (Rosa and Orey 2019). In this context, mathematical knowledge and understanding of the members of distinct cultural groups are combined with comprehension of Western mathematical knowledge systems, which may result in a dialogic (emic-etic, cultural dynamism, glocal) approach to mathematical education. An emic analysis of mathematical phenomena is based on the internal structural or functional elements of a particular cultural group. An etic analysis is based on predetermined general concepts external to the members of that cultural group (Lovelace 1984). The emic perspective provides internal conceptions and perceptions of mathematical ideas, procedures, and practices developed by these members, while the etic perspective provides the framework for determining the effects of those beliefs on the development of the mathematical knowledge. A dialogic approach includes the recognition of other epistemologies and of the holistic and integrated nature of the mathematical knowledge of the member of any given cultural group. An ethnomodeling curriculum provides an ideological basis for learning with and from the people. This curriculum that combines key elements of local knowledge with a dialogic (emic-etic) perspective is likely to produce students who can manage knowledge and information systems taken from their own reality. According to McNeil (1985), the “essence of confluent education is the integration of an affective domain (emotions, attitudes, values) with the cognitive domain (intellectual knowledge and abilities)” (p. 11). The elements of this curriculum are essential in the process of designing an emic-etic training approach that deals with integration, participation, relevance, and self as objects of learning (McNeil 1985). These elements and the nature of the previous knowledge of the students lend themselves to the principle of sequencing in curriculum development. Starting with the students’ previous knowledge, educators can move from the familiar to the unfamiliar, from the concrete to the abstract in the process of promoting the acquisition of mathematical knowledge. This dialogic perspective of ethnomodeling provides the underlying philosophy of knowledge generation and exchange within and between all subsystems of mathematics education. Key elements of this curriculum ensure the balanced integration of the affective domain of educational objectives essential to recognition and utilization of the students’ tacit knowledge. In an ethnomodeling curriculum, Rosa and Orey (2013a) state about the importance of the interaction of emic and etic knowledge. This curriculum is dynamic and continually influenced by internal creativities and experimentations as well as by contact with other external knowledge systems. The contact of local knowledge with other external knowledge systems provokes cultural dynamism that enables members of distinct cultural groups to describe, explain, understand, and comprehend the knowledge generated, accumulated, transmitted, and diffused, internationalized, and

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globalized by people from other cultures. Regarding this approach D’Ambrosio (2005) states that: (. . .) involves processes such as generation and production of knowledge, intellectual organization, social organization, and diffusion, which are usually treated in isolation, as specific disciplines such as cognitive sciences (generation of knowledge), epistemology (intellectual organization of knowledge), history, politics and education (social organization, institutionalization and diffusion of knowledge). (p. 104)

According to this context, we emphasize that current mathematics curriculum lacks a dialogic approach to prepare students for living in glocalized societies. The lack of awareness of local knowledge system, the top-down approach of traditional transfer extension models, and the alienating effects of western education indicate a need for a confluent mathematics curriculum. Elements of this curriculum include participation, integration, and relevance, which are appropriate for synthesizing local and global knowledge systems. The development process of ethnomodeling interacts with local and global knowledge by incorporating them into the mathematics curriculum in a dialogic fashion. Conclusions are based on determining whether local and global knowledge would contribute to solve existing problems and achieving the intended objectives. However, a careful amalgamation of the combination of emic and etic knowledge would be most promising, leaving the choice, the rate, and the degree of adoption and adaptation of the dialogic perspective of ethnomodeling to the members of distinct cultural groups.

Conclusions Like all human beings, researchers have been enculturated to some particular worldview. In an increasingly glocalized world, it is necessary that mindful distinctions of phenomena derived by insiders and external observers be shared. Defining both the emics and etics of a given phenomenon, while using epistemological terms, provides a reliable means toward a deeper understanding of their complementarity. This must, and can be done, carefully and by relating to and respecting local contexts in order to support the usefulness in discussions in relation to emic (local) and etic (global) mathematical practices. Researchers who come from an emic perspective believe that factors such as cultural and linguistic backgrounds, social and moral values, and lifestyles come into play when they respectfully incorporate mathematical ideas, procedures, and practices developed by members of distinct cultural groups. This context enables the recognition of emic knowledge that is not interpretable in mathematical representations or to understand that etic knowledge has no priority over other mathematical ideas, procedures, and practices. Thus, it is necessary to create collaborations between academics (etic) and communities (emic) and putting to light the articulation of the principles and priorities emanating from both sides.

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In this context, an emic approach provides diverse perceptions and alternative conceptions of common mathematical ideas and procedures. Etic approaches propitiate frameworks for determining the predominance of Eurocentric beliefs on the development of mathematical knowledge. The rationale behind the emic-etic dialogue is the argument that mathematical phenomena in their full complexity can only be understood within the context of the culture in which they occur. A combined emic-etic (dialogic) approach requires researchers to attain the emic (local) knowledge developed by members of cultural groups under study. This approach encourages them to put aside any perceived or unperceived cultural biases so that they may be able to become familiar with the cultural differences that are relevant to the members of these groups in diverse sociocultural contexts (Rosa and Orey 2016). It represents a continuous interaction between etic (globalization) and emic (localization) approaches, which offers a perspective that they are both elements of the same phenomenon through dialogue, which is inherent in the emic-etic approach related to the dynamic modification of the modeling process that strengthens an understanding of the ethnomodeling investigation for both the local (emic) and global (etic) communities through glocalization. Ethnomodeling supports the development of advanced mathematical ideas and procedures that show how powerful mathematical knowledge originated in diverse cultural contexts. According to Rosa and Orey (2007), this approach can help us to understand how we can decolonize mathematical knowledge and, most importantly, allows us to unpack ways in which sophisticated mathematical practices have been used across time and place by showing that ethnomathematics is not simplistic, folkloristic, nor primitivist translations to other mathematical knowledge systems. In this context, ethnomodeling is defined as the study of mathematical phenomena that adds cultural components to the mathematical modeling process that is supported by the ethnoscience research field. The development of this connection is conducted through the development of three cultural components used in the conduction of ethnomodeling investigations: emic, etic, and dialogic approaches. A basic tenet of ethnomodeling is that it works against colonialism, as it encourages us to be more mindful, as it values and respects sociocultural diversity of members of distinct cultural groups. And most importantly, it encourages a respectful dialogue between formal and informal scientific traditions. As an ethnoscientific approach that studies the connections between mathematics and science in the direct relation with the social, economic, political, environmental, and cultural backgrounds (Rosa and Orey 2017b), it allows us to reflect on our work, and the complementarity between emic and etic approaches must be present when conducting ethnomodeling investigations because both approaches are essential for a better understanding of human behaviors and mathematical knowledge. Those behaviors related to the development of mathematical knowledge due to a dialogic approach are related to the stability of the relations between emic and etic approaches through cultural dynamism. As researchers, we strive for a stronger sense of mindfulness between the insiders and outsiders’ worldview, perspectives, and paradigms, which is best accomplished through dialogic approaches that ethnomodeling successfully facilitates. There is a significant dialogic interface

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between ethnomodeling and ethnoscience, which should be encouraged and, when explored, leads to important interdisciplinary reflections as a consequence of diverse and distinct ways of reading and interpreting the world.

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Knijnik, G. (1996). Exclusão e resistência: educação matemática e legitimidade cultural [Exclusion and resistance: Mathematics education and cultural legitimacy]. Porto Alegre: Artes Médicas. Knijnik, G. (2006). Educação matemática, culturas e conhecimento na luta pela terra. Santa Cruz do Sul: Helga Haas. Lett, J. (1996). Emic-etic distinctions. In D. Levinson & M. Ember (Eds.), Encyclopedia of cultural anthropology (pp. 382–383). New York: Henry Holt and Company. Lonner, W. L., & Berry, J. W. (1986). Field methods in cross-cultural research. Beverly Hills: Sage publications. Lovelace, G. (1984). Cultural beliefs and the management of agro-ecosystems. In T. Rambo & P. E. Sajise (Eds.), An introduction to human ecology research on agricultural systems in South East Asia (pp. 194–205). Honolulu: East–West Centre. McNeil, J. D. (1985). Curriculum: A comprehensive introduction. Boston: Little Brown. Murray, N. (2012). Writing essays in English language and linguistics: Principles, tips and strategies for undergraduates. Cambridge: Cambridge University Press. Orey, D. C. (2000). The ethnomathematics of the Sioux tipi and cone. In H. Selin (Ed.), Mathematics across culture: The history of non-western mathematics (pp. 239–252). Dordrecht: Kluwer. Pike, K. L. (1967). Language in relation to a unified theory of the structure of human behaviour. The Hague: Mouton. Polskaya, S. (2011). Differentiating between various categories of special vocabulary (on the material of a professionals speech of English-speaking stock exchange brokers). In G. Raţă (Ed.), Academic days of Timişoara: Language education today (pp. 518–524). Newcastle: Cambridge Scholars Publishing. Rosa, M. (2010). A mixed method study to understand the perceptions of high school leaders about English language learners (ELL): The case of mathematics. College of Education. Sacramento: California State University. Rosa, M., & Orey, D. C. (2006). Abordagens atuais do programa etnomatemática: delinenando-se um caminho para a ação pedagógica [Current approaches in the ethnomathematics as a program: Delineating a path toward pedagogical action]. Bolema, 19(26), 19–48. Rosa, M., & Orey, D. C. (2007). Cultural assertions and challenges towards pedagogical action of an ethnomathematics program. For the Learning of Mathematics, 27(1), 10–16. Rosa, M., & Orey, D. C. (2010). Ethnomodelling: A pedagogical action for uncovering ethnomathematical practices. Journal of Mathematical Modelling and Application, 1(3), 58–67. Rosa, M., & Orey, D. C. (2013a). Ethnomodelling as a methodology for ethnomathematics. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 77–88). Cham: Springer. Rosa, M., & Orey, D. C. (2013b). Ethnomodelling as a research lens on ethnomathematics and modelling. In G. A. Stillman, G. Kaiser, W. Blum, & J. P. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice (pp. 117–127). Cham: Springer. Rosa, M., & Orey, D. C. (2015). Ethnomodelling as the mathematization of cultural practices. In G. A. Stillman, W. Blum, & G. Kaiser (Eds.), Mathematical modelling and applications: Crossing and researching boundaries in mathematics education (pp. 153–162). Cham: Springer. Rosa, M., & Orey, D. C. (2016). Humanizing mathematics through ethnomodelling. Journal of Humanistic Mathematics, 6(3), 3–22. Rosa, M., & Orey, D. C. (2017a). Etnomodelagem: a arte de traduzir práticas matemáticas locais [Ethnomodelling: The art of translating local mathematical practices]. São Paulo: Editora Livraria da Física. Rosa, M., & Orey, D. C. (2017b). Polysemic interactions of ethnomathematics: An overview. ETD: Educação Temática Digital, 19(3), 589–621. Rosa, M., & Orey, D. C. (2019). Ethnomodelling as the translation of diverse cultural mathematical practices. In B. Sriraman & B. (Eds.), Handbook of the mathematics of the arts and sciences (pp. 1–29). Cham: Springer Nature.

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Ethnomathematics in Education: The Need for Cultural Symmetry Tamsin Meaney, Tony Trinick, and Piata Allen

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics and Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cultural Symmetry: Blending Ethnomathematics into Mathematics Education for Indigenous Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cultural Symmetry Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wharenui/Meeting House/Longhouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orientation in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Waka Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Show-and-Tell Software for Enhancing the Teaching of Māori Language, Māori Knowledge, and Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Since it was first discussed, ethnomathematics has been promoted as a way of supporting Indigenous students to see that their heritage included mathematical ideas. However, the inclusion of ethnomathematics in formal mathematics education has been criticized as potentially reducing rather than improving Indigenous students’ possibilities to value their cultural traditions and practices for their own sake and to gain appropriate mathematical understandings. In this chapter, the cultural symmetry model is described and exemplified as a way of overcoming the issues previously identified with implementing ethnomathematics in mathematics education. The four examples focused on different Māori cultural traditions and practices, highlighting both student and teacher perspectives on the T. Meaney (*) Western Norway University of Applied Sciences, Bergen, Norway e-mail: [email protected] T. Trinick · P. Allen University of Auckland, Auckland, New Zealand e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_4

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implementation. In the conclusion, discussion of these examples provides information about both how earlier concerns were overcome by using the steps of the cultural symmetry model as reflection points for designing and implementing activities and also other issues that need to be taken into consideration. This chapter, therefore, provides insights into implementing ethnomathematics into formal mathematics education, especially for Indigenous students. Keywords

Ethnomathematics · Māori · Indigenous · Cultural traditions and practices · School mathematics

Introduction In this chapter, we describe how ethnomathematics has been integrated into school mathematics, outlining both the expectations of how it would support, in particular, Indigenous students and some of the concerns connected to this integration. We then describe the cultural symmetry model which was devised to overcome these issues. Finally, we provide examples from different mathematics learning situations to illustrate how the cultural symmetry model overcomes the concerns raised in earlier research about the use of ethnomathematics in school mathematics. Ethnomathematics is usually defined as being related to the mathematical ideas of a specific group, such as those who are nonliterate (Ascher and Ascher 1986), or vocational groups such as carpenters (Millroy 1992) or cardiovascular surgeons (Shockey 2006). D’Ambrosio, who is described as the “father of ethnomathematics” (Stillman and Balatti 2000), defined ethnomathematics “as ‘the art or technique’ (tics) of explaining, understanding, coping, with (mathema) the socio-cultural and natural (ethno) environment” (D’Ambrosio 1990, p. 22). For him, ethnomathematics, as part of mathematics education, can contribute to people becoming democratic citizens, whose individual rights for a fulfilling life should not overtake the rights of others; “thus both the social aspect and the cultural achievement are always present” (p. 23). Ethnomathematics research has always had a strong connection to education (Borba 1990), as it was often mathematics educators, rather than mathematicians, who investigated and described mathematics in the cultural practices of different groups. In Indigenous education, mathematics educators saw ethnomathematics as a way to connect aspects of culture to the mathematics their students were learning. For example, Tereshkina et al. (2015) described teacher education for an Indigenous group in northern Russia, which built on understanding the mathematics in artifacts and practices, as contributing to the valuing of the local culture. Taking a different perspective but still valuing the local culture, Ogunkunle and George (2015) used ethnomathematical ideas to support the development of secondary mathematics students’ traditional crafts skills. Nevertheless, one consequence of the majority of the research being undertaken by mathematics educators was that the inclusion of ethnomathematical practices in

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classrooms was often presumed to be about the acquisition of the standard mathematics curriculum (Pais 2011). For example, Gerdes (1996) viewed learning about ethnomathematical practices as an entry point for gaining a better understanding of school mathematics. An example of this is Shahbari and Daher’s (2020) research about how Islamic designs support students’ understandings of congruent triangles. Although students are likely to be familiar with the contexts of these tasks, the focus on school mathematics potentially “divorces the cultural practices from their context and trivializes and fragments them from their real meaning in context” (Stillman and Balatti 2000, p. 325) and maintains the hegemony of Western mathematics in schools. Therefore, if ethnomathematics is to be used in education as a resource and not as a tokenistic context, there is a need to reconsider its incorporation into education. In this chapter, we present the cultural symmetry model as a way to overcome some of the concerns, identified in earlier research, in implementing ethnomathematical practices into mathematics classrooms, particularly those with Indigenous students. Initially, we provide a rationale for incorporating cultural practices, connected to mathematics, into Indigenous classrooms, before outlining some concerns about this approach. We then describe the steps of the cultural symmetry model and provide examples of its use in Indigenous education. In the conclusion, we discuss some of the issues about incorporating the cultural symmetry model into designing and implementing mathematical activities in schools.

Ethnomathematics and Education From the end of the 1980s, ethnomathematics was adopted by mathematics educators as one of the ways of overcoming concerns about white, middle-class males, being positioned as those most likely to succeed in mathematics, with mathematics itself being situated as cultureless. Similar approaches which sought to overcome this view could be seen in other areas of educational research, including gender studies. For example, Mary Harris (1994) identified the mathematics in the everyday work of women, which had been largely ignored or misrepresented as requiring less mathematical thought than the work of mathematicians. This research sought to overcome the restrictions on who could be considered mathematicians, but not necessarily the view that mathematics was cultureless. In the nineteenth and the first half of the twentieth century, the mathematical activities of Indigenous groups were often discussed by anthropologists in disparaging ways (see Meaney and Evans 2012). Best (1907) described the numeration system of the “Neolithic Māori” and the “rudiments of modern science as observed in Māori usage” (p. 94), whereas Crawfurd (1863; cited in J. Harris, 1987) characterized Australian Aboriginal numbers as “the rudest numerals of the lowest savages of which we have any knowledge” (p. 30). These interpretations of Indigenous mathematical ideas were based on the belief in the supremacy of Western culture, which led researchers to produce the results that they expected to find (Bender and Heller 2006). For example, even when presented with a particular mathematical

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practice in a study of Aboriginal people in the 1890s, the anthropologist Haddon refused to believe his informants (J. Harris 1987). Harris suggested that Haddon was able to do this because the theory of evolution lent scientific respectability to racist beliefs, including the physical, cultural, and intellectual inferiority of Aboriginal people. Preconceptions about the type of knowledge that mathematics was and the perceived level of intellectual sophistication that Indigenous people could reach meant that Indigenous people were expected to have limited mathematical understandings, and this was what Western researchers set out to first identify and then describe (Pickles 2009). Taking the lead from these established “truths,” schooling has generally positioned Indigenous students as being “the other,” with their achievement in mathematics being viewed as unlikely to be on the same level as nonindigenous students (for a critique of this research from the Pacific region, see Meaney et al. 2008). To overcome these views, a resistance to deficit interpretation of Indigenous students began in the 1980s (Powell and Frankenstein 1997), through critiquing (neo)colonial prejudices (A. Bishop 1990) and wider Eurocentric approaches to mathematics (Joseph 1992, 1997) and mathematics education. Integrating ethnomathematics into school mathematics for Indigenous students has been situated as important in that it shows that other forms of mathematics exist (Gerdes 1985). This perspective drew on earlier work, such as that of Gay and Cole (1967) and Lancy (1978), where the education of Indigenous students in “Westernoriented” schools had been criticized and an alternative mathematics education based on Indigenous mathematics was promoted. Using the perspective of ethnomathematics, Gerdes (1985) identified situations in which mathematical elements existed in the daily life of Indigenous groups during the colonial occupation of Mozambique, but which were not recognized as such because of the colonizers’ belief in the superiority of Western mathematics. Gerdes (1986) set out to reconstruct or “unfreeze” Indigenous mathematical thinking which was “hidden” or “frozen” in traditional techniques, such as basket making, to stimulate awareness in Indigenous learners of the mathematics in their cultural practices. In these studies, ethnomathematics was advocated as a way of including aspects of Indigenous or cultural mathematics in school mathematics so that cultural and/or underachieving groups would be supported to engage, by recognizing that their culture had and continues to have used mathematical ideas. Yet, the aim to integrate ethnomathematics into mathematics education so that a broader range of people could consider their communities as using mathematics in their traditional knowledge was difficult to achieve for a number of reasons. These include that there is a long history of mathematics being considered cultureless, and when cultural practices lose their intrinsic value, they become a shell through which school mathematics is taught. Mathematics has been considered cultureless for a long time. This has led to it becoming so engrained in people’s views about the nature of mathematics that some mathematicians and mathematics educators have decried the introduction of cultural practices into mathematics lessons as misguided. For example, Cimen (2014) argued that mathematics is universal and absolute, with the structure and objects of

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mathematics, such as the Fibonacci pattern in pinecones, existing outside of human invention. From this perspective, he argued that the case for a cultural base of mathematics simply resulted in Western mathematics being given more status than “the other kinds of mathematics” that ethnomathematics identified. Thomas (1996), a mathematician, strongly argued that the contextualizing of mathematics, as was the case with ethnomathematics, needed to be resisted in order for school mathematics not to become a watered down version of mathematics. Similar concerns have been raised about ethnomathematics contributing to an apartheid system of mathematics education, where some groups of students are provided with a mathematics education that would reduce, rather than increase, the likelihood of leading to well-paid jobs or further education (Vithal and Skovsmose 1997). Thus, rather than ethnomathematics supporting Indigenous students and others to see themselves as mathematicians, there are concerns of further marginalization, through restricting access to the mathematics that counted as valuable in the dominant society. Difficulties also occurred when integrating ethnomathematical approaches into school mathematics if mathematics educators focused on the mathematics and not on the cultural practice. Consequently, when cultural practices were incorporated into mathematics lessons, they are often reduced to merely being the vehicle for transmitting (Western) mathematics. Pais (2011) suggested that although learners may engage in a range of activities, it is not until these activities are recognized as mathematics that they “become” valuable in classrooms. Therefore, labelling cultural practices as mathematics runs the risk that they are seen as having no intrinsic value in their own right (Roberts 1996). Consequently, although ethnomathematics has been touted as enriching understandings about mathematics by using contexts that Indigenous students are familiar with and which enable them to see themselves and their communities as mathematicians, concerns have been raised about how this integration could result in counterproductive outcomes. Even if Indigenous students do gain mathematical insights from interacting with familiar cultural practices, it may be that the intrinsic value of the cultural artifact is devalued if it is merely a vehicle for transmitting the mathematical ideas.

Cultural Symmetry: Blending Ethnomathematics into Mathematics Education for Indigenous Students To overcome the difficulties identified in integrating ethnomathematical approaches into school mathematics, we developed the cultural symmetry model. Building on the early work of Meaney (2002) who questioned whether ethnomathematics challenges the colonial structures imposed by the cultural imperialism of mathematics, Trinick et al. (2017) proposed a three-step approach to emphasize the sociocultural aspects of learning and teaching mathematics. The model includes recognizing the many threads—social, linguistic, cultural (knowledge and values), and mathematical—that are part of teachers’ decision-making when designing and implementing activities in their classroom.

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In this section, we elaborate on the stages in this model, before providing examples of how it has been used, and discuss these in relationship to overcoming the concerns raised in the previous section. As the model was originally developed from work done where te reo Māori, the Indigenous language of New Zealand/ Aotearoa, is used as the language of instruction, we provide examples of the framework’s three steps that draw on cultural traditions and practices from this context. The model is described as the cultural symmetry model because many things, such as language, cultural practices, and mathematics, must be balanced if mathematics education is to contribute to decolonalizing the education process. Symmetry is important in Indigenous communities in that “most designs produced by cultures throughout the world are symmetric” (Washburn 1986, p. 767). However, often, symmetrical patterns are considered just from a mathematics perspective (see Washburn 1986; Donnay and Donnay 1985), especially in school settings (Lipka et al. 2019). In contrast, Māori utilize aspects of symmetry in many cultural practices, because only if items are part of a pair—that is, with a partner on the complementary side—are things considered useful (Trinick et al. 2015). This can be seen in a variety of terms for pairs and in the dual system used for quantifying objects in Māori (Best 1906). In Māori architecture, there is also a preference for symmetry (Ascher 1991; Donnay and Donnay 1985; Hanson 1983; Meaney et al. 2008), which is reflected in the placing of an even numbers of rafters on either side of a roof on a traditional Māori meeting house, wharenui. This focus on symmetry is so predominant in Māori culture that it can be called an “organizing principle... in much of Māori stories, religion, social life, and economics” (Ascher 1991, p. 171). From this perspective, asymmetry is used to highlight a particular issue in that frequently very deliberate, very indistinct disruptions were made to the symmetry (Witehira 2013). Hanson (1983) believed disrupted symmetry in Māori art reflected tension from the real world, while Jackson (1972) proposed that symmetry was used to express resolution and unity. While it is not known for certain why Māori used design elements to disrupt the symmetry of bilateral structures, the consistent use of asymmetry demonstrates that it was and continues to be significant. The first step in the cultural symmetry model is to describe the cultural knowledge and identify the cultural values connected to the practices and artifacts under investigation, which is best done in collaboration with elders, who are knowledgeable about these practices and artifacts (see, e.g., Lipka et al. 2019). This step is included so that the intrinsic value of the practices or artifacts is front and center in any investigation, overcoming the issues raised by Pais (2011), Roberts (1996), and Stillman and Balatti (2000), among others. Ideally, the Indigenous language that would have been traditionally used to discuss this cultural knowledge is the most appropriate one for classroom discussion. Recognizing the importance of the Indigenous language in the cultural symmetry model acknowledges the close relationship between mathematical activity, language, and thought. This relationship has been discussed for some time, for example, in the work of Pixten et al. (1987) with the Navajo and of Cooke (1990) with Australian

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Aboriginal people. In New Zealand, Barton (2008); Meaney et al. (2008); and Trinick et al. (2016) have highlighted the importance of using mathematics education to revitalize and sustain the Māori language, which has been a repository of cultural knowledge about the world, built up over many thousands of years of observations and experience (Trinick et al. 2016). Culture is expressed through language, “take language away from the culture, and the culture loses its literature, its songs, its wisdom, ways of expressing kinships relations and so on” (Fishman 1991, p. 72). Language is also closely linked to issues of identity, “if we want to make sense of a community’s identity, we need to look at its language” (Crystal 2000, p. 39). To see oneself as a mathematician within a community also requires recognizing that the language of that community has the capability to discuss mathematics. If a community loses its language, it can lose much of its cultural identity, including for school students the possibility of seeing themselves as Indigenous mathematicians. Especially when an Indigenous language is in a tenuous state, there is a need to provide opportunities to sustain it, through, for example, using it in mathematics education. With many students and families in Māori-medium education being second-language learners of Māori, because of the long suppression of the language, maintaining the Māori language as part of valuing cultural practices is important in the teaching of any subject, including mathematics. Similar concerns arise in other Indigenous communities. For example, when developing a unit of work for Sámi students on the traditional tent, the lávvu, Fyhn et al. (2016) identified the terms for aspects of the lávvu that were necessary for the students to know in order to be able to talk about it appropriately but which were no longer in common usage. Although many Indigenous groups have lost their languages due to the imposition of Western colonization, working with the community to know how to respectfully discuss traditional artifacts and practices remains an important component of this first step. The second step of the cultural symmetry model is to examine the cultural practices and discuss them from a range of perspectives, of which mathematics would be one. In this way, the practices and artifacts have the possibility to be valued in multiple ways but with step one ensuring that cultural understandings have precedence. The second step can also be used to support students’ language development so that they increase their possibilities to talk about a range of topics in the Indigenous language, particularly important when the students are second-language learners of that language. Different approaches to exploring the cultural traditions and practices allow for a more nuanced understanding about how artifacts and practices come to be valued within a society and to problematize the colonialization of knowledge more generally. For many years in New Zealand educational discourse and in school curriculum, Māori culture was restricted to a recognition of visual elements, such as the signs, images, and iconography, that are immediately identifiable as representing the culture and books of Māori myths, often written by Europeans. The invisible sociocultural aspects—the values, the relationships, problem-solving processes, and knowledge—that assisted Māori with meaning-making were typically ignored

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(R. Bishop et al. 2007). This was based on dubious epistemological assumptions that presupposed a hierarchy of school subjects which valued mathematics and physics more highly than subjects such as the arts (Bleazby 2015) and, as noted earlier, the delegitimization of Indigenous knowledge as primitive and thus irrelevant to the school curriculum. Consequently, the second step of the cultural symmetry model provides opportunities for sociopolitical discussions between teachers and students, about the colonizing nature of Western mathematics (Bishop 1990). By viewing cultural practices from multiple perspectives, teachers and students can problematize assumptions about how contexts and knowledge are related, with the intention of decolonizing their education. The third step involves considering how mathematics can add value to cultural artifacts and practices, without detracting from the cultural understandings. In this way, the mathematical understandings, whether they are derived from Indigenous mathematical practices or from Western mathematics, should deepen and enrich the cultural meanings already present. For example, discussions of Māori land divisions using fractions provided parents and their children with insights into how they came to own their share of a particular block of land (Meaney et al. 2008). Similarly, Fyhn et al. (2017) provided an example where traditional braiding patterns were discussed from a variety of different perspectives. One of these was about how algebra provided insights into the weaving of a large number of threads, inspired by the traditional patterns, but which when enacted produced a new product. Lipka et al. (2019) showed how mathematical ways of describing practices connected to symmetry provided extra information about, for example, why some folding was more complex than another kind of folding. Nevertheless, cultural traditions and practices should be valued in their own right, first and foremost. In regard to understanding symmetry in Hopi ceramics, Zaslow (1986) investigated whether the description using mathematics could result in “improved organization, less ambiguity, and an ability to identify speculative correlations” (p. 234). However, his interest in doing this was without input from community members, resulting in his claims about the advantages of using mathematical ways to describe the patterns being problematic in regard to overcoming the colonization of Indigenous cultural practices and artifacts. It is Indigenous students who need to view the mathematics as adding value to Indigenous practices and artifacts, not mathematicians. Implementing the cultural symmetry model in mathematics classrooms is complex in that the aspects raised in each of the steps need to be considered simultaneously. Mathematical understandings can contribute to cultural understandings, but if they are merely presented as representations of Western mathematics, then the possibilities for using them to discuss Indigenous cultural artifacts and processes are likely to result in cultural imperialism (Bishop 1990). Instead, finding a balance between Indigenous cultural knowledge, including language, and mathematical cultural knowledge involves reflecting on the aspects highlighted in the cultural symmetry model.

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Cultural Symmetry Examples In the next sections, we provide examples of cultural practices that have been incorporated into mathematics education, the designs of which followed either explicitly or implicitly the cultural symmetry model. These examples provide insights into the possibilities that the cultural symmetry model can provide for especially Indigenous education. As is discussed in the conclusion, the examples highlight some of the challenges that occur when trying to use the cultural symmetry model in designing and implementing activities.

Wharenui/Meeting House/Longhouse The first example describes how the cultural practices in the Māori meeting house, known as a wharenui (see Trinick et al. 2017), could be integrated into a set of mathematics education activities using the cultural symmetry model. Wharenui are predominately rectangular, with a gabled roof and a front veranda. As with every tribal group, each Māori-medium school in Aotearoa/New Zealand has its own wharenui of some form. Wharenui are also in many English-medium schools, particularly secondary schools, and in higher institutes of learning, such as universities. These meeting houses are similar in design to the longhouse of other Indigenous groups such as the North America Iroquois (Kapches 1990) and the Siraya, the Indigenous community of Taiwan. Building traditions reflect important aspects of Indigenous peoples’ cultures, societies, geographies, environments, and spiritual beliefs (see, e.g., Hanson 1983) and have the potential for being integrated into discussions about school mathematics, if done respectfully. Step 1 of the cultural symmetry model is to highlight the cultural practices and artifacts in relationship to their significance to the community. This is important because wharenui represent an ancestor, of either gender, with their structure representing the body of that ancestor. The ridge beam (tāhuhu) represents the backbone, the rafters (heke) the ribs, and the barge boards (maihi) the arms. Wharenui are highly decorated with different components, incorporating a range of symmetrical patterns (see Fig. 1). Bilateral symmetry, the one most commonly used in wharenui designs, is where an axis of symmetry divides a shape into equal halves (Booker et al. 2010) and is a common component of school curricula. From a cultural perspective, the wharenui and the various symmetrical artifacts that adorn it generally represent a family’s links to an ancestor (Salmond 1978). Figure 1 shows the inside of Kahurautao, the ancestor of the Whānau-ā-Kahu (family of Kahurautao). Kōwhaiwhai are the red-, white-, and black-colored patterns found on the ridgepole or rafters in wharenui (see Figs. 2 and 3 for two common examples) and express important cultural values such as unity, genealogy, and family interconnectedness (Witehira 2013). The patterns differ from tribe to tribe, with many having kōwhaiwhai unique to their particular areas, defining the environment where the tribe exists. This, thus, has strong connections to an individual’s Māori identity.

200 Fig. 1 The outside and inside of the wharenui (meeting house) at Pahaoa Marae, belonging to one of the authors

Fig. 2 Pūhoro

Fig. 3 Pātiki

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Fig. 4 One of the traditional stylized carvings of Tene Waitere in Rauru at Hamburgisches Museum für Völkerkunde

Tukutuku panels sit between the posts (see Fig. 4) and are an integral part of the meeting house. As Jackson (1972) stated, there are often distinctions, based on who produces the different artifacts for the group whose wharenui it is, as “the house presents time past and present in a totality and a unity and it also effects a unity, through its symbolic design, among human events” (p. 64). As part of the first step of the cultural symmetry model, the stories and cultural knowledge that the wharenui represents need to be discussed first in mathematics classrooms, before other knowledge, so that there is a shared understanding that the cultural understandings are the basis for all other discussions. The second step is to discuss the different artifacts, such as the kōwhaiwhai and tukutuku, in a range of ways. For example, identification of the different design elements allows students to recognize their use in other Māori designs. Students and teachers can also discuss how Māori designs have been represented and used in mathematics education previously and how this could have resulted in a devaluing of the cultural knowledge that they represented. In the 1980s, with the first endeavors to make connections to Māori culture in mathematics, some of the symmetrical patterns within the wharenui were identified and used as examples of cultural mathematics (Knight 1984). However, a focus only on the patterns, without a connection to the cultural knowledge, came to be seen as inappropriate (Barton 1993). Some of the earlier dissatisfaction was from the focus on abstraction and de-contextualization, which enabled students to gain mathematical understandings without needing

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knowledge of the cultural significance of those patterns. Thus, there is a need to discuss the wharenui in different ways to consider how cultural practices should not be merely valued because of the mathematics that they illustrate. The third stage involves considering the patterns in the marae and the artifacts in it in ways where mathematics adds value to understanding the cultural knowledge. Although Donnay and Donnay (1985) were able to describe the symmetrical patterns in cultural artifacts, using the notation of crystallographers, they were not interested in determining how this way of describing the patterns could add value to the Indigenous culture. Yet, in school settings, making links to transformation concepts of translation, reflection, and rotation provides opportunities for reproducing the patterns in other media, such as a digital environment. Figure 5 shows an example of how students make sense of traditional kōwhaiwhai patterns using mathematical language so that the process could be shared with others.

Orientation in Space The next example draws on understandings about how Māori orientated themselves in space. Visuospatial skills have been part of mathematics education research, for many years (Clements 1998), and include knowing how to describe a position of someone or something in relationship to something else as well as how to get around

Fig. 5 Student example of transformations with traditional cultural patterns

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in the world—at first, from an individual’s own perception and then from a more abstract perspective, which includes maps and coordinates. Spatial thinking has often been assumed to be based on a natural, innate perception of the world (e.g., Piaget and Inhelder 1956). However, models and maps of spatial environments are sociocultural tools (Gauvain 1993), as is the language which describes spatial orientation. For example, Edmonds-Wathen (2011) noted that crosslinguistic research showed differences in the ways that specific groups of people communicated about space and location, to do with their spatial frame of reference, or the conceptual basis for determining where one thing is located in relation to another. Step one, in this example, involved investigating the linguistic and cultural elements, both generally and in relationship to local elements of specific areas. Māori used a variety of techniques to orientate themselves to the cardinal points— east, west, north, and south—and intermediate directions that are similar to the Western orientation system. These techniques included phenomena, such as the actions of the sun and wind and the positions of particular geographical landforms (Trinick 1999). One important spatial framework is from the shape of the North Island (Te Ika-ā-Māui—the fish of Māui) of Aotearoa/NZ. According to legend, one of the many great feats of Māui, a famous if mischievous hero, was to pull up a great fish from the depths of the ocean. This narrative is shared throughout the Pacific. This fish became Te Ika-ā-Māui (the fish of Māui—the North Island) (see Fig. 5). Te Upoko-o-te-Ika (the head of the fish) is in the south at Wellington, and Te Hiku-o-teIka (the tail of the fish) refers to Northland (also referred to as Murihiku) (Auckland Museum 2001). Figure 6 shows a sting ray superimposed onto a map of the North Island with its tail in the north and head in the south. The spatial ability of Māori to Fig. 6 Te Ika-ā-Māui (the fish of Māui)

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use their navigation of the island to form a mental map and recognize its resemblance to a fish is extraordinary. When viewed this way, the head of the fish is runga—up—and the tail of the fish is raro—down. This is the opposite of Western spatial conventions, where up is usually linked to north and down to south. From the Māori perspective, the North Island is a mental image, orientated to the shape of the fish. However, with the dominant frame of reference being that from Western culture, Te Ika-ā-Māui is presented now as having a north-up orientation (Fig. 6). Other directional terms refer to winds. Over time, if the winds come from a consistent direction, they have become directional terms. One of the most common is Te Hau-ā-uru, which refers to the west, as the prevailing wind for most of the year for the whole of Aotearoa comes from the west. However, other directional terms make use of localized winds. These form a shared understanding of wind terms among tribal groups, hapū or iwi, that become local direction markers. However, outside that group’s local area, the same term could refer to a different direction. For example, the term marangai variously means east wind, east, northeast, north, and north wind, depending on the winds in the areas where a particular hapū and iwi lived (Trinick 1999). Wind names have cultural significance to Māori, because they also connected to fishing and planting times. When a particular wind blew, it indicated that it was time to fish for a specific species. Knowledge of the land and sea breezes was important when fishing some distance from the shore (Trinick 1999). For example, on the east coast, a southeast breeze (māwake) blew fishermen offshore for several miles to desired fishing grounds, and in the evening, another sea breeze brought them back in (Pohatu in Trinick 1999). Māwake is also a wind term that occurs in the prayer, or karakia, used by the hero Māui (Anderson 1969) to assist him in catching his fish (Te Ika-ā-Māui) (Fig. 7). Understanding these local differences is important, so an implemented classroom activity involved the students investigating the origins of the spatial orientation terms for their area, in particular, the directional terms from their iwi and hapū, by interviewing and talking about spatial orientation with their elders. In doing so, the students learnt not just the directional language but also the legends and background knowledge that were linked to the different terms. Step 2 involves considering how these spatial phenomena could be discussed in other ways. For example, wharenui are situated on an east-west orientation. The back of the building is generally regarded as representing the ancestral past and the front the present and future. This arrangement is reinforced with the front of some houses facing east and to the sunrise, which is associated with renewal (McKay 2004). In an extension of the classroom activity in step 1, students were asked to create a map of their area from memory, adding significant cultural sites, place names, places of significance, and a scale (see Fig. 8). This was designed to support students to consider how they orientated themselves culturally. Consequently, they were asked not to add roads, railways, and so on, but to add other references that provided a sense of direction, such as winds. When the map creation was completed, students compared the differences and similarities between their maps with a topographical or

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Fig. 7 Wind compass as a spatial framework (developed by T. Trinick, in Trinick et al. 2015)

Fig. 8 Student showing her map minus roads, etc. (from Trinick et al. 2015)

a satellite map of the region. In discussing the outcome of the activity, the teacher reflected, “What students drew was highly influenced by cultural considerations. The most important things on their maps were the tribal landmarks.”

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Identifying how cultural considerations affect spatial orientation provides students with an understanding that Western conventions provide just one way to do this. It also allows them to consider how cultural practices have to do with spatial orientation that is connected to specific purposes such as fishing or planting. Step 3 is focused on how mathematics can add value to investigations of spatial orientation. For example, this can be done after students discuss the relationships between purposes and different ways to orientate themselves. Wayfinding is one example of spatial orientation as it involves being able to navigate in an environment; it is the cognitive element of navigation which guides an individual’s movement. An essential part of wayfinding is the development and use of a cognitive or mental map. To support the cognitive map (picture in the head) to navigate across large distances, groups such as Māori developed stories as mnemonic aids (e.g., the story of Poutini1). Traditional stories describe how an individual, generally a hero, goes from place to place, with the landmarks documented in the story. School mathematics can be used to support the students to replicate the journey without having to redo the days or weeks of walking that most of the traditional stories required of their heroes. In the implemented activity, the students found the GPS coordinates of major cultural sites/ landmarks by using Google Maps, for example, which give latitude and longitude. Students then calculate the direction from waypoint (cultural site) to waypoint. By looking at the topology of the landscapes through changing the type of map, students gain insights into the difficulties that might be encountered if the individual tried to walk only in a straight line. This can be reinforced by having them walk one or more sections of the route. After this activity was completed, the teacher reflected, “because their route from landmark to landmark was obstructed by objects such as tall trees, students found this activity difficult without the aid of electronic devices. One student had travelled most of the route before in a training run, so he was the best at finding his way.” Using school mathematics understandings about scale and locating oneself on a map can add value to traditional cultural knowledge as it provides background into the challenges experienced by their ancestors.

Waka Migration The third example used the context of traditional migration stories about Māori ancestors’ arrival in Aotearoa/New Zealand. The cultural symmetry model was used to re-examine assignments completed by Indigenous preservice students in a Māorimedium teacher education program from 2017 to 2019. The assignment required the preservice teachers to combine investigating cultural understandings about early migration stories with statistical inquiry (see Trinick and Meaney 2017; Meaney et al. 2019; Trinick and Meaney 2020). Although not all Indigenous groups have

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http://www.teara.govt.nz/en/pounamu-jade-or-greenstone/page-2

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similar migration stories, this example considers how statistical enquiry can be linked to cultural aspects and as such provides a different view of how mathematics can be integrated into the investigations and tensions that arise in the process. Almost all the preservice teachers in this Bachelor of Education (teaching) program are the descendants of families who were part of the extensive urban migration after the Second World War, when Māori shifted from socially isolated Māori-speaking communities into English-language-dominated urban areas and English-language-only schooling systems and workplaces (Spolsky 2005). Consequently, many of the younger generation including the preservice teachers either have lost their connection to their tribal roots or are struggling to reconnect. It was, thus, important to provide a context, such as early canoe migrations, that could act as a bridge between their cultural heritage and mathematics. Step 1 of the cultural symmetry model involved the preservice teachers investigating the linguistic and cultural aspects of canoe migrations as a group. Alongside tribal dialects, a person’s waka is a significant identity marker for Māori. Canoe traditions explain the origins of different Māori tribes and so provide authority and identity. They also define tribal boundaries and relationships. Genealogical links (whakapapa) back to the crew of founding canoes have served to establish the origins of tribes and define relationships with other tribes. For example, several tribes trace their origin to the Tainui canoe, while others such as Te Arawa take their name from their ancestral canoe. When identifying themselves on a marae (meeting house) outside their tribal area, people refer first and foremost to their waka. These traditions, therefore, “merge poetry and politics, history and myth, fact and legend” (Taonui 2006). The canoe (waka in Māori) traditions or stories describe the arrival in Aotearoa/New Zealand of Māori ancestors from a place, which is usually known as Hawaiki. The exact location of Hawaiki has been lost in the midst of time. With the advent of technology such as DNA mapping, it is clear that Māori migratory canoes came from different places in East Polynesia, specifically, Raiatea, Tahaa, Porapora, Tahiti, and some of the islands of the Cook Group (Underhill et al. 2001). The migration stories refer to the construction of canoes, conflicts before departure, voyaging at sea, landing, inland and coastal exploration, and the establishment of settlements in new regions (Orbell 1975). As such, these stories contain both fact and legend. The preservice teachers found the discussions about the migration stories to be valuable in that they helped them challenge European versions of the migration stories as being accidental and understanding that the waka migrations were huge achievements: I learnt that our people had the navigation sophistication to travel all over the Pacific. The journey to Aotearoa was planned and not by chance—I feel proud about our ancestors’ ability to do so.

The preservice teachers also commented on how this discussion contributed to their language awareness and learning, with one preservice teacher noting how it contributed to “being able to talk about things more effectively in te reo Māori.”

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In step 2, the cultural practices were discussed in a range of ways. However, the very pride that had been instilled in step 1 caused tensions when different perspectives were brought into the discussion. This was because these stories were traditionally considered a “taonga tuku iho,” a treasure handed down from the ancestors and thus not generally considered open to question and enquiry. This can be seen in the following response from one preservice teacher when considering how factual some aspects of their tribe’s stories were: Each tribe or hapu have their own pūrakau (stories), hitori (history) and kōrero (talk) pertaining to Mahuhu-ki-te-rangi. Who am I to question it? I can’t tell the people of Te Roroa that Rongomai is the captain because he is most likely according to the data, therefore their history is koretake (useless).

In this response, there is a questioning of how appropriate it is to use an enquiring gaze on the knowledge (Trinick and Meaney 2017). Although this is an issue that has sometimes impinged upon the revitalization of mātauranga Māori (Māori knowledge) in schooling, we anticipated that completing this assignment would allow for a discussion of these aspects and thus broaden the preservice teachers’ understandings about how they themselves could handle similar discussions in their future classrooms. This questioning also arose when differences in tribal stories became clear, as seen in the following quote from one preservice teacher: “The only thing I found challenging to tikanga (cultural practices) was the fact that each iwi (tribal group) has different ideas about nga Hekenga waka (waka migrations).” This also brought into the discussion issues to do with differences with Māori cultural traditions as well as with Western perspectives on the migration stories. However, when the preservice teachers were able to remain proud of their ancestors’ journeys, they were more willing to discuss the journeys from other perspectives, such as “Our ancestors were awesome navigators-using currents, winds speed to calculate direction and speed.” The final step of the cultural symmetry model required mathematics to add value to the cultural knowledge. Most preservice teachers identified measurement concepts as important in the construction of the waka, as well as in determining the length of the migration journeys. To do this, many preservice teachers used information from the earlier discussions. An example was as follows: I estimated the time for the voyage of my ancestral waka at 36.7 days from Raiatea to Aotearoa. This was based on contemporary waka voyagers by Hekenukumai Busby which took him 30 days to travel 3233.73 km from Aotearoa to Rarotonga, travelling around 146.98 km per day.

This response identified the relevant elements connected with waka migrations and then described how they were calculated. Another preservice teacher highlighted the measurement of the waka by discussing the use of the ethnomathematical practice of body measurements:

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Heoi, mā te kaihanga, me ōna whatu, me ōna wheako e whakaioro i ia o ngā kōko o te waka (It was the principal builder, who used his own body as the measuring tool to determine the dimensions of the waka).

A preservice teacher wrote, “the relationship between the shape and size of the waka determined how many people could go on board.” Yet, when the preservice teachers were uncertain about what was relevant mathematics to use, rather than adding value, the mathematics detracted from the waka migration stories. For example, one preservice teacher who surveyed their relatives to gain information about the ancestral journey wrote, “according to the data I collected, the journey ranged between 5 days to 76 days, so I chose somewhere in the middle.” Such a response indicated a lack of understanding about the knowledge that surveys can collect as well as how to make valid calculations of journey times, resulting in misunderstandings about these journeys being linked to the traditional stories. Inappropriate use of mathematics is disrespectful of the traditional stories because it decreases, rather than increases, understanding about these traditions and practices. In the teacher education setting, such responses became openings for discussions about how being respectful of traditional practices and artifacts demands that Western mathematical knowledge be used correctly and appropriately.

Show-and-Tell Software for Enhancing the Teaching of Ma¯ori Language, Ma¯ori Knowledge, and Mathematics While digital technology is not a distinct Māori cultural practice like the previous examples, it has been employed for more than two decades in Māori-medium schools. As Māori-medium schools are generally small and geographically or linguistically isolated (located in the English-language-dominant urban areas) and generally draw students from lower socioeconomic communities, they have often utilized digital technology. However, there is minimal research on the efficacy of digital tools to support the Māori-medium pāngarau (mathematics) context (Allen 2015; Christensen 2004; Meaney et al. 2012; Trinick 2015). There are even fewer studies examining how digital technology supports the acquisition of mathematics and Māori language and culture. Historically, mathematics education practices often default to those used in English-medium mathematics education and fail to address the unique linguistic challenges of the Māori-medium mathematics learning environment (Allen 2015; Murphy and Reid 2016; Tiakiwai and Tiakiwai 2010). Mathematics resources that are readily available in the Māori language and contain learning contexts that reflect the realities of Māori-medium students’ lives are mostly textbook-based and not digital. Yet, Māori-medium teachers have expressed a desire for Māori-medium students to have access to dynamic online content (Murphy and Reid 2016). The challenge for the Māori-medium mathematics sector is to gain access to digital learning resources that are culturally and linguistically appropriate, rather than

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defaulting to the international pool of English-language resources which are widely available. In this example, the cultural symmetry framework is used to examine the affordances of a show-and-tell software application (app) for developing Māori language, Māori cultural knowledge, and mathematical understanding. Show-andtell apps provide opportunities for users to capture their mathematical communication using multiple representations (text, video, audio, and written or drawn diagrams) (Williamson-Leadley and Ingram 2013) and are distinct from drill and practice or game-based apps that do not address L2 language development (Allen 2017). There are cost barriers associated with using drill and practice apps that have been translated into te reo Māori from English, and more often than not, the visuals are not contextualized and so do not reflect Māori culture and Māori identity (Trinick et al. 2016). In contrast, show-and-tell apps often have free versions, which allow students and teachers to capture their mathematical interactions and share these with others, providing opportunities to deepen mathematical understanding (Boaler 2006; Ball and Barzel 2018). Video presentations created with show-and-tell apps provide the opportunity to watch a presenter work through a problem, giving insights into their mathematical thinking processes (Larsen et al. 2018), language use, and cultural representations. This example illustrates how the steps of the cultural symmetry model do not need to be completed in a specific order. The different activities combine aspects of the individual steps, so at the end, all three steps have been included into the designing and implementing. In the first activity, second-language (L2) students of te reo Māori (Allen 2015; Allen 2017) were introduced to a show-and-tell app on tablets. Using the app, the students were encouraged to create presentations explaining their thinking in order to increase the need for utilizing the specialized language of pāngarau (mathematics). This included the fraction terms (quarter, half, thirds), comparison words (bigger, smaller), and Māori language structures used for negotiation (when working in pairs) and justification (when a solution was reached). In the cultural symmetry model, this specialized language was the cultural practice in focus, but rather than being something that was discussed as in step 1, as had been done in the previous examples, the language was introduced with the mathematical ideas. The mathematics added value to the language (step 3) by providing the context in which the language was to be used. The app alleviated the students’ linguistic challenge of communicating mathematically in te reo Māori and supported the students use of the language by capturing, revisiting, and revising their presentations. The students worked in pairs to create representations of unit fractions to show which fraction was a bigger portion of the whole. A screenshot of a student-created, show-and-tell presentation is provided in Fig. 9. Students used the grid feature in the app to create representations that could be compared easily and added symbolic notations by drawing on the screen. One of the key ideas in this activity is to show students that representations can have both a mathematical and cultural component, as well as being described with specialized language. As such, this is part of step 2 of

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Fig. 9 Show-and-tell app showing a comparison between unit fractions

the cultural symmetry model because it illustrated how the mathematical aspects of te reo Māori can be represented in a variety of ways. Figure 9 shows that the students captured the visual and symbolic representations as they were creating them. The recording also included any incorrect representations which were erased and redrawn and captured the student’s discussion as they worked on the problem. This gave the teacher the possibility to address any specific language issues, at a later time, without interrupting the fluency of the students’ interaction in the moment. At the end of the recording, the students concluded that the half was the “biggest” fraction because it represented a bigger portion of the whole, which was an area equivalent to 12 grid squares. The student work, in Fig. 9, exemplifies the affordances of the show-and-tell software to “show” the multiple representations created as part of the solution method and to “tell” what the problem was and how it was solved. One of these representations also provided opportunities to discuss similarities to the poutama or staircase design used in tukutuku panels (see Fig. 10), also discussed in the first example. In some tribal narratives, the poutama pattern represents the ascent of Tāne-o-tewānanga to the topmost realm in his quest for superior knowledge and religion (Moorfield 2020). It also symbolizes growth, aspirations, and the honoring of wisdom (Paama-Pengelly 2010). In this example, the student-designed representations of unit fractions (Fig. 9) provided an opportunity to discuss Māori design conventions and their related Māori cultural knowledge. This is an important part of step 1 of the cultural symmetry model and ties in with the focus on mathematicsspecialized language of te reo Māori. A Māori-medium teacher interviewed as part of this research project discussed correlations between mathematical ideas and the Māori world view: Kaiako 1: Ko tētahi āhuatanga o te ao ... ko tēnei mea te whakapapa. Ko tā te whakapapa, he tūhono i te tangata ki tōna ao. Ki ngā tāngata kei ōna taha, ki te ao tūroa, ki te hītori, ki ngā wā o uki. Nō reira ko tēnei mea te whakapapa, he honohono kia mōhio ai te tangata ki ōna pānga. Ki tērā āhuatanga, ki tērā āhuatanga o te ao. He pērā anō te pāngarau. Ko te pānga

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Fig. 10 Poutama tukutuku panel hanging in Te Puna Wānanga, University of Auckland

maha, ko ngā pānga maha, ko ngā pānga rau i roto i taua kaupapa rā. Nō reira, he hononga tērā, ā-hinengaro nei o te ao Māori me ngā mahi pāngarau.

One defining characteristic of the [Māori] world . . . is this concept of whakapapa [genealogy]. Whakapapa connects people to their world. To the people that surround them, to the physical world, to their history, to ancestral times. Therefore, this concept of whakapapa is a conduit for people to know how they are related to the various phenomena of this world. Pāngarau (mathematics) is similar in this way. The interconnectedness, the many relations and connections within the discipline. Therefore, that is a correlation, an intellectual connection between the Māori world and mathematical pursuits.

In this example, the three aspects of the cultural symmetry framework provided opportunities to explore the interconnectedness of Māori design conventions, Māori cultural knowledge, and the mathematical representations created by Māori-medium students using show-and-tell apps.

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While the Māori-medium sector has been lobbying for adequate resourcing since its inception in the early 1980s, the sector also has a legacy of providing for its own needs, particularly if the state response is inadequate. The ongoing research into Māori-medium mathematics teaching and the use of digital technology could provide another example of Māori-medium education’s self-determining approach to mathematics resource creation, despite the state’s lackluster response to providing digital mathematics resources that address, language, culture, and mathematics simultaneously.

Conclusion We argue that sociocultural aspects need to be considered in the learning and teaching of mathematics, especially when considering how ethnomathematical activities are included in mathematics education for Indigenous students. In this chapter, we have discussed how the cultural and linguistic elements should be honored in the teaching of mathematics because these are fundamental educational goals for many colonized Indigenous groups. For these groups, the learning of mathematics is not an end in itself, but a vehicle and opportunity to support more holistic educational outcomes, such as the revival and maintenance of languages and cultural knowledge. Mathematics is a high-status subject in schools and tertiary institutes, in many parts of the world, because of its link to the economy. However, positive outcomes in mathematics do not necessarily lead to positive health and wellbeing outcomes for Indigenous communities; more is needed. The cultural symmetry model provides opportunities to support teachers to design and implement tasks which contributes to students forming bridges between mathematics and cultural traditions and practices. As such, it moves ethnomathematics beyond just identifying mathematical aspects within cultural practices and traditions. If the use of ethnomathematics is not integrated into mathematics lessons thoughtfully or is done in a tokenistic way, then there is a risk that the value to Indigenous students’ identities and cultural traditions and practices are trivialized (Pais 2011). Consequently, rather than decolonizing the role of mathematics in Indigenous students’ lives, it remains a colonizing force (Bishop 1990). In order to overcome these risks, the three steps of the cultural symmetry model highlight the value that cultural traditions and practices provide to the Indigenous community, as well as more generally. By highlighting the cultural value of the tradition or practice as the first step, a foundation is provided for considering the traditions and practices in other ways, which allows for critical reflections about some of these aspects in the second step. Mathematical practices can then also be seen as being part of sociocultural situations and contexts. The third step of the cultural symmetry model situates the mathematics, not as the purpose for introducing the cultural traditions and practices, but as something that can contribute a different perspective, which can increase the value of the cultural tradition or practice. The four examples in this chapter provide ideas about how to use the cultural symmetry model to design and implement mathematics education activities in

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Indigenous context. Although the examples all come from Māori settings, there may well be equivalent possibilities in other Indigenous cultures. We argue that this model is transferable for designing and implementing mathematics activities in other contexts. The first example was about using traditional patterns, found in wharenui. Māori patterns have been used as a context for discussing transformations since the 1980s (Knight 1984). However, the cultural symmetry model included possibilities for opening up discussions with students, about how and which knowledge comes to be valuable in order to overcome the presentation of Māori patterns as only being valuable if they are linked to mathematics. Nevertheless, when implementing these activities, finding a way to incorporate mathematics as adding value to the cultural practices as required by step 3 was difficult to do well. The students were able to write about transformations using te reo Māori, but a further task is needed to utilize these language skills so that the students can add value to the traditional Māori patterns and understandings about the wharenui through their descriptions of them. The second example used traditional practices to do with spatial orientation. The tasks provided opportunities for students to find out, from the elders in their iwi (tribes), relevant, local knowledge about spatial direction as well as how these connected with traditional practices such as food harvesting. Nevertheless, the activities did not contribute to sociopolitical discussions about the valuing of different views of knowledge. It would be possible to do this if it was integrated into discussions about what the students learnt from the elders. Although using mathematical ideas to explore the challenges of their ancestors’ wayfaring can add value to cultural stories, it would also be possible to open up the discussions to consider how representations could be used to reflect the purposes connected to different forms of spatial orientation. In this way, mathematics could be used to add a different kind of value to the cultural knowledge about spatial orientation. The third example about waka migrations did raise sociopolitical discussions but also brought out tensions about examining traditional cultural knowledge with an inquiring gaze. Such tensions are not usually raised in mathematics teacher education courses, and this lack could restrict preservice teachers from feeling comfortable about raising similar discussions in their own classrooms. Raising these issues challenged the preservice teachers to think about how different knowledge is valued between Māori tribes, which could lead to discussions about how Western mathematical knowledge has come to be accepted as that which is taught and learnt in school. However, the preservice teachers’ responses to the tasks also showed that although they had statistical knowledge, some of them struggled to apply it appropriately with regard to the migration stories. As a result, rather than adding value to the stories, the inappropriate use of statistical techniques could have lessened the cultural knowledge contained in the stories. Therefore, there is a need to ensure that step 3 results in learners having explicit discussions about what adding value means and how it could be achieved in relation to the specific traditions and practices. The final example included the use of digital technologies to support school students to develop their te reo Māori skills. The findings of a previous study showed

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that students who are L2 learners of the language of instruction struggled to fully “tell” their mathematical understanding of a concept when restricted to one representation or language mode (see Allen 2015). Thus, show-and-tell software with the possibilities for multiple representations provided opportunities to reinvigorate te reo Māori and to link those discussions to traditional patterns such as poutama. The next challenge is to find ways where cultural traditions and practices can be integrated from the start into digital technologies which will require funding as well as input from community elders to ensure that what is produced is done respectfully. The cultural symmetry model can support task designers and teachers to provide learners with tasks that challenge them to consider how to respectfully link cultural traditions and practices to mathematics learning. In so doing, we anticipate that the problems identified with earlier approaches to using ethnomathematics in school classrooms could be overcome. This would provide Indigenous learners with possibilities to support their cultural identities and languages as well as seeing themselves and their community as mathematicians. However, as the discussion of the four examples highlights, using the cultural symmetry model to design tasks does not in and of itself solve all the issues. Rather, it highlights the need to broaden considerations about engaging with cultural traditions and practices alongside Western mathematics, so that other issues do not detrimentally affect the cultural knowledge in the traditions and practices. Thus, the cultural symmetry model is a start but not the end in ensuring that ethnomathematics is integrated appropriately into school mathematics.

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Ethnomathematics Affirmed Through Cognitive Mathematics and Academic Achievement: Quality Mathematics Teaching and Learning Benefits Jenni L. Harding

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connection Between Learning Theories and Ethnomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connection Between Cognitive Mathematics and Ethnomathematics . . . . . . . . . . . . . . . . . . . . . . . . . Connections Between Pedagogy and Ethnomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connections Between Culturally Responsive Teaching and Ethnomathematics . . . . . . . . . . . . . . Connection Between Instruction and Ethnomathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instruction Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recognize and Honor Students’ Cultural Experiences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classroom Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Teaching and Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Establishing Cultural Experiences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concrete Real-World Ethnomathematics Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metacognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Math Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benefits of Ethnomathematics Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics Enhances Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics Elevates Guided Inquiry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics Emphasizes Pride in Cultural Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics Esteems Cultural Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics Empowers Engagement and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics Enriches Academic Achievement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics Obstacles Explained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ethnomathematics Philosophy Elucidated to Teachers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Ethnomathematics Expanded in the Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Abstract

Ethnomathematics is the intersection where mathematics and culture meet. It is a teaching philosophy that directs and guides mathematics learning practices within the classroom. Ethnomathematics pedagogy infuses real-life cultural knowledge students bring into the classroom with mathematics curriculum. Cognitive mathematics and academic achievement affirm ethnomathematics as an effective way for students to learn mathematics. Ethnomathematics establishes its connection to learning theories (gestalt, situated cognition, and constructivism), cognitive mathematics, pedagogy, and culturally responsive teaching. Utilizing an ethnomathematics instructional philosophy includes recognizing and honoring students’ cultural experiences, creating an effective classroom environment, implementing specific mathematics teaching and learning principles, establishing cultural classroom experiences, understanding concrete real-world ethnomathematics approaches, grasping metacognition, employing math groups, differentiating instruction, assessing what is valued, and implementing curriculum. Furthermore, how to support teachers in their move to ethnomathematics curriculum is expounded. Ethnomathematics benefits the classroom by improving student mathematics knowledge through focused conversations, productive mathematics struggles, guided inquiry, esteeming cultural knowledge, and dispositions of engagement and motivation. Ethnomathematics is worth the time, effort, and thought because it perpetuates mathematics learning within the classroom. Keywords

Ethnomathematics · Pedagogy · Culturally responsive teaching · Mathematics teaching and learning · Academic achievement · Cognitive mathematics · Guided inquiry

Introduction Ethnomathematics is the intersection where mathematics and culture meet. Teachers can use ethnomathematics as a pedagogical lens to link math learning with students’ lived experiences. Mathematics instruction then is embedded in context and focuses on real-world math learning that strengthens connections for students. When students learn through their cultural backgrounds, they form positive identities with math (Abdulrahim and Orosco 2019). Learning about mathematics from this pluralistic view: • Values cultural mathematics understanding. • Creates a sense of student curiosity (Kusuma et al. 2019).

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• “Open[s] new horizons for enhancing the quality and relevance of mathematics learning” (Jurdak 2016, p. 125). The purpose of this chapter is to recognize schools have shifted toward more diversity in terms of cultural, ethnic, and racial demographics where the teacher has a different cultural background from the students. Ethnomathematics may be implemented to bridge this cultural divide. When teachers adopt an ethnomathematics philosophy, they create classroom learning environments where culture is valued and real-world mathematics connections are established. Ethnomathematics is more than cultural activities or a one-time learning experience taught in the classroom. It is a teaching philosophy that directs and guides the learning practices within the classroom. Ethnomathematics connects the how, what, and why of mathematics, causing mathematics to be better understood as it is taken from the classroom context into the cultural context and vice versa. This wholistic type of learning renders the mathematics to be more than something in just a textbook but a real-world practice that comes alive in the students’ world. Students create deep connections to and understanding of mathematics concepts based on their knowledge and experiences. An example of this mathematics coming alive is explained through the following: Sensing the feel of the swell of the sea may be learned by lying in the hull as well as by paddling and being out on the canoe feeling the wind and noting the impact on the sail also helps generate embodied visuospatial reasoning. Thus, selecting the angle of a paddle, setting the position of the outrigger of a canoe, knowing the distance between places by the amount of time experienced by the body in moving between the places, assessing angles and slopes by gesturing with the hand, stretching out arms or parts of arms to assess lengths, will all be spatial decision-making times about objects in space, supported visually (Owens 2017, p. 215).

This ethnomathematics learning encompasses the senses, world, and mathematics present in order to place geometric concepts within perspective. This is the beauty of mathematics. Ethnomathematics as a teaching philosophy values complex understanding beyond the four walls of the classroom. This research chapter is organized from the macrolevel to the microlevel by defining the construct of ethnomathematics and then establishing its connection to learning theories, cognitive mathematics, pedagogy, and culturally responsive teaching. Then, ethnomathematics instruction gives attention to students’ cultural experiences, classroom environments, specific mathematics teaching and learning principles, cultural classroom experiences, ethnomathematics approaches, metacognition, math groups, differentiating, assessment, and curriculum. The chapter then focuses on the researched benefits of ethnomathematics teaching through communication, guided math inquiry, pride in culture/identity, valued cultural math knowledge, engagement and motivation dispositions, and improved academic achievement. Finally, the chapter concludes with obstacles to overcome for implementation of ethnomathematics into classroom learning, how to support teachers in this move, and future research.

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Ethnomathematics Defined To define ethnomathematics requires an understanding of the relationship between culture and mathematics. Culture is composed of a community’s shared ideations, symbols, behaviors, values, knowledge, and beliefs (Banks and Banks 1997). Therefore, culture is the manifestations of human intellectual achievement regarded collectively through customs, arts, and achievements. Culture is complex, dynamic, and ever-changing. Mathematics is a product of culture because it represents how that community approaches mathematics while thinking about their world (Bishop 1988; D’Ambrosio 1985). Orey (2017) concluded, “culture is essential to how we think, apply and use and even develop new forms of mathematics” (p. 334). Mathematics can be understood, accessed, and thought about in a multitude of ways through culture. Ethnomathematics is the study of mathematics based on cultural practices and different ways of knowing/thinking (Albanese et al. 2017). The term ethnomathematics was coined by the Brazilian educator and mathematician D’Ambrosio and in broad terms is the study of the relationship between mathematics and culture. His specific definition is using the word “ethnomathematics as modes, styles, and techniques (tics) of explanation, of understanding, and of coping with the natural and cultural environment (mathema) in distinct cultural systems (ethnos)” (D’Ambrosio 1999, p. 146). Taking these concepts and rearranging the aspects into an equation, “ethno + mathema + tics one gets ethnomathematics” (D’Ambrosio and Rosa 2017, p. 288). Further explained by Albanese et al. (2017), ethnomathematics is a bridge between the emic and etic elements. The emic perspective is geared toward studying the perspective of the culture and respecting different ways of knowing. The etic perspective brings together the culture and academic language. In the classroom context, D’Ambrosio (2006) characterizes ethnomathematics as “a pedagogical tool that helps teachers and students to understand both the influence that culture has on mathematics and how this influence results in diverse ways in which mathematics is used and communicated” (p. 287). Ethnomathematics allows one to learn about their own culture or other cultures through mathematics learning in the classroom. It provides social awareness, reinforces cultural respect, and demonstrates a cohesive view of cultures. Ethnomodeling is the next extension to this pedagogical approach establishing effective paths to reach mathematical concepts, develop intercultural classroom activities, and transform relationships between mathematics and society (Rosa and Orey 2011). Ethnomodeling connects the cultural and academic aspects of mathematics through “integrative, participative, relevant, and use of self or community as an object of learning” based on “ideas, notions, procedures, and mathematical practices developed by the members of distinct cultural groups” (Orey and Rosa 2015, p. 378). Based on a meta-analysis of the existing ethnomathematics, ethnomathematics can be defined as a “framework for conceptual understanding of culturally-related aspects of mathematics where understanding and exposure of the mathematics of different cultures are examined in order to bring value to the culture within an

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educational setting and extend mathematical concepts” (Harding et al. in press, p. 33). Therefore, ethnomathematics is a purposeful structure implemented into learning to analyze mathematics from multiple entry points where students gain profound math understanding.

Connection Between Learning Theories and Ethnomathematics The theories of learning portray how students receive, process, and retain knowledge. These are important theories for teachers because transfer of learning from one situation to another is the goal of teaching in the classroom (Woodworth 1950). Cognitive, emotional, and environmental influences, as well as prior experience, all play a part in how mathematics understanding is acquired or changed and knowledge and skills retained (Illeris 2017). Through transfer of learning, students apply their background knowledge to new situations which requires higher levels of cognitive thinking (Yang et al. 2013). Students need to understand math beyond its abstract concepts, memorized ideas, and isolated lessons through authentic activity and culture (Brown et al. 1989). Several specific learning theories support the teaching of ethnomathematics in the classroom. The gestalt theory describes learning as making sense of the relationship between what’s new and old (Boeree 2000). Students filter their individual learning experience through their unique lens built on the fusion of previous knowledge and new information. Gestalt views of learning have been incorporated into what have come to be labeled as cognitive theories. Two key assumptions underlie the cognitive approach: that the memory system (short-term and long-term memory) is an active organized internal processor of information and that prior knowledge plays an important role in learning. Once such cognitive theory is situated cognition that recognizes current learning as primarily the transfer of decontextualized and formal knowledge. Situated cognition is depicted as “shifting the focus from individual in environment to individual and environment” (Bredo 1994, p. 29). Therefore, the individual is no longer limited by their embodied mind, but their knowledge is distributed across people within their environmental experience (Pea 2004). This places individual cognition within the context of social interactions and culturally constructed meaning where knowing and doing become inseparable. Curricula framed by situated cognition can bring knowledge to life by embedding the learned material within the culture students are familiar with. For example, formal and abstract syntax of math problems can be transformed by placing a traditional math problem within a culturally contextualized practical story problem. Piaget’s constructivism theory emphasizes the importance of the active involvement of learners in constructing knowledge for themselves (Gardner 1981). Students use background knowledge and concepts to assist them in their acquisition of novel information to form an improved cognitive schema. To design effective learning environments based on constructivism theory, teachers need to understand what children already know when they come into the

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classroom even if that knowledge does not transfer directly to standard mathematics assessments. To maximize cognitive schema, the curriculum should be designed in a way that builds on the pupil’s background knowledge and is allowed to develop with them. Ethnomathematics creates a framework of learning from the gestalt, situated cognition, and constructivism theories. Within this framework, students make connections to their culture and lives outside of the classroom in order to understand the mathematics they are learning. The mathematics is then investigated and scrutinized through these unique approaches establishing a wealth of knowledge. This way of learning values the knowledge and experiences students bring with them into the classroom and allows them to connect mathematics academic learning to already established experiences.

Connection Between Cognitive Mathematics and Ethnomathematics Cognitive mathematics is a field of research seeking to answer questions about mathematical understanding through interdisciplinary fields of psychology, education, and neuroscience. The purpose of cognitive research is to harness “brain processes, cognitive systems and their development, and the formal and informal activities that individuals engage in when learning mathematics” (Gilmore et al. 2018, p. 1). Primary informant to this definition is the field of neuroscience and the concept of neural plasticity showing how experience changes the anatomy of the brain (Diamond and Amso 2008). One such experience changing the brain within cognitive mathematics is culture. Culture shapes higher-order thinking associated with cognitive control, attention, and working memory (Hedden et al. 2008), which are key activities in obtaining, retaining, and accessing mathematical knowledge. Beller and Jordan (2018) argued the cultural dimension is indispensable to any study of mathematical cognition. Further, Han and Northoff (2008) found through transcultural neuroimaging that culture shapes the functional anatomy of the brain in both high-level (i.e., social cognition) and low-level (i.e., perception) cognitive functions. Theory of the mind research demonstrates that within mathematical understanding, culture accounts for differences even more than linguistics (Kobayashi et al. 2007). One’s brain forms differently depending on the context of where and how learning occurs. The significance culture plays in brain development, particularly in mathematical cognition, creates a clear connection to ethnomathematics. Ethnomathematics allows the cognitive brain processes to be accessed naturally for authentic learning. This type of classroom learning reduces students’ cognitive load because math is accessed in the brain through cultural systems already established. Mathematics learning is then enriched based on the way the mind processes information.

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Connections Between Pedagogy and Ethnomathematics Pedagogy is made up of teaching moves, activities, and interactions among teachers and students that are designed to further student learning. Students are active creators of knowledge, while the teacher facilities purposeful instruction (Dewey 1967). Freire and Ramos (1970) argued pedagogy should treat the learner as a co-creator of knowledge who learns with real-life associations. Furthermore, these real-life associations can be grounded within cultural math learning contexts. Classroom learning goals and objectives are “modified and shaped by the structure of cultural activities and social interactions” (Leonard 2008, p. 23) established through curriculum and instruction. Ethnomathematics curriculum provides students and teachers a learning structure that makes mathematics meaningful by valuing alternative viewpoints, cultural diversity, natural language, mathematics, and visual representations to become aware of mathematical knowledge in their own and other cultures (D’Ambrosio and Rosa 2017). Underpinning this ethnomathematics pedagogy is the knowledge that all cultures engage in rigorous mathematics. Ethnomathematics does not privilege a certain culture’s mathematics intellect over another. Mathematics curriculum has historically “neglected the contributions made by minority groups and non-dominant cultures” (D’Ambrosio and Rosa 2017, p. 286). What ethnomathematics does is it utilizes cultural frameworks “that have existed since the beginning of time. . .to help educators discover pathways that foster student engagement through conceptualizing and supporting new approaches to learning mathematics” (Furuto 2014, p. 113). Viewing mathematics as a dynamic discipline allows teachers “to consider culture and context - daily customs, language, and ideology - as inseparable from the practice of learning mathematics” (Izmirli 2011, p. 40). Ethnomathematics pedagogy establishes curriculum and instruction where mathematics experiences inside and outside of the classroom are included for learning.

Connections Between Culturally Responsive Teaching and Ethnomathematics Each student brings their own unique cultural reference, worldview, and history into the classroom. Culturally responsive teaching is a: pedagogy that empowers students intellectually, socially, emotionally, and politically by using cultural referents to impart knowledge, skills, and attitudes. There cultural referents are not merely vehicles for bridging or explaining the dominant culture; they are aspects of the curriculum in their own right (Ladson-Billings 1994, p. 18).

A model of culturally relevant mathematics instruction (Gutstein et al. 1997) includes (a) building on students’ informal mathematics knowledge and cultural experiential knowledge, (b) developing tools of critical mathematical thinking and

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critical thinking about knowledge in general, and (c) orientations to students’ culture and experience. Ethnomathematics connects explicitly to culturally responsive teaching in the area of mathematics because it is an approach constructing cultural relevant mathematics.

Connection Between Instruction and Ethnomathematics Instruction Foundation Mathematics ability is a function of opportunity, experience, and effort where effective mathematics teaching and learning cultivate mathematics abilities of every student. The National Council of Teachers of Mathematics advocates excellent mathematics programs that require “all students have access to high quality mathematics curriculum, effective teaching and learning, high expectations, and the support and resources needed to maximize their learning potential” (NCTM 2014, p. 5). In order to accomplish effective instruction, teachers understand and use the social contexts, cultural backgrounds, and identities of students as resources to foster access and learning of mathematics. Ethnomathematics is an authentic way of creating high-quality instruction in order to help students learn math through concrete real-world approaches. There are the internal ethnomathematics (math a family does) and external ethnomathematics (math every culture does). Both of these facets need space in the classroom in order for students to feel like they do not have to park their family or cultural math knowledge at the door because it is valued and solicited during instruction. Ethnomathematics helps students access rigorous math instruction, develop highlevel academic skills, and connect the relevance between what students learn at school with their lives. To begin using ethnomathematics in the classroom, teachers just need to choose one area to shift at a time, in order for all students to learn through ethnomathematics practices. Teachers can establish an ethnomathematics philosophy and pedagogy by recognizing and honoring students’ cultural experiences, creating an effective classroom environment, implementing specific mathematics teaching and learning principles, establishing cultural classroom experiences, understanding concrete real-world ethnomathematics approaches, grasping metacognition, employing math groups, differentiating instruction, assessing what is valued, and implementing curriculum.

Recognize and Honor Students’ Cultural Experiences Some teachers view Black, indigenous, and people of color (BIPOC) students’ differences in cultural background and language as weaknesses. These beliefs and attitudes we have as teachers impact our instruction in the classroom. Ethnomathematics switches this view from what students lack to the benefits of experiences, culture, and diversity brought into the classroom. Teachers need to

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have the perspective of a growth mindset where they believe students can acquire mathematics through effort, good teaching, and persistence and not that students have a fixed mindset where their abilities, intelligence, and talents are fixed traits (Dweck 2006). Teachers need to have positive perspectives on students, their parents, and cultural heritages. Teachers don’t need to become an expert on all cultural groups and languages but understand students’ cultural identities in order to create relevant learning opportunities. Every student who enters the classroom should have the opportunity to become successful with mathematics. This means having high expectations with empathy and compassion. Teachers need to learn about every student in their classroom to identify students’ strengths and areas for growth. This includes recognizing student assets, interests, and experiences. Furthermore, mathematics instruction is based around students’ culture and communities in order to make mathematics connections in the classroom, understand historical contributions to mathematics, and celebrate community accomplishments. Teachers may then align mathematics instruction with cultural experiences of their students. Some examples include literature about those who represent the class through multicultural math picture books (MMPs), scaffolding material to reflect students’ knowledge, exposing students to a diverse group of mathematicians, and including relatable explanations or examples. All students are unique, and teachers must continually adjust instructional decisions to reflect the experiences and values of those they teach. There is not a one-size-fits-all curriculum; however, ethnomathematics creates a student-centered curriculum.

Classroom Environment It is the responsibility of the teacher to develop a socially, emotionally, and academically safe classroom environment. This safe environment includes students feeling respected, heard, and included in the classroom. Within ethnomathematics, teachers believe students bring a rich store of cultural and experiential knowledge, talents, and strengths that are used as the foundation for further learning. Students feel a sense of belonging in the classroom. A mathematics learning environment is established where students feel comfortable taking education risks by figuring out mathematics problems and sharing their thinking with their group or class. Students’ ideas and contributions are valued and encouraged in order to expand and situate it within personal examples. Students are kind to each other and encourage one another. Trust is developed with both students and their families. Mistakes are viewed as a normal part of learning and used as opportunities for growth. At the end of a lesson, teachers ask student reflection questions: describe a mistake that you or a classmate had in class today. What did you learn from this mistake? This demonstrates how mistakes will happen and are opportunities for learning within the classroom. Predictable structures and routines are established in order to give students consistency. These structures are clear, modeled, practiced, and expected.

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Communication during group problem-solving is an example where teachers can model the correct ways to discuss mathematics with sentence stems such as using “I agree with ____ because. . .,” “I disagree with____ because. . .,” “explain why/how,” or “my strategy is like/dislike yours because. . ..” It cannot be emphasized enough: model, model, and model how to talk to and treat one another in the classroom. In addition to modeling, students need multiple opportunities to practice mathematics communication in order to make it their own. An ethnomathematics learning community allows everyone to be themselves and be valued as a classroom family member where their differences are celebrated. Empowering students to take ownership of not just their own learning but the classroom environment itself is a critical component of ethnomathematics.

Teaching and Learning Ethnomathematics instruction is where the teacher creates learning opportunities in the classroom with an explicit use of students’ background knowledge connecting mathematics learning to cultural foundations. The role of the teacher moves from instructor to facilitator allowing students’ experiences, perspectives, and interests to shape the curriculum. The purpose is to teach math understanding where students are challenged to think and use their own knowledge to solve problems. Ethnomathematics takes the practical appearance of a productive math struggle (Hiebert and Grouws 2007) where students grapple with mathematical ideas in groups. This student-to-student and student-to-teacher problem-solving discourse leads to higher-level learning outcomes compared to math problems being solved individually (Barron 2003). Group learning structures encompass intersubjectivity where the shared perspective (thoughts, feelings, knowledge, and empathy) is constructed in the interactions allowing the interpretation of meaning in social and cultural life. This intersubjectivity takes on the dimension of students being known, valued, and cared for. Being valued creates motivation and has direct influences on cognitive learning (Schneider and Keenan 2015). Within this collaborative space is where different orientation and cultural ideas of mathematics are negotiated and scaffolded (Donato 1994; Verenikina 2008) enabling students to arrive at a new shared understanding. Teachers achieve their learning outcomes by focusing on the following: (a) Having students understand the “how” and “why” behind math concepts. (b) Spending time to look deeply at mathematics including examples and nonexamples as well as the connections across math concepts. (c) Taking abstract concepts and making them concrete by connecting them to the cultural and lived experiences of students. Teachers engage students in purposeful sharing of mathematical ideas, reasoning, and approaches.

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Mathematics discourse is valued where students explain ideas, reason, and generate representations. Listening carefully is key in order to extend and evaluate mathematics concepts while critiquing the reasoning of others through examples and counterexamples. Multiple approaches are encouraged and valued when students are solving and thinking about problems. Students ask clarifying questions, try out others’ strategies, and articulate different approaches. Students are expected to explain, clarify, and elaborate their thinking. These mathematics conversations get into the “mess” of mathematics where conjectures are examined, strategies are scrutinized, thinking is justified, perseverance in problem-solving abounds, and sense making is cherished. This ethnomathematics discourse allows students to contextualize mathematical ideas by understanding them and connecting them to cultural and other situations through examples, illustrations, and representations.

Establishing Cultural Experiences Ethnomathematics builds upon the cultural capital students bring into the classroom in order to enhance mathematics learning. Ukpokodu (2011) explains: “providing appropriate scaffolding through the use of familiar language, metaphors, examples, and hands-on learning, thereby tapping into the ‘funds’ of mathematical knowledge students bring to the mathematics classroom” (p. 54). These mathematical “funds” can be tapped into through the following: (a) Food and recipes (fractions, measurements, mathematics teaching connections, and shapes). (b) Diverse languages encouraged in the classroom (solving of math problems, learning of math vocabulary, and parents’ participation in math learning). (c) Cultural community connections (beads in hair connection to patterns, farm unit to teach area and perimeter, and money currency). Students’ cultures are embraced and used as mathematics examples within this collaborative learning community. It is important that the cultural learning brought into the classroom does not consist of stereotypes or inauthentic learning, but an appreciation of diversity including how different people view and interpret mathematics. Through ethnomathematics, students make personal connections to math content, and these connections are “most meaningful when they are connected to the child’s cultural background” (Harding-DeKam 2014, p. 17). Student’s mathematical learning becomes personal, and it is more than just a problem in a math textbook.

Concrete Real-World Ethnomathematics Approaches Ethnomathematics learning is brought into the classroom by creating personal connections and experiences to understand math concepts. Ethnomathematics

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approaches are accomplished through connecting math to everyday experiences, developing math thinking through cultural knowledge, and using community or cultural experiences to understand mathematics. First, ethnomathematics connects math to everyday experience. Davis et al. (2009) demonstrated ethnomathematics teaching when their students were studying the idea of slope in connection to the rate of change in college calculus. Students were given graphs of distance and asked to create stories to interpret these graphs from their experiences outside of school. This produced personal math connections to classroom math study of slope and rate of change. Barton (1996) mentioned using triple weaving and sports statistics as ethnomathematics learning within everyday experiences. Moses et al. (2009) bring in culturally familiar experiences to create a stronger math conceptual foundation through their algebra project by using African drums’ connection to ratios. Mathematics becomes meaningful when it is connected to students’ everyday experiences. Second, ethnomathematics develops math thinking through cultural knowledge. A third- and fifth-grade mathematics teacher who was teaching low-income, ethnic, and language-minority students used familiar Mexican money currency in her lesson (Civil 2007). Ladson-Billings (1995) provided examples of culturally relevant teaching and how academic achievement and cultural competence can be merged. Ms. Hilliard, an African American teacher, invited her second-grade students to bring samples of non-offensive rap song lyrics and used it as a bridge to school learning. Ms. Winston, a White teacher who has taught for 40 years, involved parents in her fifth-grade classroom by creating a person-in-residence program so students learned expertise from each other’s parents and established cultural knowledge. Ms. Lewis, a White sixth-grade teacher, encouraged her students to express themselves using their home language in order to understand the content. A group of African American middle school students were involved in community problemsolving activities and then participated in a social action curriculum. Nasir (2002) communicated how math thinking develops through cultural practices for African Americans through dominos (elementary and high school) and basketball (middle school and high school). Lesser and Wagler (2019) established how statistics can be taught using a dreidel (Jewish spinning top), toma todo (Mexican spinning top), and six-sided die. Bringing in cultural activities develops high-level thinking about mathematics. Third, ethnomathematics uses community or cultural experiences to understand mathematics. Barta and Brenner (2009) contend there is a connection between ethnomathematics and the community for math learning with examples of star navigation, fish recovery, cornrow hair braiding, and housing construction. Martin and McGee (2009) asserted ethnomathematics can be taught in an African American-centered pedagogy by connecting historical figures and everyday experiences with mathematics like Dogon people of Mali, urban planning, cryptography, and Java. Varghese and McCusker (2006) shared examples from India including fractions with oral story (poetry), Kolam (rice flower drawings) for geometry, and games. Knijnik (1993) worked in a rural school in the southernmost Brazilian state and used different methods of land area measurement and estimating the volume of a

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tree trunk to bring cultural knowledge into the classroom. By bringing these community or cultural experience into the math classroom, teachers show the mathematics that exists in the world is meaningful. Ethnomathematics’ concrete approaches to instruction provide students with entry points for accessing and understanding mathematics. This not only helps reinforce the students’ own cultural knowledge with mathematics but also exposes them to other cultures and their ways of thinking about mathematics. This kaleidoscope of ideas gives context to mathematics and allows for alternative strategies of mathematical thinking to be understood and examined creating a rich mathematics interpretation.

Metacognition Metacognition is the ability to think about thinking while reflecting on the process. Cognition is the mental action or process of acquiring knowledge and understanding through thought, experience, and senses. Meta is a prefix meaning more comprehensive. Therefore, metacognition encompasses memory monitoring and self-regulation. Metacognition in the mathematics classroom includes students understanding their own capabilities with content knowledge (declarative knowledge), evaluating the difficulty of a task (procedural knowledge), and using strategies to learn information (conditional knowledge). Because metacognition is a selffunction, students may not always practice with accuracy. Additionally, students often attribute their lack of effort with not understanding the content (Lai 2011). Metacognition facilitates more effective performance on many cognitive tasks (Metcalfe and Shimamura 1994). Students may use metacognition by asking themselves the following guiding questions: What do I already know about math, and how have I solved problems like this before? Ethnomathematics’ use of metacognition takes these questions one step further by having students ask the following: What cultural or math experiences have I had that can lend themselves to this learning, and what connections can I make from this math learning to my experiences? These questions support students in their mathematics learning by helping them make connections to what they already know and thus giving them an access to solve the mathematics problem.

Math Groups The selection of the mathematics groups in the classroom is purposeful. The groups consist of three to five students and are homogeneous (same ability levels) and heterogeneous (mixed ability levels). The groups are fluid and change frequently depending on the purpose within instruction. Homogeneous groups can be the same mathematics level or the same cultural groups of students. When groups are homogeneous, the teacher can support students’ learning at their ability level by pulling a group to work with the teacher in order to support individual needs and make explicit

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mathematics cultural connections. Homogeneous groups with students who have the same language can let them think about mathematics in their first language in order to understand it before translating it into English. The heterogeneous groups are mixed-ability level where students can support and learn from each other. Other types of groups you can use are gender groups (females and males in their own groups), interest groups (where students can delve deeper into mathematics understanding by building upon personal interests such as learning about angles by researching skateboard parks, 3D sculpture, billiards, or carpentry), choice groups (where students choose who to work with), and leaderships groups (putting all of the leaders in one group giving them the opportunity to work together and allowing students to rise as leaders in the other groups). Ethnomathematics establishes groups within mathematics learning for the purpose of arranging students to discuss mathematics content/concepts and for students to get to know each other culturally.

Differentiation The ethnomathematics intent is to differentiate math instruction so that each individual in the diverse learning classroom community can access cultural mathematics knowledge and participate at their specific level of acquiring content, processing, constructing, and making sense of ideas. This allows learning to happen for students at a variety of readiness levels, interests, and experiences. Differentiation can happen through content, process, product, and the learning environment (Tomlinson 1999). Ethnomathematics differentiates math content by allowing students to bring in the knowledge they have from previous experiences to build upon or by creating cultural classroom practices to form a foundation. Ethnomathematics differentiates the math process by allowing grouping to be flexible and by teaching different learning styles (audio, visual, kinesthetic) and multiple intelligences (linguistics, mathematical, musical, spatial, kinesthetic, naturalistic, interpersonal, and intrapersonal). The product or demonstration of knowledge can be given as a choice in order for the student to show their mastery of content (such as oral presentations including rap songs, multiple ways of solving a problem diagram with the strengths and limitations of each way, menus to choose how students would like to demonstrate their learning). Differentiation within ethnomathematics allows all students to learn math at a level they are capable.

Assessment Assessment evaluates what students understand with mathematics before, during, and after instruction. It provides diagnostic feedback about the students’ mathematics knowledge foundation, performance base, and what needs to be taught.

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Assessment allows teachers to set and achieve academic goals and outcomes within standards. It can be formal in terms of quizzes, tests, presentations, and rubric assignments or informal in terms of demonstrating what you know through manipulatives, group work, investigations, etc. Mathematics assessment includes four major areas: conceptual understanding (the “how” and “why” behind the concepts), procedural understanding (the computation behind the problem), mathematical processes standards (problem-solving, reasoning and proof, communication, connections, and representation where teachers guide students in understanding mathematics content through probing questions, clarifying ideas, emphasizing reasoning, making the mathematics visible, and encouraging justification and reflection (NCTM 2000)), and attitudes (how students feel about their ability to complete mathematics and about the subject area of mathematics itself). What a teacher chooses to grade or assess tells students what is valued in the classroom. If only quizzes and tests are graded, then this demonstrates its importance in the classroom. With ethnomathematics learning, the importance is placed on the mathematics conversations themselves. These conversations can be valued by using a rubric to evaluate group work by observing students: leaning in and working in the middle of the table; equal air time (everyone takes a turn talking); sticking together discussing each problem before going to the next one; explaining how a problem is solved with justification and reasoning; listening to each other when someone is talking; asking each other questions to clarify and understand; providing solutions using multiple strategies; persevering, persists, and not giving up; following group roles/jobs; and encouraging each other (Harding 2019). What teachers grade makes a difference in the significance students place on what happens in the classroom; therefore, grading mathematical conversation places importance there. Ethnomathematics allows students the opportunity to make deep mathematical connections to concrete examples they have experience grounded within their lives, and assessment allows students to demonstrate what they learn through those connections. An example is using the Burundi drumming from African cultures as Mr. Stevens describes this mathematics activity: When I enter a classroom and begin to play [drums] I take the students to the most elemental common denominator, movement via rhythm. It’s ok to move again, to feel again: and they’re feeling math – ratios, fractions, polyrhythms. That’s math. . .So it is with math and students once they’ve felt a ratio or a fraction via my drum. It’s a whole body type of learning. It’s not cerebral it’s guttural: natural; it’s easy (Sharp and Stevens 2019, p. 450, italics original).

This classroom experience allows students a front row pass to understand the mathematics concepts of algebra including differences, patterning, ordered pairs, and function notation. This teaching achieves mathematics content standards, accomplishes conceptual and procedural understanding, uses active processing of mathematics, and creates positive attitudes about mathematics. Ethnomathematics instruction allows students to achieve mathematics understanding and, therefore, enables success with mathematics assessments.

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Curriculum Teachers need to know students, content, and curriculum in order to teach mathematics effectively. Teachers need to discern who their students are – personally, culturally, and academically – in order to effectively teach them and build upon the mathematics foundational knowledge they possess. Teachers need to understand mathematics content, use examples and models to make mathematics comprehensible, teach math in ways that make sense to students, and present content in an organized fashion where concepts connect and build upon one another. Curriculum is a planned sequence of instruction to meet educational goals and standards within mathematics. An ethnomathematics curriculum allows students to be “mindful of diversity in the context of real-world models, which are rooted in concrete situations and problems that occur throughout history” (Rosa and Orey 2015, p. 593). Ethnomathematics curriculum encourages an inquisitive mindset where cognitive math connections are connected to experiences. Ethnomathematics curriculum is designed to be supplemental to the standardsbased curriculum already present in most classrooms. The ethnomathematics learning activities or units replace other abstract textbook activities within your curriculum. A specific example of successful research-based ethnomathematics curriculum was Math in a Cultural Context for urban and rural Yup’ik elementary students (Lipka et al. 2005). This curriculum was developed by insiders and outsiders of the cultural community including elders and Yup’ik teachers. The created curriculum had students build a fish rack (a structure used to dry salmon) with mathematics investigations in proof, properties, perimeter, and area. The effects of the curriculum were as follows: (a) Altered social organization and communication in the classroom. (b) Guided inquiry to facilitate problem-solving and multiple solutions to math problems. (c) Positive changes in classroom relationships among teachers and students and between classroom and community. (d) Pride in culture and identity with ownership of knowledge. (e) Valued Yup’ik knowledge being privileged alongside traditional academic discourses (Lipka et al., p. 369). Ethnomathematics curriculum develops in-depth math learning.

Benefits of Ethnomathematics Teaching The teaching philosophy of ethnomathematics improves students’ math learning and understanding. Lipka et al. (2005) identified five learning effects realized when an ethnomathematics curriculum approach was implemented within the Yup’ik community. These positive ethnomathematics effects are established in other research

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studies proving ethnomathematics teaching and learning are effective. The next sections of this chapter validate effective math learning through ethnomathematics with enhanced communication, focus on guided inquiry, pride in culture identity, increased cultural knowledge, engagement and motivation, and improved math academic achievement.

Ethnomathematics Enhances Communication Classrooms are rich samples of diversities in societies. These diversities can be barriers to learning or catalysts for rich learning. Often times, the key to distinguish between a barrier or a catalyst is effective communication. Effective communication and discourse occur when students articulate their own ideas and seriously consider their peers’ mathematical perspectives as a way to construct mathematical understandings. Ethnomathematics enables teachers to create clear mathematics communication through discussions. Baroody and Coslick (1993) demonstrate tools for communication happening between students and students and teachers and students through discussions about problem-solving, finding patterns, and thinking. It is the responsibility of the classroom teacher to establish a learning community where space is created for mathematics discussion while valuing and including culture: Collaborative work among educators and learners makes learning more effective because it generates higher levels of engagement in mathematical thinking through the use of socially and culturally relevant activities, and this makes use of dialogical constructivism because the source of knowledge is based on social interactions between students and environments in which cognition is the result of the use of cultural artifacts in these interactions (Orey 2017, p. 343).

Ethnomathematics improves classroom mathematics communication skills. Farokhah et al. (2017) concluded that the mathematical communication ability of fifth-grade elementary students participating in an ethnomathematics-based curriculum “exceed[ed] the control class who learned using a conventional approach” (p. 542). Students from the Arab sector high schools in Israel learned ethnomathematics geometry that “inspired the students and teachers with a flow of emotions, lively discourse, and learning motivation” (Massarwe et al. 2010, p. 19). College students in Indonesia experienced an ethnomathematics learning model that positively influenced mathematical communication skills compared to students in a direct learning model that was teacher-centered where “students become lazy” (Hartinah et al. 2019, p. 808). Furthermore, a curriculum framework for sixteen Philippine indigenous populations was successful in having students bring a wide range of cultural activities and beliefs to the math classroom to spur discussion about conversational mathematics ideas (Alangui 2017). Cultivating an ethnomathematics environment enhances communication and discussions during mathematics learning which allows students to make the mathematics knowledge their own increasing learning gains.

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Ethnomathematics Elevates Guided Inquiry Guided inquiry facilitates critical, analytical, and scientific problem-solving while exploring multiple solutions to math problems. Guided inquiry is an instructional strategy where students and peers grapple with building intellectual critical thinking skills related to reflective thinking processes (Diani et al. 2019) as within the productive struggle mentioned earlier. The teacher chooses open-ended math problems and facilitates learning by listening and then asking probing questions to clarify students’ ideas, emphasize reasoning, and encourage student-to-student dialogue. Ethnomathematics supports and enhances guided inquiry. For example, students investigated fishing communities to create authentic proportional reasoning questions. The study of fishing communities contextualized mathematics for students traveling from different fishing ports and honored their background knowledge of the life cycle of the sardine (Sousa and Palhares 2019). Ethnomathematics gives the underpinnings and context for this exploration in mathematics. Research has established that critical thinking skills are enhanced when using culturally responsive math instruction (Abdulrahim and Orosco 2019). Students’ mathematical representation abilities taught through ethnomathematics inquiry learning models are higher than students taught with conventional learning (Widada et al. 2019a, 2019b). Students who utilized guided inquiry along with ethnomathematics scored higher than students just using ethnomathematics approaches (Nurdiansyah et al. 2021). In experimental and control research, academic achievement was found to be statistically significant when using ethnomathematics-based instructional approaches in geometry learning (Abdulrahim and Orosco 2019; Sumiyati et al. 2018) and problem-solving abilities (Widada et al. 2019a, 2019b; Imswatama and Lukman 2018). Ethnomathematics guided inquiry elevates effective mathematics learning.

Ethnomathematics Emphasizes Pride in Cultural Identity Cultural identity is the collective or true self which people with a shared ancestry and history hold common. Cultural identity is often hidden inside the many other, more superficial or artificially imposed “selves” (Hall 1990). In ethnomathematics teaching, the establishment of cultural connections is a fundamental aspect in the development of ethnomathematics because it allows students to perceive mathematics as a significant part of their own cultural identity (Rosa and Orey 2006). Cultural identity can be brought into mathematics learning through “the stories in their examples; the affective language that showed pride in the relationships and achievements of their culture, relatives or ancestors” (Owens 2012, p. 589). Cultural pride can be developed and traditions acknowledged and honored through a direct classroom ethnomathematics curriculum application (Alangui and Shirley 2017). Amit and Abu Qouder found “many of the participating students that we interviewed expressed the wish that the rest of the mathematics curriculum would also integrate concepts from their culture or from their daily lives, rather than just teaching from the textbook. Some were also eager to connect their heritage even further to their school experience, not just in mathematics but in other fields as well” (p. 44).

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Cultural identity might be within the same ethnic group; however, differences within cultural groups may exist depending on social economics, mathematics practiced or observed in the home or community, and experience with academic vocabulary. Two ethnomathematics teaching examples would be multicultural math picture books (MMPs) and math tool sharing. The use of MMPs in mathematics classrooms “giv[es] children a window into another culture [or their own culture] and open [s] space for children to think mathematically” (Loyd et al. 2014–2015, p. 6). This acknowledges everyone as valued mathematicians and boosts positive cultural identity and pride when the students’ culture is presented (Harding et al. 2017). An example is Gabriela’s Beautiful Carpet/La bella alfombra de Gabriela by Thompson and Thompson (2003). This story is set in Antigua Guatemala where the village is creating “carpets” in the road out of flowers, colored sawdust, and stencils (cardboard with wooden frames) for the religious Lenten procession that takes place in their community. These carpets use the mathematics concepts of patterns, symmetry, proportions, and geometry. Teaching ethnomathematics through this story creates the following: the foundational knowledge of “carpets” can help the students who have participated in these processions to connect real world mathematics knowledge to more abstract mathematics principles and the other students in the classroom, who have never participated in a “carpet” procession, will have the knowledge and information to think about these mathematics principles in new ways to extend their knowledge to the real world (Harding 2016, p. 87).

Or, teachers could bring this type of learning into the classroom by assigning students to bring a math tool from home to share with the class. A student might bring in a micrometer (device for incorporating a calibrated screw widely used in engineering and machining) that her mom uses. This classroom experiences contextualize problems, gives context to a math tool, and connects authentic math leaning from school to home. These activities are the impetus in honoring cultural math knowledge and creating pride. Abdulrahim and Orosco (2019) concluded that if instruction relates to a students’ cultural backgrounds, students form positive identities with math. Furthermore, student empowerment and activism (social justice) increased with culturally responsive pedagogy. Owens et al. (2011) posited, “One social justice issue that received acclaim during the forum was the importance of all students being exposed to mathematics of different cultures through which they obtain a more comprehensive view of culture and mathematics and become more socially aware of difference. Such approaches also reinforce cultural self-respect” (p. 258). Ethnomathematics creates and emphasizes pride in cultural identity.

Ethnomathematics Esteems Cultural Knowledge Ethnomathematics allows cultural knowledge to be privileged alongside traditional academic discourses. When students’ cultural knowledge and intellectual skills are valued as assets for mathematics learning, students gain cultural capital (Bourdieu 1995). Using cultural capital as a teaching asset, teachers bridge the gap between

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home and school mathematics knowledge. Ethnomathematics becomes “the natural domains and concrete vehicles for connecting cultural resources and values with mathematics learning” (Jurdak 2016, p. 125). Ethnomathematics pursuits in the classroom could focus on cornrow hair braiding patterns, Maya gardens plots, Ouro Preto Brazilian house numbering system, Navajo beading, Adinkra symbols creation, Klappenspiel games from Germany, graffiti geometric shapes, South Pacific Island stick charts, rangoli designs, Bulgarian embroidery patterns, or Potawatomi game Kwezage’win probability, to name a few (Barta et al. 2014). Creating ethnomathematics curriculum from the cultural communities was “based in students’ culturally acquired knowledge, valuing this knowledge” (Stathopoulou 2017, p. 120). When teachers bring ethnomathematics learning into the classroom, students can discover that mathematics is everywhere even though different cultures practice mathematics in individual ways. Ethnomathematics also broadens the field of what counts as math and who is good at math. Students can begin to view people within their cultures as mathematicians in everyday life. Davis and Davis (2016) emphasized, “ethnomathematics becomes a resource for demonstrating how certain cultural practices, especially those of the downtrodden and oppressed, can be positively valued through the revelation of the mathematical processes believed to be intrinsic to everyday practices of humans” (p. 289). Corp (2017) explained, “the data portrays all students as engaging and thinking mathematically from the story....there were also social benefits for non-Black students. These benefits include exposure, a space to ask about culture, to find commonalities, and to appreciate African American role models” (p. 49). Esteeming cultural knowledge can be accomplished by incorporating ethnomathematics curriculum that contributes to the permanence and renewal of traditional knowledge (Fantinato and Mafra 2017). Ethnomathematics values cultural knowledge and creates a space to have candid conversations to help students understand their own cultures and/or the cultures of others.

Ethnomathematics Empowers Engagement and Motivation Dispositions are tendencies for individuals to act in a particular manner under particular circumstances, based on one’s beliefs and cultural values (Villegas 2007). Dispositions are culturally formed. Teachers should acknowledge how their lifestyles, cultural values, and different worldview have shaped their dispositions toward mathematics (Orey 2017). When developing students’ mathematics knowledge, teachers need to consider the mathematics dispositions students bring from their own cultural experiences. Mathematics instruction needs to include cultural background knowledge and understanding that extend beyond the teachers’ dispositions. This pedagogy of instruction gives students multiple ways to think about and solve math problems. Ethnomathematics goes one step farther and reduces mathematics anxiety. Students have less anxiety because they can connect classroom mathematics within their own cultural practices. When mathematics includes

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cultural connections, students enjoy learning, and teaching becomes meaningful (Sharma and Orey 2017). Ethnomathematics is the conduit for students to make math connections at profound levels. Engagement and motivation are desirable dispositions for students to possess regarding mathematics learning. Engagement is a multifaced construct including affect (enthusiasm, interest, and belonging), cognition (deep learning and selfregulation), and behavior (time and effort, interaction and participation) (Kahu 2013). Ethnomathematics empowers engagement by enticing students to care about mathematics learning. The use of board games with cultural roots can trigger engagement. The African board game oware was used as a cultural instrument that immersed children in academic explorations of interesting and rich mathematical structures. “While playing games, children establish intellectual frameworks that enable them further to construct and comprehend complex mathematical ideas, strategies, and theories” (Powell and Temple 2001, p. 369): Games reveal the thoughts and lives of those who invent them. The physical structure and materials, as well as the rules of a game, reflect the culture that created it. As a result, when students play a game such as oware, they interact with aspects of the culture in which it originated” (Powell and Temple 2001, p. 373).

Another example of engagement was evident when students engaged in western mathematics as well as their own cultural understanding to build a lodge (Shockey and Mitchell 2017). Through ethnomodeling, students’ experiences were validated within efficiency and relevancy of mathematics developing a critical view of the world by using mathematics (Rosa and Orey 2013). Engagement within ethnomathematics brings math learning off the page of a textbook and connects to life experiences. Motivation is a process involving biological, emotional, social, and cognitive forces that activate a desire or willingness to do something, such as learning mathematics. Students are motivated to learn mathematics when they know the purpose and can make cognitive connections. However, when mathematics does not relate to students’ lives, their motivation to solve problems languishes. For example, in a Hawai’i classroom, students were given a subtraction problem about raccoons. One student became so frustrated trying to understand what a raccoon was, he was unable to work the problem. For many children of diverse cultural and linguistic backgrounds, school learning consists of a series of “raccoon-like” experiences where the disparities between teachers’ assumptions about what children know and what children actually know is one aspect of the mismatch between the culture of the school and culture of the home (Maaka et al. 2001). Had the problem referred to an animal familiar to the student, he may have more easily engaged with the problem and been motivated to solve it. Ethnomathematics empowers the dispositions of engagement and motivation by having students see relationships within mathematics learning.

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Ethnomathematics Enriches Academic Achievement Academic achievement is the current level of student learning and the extent to which students attain their educational goals. Ethnomathematics teaching supports academic achievement and creates student success (Alangui 2017; Corp 2017). Research has found a connection between ethnomathematics and achievement. A large, quasi-experimental study with over 400 students was conducted to look at how ethnomathematics might impact achievement. Abiam et al. (2016) found that primary school students who were taught geometry using ethnomathematics scored significantly higher on an achievement test than students taught using more conventional approaches. In another experimental study, academic achievement increased with a group of eighth-grade students who received an ethnomathematics intervention (Kusuma et al. 2019). Ethnomathematics increased persistence in problem-solving (Corp 2017; Owens et al. 2011). Irawan et al. (2018) noted sixth-grade student’s problem-solving abilities improved when teaching ethnomathematics was combined with realistic mathematics (products, illustrations, or artifacts from the real world). Furthermore, ethnomathematics instructional approaches enhanced math understanding with algebraic word problems (Kurumeh and Iji 2009). Ethnomathematics not only values the diverse ways in which students learn and understand mathematics content but also improves student achievement.

Ethnomathematics Obstacles Explained One consideration when contemplating bringing ethnomathematics into the math classroom is the implementation obstacle of budget. Schools assume that ethnomathematics requires investment in a new curriculum and other materials. Yet, ethnomathematics curriculum is not meant to replace the standards mathematics curriculum, but it is designed to “support the standard curriculum, making the material more accessible and relevant to the students” (Amit and Abu Qouder 2017, p. 46). Sunzuma and Maharaj (2019) concluded the language and examples in mathematics curriculum textbooks are foreign and unrelated to the local culture; furthermore, mathematics curriculum and textbooks lack an understanding of the indigenous knowledge of students. Teachers can incorporate local knowledge into existing curriculum to make the current materials more relevant to students in the classroom. Sharma and Orey (2017) stated for “schools with little financial resources, culturally relevant pedagogies are inexpensive yet powerful resources” (p. 176). Ethnomathematics doesn’t detract from curriculum; in fact, it gives multiple strategies for learning and provides more ways for students to be successful. Teachers reported a lack of training in ethnomathematics methods as an obstacle to implementation (Katsap and Silverman 2016). Without sufficient experience with ethnomathematics, teachers lack confidence to attempt using it in their own classrooms. However, after teachers participated in ethnomathematics trainings, they

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were willing to replicate activities they had learned (Mogari 2014). Furthermore, Naresh (2015) found ethnomathematics professional development “challenged [teacher’s] perceptions of mathematics, enhanced mathematical understandings, offered a glimpse into cultures, societies, and the mathematical activities that live and thrive in such contexts” (p. 467). Teachers can be supported in this move to use ethnomathematics in the classroom through professional development and models of how, what, and why to teach students. Obstacles to implementing ethnomathematics may be overcome by using already created ethnomathematics lessons, the cultural math knowledge your students bring into the classroom, and MMPs and by providing professional development.

Ethnomathematics Philosophy Elucidated to Teachers Current teachers in the classroom and those who are becoming teachers need to be specifically supported in how to teach using the ethnomathematics philosophy because teaching mathematics this way is different from how they learned it. Understanding how to implement effective cultural practices into mathematics may be a new way of thinking. It will take teachers time to wrestle with this ethnomathematics philosophy in order to understand what it entails, research the cultural practices that can be brought into their classroom math instruction, develop learning goals and objectives in connection to standards, create teaching strategies, and then establish curriculum for instruction. The time put into developing and implementing changes will ultimately benefit students. This support for teachers could take the form of the following: (a) Workshops brought into schools for professional development. (b) Book studies centered about ethnomathematics activities, understanding how it can be used in classrooms, etc. with specific discussions of how teachers can implement it into their own classroom. (c) Evaluating and choosing multicultural mathematics picture books (MMPs) to bring into the classroom. (d) Math coaches supporting ethnomathematics instruction by sharing activities, modeling lessons, evaluation teaching, and reflecting upon practice with classroom teachers. (e) Making connections with cultural community members and having teachers understand the math outside of their classroom in their students’ lives. (f) Taking university ethnomathematics courses. (g) Exploring ethnomathematics websites and using them in the classroom (https:// sites.google.com/site/ethnomathematics/ or https://www.todos-math.org/index. php?option¼com_dailyplanetblog&view¼entry&year¼2019&month¼05& day¼22&id¼11:ethnomathematics-mathematics-de-todos). (h) Having teacher grade-level meetings to discuss, evaluate, and reflect upon ethnomathematics teaching within their classroom.

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Teachers can make the philosophy leap to ethnomathematics with content and pedagogical support.

Ethnomathematics Expanded in the Future Ethnomathematics research has demonstrated effective mathematics learning and achievement for students. Research provides rich examples of ways to establish and maintain an ethnomathematics philosophy while teaching. In order to fully imagine ethnomathematics instruction, further research is needed to create an inclusive ethnomathematics curriculum for students at different grade bands with a clear scope and sequence. This ethnomathematics curriculum needs to be researched to see best methods of including it into textbooks and teaching materials instead of it being add on activities. Additional research is needed with specific ways to bring community and cultural mathematics in the classroom to enhance math learning.

Conclusion Ethnomathematics is the confluence where mathematics and culture exist within the classroom learning environment. Gestalt, situated cognition, and constructivism learning theories suggest effective learning of mathematics will happen within the framework of ethnomathematics. Culture exhibits significance in brain development and cognitive mathematics learning. Ethnomathematics pedagogy infuses real-life cultural knowledge into curriculum using culturally responsive teaching methods. Establishing an ethnomathematics instructional philosophy includes recognizing and honoring students’ cultural experiences, creating an effective classroom environment, implementing specific mathematics teaching and learning principles, establishing cultural classroom experiences, understanding concrete real-world ethnomathematics approaches, grasping metacognition, employing math groups, differentiating instruction, assessing what is valued, and implementing curriculum. Communication and discussions in the classroom are enhanced through a guided inquiry and ethnomathematics learning environment. Ethnomathematics elevates guided inquiry increasing math knowledge. Ethnomathematics emphasizes and esteems cultural knowledge and affords freedom to understand your own culture or the culture of others while emphasizing pride in cultural identity. It empowers the dispositions of engagement and motivation within the classroom while enriching academic achievement. Ethnomathematics obstacles may be overcome, and teachers can be supported in order to incorporate ethnomathematics into their instruction. Ethnomathematics is worth the time, effort, and thought because it perpetuates mathematics learning within the classroom.

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Rosa, M., & Orey, D. C. (2013). Ethnomodelling as a methodology for ethnomathematics. In G. Stillman, G. Kaiser, W. Blum, & J. Brown (Eds.), Teaching mathematical modelling: Connecting to research and practice. International perspectives on the teaching and learning of mathematical modelling. Dordrecht: Springer. https://doi.org/10.1007/978-94-007-6540-5_6. Rosa, M., & Orey, D. C. (2015). A trivium curriculum for mathematics based on literacy, matheracy, and technoracy: An ethnomathematics perspective. ZDM, 47(4), 587–598. Schneider, D. A., & Keenan, E. K. (2015). From being known in the classroom to “moments of meeting”: What intersubjectivity offers contemplative pedagogy. The Journal of Contemplative Inquiry, 2(1), 1–16. Sharma, T., & Orey, D. C. (2017). Meaningful mathematics through the use of cultural artifacts. In M. Rosa, L. Shirley, M. E. Gavarrete, & W. F. Alangui (Eds.), Ethnomathematics and its diverse approaches for mathematics education (pp. 153–179). Cham: Springer. Sharp, J., & Stevens, A. (2019). Culturally-relevant algebra teaching: The case of African drumming. In T. L. Shockey (Ed.), Culture that counts: A decade of depth with the Journal of Mathematics and Culture (pp. 445–458). Galena: White Plum Publishing. Shockey, T., & Mitchell, J. B. (2017). An ethnomodel of a Penobscot lodge. In M. Rosa, L. Shirley, M. E. Gavarrete, & W. F. Alangui (Eds.), Ethnomathematics and its diverse approaches for mathematics education (pp. 257–281). Cham: Springer. Sousa, F., & Palhares, P. (2019). (Ethno)mathematical tasks in context of proportional reasoning. In T. L. Shockey (Ed.), Culture that counts: A decade of depth with the journal of mathematics and culture (pp. 437–444). Galena: White Plum Publishing. Stathopoulou, C. (2017). Once upon a time...The Gypsy boy turned 15 while still in first grade. In M. Rosa, L. Shirley, M. E. Gavarrete, & W. F. Alangui (Eds.), Ethnomathematics and its diverse approaches for mathematics education (pp. 97–123). Cham: Springer. Sumiyati, W., Netriwati, & Rakhmawati, R. (2018). Penggunaan Media Pembelajaran Geometri Berbasis Etnomatematika. Desimal: Jurnal Matematika, 1(1), 15–21. Sunzuma, G., & Maharaj, A. (2019). Teacher-related challenges affecting the integration of ethnomathematics approaches into the teaching of geometry. EURASIA Journal of Mathematics, Science and Technology Education, 15(9), 1–16. Thompson, K., & Thompson, S. (2003). Gabriela’s beautiful carpet/La bella alfombra de Gabriela. Guatemala City: Vista Publications. Tomlinson, C. A. (1999). The differentiated classroom: Responding to the needs of all learners. New Jersey: Pearson Education. Ukpokodu, O. N. (2011). How do I teach mathematics in a culturally responsive way? Identifying empowering teaching practices. Multicultural Education, 18, 47–56. Varghese, T., & McCusker, D. P. (2006). On globalization and ethnomathematics. Canadian and International Education, 35(1), 1–11. Verenikina, I. (2008). Scaffolding and learning: Its role in nurturing new learners. In P. Kell, W. Vialle, D. Konza, & G. Vogl (Eds.), Learning and the learner: Exploring learning for new times (pp. 161–180). Wollongong: University of Wollongong, Australia. Villegas, A. M. (2007). Dispositions in teacher education: A look at social justice. Journal of Teacher Education, 58(5), 370–380. Widada, W., Herawaty, D., Anggoro, A. F. D., Yudha, A., & Hayati, M. K. (2019a). Ethnomathematics and outdoor learning to improve problem solving ability. In International Conference on Educational Sciences and Teacher Profession (ICETeP 2018). Atlantis Press. Widada, W., Herawaty, D., Jumri, R., Zulfadli, Z., & Damara, B. E. P. (2019b). The influence of the inquiry learning model and the Bengkulu ethnomathematics toward the ability of mathematical representation. Journal of Physics: Conference Series, 1318(1), 1–5. IOP Publishing. Woodworth, R. S. (1950). Edward Lee Thorndike: 1874–1949. Science New Series Journal, 111(2880), 250–251. Yang, L., Hanneke, S., & Carbonell, J. (2013). A theory of transfer learning with applications to active learning. Springer, 90, 161–189. https://doi.org/10.1007/s10994-012-5310-y.

Section III Cognitive Neuroscience of Mathematics Roland H. Grabner

Abstract

In the past two decades, much of the psychological research on mathematical cognition, learning, and development has pursued a neurocognitive approach combining behavioral methods with neuroscientific techniques. In this way, cognitive neuroscience research on mathematics has also contributed to the development of the interdisciplinary and multi-methodological field of educational neuroscience. The aim of this section was to provide state-of-the-art reviews of five well-established and important lines of this research, covering the early development of mathematical skills, fraction processing as specific obstacle in mathematics education, factors accounting for individual differences in mathematical abilities and competencies, the critical phenomenon of math anxiety, and behavioral as well as neural interventions to foster mathematics learning. The authors of the five chapters not only summarize current theories and evidence on these topics but also highlight critical knowledge gaps and describe promising avenues for future research. Keywords

Cognitive neuroscience · Educational neuroscience · Brain development · Quantity processing · Arithmetic · Fraction learning · Individual differences · Math anxiety · Cognitive training · Brain stimulation

There is increasing awareness and empirical evidence that mathematical competencies are key cognitive abilities in our modern, technological and informed societies. Longitudinal research has demonstrated that mathematical competencies are equally important for educational and vocational success as literacy (Parsons & Bynner, 2005), and that deficits in these competencies place a heavy burden on the individual. For instance, poor mathematical skills have been found to be associated with lower socioeconomic status, higher rates of unemployment, higher risk of delinquency, and poorer physical as well as mental health (Litster, 2013; Vignoles et al.,

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2011). At the societal level, these deficits profoundly affect the productivity of the workforce and explain a significant proportion of the differences in economic performance between nations (Gross et al., 2009; Vignoles, 2016). This situation is particularly critical as the number of individuals suffering from poor mathematical competencies is alarmingly high. The diagnosed mathematical learning disorder – developmental dyscalculia – has a high prevalence of 5–7% (Butterworth et al., 2011), which is about the same as for dyslexia (Peterson & Pennington, 2015). In addition, data from an OECD survey on adult competencies has revealed that around 20% of the population show mathematical difficulties imposing practical and occupational restrictions and that in virtually all OECD countries the number of individuals with poor mathematical competencies is considerably higher than those with poor literacy (OECD, 2016). Despite the paramount importance of mathematical competencies in our everyday lives, mathematical cognition, learning, and development had not been within the focus of research for quite a long time. Much more attention had been drawn on language development (reading and writing) and on deficits within this domain, i.e., dyslexia. Fortunately, this situation has considerably changed in the past two decades. For instance, a Web of Science™ literature research (on 4th May 2022) revealed that between 1900 and 2000 about 20 times more publications on dyslexia than on dyscalculia can be found and that this ratio decreased to 13:1 between 2000 and 2010 and to less than 9:1 between 2010 and 2020. Much of the recent research has pursued a neurocognitive approach by combining behavioral methods with neuroscientific techniques such as functional magnetic resonance imaging (fMRI) or electroencephalography (EEG). In this way, research on mathematics has substantially contributed to the growth and success of the research field educational neuroscience (e.g., Ansari et al., 2012; De Smedt et al., 2010; Grabner & Ansari, 2010). This research field can be characterized by an interdisciplinary (involving educational sciences, psychology, neuroscience, and other disciplines) and multi-methodological (applying behavioral as well as neuroscientific techniques) research approach. Despite repeated criticism on the feasibility and the benefits of investigating educational research questions with neuroscientific methods (e.g., Bowers, 2016; Howard-Jones et al., 2016), educational neuroscience has experienced an unprecedented growth that is also reflected in an increasing number of dedicated scientific journals, funding programs, research groups, and conferences. From the beginning of educational neuroscience, among all school-related topics and issues, mathematics seemed to have a very special status. For instance, in the first meeting of the Special Interest Group “Neuroscience and Education” of the European Association for Research on Learning and Instruction (EARLI) in 2010, the number of research contributions on mathematics already was higher than that on language and reading (https://www.frontiersin.org/events/EARLI_SIG22_-_Neuro science_and_Education/599, retrieved on 4th May 2022), despite an even larger research gap in favor of literacy at that time. One reason for this particular interest into mathematics has been very likely the increasing awareness that our knowledge on this key competence and on how to foster it (in typical as well as atypical development) is very limited. Another reason may have been that mathematics has been discovered as an ideal domain of cognitive neuroscience research. A typical challenge

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in studies using cognitive neuroscience techniques is the requirement of a large number of similar stimuli (e.g., problems) so that through averaging the brain signal across many trials the measurement error can be reduced. Unlike other school-related domains, mathematics offers the advantage of a large and highly structured problem space that allows developing many problems of a similar type. Additionally, mathematical problems often presented in neuroscientific studies (e.g., arithmetic problems) can be perceived quickly (without much reading time), answered easily (e.g., through entering numbers on a keyboard), and scored unambiguously (as right or wrong). To date, cognitive neuroscience research on mathematics has accumulated a large body of knowledge on the architecture of the human brain supporting the presentation and processing of numerical information, on the neurocognitive mechanisms underlying typical and atypical development of mathematical competencies, and on the neural correlates of individual differences in these competencies. This knowledge has established and broadened the foundation for the development of interventions to improve mathematical skills in general and to remediate mathematical deficits in particular. The aim of the present section in the Handbook of Cognitive Mathematics is to provide a state-of-the-art review of these important lines in cognitive neuroscience research on mathematics. It consists of five chapters that shall give readers an overview of previous and current research endeavors and of central findings. In the first chapter of this section, Stephan E. Vogel addresses basic numerical competencies that are critical in the early development of mathematical skills. There is already wide consensus that our brains are endowed with a fundamental evolutionary system that enables us to estimate the numbers of objects in a set (the “approximate number system”). Interestingly, such a system has not only been found in humans but also in several other species. This raises the critical question of whether we build our symbolic numerical understanding on this primitive system. Against this background, the author summarized our knowledge on how the human brain represents and processes non-symbolic and symbolic numerical information. In addition, the development from these basic numerical competencies to more complex arithmetic skills is discussed. Overall, this chapter demonstrates that (even early) mathematical development is a multifaceted process that involves different cognitive abilities and brain regions. The second chapter by Silke M. Wortha, Andreas Obersteiner, and Thomas Dresler deals with neurocognitive foundations of fraction processing. In the development of mathematical competencies in school, the mastering of fraction problems seems to be one of the largest obstacles. Many students (and even adults) show large difficulties and typical errors when presented with fraction problems. These difficulties are critical because fractions are important for understanding many other mathematical concepts (e.g., algebra, probability, geometry), which is also reflected in the finding that fraction understanding is a unique predictor of higher-order mathematical achievement. The cognitive mechanisms of fraction processing and learning, however, are still not fully understood. In this chapter, the authors review the findings of research using different behavioral and neuroscientific methods. Surprisingly, the two most often applied cognitive neuroscience techniques, fMRI and EEG, which differ in their temporal and spatial resolution, yielded slightly different answers to the question of how fractions are processed. At the end of this

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chapter, the authors integrate the research findings and propose a tentative temporal model of fraction processing. In the third chapter, Sara Caviola, Irene C. Mammarella, and Denes Szűcs focus on individual differences in mathematical competencies and summarize our knowledge about potential sources for them. At practically all stages in mathematical development, there are large individual differences in performance. These differences, however, have mostly been examined using a categorical approach by comparing individuals showing low or deficient performance (e.g., dyscalculics) with those in the average (typical) performance range. Research on the upper end of the performance spectrum (high achievers or experts), in contrast, has been very scarce. Both lines of research suggest that the successful acquisition of mathematical competencies is a result of a complex and dynamic interplay of individual and contextual factors. Beyond domain-specific abilities (e.g., basic numerical abilities as presented in the first chapter of this section), domain-general cognitive abilities (e.g., working memory, executive functions) have been found to loom large in mathematical development. These cognitive factors are complemented by non-cognitive factors including affect, beliefs, or motivation. Among the contextual factors are cultural/language differences, parental support, and the educational system. The complexity of this interplay highlights the need to abandon the research approach of merely comparing two groups and to investigate individual differences in a multidimensional way including multiple measures and considering the entire performance range. The next (fourth) chapter by Rachel Pizzie is dedicated to one individual factor of particular importance for mathematical development – math anxiety. Mathematics can be considered as a very special school subject as there is a specific type of anxiety related to doing mathematics or anticipating such situations. Math anxiety has been found to be associated with lower achievement in standardized mathematics tests, poorer school grades, and the avoidance of mathematics-related careers. A still unresolved question is the directional influence in this relationship. Does math anxiety develop because of deficits in (basic) mathematical abilities, or do poor mathematical competencies result from cognitive impairments and avoidance of learning situations because of math anxiety? In this chapter, not only the evidence from longitudinal studies on this question is presented but also a comprehensive summary of our knowledge about the relevance as well as interplay of cognitive and affective factors, which are essential for an understanding of how math anxiety may impair performance. In addition, different types of interventions are described that go beyond typical psychotherapeutic approaches and that can be scaled up in educational environments (e.g., emotion regulation through cognitive reappraisal). Finally, in the last chapter of this section, Karin Kucian and Roi Cohen Kadosh review the research on neurocognitive interventions to foster mathematics learning. In line with the previous chapters, they emphasize the importance of early numerical competencies in the identification of children at risk for an atypical mathematical development and raise the question of whether a training of these competencies in poorperforming children could prevent later learning problems. In this context, the scarce evidence from short- and medium-term longitudinal studies is summarized. In addition, the authors address behavioral interventions that have been developed to remediate

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mathematical deficits in children with poor numeracy (e.g., those suffering from developmental dyscalculia) and highlight important criteria for their effectiveness. In the second part of this chapter, the authors present evidence regarding the effects of behavioral as well as neural interventions on brain functioning in mathematics. Of increasing research interest among the neural interventions is transcranial electrical stimulation (tES), in which a weak electrical current is noninvasively applied to the human cortex. The first studies applying this technique in the domain of mathematical cognition and learning yielded promising results but there are still several challenges and open questions before tES can be transferred to the educational practice. The five chapters in this section illustrate the added value of combining behavioral and neuroscientific methods in research on mathematical cognition, learning, and development. This holds true for all three types of added value that have been distinguished in educational neuroscience research (De Smedt & Grabner, 2015): “neurounderstanding,” “neuroprediction,” and “neurointervention.” Neurounderstanding refers to the idea that the neuroscientific level of investigation can yield results that foster our understanding of involved cognitive processes. Neuroprediction indicates that neuroscientific measures can improve the prediction of individual differences in learning outcomes or development beyond behavioral measures. And, neurointervention describes either that neuroscientific results can inform the development of educational interventions or that neuroscientific techniques (such as noninvasive brain stimulation) amplify the effects of behavioral interventions. Despite the tremendous research progress in the cognitive neuroscience of mathematics summarized here, the five chapters also point out that critical questions related to mathematical development and learning are still unresolved and that further longitudinal research is needed. In fact, much of the current evidence (especially from cognitive neuroscience studies) is based on cross-sectional studies comparing only few age groups (e.g., children vs. adults). This obviously does not provide enough temporal resolution to track the massive developmental changes from early childhood to adolescence. In addition, many studies have so far focused only on a limited set of variables and, thus, could not elucidate the complex and dynamic interactions between different factors along different developmental trajectories. Due to these limitations in research, there is still no comprehensive neurocognitive model of mathematics development from early childhood to adolescence (or even adulthood), which could serve as a common theoretical framework for further research and as a scientific basis for practice and policy-making. The present section in the Handbook of Cognitive Mathematics may have the potential to further promote the relevant research for the establishment of such a comprehensive neurocognitive model of mathematics development. It does not only review the state-of-the art and highlight research gaps but could also arouse interest in researchers from different disciplines and in experts from practice to (further) collaborate with each other. In line with the basic idea of educational neuroscience, substantial research progress requires crossing the boundaries of disciplines, integrating different levels of investigation, and including experiences from educational practice.

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References Ansari, D., De Smedt, B., & Grabner, R. H. (2012). Neuroeducation – A critical overview of an emerging field. Neuroethics, 5, 105–117. https://doi.org/10.1007/ s12152-011-9119-3 Bowers, J. S. (2016). The practical and principled problems with educational neuroscience. Psychological Review. https://doi.org/10.1037/rev0000025 Butterworth, B., Varma, S., & Laurillard, D. (2011). Dyscalculia: From brain to education. Science, 332(6033), 1049–1053. Retrieved from http://www. sciencemag.org/content/332/6033/1049.abstract De Smedt, B., & Grabner, R. H. (2015). Applications of neuroscience to mathematics education. In R. Cohen Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition. Oxford University Press. https://doi.org/10.1093/oxfordhb/ 9780199642342.013.48 De Smedt, B., Ansari, D., Grabner, R. H., Hannula, M. M., Schneider, M., & Verschaffel, L. (2010). Cognitive neuroscience meets mathematics education. Educational Research Review, 5(1), 97–105. https://doi.org/10.1016/j.edurev.2009.11.001 Grabner, R. H., & Ansari, D. (2010). Promises and potential pitfalls of a ‘cognitive neuroscience of mathematics learning’. ZDM. The International Journal on Mathematics Education, 42(6), 655–660. https://doi.org/10.1007/s11858-010-0283-4 Gross, J., Hudson, C., & Price, D. (2009). The long term costs of numeracy difficulties. Every Child a Chance Trust and KPMG. Howard-Jones, P. A., Sashank, V., Ansari, D., Butterworth, B., De Smedt, B., Goswami, U., et al. (2016). The principles and practices of educational neuroscience: Commentary on Bowers (2016). Psychological Science. Litster, J. (2013). The impact of poor numeracy skills on adults. National Research and Development Centre for Adult Literacy and Numeracy, 44, 1–50. OECD. (2016). Skills matter: Further results from the survey of adult skills, OECD skills studies. OECD Publishing. https://doi.org/10.1787/9789264258051-en Parsons, S., & Bynner, J. (2005). Does numeracy matter more? National Research and Development Centre for Adult Literacy and Numeracy, (January), 1–37. https://doi.org/1905188090 Peterson, R. L., & Pennington, B. F. (2015). Developmental Dyslexia. Annual Review of Clinical Psychology, 11(1), 283–307. https://doi.org/10.1146/ annurev-clinpsy-032814-112842 Vignoles, A. (2016). What is the economic value of literacy and numeracy? IZA World of Labor, (January), 1–10. https://doi.org/10.15185/izawol.229 Vignoles, A., De Coulon, A., & Marcenaro-Gutierrez, O. (2011). The value of basic skills in the British labour market. Oxford Economic Papers, 63(1), 27–48. https://doi.org/10.1093/oep/gpq012

Developmental Brain Dynamics: From Quantity Processing to Arithmetic

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Stephan E. Vogel

Contents General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Foundation: Representations of Quantities and Numerical Order . . . . . . . . . . . . . . . . . . . . . . . . . The Representation of Numerical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Numerical Meaning of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The development of mathematical abilities constitutes a crucial foundation in our modern and educated societies. In the past decades, neuroscientists have begun to investigate the neurocognitive mechanisms associated with the development of these abilities. The present chapter summarizes our current knowledge about the functional brain organization related to the processing of basic numerical information and arithmetic. Relevant neurocognitive models and brain networks associated with the processing of non-symbolic numerical quantities, symbolic numerical representations – such as numerical order – and arithmetic will be discussed in detail. The presented evidence demonstrates that the development of these abilities cannot be restricted to a single cognitive mechanism or to a single brain region. It rather constitutes complex and multidimensional concepts that incorporate multiple cognitive abilities, representational dimensions, and brain regions.

S. E. Vogel (*) Section of Educational Neuroscience, Institute of Psychology, University of Graz, Graz, Austria e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_26

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Keywords

Brain activation · Representation of quantities · Representation of numerical order · Developmental changes

General Introduction The development of mathematical abilities constitutes a crucial foundation in modern and educated societies. Research has shown that mathematical abilities are equally important for life success as literacy and that deficits in this domain can have severe effects on individuals’ well-being and nations’ economy (Parsons & Bynner, 2005). Current estimates have shown that around 20% of the population in OECD countries have difficulties within mathematics, imposing great practical and occupations restrictions (OECD, 2016). Around 5–7% of the population suffer from developmental dyscalculia (DD), a severe mathematical learning disorder (Butterworth et al., 2011). In the past decades, neuroscientists have begun to investigate the neurocognitive mechanisms associated with these crucial abilities. With the help of different neuroimaging methods – such as functional magnetic resonance imaging (fMRI) or electroencephalography (EEG) – researchers have started to unravel the brain networks correlated with mathematical abilities. And although our current understanding of the neurocognitive mechanisms is still limited, key principles of the functional and structural brain organization have emerged. Insights from this research have theoretical as well as practical implications. In this chapter, I will summarize our current knowledge about the functional brain organization related to basic numerical and arithmetic abilities. In the first part, I will provide an overview of the relevant neurocognitive models and brain regions associated with the processing of non-symbolic numerical quantities (i.e., the number of items in a set) and symbolic numerical representations (e.g., knowledge about the meaning of numerals). After discussing these foundational skills, I will summarize the brain networks associated with arithmetic abilities, with a special focus on arithmetic fact retrieval. After reading this chapter, the reader should have acquired a basic knowledge of how the human mind and brain represents and develops these foundational skills.

The Foundation: Representations of Quantities and Numerical Order Basic numerical representations, which I define as the semantic knowledge about numbers, build a crucial foundation for arithmetic and mathematical abilities. Over the past decades, two central numerical dimensions have been proposed to be of great importance: the representation of numerical quantities and the representation of numerical order (see also Fig. 1).

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While the representation of numerical quantities refers to the knowledge about the total number of distinct objects within a given set of items (i.e., the set size, or numerosity: e.g., four apples), the concept of numerical order refers to the knowledge that a given object occupies a relative rank or a position within a given set of items (Lyons et al., 2016). The latter concept corresponds to the knowledge about ordered lists or sequences. For instance, the notion of numerical order allows us to infer with relative ease that 1002 comes right after 1001. Such an efficient judgment would be difficult, if numerals would only convey information about numerical quantities. Because of its obvious relevance, especially for symbolic number processing, investigations to better understand the brain networks of these two dimensions have significantly increased over the past years. In the next sections, I will discuss these concepts and the related behavioral and brain patterns in more detail.

The Representation of Numerical Quantities An ever-growing number of studies has shown that insects, fish, birds, lions, nonhuman primates, and preverbal infants possess the ability to perceive and to discriminate the number of different elements, i.e., their set size (Nieder, 2016). These findings indicate the existence of a primitive, possibly evolutionary, system

Fig. 1 Schematic illustration about two important concepts numbers refer to. Numerical quantity indicates the number of elements in a set, numerical order indicates the serial order, position, or rank of an item

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that enables the perceptual quantification of elements in a set. A typical task to investigate this neurocognitive mechanism in humans is the non-symbolic numerical discrimination task. In this task, participants are asked to decide as fast as possible, and without making mistakes, which of two presented item sets (often two arrays of dots) contains more or less items. Reaction times and error rates are recorded while participants perform this task. An abundant amount of research has shown that the behavioral measurements of this task (i.e., reaction times and error rates) show a distinct and reliable pattern. First, reaction times and error rates are lower when participants discriminate item sets that express a small numerical ratio (smaller number of items/larger number of items) compared to item sets that express a large numerical ratio (see also Fig. 2a). For instance, the decision that a set of 16 elements is more than a set of 8 elements (numerical ratio of 8/16 ¼ 0.5) is easier than the decision that a set of 12 elements is larger than a set of 8 elements (numerical ratio of 8/12 ¼ 0.67). Another, yet similar metric, is the numerical distance effect. The numerical distance effect indicates that reaction times and error rates are lower when participants discriminate item sets that express a large numerical distance compared to items sets that express a small numerical distance (the above example converts to 16  8 ¼ 8 vs 12  8 ¼ 4). The numerical ratio and the numerical distance effect are highly correlated. However, the numerical ratio effect explains a little more variance in the observed reaction time patterns. Second, the numerical ratio and distance effect is much smaller (sometimes even absent) in the small number range (i.e., numerical quantities up to three or four elements; Feigenson et al., 2004). The distinctive patterns in the small and large number range indicate the involvement of at least two neurocognitive systems: the approximate number system (ANS) and the object tracking system (OTS; Feigenson et al., 2004). While the ANS has been related to the numerical ratio and distance effect in the large, and possibly the small number range, the OTS has specifically been related to the observed effects in the small number range. Due to their primitive nature, both systems have been proposed to build a biological foundation for the representation and the development of symbolic numerical and arithmetic abilities (Feigenson et al., 2004). The following sections will review the scientific literature associated with these two systems.

The Approximate Number System (ANS) The ANS reflects an intuitive sense to nonverbally perceive the number of elements within a set, especially in the large number range (e.g., perceiving that a set of 16 dots differs from a set of 8 dots). This neurocognitive model proposes that the above discussed behavioral effects (i.e., numerical ratio and numerical distance effect) are the consequence of a neural representation that encodes numerical quantities as a function of Weber’s law (Nieder, 2016). In other words, the representation of numerical quantities is understood as a linear expression in which each numerical quantity is represented as a Gaussian distribution with a scalar variability; often called tuning curves (see Fig. 2b). Since the width of the distribution increases with the objective number of items, the precision of the approximate number system decreases as the numerical quantity increases (e.g., 8 elements are represented with a higher precision than 16 elements). As a consequence, the behavioral discrimination

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of two quantities (e.g., 8 vs 16) is determined by the relative overlap of the two representational distributions. Numerical quantities are therefore easier to discriminate when there is less overlap between the distributions. Much effort has been devoted to identifying the neural signature of the ANS in nonhuman primates and humans. In their seminal work, Nieder and Miller (2003, 2004) demonstrated number sensitive neurons in homolog brain regions of the human intraparietal sulcus (IPS) and the lateral prefrontal cortex (LPFC) using single cell recording in nonhuman primates (for a detailed review see Nieder, 2016). The findings from this work showed that the response profiles of specific neurons match the predicted tuning curves from the above-described model. More specifically, the recorded neurons showed a systematic decrease in neural firing rate as the presented numerosities deviated from the preferred numerical quantity of a specific neuron (see Fig. 3a). For instance, a neuron that is tuned to represent six items shows a maximum firing rate when six items are shown. However, the neuronal activity systematically decreases as the number of items increases (e.g., from six to nine items) or decreases (e.g., from six to three items). More recent studies were able to demonstrate number sensitive neurons within the medial temporal lobe (MTL) of humans with similar tuning characteristics as described above (Kutter et al., 2018). These findings provided strong evidence for a biological implementation of the proposed approximate number system in the human brain. The brain location of the identified number neurons fits well with brain activation patterns observed in humans using fMRI-adaptation (also called habituation). This specific experimental paradigm draws upon the natural property of neural populations to change their neural response in relation to the repeated exposure of a specific stimuli (also known as the refractory effect). More specifically, during the so-called adaptation phase a specific dimension of interest (e.g., 16 dots) is repeatedly presented on the screen. After this habituation phase a new stimulus (e.g., 8 or 32 dots) is presented, which deviates from the previous stimuli in the dimension of interest (i.e., numerical quantity). If a certain neural population is sensitive to this change, a significant recovery signal (i.e., an increase in signal strength) from habituation can be measured. This increase in activation is good evidence that a specific brain region is involved in processing the manipulated stimulus dimension (e.g., numerical quantity). Using this technique, fMRI-adaptation studies (Piazza et al., 2007) have demonstrated that regions of the human IPS show a numerical ratio dependent brain signal recovery. In other words, when a close deviant number (e.g., 12 dots: 12/16 ¼ ratio of 0.75) is presented then the increase in the signal is smaller as when a distant deviant number (e.g., 8 dots: 8/16 ¼ ratio of 0.5) is presented. This recovery pattern can be explained by the tuning curves of number sensitive neurons: a larger signal recovery is expected for greater distances/ratios, since the representational overlap between number sensitive neurons decreases with distance/ratio. Although the above findings provide strong evidence for the existence of an approximate number system, there is still an ongoing debate whether it is limited to a single brain region – the IPS. Recent work has shown that refractory signals, as measured with fMRI-adaptation, depend on a convolution of multiple neuronal effects (response fatigue, altered response dynamics, response facilitation) and

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Fig. 3 (a) Anatomical location of the intraparietal sulcus (IPS) in blue on a 3D visualization of the human cortex. Additional regions of the frontal cortex – middle frontal gyrus (MFG) and premotor cortex (pmC) – and the medial temporal lobe (MTL), which have also been observed during numerical quantity processing are shown in green. Note that the MTL is not visible from a lateral view: this is indicated by the dashed line. (b) Schematic illustration of the response profiles of number sensitive neurons of monkeys (Nieder & Miller, 2004) and humans (Kutter et al., 2018). These neural tuning curves demonstrate stronger activation for a preferred number of items, which decays as the number of items deviate. (c) Visualization of the numerical ratio dependent BOLD signal recovery (refractory) effect in the human IPS. The brain responses increase when the distance of the deviant to the adaptation number increases. The inverted tuning curve can be explained by the response profiles of number sensitive neurons displayed above

neurophysiological processes (e.g., neurovascular coupling; for a detailed discussion see Harvey et al., 2017). Using comprehensive computations to model these complex dynamics, studies were able to identify different topographical organizations of number sensitive brain regions in the superior parietal lobule (SPL) and the postcentral cortex (Harvey & Dumoulin, 2017). The involvement of different brain regions and the topographical organization challenge the notion that the IPS is the only region to be involved. In addition to these findings in adults and nonhuman primates, an increasing number of neuroimaging studies have investigated the neural development of the approximate number system in children. Using different study designs, researchers have found converging evidence that the brain activation of infants and young children is modulated by the systematic manipulation of non-symbolic numerical quantities (e.g., Hyde et al., 2010; Izard et al., 2008). For instance, Izard and colleagues (2008) found a significant neural modulation in response to the manipulation of numerical quantities in 3-months-old infants using event-related-

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potentials (ERP). A source localization (a technique to reconstruct the origin of the EEG signal using mathematical models) indicated the right inferior parietal cortex as well as the right inferior frontal cortex and the left and right anterior temporal cortex as potential regions for the origin of this number sensitive activation. The brain activation identified by this study converges with another study in which a significant modulation of the right parietal cortex was detected in 6-months-old infants using functional near-infrared spectroscopy (fNIRS; Hyde et al., 2010). (FNIRS is another neuroimaging tool that infers brain activation from changes in the oxygenation of hemoglobin. In contrast to fMRI, this method uses near-infrared light to measure these changes. Multiple wavelengths of near-infrared light are emitted via “light-emitters” mounted on a head-cap. Because oxygenated and deoxygenated hemoglobin differ in their near-infrared absorption spectra, relative differences in hemoglobin concentrations can be measured via detectors, which are also mounted on the fNIRS head-cap. This relative change in oxy- and deoxygenated hemoglobin is then related to brain activity.) In both studies activation in the right IPS could be the source of number sensitive processing. Although the precise location of the signals remains unknown, the results clearly suggest that the encoding of numerical quantities can be already detected in infants. Using an fMRI adaptation study with 3-to-6-year-old children, Kersey and Cantlon (2017) demonstrated a significant number sensitive adaptation effect within the right and the left parietal cortex close to the IPS. This brain activation pattern showed similar neural tuning profiles as in adults (Piazza et al., 2007). A particular interesting finding of this study is that children’s ability to discriminate numerical quantities outside the scanner was associated with the neural tuning curves of the right parietal cortex. Again, indicating that especially the right IPS is sensitive to the processing of non-symbolic numerical quantities. While there is emerging insights from studies with infants and young children, most of the existing evidence comes from developmental neuroimaging studies with older children (for a meta-analysis see Arsalidou et al., 2018). These studies have revealed three central findings: First, regions of the IPS, either bilateral or on the right, are reliably activated when non-symbolic numerical quantities are processed. Second, as with infants and younger children, the brain activation patterns are not solely restricted to the IPS. Significant brain activations have also been found in the frontal cortex, the premotor cortex (pmC), the middle frontal gyrus (MFG), and the visual cortex (extrastriate cortex and the lingual gyrus; Arsalidou et al., 2018). The involvement of these additional brain regions indicates that children recruit multiple networks and resources to process numerical quantities. Third, the relative contribution of the involved brain regions changes with age and/or experience. While brain activation in the prefrontal cortex often shows a negative correlation with age, the brain activation in the parietal cortex, especially the IPS, often shows a positive correlation with age. The developmental shift in brain activation (from frontal to parietal) is often interpreted as a functional specialization to process numerical quantities more effectively. In other words, as children gain more experience with numerical quantities the processing becomes more automatic.

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Although the above discussed findings confirm the functional relevance of several brain regions, especially the right (bilateral) IPS, to process numerical quantities, the nature of the observed effects is far from conclusive. Non-symbolic number comparison tasks are often confounded by additional non-numerical dimensions (e.g., response selection or stimuli properties such as surface are or the size of the dots) that make the interpretation about the function of observed brain activations difficult. Indeed, different lines of research have indicated that the estimation and comparison of non-symbolic stimuli might be significantly influenced by these non-numerical dimensions. For instance, Leibovich and colleagues (2015) investigated the brain correlates of 19 right-handed students to better understand whether numerical quantities (i.e., the number of dots) or non-numerical quantities (i.e., surface area) are more salient. Participants performed a comparison task and were instructed to either compare numerical quantities (i.e., the dot array that has more dots) or non-numerical quantities (i.e., the dot array that containing more surface area). In addition, the correlation between surface area and numerical quantities was manipulated. In one condition both dimensions were positively correlated (i.e., congruent condition), in the other condition both were negatively correlated (i.e., incongruent condition). The results of this work showed a significant interaction, especially in the right temporal parietal junction (TPJ). Specifically, greater brain activation was observed during the numerical condition when the non-numerical quantities were negatively correlated with numerical quantities (incongruent trials). Because the TPJ plays a significant role in controlling stimulus-driven attention (i.e., bottom-up attention), the activation differences indicate a more automatic processing of non-numerical quantities compared to numerical quantities. As such it is possible that previously observed brain activation effects are not exclusively related to numerical quantity processing (e.g., increase in the precision of the ANS). One argument is that brain activation in response to non-symbolic numerical quantities might reflect the inhibition of irrelevant stimuli dimensions such as non-numerical quantities. These confounding dimensions need also to be evaluated when one considers the association of non-symbolic numerical quantities and arithmetic abilities. Several behavioral studies have reported a significant, albeit correlational, link between the processing of numerical quantities and arithmetic performance (Schneider et al., 2017). This link is further substantiated by neuroscientific evidence that has shown that individuals with brain lesions (Delazer et al., 2006) and developmental dyscalculia exhibit deficits in non-symbolic numerical quantity processing (Butterworth et al., 2011). Functional differences in task-related brain activation between children with and without arithmetic difficulties have been found across different brain regions, including the IPS, the parieto-occipital cortex, and the fusiform gyrus (Arsalidou et al., 2018). While these brain activation differences are often interpreted as evidence for a deficient representation of non-symbolic numerical quantities in children with developmental dyscalculia, the findings could also be interpreted as difficulties to inhibit or to suppress incongruent non-numerical dimensions (Bugden & Ansari, 2016). For instance, Wilkey et al. (2017) tested the influence of congruent (i.e., correlation between surface area and

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numerical quantities) and incongruent (i.e., no correlation between surface area and numerical quantities) task conditions on brain activation and whether these different conditions relate to individual differences in students test scores of the Preliminary Scholastic Aptitude Test (PSAT). (A nationally administered test that is used as mathematical entry exam in US high schools.) The neuroimaging results provided significant evidence that congruency did not influence the general brain activation pattern during numerical quantity discrimination. However, a significant difference in the association between congruent and incongruent trials with mathematical abilities was found. While activation in congruent trials demonstrated a significant positive relationship with PSAT scores in the right supramarginal gyrus (SMG), activation in incongruent trials showed a negative correlation with PSAT scores in the left angular gyrus (AG). These findings indicate (a) that the correlation between brain activation and mathematical abilities differs as a function of congruency, and (b) that associations with arithmetic cannot be reduced to one single dimension (i.e., the representation of numerical quantities): it might also involve the inhibition of non-numerical dimensions. Together, there is accumulating evidence that several regions of the human brain, especially within the parietal cortex, constitute a biological substrate for processing non-symbolic numerical quantities. Brain activation in response to non-symbolic numerical quantities can be found in infants as well as in young children. Although the precise locations are unknown, the right IPS might play an instrumental role in the representation of these quantities. There is also evidence for a relative shift in brain activation from frontal to parietal brain regions that is correlated with age, indicating experience-dependent changes in how non-symbolic numerical quantities are processed. Despite this convincing evidence, a number of unknowns remain. For instance, the parietal cortex consists of a patchwork of different subregions that are involved in numerous cognitive functions – ranging from perception, control of action, visual-spatial attention to higher-order cognitive processes. This mosaic of brain functions make it extremely difficult to provide conclusive information about the precise neurocognitive mechanisms that are engaged during numerical quantity processing (Vogel et al., 2015a, 2017a). One potential confound is that some of the observed brain activation might be related to the inhibition of non-numerical dimensions during numerical quantity processing. It is also plausible that observed developmental changes during these tasks constitute a dynamic integration of domain-general and domain-specific resources to efficiently act upon numerical quantities (i.e., to efficiently access the relevant semantic information). As such it remains an open question, up to this day, what the neural responses during non-symbolic numerical quantity discrimination in children reflect and how they develop. An interesting question for future research is to investigate the developmental integration of functional networks that are engaged during the perception of non-symbolic quantities and the networks that are engaged during the active discrimination process (i.e., from perceiving numbers to acting on numbers). This question goes hand in hand with a more detailed analysis of the developmental trajectory in specific age ranges. Whether the ANS constitutes a causal foundation for the development of symbolic numerical abilities is still heavily debated. But as

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we will see in the later sections, a number of reasons indicate that this may not be the case.

The Object Tracking System The second cognitive system that is implicated in the early development of non-symbolic numerical abilities is the object tracking system (or parallel object individuation system). This system enables the precise individuation of three up to four elements within a visual scene (Fig. 4). The existence of this separate system was first described by Jevons (1871) who demonstrated a dissociation between the discrimination of small (4) quantities. When participants were asked to count the number of beans tossed into a pan, he observed that the enumeration of one up to four elements was extremely precise, while errors systematically increased as the number of elements moved beyond four. This phenomenon was termed subitizing (Kaufman et al., 1949). The neurocognitive mechanisms of subitizing have been predominantly associated with a domain-general, visual-spatial system that processes object boundaries, predicts object movements, and retains a small number of objects in working memory (e.g., Hyde, 2011). The individual object tracking system develops rapidly. Whereas 6-months-old infants display a capacity limit of one item, 12-months-old infants show adult like abilities (i.e., three-to-four; Oakes et al., 2006). Because of its properties to perceive a limited number of individual objects with a high precision, it is thought to function as an important biological primitive for the development of the so-called successor function (n þ 1). This function might constitute an important foundation for the development of arithmetic computations, as it reflects the mental

Fig. 4 (a) Schematic illustration of the subitizing effect. While the error rate to enumerate objects increases in the large number range (>4), the enumeration of objects in the small number range is quite accurate. (b) Location of brain regions that have been associated with the enumeration of small numbers

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understanding that adding one element to a given quantity increases the quantity exactly by one. Neuroimaging studies with adults have identified brain regions related to the object tracking system in areas of the inferior parietal cortex, the posterior parietal cortex and the occipital cortex (e.g., Piazza et al., 2003), as well as the TPJ (Ansari et al., 2007). For instance, Ansari and colleagues (2007) used fMRI to explore the neural activation associated with the processing of small quantities in the subitizing range and of large quantities that require estimation. The results of this study showed that the brain activation of the TPJ was moderated by the presentation of small or large quantities. While the brain activation in this region was suppressed for the discrimination of large quantities, brain activation increased for the discrimination of small quantities. This differential engagement might be reflective of a stronger reliance on stimulus-driven attention during the processing of small quantities. An interpretation that is in line with the proposition that the enumeration of small quantities connects to the visual (stimuli-driven) properties of the object tracking system. Existing evidence on the brain development of the object tracking system is extremely sparse. To the best of our knowledge only one neuroimaging study has investigated the object tracking system in infants (Hyde & Spelke, 2011). The results of this electroencephalography (EEG) study showed an early modulation of the N1 component (~150 ms post stimuli onset) in relation to small sets (i.e., object tracking system), and a later modulation of the P2p component (~250 ms post stimuli) in relation to the enumeration of larger sets (i.e., ANS) over parietal regions. These findings indicate a temporal differentiation between the two systems that can be measured on electrodes over parietal regions. However, since a source detection (i.e., the exact anatomical location of the signal cannot be directly inferred from the location of the electrodes on the scalp) was not performed, the specific localization of these effects in the infant brain is not possible. As such it remains an open question, whether the object tracking system and the ANS reflect two coins of the same brain mechanisms (i.e., the single system view) or whether they are entirely distinct (i.e., the double system view; (i.e., the double system view; Hyde, 2011). The sparse availability of developmental neuroimaging studies highlights the need for further investigations to better understand the neurocognitive mechanisms of the object tracking system and its interaction with the ANS over developmental time.

The Numerical Meaning of Symbols Different theoretical models have been proposed to explain how symbolic representations (such as Arabic numerals) develop in the human brain and how it provides the semantic foundation for the acquisition of arithmetic and more complex mathematical skills. One central framework is the Triple code model by Dehaene (1992). In this well-known model (see Fig. 5a), three representational codes for numbers were proposed: an approximate code of quantities (originally named analogue magnitude code), a code for the representation of visual information (originally named Visual-

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Arabic code), and an auditory verbal code to represent number words. The three numerical representations were proposed to be located in different brain regions in the parietal, temporal, and occipital cortex (see Fig. 5b) and to build the semantic foundation for number processing. As such it has been argued that the quality of these codes (e.g., the precision with which numerical quantities are represented) and the ability to efficiently transfer numerical information between these codes (e.g., the Arabic digit 6 can be transcoded into the number word /six/) are directly related to individual differences in numerical and arithmetic skills. A central question that arises from this framework is how the encoding between symbolic and non-symbolic codes is established?

Mapping Numerical Symbols onto Quantities? Two central frameworks have been proposed to explain the mapping between numerical symbols and the quantities they represent. The first idea – the symbolicquantity mapping – suggests that symbolic representations (visual or verbal) are

Fig. 5 (a) Illustration of the Triple code model proposed by Dehaene (1992). (b) The Triple code model visualized on a 3D model of the human brain. (c) The left side illustrated the neural network model in which non-symbolic and symbolic input fields map onto a number field of numerical quantities. The right side shows the tuning curves of non-symbolic and symbolic inputs estimated by the neural network model. Note that the representational precision (i.e., the width of the tuning curves) is greater for symbolic than for non-symbolic representations

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directly mapped onto the ANS via associative learning. It argues that a link between the preexisting biological foundation that represents numerical quantities (i.e., the ANS) and number symbols is established. In other words, children learn to associate a set of symbols (e.g., “6”) with a preexisting code of numerical quantities (e.g., “the sense of sixness”). The second idea suggests that an abstract code for number representations emerges as a function of a joint activation between symbolic and non-symbolic information. This framework does not assume a preexisting biological foundation but argues in favor for a mental construction of number representations. Supporting evidence for this hypothesis comes from a neural network (Fig. 5c) model that has simulated the joint activation of non-symbolic and symbolic number stimuli (Verguts & Fias, 2004). The model consists of two input fields: the symbolic field, which simulates the visual/auditory activation of number symbols/words, and a non-symbolic field, which simulates the activation of non-symbolic numerical quantities. Each node of the symbolic input field represents a discrete symbol (e.g., an Arabic numeral numeral). Numerical quantities are expressed as the summation of the activated nodes (e.g., the activation of four nodes represent four objects) in the non-symbolic input field. The results of the simulation showed that the neural network was able to learn a semantic link between the symbolic and non-symbolic input fields via a third associative field – the number field. A central finding of this work is that this number field generates similar neuronal response profiles (i.e., tuning curves that were discussed further above) without assuming a preexisting representation of numerical quantities as proposed in the “symbolicquantity mapping” account. This finding demonstrates that number-specific properties can emerge from associative learning without assuming a preexisting system to process numerical quantities. Thus, challenging the idea of an innate approximate number system. Another observation is that the number field demonstrates formatspecific differences. More specifically, the simulated width of the symbolic tuning curves was smaller compared to the width of non-symbolic tuning curves. This format-specific difference indicates that symbolic inputs represent numerical information with a greater precision compared to non-symbolic inputs. This precision might be a central feature of symbolic representations that allow an accurate understanding of numerical quantities. To explore the brain mechanisms associated with the above mentioned frameworks, researchers have investigated the brain correlates associated with the development of symbolic and non-symbolic encoding in adults (e.g., Holloway & Ansari, 2010; Piazza et al., 2007; Vogel et al., 2017a) and children (Emerson & Cantlon, 2015; Park et al., 2014). Several different questions have been addressed in adults. First, does the brain activation of symbolic and non-symbolic formats overlap? Or do they activate distinct brain regions? Holloway et al. (2010) used a symbolic and non-symbolic comparison task to investigate this question. The results of this fMRI study showed that the right IPS was activated during both conditions (symbolic and non-symbolic). A direct contrast between the two formats further revealed the engagement of the right superior parietal lobe during non-symbolic processing, and regions of the left temporo-parietal cortex – in particular the AG and left superior

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temporal gyrus (STG) – during symbolic processing. These results indicate that the two notations might activate distinct encoding pathways that converge upon a common representation within the IPS. This convergence was further confirmed by another neuroimaging study that used functional connectivity analysis (i.e., structural equation modeling) to directly investigate the processing pathways of symbolic and non-symbolic quantities (Santens et al., 2010). The results of this analysis indicated two distinct processing pathways for symbolic and non-symbolic quantities that converge within regions of the IPS. While the pathway for processing non-symbolic quantities included brain region of the superior parietal cortex, the pathway for processing symbolic quantities did not. These findings indicate that symbolic and non-symbolic processing activates common as well as distinct brain regions. It seems that two encoding pathways, a symbolic and a non-symbolic, converge onto a common brain region within the parietal cortex, as suggested in the neural network model of Verguts and Fias (2004). A slightly different question was addressed by Piazza et al. (2007). Do the response profiles (i.e., Tuning curves) between symbolic and non-symbolic brain activation differ? To answer that question, the authors used fMRI-adaption to habituate the brain response of adults to dot-arrays or to Arabic numerals. After this adaptation phase, close or far numerical quantities in the same or different notation were presented. The analysis of this study showed similar response profiles for symbolic and non-symbolic stimuli in the right IPS; however, format-specific response profiles also emerged in the left IPS. In line with the above discussed neural network model, the results showed a greater representational precision for symbolic numbers than for non-symbolic quantities. Especially the left IPS might be important for the encoding of symbolic representations, as the symbolic representations showed smaller tuning curves (smaller width) and therefore a higher precession to encode numerical quantities compared to non-symbolic representations in the right IPS. The distinctive involvement of the left IPS in symbolic, and the right (bilateral) IPS in non-symbolic number processing, might indicate a dynamic interaction between these regions to construct numerical quantity representations. The question whether the left IPS shows similar response profiles for different symbolic formats (i.e., Arabic digits and spoken number words) was tested in another fMRI-Adaptation study. In this work, Vogel and colleagues (2017a) tested the brain activation in response to the visual presentation of Arabic numerals and the auditory presentation of number words (both are symbolic representations). The results of two experiments showed that the left IPS was the only region of the brain that demonstrated brain activation in response to both stimuli formats, indicating that symbolic processing converges to an abstract (i.e., format independent) representation of symbolic quantities. Another neuroimaging study, which investigated the brain response of bilingual Chinese-English participants in contrast to an English-speaking control group, found converging evidence for a specific knowledge-dependent symbolic representation (Holloway et al., 2012). The crucial manipulation in this study was the presentation of two different symbolic numerical formats: the Arabic numerals that both groups could read and Chinese ideographs that only the bilingual Chinese-English group could read. As expected, the

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presentation of Arabic numerals elicited brain activation in the left IPS in both groups. However, the presentation of Chinese ideographs only activated the IPS of the bilingual group. The parietal cortex of the English-speaking participants, who were not familiar with the Chinese ideographs, was not activated. Together, these findings indicate that the processing of numerical symbols might be related to brain activation in the left IPS (in contrast to a greater engagement of the right/bilateral IPS during non-symbolic processing) and that the encoding of symbols is realized with a higher representational precision compared to non-symbolic numerical quantities, possibly via experience and education. These above discussed results of hemispheric (left/right) differences in the IPS are also supported by developmental neuroimaging studies with children. Emerging evidence indicates that the precision of symbolic representations change with age (Emerson & Cantlon, 2015; Vogel et al., 2015a) and can be associated with behavioral performance skills. For instance, Vogel and colleagues (2015a) used fMRIadaption to investigate age-related changes in brain activation in response to the presentation of number symbols in 6-to-14-year-old children. The analysis demonstrated a systematic increase of brain activation (i.e., recovery effects) in the left IPS with age. The precision of the left IPS to differentiate symbolic numbers increased with age. This change might reflect a developmental refinement (i.e., functional specialization) of the left IPS to accurately represent numerical quantities with symbols. A finding that was supported and further extended by another fMRI study with children. In this longitudinal fMRI study, Emerson and Cantlon (2015) investigated the brain correlates of 4-to-9-year-old children. The imaging results showed a significant age-dependent association between the brain activation of the left IPS and children’s symbolic numerical discrimination performance measured by the numerical distance effect. Thus, indicating that the neural precision of encoding symbolic numerical quantities in the IPS can be directly linked to behavioral performances outside the scanner. Nevertheless, it appears that the left/right distinction is too simplistic to explain the construction of symbolic representations and their associations to behavioral performances. Additional research indicates that there is a complex developmental interplay that underlies the construction of symbolic numerical knowledge. For instance, in a functional connectivity analysis, Park et al. (2014) investigated task-related functional connectivity patterns in relation to symbolic and non-symbolic comparison tasks and their associations with mathematical achievement and age in 4.5-to-6.5-year-old children. Interestingly, the functional connectivity analyses revealed task-related functional associations between the right IPS with two other brain regions: the connection with the left SMG was correlated with age, and the connection with the right precentral sulcus was correlated with individual differences in mathematical achievement. These findings further indicate that the functional integration of multiple brain circuits underlies the construction of symbolic representations and that learning of symbolic numerical information cannot be restricted to a simple mapping between symbolic and non-symbolic information. Together, the discussed evidence indicates that symbolic numerical knowledge is constructed in the human brain and that the left IPS might play a crucial role for representing symbolic representations with a higher precision than non-symbolic

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quantities. This process is likely the outcome of education and experience as well as a construction process that might include a number of different brain regions, extending beyond the classical suspects of the IPS and the visual cortex. These findings challenge the notion of a direct mapping account.

Additional Challenges to the Mapping Account Although the argumentation of a mapping account is appealing, several neuroimaging and behavioral findings have been reported that are inconsistent with the direct symbolic-quantity mapping account. (a) The classification and the adaptation of brain activation patterns leads to modest effect sizes or even fails between symbolic and non-symbolic formats (digits ! dots; dots ! digits). In contrast, large and significant effects have been reported for within formats classifications or adaptations (digits ! digits; dots ! dots; Bulthé et al., 2014). If symbols are tightly connected to non-symbolic quantities, a high classification accuracy between symbolic and non-symbolic formats (digits ! dots and dots ! digits) should occur; (b) the neural activation patterns of symbolic representations have been shown to be associated with one another, while the activation pattern between symbolic and non-symbolic representations seem to be unrelated to one another (e.g., Lyons et al., 2015). The mapping account predicts a significant correlative association between symbolic and non-symbolic representations; (c) an emerging body of evidence suggests that reported behavioral associations between non-symbolic processing and symbolic number processing are driven by individual differences to inhibit non-numerical dimensions rather than due to variations in numerical quantity processing (Wilkey et al., 2017). These findings indicate that observed links between symbolic and non-symbolic formats are established via a third variable; (d) a recent longitudinal study found evidence that early symbolic processing predicts later non-symbolic numerical quantity processing abilities. This was not the case for the other direction (Lyons et al., 2018). Inconsistent with the mapping account, this data suggest that symbolic knowledge may drives the development of non-symbolic processing. It further suggests that non-symbolic quantity representation may play a rather subordinate role for predicting symbolic math, and that there is no unidirectional link from non-symbolic numerical quantities to symbolic numerical quantities; (e) behavioral evidence has demonstrated that symbolic number representations do not solely rely on non-symbolic numerical quantity processing but are also associated with another important dimension – numerical order – that shows unique behavioral effects that differ between symbolic and non-symbolic numerical processing (e.g., Vogel et al., 2017b). Mapping Symbols to Symbols: The Case of Numerical Order Several behavioral studies with children and adults have demonstrated that the processing of numerical order (i.e., the knowledge that a given object occupies a relative rank or a position within a given set of items) elicits a different behavioral pattern of reaction times and accuracy rates compared to symbolic numerical quantity processing. In humans, the processing of numerical order is often investigated with a numerical verification task (see also Fig. 6a). In this task, children or

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adults have to verify as fast as possible whether the sequential order of three presented digits is correct (e.g., 2 3 4 for an ascending correct sequence) or incorrect (e.g., 3 4 2). Several behavioral and neuroimaging studies have shown that this numerical ordinal judgment tends to be faster for adjacent numbers (e.g., 2 3 4) compared to distant numbers (e.g., 2 4 6) in the correct order condition (i.e., numbers that are in correct ascending or descending order). Because this reaction time pattern is the opposite of the above discussed distance effect, it has been labeled as the reverse distance effect (Lyons & Beilock, 2011). The existence of the reverse distance effect has been confirmed in children (Lyons & Ansari, 2015) as well as in adults (Lyons & Beilock, 2011; Vogel et al., 2017b, 2019), and it suggests that numerical order constitutes a unique dimension of numerical representations. The predictive value of numerical order processing for arithmetic abilities was demonstrated in a large cross-sectional study with 1391 children (Lyons et al., 2014). In addition to a battery of different non-numerical tasks the children also performed a numerical order task, a numerical magnitude task and a test of arithmetic abilities. The results of this study demonstrated a significant relationship between numerical

Fig. 6 The upper panel shows a schematic illustration of the ordinal verification task. ISI, interstimulus interval of 1500–2500 ms. The lower panel shows a graphical representation of the canonical numerical distance effect and the reverse distance effect typically observed in quantity processing tasks and ordinal verification tasks, respectively

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order processing and arithmetic and that the predictive value of numerical order processing progressively increased from grade 1 to grade 6. An opposite pattern was observed for numerical magnitude processing, whose predictive value decreased with age. Specifically, while numerical magnitude was the best predictor of arithmetic performance in grade 1 and 2, numerical ordinal processing became the best predictor in grade 6, suggesting an important developmental interaction between processing numerical magnitude, numerical order, and arithmetic abilities during the first years of formal education. Despite this clear association between numerical ordinal knowledge and arithmetic, not much is currently known about the neurocognitive mechanisms involved during numerical order processing. Some evidence indicates that the reverse distance effect relates to an efficient retrieval of adjacent items (e.g., chunks) stored in long-term memory (Vogel et al., 2019). Interestingly, the reverse distance effect has also been demonstrated for non-numerical ordinal judgments such as with letters of the alphabet or the months of the year (Vogel et al., 2017b). For instance, in a behavioral study with adults, Vogel and colleagues (2017b) collected behavioral data from a group of participants who performed different ordinal verification tasks: a symbolic ordinal verification with numerals (e.g., 2 3 4), a symbolic ordinal verification with letters of the alphabet (e.g., B C D), and a non-symbolic ordinal verification with dot arrays (e.g., oo ooo oooo). The reaction time results of this study demonstrated a systematic and reliable reverse distance effect in the symbolic conditions with numerals and letters, but not with the non-symbolic dot arrays. This finding suggests (a) that the reverse distance effect is linked to the processing of ordinal knowledge across different domains (numbers and letters) and (b) that the reverse distance effect is indicative of symbolic numerical order processing and not of non-symbolic numerical order processing. Since letters of the alphabet have no existing correspondence to numerical quantities, it suggests additional sources from which number symbols derive their meaning. Investigations into the brain mechanisms of numerical order processing and its development is restricted to a handful of studies. The first study to investigate numerical order and numerical quantity processing found significant differences in the latency and the magnitude of two different event-related-potentials (ERP) in a group of adults (Turconi et al., 2004). Compared to numerical quantity task, the results of the numerical order task showed a delayed and bilateral response of the P2 component at parietal electrodes, and a greater response of the P3 component at frontal electrodes. This finding strengthens the evidence for a dissociation of the two dimensions. Although the anatomical source of the ERP components is ambiguous, the involvement of parietal and prefrontal regions during numerical ordinal judgments has also been reported in fMRI studies. In an fMRI study with adults, Lyons and Beilock (2013) investigated the neural correlates of numerical order processing and numerical quantity processing. Participants were asked to perform a symbolic numerical order task (i.e., are the numbers in increasing/decreasing order or in mixed-order), a symbolic numerical comparison task (i.e., which of two numerals is larger), a non-symbolic order task (i.e., are the dot arrays in increasing/decreasing order or in mixed-order), and a non-symbolic comparison task (i.e., which dot-array contains more dots). The analyses revealed the involvement of the IPS during

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non-symbolic ordinality and cardinality judgments, possibly indicating a common mechanism during non-symbolic ordinal processing and numerical quantity judgments (e.g., a decomposition of the problem into an iterative comparing of quantities; e.g., oo ooo oooo ¼ oo < ooo and ooo < oooo). The ordinal processing of number symbols, however, engaged a distinct brain network encompassing frontal brain regions, including the rostral supplementary motor area (PreSMA), the left dorsal premotor cortex (PMd), and the left ventral premotor cortex (PMv). The authors suggested that this activation pattern could be indicative for the retrieval of sequential visuomotor associations from long-term memory. The observed differences between symbolic and non-symbolic ordinal processing also indicated that numerical order and processing may differ as a function of format (symbolic vs non-symbolic). While number symbols may be related to a sequential retrieval of associations, the processing of dot-arrays may be linked to numerical quantity processing. Two imaging studies have contrasted the neural correlates of numerical order and numerical magnitude processing in typical developing children (Matejko et al., 2019; Sommerauer et al., 2020). Matejko et al. (2019) investigated the brain response of children and adults who performed a symbolic number comparison and an ordinal verification task. The results of this neuroimaging study showed that adults engaged the left inferior parietal cortex during numerical order processing, while children exhibited brain activation in the right lateral orbital and inferior frontal gyri (IFG) during both numerical order and numerical magnitude processing. The authors interpreted these age-dependent differences as evidence for a developmental differentiation of numerical order and numerical magnitude processing in the inferior parietal cortex – especially in the left IPS. Sommerauer and colleagues (2020) used similar tasks to investigate developmental changes across both numerical dimensions and their associations with arithmetic performance in children attending elementary school. In line with the above findings, the results showed a developmental increase in the activation pattern of the left IPS in response to numerical order but not in response to numerical magnitude processing. A significant association with arithmetic was found in two brain regions of the semantic control network: at the right posterior middle temporal gyrus (pMTG) and at the right inferior frontal gyrus (opercular part; IFGOp). Consistent with the behavioral literature, this finding indicates that individual differences in the neural correlates of numerical order processing map onto individual differences in arithmetic abilities. A finding that was also confirmed in two neuroimaging studies that investigated individual differences in brain activation pattern of numerical order processing between typically and atypically developing children (Kaufmann et al., 2009; McCaskey et al., 2018). Kaufmann and colleagues (2009) compared the neural correlates of numerical order processing in typically and atypically developing children, the latter identified to have mathematical learning difficulties (MD). Results of this study showed stronger activations in the anterior cingulate gyrus, the right inferior parietal regions (including the IPS), and the supramarginal gyrus (SMG) in children with MD. The authors interpreted this stronger activation in children with MD as compensatory

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mechanisms to perform the task. In a longitudinal study, McCaskey et al. (2018) found an age-dependent (8–11 years) activation increase in the IFG, the middle frontal gyrus, and the left IPS in children with MD (consistent with a compensatory account). However, no significant age-dependent changes in brain activation were found in the control group of typically developing peers. The increase in brain activation of children with MD might suggest a greater engagement of additional cognitive control mechanisms to compensate for their ability to access relevant numerical information. Together, the evidence from this novel research has demonstrated that (a) different and overlapping brain activation patterns are engaged when participants process the order of symbolic numbers compared to when they process the quantity of non-symbolic and symbolic numbers (e.g., Lyons & Beilock, 2013), (b) the brain correlates of numerical order processing change over developmental time (Matejko et al., 2019; Sommerauer et al., 2020), and (c) individual differences in the neural correlates of numerical order processing map onto differences in arithmetic processing (Kaufmann et al., 2009; McCaskey et al., 2018; Sommerauer et al., 2020). The findings further suggest that numerical order constitutes a unique dimension that activates regions of the parietal cortex and frontal cortex. Consistent with a multidimensional representation account, this evidence indicates that the IPS is not solely responsive to numerical quantities, but rather encodes a multidimensional construct of symbolic knowledge. A simple mapping account fails to distinguish between the rote understanding of quantities, which animals possess, and the semantic understanding that humans exhibit. It is therefore possible that the correspondence between symbols and quantities is subordinate to a rich web of symbolic association that is established over developmental time and experience. The precise interactions of how these dimensions develop in the human brain, how these brain activations are linked to each other, and how a multidimensional representation of number symbols is linked to arithmetic needs to be further tested. The acquisition of this associative symbolic representation might build an independent foundation for the rich and more complex arithmetic operations of the human mind.

Arithmetic Symbolic numerical knowledge enables the use of sophisticated arithmetic skills. The previous sections discussed several lines of research, which indicate that the representations of numerical-quantity and numerical-order constitute important semantic information that predicts arithmetic performances. Whether theses dimension represent a causal foundation for the development of arithmetic skills needs to be further determined. But it appears that a constructive account of symbolic information is the key to unlock causal mechanisms. Nevertheless, the last decade of research has also seen a remarkable growth in our understanding of how the human brain processes arithmetic problems such as addition, subtraction, and multiplication. Our knowledge about these neural correlates is tightly connected to different behavioral indices of arithmetic problem solving.

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A well-known index of arithmetic problem solving is the problem size effect (also called difficulty effect). This effect describes the basic observation that arithmetic problems with smaller operands are solved faster (and with less errors) compared to problems with larger operands. For instance, participants solve the multiplication problem 3  4 ¼ 12 faster, and with fewer errors, than the problem 7  9 ¼ 63. The effect is very robust and holds for all arithmetic operations – additions, subtractions, multiplications, and divisions. It can be observed in children as well as in adults and it occurs in production tasks (i.e., participants have to generate the correct answer) as well as in verification tasks (i.e., participants have to verify whether a presented answer to a problem is correct or incorrect, for a detailed discussion see Ashcraft, 1992). Different accounts have been proposed to explain the nature of the problem size effect. Some argue that it reflects differences in the frequency of exposure when arithmetic problems are learned. Other proponents argue that “structural characteristics” of numerical relations differ between small and large problems. For instance, larger problems are more difficult to solve than smaller problems because their mental representations are less differentiated (Campbell, 1995). The most prominent and contemporary account argues that the problem-size effect relates to different arithmetic strategies: small problems are considered to be solved via fact retrieval, whereas large problems are thought to be solved via error-prone and time-consuming quantity-based procedural strategies (such as counting or decomposing a difficult problem into smaller sub-problems; e.g., Campbell & Xue, 2001). The latter explanation is also in line with developmental perspectives. Several studies have reported an age-related transition from slow and error-prone procedures to fast and accurate retrieval strategies (Lemaire & Siegler, 1995). This transition seems to be not stage like, but rather constitutes a gradual (linear) shift in the frequency with which different strategies (e.g., procedural or fact retrieval) are used to solve the problem (for more details see the overlapping waves theory from Siegler, 1996). Through repeated practice, or direct memorization, children progressively build an arithmetic fact network that enables the efficient retrieval of answers from long-term memory (Lemaire & Siegler, 1995). Over the past decade, several brain imaging studies have investigated the neural correlates associated with the problem-size effect in adults (e.g., De Visscher et al., 2018) as well as in children (e.g., De Smedt et al., 2011). Consistent with the view of different processing routes (i.e., procedural and fact retrieval strategies), the results of these studies have identified the engagement of different brain networks (see also Fig. 7). The smaller network encompasses the angular gyrus (AG) and the supramarginal gyurs (SMG), the larger network encompasses brain regions of the inferior and superior frontal cortex, the IPS and the FG. These two networks show opposite effects. While the former demonstrates higher brain activation (i.e., less deactivation) for small compared to large problems (e.g., Polspoel et al., 2017), the latter network demonstrates higher brain activation for large compared to small problems (e.g., Polspoel et al., 2017). In line with the introduced strategy account, the opposed engagement of these brain networks is linked to differential processing mechanism associated with the

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Fig. 7 Visualization of the brain regions engaged during arithmetic in children and in adults. The relative contribution of these regions depends on the task, operation, experience, and age. In blue: regions of the frontal cortex. In green: regions of the parietal cortex. In red: regions of the occipital cortex. And in yellow: regions of the medial temporal cortex. Note that the Hippocampus, which is located in the medial region of the temporal lobe, is not visible from the lateral surface of the brain, hence the dashed visualization of this structure

retrieval of arithmetic facts and the execution of procedural operations. While the functional modulation of the first network (i.e., AG and the SMG) is typically associated with phonological processes engaged during fact retrieval (i.e., a verbal code to retrieve the answer from long-term memory), the activity of the second network is related to a mosaic of domain-specific and domain-general processes that are engaged during procedural arithmetic operations. Brain activity in the IPS is typically linked to quantity-based operations (used during calculation), activity in the FG is often linked to the visual processing of symbolic numbers, and brain activity in the prefrontal cortex is associated with additional executive resources (e.g., working memory). Although the activation patterns are well established, some of the interpretations have been recently questioned (for a more detailed discussion see Menon, 2014). For instance, neuroimaging studies have indicated that the AG and the SMG may play an instrumental role in an automatic mapping of specific properties of an arithmetic problem (its visual characteristics) to the corresponding semantic answer in long-term memory. Developmental changes in the neural correlates associated with arithmetic have also been reported in several studies. The majority of the existing work has characterized these changes by comparing different age groups (e.g., Chang et al., 2016; Qin et al., 2014). Only a handful of studies have correlated brain activations with age (e.g., Prado et al., 2014), and even fewer studies have used a longitudinal approach (Qin et al., 2014). Findings of these studies indicate that several brain regions show

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an age-related increase and decrease in brain activation. An increase in brain activation has been primarily reported for posterior regions of the brain such as the AG, the SMG, and the IPS. Age-related negative associations have been reported for anterior regions such as the DLPFC and the VLPFC. The opposing effects have been interpreted as a functional specialization of arithmetic problem solving, which converges with the evidence of a shift in the frequency with which strategies are used to solve these problems (Siegler, 1996). While younger children rely more heavily on procedural strategies, associated with regions of the prefrontal cortex, older children rely more often on fact retrieval, associated with posterior regions of the parietal cortex. More recent work has explicitly tested this linear shift in brain activation. For instance, Chang and colleagues (2016) investigated changes in brain activation in cytoarchitectonically predefined regions within the parietal cortex. Children (7–10 years old), adolescents, and adults performed a subtraction task inside an MRI scanner. Results showed a linear increase in brain activation within the anterior IPS (IPS-hIP1), the posterior section of the SMG (SMG-PFm), and the anterior AG (AG-PGa). A nonlinear change (inverted U shape) was found in the middle portion of the SMG (SMG-PF). These findings suggest subtle developmental changes in subregions of the parietal cortex, which might be related to the gradual (linear) shift in strategy change. However, strategy change was not explicitly assessed in this work, and the connection between brain activation and actual changes in strategies remains to be verified. Further evidence comes from the longitudinal work by Qin et al. (2014). The authors demonstrated a brain activation increase in the Hippocampus (HC) and a brain activation decrease in prefrontal regions in children (7–9 years), who solved addition problems at two different time points 1.2 years apart. Importantly, collected verbal reports indicated a significant increase of fact retrieval strategies during this time. This cooccurrence provides further evidence for a gradual shift in functional brain specialization. It also indicates an important role of the HC for arithmetic fact consolidation. The additional finding that adolescents and adults showed lesser brain activation in the HC compared to children indicates a time-sensitive phase of arithmetic fact consolidation. Several neuroimaging studies have also reported different brain activation patterns in relation to different arithmetic operations (e.g., Zhou et al., 2007). For instance, Zhou et al. (2007) found overlapping but also different brain activation patterns for small addition and multiplications problems in the parietal cortex. Both operations engaged a large network of brain regions, including the supplementary motor area, regions of the inferior and superior parietal cortex, precentral gyrus, middle frontal gyrus, insula, and inferior occipital cortex. However, additions showed a relative larger engagement of the intraparietal sulcus, while multiplications showed a greater activation in regions of the precentral gyrus, supplementary motor, and posterior and anterior parts of the temporal gyrus of the left hemisphere. The authors interpreted this finding as evidence that the retrieval of addition facts and multiplication facts engages different processing mechanisms to a different degree. There might be a greater reliance on visuospatial processing mechanisms (including quantity-based procedures) for additions, and a greater reliance on verbal processing

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mechanisms for multiplications – the observed brain regions are often found to be activated in language-based operations. In children, operation effects have also been reported (De Smedt et al., 2011; Prado et al., 2014). In a cross-sectional fMRI study, Prado et al. (2014) found an age-related increase of brain activity for multiplication problems in the left temporal cortex, and an age-related increase in the right parietal cortex for subtractions. Similar to the study of Zhou et al. (2007) in adults, these differences might be explained by the engagement of different processing mechanisms. The solution of multiplications might engage additional verbal processes, while subtractions might engage quantity-related processes. However, the reported studies did not always control for differences in participants’ strategy use. It is often assumed that all problems of the same operation are solved with the same strategy. An fMRI study by Polspoel et al. (2017) controlled for these arithmetic strategy differences in children and could not find significant brain activation differences between subtraction and multiplication problems. Although fMRI has excellent spatial resolution, it could be the case that other neuroimaging methods – such as EEG are more sensitive to subtle operation differences. Brunner and colleagues (2021) used the exact same stimuli and a similar procedure to control for strategy differences as Polspoel et al. (2017), but investigated differences in specific frequency bands in the EEG. The results of this EEG study not only showed a significant problem size effect in the theta band greater event-related synchronization (ERS) (Event-related desynchronization and synchronization (ERD/ERS) describe induced oscillations in predefined frequency bands of the EEG signal (Pfurtscheller & Lopes da Silva, 1999). Previous studies have associated theta ERS (around 3–6 Hz) with information retrieval from memory (Bastiaansen & Hagoort, 2003) and working memory (Sammer et al., 2007).) for large problems compared to small problems, especially at parieto-occipital electrodes, but also a significant difference between operations in the theta band. In this frequency band, retrieved multiplication problems showed a greater ERS in the signal compared to retrieved subtraction. Although the reason for this dissociation is not clear, the finding indicates that subtle operation differences exist, even when individual differences in strategies are controlled. This question needs to be further investigated. Another important behavioral effect, the so-called arithmetic interference effect, has received much attention in the last years. The main principle of the interference effect is that the compositions of arithmetic problems are restricted to the same visual elements: the digits from 0 to 9. For instance, the arithmetic problem 3  9 ¼ 27 shares 3 digits with the arithmetic problem 3  7 ¼ 21 (the digit 3, 2, and 7). Interference occurs especially when children learn to memorize new arithmetic problems (such as multiplications) that share features with previously learned ones. In other words, if a child memorizes a new arithmetic problem, and this new problem is very similar in its visual features to an already learned arithmetic problem, high interference is induced. This process is called proactive interference: the memorization of a new problem interferes with a similar problem that has already been memorized. Behavioral work has indicated that individuals differ in their level of sensitivity-to-interference and that this individual difference explains arithmetic performance – in particular the retrieval of arithmetic facts. Individuals who

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demonstrate less sensitivity-to-interference are more efficient in retrieving arithmetic problems, compared to individuals who have a high-sensitivity-to-interference. Three neuroimaging studies have investigated the interference effect in adults (De Visscher et al., 2015, 2018; Heidekum et al., 2019), and one study has investigated the effect in children (Polspoel et al., 2019). The first of these studies investigated the neural correlates of the arithmetic interference during the solving of multiplication problems (i.e., a multiplication verification task; De Visscher et al., 2015). The presented multiplications were categorized into high- and low-interfering items based on an interference parameter developed by De Visscher and Noël (2014). The results of this fMRI study showed that the verification of highinterfering problems was associated with greater brain activation in frontal brain regions, in particular the left and right inferior frontal gyrus (IFG) and the insula. The second fMRI study investigated individual differences in multiplication abilities and their association with the neural interference effect (De Visscher et al., 2018). The analyses revealed a neural interference effect in the left IFG that was negatively related to individual differences in arithmetic fluency. The neural interference effect was higher for low performers compared to high performers. Asking a slightly different question, Heidekum et al. (2019) investigated the involvement of semantic control processes to overcome this type of interference. Using an arithmetic interference task and a lexico-semantic interference task, the results of this neuroimaging study showed that resolving interference in these two tasks engaged brain regions of the left and right IFG and the left IPS. The engagement of these brain regions across different domains indicates the involvement of domain-general mechanisms in the processing of arithmetic problems. Thus far, only one neuroimaging study contrasted the interference effect to the problem size effect in children (Polspoel et al., 2019). The results of this study showed significant behavioral interference and problem size effects; however, on the neural level, only a problem size effect was observed. The reasons for this null effect are still opaque. Nevertheless, there is now good evidence that a fronto-parietal network is associated with the interference effect, its resolution, and its association with individual differences in arithmetic performance.

Conclusions A core insight that has emerged from the last decades of brain research is that the development of numerical abilities and arithmetic cannot be restricted to a single cognitive mechanism or to a single brain region. They constitute a complex and multidimensional concept that incorporates multiple cognitive abilities, representational dimensions, and brain regions. The neurocognitive networks and its associated functions interact in various complex ways to enable an efficient and flexible processing of the relevant information. The relative engagement of these brain regions is modulated by age (e.g., Vogel et al., 2015a), ability level (e.g., Sommerauer et al., 2020), and certain task constraints (e.g., Wilkey et al., 2017). The involved mechanisms can be described as a functional interaction between domain-specific and domain-general brain regions. Domain-specific brain functions

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are those that are largely restricted to the domain of interest. In the case of arithmetic, this involves certain aspects of basic number processing that are not relevant in other domains. It includes the representation of numerical quantities, the knowledge about ordinal relationships, or the knowledge of arithmetic facts. Domain-general functions are not specific to the domain of interest. They rather reflect mental operations that are important for learning and information processing more generally. This includes cognitive functions such as working memory (i.e., the ability to temporally hold information in our mind) or spatial reasoning (i.e., the ability to mentally manipulate and understand the spatial relation between and within objects). Over development time, these domain-general and domain-specific brain functions interact in various ways to enable mathematical thinking.

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Neurocognitive Foundations of Fraction Processing

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Silke M. Wortha, Andreas Obersteiner, and Thomas Dresler

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key Findings from Behavioral Research on Fraction Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eye-Tracking Research on Fraction Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neuroscientific Research on Symbolic and Non-symbolic Fraction Processing . . . . . . . . . . . . . . fMRI Studies on Fraction Magnitude Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EEG Studies on Fraction Magnitude Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fMRI Studies on Fraction Processing Not Specific to Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . A Tentative Temporal Model of Fraction Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Many children and adults experience fractions as a challenging mathematical content. Behavioral studies have extensively documented typical errors in fraction tasks and identified various factors that contribute to their occurrence. However, the cognitive mechanisms of fraction processing and fraction learning are still not fully understood. In recent years, brain imaging studies have begun to unravel the neural underpinnings of fraction processing. This chapter briefly summarizes key findings from behavioral reaction time and eye-tracking studies S. M. Wortha (*) Deparment of Neurology, University Medicine of Greifswald, Greifswald, Germany LEAD Graduate School & Research Network, University of Tübingen, Tübingen, Germany e-mail: [email protected] A. Obersteiner TUM School of Education, Technical University of Munich, Munich, Germany T. Dresler LEAD Graduate School & Research Network, University of Tübingen, Tübingen, Germany Department of Psychiatry and Psychotherapy, Tübingen Center for Mental Health, University of Tübingen, Tübingen, Germany © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_27

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and reviews more extensively the available neuroscientific studies on fraction processing. Research using functional magnetic resonance imaging (fMRI), which has high spatial resolution, suggests that fractions can be processed holistically as whole numerical magnitudes and that the intraparietal sulcus (IPS) plays a key role in such processing. On the other hand, studies that used electroencephalography (EEG), which has a high temporal resolution, provide a more differentiated picture. In line with reaction time and eye-tracking studies, these studies suggest that fractions can be processed holistically or componentially, depending on task requirements. Based on the reviewed literature and previous models on number processing, we propose a tentative temporal model of fraction processing. We conclude that further research should focus specifically on the temporal characteristics of fraction processing during problemsolving to better understand how the brain constructs and represents holistic fraction magnitude. Keywords

Fraction processing · Rational numbers · Cognitive processing · Eye tracking · fMRI · EEG

Introduction Research over several decades has found that both students and adults often encounter difficulties in understanding and working with fractions (Bailey et al. 2015; Behr et al. 1983, 1985; Stigler et al. 2010). Behavioral studies have extensively documented typical errors in fraction tasks and identified various factors that contribute to their occurrence. In recent years, a special focus has been placed on the cognitive mechanisms involved in fraction processing and fraction learning. This research has suggested that while humans are in principle able to mentally process fraction magnitude, ratios, and proportions, symbolic fractions are particularly effortful to process. Specific features in symbolic fraction representations seem to provoke systematic errors and biases in students and adults. However, it is not clear yet on which stages of processing such difficulties occur. Some studies used online measures of cognitive processes, such as reaction times, eye tracking, and, most recently, brain imaging, to tap closer into the cognitive mechanisms and the neurocognitive foundations of fraction processing. This chapter briefly summarizes key findings from behavioral reaction time and eye-tracking studies and reviews more extensively the available neuroscientific studies on fraction processing. From a neuroscientific point of view, investigating the neural correlates of fraction processing is interesting in itself because it provides a deeper understanding of the underlying biological mechanisms involved in numerical thinking and problem-solving. In addition, investigating the neural correlates of fraction processing can also add to previous behavioral research to provide a more integrative picture of students’ and adults’ difficulties with fractions.

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The first section provides a brief overview of key findings from behavioral studies that assessed error patterns and reaction times in fraction problems. The second section focuses on research that used eye tracking to assess individuals’ eye movements during fraction problem-solving. The third section reviews more systematically the available neuroscientific studies on symbolic and non-symbolic processing of fractions, decimals, and proportions. The results from these studies are the basis for a tentative temporal model of fraction processing presented in the fourth section. Finally, the conclusion in the fifth section includes suggestions for further research.

Key Findings from Behavioral Research on Fraction Processing Fractions are challenging for many people, not only children or school students (e.g., Bailey et al. 2015; Behr et al. 1984, 1985; Carraher 1996; Lortie-Forgues et al. 2015; Stafylidou and Vosniadou 2004). This is unfortunate, since understanding fractions is important for understanding mathematical concepts in many areas, including algebra, probability, functions, and geometry. There is empirical evidence that fraction understanding is a unique predictor of future achievement in higher mathematics, above and beyond several other influential variables (Bailey et al. 2012; Siegler et al. 2012). Studies found that difficulties with fractions are not limited to difficult fraction arithmetic but instead pertain to simple fraction arithmetic and fundamental understanding of fraction concepts. In fact, students can have relatively high procedural skills and perform well on fraction arithmetic problems without understanding fraction concepts (Hallett et al. 2010). Research has identified various reasons why fractions are difficult for many people. While many of these reasons have been discussed elsewhere (e.g., LortieForgues et al. 2015; Obersteiner et al. 2019), this chapter assumes a cognitive perspective. From this perspective, one reason why learning of fractions is difficult is because fractions differ in many ways from natural numbers and integers, so that learning of fractions requires some conceptual change (Vamvakoussi et al. 2012; Vamvakoussi and Vosniadou 2004, 2010). This means that within the set of natural numbers, there are several properties that do not hold for the set of rational numbers. For example, for positive natural numbers, multiplication with any number except 1 always makes a number larger, and division always makes a number smaller. Neither is generally true for rational numbers. Natural numbers can be used for counting, and there are no or only finitely many numbers between any two natural numbers. In contrast, there are always infinitely many numbers between any two rational numbers. Each natural number value is typically represented in a unique way (e.g., 2), while there are infinitely many different ways to represent any rational number (e.g., 0.5 ¼ 1/2 ¼ 2/4 ¼ 3/6, etc.). Finally, fractions differ from natural numbers in conveying numerical magnitude as the relation between two natural numbers. Learners have typically developed a firm understanding of the concept of numbers as natural numbers long before they learn about fractions. When learning fractions, learners sometimes apply reasoning based on natural numbers, which is not always appropriate to solve fraction problems. Such reasoning can lead to

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systematic errors and whole number bias (Alibali and Sidney 2015; Ni and Zhou 2005; Van Hoof et al. 2017a, b). A well-documented example is biased reasoning about fraction magnitude (i.e., the numerical value represented by a fraction symbol such as 3/7). Reasoning about fraction magnitude is, for example, required to estimate the result of an addition problem (e.g., 3/7 + 4/9 must be smaller than 1 because both addends are smaller than 1/2), or to decide quickly which of two fractions is larger (e.g., 3/5 > 4/7). Biased reasoning includes considering fractions as two distinct numbers rather than as one integrated number. In fraction comparison problems, some students have been found to rely on componential comparison (e.g., 4/7 > 3/5 because 4 > 3 and 7 > 5). Such reasoning leads to systematic error patterns, namely, to incorrect responses in incongruent fraction comparison problems (such as in the given example) but to correct responses in congruent comparison problems (e.g., 8/9 > 1/3 because 8 > 1 and 9 > 3). Persistent natural number-based reasoning also prevents students from processing overall fraction magnitude. This is a key problem because the ability to reason about fraction magnitude is considered important from both an educational and a developmental perspective (Siegler and Lortie-Forgues 2014). In experimental research, the most prominent indicator of number magnitude processing is the distance effect in a number comparison task. It was first described by Moyer and Landauer (1967) and since then has become a hallmark effect of numerical cognition. The distance effect describes the phenomenon that the magnitudes of two numbers that are numerically closer (e.g., 3 vs. 4) are compared more slowly and the comparison is more error-prone than comparing two numbers that are further apart (e.g., 3 vs. 8). The distance effect can be explained by a transformation of the numbers into an internal analogous magnitude interpretation, where two close numbers overlap more and therefore show more similarities and are more difficult to distinguish than two more distant numbers (Moyer and Landauer 1967). With the distance effect, it is possible to investigate one of the key questions in fraction processing research: are the fractions in a fraction comparison task processed holistically (i.e., as one numerical value) or componentially (i.e., the numerical values of the numerator and denominator are processed separately)? Initial studies showed mixed evidence regarding the distance effect in fraction comparison problems (Bonato et al. 2007; Ganor-Stern et al. 2011). However, the way participants process fractions seems to depend on the type of fraction comparison and on the strategies they use to solve these problems (Meert et al. 2010a, b; Obersteiner et al. 2013). For instance, Obersteiner et al. (2013) found that when academic mathematicians solved fraction comparisons, there was a distance effect of overall fraction magnitude only for fraction pairs that did not have common components (e.g., 11/18 vs. 19/24). However, when fraction pairs did have common components (e.g., 17/23 vs. 20/23, or 12/13 vs. 12/19), there was no effect of overall distance. This result could also explain diverging findings in earlier studies, in which problem type was not always varied systematically within the same experiment: In experiments in which fraction comparison problems did not have common components (e.g., Meert et al. 2010a), participants were more likely to exhibit a distance

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effect than in experiments in which fraction pairs did have common components (e.g., Bonato et al. 2007). Taken together, this line of research suggests that adults rely more strongly on componential comparison strategies in comparison problems with common components (with less activation of holistic overall fraction magnitudes). A method that can provide more insight into the cognitive processes during fraction problem-solving beyond findings from response time and accuracy studies is eye tracking.

Eye-Tracking Research on Fraction Processing Several studies have used eye tracking to assess fraction processing. The eye-tracking methodology has been used for decades in numerical cognition research (Mock et al. 2016) and, with increasing frequency, also in educational research (Lai et al. 2013), particularly in mathematics education (Strohmaier et al. 2020). Although different eye-tracking techniques exist, the most common one is a video-based one, in which an infrared light source that is placed next to a camera is used. The camera detects the reflections of the infrared light from the eyes, which then allows extracting eye fixations and eye movements. An advantage of eye tracking is its noninvasiveness, and it allows assessing individuals’ eye movements in real time. One basic assumption is that individuals process the information that is within their visual attention at a certain moment (the eye-mind hypothesis, Just and Carpenter 1980). Accordingly, eye movements and eye fixations are thought to provide information about individuals’ strategies and the cognitive processes during problem-solving (Holmqvist et al. 2011). Although this assumption does certainly not hold in all situations (see Carrasco 2011), it seems reasonable in situations in which individuals are asked to quickly solve problems that are presented visually, which is typically the case in studies on fraction processing. Similar to behavioral research on reaction time and accuracy, a key question in eye-tracking research on fraction processing is how participants process fraction components (i.e., fraction numerators and denominators) to make inferences about the overall fraction magnitudes. To address this question, most studies used a computerized fraction comparison task. However, studies differed, among other factors, in the type of fraction comparison problems and in the eye-tracking measures used for analyzing fraction processing. Regarding the type of fraction comparison problems, studies presented either simple problems, in which fraction components were small (e.g., one-digits) or the presented fraction pairs had common numerators (e.g., 11/17 vs. 11/13) or common denominators (e.g., 11/17 vs. 13/17), or more difficult problems, in which fractions had larger components (e.g., two-digits) or the fraction pairs did not have common components (e.g., 28/43 vs. 19/37). Regarding eye-tracking measures, studies analyzed either the number of fixations, the fixation time on the fraction components, or the saccades (i.e., rapid eye movements between two fixations) between fraction components. In an initial study, Obersteiner et al. (2014) used eye tracking with a small sample of eight adults, who were asked to solve fraction comparison tasks on a computer

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screen. The stimulus set included fractions with one- and two-digit components and fraction pairs with and without common components. The authors examined participants’ fixation times on the fraction numerators and denominators. They found that these fixation times depended on the presented fraction type. In fraction pairs with common numerators or common denominators, participants fixated longer on the unequal denominators or numerators, respectively. When fraction pairs did not have common components, there were no significant differences in fixation times between numerators and denominators. This finding supported the assumptions from earlier reaction time studies (see section “Key Findings from Behavioral Research on Fraction Processing”), suggesting that people rely on simple componential comparison strategies in problems that can be solved correctly with such strategies (i.e., problems with common components) and that they rely on more demanding holistic comparison strategies when simpler strategies are not successful. These initial findings were largely replicated in a study by Obersteiner and Tumpek (2016), who used a more systematically controlled set of fairly complex comparison problems with two-digit components. In their study with 25 adults, they examined, in addition to the number of fixations on fraction components, the saccades between these components. Analyzing saccades may provide better information about which fraction components participants integrate during the problemsolving process. Although the results were less clear-cut than one may have expected, there were relatively more saccades between the unequal fraction components in problems with common components. In problems without common components, there was no such difference, and the relative number of saccades between the numerators and denominators of each fraction was highest among all relevant saccades. It seems plausible that people switch between the numerator and the denominator of a fraction to determine the fractions’ overall magnitude. Such processing is required to a larger extent to compare fractions without common components than to compare fractions with common components. Ischebeck et al. (2016) investigated not only saccades between two fraction components but sequences of three consecutive fixations. Their study with 20 adults again confirmed earlier findings. Sequences of three consecutive fixations on numerators were more frequent in fraction pairs with common denominators, while such sequences with fixations on denominators were more frequent in fraction pairs with common numerators. No systematic differences in the frequencies of these sequences were found for fraction pairs without common components. A study by Huber et al. (2014) suggested that it is not only the problem type that influences the processing of fraction components but also the way in which these problem types are combined to experimental blocks. The authors studied a larger sample of 36 adults and used simple fraction comparison problems with one-digit components. The analysis of fixation times confirmed that participants adapted their comparison strategies (componential or holistic) to the type of the comparison problem (i.e., more componential processing in common component problems than in problems without common components). However, the results also showed that these differences were more pronounced in a blocked condition, in which problems of different types were presented in separate experimental blocks, than

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in a randomized condition in which problems of all types were presented intermixed in the same block. Presumably, if problems were presented in blocks, participants were able to anticipate which type of strategy would be successful to solve the fraction comparison problem and then adapted their strategy accordingly. Unlike in the studies mentioned above, the study of Huber et al. (2014) found that, regardless of problem type, fraction denominators required particularly many fixations, suggesting that denominators are generally more demanding to process than numerators. While other studies on fraction magnitude comparison did not document such a general difference in the processing of numerators and denominators, the important role of the denominator was also found in a study on fraction addition. Obersteiner and Staudinger (2018) presented fraction addition problems to 28 adults who were asked to solve these problems mentally. Fraction addition problems were categorized into four different types, depending on whether the two fraction denominators were equal, multiples of one another, prime numbers, or did not fall in any of these categories. In fraction addition problems, these features of the fraction denominators determine whether or not specific shortcut strategies are applicable to solve the problem. Analyzing the numbers of fixations and the numbers of saccades between fraction components revealed that problem difficulty was particularly related to more extensive processing of fraction denominators. When items required multiplying fractions to get a common denominator, the numbers of saccades between the denominators were particularly high. Finally, Hurst and Cordes (2016) investigated 62 adults who solved rational number comparison problems in which numbers were presented in varying formats, including fractions and decimals. In problems involving fractions, the fractions did not have common components, and fraction pairs varied in whether or not simple comparison of the numerators led to the correct response. Unlike earlier studies, the authors found that fixation times were larger for numerators than denominators. An explanation for this diverging finding could be that some participants in this study may have had less mathematical experience than those in other studies, who were often mathematically skilled adults (e.g., Obersteiner and Tumpek 2016). The less competent participants may have relied more strongly on natural number-based reasoning (see section “Key Findings from Behavioral Research on Fraction Processing” ). This explanation seems reasonable because fixation times on numerators were related to performance on a fraction procedure test. Additionally, this study documented an overall distance effect on certain eye-tracking parameters: fixation times decreased with the ratio between the two numbers in a comparison item, suggesting that comparison problems with larger ratios were easier to compare than those with smaller ratios, a finding that has so far only been reported for reaction times (e.g., Obersteiner et al. 2013) but not eye movements. To summarize, eye-tracking research largely confirmed and extended previous research of individuals’ accuracy and response times in fraction processing tasks. Studies suggest that people are able to quickly process fraction magnitudes by integrating the information from numerators and denominators. Eye movements clearly show that participants adapt their strategies and use simple heuristics, such as comparison of fraction numerators or fraction denominators, in special cases of

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fraction comparison tasks that do not require processing the overall fraction magnitudes. Fraction magnitude processing, on the other hand, seems to be related to more extensive switching between the numerator and denominator of each fraction.

Neuroscientific Research on Symbolic and Non-symbolic Fraction Processing There is ample evidence from neuroimaging studies investigating natural number processing in the developmental and adult brain showing that the intraparietal sulcus (IPS) is the key area for representing symbolic and non-symbolic numerical magnitudes (Emerson and Cantlon 2015; Lyons et al. 2015). This was especially supported by the finding of a neural analogue to the behavioral distance effect: IPS activation is inversely related to the numerical distance between two numbers (Cohen Kadosh et al. 2005; Kaufmann et al. 2005). However, not only the IPS is involved in number processing; a variety of studies also report activation of frontal brain areas, resulting in the so-called frontoparietal network underlying numerical processing (Arsalidou et al. 2018; Emerson and Cantlon 2012). The neural correlates of natural number representation have been investigated extensively for about three decades now. Unfortunately, there is considerably less research on the neural mechanisms underlying the processing of fractions and proportions. To date, there exist only a couple of studies that investigated the neural correlates of fraction processing in adults. Some studies used functional magnetic resonance imaging (fMRI) to investigate the neural correlates of general proportion (Jacob and Nieder 2009b; Mock et al. 2018, 2019) and fraction processing (DeWolf et al. 2016; Ischebeck et al. 2009; Jacob and Nieder 2009a) and training studies on fraction processing in adults (Wortha et al. 2020). fMRI is a noninvasive and popular neuroimaging method to measure brain activity. One advantage of fMRI is its high spatial resolution, which enables researchers to investigate even small structures (below 3 mm). Another advantage is that the whole brain can be measured, from cortical areas just below the skull up to subcortical areas deep in the brain. In contrast to its high spatial resolution, temporal resolution of fMRI is rather poor, as it technically takes some time to measure the whole brain (about 2 seconds). If only specific parts of the brain are measured, duration decreases. In addition, the sluggishness of the BOLD signal does not allow investigating fast processes and their timing characteristics. Additionally, a couple of electroencephalography (EEG) studies addressed the temporal processes when dealing with fractions (Barraza et al. 2014; Fu et al. 2020; Rivera and Soylu 2018; Zhang et al. 2012, 2013). EEG is also a noninvasive neuroimaging method which can be used as a diagnostic tool in clinical settings but also as a neuroscientific research tool. Compared to fMRI, one advantage of EEG is its high temporal resolution in the millisecond range. One of the biggest disadvantages is its low spatial resolution. The derived signal from one electrode is not the signal from a single nerve cell or a specific brain area. Instead, the derived signal comes from many different neurons and areas whose electric fields overlap.

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There are mainly two different types of activity that can be measured: The spontaneous EEG activity that is reflected by different frequency bands (e.g., α, β, γ, and δ) and so-called event-related potentials (ERPs). Information about the current state of consciousness (e.g., awake vs. asleep state) can be derived via the frequency bands. ERPs are voltage fluctuations (positive or negative) that are timelocked to the onset of a sensory, motor, or cognitive event. This means that they reflect neural responses specifically related to a certain stimulus (e.g., a sound) or behavioral response (e.g., pressing a button). ERP components are usually named in terms of their peak polarity (N ¼ negative deflection and P ¼ positive deflection) and peak latency (in milliseconds). Table 1 provides an overview of the most common ERPs and specifies their role in fraction processing. The table also describes their general characteristics, location and time window, and attributed cognitive functions. Like in behavioral studies, one key question in most of these studies was if fractions are processed holistically or componentially. In the following subsections, we will provide an extended summary of all neuroscientific studies addressing fraction magnitude processing (subsection “fMRI Studies on Fraction Magnitude Processing” and “EEG Studies on Fraction Magnitude Processing”) and fMRI studies on fraction processing not specific to magnitude (subsection “fMRI Studies on Fraction Processing Not Specific to Magnitude”).

fMRI Studies on Fraction Magnitude Processing Using an adaptation paradigm, Jacob and Nieder (2009a) showed that the anterior IPS is involved in processing symbolic fractions and fraction words. Twelve participants were visually adapted to a specific fraction magnitude (e.g., 1/6) by being presented with fractions of the same magnitude continuously (i.e., 1/6, 5/30, 2/12). After adaptation, deviant symbolic fractions (e.g., 4:12) or fraction words (e.g., “one third”) were presented. During the adaptation part of the experiment, the BOLD signal decreased. After presenting the deviants, signal recovery was found as a function of numerical distance between deviant and adapted fraction magnitude in the bilateral IPS, bilateral prefrontal cortex, and the right cingulate cortex. This effect was independent of presentation format (i.e., symbolic fractions and fraction words). In a second similar adaptation experiment with 15 adult participants, Jacob and Nieder (2009b) investigated the neural correlates of non-symbolic proportion processing (i.e., line proportion and dot proportion). Similar to the first adaptation experiment, after presenting the deviant stimulus, the BOLD signal recovered as a function of the distance between the deviant proportion and the adapted proportion with strongest effects in bilateral anterior IPS. Additional activation clusters were found in bilateral prefrontal and precentral regions with right lateralized dominance. Both adaptation experiments provide evidence that magnitude representation of symbolic fractions, fraction words, and proportions activate more or less the same key area (i.e., the IPS) involved in the magnitude representation for natural numbers. Additionally, the experiments by Jacob and Nieder showed that fraction and

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Table 1 Different ERP components important for fraction processing and their general characteristics (for the role of ERPs in number comparison, see also Dehaene 1996) ERP N100

N200

P300

General characteristics Usually elicited by an unpredictable auditory stimulus in the absence of task demands. However, it can be triggered by any novel stimulus, regardless of its modality. The amplitude increases with stimulus intensity and decreases with the length of the interstimulus interval (ISI)

Location and time window Fronto-central maximum; peaks between 80 and 120 ms after stimulus onset

Cognitive function Associated with an orientation reaction of the brain by comparing an incoming stimulus with previously stored stimulus characteristics (¼ physical stimulus properties)

Usually elicited by a Go/NoGo paradigm: Subjects are asked to respond to certain stimuli (¼ Go stimuli) by pressing a button, and to suppress this response for other stimuli (¼ NoGo stimuli). The amplitude increases for NoGo stimuli compared to Go stimuli Usually elicited by an oddball paradigm: a series of equal stimuli (¼ standard) is interrupted by an unequal (¼ deviant) stimulus. The P300 only occurs if the subject is actively engaged in the task (e.g., pressing a button whenever the target

Fronto-central maximum; peaks between 200 and 400 ms after stimulus onset

Often associated with motor inhibition; plays an important role in conflict monitoring

Broadly distributed on the head surface with a maximum amplitude at parietal derivation sites; peaks between 300 and 500 ms after stimulus onset

Index of brain activity to measure the processing of incoming information and its integration into current memory representations

Role for fraction processing Larger in congruentincongruent condition; detection of conflict of past stimulus experience (Fu et al. 2020); Task-specific identification (non-symbolic fractions) and stimulus-specific identification (symbolic fractions; Zhang et al. 2013) Inhibitory control to overcome whole number bias; amplitude increases for incongruent fraction pairs compared to congruent fraction pairs (Fu et al. 2020); Cognitive control, complex fractions (Zhang et al. 2012)

Inhibitory control to overcome whole number bias; amplitude decreases for incongruent fraction pairs compared to congruent fraction pairs (Fu et al. 2020); Componential processing, simple fractions (continued)

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Table 1 (continued) ERP

N400

General characteristics stimulus occurs). The latency varies with difficulty of stimuli discrimination. The amplitude varies with unlikelihood of occurrence of the target Usually elicited by a semantic context that makes little or no sense. (i.e., He spread jelly on his bread vs. He spread jelly on his bed). The amplitude of the N400 depends on how well the presented word fits into the context: The lower the semantic context, the greater the amplitude of the N400

Location and time window

Cognitive function

Role for fraction processing (Zhang et al. 2012); Stimulus-specific semantic processing of fractions and decimals (Zhang et al. 2013)

Centro-parietal derivation sites, maximum slightly lateralized to the right; peaks between 250 and 500 ms after stimulus onset

Related to the processing of the semantic content of information; index of the perceived incongruency (¼ semantic evaluation) of a word

During holistic processing strategy more pronounced than during componential processing strategy (Barraza et al. 2014); Semantic congruency during a match/mismatch fraction task (Rivera and Soylu 2018)

proportion activation in the IPS was modulated by the holistic distance effect (i.e., activation changes as a function of the distance between two fractions). Ischebeck et al. (2009) observed that in a sample of 20 adults who solved a fraction magnitude comparison task, neural activation within the right IPS was modulated by the holistic distance between the to-be-compared fractions, but not by the componential distance between numerators or denominators. This finding suggests that in a fraction comparison task, the IPS processes fractions in a holistic but not in a componential way. DeWolf et al. (2016) were the first to investigate whether magnitude representation in the IPS differed between rational numbers (i.e., fractions, decimals) and natural numbers. Sixteen adult participants conducted magnitude comparison tasks in the different notations. The numbers in all comparison tasks had two digits, meaning that single-digit fractions (e.g., 1/9 vs. 3/7), double-digit decimals (e.g., 0.11 vs. 0.43), and double-digit natural numbers (e.g., 11 vs. 43) were presented. Additionally, the numbers within a trial were always of the same type (i.e., there were no cross-format comparisons). Results showed that while in

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comparison tasks, all three number representations showed a holistic distance effect, the activation pattern evoked by fractions within the IPS was different compared to the activation patterns evoked by decimals or natural numbers. For the latter two representation formats, activation patterns did not differ regarding IPS activation. Although fractions and decimals are both rational numbers, they showed distinct neural magnitude representations within the IPS. Decimal representation was closer to natural number representation. This result suggests that although the IPS is involved in the magnitude representation of all numbers, only fractions activate a distinct subarea within the IPS. In two studies with a sample of 24 adult participants, Mock and colleagues investigated domain-specific and domain-general processing (e.g., cognitive control and working memory) of proportions. In the first study (Mock et al. 2018), participants conducted a proportion magnitude comparison task with four different notations (i.e., fractions, pie charts, dot patterns, and decimals). Magnitude-related processing, modulated by a holistic distance effect, showed a shared neural correlate of specific occipito-parietal activation including the right IPS (Mock et al. 2018). In their second study (Mock et al. 2019), they examined brain activation of proportion processing (e.g., fractions, pie charts, dot patterns, and decimals) independent of the holistic distance effect. They found a shared neural substrate in the bilateral inferior parietal cortex. Additionally, activation in a frontoparietal network was found for part-whole processing of proportions. Both studies indicate not only an involvement of domain-specific areas (i.e., IPS) but also domain-general areas (i.e., bilateral inferior parietal cortex) for proportion processing and again a modulation of the IPS via the holistic distance effect. In a recent study, Cui et al. (2020) examined if fraction processing was dependent on areas associated with semantic processes of mathematics. In their study, 68 adult participants performed a magnitude comparison task either with natural numbers or fractions. Additionally, all comparison tasks had different levels of difficulty, defined by the numerical fraction distances (i.e., short distance pairs vs. long distance pairs). Cui et al. examined whether brain regions responsible for semantic processing (e.g., middle temporal gyrus) are more involved in processing fractions than processing natural numbers. Fractions were proper fractions with single-digit numerators and denominators. Denominators ranged between 2 and 9 and numerators ranged between 1 and 8 (e.g., 4/7). Natural numbers ranged between 10 and 99 and corresponded closely to a fraction converted into percentage (i.e., 1/2 and 50 for 50%). Cui et al. found that activation in the left middle temporal gyrus (MTG) differed between fraction and natural number comparison. Moreover, they examined functional connectivity from the left MTG to the IPS and found that connectivity was stronger when processing fractions than when processing natural numbers. Additionally, connectivity strength correlated positively with accuracy in comparing fractions. The results suggest that the left MTG – a brain area which has been shown to be associated with semantic understanding – was further associated with fraction processing. Finally, Wortha et al. (2020) conducted a first training study to investigate changes before and after a specific training of fraction magnitude processing in

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48 adult participants. Participants were trained on 5 consecutive days on a number line estimation task with feedback (positive and negative; 5% around the correct position) to improve their magnitude processing of fractions. In addition to the fraction comparison task, the number line estimation task is often used in numerical cognition to assess number magnitude understanding (Berteletti et al. 2010; Geary et al. 2008; Siegler and Opfer 2003). In this task, participants had to indicate the correct position of a given fraction on an empty number line ranging from 0 to 1. Each training session consisted of 96 items in 12 runs containing 8 items each. Each item from the trained stimulus set was repeated twice within a training session. Before and after the training, neural correlates of fraction magnitude processing were measured with a symbolic fraction magnitude comparison task, a line proportion comparison task, and a fraction-line proportion matching task (i.e., a fraction and a line proportion were shown, and participants had to indicate whether both magnitudes were identical or not). Results showed that prior to the training, the holistic distance effect modulated IPS activation in the line proportion comparison and the fraction-line matching task, but not the fraction comparison task. Pre-post comparisons revealed no differences in brain activation for the line proportion comparison task and the fraction-line matching task. However, for the fraction magnitude comparison task, a holistic distance effect on IPS activation was observed after the training. Thus, this study provides a first neurophysiological hint that a number line estimation training elicits changes in brain activity for magnitude processing of symbolic fractions. The study therefore supports the assumption that number line estimation training can support fraction magnitude processing. To summarize the results from the fMRI studies on fraction processing described above, Fig. 1 illustrates the most relevant brain areas. Table 2 provides an overview of the described studies. The table details the experimental tasks, the effects of interest that were assessed, the type of fractions used in the studies, and the main conclusions. All these studies provided support that during fraction magnitude tasks, fractions are more likely to be processed holistically rather than componentially on a neurophysiological level.

EEG Studies on Fraction Magnitude Processing Other studies used EEG to assess the temporal characteristics of brain activation during fraction processing (see Fig. 2 for a schematic illustration of the most prominent ERPs for fraction processing). For instance, Zhang et al. (2012) investigated how 17 adult participants compared fractions and decimals. In a simple magnitude comparison condition, all items were presented as fractions and had common numerators (e.g., 1/4 or 1/8). In a complex magnitude comparison condition, items were presented as fractions with common numerators or as decimals (e.g., 0.25 or 0.125). Items in the complex condition were presented in random order, requiring participants to switch flexibly between the two representation formats (fractions and decimals). In both conditions, the target fraction had to be compared with a standard fraction (i.e., 1/5). Behavioral results showed that participants

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Fig. 1 Schematic illustration of the most relevant brain areas involved in fraction magnitude processing (IPS intraparietal sulcus, MTG middle temporal gyrus; prefrontal cortex ¼ belongs to the frontal lobe and is associated with executive functions like working memory and inhibitory control)

processed fractions componentially but not holistically in both conditions. Additionally, for the simple condition, electrophysiological results linked the P300 to componential processing. In contrast to behavioral data, electrophysiological data acquired during the complex condition did not suggest that fractions were processed componentially. Nevertheless, for the complex condition, a N200 with a longer latency and increased negative amplitude was found compared to the simple condition. The authors linked this ERP to the task switching requirements (i.e., switching between common fractions and common decimals) of this condition. In a second study, Zhang et al. (2013) more specifically investigated event-related potential associated with processing of fractions with common numerators and decimal fractions. Thirteen participants were asked to perform a magnitude matching task. The task included two conditions, a common fraction condition and a decimal fraction condition. In the common fraction condition, participants were asked to match a non-symbolic fraction (i.e., line bar) with a symbolic fraction. In the decimal fraction condition, participants were asked to match a non-symbolic fraction with a decimal fraction. This study found that comparing non-symbolic fractions to symbolic fractions elicited larger N100 and P300 amplitudes than when comparing non-symbolic fractions to decimals, indicating that the visual identification of the non-symbolic fractions differed between conditions. Moreover, the study found that holistic processing of fractions was linked to the P200, N300, and P300 ERP components. Thus, this study provided evidence that fractions and decimals were processed holistically. Barraza et al. (2014) examined more closely whether participants processed fractions componentially or holistically and if neural synchronizations measured by spontaneous EEG could provide information about the used processing strategy.

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Table 2 Summary of all fMRI studies on fraction processing Reference Schmithorst and Brown 2004

Task Mental addition and subtraction of fractions

Effect of interest Validation of the triple-code model for complex numerical tasks

Fraction type Symbolic fractions; singledigit fractions

Jacob and Nieder 2009a

Adaptation task

Holistic distance effect and adaptation effect

Jacob and Nieder 2009b

Adaptation task

Holistic distance effect

Ischebeck et al. 2009

Fraction magnitude comparison

Distance effect (holistic vs. componential vs. cross product)

Symbolic fractions, fraction words; single- and two-digit fractions Non-symbolic proportions (i.e., line proportions and dot proportions), numerosity; line proportions: single-digit fractions; dot proportions: two-digit fractions Symbolic fractions; singledigit fractions

Klabunde et al. 2015

Two days of matching training (i.e., matching fractions to pie charts and pie charts to decimals); subsequently test of trained relations, symmetry and transitivity

Stimulus equivalence

Symbolic fractions, pie charts, decimals; single-digit fractions

Main conclusions The triple-code model is also suitable for complex numerical tasks Fractions are represented in the anterior IPS analogue to natural numbers Overlap of frontoparietal areas involved in natural number and proportion processing; proportions are processed in the anterior IPS

IPS activity is modulated by the holistic distance effect, but not the componential distance effect Increased brain activation in areas involved in equivalence relations. Significantly greater activation in participants with fragile X syndrome indicating a different neural execution for these participants during performance in equivalence relations (continued)

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Table 2 (continued) Reference DeWolf et al. 2016

Task Magnitude comparison task

Effect of interest Neural representation of magnitudes in the IPS across notations

Fraction type Natural numbers, decimals, fractions; singledigit fractions

Mock et al. 2018

Proportion magnitude comparison task

Holistic distance effect

Mock et al. 2019

Proportion magnitude comparison task

Domain general processes

Symbolic (i.e., fractions and decimals) and non-symbolic (pie charts and dot patterns) proportions; single-digit fractions Symbolic (i.e., fractions and decimals) and non-symbolic (pie charts and dot patterns) proportions; single-digit fractions

Cui et al. 2020

Magnitude comparison task

Holistic distance effect

Natural numbers, fractions; singledigit fractions

Wortha et al. 2020

Five days of number line estimation training; pre-post fMRI session: Magnitude comparison task and matching task

Holistic distance effect

Symbolic fractions and line proportions; single- and two-digit fractions

Main conclusions Fractions and decimals show distinct neural magnitude activation patterns in the IPS; decimals and integers do not differ in their activation patters in the IPS IPS is involved in processing of relative (proportion) magnitudes

Activation of a general frontoparietal network independent of the distance effect for partwhole processing of proportions (fractions, pie charts, and dot patterns vs. decimals) The left middle temporal cortex was identified as a region important for conceptual knowledge of fractions Modulation of IPS activation through distance effect for symbolic fractions was only observed after the training but not before

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Fig. 2 Schematic illustration of the most prominent ERPs found in fraction magnitude processing (time course and scalp distribution). For instance, the P300 reflects a positive deflection with a maximum peak around 300– 500 ms after stimulus onset. A maximum amplitude can be derived over parietal areas

Twenty adult participants completed a fraction magnitude comparison task in two blocks, consisting of 78 fraction pairs per block. One block included fraction pairs with either common numerators or common denominators (encouraging componential fraction processing). The other block included fraction pairs with no common components (encouraging holistic fraction processing). Fractions were always proper fractions with numerators and denominators ranging from 1 to 9. The authors found that componential processing was associated with left frontal-parietal alpha desynchronization, whereas holistic processing was linked to an increased phase synchrony on theta and gamma bands. Moreover, the N400 was linked to holistic processing of fractions. Thus, this study provided for the first time electrophysiological evidence on how oscillatory brain activity (¼ spontaneous brain activity) is reorganized depending on whether fractions are processed holistically or componentially (theta and delta band changes vs. alpha band changes). Moreover, it provides evidence for the existence of both strategies on a neural level. Rivera and Soylu (2018) specifically investigated the role of N400 for fraction processing. The N400 is usually linked to the semantic processing of information (see also Table 2) and therefore could provide information whether fractions are processed holistically or componentially. Twenty-four adult participants were asked to conduct a fraction matching task. They were presented a fraction and were asked whether the magnitude of this fraction was identical to (¼ match) or different from (¼ mismatch) a target fraction (i.e., 1/2, 1/3, and 1/4). The results showed that the

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N400 was suitable to detect semantic congruency while comparing fractions. Especially fraction pairs with shared components modulated the latency of the N400: fraction pairs without shared components showed a match/mismatch difference. The authors concluded that the missing modulated latency for shared components hints to an inhibition process necessary to solve the comparison task. This might be linked to inhibitory control being required for fraction magnitude processing depending on the properties of the fractions. Finally, Fu et al. (2020) were the first to investigate ERP correlates of inhibitory control necessary to suppress the whole number bias (see section “Key Findings from Behavioral Research on Fraction Processing”). Twenty-eight adult participants were asked to perform on a negative priming paradigm. Three types of fraction items were used, congruent items (e.g., 2/5 > 1/5 because 2 > 1), incongruent items (e.g., 1/4 > 1/5, although 4 < 5), and neutral items (e.g., identical fractions 1/3 vs. 1/3). The negative priming paradigm consisted of test trials and control trials. During test trials congruent and incongruent fraction pairs were shown, whereas during control trials neutral fraction pairs and congruent fraction pairs were shown. Behavioral results revealed a negative priming effect: participants responded faster to congruent items after completing the neutral items than after completing the incongruent items. Additionally, EEG results showed that increased N100 and N200 amplitudes and a decreased P300 amplitude were found for the test trials compared to the control trials. Therefore, this study provides electrophysiological support for the assumption that adult participants have to rely on inhibitory control to overcome the whole number bias in fraction comparison. Table 3 provides an overview of the EEG studies described above. Consistent with the fMRI studies, the table shows the tasks, effects of interest, the fraction type, and the main conclusions. Compared to the fMRI studies described earlier, the EEG studies provide a mixed picture on how fractions are processed. Similar to behavioral studies EEG studies hint to the existence of both componential and holistic processing strategies.

fMRI Studies on Fraction Processing Not Specific to Magnitude Not all neuroscientific studies on fraction processing focused specifically on fraction magnitude. Some studies addressed other research questions related to fraction processing. For instance, Schmithorst and Brown (2004) were the first to investigate the neural correlates of mental fraction arithmetic with fMRI. Their aim was to investigate whether the triple-code model (TCM, Dehaene and Cohen 1995, 1997; Dehaene et al. 2003) was also suitable for complex mathematical problems. The TCM is the most influential model for number representation in the adult brain in numerical cognition. It assumes the existence of three distinct codes of number representation, which are task specific: First, a visual Arabic code in which numerals are represented by strings of Arabic digits. Bilateral fusiform and lingual regions were identified to be associated with activation of the visual Arabic code.

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Table 3 Summary of all EEG studies on fraction processing Reference Zhang et al. 2012

Task Comparison decision with a fixed reference fraction; simple (only common fractions) and complex (common fractions and decimals) condition Matching task; common fraction and decimal fraction condition

Effect of interest Holistic and componential processing

Barraza et al. 2014

Fraction magnitude comparison task

Fraction processing strategies (holistic processing strategy vs. componential processing strategy)

Rivera and Soylu 2018

Matching task

Semantic congruency

Fu et al. 2020

Fraction comparison task with negative priming paradigm (congruent vs. incongruent vs. neutral)

Negative priming effect

Zhang et al. 2013

Semantic processing; distance effect (close vs. far)

Fraction type Symbolic fractions and decimals; single-digit fractions

Nonsymbolic fractions/ symbolic fractions/ decimals; single-digit fractions Symbolic fractions; single-digit fractions

Symbolic fractions; single- and two-digit fractions Symbolic fractions; single-digit fractions

Main conclusions Componential processing of fractions in both conditions; P300 correlates in simple processing with componential processing

Holistic processing of fractions was linked to the P200, N300, and P300 ERP components

Componential processing induces a left frontal-parietal alpha phase desynchronization; holistic processing induces an increase of phase synchrony on theta and gamma bands; N400 more pronounced for holistic processing N400 was linked to semantic congruency while comparing fractions Larger N100 and N200 amplitudes and a smaller P300 amplitude as electrophysiological correlate for inhibitory control of the whole number bias

Second, a quantity and magnitude code, in which size and distance relations between numbers are represented nonverbally and in a category specific way. The bilateral hemispheric number magnitude representation is associated with brain areas around the bilateral IPS.

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Third, a verbal code, in which numbers are represented by words. The verbal representation is associated with left perisylvian areas like middle MTG and superior temporal gyrus (STG), supramarginal gyrus (SMG), and the angular gyrus (AG). Fifteen adult participants were presented with three fraction addition or subtraction problems. The problems consisted of single-digit fractions and improper fractions, and the arithmetic operation could also result in a negative number. The results supported the assumption that the TCM is indeed a suitable framework not only for simple natural number processing but also for more complex fraction arithmetic because task-specific activation was found in the bilateral IPS, left perisylvian, and ventral occipitotemporal areas. Klabunde et al. (2015) conducted a first fMRI training study on equivalence relations of proportions that were represented as fractions, pie charts, or decimals, in eight adolescent participants with fragile X syndrome and a control group including ten matched (i.e., age and IQ) participants with intellectual disabilities. Participants were trained for two days in 10 min sessions until they were able to have over 80% accuracy on matching fractions to pie charts (A ¼ B relation) and pie charts to decimals (B ¼ C relation). Neurofunctional changes from before to after the training for symmetry relations (B ¼ A and C ¼ B) indicated significantly increased brain activation in the left inferior parietal lobule, left postcentral gyrus, and left insula for both groups. Moreover, for equivalence relations (A ¼ C and C ¼ A), fragile X syndrome participants showed greater activation in the right middle temporal sulcus, left superior frontal gyrus, and left precuneus compared to the control group. In summary, neuroimaging studies (see Tables 2 and 3 for a summary of all studies) show mixed results on whether people typically process fractions holistically or componentially. Interestingly, experimental results seem to be dependent on the neuroimaging method used: All fMRI studies indicate that the holistic distance effect modulates IPS activation of the to-be-compared proportions/fractions, but not the componential distance effect. EEG studies, however, show that holistic as well as componential processing is used depending on task requirements. One possible explanation for this discrepancy could be that EEG is more suitable to depict the temporal characteristics of fraction magnitude processing and may therefore be more sensitive to fine granular distinctions between both processing strategies. Another explanation provided by Zhang et al. (2012) is that both processing strategies might be used sequentially. Thus, participants might first prefer the holistic processing strategy (Ischebeck et al. 2009; Jacob and Nieder 2009a; Kallai and Tzelgov 2009) and later in the process switch to a componential strategy. This sequential execution of strategies cannot be detected via fMRI because of its poor temporal resolution.

A Tentative Temporal Model of Fraction Processing An important question that is not yet completely resolved concerns the temporal course in which people process the magnitude information represented by a symbolic fraction. The following preliminary model (see Fig. 3), which could be explored in further research, summarizes our current knowledge and is based on

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Fig. 3 Model proposal for the temporal course of fraction magnitude processing during a magnitude comparison task. This tentative model consists of four stages: 1. Visual indentification of the presented fractions. 2. Extracting the magnitudes of the numerator and denominator seperately and bulding a magnitude representation for each presented fraction. 3. Comparing the magnitudes of both fractions. 4. Coming to an conclusion on which fraction is larger or smaller and responding by pressing the respective botton. Please not that for simplicity, only one fraction is shown. Therefore, in a fraction magnitude comparison task stage 1 and 2 are performed twice

the literature review above as well as on previous models of number processing (e.g., Dehaene 1996): In an initial stage, the input (symbolic fraction) needs to be perceived. Depending on participants’ mathematical knowledge and experience, the two natural number components (i.e., the numerator and the denominator) may be processed automatically and generate two initial magnitude representations. Next, these magnitude representations are used to create an integrated magnitude representation of the whole fraction. This step is effortful and less automatic, unless people have a certain amount of experience with the specific fraction. Note that the process how such integrated magnitude representations are created is not well understood (e.g., division, estimation, visualization, etc.). In fraction comparison tasks, this process occurs twice, once for each fraction. Finally, the two fraction magnitude representations are compared in relation to each other, and the response is created. While the model describes this process in theory, there are several reasons to assume that this process will often deviate in concrete experimental settings. One reason is that it will depend on the specific fractions in a comparison task whether or not people actually engage in (more effortful) activation of fraction magnitude, or whether they can solve the problem using shortcut strategies. Research has documented the large variety of strategies people, including primary and highschool students, can use to solve fraction comparison tasks (Clarke and Roche

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2009; Fazio et al. 2016). These strategies include simple comparison of individual fraction components but also more sophisticated ones such as the use of benchmarks, cross-multiplication, or approximations based on fraction components. Specific pairs of fractions may allow for simple strategies based on visual comparison of number symbols (e.g., 4/11 vs. 16/3). In studies that aim at assessing fraction magnitude processing, it is therefore important to create comparison problems in such a way that item feature discourages the use of shortcut strategies that do not include reasoning about fraction magnitudes. Another factor that may influence the process of creating mental representations of fraction magnitude is the varying degree to which people rely on the componential natural number magnitudes when making decisions in fraction magnitude comparison tasks. While the phenomenon of whole number bias is well documented, conflicting evidence regarding the strength and the direction of the bias (larger components mean larger fraction, or the reverse) has led to the assumption that biased reasoning may depend on many factors. These factors include task characteristics, individuals’ strategy use, their mathematical experience, as well as the interaction between these factors (Alibali and Sidney 2015; Obersteiner et al. 2020). In this regard, however, neuroscientific studies have the potential to contribute to the open question when exactly in the process potential interference with natural number magnitudes occurs. If our brain activates natural number magnitudes automatically whenever processing a symbolic fraction, an initial whole number bias (not necessarily observable on the behavioral level) may be unavoidable. For highly trained fraction symbols (e.g., 1/2), an automated activation of magnitude is possible, but this seems unlikely for most fractions. In fact, quick responses in fraction magnitude comparison tasks have been found to require inhibition processes (e.g., Fu et al. 2020; Gómez et al. 2015). This has educational implications because the effect of instruction will depend on the question how automatic biased reasoning occurs.

Conclusion Difficulties with fractions have been documented for decades, but research has only recently addressed the cognitive mechanisms underlying fraction processing. In this chapter, we summarized state-of-the-art eye-tracking and neuroscientific research on cognitive mechanisms of fraction processing. This research includes neuroscientific studies that assessed brain function during fraction problem-solving. Functional MRI studies suggest that the IPS plays the dominant role in fraction magnitude processing and largely confirm earlier models that describe number processing in the brain. EEG studies addressed the temporal characteristics of fraction processing. Overall, the question whether fractions are processed holistically or componentially seems to depend on many factors. Eye-tracking and EEG research shows evidence for both processing strategies depending on task requirements. EEG studies showed that different ERPs are linked to holistic (e.g., P200, N400) or componential fraction processing (e.g., P300). Additionally, different patterns in oscillatory brain activity

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were linked to either holistic (e.g., increased phase synchrony on theta and gamma bands) or componential (e.g., alpha phase desynchronization) processing. In contrast, fMRI research provides evidence that fractions are solely processed holistically within the IPS. A sophisticated combination of methods could perhaps better elucidate the underlying processes. To increase the potential relevance of neuroscientific studies for educational contexts, future studies should include younger participants and particularly school students who are just beginning to learn fractions in school. These studies should be complemented by behavioral studies investigating the use of fraction concepts in young children before fractions are introduced in school. The seminal investigations by Nieder and colleagues in the human and the monkey brain, for example, show that fractions and proportions can be processed by the brain across species – a capacity that may be used from early on (Jacob et al. 2012; Vallentin and Nieder 2008, 2010). Moreover, to allow for causal inferences, it would be necessary to use longitudinal designs and experimental intervention studies. For example, one could use typical fraction magnitude tasks (i.e., fraction comparison, number line estimation) to systematically train participants in quick processing of fraction magnitude. Three current studies address these issues: Hubbard and colleagues are currently investigating fraction learning using a longitudinal design in which children from second to fifth and fifth to eighth grade are assessed across a period of four years (for more information see also: https://web.education.wisc.edu/lambda/). In the current project FracMag (Fraction Magnitude; Rosenkranz et al. 2019), an intervention to foster fraction magnitude representation has been implemented in secondary school children, which is complemented by fMRI measurements. Finally, in an fMRI study, Wortha et al. (2020) implemented a computer-based number line estimation task to train fraction magnitude processing. Future studies should aim for better controlled experimental settings and include younger participants, to increase the potential relevance of these studies for learning fractions at school. Furthermore, it would be desirable to use intervention or longitudinal designs to better understand the causal effects of specific training and the development of fraction magnitude processing in the brain over time.

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Individual Differences in Mathematical Abilities and Competencies

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Sara Caviola, Irene C. Mammarella, and Denes Szűcs

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From a Categorical to a Dimensional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Importance of Domain-General Cognitive Factors as Sources of Individual Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working Memory and Mathematical Achievement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Executive Functions and Attentional Control in Mathematical Learning . . . . . . . . . . . . . . . . . . Beyond the Purely Cognitive: Metacognition, Affect/Beliefs, and Motivation as Sources of Individual Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metacognitive Abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Negative and Positive Attitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Cultural and Language Differences to Contextual Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linguistic Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contextual Factors: From Parental Support to Educational Systems . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Mathematics is one of the core subjects in education and a critical factor in driving future life success. Thus, understanding how children learn to master S. Caviola (*) Department of Developmental Psychology, University of Padova, Padova, Italy School of Psychology, University of Leeds, Leeds, UK e-mail: [email protected] I. C. Mammarella Department of Developmental Psychology, University of Padova, Padova, Italy e-mail: [email protected] D. Szűcs Department of Psychology, University of Cambridge, Cambridge, UK e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_28

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numerical concepts and mathematical skills is of vital importance in the education domain. Behavioral outcomes on tasks are influenced by a variety of individual and shared (for example, contextual) factors. Individual factors, such as cognitive (for example, working memory), affective (for example, mathematical anxiety), and motivational aspects have been extensively studied in relation to children’s mathematical competencies. Similarly, contextual factors related to the perceived classroom environment, as well as cultural aspects (for example, exposure to early home activities) can also have serious impact on school performance in general, and specifically on mathematics achievement. In this chapter, we review some main sources of individual variability considered in the literature. We argue that multidimensional large scales studies are necessary to move the field forward. Keywords

Mathematical learning · Dimensional approach · Variability · Cognitive and affective aspects · Contextual factors

Introduction The psychological and educational literature acknowledges the universal importance of mathematics as one of the core subjects in education and a critical factor in driving future life success (Ritchie & Bates, 2013). Thus, understanding how children learn to master numerical concepts and mathematical skills is of vital importance in the education domain (Dowker, 2019; LeFevre et al., 2017). Mathematical learning encompasses a complex set of academic skills, and it often does not have the immediate appeal of other subjects, causing stress in both pupils and teachers alike (Schaeffer et al., 2021). Children may show significant individual differences in their ability to perform even simple mathematical tasks. More importantly, differences in both conceptual understanding and procedural knowledge during the early stages of learning can lead to subsequent difficulties in academic skill acquisition during later grades (Cargnelutti et al., 2017; Peng et al., 2016). A wide body of research, albeit with different aims, has tried to account for the differences often encountered in pupils’ maths achievement, resulting in many, multilayered potential sources of diversity, including cognitive, environmental (or contextual), and cultural factors (Devine et al., 2018). After a brief overview of the main methodological approaches, we begin this chapter by covering the several cognitive factors thought to be crucial to mathematics performance. We then address the contribution of affective, emotional, and belief schemes that have been investigated in the domain of mathematical learning also considering the influence of environmental and contextual aspects. Following this, the chapter turns to research that examines cross-cultural differences in mathematics performance considering the different perspective aspects, from linguistic factors to parental support and educational systems.

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From a Categorical to a Dimensional Approach Research into individual differences in psychology is diverse and continually evolving. The methodologies and analyses in studies vary considerably in their complexity – from simple correlation design, through multiple regression, structural equation modeling, to longitudinal methods that examine change over time (i.e., growth curve modeling). Yet, other studies adopt an extreme-groups design, common in psychopathology research (Preacher et al., 2005). The use of the extreme-groups approach is linked to a taxonomic conceptualization of phenomena, in which low (or high-)-scoring individuals are considered to have a condition that is disconnected from normal function (Fisher et al., 2020). Thus, in the field of mathematical learning, this approach led to research focused on understanding mildly and moderately impaired mathematical skills. When looking into individual differences in mathematical difficulties, researchers usually select children with a low-achievement profile from larger typically developing samples (i.e., by using only psychometric cutoffs according to a categorical classification scheme), in an attempt to target and isolate specific core deficit aspects. The widespread use of this methodology, partially justified by its relative simplicity, hides flaws and limitations in participant selection, sample sizes, statistical methods, and use of a restrictive range of measures (Astle & Fletcher-Watson, 2020). Starting from the sampling strategies, the selection criteria implemented to allocate participants into high versus low-scoring groups are extremely inconsistent between studies (Murphy et al., 2007). For example, many studies that define low achiever children as “dyscalculic” or with “mathematical learning disorder (MLD)” had not actually tested children with a previous clinical diagnosis. This misleading use of diagnostic labels conveys the message that these children might be qualitatively different from those that get higher scores. As Peters and Ansari (2019) commented, “such reasoning is unwarranted because selection is based on an arbitrary point along the normal distribution of ability scores on a particular measure” (p. 2). Even though the diagnostic process for a severe mathematical impairment is clearly described in the main classification manuals (e.g., the Diagnostic and Statistical Manual of Mental Disorders [DSM 5]; American Psychiatric Association [APA], 2013), several studies simply selected children with a low-achievement profile from larger samples by using only psychometric cutoffs that often did not match those for a clinical diagnosis of specific learning disorders. In fact, even studies measuring the prevalence of mathematical learning difficulties used widely varying cutoffs for diagnosis, ranging from the 2nd to the 25th percentile (Devine et al., 2013). Recently, Mammarella et al. (2021) considered the above problems of the literature. Based on studies reported in two recent meta-analyses (Peng et al., 2018; Schwenk et al., 2017), it was observed that 88% of relevant studies adopted widely varying cutoff points – not supported by a clinical diagnosis – to select MLD children from larger typically developing samples. The study examined whether differences between typically developing children and those with MLD (only identified by using cutoffs commonly adopted in the literature) would reflect global characteristics of the

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population considered as a whole (a), or whether the MLD group would be characterized by specific core deficits that cannot be inferred from parameters that describe the rest of the population (b). The former option was named the dimensional hypothesis and the latter the core deficit hypothesis. In particular, according to the dimensional hypothesis, children with MLD should be placed at the most extreme coordinate positions along some dimensions of a multidimensional space, without a marked discontinuity relative to the entire population. On the contrary, according to a core deficits hypothesis, children with MLD should be markedly distinct from the characteristics of the sample population. In our study, we identified 47 children with an MLD profile and 895 control children from a sample of 1303 children, by using widely accepted cutoffs and criteria (Murphy et al., 2007), and then implemented a simulation procedure. For both the observed sample and the simulated population, we computed Cohen’s d expressing the difference between the “MLD” and “typically developing” children for several variables of interest, including measures of basic numerical knowledge and cognitive abilities. As highlighted in Fig. 1, our findings suggested that none of the measures of basic number processing or domain-general abilities could identify specific core deficits in our “MLD” children. Rather, all differences between the MLD and control groups were more likely to reflect global characteristics of the population sampled. This is evident by the fact that for almost all variables, the observed (black dots with error bars) and simulated (red circles) Cohen’s d values were remarkably similar. In other words, practically every single standardized difference between the MLD and control group reflected how the cognitive variables are related to mathematics score in the non-MLD population. Even if this study is limited by not having included a real clinical MLD sample, the final message is clear and replicates Szucs’ (2016) conclusions: “it is much more useful to aim to position children in a multidimensional measurement space rather than distribute them into artificially separated sub-groups” (p. 297). According to

Fig. 1 Observed Cohen’s d (black dots) between children with MLD and controls, for all variables of interest. Error bars represent 95% BCI of the observed Cohen’s d. Violin plots represent their posterior distributions (using default priors). Red circles represent Cohen’s d simulated under the dimensional hypothesis. (Adapted from: Mammarella et al., 2021)

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this view, MLD children would occupy different moderate to extreme positions in a multidimensional space without blunt changes expected by core deficit models. The main challenge is to understand how position along one dimension may affect positions along other dimensions. For example, different cognitive impairments in children with MLD could vary as a function of comorbidity, severity of the disorder, age, and possibly other factors, too (Peng et al., 2018). A further factor supporting the validity of the multidimensional approach to individual variability in mathematics is the influence of intelligence as a crucial aspect for the diagnosis of specific learning disorders. Indeed, the traditional discrepancy hypothesis (i.e., the divergence between normal to high general intelligence and poor academic achievement) has been largely criticized as mainly considering intelligence as a unitary construct. In contrast, the dimensional distribution of academic and intellectual achievement (i.e., a continuum of smoothly changing values with no break points) has gained support, raising further doubts on the use of specific cut points (Branum-Martin et al., 2013). In a recent work, Toffalini et al. (2017) examined the intellectual profiles of a large sample of children (N ¼ 1049) with a diagnosis of specific learning disorders. According to the ICD-10 coding system (WHO, 1993), the international alternative to the American DSM-5 (APA, 2013), cases were classified as specific reading disorder (F81.0), spelling disorder (F81.1), disorder of the arithmetical skills (F81.2), or mixed disorder of academic skills (F81.3). Unsurprisingly, the majority of children had more than one academic impairment, confirming that co-occurring difficulties are the norm rather than the exception (Astle & Fletcher-Watson, 2020; Crisci et al., 2021). The specific arithmetic disorder was the smallest subgroup in the sample, consistent with the rarity in having selective math impairments. Indeed, the majority of children with math difficulty also show associated literacy-related disorders (Szűcs, 2016). Regarding the intellectual profiles, the four identified subgroups demonstrated many similarities, supporting the more recent DSM-5 classification (APA, 2013) leaning toward a single learning disorder category. In particular, the most evident weakness shared across all subgroups was the cognitive proficiency index (which is a composite of working memory and processing speed indices of the Wechsler’s Intelligence scale) below the general ability index by around 1 standard deviation, with the working memory index reaching the lowest level in all subgroups. In particular, the most evident weakness shared across all subgroups was the cognitive proficiency index (which is a composite of working memory and processing speed indices of the Wechsler’s Intelligence scale, probably the most famous and widespread intelligence battery) below the general ability index by around 1 standard deviation, with the working memory index reaching the lowest level in all subgroups. This is in line with the observation that working memory and processing speed are crucial for the successful acquisition of skills in different areas of academic learning (Peng & Fuchs, 2016). While it appears that all subgroups are characterized by a heterogeneous intellectual profile, they also have some specific properties. Children with mathematics impairments are somewhat better on verbal tasks but tend to perform worse in visuospatial processing tasks (i.e., the majority of tasks included in the perceptual reasoning index). This

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observation is in line with findings indicating that mathematical difficulties are characterized by a general weakness in visuospatial abilities, including visualspatial problem solving, visual perception, and even visuomotor integration (Pieters et al., 2012; Szucs et al., 2013). Here, up until now, we have looked at only one end of the dimensional continuum, the lower extreme of mathematical achievement. So far, the above outlined categorical approach to studying mathematical achievement meant that gifted children have been particularly neglected by research. To date, most psychological research has indeed studied (potentially) important aspects mainly in children and adults with normal as well as poor mathematical abilities. In contrast, systematic research on gifted/excellent mathematical knowledge in both adults and children is relatively rare and was almost completely overlooked for several decades (Leikin, 2020), even though the past decade recorded increased attention on the construct of high mathematical potential (Myers et al., 2017; Zhang et al., 2017). On the one hand, research findings in studying high-functioning individuals may not be generalized to other populations. However, on the other hand, the analysis of such “outliers” can provide new perspectives into so far neglected mechanisms. Exceptionality in mathematical knowledge is considered as a bidimensional construct composed of specific mathematical skills blended with creativity (e.g., a combination of logic, imagination, and intuition; Livne & Milgram, 2006). Zhang et al. (2017)’s review focused on current evidence of neural mechanisms, from structural and functional perspectives, in mathematically gifted individuals. In particular, they highlighted superior cognitive control functions in math prodigies, supported by an enhanced fronto-parietal network and strongly activated posterior parietal cortex. Another quite recent review looked at cognitive, motivational, and social factors associated with gifted mathematics (Myers et al., 2017). They found that relatively few psychological and cognitive neuroscience studies have focused on gifted students and the variables measured in each study were very diverse. Studies tested various categories of mathematically talented individuals ranging from exceptional calculators to math prodigies skilled in mathematical reasoning. Studies also used a range of very different mathematical tests. Methodological variation probably contributes to diverse findings. Myers et al. (2017) concluded that several cognitive variables seemed to be associated with mathematical giftedness, such as spatial processing, working and short-term memory, motivation/ practice time, reasoning, general intelligence, speed of information processing, and executive functions. Both literature reviews emphasized the fragmented nature of evidence, together with some problematic methodological aspects that future research needs to correct, for example, the frequent absence of control groups (Fehr et al., 2011), participant selection, and low-statistical power to detect small effects. Overall, the relationships between various factors linked to mathematical giftedness have not been explored systematically. There is a clear need for large multifactor studies that could test domain-general functions together with environmental, motivational, and emotional factors, rather than looking at them singularly.

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The Importance of Domain-General Cognitive Factors as Sources of Individual Differences Mathematics is a very complex subject, comprising of multiple areas of expertise, like arithmetic, geometry, algebra, and calculus. Each of them builds on interlinked topics and overlapping concepts, some of these hierarchical in nature. For example, learning how to solve arithmetic problems such as 34–28 first requires the understanding of what Arabic symbols represent (e.g., the quantity behind the digits and the meaning of arithmetical signs), proficiency in retrieving arithmetical facts, and the application of concepts and procedural knowledge. All these competencies are defined as domain-specific abilities in the cognitive and educational psychological research fields, ranging from more basic ones (i.e., ability to discriminate magnitudes) to more advanced (i.e., problem-solving skills). Mathematical processing is supported by multiple brain regions matching specific perceptual and cognitive processes, including visual and auditory processing, working memory, executive functions, attention, and cognitive control (Iuculano et al., 2018), as the topology of brain systems engaged in different mathematical tasks varies considerably not only across individuals but also with learning and development. Similarly, cognitive processes involved in mathematical tasks may vary according to individuals’ proficiency in mathematical skills and several other factors. Thus, the acquisition of each level of mathematical competence builds on interactions of a range of cognitive and noncognitive skills (Szűcs et al., 2014; Xenidou-Dervou et al., 2018). A wealth of correlational and longitudinal studies consistently demonstrated the involvement of an extended network of cognitive skills in the overall mathematics attainment. Such research has mainly focused on working memory processes and, to a slightly lesser extent, on executive functions (e.g., the set of processes that control and guide our thoughts and behavior). These nonnumerical skills have been usually labeled with the more general term of domain-general abilities. Different types of studies have attempted to describe and understand the role of domain-general abilities in mathematics, answering quite different research questions. Correlational studies aim to examine the relationship between mathematics and domain-general abilities at a single time point, experimental studies to explore the role of domain-general skills in performing specific mathematical tasks, while learning or training studies aim to identify how specific cognitive factors support the acquisition of new mathematics knowledge. Thus, only by investigating the differential role of different cognitive processes in several mathematical components, in different age groups, as well as distinguishing between learning processes and experienced application of mathematical knowledge, we can disentangle the relationship between domain-general skills and mathematics.

Working Memory and Mathematical Achievement Working memory (WM) can be defined as a set of processes that hold in mind a limited amount of information, keeping it available and allowing its processing

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while performing other concurrent tasks. Although this WM definition is quite frequently used, WM structure and capacity are still controversial topics in the literature. Componential models (e.g., Baddeley, 1986) posit different temporary storage mechanisms, each with its own limited capacity, whereas attention-based models (e.g., Cowan et al., 2005) stress the role and the limited capacity of a single, central attentional resource. The greater proportion of research that has investigated the role of WM in maths achievement has been based on Baddeley (1986)’s multicomponential model. According to this model, WM is a three-way system composed of the central executive (assumed to be an attention-controlling system), and two slave-systems, the visuospatial sketchpad and the phonological loop, which manipulates and stores visual images and speech-based information, respectively. The distinctions between the central executive system and specific memory storage systems in some ways parallel the distinction between WM and short-term memory. Short-term memory tasks only require recalling information previously presented (i.e., passive recall). For example, the classical Digit span task requires to recall of a series of digits of increasing length (generally, from 3 to 8). Differently, WM tasks require simultaneously processing and recalling information (i.e., active recall). For example, in the listening span task, a series of sentences of increasing length are presented, and participants are required to confirm after each sentence whether its general meaning is true or false, and then to recall the last word of each sentence. Similarly to the Digit span, the series of sentences increase in number. Thus, initially for a series of two sentences (i.e., length 2), two final words have to be remembered, for a series of three sentences three final words have to be remembered, then four, and so on. Relying on this model, research has demonstrated that WM is a strong predictor of individuals’ mathematical competences, both considering normal (Friso-van den Bos et al., 2013; Peng et al., 2016) and impaired mathematical competences (Peng & Fuchs, 2016; Szűcs, 2016). A particularly prominent feature of research in this field is the attempt to pinpoint which specific WM components (i.e., verbal and visuospatial short-term and WM) are most critical for explaining the relationship between cognitive and mathematical abilities (Caviola et al., 2014; Szucs et al., 2013, 2014). The verbal short -term and WM components seem more involved in the earliest stages of learning, such as counting, and the verbal mapping of quantity representations (Menon, 2016). Visuospatial short-term and WM seem to provide a mental workspace for manipulations and are often found to be weak in children with mathematical learning disabilities (Mammarella et al., 2013, 2018; Peng et al., 2016; Szucs et al., 2013). In a recent large cross-sectional study, we examined the specific influence of both domain-specific magnitude comparison tasks (i.e., basic numerical processing tasks rely on the different types of numerical representation) and domain-general measures (verbal and spatial short-term and WM) on children’ mathematical achievement (Caviola et al., 2020). We tested a very large sample of primary school children attending 2nd, 4th, and 6th grades (N ¼ 1254). Moreover, we also used reading decoding as an outcome measure to determine whether findings were specific to

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mathematical achievement. Due to high statistical power, we were able to estimate effects, their relative importance, and specificity of each predictor with also a high time resolution across development. As for domain-specific skills (i.e., basic magnitude comparison task), results were quite overlapping with previous findings: Numerical comparison ability involving Arabic digits proved to be a reliable and largely specific correlate of mathematical achievement; vice-versa, magnitude comparison ability involving analogue stimuli only (i.e., set of dots) never became a significant predictor of maths when other variables were included in regression models (for similar results, see also Schneider et al., 2017). We also found that verbal WM performance and short-term memory components support both reading and math achievement; thus, verbal short-term and WM were not specific predictors of mathematical attainment. In contrast, visuospatial WM appeared to be an increasingly specific correlate of mathematical outcome, and this specific relation became stronger in older children (Grade 4 and 6 here). Indeed, visuospatial WM likely provides a mental workspace utilized in maths but not in reading performance (Szűcs et al., 2014). Thus, these results are in line with previous research outcomes. For example, Szűcs et al. (2014) used robust bootstrap statistics with 98 10-year-old children with normal reading skills, to examine the involvement of different domain-general cognitive tasks and a series of domain-specific measures related to basic numerical processing. Their findings falsified again the importance of basic magnitude abilities and further highlighted the importance of visuospatial short-term and WM, verbal knowledge, and general executive functioning in explaining the success of mathematical performance. In addition, the authors only found a weak correlation between mathematical competence and verbal short-term and WM, and other cognitive components such as sustained attention and inhibition. Looking at meta-analytic results, Peng et al. (2016) analyzed 829 effect sizes from 110 studies showing a stronger association between overall composite WM scores and mathematical performance (r ¼ 0.38) rather than single verbal and visuospatial components of WM, respectively, r ¼ 0.30 and r ¼ 0.31 (for similar results, see also Friso-van den Bos et al., 2013). The small variation in the strength of the relationships could be explained by the different WM tasks used (verbal and visuospatial) and by the developmental progression of general cognitive resources (Meyer et al., 2010). Peng et al. (2016) also confirmed how different types of mathematics skills and level of mathematical competences are sources of variation in determining the strength of the math-WM relationship. Different mathematical measures can draw on a constellation of mathematical subskills that may differ from test to test, from individuals’ level of expertise and strategies repertoire they use, resulting in different degrees of cognitive load. Likewise, after controlling for age, domains of WM, and type of mathematics skills, individuals with mathematical difficulties in comorbidity with other disorders seem to show a stronger association between WM and mathematics compared with typically developing samples. However, the authors did not report the coding strategies adopted to classify their sample types as having mathematical difficulties or not, thus resulting in a weakly reliable moderator.

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Table 1 Standardized effect sizes in studies which matched versus which did not match reading achievement. (Adapted from Szűcs, 2016) Verbal short-term memory Visuospatial short-term memory Verbal WM Visuospatial WM

Matched for reading 0.37 0.81 0.59 0.90

Nonmatched for reading 0.86 0.63 0.94 0.61

A more systematic analysis on this topic had been conducted by Szűcs (2016). In his meta-analysis, Szűcs (2016) included 36 studies and summarized the data collected on 665 children labeled with developmental dyscalculia and 1049 controls. Studies that matched reading in children with and without math-learning disability reported smaller absolute differences in effect sizes, between children with mathlearning disability and control groups in verbal short-term and WM performance. In contrast, the difference between groups was larger in visuospatial short-term and WM performance in matched than in nonmatched groups (see Table 1). The results highlighted the existence of two possible cognitive profiles: a subtype associated with reading problems and weaknesses in the verbal component of WM and shortterm memory; conversely, the second profile, without any associated reading problems, showed selective drops in visuospatial WM and short-term memory. Taken together, this evidence suggests the importance of considering multiple cognitive variables simultaneously, especially when we want to investigate their weight in a complex domain such as mathematical development that likely relies on an extended network of cognitive skills (Szűcs et al., 2014; Xenidou-Dervou et al., 2018).

Executive Functions and Attentional Control in Mathematical Learning Executive functions (EFs) skills are particularly important when individuals are dealing with novel, rather than routine, situations or activities. One of the more diffuse EFs models is Miyake’s (Miyake et al., 2000), which identifies three basic EFs: (a) inhibition, or the ability to deliberately inhibit dominant, automatic responses when required; (b) shifting (also called cognitive flexibility), which is the ability to switch between tasks, operations, or mental sets to adjust to changed priorities; and (c) updating, or the ability to update and monitor information in WM, replacing old and no longer relevant information with more recent and relevant input, and translating instructions into action plans. While there is no doubt that WM is crucial for holding interim computation in mind during a mental calculation task, the role of inhibition and switching is less clear in supporting mathematics achievement. For example, inhibition may be needed to suppress the unwanted activation of number facts during a fast-task retrieval, and shifting may be involved when individuals move between different

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arithmetical problems and consequently different procedural rules. Thus, compared to WM, fewer studies have investigated the role of inhibition and shifting in mathematics performance, and the findings are overall mixed. Inhibition of irrelevant information in mathematical abilities has manifested in different ways. It may, for example, involve the suppression of immature or inappropriate strategies, such as addition when multiplication is required, or suppression of irrelevant information during the execution of word-problem solving (Bull & Scerif, 2001). The majority of studies seem to suggest that inhibitory control abilities do predict performance in mathematics (Bull & Scerif, 2001; Gilmore et al., 2013). In their broad meta-analysis on WM and math relationships, Friso-van den Bos and colleagues extended their analysis to inhibitory functions as well (Friso-van den Bos et al., 2013). They collected 131 correlations drawn from 29 studies revealing a medium effect size (r ¼ 0.27) moderated by the type of mathematics measures (e.g., correlations were higher when national curriculum tests or composite scores were considered than when more specific tests). Some studies have identified deficient inhibition skills (Szucs et al., 2013) and deficient attentional skills (Soltész et al., 2007) when concerning mathematical difficulties: Inhibitory and attentional processes, in fact, serve to select the elements to be processed in the correct order, and consequently to inhibit interference from irrelevant elements. For example, both children with typical and atypical development have consistently presented impairment in these mechanisms (Bull & Scerif, 2001; Lee & Bull, 2016). Several studies have also highlighted how children’s ability to focus on the task and control their attention predicts their long-term success in mathematical learning beyond the influence of domain-specific skills. Indeed, attention control includes the ability to exclude irrelevant information while processing information relevant to the task itself, as well as the ability to ignore external distractions and to be able to remain focused even in sometimes chaotic situations, such as a classroom context (Geary et al., 2012). As far as the relationship between mathematical achievement and shifting is concerned, evidence is sparser (Bull & Lee, 2014). Only two meta-analyses focused on shifting, and academic achievement found substantial evidence in favor of a relationship (0.26 < rs > 0.28; Friso-van den Bos et al., 2013; Yeniad et al., 2013), but these results should be interpreted with caution. Friso-van den Bos and colleagues collected 94 effect sizes from 18 studies (Friso-van den Bos et al., 2013). Although with a significant variance across studies, the analyses showed a variation in effect sizes due to maths measures considered and individuals’ characteristics (e.g., age and sample type). Similarly, Yeniad et al. (2013)’s meta-analysis, based on 18 studies (N ¼ 2330), concluded that shifting abilities do predict performance in mathematics. However, intelligence was found to be a stronger predictor of academic performance (for both maths and reading) than shifting. Moreover, shifting was substantially associated with intelligence, thus leaving an open question whether shifting can be considered an independent contributor of mathematics over and above general intelligence. Among other variables, the authors coded the heterogeneity of shifting procedural rules, type of scoring, and sample characteristics (e.g.,

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children’s age, grade level, SES, or gender), but these moderations could not be tested due to an insufficient number of studies per subset. In two recent reviews of the literature, Gilmore and Cragg (2018) provide a series of compelling reasons explaining the inconsistency of the results when it comes to EFs and maths relationships. Specifically, they suggested that EF skills are important for learning and performance across all academic subjects; thus, it is possible that the relationship between EF and mathematics is not specific. In addition, starting from the developmental nature of both EFs and mathematics, the complexity of the relationships is further increased by the nature of tasks. Different types of tasks may tap into different aspects of inhibition skills (e.g., resistance to proactive interference and prepotent response inhibition), with different developmental trajectories. Similarly, shifting tasks may vary with the level of complexity, and consequently, also the additional cognitive processes may vary as well, affecting the relations with attainment scores. In conclusion, to better understand the nature of the relationship between EF and mathematics, it is important to explore the differential role of EF skills in multiple components of mathematical knowledge in different age groups, as well as considering the complexity of EF processes.

Beyond the Purely Cognitive: Metacognition, Affect/Beliefs, and Motivation as Sources of Individual Differences As reviewed above, a wide body of literature reveals the importance of individual differences in cognitive factors, both specific (i.e., basic and conceptual numerical knowledge) and general (i.e., cognitive resources). However, several other sources of variability contribute to account for the complex nature of individual differences in mathematics. In particular, a growing amount of research is recognizing the importance of metacognitive skills as well as affective and motivational aspects.

Metacognitive Abilities Metacognition – the knowledge, monitoring, and regulation of cognition (Flavell, 1976) – involves active control over the thinking processes involved in learning. It is essentially a twofold concept that can be summarized as the sum of metacognitive knowledge and metacognitive regulation processes (Flavell, 1976). The former involves declarative knowledge individuals have about their own cognitive abilities and cognitive strategies; the latter refers instead to procedural knowledge aimed to monitor, coordinate, and control one’s cognition and actions. Based on this general definition, it appears clear why metacognitive skills are important for any type of school learning, but with a particular focus on mathematical learning, as children acquire knowledge about how their minds work, as well as strategies to properly master mathematical concept and procedures (Cornoldi & Lucangeli, 1997). Yet, research in this field is sparse in comparison to other domains, and the existing evidence is mixed although converging to the conclusion that individual differences

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in metacognitive skills are quite relevant for mathematical learning (Desoete & De Craene, 2019). Metacognition seems to be one of the most important predictors of mathematical performance as it substantially represents an important “predisposition” that enables children to learn (Ohtani & Hisasaka, 2018). In their review of the literature on the role of metacognition in mathematics education, Schneider and Artelt (2010) confirmed the extensive influence of declarative knowledge on mathematics performance, concluding as all learners benefit from metacognitive instruction procedures. This finding was consistent with the meta-analytic review of Ohtani and Hisasaka (2018), who found a positive correlation between metacognition and mathematics, however moderated by the choice of measurement tools. A large number of measures are indeed used to assess metacognition, and these can be distinguished between off-line/self-report measures – mainly applied to assess declarative knowledge and with the form of questionnaires – and online measures used to tap procedural metacognitive processes and investigated by means think-aloud or log-file protocols (Veenman & van Cleef, 2019). Between those, unsurprisingly, online tools have shown stronger correlations with academic performance (r ¼ 0.53), compared to the self-report questionnaire which exhibited lower correlation coefficients (r ¼ 0.19; Ohtani & Hisasaka, 2018). Interestingly, Bellon, Fias, and De Smedt (2019) assessed 7 and 8 year olds in arithmetic, as well as in EF and metacognitive tasks (both general metacognitive knowledge and on-task metacognitive monitoring). They found that both updating and on-task metacognitive monitoring are important unique predictors of arithmetic; however, the specific metacognitive task proved to be more important than updating. The authors concluded that it is important to include metacognition in cognitive research to better understand individual differences in arithmetic. A large body of the literature also considered evidence from instructional behavior and/or intervention studies associated with a wider definition of metacognition including self-regulated learning strategies which comprises both the knowledge and control of not only cognition, but also of motivation. In their metaanalysis, Donker and colleagues (Donker et al., 2014) included intervention studies on learning strategy instruction and strategies application focused on improving self-regulated learning. They addressed the question which learning strategies are the most effective in enhancing the academic performance of students. Three types of learning strategies were included in this meta-analysis: cognitive, metacognitive, and management strategies, and their related motivational aspects and metacognitive knowledge. Different academic performances were considered; however, metacognitive knowledge instruction appeared to be valuable in all of them. Specifically, 180 effect sizes reported substantial effects in writing (Hedges’ g ¼ 1.25), science (0.73), mathematics (0.66), and comprehensive reading (0.36). These academic skills differed in terms of which strategies were the most effective in improving academic performance. For example, in mathematics, elaboration proved to be very effective. This strategy entails attaching meaning to new information by connecting the old to the new material, explaining step-by-step choices for solving a problem.

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More recently, Hacker and colleagues (Hacker et al., 2019) examined two intervention studies of teaching fractions. In the first study, children at risk of mathematical difficulties, attending grades 4th to 6th, were randomly assigned to treatment and control conditions. The intervention program consisted in 6 lessons of a language-based, metacognitive instructional intervention for teaching foundational concepts of fractions. Teachers in the control condition provided supplemental instruction using the regular mathematics curriculum. The results showed that students in the intervention group exhibited statistically significant gains over peers in the control group in a range of different skills related to fraction knowledge (Hedges g ranged from 0.60 to 3.20). The second study used a single-case multiplebaseline design aimed to boost computational performance with fractions. In this study, results, although not statistically significant for all the near-transfer measures, showed that metacognitive intervention had positive effects on students’ pre- and postintervention gains (d ¼ 0.70). Taking together, and in line with Gascoine and colleagues’ systematic review, there is strong support that metacognition plays a significant role in mathematics performance (Gascoine et al., 2017). However, future research should aim to systematically distinguish between the assessment tools, and, in line with the componential nature of mathematics (Dowker, 2019), also control for students’ level of mathematical competencies and their cognitive skills. Moreover, while several studies explicitly or implicitly assume a positive mediating role of metacognitive skills in mathematics learning, quite remarkably, no research explicitly verified this mediating effect (Verschaffel et al., 2019). Given the assumed positive role of metacognition in mathematics performance, it is highly recommended that this mediating hypothesis is properly tested. Finally, given the well-established and enduring problems with the emotional and motivational aspects, more research should also look at the effect of metacognitive strategies on motivational-affective outcomes, such as students’ interest or self-esteem in mathematics, since both are quite important attitudes for successful academic attainment.

Negative and Positive Attitudes Bourgeoning research on cognitive factors often led to the neglect of the impact of individual differences related to personal factors, implicitly suggesting that such variables have tangential or negligible effects on math performance. A consistent body of research showed indeed how emotional states and beliefs may deeply impact learning and performance in mathematics. These affective factors can be described along a continuum of negative versus positive attitudes toward mathematical situation. Among the former, the most studied in relation to mathematics attainment is an unpleasant emotional response associated with the execution of numerical and arithmetic tasks. This negative feeling, defined by researchers with the term mathematical anxiety, often interferes with mathematical performance since the beginning of school years (Mammarella et al., 2019). A more comprehensive overview of the literature of this construct and its (negative) relationship with individual

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differences in mathematical achievement is provided in ▶ Chap. 13, “Mind, Brain, and Math Anxiety,” of this book. Other negative effects, defined as “choking under pressure,” refer to stressinduced situations while performing mathematical tasks which trigger a worsening in performance expected for an individual’s level of ability (Caviola et al., 2017). In the same way as math anxiety phenomenon, the main hypothesis of this research field is that contextual/social pressure interferes with limited cognitive resources (Beilock & Carr, 2001). Beilock and colleagues (Beilock & Carr, 2001) investigated how individual differences in WM capacity relate to high- and low-pressure conditions defined by different social scenarios, such as monetary incentives or peer pressure, while subjects solve modular arithmetic problems (a complex task that can be solved by computation or estimation strategies). They found that participants with higher WM capacity were more disturbed by the pressure constraint than the subjects with less WM capacity, leading to a more significant lowering of arithmetic performance in those with higher WM capacity. According to this perspective, pressure is assumed to cause worrying off-task thoughts that result in overload of WM already engaged in the math task, indeed similarly to math anxiety. However, these phenomena differ according to the source of the negative affective reaction: In the case of math anxiety, the negative reaction is linked to exposure to mathematical content, possibly since primary school years. For choking under pressure, no personal susceptibility to anxiety traits is indicated – it is often individuals who desire to perform well (and have also shown to be especially likely to do well) who show a more consistent performance decrement. Finally, another possible source of performance decrement in mathematics is stereotype threat (Moore et al., 2014). The key factor behind this phenomenon is membership of a social/gender/ethnic group to which some negative stereotype is attached about a specific domain of knowledge, such as mathematics. Its negative effect is strengthened for those individuals who either value success more strongly in this particular subject or for those who recognize themselves more strongly with the stereotyped group. A clarifying example is provided by the gender stereotype in mathematical performance in stressful contexts. Research in this specific field suggested that girls’ mathematical performance is disrupted under threat, not because of their scarce numerical abilities but, as a sort of vicious circle, because girls feel threatened by the possibility that their performance will confirm the negative stereotype associated with their belonging to the female gender. However, as the next studies will show, evidence for stereotype threat effects in mathematics for primary, middle, and high school girls is mixed. Some studies report evidence of stereotype threat in girls as young as kindergarten age (Tomasetto et al., 2011), whereas others have found no effect even for high school girls (e.g., Cruz-Duran, 2009). For example, Tommasetto et al., (2011) found that stereotype threat impaired girls’ performance on mathematical tasks among pupils from kindergarten through 2nd grade. Their results showed girls’ performance declined after their gender identity was made salient, and above all, this negative relation was moderated by mothers’ endorsement of a gender stereotype threat. Thus highlighting the importance of improving girls’ attitudes or beliefs toward

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mathematics (and science in general). Differently, this effect seems to disappear when considering older students. Ganley and colleagues (Ganley et al., 2013) did not find clear evidence that mathematics performance of school-age girls (from 4th to 8th grades) was impacted by stereotype threat, either explicitly or implicitly activated. Across two out of three studies, girls reported lower mathematics attainment regardless of whether stereotype threat was activated. In their conclusions, the authors strongly encouraged that more nuanced research needs to be done to deepen the role of stereotype threat in this domain, by following a multidimensional perspective and thus considering an increased number of other factors involved. Finally, in a recent meta-analysis, Flore and Wicherts (2015) found evidence, although small (g ¼ 0.22), for a negative effect of the stereotype threat on girls’ math, science, and spatial skills. Although, the robustness of the stereotype threat effect can be questioned by the presence of publication bias. Looking at the brighter side, students are equipped with such personal assets or individual resources like resilience, self-concept, and self-efficacy, among others, that act as protective factors when they face difficulties or stressful situations (Windle, 2011). Unlike negative factors, these several personal assets have been found to sustain positive life outcomes and academic success (Masten, 2001). Resilience (often called ego-resiliency) is a pattern of individual features, such as general resourcefulness, strength of character, and flexibility of functioning, that enables people to recover quickly from difficulties and day-to-day challenges (Fletcher & Sarkar, 2013). At school, ego-resiliency helps children to cope with potentially stressful situations in their academic lives by fostering their competence and enabling them to focus their efforts, with a positive effect on the outcome (Swanson et al., 2011). Children who are more ego-resilient adapt more quickly, are more flexible in using problem-solving strategies, and are more persistent in achieving their academic goals (Alessandri et al., 2017). In a recent work, Donolato and colleagues (Donolato et al., 2020) examined the specific contributions of both negative (i.e., school anxieties) and protective factors (i.e., ego-resiliency) to mathematics performance in primary- and middle-school children (N ¼ 274). Children were tested with several self-report tools measuring both mathematical anxiety and other forms of anxiety, as well as ego-resiliency. Children also completed standardized intelligence and mathematical tasks. After accounting for other forms of anxiety (i.e., test and general anxiety), mathematical anxiety showed its negative effect on children’s mathematics performance, strengthening the previously reported evidence of a specific association between these variables (Devine et al., 2018). In other words, although mathematical and test anxiety overlap to some extent, mathematical anxiety remained more strongly (negatively) associated with mathematics performance (Fig. 2). Thus, it may be that being generally more anxious and worried about academic performance are risk factors that may contribute to the development of a more specific form of anxiety – mathematical anxiety – and therefore be indirectly related to mathematical performance (Mammarella et al., 2019). As regarding the role of ego-resiliency, a positive effect on mathematics performance was found. This is in line with previous studies reporting a positive association between ego-resiliency and academic achievement and supports the conviction that

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Fig. 2 Final model with standardized coefficients. All reported paths are statistically significant ( p < 0.05). RES ego-resiliency factor (ER ego-resiliency scale), GA general anxiety (RCMAS-2 general anxiety scale, PA physiological subscale, WO worries subscale, SO social subscale), TA test anxiety (TAQ-C test anxiety scale, PA physiological subscale, THO thoughts subscale, OFF off-task behavior, SO social subscale), MA mathematics anxiety (MLA math learning anxiety, MTA math testing anxiety), gF fluid intelligence (Cattell intelligence scale), MATH mathematics literacy (N numbers, SF space and figures, DP data and prediction, RF relations and functions). (Adapted from: Donolato et al., 2020)

ego-resiliency can be an important personal asset in relation to children’s mathematical achievement (Alessandri et al., 2017). This study also aimed to see whether ego-resiliency correlated with different forms of anxiety, and whether the associations between these variables contributed to explaining the relationship between ego-resiliency and mathematics. The results highlighted that ego-resiliency was negatively associated with general anxiety only, but not with other forms of school anxiety (i.e., test or mathematical anxiety). These results further point to a protecting factor role of ego-resiliency, suggesting how this factor may help to balance individuals’ adaptive capacities on an emotional level in stressful situations. In sum, the

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results of structural equation models showed that mathematical anxiety had a main negative effect on mathematics performance, over and above the effect of other forms of anxiety. Vice versa, and this is the real key aspect of this work, ego-resiliency had a direct positive effect on mathematics performance: Higher levels of ego-resiliency would boost students’ ability to manage any worries they might have about their performance, thereby reducing any school anxieties. In a similar vein, self-efficacy represents individuals’ expectations and convictions about their own abilities, and what they can accomplish in a given situation (Tsang et al., 2012). As defined by Ashcraft and Rudig (2012), who nicely readapted the original Bandura’s (1977) definition, self-efficacy encompasses elements of prior success, interest, and motivation to purse mathematics. Meta-analytic evidence showed high positive correlations between self-efficacy and vocational interest in math (r ¼ 0.73; Rottinghaus et al., 2003). However, an opposite relation has been found between mathematical anxiety and mathematics (r ¼ 0.28; Barroso et al., 2021). Based on the social cognitive perspective (reciprocal determinism), self-efficacy and achievement should mutually reinforce each other. Within this view, academic self-efficacy influences students’ level of effort, persistence, and their choice of activities (Bandura, 1977). As a consequence, higher academic achievement can be expected from students with higher self-efficacy. At the same time, positive mastery experiences are considered to be the strongest source of self-efficacy (Bandura, 1977); thus, also successes in academic achievement more likely strengthen students’ judgments of their capability to cope with future academic requests. However, the empirical support about this reciprocal relationship is sparse because existing results either derived from cross-sectional data (e.g., Skaalvik et al., 2015) or isolated measures of self-efficacy and achievement, respectively (e.g., Parker et al., 2014), further highlighting the need of more longitudinal research outcomes. As an example, Williams and Williams (2010) tested the reciprocal relationship of math self-efficacy and mathematics achievement in 33 countries. Data were gathered from the 2003 Program for International Student Assessment (PISA; OECD, 2004) international survey assessing 15-year-old students. The results provided supporting evidence that self-beliefs and performance iteratively modify each other in 26 out of 30 countries. Moreover, the authors found a consistent negative effect of gender on self-efficacy: Females showed consistently lower levels of math self-efficacy than did males. This seems to support the use of gender as an instrumental variable to generate stereotype threat, as well as the self-fulfilling prophecy, namely when socialization works against girls concerning mathematics attainment. Similarly, Louis and Mistele (2012) replicated the same results with a large cohort of about 7400 eighth grade students through the 2007 Trends in International Mathematics and Science Study (TIMSS) national US survey. The results showed that the mathematics self-efficacy scores were statistically significantly different by gender, and in particular males reported higher self-efficacy levels than females in mathematics. No gender difference emerged when looking at mathematics proficiency though. However, due to the cross-sectional nature of both data sets used for these international comparative studies, cross-lagged effects could not be considered.

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Stronger evidence of this reciprocal effect self-efficacy/math performance flowed from longitudinal studies, however still few in number (Hannula et al., 2014; Schöber et al., 2018). Hannula and colleagues presented a longitudinal survey of Finnish students, tested three times over 7 years, from grade 3 until grade 9 (Hannula et al., 2014). The results of this longitudinal study support the view that math selfefficacy and attainment are mutually linked, considerably stable over time and with a dominant effect from achievement to self-efficacy (see also Schöber et al., 2018 for different results). They also included a second affective measure more closely related to students’ individual interests, mathematics enjoyment, reflecting more enduring intentions for specific math-related scenario. The two affective measures were highly correlated, showing an important amount of shared situation-specific effects.

Motivation Upholding this brief overview, it is quite evident that several other components would explain variation in mathematical skills. For example, the abovementioned self-efficacy and mathematics relationship can be also explained through a mediating or moderating effect of other variables, like home math environment (e.g., all mathrelated activities and parents expectations; Daucourt et al., 2021) or students’ intrinsic motivation (e.g., their interest in peruse mathematics; Ganley & Lubienski, 2016). Mathematical motivation captures the extent to which individuals embrace challenges, value the importance of mathematical abilities, and are motivated to perform well in this academic domain. As mathematics, motivation is a multifaceted construct where self-perceived abilities (i.e., individuals’ perception of their own competences), interest (i.e., gratification students gain from learning and doing math-related activities), and the perceived importance of success in mathematics are the three most investigated dimensions (Wigfield & Eccles, 2000). Needless to say, the positive effect of high motivation depends on whether the motivation is intrinsic or extrinsic. That is, intrinsically motivated students generally engage in a mathematical task (either for learning or applying maths concepts) for the sake of interest in increasing their own understanding and expertise in maths, thus perceiving maths as important, and enduring in pursuing math activities. In contrast, those with extrinsic motivation wish to gain external recognition (Deci & Ryan, 1985), such as good grades or praise from teachers and parents. Consequently, they tend to seek less help from teachers (and parents), thus avoiding situations in which they might experience negative feedback or consequences – when they experience such negative judgments, they tend to disengage their pursuit of maths (Murayama et al., 2013). The studies presented here may offer some new insights on metacognitive skills as well as affective and motivational aspects, but more research is still needed, especially from a longitudinal perspective. First, future investigations should evaluate and compare the contribution of specific protective factors (i.e., resilience) together with other personal assets that may contribute to success in mathematics, such as self-efficacy and motivation (Grigg et al., 2018). Second, studies should

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adopt a longitudinal design to examine whether these personal assets can really prevent or modulate the development of mathematical anxiety, and to what extent these factors could support or hinder mathematical achievement over time. Last but not least, research should also consider more extensively external factors, such as teachers’ and parents’ expectations regarding children’s academic achievement, which are known to be related to children’s anxiety (Gunderson et al., 2012) or also different contextual conditions through which children experience mathematical learning.

From Cultural and Language Differences to Contextual Factors Average mathematics achievement varies greatly among countries (Gonzales et al., 2008; OECD, 2010). The systematic, large-scale investigations of cross-national disparities in students’ mathematical achievement have been born to explain the magnitude of these learning gaps. The first outsized cross-cultural study began more than 50 years ago to examine potential influences of contextual factors, such as the amount of homework and family background accounted for a broad range of mathematical disparities. These cross-national studies have quickly increased over the years, and they are now regularly conducted through the Program for International Student Assessment (PISA; https://www.oecd.org/pisa/aboutpisa/) and the Trends in International Mathematics and Science Study (TIMSS; http:// timssandpirls.bc.edu/), two of the most widely cited cross-national surveys. As regarding cultural differences, unfortunately, the story is not changed either, and the difference in maths scores between the top-ranking nations and the bottomranking nations is quite consistent (about 3 SD). Several sources of cultural differences can be listed to explain the observed discrepancies in mathematics performance. Previous studies tap into linguistic factors; parental support; school climate; educational systems; cultural beliefs; school readiness; and socioeconomic status. Due to the vastity of these sources of heterogeneity, in this last section of the chapter we will focus on linguistic and contextual factors encompassing parental support and school climate.

Linguistic Factors Children’s language abilities are known to contribute to their mathematical performance (Peng et al., 2020). Yet, findings about the relation of language and mathematics are diverse: Some studies have reported strong correlations (rs. > 0.70; e.g., Merz et al., 2015), and others have found weak to nonsignificant correlations (rs. > 0.20; e.g., Mellard et al., 2015). To date, only four reviews have specifically investigated this relationship, each focusing on specific characteristics of both mathematics and languages elements (Chow & Jacobs, 2016; de Araujo et al., 2018; Peng et al., 2020).

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Although based on three studies only, Chow and Jacobs (2016) concluded about the importance of oral language abilities in fraction performance and understanding for school-age children. de Araujo et al. (2018) provide an extensive qualitative summary of the literature, with a focus on how individuals learn mathematics in a multilinguistic educational setting. They suggested that language is a primary medium for individuals’ acquisition of mathematics knowledge, in line with more recent research (Martin & Fuchs, 2019). Koponen and colleagues focused on the role played by a basic phonological processing competence – the rapid automatized naming skill – in mathematical learning (Koponen et al., 2017). The authors found a significant correlation between this component of phonological processing and mathematics across 38 studies (r ¼ 0.37). This effect was moderated by the type of mathematical tasks considered. Similar results were highlighted by Peng and colleagues’ meta-analysis (Peng et al., 2020) who found a moderate relation between language and mathematics. Compared to previous reviews, Peng extended the analysis to different types of language skills as well as different types of mathematics skills, also controlling for cognitive skills such as working memory and intelligence. Taken together, the findings of these reviews shed light on some relations between language and mathematics. However, a recent line of research suggests looking at the role of slightly different kinds of language, in particular the mathematical language of preschool children and how its mastery influences mathematical development (e.g., Kung et al., 2019; Purpura et al., 2017). Research on the role of children’s general language skills and their mathematical proficiency had tended to focus on early grades, confirming the importance of linguistic abilities for mathematical skill development. In addition, rising research advocates that mathematicsspecific language also matters (Purpura et al., 2017), showing as mathematical language resulted in being a significant predictor of later mathematical abilities over general language skills and initial mathematical knowledge. Mathematical language is an interesting construct and may be defined in different ways. For example, some studies consider the verbal labels associated with each number such as not only words related to counting skills and cardinality principle (e.g., one, two, and three), but also words referring to equivalence concepts (e.g., same, equal), as part of mathematical language (e.g., Klibanoff et al., 2006). Purpura et al. (2017) proposed a different definition of mathematical language referring to more specific quantitative (e.g., more, fewest, and a little bit) and spatial (e.g., before, after, and nearest) words. Similarly, research on text problem solving shed some light on the context-dependency of certain mathematical-related words. For example, it seems that when a text problem presents words such as “more,” “buy,” and “get,” those facilitate the processing of requiring addition to reach a solution; conversely, if the same text problem also conveys words like “less” and “sell,” they seem to hinder the solution process. Next to mathematical-related type of words, a large body of studies focused on the way number words are formed and how this may determine different number word-learning trajectories. Indeed, number word systems vary considerably with respect to their transparency – or lack of – by which the place-value structure of the Arabic number system is reflected in number word formation. Indeed number words

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do not always reflect the place-value structure properly (e.g., German: The number word for 23 is “dreiundzwanzig,” literally three-and-twenty), which further complicates the learning process. From a lexical point of view, differences between language groups in number word acquisition should be fairly trivial for numbers up to ten. This is unsurprising since, in most languages, numbers from 1 to 10 are labeled by exactly ten arbitrary number words that kids learn and map to the respective numerical magnitude within an ordered sequence. And indeed, crosscultural studies examining counting abilities in children aged between 3 and 6 found trivial differences across language groups for numbers up to 10 (LeFevre et al., 2002). Taken together, a variety of different linguistic influences seem to affect the achievement of numerical and, consequently, mathematical competencies. Importantly, some linguistic aspects seem to impact specific numerical and/or mathematical skills early on while others might affect the acquisition of different abilities, which only follow later. Given that early mathematical abilities are strong predictors of later mathematical success in primary school and that associations between language and mathematics are the strongest at the youngest ages (Peng et al., 2020), future studies aimed to disentangle the role of language in mathematical competence should focus on the role of specific mathematical language in preschoolers rather than focus on general linguistic competences. Notably, however, others have questioned specificities in number word systems as a sole contributing factor to observed cross-cultural differences and argue that differences in, for instance, approaches to teaching and learning as well as differences in home experiences (e.g., LeFevre et al., 2002) need to be considered as plausible additional (or alternative) explanations for the observed individual differences.

Contextual Factors: From Parental Support to Educational Systems Similarly, environmental (intended as social and contextual) factors play an important role in the developmental process of mathematical learning. These additional sources of heterogeneity may indeed impact the relative mastery of mathematical competencies, such as different contextual conditions through which children experience mathematical learning. The ways in which children are taught, the types of training teachers get, curricular contents can vary substantially, across cultures and different socioeconomic status. Although supporting families’ participation in their children’s education may serve as a valuable approach to boost children’s success and minimize the achievement gap that exists between ethnic minorities and in low-income communities (Wong & Hughes, 2006), little is known about the underlying mechanisms through which parental involvement influences their children’s academic achievement. Parental support, or parental involvement, is usually defined as motivated parental attitudes and actions intended to impact children’s educational well-being that in turn positively influence school outcomes (El Nokali et al., 2010). In a recent and quite extensive meta-analysis, Barger and colleagues (Barger et al., 2019) showed small

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positive associations (rs.13 to 0.23 – stable over time) between parents’ natural involvement in children’s schooling and children’s academic engagement. Parental support has traditionally been outlined by parents engaging in school-based activities (e.g., attending parent-teacher conferences and school events), but recent studies have shown that a much broader conceptualization of parent involvement includes activities and interactions between parents and their children at home and in their communities (e.g., supervision and monitoring of homework activities, daily conversations about school, and visiting local community institutions for learning purposes). A growing body of literature demonstrates indeed how early experiences, mainly informal activities carried out with parents in the home environment, can shape children’s developmental trajectory and can contribute to explaining individual differences in mathematical knowledge. Children’s early home numeracy experiences – both in terms of kinds of informal learning activities and variation in exposure to those activities – can vary according to different cultural and socioeconomic status. Self-report studies indicate that parents involve their children in both formal and informal numerically relevant activities. The former activities imply explanations of number concepts, practicing number skills like identifying or writing Arabic digits; the latter refers to more engaging activities that are not focused on directly teaching or explaining numerical/mathematical concepts (e.g., playing board games, measuring ingredients while cooking, or setting the table). Although some studies have not found relations between the home numerical environment and children’s mathematical achievement, a recent meta-analysis reported an effect size of r ¼ 0.13 (Daucourt et al., 2021). Following the sociocultural perspective, several studies highlighted how children from different cultures can show a considerable disparity in numerical knowledge and how these diversities may reflect differences in their early home numeracy activities. For example, the superiority of Chinese children in mathematical skills over Western peers is well established. These differences are in part explained by the greater amount of time Chinese parents spend with their children doing numericalrelated activities and arithmetic tasks. A similar pattern emerged when comparing numerical knowledge of young children from low-income versus middle-income backgrounds living in the same country. Differences in mathematical knowledge of children from lower- and higher-income backgrounds may also reflect differences in the more general environmental support for mathematical learning, such as schools. With regard to educational systems, several researchers looked at the interplay between different aspects related to school and classroom climate (or environment) and how they may affect academic achievement. Current conceptualizations of school and classroom climate encompass numerous contextual and/or environmental factors (Zullig et al., 2010), making it difficult to converge to a unified definition of these rather comprehensive constructs (Wang & Degol, 2016; Wang et al., 2020). For instance, school climate comprises the whole set of social and physical aspects that broadly define school life: In brief, it defines its quality and character, as it is perceived by students, teachers, and the broader school entourage (Thapa et al., 2013). Regardless of any specific theorizations, extant reviews point out that a

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dimension that has attracted attention in both school and classroom climate is the quality of interpersonal relationships, both between peers and between students and teachers, individually (Wang et al., 2020). A positive student-teacher relationship has been shown to directly affect students’ motivation to learn (Davis, 2006) and cover a prominent role in their competencies development and academic achievement (Lei et al., 2018). A relatively consistent body of literature confirmed the positive impact of supportive school climate on students’ academic experiences, both from an emotional and academic point of view (Thapa et al., 2013). Those findings entail that an inclusive learning environment is characterized by teachers that are aware of their students’ attitudes and metacognitions toward math and may, thus, engage in educational practices that encourage their learning and school well-being. In their review of the literature on the role of metacognitive beliefs in mathematics education, Schneider and Artelt (2010) confirmed indeed the extensive influence of declarative knowledge on mathematics performance, concluding as all learners benefit from metacognitive instruction procedures. However, this relationship could also work on the contrary. In fact, teachers’ own negative emotion toward mathematics may negatively affect their expectations about students and, as a consequence, impact their ability to create an inclusive learning environment, thus impairing students’ learning processes (Mizala et al., 2015). Moreover, teachers’ mathematical anxiety could also negatively influence their students’ mathematical achievement, especially true for girls. Regarding school climate’s multifaceted structure, relatively recent literature reviews (Thapa et al., 2013; Wang & Degol, 2016) suggested the existence of four areas, specifically including “safety,” “institutional environment,” “academic climate,” and “community.” “Safety” stresses out the emotional and physical security, with the former referred to the degree to which school personnel is emotionally supportive or the presence of a counseling system, and the latter to the degree of overall security as it is perceived by all the stakeholders. “Institutional environment” refers to physical aspects that characterize schools, such as availability of resources and building’s features (Wang & Degol, 2016). “Academic climate,” instead, focuses on learning and teaching practices that define academic atmosphere, involving also academic support. This latter is defined as the combination of methods that foster students’ positive learning experience and might have an impact on their educational outcomes (Thapa et al., 2013). In this regard, receiving positive support from teachers is associated with enhanced social and academic competencies (Koca, 2016). Finally, “community” pinpoints aspects that are more strongly related to the quality of interpersonal relationships, both between peers and between students and teachers, individually. In particular, alongside academic support, a positive studentteacher relationship has been shown to directly affect students’ motivation to learn (Davis, 2006) and covering a prominent role in their competencies development (Hamre & Pianta, 2001). To sum up, research upon school climate highlights that a positive school climate nurtures students’ social, emotional, and academic/cognitive development, leading to students’ well-being on the short-term period, and successful academic attainment on the long term (Thapa et al., 2013).

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Conclusion This chapter provides a comprehensive summary of the possible sources of individual differences in mathematical learning. Given that there is considerable variability among children with regard to cognitive and personal variables, and that mathematical abilities vary considerably in terms of their cognitive demands and how they are perceived, the degree to which cognitive, personal, and contextual factors explain individual differences varies across mathematical skills and individuals’ experience/mastery. Therefore, a large number of variables must be considered if we aim to understand individual differences in mathematical development. The literature primarily embraces research questions targeting specific relationships between a few measures of one or two domains only, for example, showing that socioeconomic status is linked to educational outcomes, or beliefs linked to educational outcomes. Few studies have attempted to explore how different factors might influence each other, for instance, analyzing the shielding effect of personality characteristics (e.g., resiliency or motivation) on the relationship between contextual factors and educational attainment, and the interactions between socioeconomic factors, educational outcomes, and cognitive abilities. As we have summarized, research into individual differences in children’s mathematics is diverse, sometimes sparse, and contradictory, but continually evolving. Some important trends and new directions are evident. First, there is a growing consensus in recognizing the need (and advantage) to apply a multidimensional approach. This marks a departure from the extreme-groups approach which reflects a taxonomic conceptualization not always respected across studies (i.e., misleading use of diagnostic labels). Another important insight concerns the necessity to include multiple measures which reflect the complexity of mathematical development. This call asks for increasing the number of large multifactor studies that could consider multiple domain-general functions together with environmental, motivational, and emotional factors, looking at their interaction rather than considering singularly. This new direction should take into account variation in school systems and demand a more rigorous and shared use of terms. Many overlapping terms are used when studying how children learn and apply mathematical concepts. For example, a number of different terms refer to beliefs about students’ math abilities, and there are some nuances in the definitions of these constructs (e.g., confidence, perceived competence, self-concept, and self-efficacy), similarly for what research defines as either contextual or environmental aspects. Moreover, many of the items used to measure these aspects are similar and tap into very similar underlying constructs. To fasten the building up of consistent results (e.g., replication studies), agreement on terminology and the measures seems desirable.

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Mind, Brain, and Math Anxiety

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Rachel Pizzie

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Is Math Anxiety? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How Is Math Anxiety Identified? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Who Develops Math Anxiety? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding Math Anxiety Through General Anxiety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Background: Cognitive Interference Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Background: Processing Efficiency Theory and Attentional Control Theory . . . Math Anxiety, Processing Efficiency Theory, and Attentional Control Theory . . . . . . . . . . . . Neuroimaging, Math Anxiety, Processing Efficiency Theory, and Attentional Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Math Anxiety and Emotional Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interventions and Emotion Regulation in Math Anxiety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Math Anxiety and Mathematical Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Math Anxiety and Numerical Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interventions and Math Competency in Math Anxiety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Math anxiety refers to the feelings of anxiety, tension, and negativity associated with mathematical calculations, or anticipating mathematics. Math anxiety is not only associated with negative experiences with math, but is also inversely related to math performance and achievement. Math anxiety is associated with avoidance of mathematics, such that highly math-anxious individuals not only avoid completing math problems, but also avoid math classes, majors, and quantitative careers. Avoidance is an impediment to success in an increasingly quantitative and technological society. Math anxiety provides an interesting perspective from R. Pizzie (*) Gallaudet University, Washington, DC, USA e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_29

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which scientists can explore how cognition and affect come together within one’s educational experience. From the perspective of affective science, math anxiety finds its theoretical grounding in the relation between anxiety and working memory processing. Anxiety theories hypothesize that anxiety about mathematics negatively influences inhibition, disrupting working memory processes and resulting in deficits in processing efficiency of mathematics. In addition to negative emotional experiences and physiological sensations, math anxiety is associated with deficits in mathematical cognition and fundamental numerical skills. Math anxiety is also related to deficits in perception of numerical magnitude, counting, and simple arithmetic processes. Math anxiety represents the study of individual differences in emotional experiences, cognitive processes, and biological mechanisms. Math anxiety brings together the study of minds, brain processes, and educational outcomes. Addressing the negative association between anxiety and education represents an important challenge in improving mathematics education. Keywords

Math anxiety · Anxiety · Affect · Mathematics · Numerical cognition

Introduction Imagine this familiar educational scenario: You are a high school student about to take a test in your high school math class. You take your seat at your desk, and as other students enter the classroom, you begin to feel some familiar and unpleasant sensations: Your heart begins to race, your mouth goes dry, and your palms begin to sweat. As your instructor makes announcements and distributes the test, negative feelings continue to mount, and your mind begins to race with worry. Finally, it is time to begin the test, and you begin the problems. As you continue to feel anxious and uncomfortable, perhaps your mind goes blank, or perhaps your thoughts continue to race, constantly returning to the consequences of failing the test, and intruding on your ability to think through each problem. You glance around the room and see other students struggling, or looking calm and relaxed. As you continue to feel negative and unsure of your own performance, watching other students only serves to make you feel increasingly nervous. Although you studied for the test, now you feel like you cannot access the material, or did not study in the right way. Finally, you finish the test, but perhaps not having performed as well as you had hoped, and ultimately earning a lower mark on the test. Although students may have a wide variety of experiences in mathematics learning, many students experience extreme amounts of negative emotion associated with mathematics, an experience known as mathematics anxiety, or math anxiety (Ashcraft 2002; Dowker et al. 2016; Hembree 1990; Ramirez et al. 2018; SuárezPellicioni et al. 2015). Math anxiety refers to the feelings of anxiety and negativity felt in association with doing mathematics, as well as anticipation of doing

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mathematics. Negative emotions, negative attitudes toward math, and self-reports of negative experiences with math are all important components of math anxiety and are related to additional negative math outcomes. More math-anxious individuals tend to have poorer performance on math tasks in the lab, standardized math tests, have lower grades in math classes, and tend to avoid further math classes and mathrelated careers (Ashcraft 2002; Beilock and Maloney 2015; Dowker et al. 2016). Math anxiety is broadly associated with not only avoidance, observed as disengagement from mathematical calculations, but also avoidance of math classes and careers (Ashcraft 2002; Ashcraft and Faust 1994). In today’s increasingly technological society, shying away from learning mathematical or quantitative skills may have detrimental effects on educational outcomes and career prospects (Beilock and Maloney 2015). Since the 1970s, research on math anxiety has continued to develop our understanding of the experiences and deficits related to math anxiety. In older conceptualizations of affect and cognition, it was often the case that “cold” cognitions were pitted against “hot” emotional processes. However, both perspectives represent important features of math anxiety, one representing the emotional reactions, physiological sensations, and affective components, and the other representing the “cold” cognitions, and numerical calculations, all of which are unified in the experience of math anxiety. Although often considered separate processes, the cognitive and affective components of math anxiety are inextricably linked. Math anxiety provides an excellent test case to study cognitive and affective processes together in a context that has real-world educational implications. The following chapter begins with an introduction to math anxiety, discussing its origins. Then the connections between general anxiety and math anxiety are discussed, first exploring how these foundational theories such as Cognitive Interference Theory, Processing Efficiency Theory, and Attentional Control Theory provide a theoretical basis for math anxiety. Then, the chapter explores more literature regarding emotional reactions associated with math anxiety. Finally, the chapter explores the connections between math anxiety and mathematical cognition. Ultimately, in studying the connections between the mind, the brain, and education, this research considers the links between emotion and cognition, and how these experiences ultimately shape learning and educational outcomes.

What Is Math Anxiety? How Is Math Anxiety Identified? Mathematics anxiety occurs when individuals experience feelings of fear, apprehension, tension, and increased arousal when engaging with math, whether that may occur in an academic context or in everyday life (Ashcraft 2002; Ramirez et al. 2018). Math anxiety was originally conceptualized as an extension or subtype of test anxiety (Ashcraft 2002; Richardson and Suinn 1972), or situational anxiety associated with testing environments (Alpert and Haber 1960). For both situational

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anxieties, they were conceptually differentiated from general or generalized anxiety. Even though individuals may experience intense anxious emotion in specific scenarios, they may not consider themselves to regularly experience a great deal of anxious emotion. Math anxiety is thought to be a “trait” level characteristic, similar to a personality characteristic and relatively stable over time (Hembree 1990; Ramirez et al. 2018). Math anxiety has not been established as a clinical “disorder” in the same way that generalized anxiety has been classified. There are no officially agreed upon norms by which math-anxious individuals are categorized as being “high” or “low” in math anxiety. Generally, math-anxious individuals are characterized as “highly” math anxious due to self-report alone, and compared to the context of the population of the specific research sample. Math anxiety is defined and identified using self-report questionnaires. Individuals who are high and low in math anxiety are identified as such because they endorse self-reported survey items that illustrate attitudes, experiences, and beliefs. In other words, math-anxious individuals are identified to be high in math anxiety because they tell us about these experiences. Math anxiety is related to other experiences of anxiety and is associated with “general” or generalized anxiety (correlated at r ¼ 0.3), and test anxiety (generally correlated at r ¼ 0.5–0.8; Ashcraft 2002; Hembree 1990). Although math anxiety is related to additional experiences of anxiety, it is still thought to be its own separate construct (Pizzie and Kraemer 2019). Scientists have developed a variety of questionnaires and surveys designed to identify individuals along the spectrum of math anxiety, including the Math Anxiety Rating Scale (Richardson and Suinn 1972), The Abbreviated Math Anxiety Scale (Hopko et al. 2003), or even a single question that can identify one’s level of math anxiety (NúñezPeña et al. 2014).

Who Develops Math Anxiety? Math anxiety does not have any one specific trigger or etiology. Instead, math anxiety likely results from a number of different experiences, attitudes, and contributing factors. As a result, it can be difficult for researchers to study because it cannot be systematically manipulated (Ramirez et al. 2018). Math anxiety seems to develop in association with increased pressure to perform, such as high expectations, time pressure, or social pressure (Beilock et al. 2004). Anecdotally, math-anxious individuals have reported triggering experiences such as being called to the board in front of the class to solve a problem they did not understand, having to take difficult, high-stakes math tests, or being derided by the teacher for requesting additional assistance. Many individuals have reported that negative and hostile reactions to students’ difficulties with mathematics are associated with increased negative attitudes (Jackson and Leffingwell 1999). These experiences are characterized by social influence, as well as increased pressure to perform.

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Math anxiety is related to decreased math competency (Ashcraft and Kirk 2001; Ramirez et al. 2018), although the directional influence between math anxiety and numerical/mathematical skill is still up for debate. This directional influence presents a “chicken and egg” problem – do math-anxious individuals develop anxiety because they sense deficiencies in mathematical understanding and skill, or does anxiety disrupt mathematical learning and knowledge through avoidance? From the first perspective, deficits in math understanding and knowledge are the “root cause,” leading to later anxiety. In this view, instead of causing reduced mathematical skill and achievement, math anxiety may stem from decreased math competency. For example, one cross-sectional study reported disruptions in the perception of numerical magnitude, and less precise representations of numerical magnitude in individuals with math anxiety (Maloney et al. 2011). There may be underlying differences in fundamental math skills and perceptions that are associated with math anxiety. In addition, some longitudinal studies have suggested that deficits in math achievement are associated with later increases in math anxiety. These effects are stronger than the relationship between earlier math anxiety and later deficits in mathematics achievement (Ramirez et al. 2018). Disruptions in math achievement are associated with later increases in anxiety about mathematics. Perhaps some of the fundamental aspects of numerical cognition may be distorted in math anxiety, leading to negative associations with mathematics. However, from another perspective, increased anxiety about mathematics is the root cause, leading to underperformance in mathematics. In this view, it is too simple to assume that math-anxious individuals are simply “bad at math,” and therefore increasingly anxious about mathematical experiences. As math anxiety is associated with avoidance, reduced engagement with mathematical material over time can result in reduced mathematical achievement. In other words, although math-anxious individuals may not start off with deficits in performance, by avoiding math studying, and avoiding further math classes, math-anxious individuals may end up with reduced math skill because they have not been able to adequately engage with or effectively learn the material due to avoidance. Furthermore, some research suggests that when additional anxiety and pressure to perform are removed, math-anxious individuals perform mathematics at the same level as those low in math anxiety (Ashcraft and Faust 1994). This suggests that math-anxious individuals may not lack in skill or understanding, but that the anxious emotion they experience interferes with the cognitive processes needed to perform math calculations. When the anxiety is reduced, either by decreasing performance pressure or providing an additional method of reducing anxiety (Hembree 1990), math performance is no longer decreased. However, there are many aspects that may contribute to the development of math anxiety, and both math anxiety and math achievement likely have reciprocal effects on one another. The origin of this “chicken-and-egg” problem is difficult to discern but potentially has important implications for understanding how to intervene in the development of math anxiety and deficits in math performance. Further longitudinal work needs to elucidate the directionality of the relationship between math anxiety and mathematical skill and achievement.

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There are additional demographic factors that are also associated with math anxiety, such as age or gender. Increased math anxiety is recorded even in young children as young as 7 years old (Young et al. 2012), seems to reach a peak and plateau at around age 15 (Hembree 1990), and continues to affect individuals into young adulthood and beyond. Though less is known about the influence of math anxiety on individuals after young adulthood, research suggests that math anxiety continues to persist into adulthood after formal schooling has been completed (Douglas and LeFevre 2017; Pizzie and Kraemer 2019). With respect to gender, females tend to report higher math anxiety compared to males (Dowker et al. 2016; Hembree 1990; Suárez-Pellicioni et al. 2015). Increased math anxiety for women and girls is tied to societal stereotypes that women and girls perform worse in mathematics (Gunderson et al. 2011). As women and girls internalize stereotyped societal viewpoints, they may be more likely to experience increased anxiety associated with mathematics. In addition, the anxious attitudes of those who are close to us may in turn affect our own anxiety (Beilock et al. 2010; Maloney et al. 2015). For example, teachers who are more math anxious tend to have more mathanxious students (especially if they are female, Beilock et al. 2010). Parents who are more math anxious tend to have children who also have increased anxiety about mathematics (Maloney et al. 2015). Parental involvement may also play a key role in development of math anxiety, as parents who are able to scaffold more supportive experiences associated with mathematics show decreased math anxiety in children (Vukovic et al. 2013). There is a myriad of factors that may contribute to one’s level of anxiety regarding mathematics, but it is important to understand that each individual may experience anxiety due to a different etiology, and may experience different aspects of negativity associated with mathematics. To understand the nature of the deficits created by math anxiety, researchers also must consider the context and the population being studied. In many research studies focused on exploring math anxiety in laboratory settings, research involves recruiting a sample of individuals from a population of convenience. These samples frequently include young adults who are academically focused and are already academic high achievers, such that they achieved test scores and grades that were sufficiently high enough to warrant acceptance in the competitive college admission process. However, this population of young adults may not reflect the full spectrum of deficits created by math anxiety. Students enrolled in a remedial math class at a community college (e.g., Jamieson et al. 2016) may be experiencing more severe anxiety-related deficits in their math performance compared to students who may have been able to “overcome” math anxiety in order to gain admission at an institution that emphasizes high academic achievement. Nevertheless, math anxiety seems to reach across institutions, and math anxiety is consistently associated with deficits in math performance in lab tasks, as well as broader academic achievement (Beilock and Maloney 2015). In considering the sample of students and participants in which math anxiety is studied, it is also important to consider what participants are being included and compared. For example, many research studies dichotomized math anxiety, comparing those who score higher on a measure of math anxiety to those who score lower

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on the measure of math anxiety. Researchers have occasionally split groups at a midpoint (e.g., Young et al. 2012) or recruited individuals at extreme ends of the spectrum of math anxiety (e.g., Lyons and Beilock 2012a, b; Pizzie and Kraemer 2017). Whereas these are techniques that can identify important differences associated with math anxiety, dichotomies do not necessarily accurately represent the full spectrum of individual differences in math anxiety and variability therein. For example, one population that may be relatively understudied is those who experience moderate math anxiety, or those who score in the midrange of math anxiety scales. Individuals may still experience a significant amount of anxiety or antipathy toward mathematics, but they do not necessarily experience the same degree of deficits as those who score more highly and experience more drastic differences in performance. Given how math anxiety is associated with math experiences and academic achievement, considering individuals who score within the middle range on math, anxiety measures may be interesting or important. For example, it is possible that individuals show differences in their ability to regulate their emotions and their resilience to stress (Jamieson et al. 2013), or perhaps they received excellent math instruction that helped to bolster their understanding of math concepts, and so are still able to perform at a high level despite experiencing increased anxiety (Iuculano et al. 2015). In considering who develops math anxiety and how math anxiety is developed, it is essential to think about how this phenomenon is measured, and how this population is being studied in order to understand the limitations of each set of research findings. Math anxiety provides an interesting vantage from which to study the interactions between cognition and emotion, such that processes cannot inherently be separated from one another. Although the primary perspectives from which math anxiety is examined may change, ultimately it is the confluence of all factors that come together to create a unified academic experience within each individual.

Understanding Math Anxiety Through General Anxiety This section explores previous literature on generalized or general anxiety (Eysenck 2010). General, generalized, or trait anxiety are defined as the persistent, worrisome, and anxious feelings that are not necessarily focused on a specific cause, but instead refer to broader feelings of negativity and unrest (Eysenck 2010; Eysenck et al. 2007; Mogg and Bradley 1998). Math anxiety draws on some of the fundamental theories regarding general anxiety in order to form the basis of the theoretical grounding of math anxiety. General anxiety can be conceptualized in two ways: “state” general anxiety is transitory or related to a particular state. Or, general anxiety can be thought of as a personality characteristic: a pattern of emotional experiences that is more stable and representative of numerous experiences over time (“trait anxiety”). Surprisingly, many of the findings related to general anxiety do not differentiate between anxiety as a state- or trait-level factor, although ratings of state and trait ratings of anxiety are moderately correlated (approximately r ¼ 0.5; Eysenck 2010).

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In general, the relationship between anxiety and performance can be conceptualized through disruptions in executive function created by anxiety. Working memory is a component of executive function and refers to a short-term memory system that combines the cognitive skills, manipulations, and thoughts needed for the present task at hand (Friedman and Miyake 2017). One component of executive function that is also key for successful task completion is to inhibit negative or distracting thoughts, or inhibition of task-irrelevant information. Anxiety influences attentional control, disrupting working memory and altering inhibitory control (Eysenck 2010). In other words, anxiety functions almost like a secondary task, taking up cognitive resources through worry, perseverative thoughts, or intrusive and task-irrelevant processes. This disruption account of general anxiety is thought to be mirrored in math anxiety. There are several explanations for “disruption,” “dual-task paradigm,” “interference,” or “inefficiency,” and largely these conceptualizations are consistent with one another, although provide more or less specificity in the explanations of exactly how anxiety influences the execution of a cognitive task. Broadly, this section presents explanations of the relationship between anxiety and cognitive performance, including “Cognitive Interference Theory” which generally refers to a disruption of anxiety and working memory. Cognitive Interference Theory hypothesizes that anxiety interferes with attentional control through negative intrusive thoughts, disrupting cognitive performance. Then, the following research explores “Processing Efficiency Theory and Attentional Control Theory,” which are two related theories that explore how effort and expediency of cognitive processes in anxiety influence cognitive outcomes (Eysenck et al. 2007; Mogg and Bradley 1998). Processing Efficiency Theory and Attentional Control Theory together hypothesize that anxiety has a negative influence on two components of the “central executive” within working memory: inhibition and shifting, which create deficits in the efficiency of task-related cognitive processing (Eysenck et al. 2007). These two prominent explanations explore the connections between general anxiety and cognitive performance, and the following section discusses how each of the theories may be used as a theoretical model to understand math anxiety.

Theoretical Background: Cognitive Interference Theory In Cognitive Interference Theory, worry over evaluation forms a signature component of the link between anxiety and decrements in performance, as in test anxiety (Eysenck 2010; Sarason 1988). Cognitive Interference Theory posits that self-related ruminations and internal stimuli (e.g., thoughts about failure) impair attentional control in anxiety by taking away resources from task-relevant stimuli. In other words, attention becomes impaired when an individual can no longer control their attention to focus on the task at hand, and instead the individual ends up being distracted by negative self-talk and ruminations. In addition, the effects of interference should be exacerbated when evaluative instructions are used, increasing pressure and self-related thoughts. Cognitive Interference Theory also proposes that effects of interference should be greater for more difficult tasks than easier tasks,

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as the more difficult tasks increase demands on attention. This theory could also be considered essential to understanding the “disruption account” of math anxiety (Ramirez et al. 2018). The “disruption account” posits that deficits in mathematics are associated with math anxiety through a transitory reduction in working memory resources, reducing and disrupting the cognitive resources needed to complete the task at hand. Much of the foundational research on math anxiety is consistent with the perspective of Cognitive Interference Theory. Similar to test or general anxiety, math anxiety researchers have hypothesized that the negative association between math anxiety and math performance can be attributed to interference in working memory processes associated with negative self-talk (Beilock et al. 2004). One of the mechanisms theorized to unify choking under pressure, math anxiety, and stereotype threat is that all of these phenomena are unified by working memory deficits created by negative self-related thoughts (Beilock et al. 2004). In anxiety, individuals may experience negative, distracting, and intrusive thoughts that may distract and interrupt the attention and working memory resources needed to focus on the task at hand (Eysenck 2010; Eysenck et al. 2007). Worry is thought to be especially disruptive to task performance by interrupting and disrupting working memory resources. One study hypothesized that negative self-talk would be associated with working memory deficits by capitalizing on verbal working memory resources (DeCaro et al. 2010). In this study, anxious individuals showed deficits on math problems under increased pressure, especially problems that relied heavily on verbal processes. In order to alleviate increased pressure on working memory resources, participants were instructed to talk through their strategy aloud, which would increase focus on verbalizing the task-related instructions and reduce the influence of distracting negative self-talk. The results illustrate that talking through the problem aloud helped to keep worries at bay, and more anxious individuals no longer showed a negative association between anxiety and performance. This study (DeCaro et al. 2010) also demonstrates support for the idea that increased pressure, and evaluative instructions should create greater interference within cognitive resources (Eysenck 2010; Sarason 1988). The study created increased pressure to perform, in which the experimenters induced monetary, social, and time pressure, and stated that participants’ performance would be reviewed and judged by a panel of professors. High-pressure problems were effective at inducing feelings of pressure and worry, and accuracy decreased with increased pressure, especially for difficult problems that relied on verbal working memory. Results also indicate that the greater deficits in performance are observed for more difficult problems (DeCaro et al. 2010), which has also been illustrated across a wide variety of other studies on math anxiety (Ashcraft 2002). Cognitive Interference Theory suggests that as problem difficulty increases, additional resources are needed within attention and executive function, and because anxiety already detracts from cognitive resources, there are fewer resources that remain to effectively solve the problem (Eysenck 2010; Sarason 1988). Ashcraft and colleagues have suggested that simple problems are less likely to be disrupted by anxious thought-process because many simple math processes are well rehearsed,

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and rely on recall of “math facts” that require fewer cognitive resources (Ashcraft 2002). However, when problems are increasingly difficult, such as those that involve carry operations or multiple steps in order to successfully solve the problem, more anxious individuals demonstrate poorer performance. The researchers suggest that the explanation for deficits in performance associated with increased anxiety is because anxiety is competing for the same cognitive resources which would otherwise be utilized to successfully solve the task at hand. However, there are some limitations of the disruption account as explained by Cognitive Interference Theory (Sarason 1988). This theory does not specifically explain how anxiety interacts with the processes of working memory, or specifically how anxiety or worry interacts with the attentional system, and provides no theoretical grounding for this mechanism (Eysenck 2010; Eysenck et al. 2007). Cognitive Interference Theory also assumes a particular directionality by which anxiety and performance deficits occur, with worry occurring as an antecedent to disruption of task performance. In reality, it is likely the case that the relationship is bidirectional, with worry and poor task performance interacting with one another in a dynamic manner to amplify the negative relation between anxiety and task outcomes. In addition, this theory also exaggerates the role of self-related or taskirrelevant thoughts as the primary mechanism by which performance and achievement are impacted. In some tasks, elevated worry is reported, but no task-related deficits is observed. Although worry may interfere with task processing, there may be compensatory behaviors or increased effort which mitigates the negative impact of worry on performance. Overall, Cognitive Interference Theory may be consistent with many of the results observed for math anxiety. Theories such as Processing Efficiency Theory and Attentional Control Theory provide a more complete picture and theoretical grounding for the relationship between math anxiety and math performance by further explaining the cognitive processes and mechanisms of the “disruption account” (Ramirez et al. 2018). Whereas Cognitive Interference Theory hypothesizes that anxiety influences working memory more generally, Processing Efficiency Theory and Attentional Control Theory together make more specific hypotheses about the components of working memory that are affected by anxiety and provide a more mechanistic explanation for why anxiety impacts task performance.

Theoretical Background: Processing Efficiency Theory and Attentional Control Theory Two of the major theoretical perspectives that link anxiety and deficits in cognitive performance are Processing Efficiency Theory and Attentional Control Theory (Eysenck 2010; Eysenck et al. 2007). By comparing math anxiety to more generalized forms of anxiety, understanding key theories in anxiety research aids in our understanding of how math anxiety may influence cognitive resources and task performance. This section surveys key theories of working memory in order to illustrate the theoretical basis by which anxiety disrupts cognitive processes and is

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Inhibitory Control

Negative Attentional Control Attentional Control Theory

“Central Executive”

Working Memory

Phonological Loop

Shifting (TaskSwitching)

Positive Attentional Control

Updating

Visuospatial Sketchpad

Fig. 1 Working memory, processing efficiency theory, and attentional control theory Note: The left part of the figure depicts a common model of working memory structure (Baddeley 1996; Friedman and Miyake 2017; Miyake and Friedman 2012). In this model (solid lines), working memory is composed of three domains, (1) the “central executive” (2) the phonological loop, and (3) the visuospatial sketchpad. In performing cognitive tasks, Processing Efficiency Theory (gray, dotted lines; Eysenck 2010; Eysenck et al. 2007) hypothesizes that anxiety has the greatest effects on the “central executive,” disrupting the efficiency of these cognitive processes. The “central executive” is also comprised of three components: (a) inhibitory control or inhibition, (b) shifting or task-switching, and (c) updating. Attentional Control Theory (gray, dashed lines; Eysenck et al. 2007) proposes that anxiety influences cognitive performance through deficits in two processes: negative attentional control, which impacts inhibition, and positive attentional control, which impacts shifting. In negative attentional control, anxiety detracts from the ability to inhibit task-irrelevant thoughts. In positive attentional control, anxiety detracts from the cognitive mechanism that allows us to switch between cognitive tasks, maintaining focus on the task at hand, or shifting. Together, Processing Efficiency Theory and Attentional Control Theory propose a specific explanation of how anxiety impacts working memory capacity and cognitive task performance

negatively associated with task performance. Both Processing Efficiency Theory and Attentional Control Theory posit that anxiety disrupts specific components of working memory and provide a specific explanation of how anxiety disrupts task performance. In Fig. 1, one of the popular theories of the hypothesized components of working memory is presented and illustrates how Processing Efficiency Theory and Attentional Control Theory affect these components. In explaining the relationship between anxiety and cognitive performance, first it is essential to explain a model of working memory structure and function (Baddeley 1996). This popular theory of working memory identifies three main components (see Fig. 1): (1) the central executive, a domain-free cognitive system involved with planning, strategy selection, and attentional control, (2) the phonological loop, which refers to a system of verbal rehearsal and self-talk, and (3) the visuospatial sketchpad, which involves storage and simulation of transient visuospatial

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information function (Baddeley 1996). The relationship between working memory and anxiety focuses on the “central executive,” as this domain of working memory governs many of the essential functions necessary for cognitive task performance, and this is the component of working memory impacted by Processing Efficiency Theory. Within this model of working memory and the “central executive,” there are three main cognitive functions or processes (Friedman and Miyake 2017; Miyake and Friedman 2012). In this view, the “central executive” is associated with three main cognitive processes or functions: (1) inhibitory control, or inhibition, (2) shifting, or task-switching, and (3) updating. Inhibition, or the ability to resist disruption or inhibit task-irrelevant processes, can also be thought of as one’s ability to intentionally avoid making dominant, automatic, or prepotent responses (Eysenck 2010). In math anxiety, inhibition might involve trying to avoid disruption from distracting negative thoughts about math while still maintaining focus on solving a difficult math problem. Shifting, or shifting set, involves the ability to switch or flexibly change between multiple tasks or cognitive processes while still maintaining focus on the task at hand. This process might involve rapidly changing between two different tasks or mental sets, keeping demands, attributes, or rules associated with each task separately, and maintaining focus on the preferred task. In math anxiety, it might look like switching back and forth between a math task and another difficult cognitive task, such as analogies (Pizzie et al. 2020b). Or one can imagine a hypothetical “real-world” example, rapidly switching back and forth between one’s math homework and checking a text message conversation with a friend. The updating function, or “information updating” is associated with the maintenance of working memory representations. The updating function is associated with replacing information that may be outdated, or inhibiting or prioritizing information that is relevant to the demands of the task at hand (Carriedo et al. 2016). In math anxiety, updating is essential for something like completing an order-of-operations task, simultaneously keeping in mind the rules for sequencing the actions for each operand, as well as making the calculations and keeping them in mind to correctly solve the problem. Processing Efficiency Theory identified that the “central executive” within the working memory system was specifically impacted by anxiety. By disrupting the “central executive,” anxiety impacts processing efficiency of the task (Baddeley 1996; Eysenck 2010; Eysenck et al. 2007). Processing Efficiency Theory emphasizes two components of completing any cognitive task: performance effectiveness and performance efficiency. Performance effectiveness is identified as the quality of the task performance, usually indexed by standard metrics of how well a task is performed, such as accuracy or speed of responses. In the performance of mathematics, this refers to the number of problems in which the individual achieved a correct answer, answered in a certain period of time, or perhaps solved using a particular strategy. Performance efficiency refers to the relationship between the performance effectiveness, and the cognitive resources or effort that was required in order to complete that task. Efficiency of the task works as a balance; if increased performance is desired, one might need to invest increased cognitive resources and

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effort. However, efficiency is decreased if one must increase the amount of resources and effort in order to attain a level of performance (Eysenck et al. 2007). Anxious individuals often expend great effort to complete a cognitive task, often without great or improved output (Eysenck 2010; Eysenck et al. 2007). For example, if a highly math-anxious individual must exert an increased amount of cognitive resources and great effort in order to solve the same math problems as an individual with decreased math anxiety, the processing efficiency for the highly anxious person is said to be decreased. This is one of the central tenets of Processing Efficiency Theory: Anxiety is likely to have a greater negative impact on processing efficiency than on processing effectiveness. Previous research on anxiety within the framework of the Processing Efficiency Theory suggests that within components of working memory, anxiety has the strongest negative effects on the “central executive” (see Fig. 1; Eysenck 2010; Eysenck et al. 2007). When anxious individuals were asked to perform in a dual-task paradigm, when both tasks relied on the central executive, anxious individuals showed the most pronounced deficits. When the secondary task involved alternative components of working memory, such as relying on the phonological loop or visuospatial sketchpad, the deficits on the primary task were not as pronounced. Results suggest that anxiety also utilized resources of the central executive, but anxiety was less reliant upon resources within the phonological loop and visuospatial sketchpad. Furthermore, understanding key processes within the “central executive” is essential to understanding the relationship between anxiety and Attentional Control Theory (Eysenck et al. 2007). In Attentional Control Theory, anxiety impairs working memory performance through the disruption of two kinds of attentional control: negative attentional control and positive attentional control. In negative attentional control, inhibitory control is used to prevent disruptions from taskirrelevant internal or external stimuli from interrupting task-relevant processes (Miyake and Friedman 2012). Negative attentional control prevents interruption from the task at hand by inhibiting other cognitive processes or responses. In math anxiety, negative control would involve inhibiting negative self-talk during a math task. On the other hand, positive attentional control is thought to be related to the shifting function, or the ability to flexibly attend to changing demands of tasks as in switching or juggling multiple tasks. In math anxiety, this might look like having difficulty switching to focus on a math task after focusing on a different task (Pizzie et al. 2020b). General anxiety impairs the efficiency with which individuals are able to engage in cognitive processes, impairing the efficiency of negative attentional control (inhibition), and positive attentional control (shifting). Cognitive Interference Theory and Attentional Control Theory and Processing Efficiency Theory are not mutually exclusive theories of the cognitive processes involved in anxiety and can be considered complementary approaches that act on various components of working memory (see Fig. 1). Processing Efficiency Theory and Attentional Control Theory are built on some of the tenets and theoretical explanations provided by Cognitive Interference Theory. Whereas Cognitive Interference Theory illustrates that working memory is disrupted by anxiety, Processing

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Efficiency Theory and Attentional Control Theory elaborate and specify the cognitive processes that are disrupted by anxiety within working memory. In math anxiety, Processing Efficiency Theory and Attentional Control Theory suggest that anxiety impacts working memory processes related to the central executive. Within this domain, math anxiety negatively influences both positive and negative attentional control mechanisms, impacting the cognitive efficiency with which these processes occur in math, and ultimately resulting in deficits in math performance (Hopko et al. 1998).

Math Anxiety, Processing Efficiency Theory, and Attentional Control Theory The following section brings together the aforementioned foundational theories and their utility in understanding math anxiety, processing efficiency, attentional control, and cognitive performance. In math anxiety, anxious thoughts, feelings, and physical sensations are thought to interfere with the efficiency of the mathematical process (Ashcraft 2002; Ramirez et al. 2018; Suárez-Pellicioni et al. 2015). Anxious responses disrupt mathematical cognition by disrupting cognitive processes related to the “central executive,” creating a negative association between anxiety about math and mathematical performance and achievement. From the perspective of Processing Efficiency Theory, it is important to consider that math anxiety does not always need to fully disrupt the performance effectiveness of mathematics, but disrupts the efficiency of mathematical cognitions. Even if math-anxious individuals are able to successfully complete mathematical computations, behind the scenes, this might require additional effort or cognitive control in order to complete the same task as those who are low in anxiety. Much of the research on math anxiety is related to Attentional Control Theory, specifically negative attentional control and decreased mathematical processing efficiency (Ashcraft 2002; Ashcraft and Kirk 2001). Negative attentional control is at the root of the problem: Worrying and intrusive thoughts consume limited working memory resources of the central executive, with anxiety limiting the ability to control or inhibit the disruption created by negative thoughts. In a reading task that included either math-related or neutral distractors, individuals with greater math anxiety had greater difficulty inhibiting math-related distractors, had longer reading times, and made increased errors (Eysenck 2010; Hopko et al. 1998). Results suggested that more math-anxious individuals have difficulty inhibiting attention from math-related distractors, indicating insufficient negative attentional control. With regard to Processing Efficiency Theory, decreased efficiency also supports the idea that math-anxious individuals do not necessarily have underlying misunderstanding of mathematical material, but that anxiety impedes the ability to access and demonstrate computations and abilities by impacting the “central executive.” Indeed, the majority of the tasks chosen for laboratory tasks are not limited by mathematical knowledge. Tasks for lab use are chosen because all participants are likely to know how to complete the majority of the arithmetic tasks that are used for

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test cases and stimuli. However, in keeping in mind the perspective of Processing Efficiency Theory, the efficiency with which highly math-anxious individuals are able to complete mathematical tasks is reduced, either by increased effort or investment of cognitive resources, or by reduced effectiveness and decreases in mathematical performance. Because anxiety influences the central executive, cognitive tasks that also utilize this component of working memory will show deficits in efficiency in those who are highest in math anxiety (Suárez-Pellicioni et al. 2015). In a working memory span task (assessing how many items can be held and operated on at the same time), high and low math-anxious individuals were asked to complete a verbal or numerical span task (Ashcraft and Kirk 2001). Participants were asked to either hold in mind the last words in a series of sentences, or hold in mind the last digits of a series of arithmetic verification tasks, ultimately having to report words or numerals in serial order. The results indicated that increased math anxiety was associated with poorer performance in the arithmetic working memory span task, but not necessarily the verbal working memory task. Increased anxiety associated with manipulating mathematical information was utilizing central executive working memory resources, and therefore resources could not be devoted to the mathematical task at hand, leading to decreases in efficiency. Further, mathematical processes that are more difficult and rely more on working memory resources are also competing for resources utilized by the maintenance of math anxiety. For example, more math-anxious individuals show increased math errors and working memory performance deficits when problems involve “carry” operations (Ashcraft and Kirk 2001). The additional cognitive steps needed for “carry” operations are taxing for working memory resources, as it involves keeping an extra amount in mind while also completing additional calculations. When anxiety additionally detracts from cognitive resources, researchers observed that individuals who reported more math anxiety showed the most exaggerated deficits in performance. Interestingly, in this experiment (Ashcraft and Kirk 2001), two dualtask paradigms were used: one in which the cognitive load was lower in which participants had to hold two letters in mind, and one in which they had to hold 6 letters in mind, all while completing simple arithmetic or arithmetic with carry operations. If it were merely the dual-task paradigm that was problematic, or perhaps that the task provoked anxiety, it would be expected that both the low- and high-load conditions would have resulted in increased errors. Instead, it seems to be the interaction between increased working memory load that results in decreased performance by taxing executive control. Importantly, when some of the additional pressures and task constraints that may be additionally anxiety provoking are removed, such as eliminating the time constraints and allowing participants to solve the problems on their own time, the anxiety-related deficits in performance are no longer observed (Ashcraft and Kirk 2001). This result speaks to the idea that anxiety primarily impacts performance efficiency, resulting in increased effort or decreased cognitive resources to perform the task when anxiety is elevated. In other words, the anxiety interferes with the performance of mathematics. The performance deficits observed for simple arithmetic tasks like this (even more complicated

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arithmetic using carry operations) are not necessarily due to a fundamental misunderstanding of the concepts needed to solve the problem. Although much of this work had addressed how working memory changes in response to the state of being anxious about mathematics, there are also individual differences in dispositional working memory. In other words, researchers can not only examine how working memory resources are utilized by specific tasks, but can also measure differences across dispositional working memory abilities in each individual. In a series of studies (Ramirez et al. 2016; Vukovic et al. 2013), research investigated the relationship between math anxiety and individual differences in working memory capacity. Given the processing efficiency hypothesis, one might expect that the greater one’s capacity for increased working memory resources (i.e., greater dispositional working memory span), the less anxiety might be able to detract from these resources. In fact, the results suggest that increased working memory capacity is not an aid to math performance for those who are high in math anxiety. On a standardized test of math achievement, those who were highest in working memory capacity were most negatively impacted by math anxiety (Ramirez et al. 2018). The researchers explained the seemingly paradoxical effect by considering the differences in cognitive strategies used by those low or high in dispositional working memory. Individuals higher in working memory capacity tended to use mathematical strategies that were higher in working memory load. The individuals who were lower in working memory capacity utilized strategies that were less taxing on cognitive load, and therefore the strategies were more robust to the detrimental effects of anxiety on test performance.

Neuroimaging, Math Anxiety, Processing Efficiency Theory, and Attentional Control Theory The following section explores how results from behavioral studies on Attentional Control Theory and Processing Efficiency Theory are supported by neuroimaging findings. Neuroimaging results provide additional support and explanation for the interaction between working memory and math anxiety. For example, in a study by Pletzer et al. (2015), the authors compared neural activity during numerical tasks across participants who were high and low in math anxiety. They suggest that processing efficiency in working memory related to the numerical tasks is related to decreased activity in the default mode network (Pletzer et al. 2015). The default mode network has been associated with introspective or task-irrelevant thoughts and shows deactivation when individuals are engaged in a specific goal-directed external task. This deactivation of the default mode network during a task is also associated with increased performance of the directed task (Pletzer et al. 2015). The results demonstrate that those who were lower in math anxiety showed greater deactivation of the default mode network in comparison to those higher in math anxiety (Pletzer et al. 2015). This difference associated with math anxiety was most pronounced in tasks where inhibition was required, providing further support using neuroimaging data that negative attentional control may be disrupted in math anxiety, drawing from Attentional Control Theory.

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In addition to differences in negative attentional control associated with math anxiety, neuroimaging results have also provided support for the idea that positive attentional control, or shifting as a working memory function, may be disrupted in more math-anxious individuals. In a recent study, Pizzie et al. (2020b) examined how math anxiety might interact with positive attentional control, or shifting (Eysenck 2010), in a task-switching paradigm. High and low math-anxious individuals were asked to complete a task wherein they completed a series of problems in the fMRI scanner, unpredictably switching back and forth between analogies and order-of-operations arithmetic problems (Pizzie et al., 2020b). Both tasks would be cognitively challenging, but only the math task would additionally tax executive function control by increasing anxiety for those who were high in anxiety about math. In a normative “switch cost,” when individuals switch into a new paradigm, one would expect an increase in working memory load, as individuals need to recruit working memory resources in order to consider new task demands. In switch trials, it would be expected that individuals would respond more slowly, or would show increased neural activity associated with increased effort of adopting the new task set. Indeed, this is what was observed in low math-anxious individuals. Low mathanxious individuals showed a normative switch cost, slowing down their responses on “initial” problems in a sequence of math problems, and showing increased neural activity in a set of brain regions associated with arithmetic processing. Those low in math anxiety recruited regions of the brain related to the task at hand. These individuals showed increased activity in neural regions associated with increased effort in arithmetic processing, here created by increased “switch cost” where working memory is taxed by having to change set. Previous research on math anxiety shows that cognitive processes related to Attentional Control Theory are disrupted, and positive attentional control is also likely to be disrupted in highly math-anxious individuals. However, for this study, it was uncertain what pattern of disruption would be shown by highly math-anxious individuals: (a) Math-anxious individuals might show increased engagement with the task at hand, showing an exaggerated switch cost due to increased effort in task switching (i.e., longer reaction times, increased neural processing), or (b) highly math-anxious individuals might show a pattern of disengagement, speeding through trials (faster RTs) and decreased processing. Indeed, for the highly math-anxious individuals, they show a pattern of results consistent with avoidance and “speeding through” when they must switch into a sequence of math problems. In switching into a sequence of math problems, more math-anxious individuals disengage from problems, showing decreased reaction times, and importantly, showing decreased neural activity in regions of the brain that would support arithmetic processing. Although significant differences in accuracy were not observed, if one considers these results with respect to Processing Efficiency Theory, even if overall processing effectiveness (i.e., accuracy) is not impacted, one can still observe math anxietyrelated differences in the processing efficiency (i.e., changes in RT or cognitive effort operationalized by neural processing). More math-anxious individuals are showing difficulty with positive attentional control and are not able to effectively recruit the cognitive and neural resources necessary to support increases in working memory

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load associated with switching or shifting. Indeed, this study may also show additional evidence that highly math-anxious individuals show difficulties with negative attentional control as well. If highly math-anxious individuals are having difficulty inhibiting their automatic response to disengage from mathematical material, this could additionally explain why highly math-anxious individuals initially try to avoid math problems by speeding through and under-recruiting neural resources to support arithmetic processing. Both studies illustrate that Processing Efficiency Theory and Attentional Control Theory play an important role in understanding working memory deficits in math anxiety. As suggested by the original literature concerning trait anxiety, working memory, and attentional control, math anxiety is also associated with similar disruptions in working memory, altering the cognitive efficiency with which individuals process mathematical information.

Math Anxiety and Emotional Responses Although much of the previous work on math anxiety has specifically focused on interruptions in attentional control in mathematics anxiety, there are multiple emotional reactions that one must consider with respect to math anxiety. Indeed, much of the research on math anxiety has been predicated on the theory that ruminations and distracting self-related thoughts are the main mechanism of disruption of negative attentional control (Ashcraft 2002). In this view, the ongoing suppression or inhibition of intrusive thoughts reallocates limited resources away from mathematical cognitions and computations. This perspective is consistent with both Processing Efficiency Theory and Attentional Control Theory and is the main focus of Cognitive Interference Theory, as discussed previously. In addition to the disruptions in working memory, math anxiety is also associated with increased negative experiences of mathematics, not only indicated by self-report (Pizzie et al. 2020a), but also recorded in biological correlates of negative affect (Lyons and Beilock 2012b; Pizzie and Kraemer 2017; Pletzer et al. 2015; Suárez-Pellicioni et al. 2014; Young et al. 2012), such as increased reactivity in the amygdala, a structure associated with reactivity to increased vigilance, attention, and threat-related processing (LeDoux 2014). In young children, high math anxiety is associated with increased amygdala reactivity during mathematics, and decreased recruitment of the intraparietal sulcus (IPS), a region associated with numerical and mathematical processing (Dehaene et al. 1999). The results of Young et al. (2012) are especially illustrative of the hypothesized increase in negative reactivity associated with increased amygdala activity. The results also show decreased mathematical processing, associated with decreased neural activity observed in the IPS. Math anxiety was associated with increased amygdala reactivity during mere presentations of math during an attentional task (Pizzie and Kraemer 2017). Reactivity was also observed when young adult participants were presented with mathematical equations, as part of an attentional task, and demonstrates that increased

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vigilance and reactivity are associated with math anxiety even when individuals are not performing math computations. Anticipation of an upcoming math task was also associated with elevated neural reactivity associated with processing of negative affect (Lyons and Beilock 2012b). The results showed that increased math anxiety was associated with increased activity in bilateral dorso-posterior regions of the insula, which have previously been associated with processing negative physical sensations such as pain. Increased math anxiety is also associated with increased reactivity to feedback in mathematics (Suárez-Pellicioni et al. 2014). More mathanxious individuals show exaggerated reactivity to negative feedback, suggesting an extension of increased vigilance or phobic response. The results also suggest that exaggerated negative emotional reactivity to feedback may detract from math learning in math anxiety. Although some work suggests that it is specifically ruminations and intrusive thoughts that interrupt the performance of math skills (DeCaro et al. 2010; Hopko et al. 1998), the rumination explanation cannot necessarily explain all the emotional or cognitive reactions observed in math anxiety. This is also a common explanation of the effects seen more broadly in general anxiety (Eysenck 2010; Mogg and Bradley 1998), but it is unlikely that some of the low-level cognitive and attentional effects observed in anxiety can be fully attributed to rumination, which is a major criticism of Cognitive Interference Theory. Even though there was an attentional bias for anxious individuals to negative information like negative facial expressions (Bishop 2008), it seems unlikely that participants are internally ruminating or having negative intrusive thoughts about the stagnant picture of the fearful face they saw for 1–2 s. It seems that not all the emotional reactions in math anxiety can be attributed to worry or rumination, and an important emotional reaction and key feature of math anxiety is avoidance (Ashcraft 2002; Ramirez et al. 2018; Suárez-Pellicioni et al. 2015). Math anxiety is often characterized by avoidance of mathematics in the short term, avoiding math problems as a local avoidance strategy (Choe et al. 2019; Pizzie and Kraemer 2017). Math-anxious individuals also show long-term avoidance, or a global avoidance strategy, wherein math-anxious individuals tend to avoid taking further math courses, college majors, and careers (Ashcraft 2002; Beilock and Maloney 2015; Hembree 1990). For example, more math-anxious individuals show a speed-accuracy trade-off when solving math problems – sacrificing accuracy in order to speed through and avoid spending more time on problems (Ashcraft 2002). Local avoidance also manifests as avoidance of investing cognitive energy in math calculations, such as gravitating toward problem-solving strategies that rely on memorization rather than working-memory-intensive strategies (Ashcraft and Faust 1994; Ramirez et al. 2016). Math-anxious individuals chose easier, low-reward calculations when compared to more difficult calculations associated with a higher reward (Choe et al. 2019). Math-anxious individuals avoid more working-memoryintensive problems even when a potential reward is present. Combined with global patterns of avoiding math classes, majors, and careers (Hembree 1990), the results indicate that math-anxious individuals show a pattern of avoidance across many math-related contexts.

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However, taking into account these tendencies as patterns of emotional behavior, anxious attentional engagement, and rumination, and avoidance represent two different emotional and behavioral tendencies. Intrusive negative thoughts related to a specific topic like math actually represent a tendency to approach, or perseverate when engaging with that material (Mogg and Bradley 1998). Anxious individuals engage too much with material that is negative or threatening, to an extent that they have difficulty disengaging from that material (i.e., stopping intrusive negative thoughts about math). On the other hand, avoidance tendencies represent the opposite behavioral pattern, such that people show a disengagement bias, and have trouble approaching something like mathematics. Disengagement bias is a pattern of behavior observed in specific phobia (Pizzie and Kraemer 2017), where individuals show increased threat-related arousal, avoidance, and disengagement, much like a phobia (i.e., a spider phobic quickly identifying a spider and then actively avoiding and disengaging from said spider). Both patterns of behavior, anxious engagement and phobic vigilance and disengagement, could potentially be related to behaviors observed in math anxiety. Pizzie and Kraemer (2017) tested this experimental question: Does attention allocation in math anxiety more closely resemble general anxiety or phobia? High- and low-mathanxious individuals performed a dot probe paradigm in the fMRI scanner in order to assess attentional allocation. The results of this study indicated that the more mathanxious individuals showed an engagement bias for negative information (negative pictures), but showed the opposite pattern of behavior for mathematical information (mathematical expressions), demonstrating a disengagement bias, or avoidance, associated with increased anxiety. During the math trials, there was also a positive association between math anxiety and amygdala reactivity, a region of the brain associated with increased vigilance and attention. Overall, the results indicate that math anxiety may more closely resemble math phobia, showing a pattern of increased vigilance, behavioral disengagement, and avoidance. The results of Pizzie and colleagues (Pizzie and Kraemer 2017) indicate that math-anxious individuals show exaggerated reactivity to mathematics, or even the mere thought or presentation of mathematics. Participants were never asked to solve any math problems, and yet the mere presentation of problems resulted in hypervigilance and avoidance. Similarly, in a stroop task utilizing numbers instead of letters, more math-anxious individuals show exaggerated reactivity to even the mere presentation of numbers, even when no calculations is required (Hopko et al. 2002). Reading mathematical terms and words was associated with disruptions in working memory for those higher in math anxiety (DeCaro et al. 2010). Even more mathanxious individuals who are anticipating the appearance of math calculations, but are not currently attempting to solve any problems, show increased activity in regions of the brain associated with negative reactivity and pain (Lyons and Beilock 2012b). Although all these studies may utilize slightly different stimuli, they show that even in the absence of currently having to perform calculations, the mere thought or subtle presentation of numbers is enough to disrupt the efficiency of processing in working memory or attentional allocation. With these stimuli, which are briefly presented and fairly innocuous (i.e., numerals), it seems unlikely that math-anxious individuals

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would report increased negative self-talk or ruminations related to encountering the stimuli. Changes in processing efficiency may operate outside of conscious attention and may not involve the kinds of ruminations or negative intrusive thoughts that are usually thought to drive many of the deficits in working memory that are attributed to math anxiety. Although it may be the case that math-anxious individuals do experience negative intrusive thoughts related to mathematics, results suggest that not all of the deficits observed can be wholly attributed to that mechanism of disruption. If it seems unlikely that many of the deficits in working memory function can be attributed to intrusive thought, as many studies have hypothesized, does attentional control theory still provide a theoretical grounding for math anxiety? In this respect, there may still be an important role for Processing Efficiency Theory and Attentional Control Theory in math anxiety (Eysenck 2010; Eysenck et al. 2007). In Attentional Control Theory, much of the focus on negative attentional control has been on inhibiting intrusive negative thoughts. However, another function of inhibition in negative attentional control is inhibiting prepotent or automatic responses. In the case of math anxiety, disengagement from mathematics and allocating attention away from math is a prepotent or automatic response, evident by persistent avoidance. Negative attentional control in math anxiety may be the push-pull created by monitoring and regulating the automatic or prepotent response to disengage, and instead attempting to regulate and maintain focus to approach the math task at hand. Such an ambivalent simultaneous response to approach and avoid would still represent disruptions in inhibition and negative attentional control. In addition, because this process would encompass two simultaneous response tendencies, both to disengage from mathematics and also to try to remain engaged with the information in order to solve the computation, this would be incredibly taxing on working memory resources (Cacioppo and Berntson 1994). Attempting to inhibit disengagement or avoidance in math anxiety might still result in disruptions in negative attentional control, and lead to disruption in the efficiency of working memory. Overall, whether individuals who experience anxiety about mathematics experience negative affect through the disruption of task-related thoughts via intrusive negative thoughts, or through persistent disengagement from mathematics, it is sure to result in a negative emotional experience that is detrimental to success in mathematics.

Interventions and Emotion Regulation in Math Anxiety Thus far, this chapter has explored studying math anxiety using previous theories and findings from the research on general anxiety. The following section shifts focus to math anxiety interventions that leverage emotionality in math anxiety, illustrating that decreasing negative affect and anxiety ameliorates the negative effects of math anxiety. Targeting emotionality in interventions to ameliorate the deficits observed in math anxiety is an important avenue in improving negative affect and increasing math performance. Although in a math class, it may seem counterintuitive to focus on

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decreasing anxiety and negative affect associated with mathematics, previous literature has suggested that interventions that focus on improving negative affect also result in increases in math performance without having to provide further instruction in mathematics (for review, see Hembree 1990). Previous treatments have included relaxation therapies, systematic desensitization, and cognitive behavioral therapy, resulting in significant reductions in the effects of math anxiety. However, providing individualized therapy, such as systemic desensitization or cognitive behavioral therapy, can be time-consuming, expensive, and does not necessarily scale up to an intervention that would be amenable to inclusion in educational environments such as schools. Focusing on ameliorating math anxiety via mechanisms that change the emotional experience through increasing cognitive control has been a major focus of research in this area. Previous work has suggested that individuals who are able to increase cognitive control and increase brain activity in regions of the prefrontal cortex in anticipation of an upcoming math task subsequently showed increased performance on the task (Lyons and Beilock 2012a). This intervention framework is largely based on the appraisal theory of emotion (Blascovich 1992), such that the thoughts and cognitive “labels” of an emotional stimulus contribute to the resulting emotional response to the situation. Cognitive reappraisal is an emotion regulation strategy focused on these appraisals and is a method by which math-anxious individuals might decrease anxiety and improve mathematics performance (Jamieson et al. 2010, 2012, 2016; Park et al. 2014; Pizzie et al. 2020a; Pizzie and Kraemer 2021; Ramirez and Beilock 2011). Cognitive reappraisal is a method by which one can alter an emotional experience by reframing or taking a more objective perspective on the emotional experience (Buhle et al. 2013; Gross 1998). By changing the cognitive appraisals and thought processes assessing the emotional situation, one changes the experiences of physiological sensations and updates the context in a dynamic fashion, thereby changing the experience of the emotion. In the affective science literature, emotion regulation and cognitive reappraisal are associated with decreasing self-reported negative affect, as well as reducing amygdala reactivity and increasing activity in regions of the brain associated with cognitive and affective control (for review, see Buhle et al. 2013). The ability to gain perspective and regulate one’s emotions is perhaps the key aspect of this kind of intervention. One way in which emotion-targeted intervention has been introduced to educational environments is through the use of expressive writing interventions (Park et al. 2014; Ramirez and Beilock 2011; Rozek et al. 2019). Compared to a control writing condition, students were asked to write down their test-related thoughts and worries a few minutes before taking an upcoming test. Students who were able to write and express their worries about the upcoming exam showed increases in math test performance in the lab (Park et al. 2014), as well as in real-world classroom environments such as math and science classes (Ramirez and Beilock 2011). For those highest in anxiety (test anxiety), introducing the expressive writing intervention significantly improved performance on classroom tests. Although this kind of emotional expressivity may free up working memory resources by “dumping” worrisome thoughts, one of the hypothesized mechanisms

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that expressive writing is helpful is that it provides an environment in which to review worries and reappraise these concerns. The expressive writing intervention provides an opportunity for cognitive reappraisal, reframing the testing experience, reducing negative affect, and resulting in improved performance. This kind of emotion regulation intervention may be especially helpful for students who come from backgrounds characterized by socioeconomic disparities (Rozek et al. 2019); low-income students who were enrolled in the expressive writing intervention had a significantly lower rate of failed courses. This intervention is fairly simple, easy to implement, inexpensive, and scales to large populations of students. However, the research examining expressive writing does not necessarily provide any further details about the process by which reappraisal might influence academic performance, especially for those who might experience increased anxiety. Cognitive reappraisal, and reframing one’s stress response, was also connected to physiological responses to stress and demonstrated both short- and long-term increases in performance (Jamieson et al. 2010, 2012, 2013, 2016). In these interventions, individuals were encouraged to reframe any feelings of stress that they might experience. Instead of viewing physical feelings of stress and thoughts about increased stress as negative, individuals were encouraged to think about the stress response as an adaptive way that the body readies itself to respond to challenges, and that this increased arousal would help individuals to perform better. In response to stressful situations, reappraisal is associated with improved cardiovascular responses, indicating that participants are reacting to increased stress and arousal as though it is a challenge instead of a threat, by increasing cardiac efficiency and decreasing vascular resistance (Jamieson et al. 2012). Utilizing reappraisal may decouple the relationship between increased arousal and decreased performance, such that increased arousal is no longer detrimental to math performance for those who are high in anxiety (Pizzie and Kraemer 2021). In an fMRI study, participants were taught to utilize the reappraisal technique and apply it to math problems and analogies (Pizzie et al. 2020a). For those higher in math anxiety, the reappraisal technique was not only associated with decreases in self-reported negative experience, but reappraisal was also associated with increases in performance. When examining the neural effects, for highly math-anxious individuals, increased accuracy in the reappraisal strategy was also associated with increased neural activity in a network of brain regions associated with arithmetic processing. Not only does reappraisal decrease negativity and improve performance for those higher in math anxiety, but also improvement in performance is associated with increased activity in neural substrates that support increased arithmetic computation. The reappraisal intervention provides no additional instruction for mathematical learning, and yet results suggest increased support for math computation in the brain. In examining stress responses to mathematics, utilizing reappraisal was associated with increased measures of sympathetic nervous system activation (measured by salivary alpha amylase) and was associated with increases in performance on a GRE math test. The results demonstrate that reappraising increased arousal in a positive matter which was associated with improved performance on mathematics in a stressful situation. Moreover, participants in the reappraisal condition showed not

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only increased math performance in the lab, but also in their real-world GRE math performance months later (Jamieson et al. 2010). In a real-world classroom environment, introducing the reappraisal strategy to community college students in a remedial math class was associated with more positive appraisals of mathematics (i.e., students felt they had adequate knowledge and resources in order to succeed in mathematics) and with improvements in grade performance (Jamieson et al. 2016). Overall, not only more math-anxious individuals experience increased negative emotional experiences when making math computations, but also the mere exposure to mathematical information can result in an automatic negative affective response. In addition to self-report, biological indices of emotionality, such as amygdala reactivity, or cardiovascular responses, are also altered in the face of stress associated with mathematics. Many have proposed a variety of interventions in order to reduce the negative association between math anxiety and negative emotional experiences and mathematical performance. Interventions that specifically focus on the cognitive appraisals made during the emotional experience are a promising technique for more anxious individuals. Using these techniques, more math-anxious individuals can reframe or rethink their experiences in order to reduce negativity and improve mathematical performance. The resulting reappraisal and expressive writing interventions have been tested in both laboratory and educational settings and provide some promising results that such simple interventions may provide a method by which those highest in math anxiety can avoid anxiety-related decrements in math performance.

Math Anxiety and Mathematical Cognition Another major focus of math anxiety research has been from the perspective of mathematical cognition, evaluating how math function is altered in math anxiety. In the previous section, much of the research posits that math anxiety disrupts working memory resources, altering math processing. However, considering the negative association between math anxiety and measures of mathematics achievement, one must also consider the idea that math anxiety may stem from fundamental alterations in math processing. This relationship between mathematical deficits and math anxiety is highlighted by Ramirez et al. (2018), and referred to as the reduced competency account. One reason for the negative relationship between math anxiety and math performance or achievement is that due to avoidance behavior, math-anxious individuals gain less practice with mathematical material, and therefore underperform (Ashcraft 2002; Dowker et al. 2016). However, it is also plausible that the direction of the relationship may be reversed, such that decreased math achievement and competence may lead to subsequent increases in anxiety associated with mathematics. In other words, math anxiety may stem from poorer math skills. Using structural equation modeling, one study sought to disentangle the relationship between math anxiety and math achievement over time across middle and high school students (Ma et al. 2004). Using longitudinal anxiety and achievement data

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from the Longitudinal Study of American Youth, as students progressed through their adolescent years of schooling, poorer math achievement was more likely to be associated with later increases in math anxiety. The results of this study demonstrated that initial lower mathematics achievement was related to later increases in anxiety about mathematics. However, the relationship between initial math anxiety and subsequent math achievement was not as strong. These results suggest that anxious attitudes may result from decreases in math performance. However, it is important that these results are viewed with a critical eye. For example, although it is not uncommon to measure math anxiety with self-report of a small number of questions (Núñez-Peña et al. 2014), Ma et al. (2004) only used the Likert-scale responses of two questions in order to measure anxiety about mathematics (“Mathematics often makes me nervous or upset,” “I often get scared when I open my mathematics book”), which may not be an optimal measure of math anxiety (Pizzie and Kraemer 2019). In addition, although math achievement may have been significantly statistically associated with subsequent anxiety across each academic year from grade 7–12, the effect sizes are still relatively small, especially compared to other effects observed in the study. For example, math achievement in grade 7 was significantly associated with math anxiety in grade 8 but only accounted for approximately 4% of the variance of that effect (Ma et al. 2004). In comparison, from year to year, math achievement is strongly associated with the previous year’s math achievement and accounts for 82–96% of the variance in achievement scores. Despite the small effect sizes, this study provides one of the few longitudinal measures of math anxiety and math achievement, in the literature, and gives important insight about the importance of considering math performance as an antecedent of anxiety about mathematics. Still more work is needed in order to gain a better understanding of the dynamic relationship between math anxiety and mathematics achievement and performance.

Math Anxiety and Numerical Processing Research has suggested that there may be underlying or preexisting difficulties in numerical processing that underlie math anxiety (Gunderson et al. 2011). Measurements of math anxiety and math achievement illustrate that even young children experience math anxiety and deficits in math achievement in early elementary school. In the numerical/spatial difficulties framework, the reduced numerical or mathematical abilities may be a fundamental cause of math anxiety (Beilock and Maloney 2015; Ramirez et al. 2016). In general, research suggests that not only is math anxiety associated with more difficult mathematical calculations, but may also be associated with disruptions in more fundamental mathematical and spatial skills. Although some of the early research on math anxiety using college-aged young adults suggested that “easy” math skills were less impacted by anxiety-related deficits, research with younger children shows that anxiety is negatively associated with early math skills, with potentially long-range effects throughout one’s educational career. Focusing on the development of anxious attitudes associated with

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fundamental math skills and numerical understanding provides insight into how and why math anxiety develops (Beilock and Maloney 2015). In the numerical/spatial difficulties framework, math anxiety is associated with decreased efficiency in processing numerical and spatial skills (Ramirez et al. 2018). For example, more math-anxious individuals show increased reaction times (decreased processing efficiency) in a visual enumeration task (Maloney et al. 2010). Math-anxious individuals may show deficits in more “basic” mathematical skills than previously hypothesized, as they demonstrated increased processing time on a counting task (although numbers in the subtilizing range did not show the same delay). For example, when young adult undergraduates were presented with a number of objects (i.e., 5–9 squares) on a computer screen, the longer the more math-anxious individuals took to count the objects, the more errors they made. Even rapid processing of numerical magnitudes may be less efficient in more mathanxious individuals. Some work also suggests that spatial skills and visuospatial processing may be altered in math-anxious individuals (Sokolowski et al. 2018), and that math anxiety may also be related to spatial skills and spatial anxiety (Lyons et al. 2018). Math anxiety is also associated with decreased efficiency in processing numerical magnitude by examining numerical distance effects (Maloney et al. 2011). The numerical distance effect posits that on a mental representation of a number line, numbers that are positioned close together (i.e., 7 and 8) may share some overlap in representation, and may be more difficult to distinguish from one another than numbers that have more space between them and more distinct representations (i.e., 5 and 9). In an initial task, participants judged whether single numerals (1–9) were above or below 5. Compared to those low in math anxiety, highly math-anxious individuals were slower to respond and showed an increased numerical distance effect, such that they had increased responses for numerals that were closer to 5. Similarly, in comparing two numerals that were simultaneously presented, math-anxious individuals were significantly slower to decide which of the numerals was larger when the numerals were closer together on the number line. Deficits or inefficiency in processing of symbolic numerical magnitude is associated with increased math anxiety. This was further replicated by research suggesting that both numerical size and distance effects are exaggerated in more highly math-anxious individuals, showing differences in event-related potentials (ERPs) related to cognitive processes (NúñezPeña and Suárez-Pellicioni 2014). More math-anxious individuals showed slowed responses in making numerical distance judgments, and ERPs associated with numerical distance showed exaggerated responses and greater amplitudes (NúñezPeña and Suárez-Pellicioni 2014). Exaggerated numerical distance effects were observed for more math-anxious individuals for symbolic (i.e., Arabic numeral) presentations (Dietrich et al. 2015; Maloney et al. 2011; Núñez-Peña and SuárezPellicioni 2014). However, results of another study suggest that the effects cannot necessarily be attributed to anxiety-associated differences in the approximate number system, which is a cognitive system associated with processing general “number sense” (Lyons 2015). Nonsymbolic dot comparison tasks were not associated with

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variation in math anxiety, indicating that processing of symbolic representations of numerical magnitude may be more strongly related to variation in math anxiety (Dietrich et al. 2015). Math anxiety may be associated with differences in the processing efficiency of symbolic numerical magnitude, illustrating that the processing deficits associated with math anxiety go beyond the processing of mathematical calculations. However, these results also indicate that the differences in processing may not impact processing of magnitude, indicating that the representation of symbolic numerals may be the source of differences in processing. In examining the relationship between math anxiety and math performance, it is essential to explore how math strategies and the efficiency of math processes may be influenced by math anxiety. Math anxiety may not necessarily influence whether the individual knows or understands how to complete relatively simple calculations but may instead influence the cognitive strategies and thought processes used to complete the problem. For example, when math-anxious individuals show increased activity in a network of frontoparietal regions while anticipating an upcoming math task, increased brain activity during the anticipatory period was associated with decreased anxiety-related performance deficits in a subsequent math task (Lyons and Beilock 2012a). Recruitment of frontal regions is thought to represent the ability to anticipate an upcoming difficult task, and prepare the appropriate cognitive and emotional control resources in order to complete the task at hand. Differing cognitive strategies and cognitive control may be necessary for math-anxious individuals. Further, cognitive strategies and efficiency may differ even for easy math problems (Chang et al. 2017). In this study, high and low math-anxious individuals performed easy or simple addition and subtraction problems in the scanner. For low mathanxious individuals, problems were likely solved efficiently and using a recall strategy that operates more or less automatically. For low math-anxious individuals compared to high math-anxious individuals, performance on simple math task has a negative relationship with activity in the frontoparietal network, a network of brain regions that supports cognitive and attentional control devoted to the task at hand. For low math-anxious individuals, performance on the arithmetic task was better the less activity was shown in the frontoparietal network. Compared to high mathanxious individuals, low math-anxious individuals may show better performance when utilizing strategies that rely on automaticity and decrease cognitive control and working memory-intensive strategies. Across the spectrum of math anxiety, cognitive control and strategy use may have variable effects during the anticipation and calculation phases of arithmetic processing. Even relatively simple arithmetic problems may result in different cognitive processing strategies across the spectrum of math anxiety.

Interventions and Math Competency in Math Anxiety In the previous section, several studies that targeted emotionality as a method to ameliorate the negative relationship between math anxiety and math performance were presented. However, other intervention strategies have specifically targeted

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improved math performance and understanding as a method to reduce the negative impact of anxiety on math achievement (Agarwal et al. 2014; Supekar et al. 2015). Specifically, interventions that improve the quality of instruction or improve study techniques may be especially advantageous for more anxious students. Extensive research has suggested that encouraging students to engage in “desirable difficulties” to improve their engagement and learning of educational material may be advantageous to learning (Bjork and Bjork 2011). In both real-world and laboratory settings, utilizing self-testing, or using low-stakes tests, quizzes, or retrieval methods (e.g., flashcards) was associated with improved learning (McDaniel et al. 2007). Integrating self-testing practice in real-world classroom settings was associated with improved learning (McDaniel et al. 2007). In addition to the gains in performance, implementing a self-testing study strategy intervention was associated with decreases in test anxiety (Agarwal et al. 2014) and, similarly, ameliorated the negative relationship between math anxiety and performance (Pizzie and Kraemer 2019). Another intervention study introduced an 8-week one-on-one tutoring intervention for third grade students in which children were scanned before and after a tutoring intervention (Supekar et al. 2015). This intervention focused on improving a number of different math concepts, and encouraging students through increased exposure to high-quality mathematics instruction. At the end of the intervention, highly math-anxious children showed decreases in math anxiety. Math performance improved across both high and low math-anxious children across the study. On the neural level, the tutoring intervention was associated with decreases in aberrant brain activity in highly math-anxious individuals, including the amygdala and additional frontoparietal brain areas. Moreover, in high math-anxious children, decreases in hyperactivity in the right amygdala were also associated with decreases in selfreported math anxiety as a result of tutoring. The results of this study suggest that improving instruction and competence in mathematics is associated not only with increases in math performance, but also with reductions in math anxiety and changes in brain activity that suggest that increased math exposure reduces aberrant brain function related to anxiety. Across both intervention techniques, improving exposure to high-quality instruction and study techniques is related to reductions in anxiety about mathematics, and improvements in math performance. In addition to potentially improving mathematical understanding or conceptual development, the interventions also combat avoidance tendencies. As math anxiety is associated with both local and global avoidance of mathematics, these interventions focus on encouraging math-anxious individuals to approach mathematics more often, and to make those interactions with math as efficacious as possible. For example, instead of just rereading a chapter in their math book to study for a math test, students would be able to use that study time to actively practice calculations using self-testing, gaining more exposure and knowledge. The techniques may serve as a kind of exposure therapy (Pizzie and Kraemer 2017; Supekar et al. 2015), addressing the dual purpose of providing both improved instruction and practice, and encouraging math-anxious individuals to overcome phobic avoidance tendencies to approach mathematics more often. Future studies

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should continue to investigate the complex relation between mathematical learning processes and math anxiety.

Conclusions The present chapter has reviewed many dimensions and perspectives of mathematics anxiety. Over the past several decades of research, scientists have learned a great deal about the relationship between math anxiety and math performance and achievement. Although the connections between anxiety and performance are often posited as a kind of “chicken-and-egg” relationship, it is likely the case that both elements of anxiety and skill interact in a dynamic manner over time to influence one’s experiences with mathematics. As methods are proposed to improve experiences and understanding of mathematics, it is essential to consider the interplay between both emotional experiences and mathematical knowledge and skill. Math anxiety brings together all these elements, analyzing how the development of cognitive and affective processes creates one’s individual experience of mathematics in educational environments and beyond. The chapter began by considering math anxiety from theoretical perspectives on general anxiety to ground our understanding of math anxiety. This research points to a few theoretical perspectives highlighting the relationship between anxiety and deficits in cognitive performance, in this case, math performance and achievement. In Cognitive Interference Theory (Eysenck 2010; Sarason 1988), intrusive thoughts and worries interrupt task-related thoughts, disrupting working memory processes, and creating deficits in math performance. Cognitive Interference Theory has inspired much math anxiety research, theorizing that intrusive worries interrupt math-related thoughts. However, this theory places a reliance on ruminations and worry, as the method by which working memory resources are interrupted, and does not necessarily specify how the secondary task created by worry interferes with cognitive resources. Instead, math anxiety finds most of its theoretical grounding in Processing Efficiency Theory and Attentional Control Theory, which are two theoretical perspectives that provide a better understanding of how cognitive resources are influenced by anxiety and emotion. Processing Efficiency Theory and Attentional Control Theory build on similar perspectives from Cognitive Interference Theory, hypothesizing that anxiety detracts from the efficiency of cognitive processes. By impacting the efficiency of cognitive processes related to a specific task, anxiety increases the amount of effort, time, or cognitive control needed in order to complete a task. Further, efficiency is impacted through disruptions in inhibition and shifting functions, limiting working memory function in the task at hand. In this way, more math-anxious individuals have difficulty inhibiting prepotent responses, such as worrying thoughts or avoidance behaviors, and must engage additional effort or cognitive control in staying engaged in the math task at hand. Increased effort or cognitive control failure to inhibit prepotent responses takes away working memory resources from math computations, functioning as a competing secondary task, and

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impeding resources from going toward mathematical computations. Math anxiety is associated with deficits in working memory performance with regard to mathematics, providing evidence for both Processing Efficiency Theory and Attentional Control Theory as a perspective that connects math anxiety and math performance. Another feature of math anxiety is the local and global avoidance that characterizes math anxiety, including avoidance of math problems (Ashcraft 2002; Ashcraft and Faust 1994; Pizzie and Kraemer 2017), as well as math classes, majors, and careers (Ashcraft 2002; Beilock and Maloney 2015; Hembree 1990). Given the pattern of negative emotionality, negative neural reactivity related to increased vigilance, in addition to its resemblance to other anxious behaviors, math anxiety also resembles math phobia (Pizzie and Kraemer 2017; Ramirez et al. 2018; SuárezPellicioni et al. 2015). From the perspective of Processing Efficiency Theory and Attentional Control Theory, failures to inhibit both worries and avoidance behaviors may represent significant challenges for math-anxious individuals. As emotional behaviors are considered as targets for intervention, providing support for both decreasing worry and encouraging math-anxious individuals to decrease avoidance may prove to be an important feature for reducing the negative impact of anxiety on performance. In order to reduce negative affect, and increase math performance in math anxiety, various studies have specifically focused on ameliorating feelings of stress and anxiety (for review, see Hembree 1990). Various studies have provided some promising evidence for intervention strategies, such as relaxing and reducing stress (Hembree 1990), writing about test-related worries in expressive writing (Park et al. 2014; Rozek et al. 2019), or utilizing cognitive reappraisal strategies to rethink or reappraise one’s emotional experience (Jamieson et al. 2010, 2012, 2016; Pizzie et al. 2020a; Pizzie and Kraemer 2021). These interventions suggest that rethinking or reframing one’s emotional experiences from a different perspective may be a valuable technique for math-anxious individuals to reduce their experience of negative affect and improve math performance. Future work should continue to investigate the relationship between emotional intervention techniques with respect to other factors known to influence math anxiety, such as examining the relationship between emotional interventions and working memory function. As real-world ramifications of affective science research are considered, more work is needed to understand how cognitive processes and interventions play out in real-world classrooms and learning environments. Much of the research that considers math anxiety is relatively agnostic to the relationship between math anxiety and mathematical skill. Although some research suggests that math processing might be disrupted, research also suggests that math anxiety may stem from underlying differences in numerical understanding or perception. In other words, math anxiety may arise from some fundamental disruptions in numerical or cognitive skills that underlie even simple mathematics. These theories are not mutually exclusive, and it is important to acknowledge how fundamental differences in number sense shape the relationship between math anxiety and math performance. Several studies find anxiety-related differences in processing of simple or fundamental mathematical skills such as counting (Maloney et al. 2010),

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and differences in executive function processing in simple mathematics (Chang et al. 2017). Additional research suggests that math anxiety may be associated with fundamental processing related to number line perception, suggesting that more math-anxious individuals may have a less precise representation of a linear number line (Dietrich et al. 2015; Maloney et al. 2011; Núñez-Peña and Suárez-Pellicioni 2014). In addition to considering the ways in which emotionality interrupts cognitive processes related to cognitive processing of mathematics, scientists must consider how anxiety may arise from or disrupt some of the fundamental cognitive processes related to the perception of numerals and mathematical concepts. Interventions that have focused on providing additional support and instruction in mathematics have also been efficacious in reducing the negative relationship between math anxiety and math performance. For example, studies that provide additional support for improved instruction that emphasizes “desirable difficulties” and self-testing may be helpful for more anxious students (Agarwal et al. 2014; Supekar et al. 2015). This technique may seem counterintuitive, as encouraging more anxious students to engage more often with material that makes them feel negative, especially in ways that may be cognitively challenging (i.e., self-testing), easily could have exacerbated the negative experiences of students with mathematics. Instead, improved study habits seem to do the opposite, and encourage students to actively engage with the material in a manner that improves learning the material, and encourages students to habituate to situations similar to testing. Similarly, a oneon-one 8-week tutoring intervention designed to improve mathematical understanding was also associated with improved math performance, reductions in math anxiety, and reductions in math anxiety were also associated with decreases in amygdala reactivity (Supekar et al. 2015). In addition to providing support for better conceptual understanding of the material in a math-learning environment, what both techniques have in common is increased exposure to mathematics in a supportive manner that aids in understanding. In this way, math-anxious individuals not only are encouraged to overcome some of their tendencies to avoid mathematical material but also are provided with experiences where they can habituate to mathematical material. Both techniques allow math-anxious individuals to gain more experience with math in a manner that supports better learning. How and why math anxiety develops over the course of one’s educational experience is a multidimensional puzzle for researchers to consider. In addition to exploring the emotional and mathematical processes that occur within the mind of someone who is math anxious, scientists must also consider the educational context in which students are trying to learn. Although researchers focus on isolating emotional and cognitive processes in the lab, scientists must also consider the social and environmental context that may define one’s educational experience, from exploring mathematical instruction to evaluating the influences of one’s interactions with parental figures, peers, teachers, and the educational environment at large. Improving the quality and experience of math education is of vital importance in attracting more individuals to quantitative careers. Math anxiety represents a significant obstacle for individuals to overcome in order to excel in mathematical and quantitative subject matter (Beilock and Maloney 2015). In exploring math anxiety,

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it is essential to consider how one’s experience is created by emotion, mathematical cognition, and educational experiences, impacting mind, brain, and education. As researchers and learners continue to learn more about math anxiety, it is essential continue to explore how best to ameliorate the negative effects of math anxiety on math achievement, investigating new ways to encourage each individual to thrive.

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Neurocognitive Interventions to Foster Mathematical Learning

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Karin Kucian and Roi Cohen Kadosh

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prevention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prediction of Arithmetical Skills in School by Early Numerical Competencies . . . . . . . . . . . Does Training of Early Numerical Skills Prevent Later Math Problems? . . . . . . . . . . . . . . . . . . Longitudinal Outcome of Early Prevention Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Developmental Dyscalculia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavioral Interventions to Foster Mathematical Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavioral Interventions for Math Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Numerical Interventions on the Brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brain Stimulation to Foster Mathematical Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In our exceedingly technical world, numeracy is recognized as an essential skill to meet everyday demands of life. However, poor numeracy and severe numerical learning problems are very common in our society and imply serious obstacles in daily lives, school, or professional success. It is time to gain advantages from obtained numeracy research and neuroscientific knowledge of the recent years for affected people. First, early symbolic numerical skills have been proven being crucial for later mathematical skills and enable us to identify children at risk for mathematical learning disorders already in preschool. Second, an early support of these preschool children has the potential to enable them to catch up to the K. Kucian (*) Center for MR-Research, University Children’s Hospital Zurich, Zurich, Switzerland e-mail: [email protected] R. Cohen Kadosh Department of Experimental Psychology, University of Oxford, Oxford, UK e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_30

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mathematical levels of their peers, facilitates school entry, school development, and prevents the development of severe math learning problems. Third, general evidence-based recommendations are given that make an intervention particularly effective for people with poor numeracy. Moreover, we highlight the neuronal changes that go along with successful numerical training. Finally, insights are provided into recent approaches to stimulate the human brain by noninvasive methods using low electrical currents open new venues to facilitate numerical learning. Although it is still unclear which factors best predict individual learning success through intervention. However, there is evidence that the earlier you intervene the better and that children with more severe numerical difficulties are rather dependent on intense and individualized interventions. Keywords

Intervention · Dyscalculia · Prevention · Math learning · Brain · Brain stimulation

Introduction Numbers are ubiquitous in our daily lives. Consequently, successful participation in our society is dependent on adequate numerical skills. Poor numerical abilities, in contrast, imply serious difficulties in daily or professional lives for affected persons. We should therefore aim at identifying people with specific math learning problems as early as possible and support them effectively. As will be outlined in the following chapter, prediction of later numerical difficulties would already be possible in preschool. We summarize which early numerical competencies predict best later numerical skills or particularly later math learning disorders. Furthermore, the current chapter describes evidence that preschool children who have been identified as being at risk for math learning disorders benefit significantly from an early support. Moreover, only in recent years, important efforts have been attempted to investigate the longitudinal outcome of interventions that focus on early phases of numerical competence as well as the evaluation of effects fostering numerical precursor skills in young children. The present chapter will provide an overview of the longitudinal outcome that can be expected of these early interventions. If children with serious math learning difficulties are not identified early, their chronic experience of failure can develop anxiety, depressive, or aggressive symptoms and they receive a much higher degree of negative feedback from their teachers. Following, these children often respond with avoiding all math-related topics and their downward spiral of decreasing numerical and mathematical competencies and increasing frustration begins to spin. Therefore, affected children are dependent on effective intervention. The current chapter will therefore provide evidence-based recommendations on how an intervention or therapy should look like and which are the necessitated prerequisites in general. As the focus of this chapter lies particularly on the neurocognitive aspects of intervention, an overview of effects of numerical intervention on the brain is provided. Additionally, new and

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innovative approaches to facilitate cognitive learning using noninvasive brain stimulation will be highlighted. The present chapter includes a critical overview of potentials and limitations of noninvasive brain stimulation particularly on numerical and mathematical learning. Finally, we will close the chapter by discussing aspects that are relevant for the prognosis of an intervention, as we know that numerical learning disorders persist in general into adulthood and not all people benefit from each intervention. Therefore, it is an important step to determine individualized therapy for preschool children, schoolchildren, adolescents, and adults with specific numerical and mathematical learning problems.

Prevention Numerical competencies are essential for many aspects of day-to-day living, and they are becoming even more crucial with the increasing role of technology in contemporary society. Importantly, profound difficulties with numeracy are very common with a prevalence rate around 6% (for review please see Kucian 2016). Affected children are usually diagnosed not before second or third grade, when their numerical learning problems get so severe that they can no more be compensated. By that time, affected children lack the understanding of basic numerical understanding, have often negative experiences with math related issues, and often develop specific mathematical anxiety. These negative emotional factors in numerical cognition have negative effects on mathematical learning, but can also have detrimental long-term consequences on school achievement and quality of life (for review please see Dowker et al. 2016). Therefore, a goal should be to identify children with math learning problems as early as possible and support them effectively.

Prediction of Arithmetical Skills in School by Early Numerical Competencies Early numerical skills vary greatly between children before they start school. In addition, we now know from various long-term studies that early numerical competencies such as number knowledge, verbal counting, object counting, nonsymbolic magnitude comparison, number comparison, rapid enumeration of small quantities without counting (subitizing), or simple calculations are good predictors for later mathematical achievement in primary school (e.g., Gallit et al. 2018; Jordan et al. 2009; Krajewski and Schneider 2009; Toll et al. 2016). Research has been able to further differentiate between these early numerical skills, revealing that some numerical competencies seem to be even more important than others for later numerical development. Generally, it can be differentiated between symbolic and nonsymbolic skills. Symbolic numerical abilities include, for instance, the comparison of two sets of objects and symbolic skills include the active handling of Arabic digits (e.g., number comparison, linking numbers and magnitudes, reading numbers). Particularly these symbolic skills

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have been identified as stronger predictors for later mathematical skills (e.g., Göbel et al. 2014). According to Krajewski and Schneider (2009), the link between magnitudes and Arabic digits represents the most important concept for successful mathematical learning in primary school. In line, Kolkman et al. (2013) reported that symbolic skills play a predominant role as predictors for math performance. More recent research with larger cohorts provides corroborating evidence. The examination of 670 children at the end of first year of kindergarten by Toll et al. (2016) revealed that symbolic number sense is the strongest predictor of mathematical performance in first grade. Similar results have been reported by Caviola et al. (2020) from a large sample of 1254 second graders. Nonsymbolic magnitude comparison had no association with mathematical performance. In contrast, overall symbolic number comparison accuracy was a reliable and the mostly specific predictor of math achievement. This has also been confirmed by Göbel et al. (2014), showing that the knowledge of the Arabic numerical system predicted the growth in arithmetic performance, whereas no impact of the magnitude comparison was found. Similarly, Missall et al. (2012) showed in their study from kindergarten to third grade that symbolic skills (comparing numbers, inserting a missing number in a number sequence) are the most robust factors for predicting later mathematical skills. Finally, the importance of symbolic number skills for later arithmetical competencies has been confirmed in different review works (Schneider et al. 2017; Szkudlarek and Brannon 2017). Schneider et al. (2017), for instance, investigated systematically in their meta-analyses the strength between symbolic or nonsymbolic magnitude comparison and broader mathematical competencies. They evaluated 284 effect sizes of 45 articles with 170 201 participants. Their findings revealed that effect sizes were significantly higher for symbolic (r ¼ 0.302, 95% CI [0.243, 0.361]) than for nonsymbolic magnitude comparison (r ¼ 0.241, 95% CI [0.198, 0.284]). Effect sizes decreased slightly with age, but the majority of the subjects were between 6 and 9 years old. This means that age is only a weak moderator of the association between magnitude comparison skills and mathematical competencies. More importantly, the effect sizes were dependent on the way the magnitude comparison skills were evaluated (e.g., by solution rates, Weber fractions, distance effects, reaction times, or others) and how numerical competences were assessed. Nevertheless, the systematic analyses of effect sizes corroborated the view that the association between symbolic magnitude processes with mathematical skills are stronger compared to nonsymbolic magnitude processing. Schneider et al. (2017) conclude that symbolic magnitude processing is a more eligible candidate for diagnostics and interventions in children. Please see Fig. 1. Beside early numerical skills, results also revealed that lower general cognitive abilities in kindergarten (lower intelligence and working memory capacities) hinder magnitude and number knowledge before school entry (Gallit et al. 2018). Children with lower general cognitive skills are probably in a worse starting position for later numerical achievement in school. Taken together, children who already have difficulties in basic numerical skills or lower general cognitive abilities before starting school seem to develop their further

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Fig. 1 Early numerical competencies. Early numerical competencies (green and blue arrows) in preschool and kindergarten are good predictors of later numerical and mathematical performance in school. Current evidence suggests that early numerical skills requiring symbolic number processing (green) are stronger predictors than nonsymbolic numerical skills (blue)

numerical skills more slowly. It can be concluded that numerical problems manifest early and lead to difficulties in calculation skills. Without additional support, it is very likely that these children will always be among the worst at arithmetic during their subsequent primary school years. Based on the outlined findings, we have to be aware that the developmental course of children with deficits in basic magnitude and number knowledge in kindergarten is rather negative, although they show individual improvements. However, many children with early numerical difficulties will not be able to catch up to the mathematical levels of their peers with age-appropriate basic numerical skills in kindergarten. Therefore, particularly children with problems in specific numerical precursor skills are dependent on adequate support at an early age to prohibit further negative developments in mathematics. However, this raises the question if specific intervention in kindergarten for children with impaired numerical abilities can facilitate the acquisition of calculation skills in school and prevents them from further failure in mathematical topics.

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Does Training of Early Numerical Skills Prevent Later Math Problems? Empirical findings to evaluate preventive options of basic numerical skills are rare. Only in recent years, longitudinal studies focus on the early phases of numerical competencies and the evaluation of preventive interventions that promote numerical precursor skills. As outlined above, it is possible to predict future calculation skills already in kindergarten. Accordingly, we are also able to identify a child who will most probably have mathematical learning difficulties later in school by testing these predictive early numerical competencies. The option to identify math specific developmental risks at an early stage gives hope to counteract with appropriate intervention programs and thereby reduce the risk of school problems in math. Existing preventive support programs train numerical skills such as verbal counting or counting objects, the rapid acquisition of small quantities without counting (subitizing), addition and subtraction, numerical comparison tasks, or number-dependent logical thinking. In general, the goal of a prevention program is to foster and strengthen the early numerical competencies (please see also Fig. 1). The effectiveness of such early support could be proven in a meta-analysis (Malofeeva 2005) and a more recent review provided a differentiated overview (Mononen et al. 2014). In sum, the majority of the examined studies corroborated that children who received early preventive support did better than those without support. Accordingly, there exists strong evidence that rather than waiting to support children with poor numeracy until school, evidence-based programs should be deployed before school entry to develop early numerical skills in children at risk for numerical disorders. In their review, Mononen et al. (2014) came to the conclusion that “if the majority of children could master key early numeracy skills at the beginning of formal schooling, better mathematics learning outcomes should result later on, and reduce the need for supplemental mathematics support at school age.” However, they also stress that there will always exist a number of children who will not respond to intervention, regardless of the evidence-based interventions used. In other words, it is tempting to believe or to wish that an early intervention, which has been proven to help children at risk for math learning problems, has the capacity to support every child with reduced early numerical competencies. It would be nice to have a universal intervention for these children, but it is far from reality because mathematical learning problems are per se distinguished by a very heterogeneous character in terms of symptoms and clinical appearance. The reason for this might be the fact that numerical cognition and calculation presupposes a perfect interaction of a multitude of cognitive subcomponents, which can be individually disturbed. Consequently, the heterogeneity of the symptoms in numerical disorders suggests that different intervention programs will have a differential effectiveness. Moreover, children with serious numerical problems and, as a result, poor initial arithmetic skills seem to benefit less from standardized prevention programs and need more intensive and more individualized support in order to achieve similar successes as children with less severe difficulties.

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Another problem is the difficulty to differentiate between effects of the intervention in preschool and the possible effects simply from the additional attention given to the child. Therefore, studies, which include, in addition to a passive control group, an active group of children who receive an alternative intervention, would be important. Jordan et al. (2012), for instance, did include both an active and a passive group of children. All participants were attending schools in low-income communities and were therefore considered at risk for underachievement. The 30-minute lessons were carried out in small groups of four kindergarten children, 3 days per week for a total of 24 lessons. The numerical intervention group was trained on number recognition and base ten principles, number sequencing, verbal subitizing, finger counting, number list activities, written number activities, part-whole relationships, problem solving and operations, and linear number board games. The active control group performed for the same amount of time a language intervention, which focused on storybooks with some quantitative vocabulary. The passive control group was engaged in regular kindergarten activities. Their findings point to evidence that the numerical intervention group outperformed both the (passive) control group without any intervention and the active control with the language support. The large effect size (number intervention vs. passive control, effect size Cohen’s d ¼ 1.8) infers that the number intervention was much more effective in raising outcome scores. Hence, the extra attention given to children did not seem to be the reason for the group difference. In contrast, the children benefited specifically from the numerical intervention. The findings described above highlight the effectiveness of early support. However, despite low performing children were able to reduce their performance gap by making remarkable progress after such an intervention, these children often still lagged behind the performance of their age-related peers. Differential progress rates might partially be explained by the influence of pedagogical factors. In term of the intervention setting, the review has shown that preschool children at risk for numerical difficulties can be successfully promoted in small groups. Working with a small group of children or one-to-one, a teacher has the opportunity to pay more attention to individual children’s needs and to guide, model, and give personal feedback. However, such a personalized setting depends on the teacher’s time and resources for implementation. Regarding the intervention duration, even short prevention programs of less than 12 weeks have the capacity to significantly improve children’s numeracy skills. Explicit introductions have been found to be effective with low-performing children. The study by Malofeeva (2005) reported that also peer-assisted tutoring works well with preschool children. Most of the intervention studies included only assessments immediately after the completion of the prevention program or assessed delayed posttest measurements after a few weeks. In general, the intervention effects held when measured between three to 9 weeks after the intervention (Mononen et al. 2014). However, it would be important to know the long-term effects of the support of early numerical competencies. The longer the positive effect of early numeracy intervention remains after the intervention, the more effective it is to prevent mathematics difficulties.

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However, to date, we have only very weak evidence whether the positive intervention effects held longitudinally.

Longitudinal Outcome of Early Prevention Programs The advantage of longitudinal studies is that researchers are able to detect developments or changes in the characteristics of the target population at both the group and the individual level. However, longitudinal studies are scarce, particularly in children and in combination with interventions trying to enhance numerical thinking. As McCaskey et al. (2017) state in their longitudinal examination of children with math learning difficulties, the lack in this field of research might arise from several reasons: “Firstly, longitudinal studies in general are especially prone to high dropout rates” (McCaskey et al. 2017, p. 12). “Secondly, longitudinal study designs are very time consuming regarding (re-)recruitment and maintenance of the participant’s motivation. Thus, developmental questions are in many cases examined by more time-efficient methods such as cross-sectional designs. Importantly, cross-sectional designs do not take into account inter-individual differences to the same extent as longitudinal designs. Furthermore, most cross sectional-studies compare adults and children and might therefore miss an opportunity to capture the full developmental trajectory” (McCaskey et al. 2017, p. 13). However, particularly in terms of the evaluation of any intervention effects, it is indispensable to consider longitudinal study designs on a short- and importantly also long-term base. Only in recent years, efforts have been attempted to investigate the longitudinal outcome of interventions that focus on early phases of mathematical competence as well as the evaluation of effects promoting numerical precursor skills in young children. As Aunio et al. (2005) pointed out in their conclusion “it is essential to conduct longitudinal research with relation to interventions, as we would otherwise mistakenly judge the effects of instructional efforts.” In particular, their study investigated the possibility of enhancing the level of preschoolers’ number sense by introducing two numerical intervention programs. Promisingly, the children showed enhanced number-sense performance immediately after the instruction ended. However, the difference between the groups was not sustainable, as 6 months after the intervention the children who completed the intervention and those without intervention performed equally well. So, the effect faded after 6 months from the end of the intervention. A bit more promising are the reported findings of Krajewski (2008). The effectiveness of her training of basic mathematical skills, which was carried out in the last year of kindergarten, could be proven in the short and medium term. The German numerical intervention “Mengen, zählen, Zahlen” (Krajewski et al. 2013) includes building blocks, playing chips, and playing cards. In an interactive way, kindergarten children learn the numbers and magnitudes up to 10, or relations such as bigger or smaller than. The children train for 8 weeks, 3 days per week during 30 min intervention session of small groups. Children of the intervention group (N ¼ 71)

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improved their magnitude and counting skills significantly more compared to a group of children without training (N ¼ 108) or a group of children who completed a cognitive control training (N ¼ 45). These effects seemed to be stable after 7 months, which was the time point shortly before the children entered first grade. However, the author reported no transfer effects to mathematical school performance since there were no differences evident between the three groups at the end of first grade. In general, existing studies of early numerical intervention showed that both typically developing children and those who are at risk for numerical disorders benefit at short term from training programs that foster basic numerical skills. However, the findings on the long-term effects of early intervention and especially on transfer on not directly trained mathematical skills are still unclear and need further research. However, a recent work by Moraske et al. (2019) did a first step into this direction and provided very interesting results by examining the short- and long-term effects up to the third grade of an early intervention in kindergarten for children at risk for math learning problems. Their findings demonstrated how in practice an early diagnosed risk of developing math disorders could effectively be counteracted. From totally tested 10 897 children in kindergarten, 88 children were identified as at risk for the development of math learning disorders. A child was considered at risk for math learning disorders when her/his performance in a standardized counting and magnitude test was in the lowest 10%. The intervention was carried out under conditions corresponding to daily kindergarten routine and was based on the German training programs “Mathematik im Vorschulalter” (Rademacher et al. 2009) and “Mengen, zählen, Zahlen” (Krajewski et al. 2013). About three quarters of the tasks stem from the training “Mathematik im Vorschulalter” and included visual differentiation skills, spatial perception, magnitude estimation, concept of numbers, simple arithmetic problems, manipulation of symbols, recognition of abstract-logical associations, and causation. About one quarter was selected from the training “Mengen, zählen, Zahlen” and included counting, knowledge of Arabic digits up to 10, and the understanding of the quantity concept. The intervention lasted 11 weeks, with 2 sessions per week, each 30–40 min long. The training was carried out by kindergarten teachers who received a two-day schooling and a helpline was provided. Strikingly, children who received such an intervention in kindergarten showed longterm benefits, indicated by average calculation skills in first, second, and third grade. Moreover, preschool support of children at risk for math learning disorders has even the potential to diminish the probability to develop severe mathematical learning disorders, such as developmental dyscalculia in school (please see following section about further details on developmental dyscalculia). In contrast, kindergarten children at risk for developmental dyscalculia, who did not receive the preventive support in kindergarten, performed below average in first, second, and third grade and developed significantly more often developmental dyscalculia. In sum, results confirm long-term efficacy of a preschool training stimulating numerical competencies and point to the potential of preventing children at risk to develop severe math problems. In addition, this long-term study demonstrates that a wide use of a

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preschool intervention to promote numerical and calculation precursor skills would be possible and encourages teachers to first identify children at risk for math learning disorders early and second, offer them adequate support to facilitate school entry, school development and protect these children from failure, the development of severe math learning problems, and all related negative experiences. In line, Clements and Sarama (2020) concluded in their recent review on early childhood mathematics intervention that “preschool and primary grade children have the capacity to learn substantial mathematics, but many children lack opportunities to do so. Too many children not only start behind their more advantaged peers, but also begin a negative trajectory in mathematics. Interventions designed to facilitate their mathematical learning during ages 3 to 5 years have a strong positive effect on these children’s lives for many years thereafter” (p. 968).

Developmental Dyscalculia Developmental dyscalculia is defined as a specific learning disorder affecting the development of numerical skills. The World Health Organization (WHO) recognizes dyscalculia as a developmental disorder of school skills in which the numerical problems cannot be explained by a general intellectual disability (WHO 2009). The arithmetic difficulties usually arise from the beginning of learning to calculate and persist over a longer period of time, if untreated even into adulthood (Grond et al. 2014; Schulz et al. 2018; Shalev et al. 2005). In other words, dyscalculia does not simply grow out. Estimated prevalence rates of developmental dyscalculia vary around 6% (Morsanyi et al. 2018; Schulz et al. 2018). This means that one or two affected children sit in each classroom. Recent findings point to girls who seem to be slightly more affected by developmental dyscalculia than boys (Schulz et al. 2018). Accordingly, developmental dyscalculia is a common learning disorder, which can have great impact on people’s lives. Children with developmental dyscalculia can often be recognized by the fact that they have trouble estimating or comparing quantities, show counting difficulties, especially when counting backwards, use their fingers to help with arithmetic even after years, cannot automatically call up results from their memory, do not understand arithmetic operations or the system of decimal places, they decipher mathematical word problems incorrectly, have no understanding of time, lengths, weights and money, and sometimes have difficulty drawing figures, recognizing symmetry or mentally rotating objects. Generally, mathematical content is learned with great difficulty and is often forgotten the next day. In addition, developmental dyscalculia is often associated with additional disorders. The two most common are reading and spelling disorder and attention deficit disorder. The numbers of how often these disorders occur together with dyscalculia fluctuate between 22 and 40%, indicating that developmental dyscalculia can occur together with these deficits (Landerl and Moll 2010; Rubinsten 2009). It is not uncommon for affected children to develop psychological problems because of

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developmental dyscalculia. Many dyscalculic children show externalizing symptoms, such as aggressive behavior or social behavior disorders. At least as often, children with developmental dyscalculia also develop disorders from the internalizing spectrum with specific phobias (math anxiety, test anxiety), fears that extend to everything school-related, avoidance of school and finally depressive symptoms with unhappiness, self-devaluation, loss of drive, and social withdrawal. Dyscalculia is a complex learning disorder, the causes of which are not yet clearly understood. Today it is assumed that it is a combination of several factors that causes dyscalculia. These include genetic, environmental, and neurobiological factors. For example, if one sibling has developmental dyscalculia, the risk is increased by 5 to 10 times that the other sibling will also develop arithmetic problems (Shalev et al. 2001). We also know today that developmental dyscalculia is associated with alterations in brain morphometry or brain activation (for review please see e.g., Kucian 2016). Deficits in brain function, brain structure, functional and structural connectivity between different brain areas, or metabolism have been reported across almost the entire numerical brain network. The heterogeneous character of developmental dyscalculia possibly explains these reported differences of affected brain regions. Regarding the variety of abnormalities in neural networks for numerical cognition in dyscalculia, we cannot yet draw a clear picture. However, there is converging evidence that the activation pattern of children with developmental dyscalculia is less precise and functional and structural deficits are apparent in key regions for number processing, which mainly comprise parietal areas. However, other cortical and subcortical regions that contribute to number processing can also be affected. Moreover, stronger recruitment of supporting areas associated with general cognitive skills such as working memory, attention, monitoring, updating, or finger representation are supposed to reflect compensatory mechanisms in dyscalculic children. Next to genetic or neuronal underpinnings, also environmental factors of the child play a central role. Aspects such as the child’s self-concept, a positive relationship with parents, peers and teachers, and family security can have an impact on numeracy. Although these are not directly related to dyscalculia, they contribute to well-being and help to compensate for failures in mathematics.

Behavioral Interventions to Foster Mathematical Learning Behavioral Interventions for Math Learning In general, behavioral interventions for math learning address school-aged children with problems in numerical understanding. Accordingly, this includes children with diagnosed developmental dyscalculia, but also children who do not fulfill the diagnostic criteria but have math learning problems. Usually these children belong to the lowest 25% in numerical performance. So far, there have been only few standardized intervention programs for math learning that have been scientifically evaluated with regard to their effectiveness. This is also due to the fact that mathematical learning disorders, on the one hand,

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have very different manifestations and accordingly require flexible and individualized therapy, and, on the other hand, longitudinal intervention studies are very complex and demanding. Children who fail to learn arithmetic have multiple risks: The chronic experience of failure can develop anxiety and depressive symptoms, which in the long term often become a bigger problem than the actual school learning handicap. Pupils with special educational needs receive a much higher degree of negative feedback from their teachers, which also makes them less attractive in the eyes of their peers and therefore easier to become social outsiders (C. Huber 2009). Integrative learning therapies that combine specific support with basic psychotherapeutic elements are particularly indicated for such secondary socio-emotional disorders (Bender et al. 2017). The following recommendations for the intervention and therapy of mathematical learning disorder are based on the S3 guidelines for the diagnosis and treatment of mathematical learning disorders (AWMF 2018). The S3 guidelines derive from a detailed systematic literature review. Obtained data from the retrieved studies were evaluated in a meta-analysis, and corresponding recommendations on the diagnosis and treatment of math learning disorder were jointly issued by the 20 societies and associations that participated in the creation of this guideline (Haberstroh and Schulte-Körne 2019). Importantly, the basis for successful support and therapy is always a detailed diagnosis of the problem situation. The treatment should primarily address the problems in number processing and arithmetic identified in the diagnosis. If, in addition to the specific numerical difficulties, other secondary problems (e.g., math anxiety) or comorbid disorders (e.g., AD(H)D) occur which also have a negative impact on numerical abilities, these must be taken into account in the therapy. This means that the interventions must be adapted to the individual profile of strengths and weaknesses in the domain-specific and domain-related functional areas and, if necessary include medical-psychiatric and psychotherapeutic methods. As outlined in the previous section on early prevention of mathematical learning disorders, specific support should begin as early as possible. However, mathematical learning disorders are mostly recognized in elementary school. Regular lessons are usually too complex for children with arithmetic weaknesses or arithmetic disorders and convey learning content for which children with learning disabilities have not yet established the relevant basic knowledge or have not automated it sufficiently (Landerl et al. 2017). For this reason, children with mathematical disorder often first have to develop those basic numerical skills that were normally acquired before school entry and without explicit support. Basically, it is important to be sensitive to a possible mathematical learning disorder in order to recognize and treat it as early as possible. Intervention should be symptom-specific and, accordingly, in the basic numerical and arithmetical area. Evidence-based, disorder-specific support programs are oriented towards numerical and arithmetic content, should be carried out in units that are clearly structured in terms of time and topic, and adaptively adjust to individual learning progress. Interventions and support formats have proven to be more

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effective in an individual setting than in a group setting. The recommended duration of a session is 45 min, whereby this includes, in addition to the specific support, also addressing current experiences in school and outside of school as well as the emotional state. Successful therapy requires interdisciplinary cooperation and professional exchange between the therapist, teacher, school management, psychologist and doctor, parents and the child concerned. It is recommended to use uniformly designed mathematical-didactic illustration materials that correspond to those used in school (Landerl et al. 2017). In addition, the learning materials should generally be designed in a simple way and distracting elements should be avoided. Intervention should be appropriately frequented, should consider the child’s everyday experience, and should take place under reward conditions. In order to assess whether the applied therapy is successful, follow-up checks are recommended at least once a year. Appropriate specialist knowledge is required from the treating therapist. These should qualify through specific courses, training, and further education with a focus on school developmental disorders, in particular arithmetic disorders, and integrative learning therapy. In addition to traditional personal support and therapy, computer-aided intervention programs are increasingly being used today. Overall, computer-aided support programs often help motivate the children or enable the support units to be planned independently of time and location, which is an attractive alternative or supplement, especially for people who are involved in school and work. Computer programs can support a therapy in a meaningful way, especially for repeated practice to automate numerical fact knowledge (Heine et al. 2012). Computer programs that, thanks to the methods of artificial intelligence, are able to continuously adapt to the level of difficulty of the exercises and the individually required skill components are particularly recommended (Räsänen et al. 2015). The computer offers direct and positive feedback along with one’s own learning progress, and the practice situation is shielded from negative social cross-comparisons. Concerning the effectiveness of computer-assisted intervention, the reported findings are contradictory, either less effective than instruction given by teachers (Kroesbergen and Van Luit 2003; Malofeeva 2005) or more effective (Xin et al. 2005). A more recent meta-analysis came to the result that computer-based support programs are just as effective as support provided by a therapist (Chodura et al. 2015). Overall, such evidence-based computer-supported intervention programs are primarily to be viewed as supplementary assistance that can be used in school, at home, and in therapy, but are not intended to replace the teacher or the therapist (see meta-analyses of Ise et al. 2012; Kroesbergen and Van Luit 2003). With regard to the duration of the intervention, shorter interventions over a longer period show more positive effects than longer-lasting intervention units. The end of a therapy cannot be determined in advance. The therapy should only be ended when it is no longer necessary for the specific life situation of the child or has proven to be unsuitable. Ideally, a successful intervention enables the person to cope independently and successfully with the requirements in school, professional practice, as well as in everyday life. If this is the case, the therapy can be ended.

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Fig. 2 Intervention and therapy. Figure summarizes important points that should be considered in the intervention and therapy of math learning disorders. The recommendations are based on the S3 guidelines for the diagnosis and treatment of mathematical learning disorders (AWMF 2018) and the figure is adapted from (von Aster et al. in press)

Ultimately, for every treatment, the children and adolescents affected and their parents must be carefully informed about the examination findings and the further procedure derived from them. Accordingly, regular in-process consultations with parents and teachers are also required. Figure 2 gives an overview of important points that should be considered during intervention of math learning disorders in terms of start, the course of the intervention, and the end.

Effects of Numerical Interventions on the Brain Converging evidence is growing that math learning disorder is associated with different alterations in brain function and brain structure. Recent work in the field of developmental dyscalculia has examined the neural aspects of this math learning disorder by means of contemporary brain imaging techniques. Number processing and calculation is a demanding cognitive ability that is processed by a complex neuronal network that includes different and widespread brain areas. In addition to the key areas for numerical cognition located in the parietal lobes, the prefrontal cortices and regions associated with the dorsal and

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ventral visual pathways as well as subcortical areas and the cerebellum all play a significant role in numerical cognition (for review see Arsalidou and Taylor 2011; Kucian 2016). The evidence to date provides support that in children with math learning disorder, this neuronal network is disturbed on different levels (for review see e.g., Kucian 2016). Deficits in brain function, brain structure, functional and structural connectivity between different brain areas, or metabolism have been reported across almost the entire numerical brain network. Regarding this wide range of abnormalities in neural networks for number processing and calculation in math learning disorder, it is difficult to draw a clear picture. Differences between findings of affected brain regions are possibly explained by the heterogeneous character of math learning difficulties. However, based on the existing scientific evidence, it can be concluded that the activation pattern of children with mathematical problems is less precise, and key functional and structural deficits are apparent in core regions for number processing, which mainly comprise parietal areas. Typically, a reduced activation in the parietal cortex including the intraparietal sulcus, reduced grey matter, as well as diminished functional as well as structural connection of these core areas with other cortical and subcortical brain regions have been reported. In this respect, it is important to highlight that in the case of children with math learning disorder, other cortical and subcortical regions that contribute to numerical cognition can also be affected. In addition, a stronger recruitment of brain areas that support numerical cognition such as working memory, attention, monitoring, updating, or finger representation has been often observed in children with math learning disorder. This is supposed to reflect compensatory mechanisms in affected children, but could also reflect deficits in these domain-general skills that might contribute to the development of math difficulties. Taken together, there is convincing evidence that math learning disorders are associated with neuronal alterations (for review see e.g., Kucian 2016). Luckily, our brain is a plastic organ and constantly adapts to use and stimulation, although this may depend on different factors such as the developmental stage and environment. Hence, it can be assumed that specific and successful intervention, when carried out under recommended pedagogical conditions as summarized in the previous section, should have the capacity to normalize brain alterations or induce compensatory neuronal mechanisms. However, there is very little knowledge available since only a hand full of studies investigated neuronal changes based on numerical intervention. Moreover, it may be that those with neurodevelopmental disorder may have a reduced capacity for neuroplasticity at some stage or brain region (Krause and Cohen Kadosh 2013). The first study, which evaluated the effects of a computer-based intervention that focused on strengthening spatial number representations through a number line training on behavioral as well as neuronal level, reported general learning benefits (Kucian et al. 2011). Children with and without developmental dyscalculia were instructed to train at home 15 min a day, 5 days a week for 5 weeks with the computer game. The game consisted of a number line training with different difficulty levels and functioned on an individual adaptive way. Either an Arabic digit, an addition problem, a subtraction problem, or a number of dots appeared on a

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spaceship, which was about to land on a new planet. On the planet, a number line from 0 to 100 was displayed and the challenge was to land the spaceship at the corresponding location on the number line. The child had to steer the spaceship to the correct position on the number line, using a joystick. Results showed that children with and without dyscalculia enhanced their number representations. In addition, a positive transfer effect was demonstrated by improved arithmetic abilities, although the children were never trained in exact arithmetic. In addition to behavioral improvements, the intervention was followed by a modulation of brain activation. Functional magnetic resonance imaging (fMRI) depicted a decrease in supporting brain activation mainly in the frontal lobes after the training. Therefore, it seems that after completion of the training, the fMRI task (judgment if three presented Arabic digits are in order or not) puts less demand on quantity processing and working memory and requires less attentional effort. Hence, these findings seem to support a compensatory model of brain plasticity after intervention in children with dyscalculia. A follow-up examination 5 weeks after training revealed a significant increase of brain activity in parietal areas for the children with math learning disorder, including the intraparietal sulcus, bilaterally. Since the intraparietal sulcus is known to play a pivotal role in numerical cognition, the authors speculate that time for consolidation after training was needed to establish neuronal representation, which tends into the direction of normalization of altered brain activation. In addition, a more recent study looked into details of the identical intervention on functional connectivity in the same cohort (Michels et al. 2018). Before the training, children with math learning disorder showed hyperconnectivity of the intraparietal sulcus to different brain regions of the numerical network, including the parietal, frontal, visual, and temporal areas. This initial hyperconnectivity is in line with literature and appears to constitute a neuronal characteristic of impaired numerical cognition (Jolles et al. 2016; Rosenberg-Lee et al. 2014). After 5 weeks of numerical intervention, the abnormally high functional connectivity in dyscalculic children vanished and could be normalized on the neuronal level. As functional connectivity in children with dyscalculia was indistinguishable to control children’s connectivity after intervention, the authors concluded that training lead to a re-organization of inter-regional task engagement. The second study, which investigated the behavioral and neural effects of intervention in a group of children with math learning disabilities, used 8 weeks of 1:1 cognitive tutoring (Iuculano et al. 2015). The tutoring combined conceptual instruction with speeded retrieval of arithmetical facts. The intervention led to a normalization of numerical performance and in parallel elicited functional brain changes in children with mathematical difficulties by normalizing their aberrant functional responses to the level of typically developing peers in a distributed network of parietal, prefrontal, and ventral temporal–occipital areas that support successful numerical problem solving. Moreover, machine-learning algorithms show that brain activity patterns in children with math learning disorder were no more discriminable from typically developing peers after 8 weeks of 1:1 tutoring. In the identical sample, Supekar et al. (2015) additionally demonstrated a remediation of (childhood) math anxiety and associated neuronal circuits through cognitive tutoring.

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The outlined existing studies that examined the effects of a specific numerical intervention on brain function in children with math learning problems corroborate that the brain of affected children is able to adapt successfully by neuroplastic changes. However, it would be interesting to see which children profit from a certain intervention and react accordingly with neuronal normalization or with compensatory effects and which children do not benefit. Furthermore, it would be valuable to identify any predicting features of intervention success and neuronal changes. Finally, we gained so far just first insights of possible modulations of brain activation networks by therapy. However, we do not know yet if intervention has also the potential to normalize altered brain structure, such as grey matter volume or white matter fiber connections. This might be assumed as first findings in the field of reading have demonstrated that after 8 weeks of intensive reading intervention various white matter fiber connections in the brain of poor readers changed in concert with growth of their reading skills (E. Huber et al. 2018). Moreover, the authors identified fiber tracts whose properties predicted reading ability but remained fixed throughout the intervention, suggesting that some anatomical properties predict the ease with which a child learns to read, while others dynamically reflect the effects of experience.

Brain Stimulation to Foster Mathematical Learning One of the recent approaches to facilitate cognitive learning is using noninvasive brain stimulation (NIBS). While there are different types of NIBS, like transcranial magnetic stimulation, we will focus in this chapter on transcranial electrical stimulation (tES), due to its reduced sensation, which makes it optimal for combination with cognitive training, and its frequent use in studies focusing on numerical and arithmetic learning (C. Y. Looi and Cohen Kadosh 2016; A. Sarkar and Cohen Kadosh 2016; Schroeder et al. 2017; Simonsmeier et al. 2018). Moreover, tES studies have shown promising effect as reflected by a systematic review and metaanalysis and we refer the interested reader to this study (Simonsmeier et al. 2018). In tES, one or more electrodes are attached to the scalp, with the intention to modulate the activity in one or more brain regions, by transmitting a low current intensity (usually 1–2 mA) that does not directly lead to an action potential but can increase its likelihood (Reed and Cohen Kadosh 2018). tES includes several types of protocols. The most known ones are transcranial direct current stimulation (tDCS), transcranial alternating current stimulation (tACS), and transcranial random noise stimulation (tRNS) (Polania et al. 2018; Reed and Cohen Kadosh 2018; Santarnecchi et al. 2015). While describing the exact working mechanism of each tES type is beyond the scope of this chapter, the most notable differences are that tDCS applies a direct current to scalp, while tACS and tRNS use alternating current that can be at a specific frequency (tACS) or at random frequencies (tRNS). Currently most of the studies with tES were run on healthy adults. This was done in order to examine its suitability before proceeding to more vulnerable populations, such as children with atypical development. Studies in adults have shown that tDCS

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can facilitate numerical learning. In one study, the speed of learning of new seen (artificial) digits could be facilitated or impaired depending on the regions stimulated in the parietal and prefrontal cortices, as well as improving the automatic processing of the new learned digits, and the link between their magnitude and space (Cohen Kadosh et al. 2010; Iuculano and Cohen Kadosh 2013). Few more studies used tDCS in order to impact activity in the parietal cortex, which have shown improved behavioral performance in a numerical or arithmetic task or alteration in brain activity without behavioral improvement (Clemens et al. 2013; Grabner et al. 2015; Hauser et al. 2013; Klein et al. 2013; Chung Yen Looi et al. 2016; Mosbacher et al. 2020). Another tDCS study tried to target the prefrontal cortex in order to improve the performance of those with math anxiety, as well as altering their physiological response during this stressful scenario (Amar Sarkar et al. 2014). In this study, tDCS was used with the intention to alter the activity imbalance between the left and right dorsal prefrontal cortices (dlPFC) that is associated with those with greater anxiety (Brunoni et al. 2013; Ironside et al. 2016; Shackman et al. 2009). Compared to sham (placebo) tDCS, tDCS improved arithmetic performance and reduced cortisol level in those with high math anxiety. In contrast, tDCS impaired the performance in those with low math anxiety. Moreover, in this group, tDCS also abolished the natural reduction in cortisol level seen under sham tDCS. While it is likely that there are simpler ways to reduce math anxiety, these results highlight a neglected, but critical point; a fixed-stimulation protocol might be beneficial for some but ineffective or impairing for others. Several studies have used tRNS in order to examine whether it can maximize the benefit from arithmetic or numerical training. First it was found that tRNS over the dlPFC, but not over the parietal cortex, can facilitate arithmetic learning, modifying brain activity in the stimulated region, and have a beneficial effect compared to sham tRNS after 6 months (Snowball et al. 2013). A later study has found similar results with tRNS over the parietal cortex during numerosity training in comparison to sham tRNS or tRNS over a region that is assumed not critical for numerosity discrimination (the primary motor cortex). Here tRNS facilitated the learning to discriminate between numerosities with a beneficial effect that sustained up to 4 months (Cappelletti et al. 2013). Notably, tRNS alone had no effect on training, highlighting that tRNS per se is ineffective if it is not coupled with cognitive training/task. This idea fits with the view that tRNS induced small changes at the neuronal level that are not strong enough without the combination of cognitive training. Later studies have shown that the tRNS effect on arithmetic learning depends on task difficulty, with greater improvement on a more demanding task (Popescu et al. 2016). These findings have led to a proof-of-concept study in 12 children with math learning disorder. In this study that was ran in a school for specific learning difficulties, the children received cognitive training and sham or real tRNS twice a week for 4.5 weeks (Chung Yen Looi et al. 2017). Inspired by previous cognitive training studies that have coupled symbolic understanding with spatial cognition (Kucian et al. 2011), as well as embodiment (Link et al. 2013), the children were asked to map a given number on a physical line in a gamified ping-pong game. The

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range of the number needed to be mapped and the bat’s size played with increased and decreased, respectively, as performance improved. It was found that tRNS facilitated learning, as highlighted by a steeper learning slope, similar to what was found in previous tRNS studies in adults mentioned above. Moreover, the effect of tRNS interacted with the performance in the cognitive training to predict improvement in a standardized maths test, highlighting a successful transfer from cognitive training to improvement in standardized maths test scores for those who received tRNS (Chung Yen Looi et al. 2017). A recent tRNS study in juvenile mice used the same tRNS protocol in Looi et al. (2017), albeit without cognitive training (Sanchez-Leon et al. 2021). This study found that tRNS reduced GAD 65/67, the precursor of GABA, an inhibitory neurotransmitter that has been recently associated with mathematical education in adolescents (Zacharopoulos et al. 2021). GABA has been highlighted as in important neurotransmitter for neuroplasticity and in the modulation of sensitive periods in human development (Werker and Hensch 2015). The tRNS study by Sanchez-Leon et al. (2021) suggests that the beneficial effect found in Looi et al. (2017) maybe due to increased cortical excitation, and modulation of neuroplasticity, by removing breaks for neuroplasticity that can occur in this population, or when intervention is applied at less optimal time-window in terms of the sensitive period. At the cognitive level, another study in healthy young adults, which was based on a previous tRNS that showed improvement in sustained attention (Harty and Cohen Kadosh 2019), has suggested that the effect of tRNS over the dlPFC could be explained by improvement of neural markers that are associated with sustained attention (Sheffield et al. 2020). If correct, this finding suggests that tRNS would be most optimal when a similar protocol be applied in those who have comorbidity between attentional problems and math learning. In support of this, another study found that the same tRNS protocol as in Looi et al. (2017) improved clinical symptoms in children with ADHD when coupled with executive functions training (Berger et al. 2019). However, despite the encouraging results, we would like to highlight several challenges that prevent the translation of tES to approved treatment at this stage. First, the extrapolation from typical to atypical brain can be misleading, as was highlighted by a single-case study that has found a dissociation between effects in adults with and without math learning disorder (Iuculano and Cohen Kadosh 2014). Moreover, the differences in brain organization between children and adults are vast. Therefore, more work is needed in order to develop more effective and safer tES protocols in minors (Cohen Kadosh et al. 2012; Davis 2014; Krause and Cohen Kadosh 2013; Santos et al. 2021). Second, most of the studies in the field have a modest sample size. While the sample size has been increased in the recent years in young adults, this is not the case in atypical developing children, and it is still an open question whether the modest sample size in early phase studies has led to a type I or II error (Button et al. 2013). In this case, multiple-center studies would allow easier recruitment, and results are likely to be generalized beyond a single-site. Third, the present approach in tES is dominated by a one-size-fits-all approach. That is, a single tES protocol is applied to all the individuals in the study, without

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taking into account their uniqueness in terms of cognitive ability, brain functions, gender, symptoms, etc. We have recently suggested a sophisticated approach to overcome this challenge (Lipka et al. 2021; van Bueren et al. 2021), and similar or complimentary directions are needed in order to push this promising field forward.

Prospects An important question is how the individual prospects of such interventions are? We know nowadays that numerical learning disorders persist into adulthood without appropriate intervention (McCaskey et al. 2017; Schulz et al. 2018; Shalev et al. 2005). In other words, it is a persistent learning disorder that does not simply grow out by itself. On the contrary, the probability that a child with a numeracy disorder will suffer from this learning deficit into adulthood is very high. In addition, there are repeated negative experiences in dealing with numbers and arithmetic, which often leads to an avoidance attitude towards everything that has to do with mathematics. In turn, this favors the development of secondary symptoms and supports the downward spiral. For this reason, it is extremely important to interrupt this ever-increasing deterioration through targeted intervention. Although the available data on intervention studies are promising and report significant improvement, it cannot be assumed that all children respond equally well to a particular support program. However, there are hardly any studies on the prognosis of certain therapy programs. It is assumed that children with less severe arithmetic learning problems respond better to intervention programs and show greater improvements after training compared to children with a pronounced math learning disorder such as developmental dyscalculia (please see meta-analysis by Chodura et al. 2015). This finding is in line with the Matthew effect in the field of education (Stanovich 1986). It is possible that children with a milder form of numeracy problems may have fewer disorder-specific symptoms and are therefore more likely to respond to intervention programs. In comparison, children with severe math learning disorder often show additional comorbid problems, which can also hinder learning success. Moreover, the heterogeneity of the symptoms in math learning disorders suggests that different intervention programs will be effective for the individual profiles of problems. Accordingly, children with severe arithmetic problems and, as a result, poorer initial arithmetic skills seem to benefit less from standardized support programs and need more intensive and more individualized support in order to achieve comparable successes as children with less severe problems. First studies address the important question of differential effectiveness, that is, which child benefits more or less from a specific numerical intervention. Powell et al. (2017) found that initially higher individual performance level in arithmetic, working memory, language comprehension, and attentional performance predicted better intervention success in second grade pupils with arithmetic problems. In contrast, Clark and colleagues (2019) reported that kindergarten children with initially poorer numerical performance benefited more from a specific intervention. Another study came to the conclusion that the numerical skill before the intervention is not an

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indicator of the individual success of an intervention program (Fuchs et al. 2019). Overall, it is still unclear which factors best predict individual learning success through intervention. In accordance with the S3 guideline for the diagnosis and treatment of mathematical learning disorders (AWMF 2018), it is recommended that children with problems in numeracy be supported as early as possible through specific intervention. The meta-analysis by Chodura et al. (2015) is also optimistic for older children; however, they found positive effects due to intervention programs for children with numeracy difficulties between the first and fifth grade regardless of age. This means that children who receive help relatively late can also improve their numerical and mathematical competencies. However, it seems that intervention effects are smaller for students in secondary school compared to primary school (see meta-analysis by Stevens et al. 2018). Accordingly, intervention for secondary school students with numeracy problems seems to be more demanding. Problems in numeracy and mathematics may be more complex and more firmly rooted in adolescents and adults and therefore more difficult to treat. Furthermore, the assumption is plausible that neural plasticity (and thus also the learning potential) is smaller with increasing age (Kolb and Gibb 2011). Future research can highlight whether combining the intervention with other approaches, such as NIBS as highlighted in this chapter, can improve the outcomes from the intervention.

Conclusion The development of higher cognitive skills by learning is a highly complex development which is easily disruptive both in preschool and in school periods and can lead to a delay, weakening, or even a lack of corresponding neurocognitive maturation processes, which then leads to failure at school and often secondary emotional disorders, especially specific anxieties. The quality of pedagogy and didactics is to recognize heterogeneous learning requirements at an early stage and provide adequate support and stimulation if needed. In terms of numerical cognition, early numerical skills have the potential to make predictions about later mathematical skills and allow the identification of children at risk for mathematical learning disorders already in preschool. Particularly, numerical processes that include numerical symbols (Arabic digits) have been emphasized to be important indicators for the development of future mathematical abilities in school. Importantly, specific support and therapy can contribute significantly to improving performance, reducing anxiety, and neuronal maturation of numeracy. Here it can be said the earlier the better. Studies have shown that an early support of the development of these basic numerical abilities in preschool has the capacity to lift these children’s numerical skills to peer level and therefore facilitates school entry and further numerical learning in school. Moreover, such an early support was even able to prevent the development of dyscalculia. Developmental dyscalculia is a common and specific learning disorder of numerical skills that persist into adulthood if untreated. A combination of genetic, neurobiological, and environmental factors have been discussed as possible

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reasons of this learning disorder. Affected children are usually detected and diagnosed in primary school years. Accordingly, most available interventions target children in this age range. Some recommendations were given which make an intervention particularly effective for children with developmental dyscalculia or generally poor numerical abilities. Brain imaging techniques corroborated the effectiveness of some interventions by unveiling neuro-plastic changes of numerical brain networks that went along with the behavioral improvements. In that respect, new approaches target directly the brain by noninvasive stimulation methods to enhance numerical learning. In conclusion, poor numeracy and developmental dyscalculia are common problems in our society and imply a handicap in school, professional, or daily lives, which could be reduced by adequate support of affected people.

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Section IV Biological Approaches to Mathematics Dan Vilenchik

Abstract

Human beings may be lured by romantic and sentimental ideas about why and how they function in the world. “Spirit,” “soul,” “higher-self,” “lower-self,” “ego,” etc., are all linguistic concepts that are a product of human thought, and are too easily taken as solid facts (we have two eyes, that’s a fact). However, if one observes carefully, one is left with the only verifiable fact that the process of thought, the accumulation of knowledge, remembrance, and learning are all rooted in the biological organ that we call the brain, which is part of our biological body. Hence if one is to understand any process that involves human learning, or human thought, the path must begin in our biology. In this section we highlight different aspects of our biological “hardware” and its intimate relation with mathematical capabilities and the process of learning mathematics. Keywords

Geometry · Vision · Numerical capabilities · ADHD

Introduction The human brain is a very ancient one. Although one may naively think that a baby’s brain is tabula-rasa, this is in fact very far from the truth. Our brain dates back billions of years ago, to the early forms of cognizant life forms. Many of the mechanisms that we encounter every day, waiting in line at the post office, or having a road rage at an old lady driving 10 mph below the allowed speed, are rooted in primordial animalistic instincts. We also naively think that the society in which we live has nothing to do with our biology. Expressions like “nurture vs. nature” contribute to that confusion, as they suggest that the two things are separate and independent of each other. Well, if we think for a moment, what is nurture? what is society? Society is formed by human beings. Society and culture are a direct product of human thought and design, both of which exist due to biological faculties of the

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brain. In this section we will discover how the culture we live in, which is reflected from and reflects on the neurobiology of the brain, effects the way we compute risks and make decisions. In this section we will learn that human beings are born with innate mathematical capabilities that are shared with other animals as well, and not only with primates but even with birds. These capabilities need not be learned but are rather hard coded into our brain (the same way hunger or pain need not be learned). These capabilities may be further developed in a process of learning mathematics, which of course takes place within a culture (the education system). This same culture, a product of human thought, taking place in the physicality of our biological brain, also produces labels such as ADHD. We will discover how neurological conditions of the brain, such as ADHD, affect our capabilities of learning mathematics. What brain activity is required to allow mathematical learning (e.g., memory) and what happens when such faculties are limited to some degree. We will be surprised to discover (spoiler alert!) that unlike the popular belief that ADHD is responsible for low achievement in mathematics, it is not ADHD but rather certain faculties of the brain that account (and even that only partially) for our mathematical abilities. Finally, we will learn how our biological sensory apparatuses, in particular vision, are designed to allow geometrical perception, and how they can be trained (but also fooled) to allow abstraction and formalism that goes beyond our Euclidean reality. We will see how the combination of vision, touch, and language play together to achieve the level of mathematical knowledge that we accumulated thus far.

The Neurobiological Basis of Numerical Cognition: Decision-Making Processes as a New Line of Inquiry

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Lital Daches Cohen and Orly Rubinsten

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Domain-Specific Mechanisms Underlying Numerical Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Processing Skills and Math Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Processing Skills and Mathematical Learning Disabilities . . . . . . . . . . . . . . . . . . . . . Domain-General Mechanisms Underlying Numerical Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working Memory and Numerical Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visual Form Perception and Numerical Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inhibition and Numerical Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emotion and Numerical Cognition: Math Anxiety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decision-Making and Numerical Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

416 417 418 419 420 420 421 422 422 423 424 425 426

Abstract

Numerical cognition plays an important role in our daily life and affects personal and economic success, but there are significant individual differences in the development of math abilities. The accumulative literature indicates that domain-specific and domain-general skills, as well as emotional processes, are the mechanisms underlying numerical cognition and its developmental pathways. In an attempt to bridge the gap between psychology, neuroscience, and education, the chapter reviews the main sources of typical and atypical developments of numerical cognition. There is no consensus on these mechanisms or their relations to math skills. Against this background, the chapter proposes a relatively new line of research to investigate the neurobiological basis of numerical cognition outside the lab: decision-making processes. Decision biases that rely on L. D. Cohen · O. Rubinsten (*) Edmond J. Safra Brain Research Center for the Study of Learning Disabilities, Department of Learning Disabilities, University of Haifa, Haifa, Israel e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_34

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systematic manipulation of fundamental aspects of number processing can shed light on the genetic, neurobiological, and cognitive markers of numerical processes. The chapter presents comprehensive information on how culturally acquired capacities are learned and organized in the human brain and how these abilities may be impaired. Keywords

Numerical cognition · Mathematical learning disabilities · Numerical processing skills · Working memory · Visual form perception · Inhibition · Decision-making

Introduction Numerical cognition plays an important role in everyday life and is a key factor in personal and economic success (e.g., Gravemeijer et al., 2017). Numerical skills are not limited to humans; the ability to process numerical information is crucial for survival across species. However, not everyone has the same level of ability. The cumulative literature demonstrates large individual differences in the development of math abilities, with about 6% of the population suffering from math learning impairments of developmental origin, commonly called mathematical learning disability (MLD) or developmental dyscalculia (Rubinsten & Henik, 2009). MLD is defined as substantial underachievement in mathematics relative to the level expected according to age, intelligence, and educational opportunities (American Psychiatric Association, 2013). The symptoms of MLD are heterogenic (Kaufmann et al., 2013; Rubinsten & Henik, 2009) and include weaknesses in diverse math abilities (American Psychiatric Association, 2013). Over the past few decades, cognitive and neural research has advanced considerably in characterizing the mechanisms underlying numerical processes and their developmental trajectories (Matejko & Ansari, 2018). Explanations of individual differences in numerical cognition include difficulties with fundamental aspects of numerical processing (i.e., domain-specific cognition (Schneider et al., 2017)) and other domain-general processes (Kaufmann et al., 2013), such as working memory (Bugden & Ansari, 2016), visual perception (Zhou et al., 2015), and inhibition (Szűcs et al., 2013). Another branch of the research has emphasized the importance of attitudes toward and feelings about math (Rubinsten et al., 2018, 2019). Rubinsten (2015) suggests that by examining the implicit tools that entail the systematic manipulation of fundamental aspects of number processing, researchers can expand the neurocognitive understanding of typical and atypical numerical development. For example, the mechanisms underlying numerical cognition have been found to play an important role in risk perception and decision-making (see Schiebener & Brand, 2015 for a review). Yet the research on how numerical processing actually shapes decisions is limited (Patalano et al., 2015). Against this background and in accordance with the multidisciplinary emerging field of mind, brain, and education, this chapter aimed to bridge the gap between

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Fig. 1 Decision-making processes as a tool to measure numerical cognition outside the lab

psychology, neuroscience, and education on their respective understandings of the development of numerical cognition. As our review indicates, there is no consensus on these mechanisms or their relations to math skills. Against this background, a relatively new line of research to investigate the neurobiological basis of numerical cognition outside the lab is proposed: decision-making processes. Decision biases relying on systematic manipulation of fundamental aspects of number processing may help us identify reliable genetic, neurobiological, and cognitive markers of numerical processes (see Fig. 1).

Domain-Specific Mechanisms Underlying Numerical Cognition Two domain-specific mechanisms are commonly used in adult number processing: (1) the phylogenetically determined approximate number system and (2) the culturally acquired exact number system (Castronovo & Göbel, 2012; for instance, see Fig. 2). The approximate number system (ANS), originally called the “number sense” (Dehaene, 2011), represents non-symbolic quantities or magnitudes in an approximate manner (e.g., the quantity of dots in a dots’ array; for a review, see Leibovich et al., 2017). Research has long suggested that humans, animals, and preverbal infants share these approximate numerosities analogically represented in a compressed language-independent internal continuum – the mental number line (Dehaene, 2011). The exact number system (ENS) develops later in life and represents abstract language-based symbolic representations, including both Arabic digits (e.g., 1, 5, and 23) and numerals (e.g., “one,” “five,” “twenty-three”; Castronovo & Göbel, 2012).

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Fig. 2 Two domain-specific mechanisms in adult number processing

Converging evidence supports the notion that the processing of symbolic and non-symbolic representations relies on both common and distinct brain regions (Matejko & Ansari, 2018). These results imply distinct effects of symbolic numerals (i.e., ENS) and non-symbolic numerosity representations (i.e., ANS) on different math abilities (e.g., Schneider et al., 2017). Here, the focus is on the question of whether the processing of non-symbolic (i.e., array of dots) and/or symbolic magnitude representations (i.e., digits and numerals such as “1,” “23,” and “four”) is relevant for higher-level mathematical competence.

Numerical Processing Skills and Math Performance Some evidence suggests that exact symbolic representations are mapped onto the ANS, indicating the ability to perceive and evaluate noncountable representations is the cognitive foundation for general math skills (Dehaene, 2011). Indeed, non-symbolic processing skills have been found to predict early arithmetic capacities, including counting and mental calculations (Schneider et al., 2017), and higher mathematical reasoning (Amalric & Dehaene, 2016). For example, high-level mathematical thinking and basic number sense have been found to make minimal use of brain areas involved in the processing of language; instead, they recruit circuits initially involved in space and number (Amalric & Dehaene, 2016). In this vein, Albert Einstein stated: “Words and language, whether written or spoken, do not seem to play any part in my thought processes” (Hadamard, 1945). However, other studies have not found a link between non-symbolic numerical abilities and math skills (for a review, see De Smedt et al., 2013) and suggest that symbolic processing skills have a more prominent role in math performance (Matejko & Ansari, 2016; Schneider et al., 2017). The mixed results may reflect the respective studies’ use of different combinations of covarying physical and continuous non-numerical features (e.g., size, luminance, convex hull, and density; Clayton et al., 2015; for example, see Fig. 3). Recent studies indicated that continuous visual properties, such as spatial and temporal aspects, can better explain behavioral (e.g., Zhou et al., 2020) and neurophysiological data (for a review, see Gebuis et al., 2016) in non-symbolic comparison tasks. Previous findings indicated an overlapping parietal mechanism for processing and representing various dimensions of countable and noncountable

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Fig. 3 Example of a dot array presentation with different combinations of numerical (i.e., quantity and ratio) and covarying physical and continuous non-numerical features (i.e., convex hull, diameter, surface, density, and circumference)

magnitudes (Skagerlund et al., 2016), but a recent study (Cheyette & Piantadosi, 2020) pointed to a unified representational system for both small and large numerosities, thus contradicting the classical distinction between precise smallnumber and imprecise large-number representations (Dehaene, 2011). These results are in line with the theory of magnitude (Walsh, 2003), according to which quantity, time, and space are part of a general magnitude system that continues to evolve throughout the lifespan and contributes to the development of computation abilities.

Numerical Processing Skills and Mathematical Learning Disabilities A core domain-specific cognitive deficit associated with MLD is an impairment in the processing of numerical magnitudes (Butterworth, 2011; De Smedt et al., 2013). One main hypothesis about the origin of this impairment focuses on the imprecise representation of non-symbolic numerical magnitudes (Butterworth, 2011). The second points to inefficient access to information on magnitudes in symbolic representations (Rousselle & Noël, 2007). The cumulative evidence suggests that MLD can involve impairments in symbolic and/or non-symbolic numerical representations (e.g., Bulthé et al., 2019), but there is an ongoing debate on the core non-symbolic deficiency. For example, a recent study found that preadolescents with MLD had higher numerosity discrimination thresholds than their typically developing peers, but their size thresholds were not significantly different (Anobile et al., 2018). These results support a domain-specific deficient “number sense” in MLD (Butterworth, 2011). In line with the general magnitude processing mechanism theory, other studies have noted impairments in the processing of continuous magnitudes (e.g., Bugden & Ansari, 2016). Despite these findings, the novel assumption of deficient “magnitude

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sense” among individuals with MLD has not been fully investigated (Bulthé et al., 2019). Be that as it may, less automatic access to numerical representations results in more executive load, which, in turn, hampers inhibition of irrelevant information. For example, as mathematical proficiency levels are increased, numerical representations lack a robust spatial connotation, with the most pronounced number-space interaction appearing among young adults with MLD (Cipora et al., 2016). In short, there is no consensus on the cognitive mechanisms underlying numerical difficulties, possibly because of the heterogenic nature of MLD (Kaufmann et al., 2013; Rubinsten & Henik, 2009).

Domain-General Mechanisms Underlying Numerical Cognition More comprehensive accounts explain the heterogeneity in MLD and the contradictory findings by considering domain-general abilities (Kaufmann et al., 2013). Recent studies have revealed a whole network of brain regions involved in typical (Matejko & Ansari, 2018) and atypical numerical processing (Bulthé et al., 2019) including (1) the parietal cortex, responsible for processing numerical and visuospatial information (Rosenberg-Lee et al., 2015); (2) frontoparietal networks, crucial in the development of arithmetic and mathematical abilities (Kucian et al., 2018), and (3) frontal areas of the brain, associated with more efficient and automatized processing (Rosenberg-Lee et al., 2015). These brain regions are recruited during domain-general processes as explained below.

Working Memory and Numerical Cognition Working memory is a limited cognitive system that can temporarily store and manipulate information (Baddeley & Hitch, 1974). It has been found to be a strong predictor of math performance (e.g., Xenidou-Dervou et al., 2018). The widely used multicomponent model of working memory (Baddeley & Hitch, 1974) distinguishes between the central executive working memory system, which is responsible for the manipulation of information, as well as monitoring and control processes, and two “slave” storage systems. These “slave” systems include verbal and visuospatial working memory. While the first is involved in the processing of verbal/speech-based information, the latter deals with spatial and visual information. Behavioral (Bugden & Ansari, 2016) and neuroimaging (Ashkenazi et al., 2013) studies have noted disruptions in the visuospatial working memory of children with MLD, despite intact verbal and central executive working memory abilities. For instance, a positive correlation between activation in number-related and visuospatial working memory frontoparietal areas was found in typically developing children, but not in their peers with MLD, suggesting that MLD is characterized by inappropriate use of working memory resources during arithmetic problem solving

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(Ashkenazi et al., 2013). Based on the evidence, some argue that the parietal brain mechanisms involved in storing and manipulating magnitude representations are impaired in MLD (Bugden & Ansari, 2016). Comparatively little research focuses on the performance of advanced mathematics in adults. Recently, Hubber et al. (2019) found that adult mathematics students demonstrate superior visuospatial working memory capacity and general visuospatial skills compared with humanities students. Yet further research is required to determine whether visuospatial working memory skills support the acquisition of advanced mathematics or whether mathematics training enhances working memory skills. Nevertheless, the important role of visuospatial working memory in typical and atypical numerical cognitions accords with extensive research documenting the positive relations between visuospatial abilities and symbolic and non-symbolic processing skills, including complex mathematics, such as algebra (Landy et al., 2014). The predominant explanation for this link relies on the spatial nature of the mental number line (Dehaene, 2011). However, visuospatial working memory tasks are highly dependent on visual form perception (Zhou et al., 2015).

Visual Form Perception and Numerical Cognition Visual perception is important for non-symbolic and symbolic numerical processing skills (Zhou et al., 2015) and for advanced mathematical problem solving (Marghetis et al., 2016). In one study, after visual form perception was controlled, the correlation between non-symbolic processing skills and arithmetic fluency was no longer significant in typically developing children (Zhang et al., 2019). Children with MLD have demonstrated poorer performance on visual form perception than their typically developing peers, and these variances have explained their differences in non-symbolic processing skills (Zhou & Cheng, 2015). Similarly, in a recent longitudinal study, the predictive role of non-symbolic processing skills in math achievements was explained by visual form perception (Zhou et al., 2020). In another study, high-level algebraic reasoning was accomplished by basic visual processes adapted to represent and evaluate abstract conceptual knowledge (Marghetis et al., 2016). While some argue that mathematical training should avoid and even suppress perceptual strategies (e.g., Kirshner & Awtry, 2004), by and large, the findings suggest that educational approaches to mathematics and related fields should aim to refine students’ reliance on perceptual strategies. However, most studies in this area have included native Chinese (Zhang et al., 2019; Zhou & Cheng, 2015; Zhou et al., 2015, 2020). Cross-linguistic research suggests that visual skills have a more prominent role in Chinese than English (e.g., Bolger et al., 2005). An alternative line of research emphasizes the role of the ability to suppress cognitive or behavioral distractions and unwanted responses (i.e., inhibition) in symbolic (Gilmore et al., 2015) and non-symbolic processing (Leibovich et al., 2017).

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Inhibition and Numerical Cognition Many studies have linked inhibition of numerical information to symbolic math skills, such as procedural skills and conceptual understanding (Gilmore et al., 2015). Similarly, significant differences between typically developing children and their peers with MLD have been demonstrated in numerical inhibition tasks (e.g., Szűcs et al., 2013). Recent findings suggest that young adults who are experts in math are more efficient than non-experts in inhibiting distracting numerical information (Lubin et al., 2016). It has also been suggested that non-symbolic numerical skills are influenced by the inhibition of continuous non-numerical features (Zhang et al., 2019). For example, in one study, a tendency to spontaneously focus on numerosity positively correlated with children’s performance in non-symbolic comparison tasks and adults’ math academic achievements (Ben-Shachar et al., 2020). A theoretical model of the development of numerical cognition (Leibovich et al., 2017) posits that with experience, a child learns the correlations between numerosity and continuous visual properties. But only with the further development of cognitive control and inhibition is the child able to understand that these correlations can be violated or make comparisons between numerosities even when they do not correlate with continuous variables. Some suggest that a more general deficit in inhibitory abilities underlies the impairments in numerosity judgment among adults (Szűcs et al., 2013) and children with MLD (Bugden & Ansari, 2016). However, others argue that inhibition does not account for the link between non-symbolic processing skills and math performance among typically developing children (e.g., Malone et al., 2019).

Emotion and Numerical Cognition: Math Anxiety Attitudes toward and feelings about math, specifically math anxiety, have a strong influence on performance (for a review, see Rubinsten et al., 2018, 2019). Hence, math anxiety should not be ignored in a broad view of the neurobiological basis of numerical cognition. Math anxiety is a common phenomenon, with women reporting higher levels (Hart & Ganley, 2019). In fact, math anxiety is a leading explanation of why women are under-represented in science, technology, engineering, and math (STEM) professions (Lent et al., 2018). Math anxiety is characterized by heterogeneous symptoms (for a review, see Rubinsten et al., 2018), including negative attitudes toward math (Kucian et al., 2018) and excessive and sometimes unreasonable negative emotional reactions to situations involving numerical information (Kucian et al., 2018). Moreover, math-anxious individuals demonstrate a distinct pattern of neural activity that involves decreased activation in areas necessary for numerical computations and problem solving and increased activation in competing networks related to emotional processing (e.g., Young et al., 2012). Unsurprisingly, math anxiety is often related to poor math

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skills and reduced numerical cognitive abilities (e.g., Kucian et al., 2018), regardless of general cognitive abilities, such as working memory (Skagerlund et al., 2019). These numerical difficulties can lead to reduced socioeconomic status (Ritchie & Bates, 2013), diminished financial literacy (Skagerlund et al., 2018), and increased health costs (Reyna et al., 2009) and can have negative implications for STEM career training and success. The math anxiety score for adults with a STEM career is approximately half a standard deviation lower than the score for those without a STEM career (Hart & Ganley, 2019). Importantly, there may be a familial transmission of math anxiety (Daches Cohen & Rubinsten, 2017) through the home environment (Maloney et al., 2015) and inherited genetic influences (e.g., Malanchini et al., 2020). Thus, the salience of math anxiety continues into adulthood (Skagerlund et al., 2018), an especially problematic issue today given the growing reliance on technology and the fields of engineering and mathematics (Ashcraft & Moore, 2009). Math anxiety is associated with an attentional bias toward math-related information (e.g., Rubinsten et al., 2015). This attentional bias impairs cognitive resources; thus, performance is reduced (for a review, see Luttenberger et al., 2018). There is strong evidence that math anxiety involves attentional bias toward symbolic but not non-symbolic numerical information (e.g., Dietrich et al., 2015). Yet Lindskog et al. (2017) found that undergraduate students with high math anxiety had poorer non-symbolic processing skills than those with low math anxiety. Similarly, Braham and Libertus (2018) showed that the positive link between non-symbolic processing skills and applied problem solving was only present in undergraduate students with high math anxiety. There is neural evidence of math anxiety as early as the start of elementary school (Young et al., 2012). Taken together, the results imply the deep roots of math anxiety in the development of numerical cognition (Lindskog et al., 2017). In other words, math anxiety and non-symbolic processing skills may interact to influence math performance (Braham & Libertus, 2018).

Interim Summary Numerical cognition is not a unitary construct but a complex network that incorporates domain-specific and domain-general mechanisms (e.g., Bulthé et al., 2019; Matejko & Ansari, 2018), as well as emotional processes (i.e., math anxiety; for a review, see Rubinsten et al., 2019). The main controversy centers on the role of non-symbolic numerical skills in the development of numerical cognition (for a review, see De Smedt et al., 2013). In accordance with the theory of magnitude (Walsh, 2003), the literature suggests the need to investigate these skills by integrating countable and noncountable magnitude dimensions (for a review, see Leibovich et al., 2017), for example, in decisionmaking tasks (Khaw et al., 2018).

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Fig. 4 The motives of decision-making processes

Fig. 5 An example of a decision under uncertainty that depends on attitudes to payoffs (gains and losses; on the right) in the context of two dissociable forms of uncertainty (risk and ambiguity; on the left)

Decision-Making and Numerical Cognition Value-based decisions range from mundane (e.g., which TV show to watch) to crucial (e.g., which job to accept) situations and depend on both personal and environmental characteristics (Rangel et al., 2008; see Fig. 4). As illustrated in Fig. 5, many decisions involve choosing between uncertain outcomes when the probabilities of the possible outcomes are known (i.e., risky decisions) or unknown (i.e., ambiguous decisions; Levy et al., 2010). Behavioral and neuro-economics methods mathematically define the optimal behavior in a decision-making task and therefore enable an exact quantification of individual deviations from these norms (Tversky & Kahneman, 1992). Work specifically on decision-making is limited in the wider research on the neurobiological basis of numerical cognition (Patalano et al., 2015). However,

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empirical data support the contribution of domain-specific numerical processing, domain-general cognitive skills, and emotional factors to risk perception and decision (for a review, see Schiebener & Brand, 2015). Specifically, less precise symbolic-number mapping, rather than symbolic numerical processing skills per se, can explain risky decision-making (e.g., Keage & Loetscher, 2018). The ability to represent and manipulate non-symbolic numerosities can have compensatory effects on choice behavior in individuals with impaired symbolic numerical skills and domain-general cognitive skills (Mueller & Brand, 2018; but see Patalano et al., 2015). Interestingly, and consistent with the theory of magnitude (Walsh, 2003), Keage and Loetscher (2018) recently showed a positive correlation between subjective risk perception of everyday activities and a numerical judgment task in which participants were asked to indicate numbers’ position on a horizontal line. If risk magnitudes are indeed mapped onto the common magnitude representation system (Khaw et al., 2018), behavioral and neuroeconomics methods that implicitly assess decision-making under risk are very appropriate to investigate the neurobiological basis of numerical cognition. In this vein, one of the practices of pure mathematicians is making-meaning for decision-making in a situational context and mathematical representations of this context (Stillman et al., 2020). Thus, an examination of the integration of countable and noncountable magnitude dimensions in individuals with varying levels of math ability may shed light on the neurocognitive mechanisms underlying numerical cognition (Leibovich et al., 2017).

Conclusion Cognitive and neural research has made significant progress in characterizing the mechanisms underlying numerical cognition and their typical and atypical developmental pathways (Matejko & Ansari, 2018). These mechanisms include domain-specific (i.e., symbolic and non-symbolic numerical processing abilities; Schneider et al., 2017) and domain-general processes, such as working memory (Bugden & Ansari, 2016), visual perception (Zhou et al., 2015), and inhibition (Szűcs et al., 2013), as well as emotional factors (i.e., math anxiety; Rubinsten et al., 2019). There is evidence linking each of these mechanisms to higher mathematics. For example, as mathematical proficiency levels are increased, young adults demonstrate superior inhibition of distracting numerical information (Lubin et al., 2016) and greater visuospatial working memory capacity (Hubber et al., 2019). Risk perception and decision-making are influenced by domainspecific numerical processing abilities, domain-general skills, and affect (for a review, see Schiebener & Brand, 2015). Therefore, decision biases may have implications for the identification of reliable genetic, neurobiological, and cognitive markers of numerical processes among diverse populations of different ages and with varying levels of expertise in mathematics, ranging from people with MLD to expert mathematicians.

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Trusting Your Own Eyes: Visual Constructions, Proofs, and Fallacies in Mathematics

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Faculty of Vision and Development of Mathematical Thinking . . . . . . . . . . . . . . . . . . . . . . . . . . Vision and Imagery Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visualization and Conceptualization in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visual Fallacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visual Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometrical Constructions Related to Art and Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometrical Constructions as Solutions of an Interconnecting Problem . . . . . . . . . . . . . . . . . . . Gradual Geometrical Constructions in Dynamic Geometry Environments . . . . . . . . . . . . . . . . Visual Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visualization in Non-Euclidean Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter discusses how human biological capability of seeing contributes to the cognitive development of mathematical thinking. Thus, the phenomenon of visualization in mathematics, viewed as practice of handling visual and spatial information with the aid of physical artefacts, diagrams drawn on paper and mental imagery, is examined. It is stressed that visual mental images rely on all human senses; they develop through perception, action, and language, along with the process of conceptualization of mathematical objects. The latter are considered as distinct from the physical objects, from which their properties are abstracted and generalized. One can study mathematical objects through their various semiotic representations, including visual images. Examples of geometric constructions and reasoning with figures in the Euclidean plane are provided. M. Kondratieva (*) Faculty of Education and the Department of Mathematics and Statistics, Memorial University of Newfoundland (MUN), St. John’s, NL, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_38

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They reveal a possibility of finding and comparing multiple solutions with the aim of noticing new mathematical relations. These examples also illustrate that working with visual material requires and supports the development of mathematical knowledge. This includes knowledge of mathematical properties of figure’s elements and detection of underlying structure. As well, the study of non-Euclidean geometries, while significantly relying on mathematical formalism, calls for supporting visual aid. Geometrical solutions emerge through refocus of viewer’s attention from the external, visual features to internal logical characteristics of the figures. Examples of visual fallacies and paradoxes based on unfaithful images further support the idea of the importance of logical analysis of figures. They necessitate an alternative verification of what could be visually perceived as trustworthy, and call for a rigorous deductive approach in mathematics. While the role of figures in formal proofs may be disputed, visual imagery often empowers work on mathematical discovery. Keywords

Mathematical object · Interconnecting problem · Basic geometric configuration · Visual paradox · Mental image · Geometrical construction · Visualization · Conceptualization

Introduction Perhaps “the great book of nature is written in the language of mathematics and the characters are circles, triangles and other geometric figures” because “a picture is worth a thousand words.” However, it may be the case that “we do not know what we see, we see [only] what we know.” As well, there is advice: “if you see a ‘buffalo’ sign on an elephant’s cage, do not trust your eyes.” These and similar wisdoms attributed to great philosophers of the past, and reiterated many times after, tell us about both the significance and complexity of dealing with visual information. They apply to human daily experience, but also to scientific practice, considering that “the faculty of vision is our most important source of information about the world” (Adams & Victor, 1993, p. 207). This chapter is concerned with the question: How does the human biological capability of seeing contribute to the cognitive development of mathematical thinking, and what experiences are important to facilitate this contribution? The central role of our consideration is devoted to the phenomenon of visualization, defined broadly as “the ability, the process and the product of creation, interpretation, use of and reflection upon pictures, images, diagrams, in our minds, on paper or with technological tools, with the purpose of depicting and communicating information, thinking about and developing previously unknown ideas and advancing understandings” (Arcavi, 2003, p. 217, and references there). We start with a brief review of psychological aspects, presenting a summary of important steps in developing the ability of visualization during a life journey from

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an infant to a mathematician. Then we consider examples of visual constructions, visual reasoning, and visual fallacies with some implications for the learning of mathematics. We conclude with a touch on epistemological issues related to the use of visualization in proofs.

The Faculty of Vision and Development of Mathematical Thinking Vision and Imagery Development Let us first reexamine the biological perspective on learning as adaptation to one’s surroundings. In order to detect signals from their environment, biological creatures, even the most primitive ones, use their basic senses. These senses enable an organism to recognize either possible danger or promising targets, which is essential in defining their reaction. The need for classification of the signals (as risky, pleasing, unknown, etc.) assumes a possibility of their interpretation. For that, the ability to use (statistical) patterns is useful. In other words, organisms learn rules of advantageous behavior based on their previous experience. Leaving aside the question about which of the rules are learned and which are innate, we note that as an organism matures, it accumulates more and more rules for signal interpretations. The processes of gathering, storing, adjusting, and systematizing these rules are essential for individual and the entire population’s survival in the environment. Through these processes, which are controlled by “brain” (or less sophisticated nervous systems), organisms learn about the physical objects surrounding them, defining their (subjectively) attractive, repulsive, or neutral characteristics. Note that a signal from the environment by itself does not constitute information for a given organism unless the signal is noticed by its senses and classified by its “brain.” Humans are biological creatures and the human “brain creates flexible methods of spatial representation based on information from the senses” (Jackson, 2002, p. 1249). Vision is one of the most important human senses: “The largest part of cerebrum is involved in vision and the visual control of movement, the perception and elaboration of words, and the form and colour of objects” (Adams & Victor, 1993, p. 207). Experiments with reversed glasses show that brain effectively coordinates visual and tactile information as a part of general adaptation to the surrounding (Erismann & Kohler, 1950). The point of our particular interest is the cognitive development of the human ability to visualize. According to Piaget and Inhelder (1971), when a person creates a spatial arrangement (e.g., drawing on paper or computer screen, or 3D shape out of material objects), there is a visual image in a persons’ mind, guiding this creation. Thus, visual image is “a mental construct depicting visual and spatial information” (Presmeg, 2006, p. 207). In this sense, visualization, as defined in the section “Introduction,” involves processes of constructing and transforming both visual mental images and their representations of spatial nature (e.g., physical diagrams, graphs, figures). This includes two aspects of spatial thinking elaborated by Bishop (1983), namely, interpreting visual information and visual processing (Presmeg, 2006).

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Broadly, we are concerned with the development of mathematical thinking, in particular, with the processes of conceptualization, generalization, abstraction, and deduction. What is the role of visualization in these processes? Note that the objects of mathematical study “are not amenable to any concrete imagination or manipulation. They are . . . immaterial, not palpable or tangible, and accessible only to our thinking. Modern mathematics has developed a sophisticated methodology for creating and calling into existence those mathematical objects. [They]. . . gain the status of independent objects with specific properties and qualities” (Dörfler, 1995, p. 82–83). Furthermore, the methodology mentioned above involves development of semiotic systems in which the objects could be represented. The semiotic systems’ development is based on “semiotic means of objectification – e.g. artifacts, linguistic devises and signs that are intentionally used by the individual in social processes of meaning production” (Radford, 2002, p.14). Remarkably, “mathematical objects must never be confused with the semiotic representations used, although there is no access to them other than using semiotic representation” (Duval, 2006, p. 107). So the question is, how it is possible to work with these objects and learn their properties if they “exist” only in mathematical discourse? Take for example, geometrical figures viewed as “mental entities, which possess simultaneously conceptual and figural characteristics” (Fischbein, 1993). How do these ideal Platonic figures, which are objects of theorems in Euclidean geometry, relate to imperfect shapes we can draw and touch? Using concrete (physical or mental) images becomes problematic given the generalized character of the statements that require a conceptual grasp of the situation. Observing “no image of a triangle would ever be adequate to the concept of it. For it would not attain the generality of the concept, which makes this valid for all triangles,” Kant proposed that concrete “images must be connected with the concept always only by means of the schema that they designate.” (Kant, 1787/1998, p. 273, A141–142). Our thinking involves images because we do not have direct access to concepts. “The adult, as does the child, needs a system of signifiers dealing not with concepts but with objects as such and with the whole past perceptual experience of the subject. This role has been assigned to the image” (Piaget & Inhelder, 1969, p. 70). Further, . . . “formation of mental images cannot precede understanding” (ibid, p.72). Thus, mental images are not exact copies of perceived objects, but rather meaningful re-creations of perceptions. Concepts, metaphorically speaking, are “mental boxes” containing information obtained by the whole previous experience (perceived by senses and received verbally); mental images are “labels” or “windows” which provide access to the boxes. Thus, images (which may or may not be visual) result from our attempts to organize, classify, structure, and optimize the route of access to the information. The complexity consists in doing that when the amount of information is large. Information associated via a mental image with a particular concept could be insufficient, inadequate, or misplaced which would lead to defectiveness of the image. The process of obtaining and reorganizing information (through assimilation and accommodation) is never complete, so mental images are always subject to alteration. Researchers have hypothesized about how visual mental images may be formed. According to their observations, a very young child perceives objects entirely

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through its actions on them. At first, a child may even reject the very existence of objects which are out if their reach or view. Piaget uses the term “sensori-motor” to characterize this “hand-on” phase of a child’s representation of the world. An older learner may also use this form of representation when the task calls for it. Tactile information significantly contributes to the processes of visualization and conceptualization. Bruner (1966) describes this phenomenon as “enactive” mode of cognition, proposing that it may support both iconic and symbolic types of thinking. This idea justifies the use of manipulatives in school mathematics. A pictorial or iconic representation of an object is formed when a series of actions is organized in a summative image, which then becomes abstract and free of concrete actions. Constructing a mental plan of a labyrinth in which one repeatedly is searching for a way out (Mandler, 1962) could serve as an example of developing a summative image. However, these images constructed through actions may still depends on a learner’s specific representation of the world and may have a different degree of sophistication. The first important step in imagery development of infants is mental separation of the image from the object it signifies and recognition of an object by its image. These early-age images are static and reproductive. The focus is on external attributes. A second sign of progress is when images can be evoked without the object present and they may be expressed in a form of a naïve drawing. Third, there is a refocus on the objects’ internal properties and structure. The child becomes capable of reconstructing the physical picture from its parts and moving an image in thoughts. Forth, the ability to act on a picture (e.g., drawing additional parts) is formed. The child may form an anticipatory image, that is, may conjecture and predict certain things. Drawing pictures may now help the child to organize their observations and ideas. Finally, learners form images equipped with a network of properties and reveal conceptual grasp of objects. Thus, the iconic mode of cognition includes a range of images from naïve and concrete drawings to structured diagrams and geometrical figures. The development of imagery provides the necessary ground for the next stage, when the learner becomes capable of operating with symbols on a formal and abstract level and easily switches between visual and verbal-logical modes. These general ideas about imagery development bring us to the second part of the section that addresses visualization specifically in learning and doing mathematics.

Visualization and Conceptualization in Mathematics As illustrated above, conceptualization and its support through images widely occur outside of mathematics. But in mathematics, we have to deal with mathematical objects that are distinct from physical ones, and this presents extra difficulty. Considering that human thought is embodied through sensorimotor function and built linguistically through metaphors based on human perception and action (Lakoff, 1987; Lakoff & Núñez, 2000), Tall (2008) suggested that human genetically inherit the following three abilities that play an essential role in developing

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mathematical concepts. The abilities are: (1) recognition of similarities, differences and pattern through perception, (2) repetition of sequences of operations through action, (3) language, enabling perception and action to be expressed and conceived in increasingly subtle ways. Mathematical concept development goes through gradual construction of knowledge structures with the ultimate goal of arriving at “crystalline concept,” which “has an internal structure of constrained relationships that cause it to have necessary properties as a part of it context” (Tall et al., 2012). This can be achieved by the process of conceptualization which includes the following stages: (i) perceptual recognition of objects with simultaneous properties; (ii) verbal description of properties related to its visual or symbolic representation; (iii) definition and deduction of other properties; (iv) establishing equivalent properties by proof; (v) different ways of expressing underlying crystalline concept; and (vi) connection of several concepts in a network (Tall et al., 2012). While the sophistication level increases from one stage to the next, we should keep in mind that “all perceiving is also thinking, all reasoning is also intuition, all observation is also invention” (Arnheim, 1969). This makes concept formation to be an overall challenging adventure, in which language enables categorization of visual images, leading to definitions of concepts in terms of observed properties or proposed set-theoretical properties. Ideal Platonic figures are products of crystalline conceptualization. The key insight in developing adequate visualization of geometric figures is a move from noticing properties and attributes of a drawing to identifying relations between these properties and justifying them by means of deduction. This is in agreement with van Hiele model of learning geometry, in which learners progress from recognition of figures by naïve description to the use of definitions to make geometric constructions and applying deductive reasoning. The success of the latter stages depends on students’ relational understanding (Skemp, 1987) of the elements of the figures they work with. Formation of adequate visual images in geometry and other mathematical topics requires proper development of corresponding concepts, which, as argued in Tall et al. (2012) and Tall (2013), can be done through various modes of maturation of proof structures: geometric embodiment, algebraic symbolism, and axiomatic formalism. Arcavi (2003) provides more insightful examples illustrating how images that appear on this journey may become “an essential factor for creating the feeling of self-evidence and immediacy” (Fischbein, 1987, p. 101). They can “reveal data” (Tufte, 1983) by graphing an array of numbers, “accompany a symbolic development” by adding “meaning and conviction to a symbolic proof,” enable an alternative way to “engage with concept and meaning” in problem-solving, organize data, and even embrace an “analytical process that concludes with a solution that is general and formal” (Arcavi, 2003, p. 230). Sfard (1991) distinguished operational and structural conceptions. The first type is associated with actions and processes performed on familiar objects. It is the second type that is the most illuminating and is supported by visual imagery that allows access to mathematical objects. Based on the historical development of such

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concepts as number and function, she hypothesized that structural concepts emerge through three stages: interiorization (of a process), condensation (of operational information), and reification (of new object). As a result, “a process solidifies into object, into a static structure. Various representations of the concept become semantically unified by this abstract, purely imaginary construct. The new entity is soon detached from the process which produced it and begins to draw its meaning from the fact of its being a member of a certain category” (Sfard, 1991, p. 20). Mathematical objects form a hierarchy as new objects are built upon ones that are already conceptualized. Duval (2006), with reference to Kant and Piaget, clarifies the distinction between knowledge objects and transient phenomenological objects. The former “can be viewed as the invariant of a set of phenomena or the invariant of some multiplicity of possible representations.” In the latter case, the attention is focused “on such or such aspect (shape, position, size, succession...) of what is given.” He explains that “mathematical objects (numbers, functions, vectors, etc.) are knowledge objects, . . . while semiotic representations . . . are transient phenomenological objects. They [representation] are ‘mathematical objects’ under the condition that attention can focus on some invariant (the assumed represented relations) and not only on their visual data and their perceptual organization” (Duval, 2006, p. 129). This refocus on the invariant property is also depicted in the description of structured concepts, provided by Sfard (1991), the idea of procept (Gray & Tall, 1994) and lies at the heart of the APOS theory (Dubinsky & McDonald, 2001) of mathematics learning. All cognitive development theories emphasize the complexity of emergence of visual images that fully support working with mathematical objects. Visualization is never independent from other cognitive processes and thus requires a cooperation of them: “conceptual comprehension in mathematics involves a two-register synergy, and sometimes a three-register synergy. That is the reason why what is mathematically simple and occurs at the initial stage of mathematical knowledge construction can be cognitively complex and requires a development of a specific awareness about this coordination of registers.” (Duval, 2006, p. 126). For a mathematician, mental images emerge “from his desire for simplicity” (Nardi & Iannone, 2003, p. 369). The simplification may be achieved by the structural grasp of the several instances of the same phenomenon: “I need [an image] in order to have a simultaneous view of all elements . . . to hold them together, to make a whole of them, to achieve synthesis . . . , and give the concept its physiognomy” (Hadamard, 1949, p. 77). Many students of mathematics do not naturally possess this talent of structural grasp in the context that involves abstraction and generalization. For instance, if a new concept is introduced structurally, students may initially interpret the definition in an operational way (Sfard, 1991). In the worst-case scenario, students employ memorization without understanding. In general, faulty learning experiences lead to a discrepancy between the concept formal definition and visual image formed by a student (Tall & Vinner, 1981). Presmeg (2006) identified different types of images: concrete (picture in the mind), kinesthetic (of physical movement), dynamic (the image is transformed), memory image of formula, and pattern image. She found concrete imagery to be

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prevalent in students. She also suggested that pattern imagery, being a strong source of generalization, should be encouraged. “Students may be willing to use visual representations but have little training associated with this skill” (Stylianou, 2001). However, the use of visual images in teachings is not so straightforward as it might seem. This may be explained partly by the historical development of mathematics in the twentieth century when figural reasoning was largely suppressed by the formal symbolic set-theoretic approach. From the psychological perspective, visual reasoning, because it operates holistically and may use multifunctional registers, creates a greater cognitive load than more sequential modes of reasoning (Dreyfus, 1991). Since students might have greater difficulty carrying out processes in these multifunctional registers, they are often avoided in teaching. At the same time, visualization is used “for giving ‘meaning’ to mathematical processes that are carried out within mono-functional registers.” (Duval, 2006, p. 127). This ambiguity of teaching practices may explain student’s difficulties such as observed in Kondratieva and Radu (2009): university level students of precalculus were largely unsuccessful in matching images of basic curves (straight lines, circles, parabolas) with their algebraic and verbal expressions. Thus finding out “what aspects of instruction might encourage learners to use visualization, and what aspects might help them to overcome the difficulties and make optimal use of the strengths of visual processing” (Presmeg, 2006) remains of interest. Although the topic of mathematics teaching is outside the scope of this chapter, examples from the next sections may highlight some useful approaches to this issue.

Visual Fallacies Visual cues support our intuition. Therefore, if visual information is misinterpreted or visual images are distorted, this would influence the conclusions based on it. There are several pitfalls of this sort. We discuss some of them in this section. First, our perception of objects is relative, for example, the sense of a certain color may depend on the colors surrounding it. Similarly, the size of an object is perceived in comparison with other things positioned next to it. As well, our brain may alter or ignore some pieces of visual information in accordance with statistics of the previous experience. Thus, we may overlook an error in a word frequently seen before because our brain fixes it for us. Similarly, a careful consideration of Fig. 1 in Giaquinto (2020) reveals an unintentional inaccuracy in the drawing that can easily be overlooked if the meaning of the construction dominates its appearance. As well, if our attention initially focuses on a certain visual perspective of a drawing, it could be very difficult to switch to an alternative one. Visual illusions and ambiguities of such sorts are frequently used in psychological experiments. Second, there are problems that challenge our initial intuitive visual reaction. One example is the requirement to draw a triangle connecting all four vertices of a square. The first response most people have is that it is impossible to do because of their implicit assumption that three of the given points must be vertices of the required triangle. Once this assumption is suspended, the solution follows.

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Fig. 1 The paradox of area rearrangement: 169 ¼ 168

Let us give another example. Suppose a rope is placed on the ground around the Earth’s along its equator. How much longer would the rope be that is uniformly lifted one meter above the ground? Naturally many people would inquire for the size of the equator and intuitively predict that the difference in the ropes’ length will be large. However, assuming the Earth is a perfect sphere, simple calculations reveal that first, the answer is independent of the Earth’s radius and second, the difference is only approximately 6.3 meters. This initial visual misjudgment can also be corrected by entirely visual approach as discussed in Arcavi (2003). Third, all 2D drawings of 3D objects are only schematic images of the latter. A person may not learn all the properties of a cube just from looking at its picture or net: an experience with a bodily object is required for that. In fact, even seeing a physical object may lead to a perceptual misconception, because “the spatial ability of a sighted person is based on the brain analyzing a two-dimensional image, projected onto the retina, of the three-dimensional world” (Jackson, 2002). In addition, an object that could “exist” in drawings may violate geometry of the space of physical objects. An example of such an optical illusion is the Penrose triangle, a collection of three pairwise perpendicular sides. Under examination, the drawing makes sense locally at each corner, but can’t be united in a whole as a material 3D object (Penrose triangle, 2021). A similar phenomenon occurs in Penrose stairs, a circular set of steps that always go up locally but leads to the initial position in the end (Penrose stairs, 2021). M.C. Escher developed this idea in his famous lithographs Ascending and Descending and Waterfall. He was not a mathematician; however, he certainly had an extraordinary imagination appreciated by mathematicians, artists, and philosophers, professional and amateur. Forth, planar drawing (even if it is not related to 3D objects) may contain imprecise and misleading elements. In some cases, this leads to visual paradoxes. One example is the visual proof by area decomposition that 168 ¼ 169. Figure 1 shows a rearrangement of the parts of a 13  13 square into a 8  21 rectangle, which leads to a paradox because the result violates the area conservation principle. This very principle is used in visual explanations of various algebraic identities such as shown in Fig. 2.

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Fig. 2 A visual interpretation of an algebraic relation based on the area decomposition

Fig. 3 The unfaithful image used in the “proof” that “all triangles are isosceles”

Most such visual demonstrations are restrictive because they apply to only positive values of the variable. Even so, the paradox makes one become doubtful about the principle of area conservation. However, a careful numerical examination of the slopes of the lines presented in Fig. 1 leads one to a realization that the rearrangement is a visual illusion, because the slopes of some lines are only approximately equal. It is unlikely that the “missing” area can be revealed to the eye by either redrawing or cutting the figures, because of the roughness of our instruments. In fact, more figures of this type exist in which the effect is even stronger. This could be shown algebraically using properties of the Fibonacci numbers. Another example is a “proof” that all triangles are isosceles based on an unfaithful drawing. Assume that in a triangle, an angular bisector and the perpendicular bisector of the opposite side intersects at point M, as shown in Fig. 3. We drop perpendiculars ML and MK on the other two sides of the triangle. Since BM is the angular bisector, BLM and BKM are congruent right triangles. So BL ¼ BK and ML ¼ MK. By the property of perpendicular bisector, AM ¼ MC. Hence MCL and MAK are congruent and LC¼KA. Consequently, AB ¼ AK + KB¼CL + LB¼BC. If not for the conclusion, one may never suspect a lie because the reasoning looks very much like one of those proofs found in the books. And in fact, the reasoning itself is correct, it is the figure that makes the conclusion wrong. The resolution of the paradox requires a geometrical construction (in the sense discussed in section “Visual Constructions”) that truly corresponds to the word description provided in the accompanied reasoning. Figure 4 shows such construction. In particular, it reveals that BC¼BL + LC, as stated above. However, in contrast with previous

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Fig. 4 The faithful image of the triangle and its lines constructed geometrically

assertion, AB¼BK-AK, confirming that AB and BC have distinct lengths. The case of an isosceles triangle is degenerate: here AK ¼ LC ¼ 0. These two examples show that rigorous analysis of figures for their internal consistency is required in order to resolve a visual paradox. The process of resolution involves stages of understanding of all visual elements individually, the state of puzzlement by the fact that individual parts do not work together, and finding the reason of inconsistency (more details are found in Kondratieva, 2009a, b). Finally, we discuss examples that fail our visual intuition because they are related to the notion of infinity. We start with an historical example. Zeno surprised the ancient world by demonstrating that the concept of infinity was borne with apparent paradoxes. There is no motion, he proclaimed, because in order to move from point A to point B, one needs to cover the half way, then the half of the second half, then the half of the remaining portion, and so on infinitely many times; thus one will never get to the point B. In response, Diogenes simply made a step from A to B. This anecdote is explained by modern calculus. The infinite sum of natural powers of ½ is 1. Related visual interpretation sometimes is shown on a unit square instead of a unit segment; however, it does not change the essential question: Can we accept that the infinite sequence of segments whose length “tends to” 1 is actually equal to 1? The same tension you may sense in Example 1 (section “Visual Constructions”), which suggests that the infinite sequence of Fibonacci rectangles results in the golden rectangle. Can you visualize this process as a whole? Zeno paradoxes stimulated ancient thinkers such as Eudoxus, Archimedes, and Euclid to invent more precise concepts and methods of dealing with infinite processes. Aristotle proposed distinguishing between actual and potential infinities. For instance, for any collection of n > 0 pebbles, one can think of a collection of n + 1 pebbles. This is an example of a potential infinity. On the other hand, a collection of unlimited number of pebbles is an instance of actual infinity. Aristotle argued that mathematical methods deal with potential rather than actual infinity because perception of actual infinity by humans is problematic. In an attempt to avoid paradoxes, confusions, and perceptual doubts, the notion of limit was later invented. Treating objects as potentially infinite allows us to work with finite entities of which we are

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taking the limit. Many calculus methods adopt precisely this point of view and therefore make it possible to visualize these objects, e.g., to interpret the partial sums of series and the Riemann sums for integrals as areas. However, visualization of the limiting process is tricky, because in most cases, one will not be able to conclude by the picture alone if the limit actually exists. Mathematicians also invented ways to deal directly with actual infinity. Moreover, they discovered that there are actual infinities of different kinds, for example, natural numbers 1,2,3,4,. . . constitute a countable infinite set, while real numbers are uncountable. The notion of a set allows one to speak of an infinite collection (e.g., of all natural numbers or of all real numbers) as of an object by itself. In fact, one accepts the actual infinity by treating its occurrences as a symbol, which is a part of a deductive system. The set-theoretical view dramatically influenced the development of mathematics and led for instance to contemporary understanding of the real number system as a whole. While it is customary to visualize real numbers on the number line, some properties of infinite number sets are quite unexpected. For example, on the one hand, natural numbers form a proper subset of integers (every natural number is an integer but there are integers that are not natural numbers), but on the other hand, there exists a one-to-one correspondence between the two infinite sets. The situation when a “part is equal to the whole” is impossible for finite sets suggesting that visualization of infinite sets requires special care. The difference between actual and potential infinities leads to an epistemological distinction in perceiving a nonterminating expression (e.g., a series) or a nonterminating construct (as in Fibonacci rectangles) as object versus process. The “object” view is consistent with the idea of the metaphor of infinity in which “processes that go on indefinitely are conceptualized as having an end and an ultimate result” (Lakoff and Nunez, p. 258), as opposed to the “process” view. This explains the distinction between “tends to” and “equals” viewpoints outlined above. As Dubinsky et al. (2005) observed in relation to the infinite decimals, “An individual who is limited to a process conception of . 999 . . . may see correctly that 1 is not directly produced by the process.” It is only when “. 999 . . . is considered as an object, it is a matter of comparing two static objects, 1 and the object that comes from the encapsulation. It is then reasonable to think of the latter as a number so one can note that the two fixed numbers differ in absolute value by an amount less than any positive number, so this difference can only be zero” (Dubinsky et al., 2005, pp. 261–262). Note that the same conclusion applies to the Zeno case, which may be assisted with visualization on the number line. Adequate visualization of the encapsulated limit could be problematic because the limiting object may not possess all the properties of each individual term of the infinite sequence. Thus, in contrast with Fibonacci rectangles, each of which has integer dimensions, the sides of the golden rectangle are in irrational proportion (cf. section “Visual Constructions”). Another example is the Koch snowflake, a fractal resulting from the following infinite process (Koch snowflake, 2021). Start with an equilateral triangle of side length 1. Then, partition each side into three equal portions. On the middle portion of

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each side, place another equilateral triangle, with side length equal to the length of that portion. Now, do the same thing on each side of the resulting figure and continue indefinitely. The figure that appears as the limit of these constructions is the Koch snowflake. Note that the sequence of areas is bounded because each figure can actually be inscribed inside the same circle as the original triangle. A very special pffiffiffi property of the Koch snowflake is that while its area is finite (more precisely, 25 3), the perimeter is infinite. This could be verified by evaluations of related infinite series representing the sum of either triangular areas or the length of segments that constitute the figure and its boundary. While visualization of each finite snowflake is possible, the properties of the limiting object are counterintuitive. A similar phenomenon occurs for the Gabriel’s horn, defined as a surface of revolution about the x-axis of the hyperbola y ¼ 1/x restricted to the segment [1, 1). Evaluation of related improper integrals reveals that the area of this surface is infinite, while the volume is finite. Is has been said that the container in the shape of the Gabriel’s horn does not hold enough paint to cover its own surface! As in the previous case, the object results from the family of horns of finite length obtained by revolution about the x-axis of the hyperbola restricted to the segment [1, N] where N > 1. Each of the finite objects (for N < 1) is readily accessible for visualization and has finite volume and surface area (Gabriel’s horn, 2021). Considering all the possibilities for misinterpretation of visual information, misleading visual cues, and unexpected or contradictory outcomes based on intuitive judgment, the question is, can we trust out own eyes when doing mathematics? In fact, there are benefits of being occasionally exposed to inconsistencies, paradoxes, and flaws of various kinds that create doubts. In the learning of mathematics, these experiences call for a need to justify mathematical statements. Learners who used to make intuitive judgment about the validity of assertions often disregard necessity for their logical confirmation. In mathematics, this is not acceptable. The consistency check is required in order to separate truth from illusion, actual properties of mathematical objects from ones mistakenly attributed to them. The verification may be attempted through a combination of informal and formal proofs, reflecting the duality of intuition and rigor, the duality of human psychological and logical development. However, the awareness of possible fallacies eventually calls for rigor. Next sections give more specific examples illustrating this idea. From the cognitive perspective, apparent paradoxes foster the individual process of equilibration. Jean Piaget defines equilibration as a “series of cognitive actions performed by a knower seeking to understand [something] which is experienced as novel, resistant, perturbing, disequilibrating” (Dubinsky & Lewin, 1986). The instances when equilibration results in reconstruction of existing cognitive structures (i.e., reflective abstraction) has been associated with successful learning. This includes reviewing and restructuring existing information, accommodation of novelty, and thus refining conceptual understanding of the objects in play by establishing nonobvious relations. In addition, the sense of puzzlement triggers a peculiar kind of pleasure, aesthetics of mind (Danesi, 2002). Through socialization, these local individual processes may produce a global change. The history of mathematics

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shows that fallacies and ambiguities call for new inventions, creative solutions, and shift of existing paradigms, which “imbues mathematics with depth and power” (Byers, 2007, p. 11).

Visual Constructions Making and examining geometric constructions is a very conceptually rich activity that can involve algebra, analysis, topology, and other topics that reveal connections with geometry. This is in agreement with Arnheim’s position that interpreting visual information is a complex intellectual activity that requires abstraction, reasoning, and awareness of structure, and that drawing “is a form of reasoning, in which perceiving and thinking are indivisibly intertwined” (Arnheim, 1969, p. v). In geometric constructions, the goal is to specify how to produce figures that obey stated conditions. For constructions within Euclidean geometry, often, the only available instruments are a compass and unmarked ruler. This assumes that one can only draw straight lines and circles, while everything else should be constructed. The compass is used to choose the unit length, and other dimensions of the picture are drawn in proportional relation to it. However, the construction of figures could alternatively be completed using other instruments (Bartolini Bussi, 2000), including computer technology or described algebraically. The physical or mental act of construction (and deconstruction as a part of a drawing examination) is very important for image creation. Some examples of geometric constructions that illustrate cultural framing, interconnectedness, and gradual development of knowledge, are discussed in this section.

Geometrical Constructions Related to Art and Architecture Example 1 Visualizing the golden ratio. The golden rectangle may be defined as a rectangle with dimensions a and b (where b > a) such that if one cuts from it a square of size a  a, the remaining rectangle becomes similar to the original one. To achieve this construction in the strict mathematical sense, one may set up an algebraic equation that follows from the definition and express the similarity between the original (a  b) and truncated a (a  (b  a)) rectangles: ba ¼ ba . By solving it algebraically to obtain a numerical pffiffi value for the ratio b/a, we find the irrational number known as the golden ratio: 1þ2 5. Thus, the problem becomes to construct (and visually represent) this irrational pffiffiffi length. Noticing that 5 is the rectangle’s diagonal with sizes 1 and 2, the construction of golden rectangles follows. We only need several elementary subconstructions, such as a right angle. An example is shown in Fig. 5. Interestingly, one may also construct a sequence of rectangles with integer dimensions that approximate the golden rectangle. For that, one needs to recall the

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Fig. 5 A geometrical construction of the golden rectangle CDFE

Fig. 6 Geometrical structure of Fibonacci rectangles

connection between the golden ratio and the Fibonacci sequence 1, 2, 3, 5, 8, 13,. . . in which every term (starting from the third) is equal to the sum of two preceding terms. It appears that the ratios of two consecutive terms of the Fibonacci sequence, e.g., 8/5 or 13/8 approximate the golden ratio, and the precision of the approximation increases as the terms grow. In calculus, this could be expressed in terms of limits, but for our purpose, we observe that the sequence of Fibonacci rectangles of the sizes 1  2, 2  3, 3  5, 5  8,etc. can be geometrically constructed by attaching to a given rectangle a square of the corresponding size: 2  2, 3  3, 5  5,..., as shown in Fig. 6. This reveals a geometrical relation between the Fibonacci rectangles and the golden rectangle (as defined using similarity): two consecutive Fibonacci rectangles are not exactly similar but “approximately” they are! The possibility to construct next Fibonacci rectangle from the previous one illustrates the concept of potential infinity, while the golden rectangle constitutes an instance of actual infinity (as discussed in section “Visual Fallacies”). Kondratieva (2011a) offers more examples of constructions related to the golden ratio, such as Pythagorean pentagrams and Egyptian octagrams. These examples are

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not only mathematically rich but also have cultural and historical significance. Other uses of geometry and its relations to human practices, arts, and architecture are discussed in that paper. This suggests an interdisciplinary context and the idea of the possible affectively enhanced entrance to some pure mathematical constructs such as the golden rectangle and the golden ratio. In this case, the introduction to these topics is essentially visual.

Geometrical Constructions as Solutions of an Interconnecting Problem The interconnecting problem approach (Kondratieva, 2011b) requires learners to consider several solutions at both elementary and advanced levels and use various mathematical tools and representations. Thus, several concepts may be associated with one problem, which provides the basis for their comparison and finding possible mathematical connections. As discussed in section “Visualization and Conceptualization in Mathematics,” establishing and justifying such connections is a part of mathematical concepts’ construction. Let us consider the following interconnecting problem. Example 2 Inscribing a circle in an angle. Start with an arbitrary angle ABC and point D inside the angle. The problem is to draw a circle tangent to the sides of the angle and passing through the point D (that is, we need to construct the center and the radius of the circle). Four solutions are presented in Kondratieva (2011b): a naïve intuitive model of enactive flavor, two iconic (figural) solutions, and a symbolic one. We refer interested readers to look up the details in the paper. Here we only outline the ideas and their interplay, discussing specifically the role of visualization. The first approach replaces the given 2D problem with a 3D model of a balloon in a conical basket and demonstrates that the solution exists in 3D by a pure imaginative experiment of the balloon inflating. Note that the original problem is a plane section of this 3D model. The approach could be mimicked using a similar 2D experiment in dynamic geometry software, but it would require building an applet with specific invariants. This by itself is a good exercise, but it does not solve the problem exactly. However, it plays an important role in the exploration, visualization, and internalization of the situation. Each of the other three mathematically more advanced approaches uses the fact that the centre of the circle inscribed in an angle lies on the angular bisector. This observation is essentially based on one’s embodied knowledge because it refers to the axial symmetry of the geometrical figure and may be demonstrated to a child by folding the picture along the angular bisector. (Kondratieva, 2011b)

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Fig. 7 A similarity-based construction of the centers I and J of the circles inscribed in angle ABC passing via given point D

The second approach is pure Euclidean and starts with inscribing an arbitrary axillary circle and finding its points of intersection (H and G) with the ray BD. Then one uses similarity, specifically the fact that similarity results from uniform scaling, to produce required circles (see Fig. 7). Here KI || DJ || HE and DI || LJ || GE, so triangles KID, DJL, and HEG are similar. “Again, one can appeal to the embodied cognition, the natural sense of geometrical perspective, to view the second circle as a magnified copy of the first. This view . . . forms the basis for the construction employed” (Kondratieva, 2011b). The third approach is also visual, and it uses the geometrical definition of the parabola as a set of points equidistant from a given point (focus) and a line (directrix). The following observations are used: “the set of all circles inscribed in an angle form a family; their centres lie on the ray which is the angle bisector. Similarly, the set of circles passing through D and tangent to one side of an angle form another family; their centres lie on a parabola with focus at D and the directrix being the side of the angle. The centre of the required circle is at the same distance from the angle’s sides as it is from the given point D, thus the elements common to both families give the required circles” (Kondratieva, 2011b). This approach is illustrated in Fig. 8. The fourth approach is purely algebraic, but on inspection, it could be viewed as an algebraic representation of the previous approach in an appropriate coordinate system. However, it could be derived independently from the third solution by analyzing a sketch of the required construction using synthetic geometry. In summary, we see that various visual ideas play different roles in solving this problem. If the intuition presented by visual ideas is enhanced by appropriate mathematical apparatus, this leads to rigorous mathematical constructions. In addition, we see how a visual idea may be converted into an equivalent pure symbolic approach.

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Fig. 8 A construction of the centers E and F of the inscribed circles passing via given point D based on the geometrical definition of the parabola

From this perspective, the interconnecting problem approach can develop students’ visualization skills and advance understanding related to the concepts involved in various solutions. In particular, Kondratieva (2011b) presents a theorem that emerged from comparing two of the visual approaches that use the similarity of triangles and geometrical definition of the parabola, respectively. Example 3 Drawing a parabola. Several constructions based on geometric properties of the parabola considered in different historical, theoretical framings and aiming at “the development of representational flexibility between algebraic/functional and geometrical approaches” are discussed in Kondratieva & Bergsten (2021). One construction is based on a functional approach, where the parabola is treated as the graph of the quadratic function. Another construction is based on the focus-directrix geometric definition of the parabola (Fig. 9, left), already mentioned in Example 2. The third construction is “by application of areas” (Fig. 9, right) with reference to the definition of the parabola by Apollonius of Perga. Under what conditions exactly the same curve is produced by all of these methods? A relational bridging of the three constructions is presented in Kondratieva & Bergsten (2021). This chapter gives a possible scenario of conceptualization of the notion of the parabola with a strong emphasis on the visual component, including the animations, which are supported by the use of computer technology. The latter will be discussed in the next example.

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Fig. 9 Two dynamic constructions: as point C moves along AB, F traces a piece of a parabola

Gradual Geometrical Constructions in Dynamic Geometry Environments Experimentation as a learning approach is widely used in natural sciences. While not much developed, an “exploratory approach to experiments that includes concept formation also pertains to mathematics” (Sorensen, 2010). However, in mathematics, we deal with mathematical objects, not physical objects. Therefore, in mathematics, learning “starts with extended calculations, constructions and experimentations. In this way a ‘quasi-reality’ is created which allows for observing phenomena, discovering patterns, formulating conjectures, and last not least for explaining, that is, proving patterns” (Wittmann, 2009, p. 255). Modern technology such as computer algebra systems (CAS) and dynamic geometry environments (DGE) allows designing, performing, and validating experiments, more closely portraying the approach used in natural sciences. Availability of such technologies largely extends possibilities of mental experimentation in mathematics for both researchers and students of mathematics. The portion of “quasi-reality” displayed by the technology may reduce the cognitive load of imagining a system’s response on a specific input. In particular, dynamic geometry software allows experimentation directly with geometrical images. In Example 4 below, we discuss technologically assisted work with dynamic geometric constructions starting from the basic ones and leading to one of the most fundamental results in geometry of triangles: Euler’s line. Explanations arising from experimental mathematics often have the status of preformal proofs, defined in Blum and Kirsch (1991, p 187) as a “chain of correct but not formally represented conclusions.” Preformal proofs include visual, operative (Wittmann, 2009), and generic (Leron & Zaslavsky, 2009) proofs, which carry on the same logic as formal reasoning but reduce the level of abstraction by dealing with either visual images, “quasi-real” objects, or ideas presented in generic examples (particular cases). While not being general, preformal proofs provide an important step on the way from empirical observation towards general proofs because they essentially capture “the main ideas of the complete proof in an intuitive and familiar context, temporarily suspending the formidable issue of full generality, formalism,

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and symbolism” (Leron & Zaslavsky, 2009, p. 56). Formalization, verification, and exposition of these preformal arguments for an external reader lead to a formal proof. A process of geometrical constructions (with or without technological assistance) is often conjoint with preformal proofs because it results in concrete figures relying on general ideas. For that reason, construction tasks were traditionally used for bridging theoretical and practical aspects of geometry. For example, the theoretical notion of a circle (as a collection of all points in the plane, equidistant from a given point) reveals its practical meaning in compass constructions that involve distances, e.g., construction of a triangle with three given segments as its sides. “Any successful construction corresponds to a specific theorem” (Arzarello et al., 2012, p. 100), and it represents certain properties or relationships. Here we present a sequence of tasks that respects two principles: the complexity of tasks increased from simple to advanced, and solutions of later tasks rely on ideas and properties found in earlier tasks. All tasks require constructing an object with given properties and explaining how and why the proposed construction works. The use of dynamic geometry provides a ground for experimentation and visual confirmation of proposed ideas. All construction tasks only assume the use of a compass and unmarked ruler; that is, the tools are limited to “placing a point,” “connecting two points by a straight line,” and “drawing a circle.” If a new construction (e.g., right angle, segment’s midpoint) is achieved by means of previously used tools, a new tool can be added to the toolbox. The user can pick a scale and unit length. The main advantage of the constructions employing dynamic geometry software compare to constructions on paper is the possibility to drag points and interact with pictures while preserving certain build-in conditions (such as the proportion of segments, orthogonally of lines, etc.). The goal of such interactions is to observe new conditions that were not initially build-in and thus can be viewed as implications of them. Thus, the goal of the sequence of tasks presented below is to construct certain points in a triangle and learn Euler’s relation between them. The tasks are supplied with brief solutions partly observed in students’ work (Kondratieva, 2013a). Example 4 Constructing Euler’s line in a triangle. Task 1: Construct an isosceles triangle Solution 1a: Place a point A and draw a circle with center at A and any radius. Place two arbitrary points B and C on the circumference. Since CA ¼ BA are the radii, ABC is isosceles. Solution 1b: Draw segment AB ¼ 4. Draw circle U of radius 3 centered at point A, and circle V of radius 3 centered at point B. A point C, where circles U and V intersect, gives an isosceles triangle ABC. Here AC ¼ 3 because C is a point on circle U. Similarly, BC ¼ 3 because C is a point on circle V. While both solutions are valid constructions of an isosceles triangle, the second method appears to be more useful for solving the second task. Task 2: Describe the place of all points X in the plane equidistant from the two given points A and B, that is AX ¼ XB. Construct the midpoint of the segment AB

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Solution 2: Drawing several isosceles triangles on the same base AB (like in Solution 1b) reveals the idea that their third vertices C, D, . . . lie on a straight line. The midpoint of AB is the intersection of AB with the CD. Observe that the line CD constructed in Task 2 must be perpendicular to the segment AB because it is the axis of symmetry of each of the isosceles triangles with base AB, and thus it is the axis of symmetry of the segment AB as well. This observation provides the key to construction in Task 3. Task 3: Construct a line perpendicular to a given line l and passing through a given point X in the plane Solution 3: Draw a sufficiently large circle with center at X and mark the intersection points with line l as A and B. We have AX ¼ XB and then continue as in Task 2, constructing the perpendicular bisector of AB. Task 4: Construct a circle passing through vertices of a given triangle Solution 4: We need to find the center of the circle, that is, the point equidistant from all three vertices A, B, and C of the triangle. We find all points equidistant from A and B by constructing the perpendicular bisector of the segment AB, as in Task 2. Similarly, all points equidistant from A and C lie on the perpendicular bisector of the segment AC and all points equidistant from B and C lie on the perpendicular bisector of the segment BC. We can construct the three lines and find the center of the circle as their intersection. But do we know that they are concurrent (all three intersect at one point)? Experimentation seems to confirm the concurrency. An explanation of this fact is as follows: Denote by O the point of intersection of two perpendicular bisectors, say for segments AB and BC. Then that AO¼BO because point O is equidistant from A and B by construction. Similarly, BO¼CO by construction. Therefore, AO¼BO¼CO, and so O is equidistant from A and C. But that means, O lies on the perpendicular bisector to AC, and thus all three meet at O. This geometrical construction followed by an explanation illustrates a discovery of the fact that for any triangle, there exists a circumcircle. Its circumcenter is found at the intersection of the perpendicular bisectors of the sides. The observable concurrency of three medians in any triangle can be interpreted physically: the point of intersection is the center of mass G. The concurrency is proved by using either vectors or Ceva’s theorem. Learning about the concurrency of altitudes (at the point H called orthocenter) and the relation between points O, G, and H in any triangle is the subject of our final task. Task 5: Construct midpoints A0 , B0 , and C0 (Task 2) of sides BC, CA, and AB, respectively, and form the median triangle A0 B0 C0 . What is the transformation (of the plane) that converts one into another? Solution 5: To start with, let us find what relations exist between the medial and the original triangles. One can experimentally observe (Fig. 10) that their sides are pairwise parallel (e.g., AB || A0 B0 etc.) This fact could be confirmed by Thales’ intercept theorem. Consequently, the medial triangle is similar to the original one in the proportion 1:2. Further, note that the centers of mass of the two triangles coincide, G ¼ G0 . Then the required transformation of ABC into A0 B0 C0 consists of

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Fig. 10 The median triangle A0 B0 C0 and Euler’s line OH

180-degrees rotation around the center of mass G ¼ G0 and shrinking the object to produce a twice-smaller triangle. In other words, this transformation is a homothety with center G and coefficient k ¼ 0.5. This is what, in fact, happens with each point of the plane under the transformation. In particular, this transformation implies that H, G, and H0 are collinear (G lies between H and H0 ) and HG ¼ 2H0 G. Since altitudes in A0 B0 C0 coincide with perpendicular bisectors in ABC, the three altitudes are concurrent and H0 ¼ O, that is, orthocenter H0 of A0 B0 C0 is the same point as the circumcenter O of ABC. From the property H0 ¼ O, we immediately conclude that H, G, and O are collinear and HG ¼ 2OG in any triangle. The segment HO is referred to as the Euler’s line (Fig. 10). The geometrical fact we just learned is: In any triangle, the center of mass lies on the segment connecting the orthocenter and circumcenter, always twice closed to the latter than to the former. Note that concurrency of lines and other relations between points, if they take place, may be easy to observe in dynamic geometry but still may not be easy to prove. In this example, we demonstrated how required proofs could be supported by a sequence of construction tasks. The tasks rely on learners’ actions while focusing learners’ attention on important steps and relations. The homothety transforming a triangle into its median triangle seems to serve as an organizing factor that structurally explains many facts, including the geometry of Euler’s line. Further geometrical investigations of this line can be found in Kondratieva (2013a). This sequence of tasks supports the development of a triangle’s mathematical concept and provides dynamical images upon which the summative mental image of a triangle is gradually adjusted. Most importantly, at each step, new properties are observed and linked to already familiar ones by means of (preformal) proofs.

Visual Reasoning As it was pointed out, the process of figures’ construction discussed in the previous section relies on reasoning. Thus, we already met some ideas related to the phenomenon of visual reasoning, including the notion of preformal proof, which includes visual proof as a subcategory.

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Visual components can be present in generic proofs. “Dots or pebble diagrams of ancient math used to convince one of elementary truth of number theory. They display particular number of dots, while the truth is general (Giaquinto, 2020).” For instance, merging two rows of three dots and of two dots, in each, and obtaining two rows of five dots in each, one generically proves that the sum of two even numbers is even, while the diagram only shows 6 + 4 ¼ 10. The key observation is that all even numbers are represented by two rows of dots, whereas for an odd number’s representation, one dot always lacks a pair. Similarly, one demonstrates visually that the sum of two odd numbers is even. Visual reasoning uses external visual representations (diagrams, symbol arrays, kinematic computer images) and internal visual imagery. Often it combines with verbal and symbolic reasoning. A diagram could be a great image of a mathematical relation, however, only if you know what exactly it represents in a given context. Generally, a diagram carries the intended meaning only in combination with a formula or words, but not by itself. “Proofs without words” type of problems (Nelson, 1993) give an example of figures that are supposed to reveal for a viewer a mathematical fact without a substantial verbal or symbolic explanation. However, as we saw in the “Visual Fallacies” section, pictures could be deceiving or misinterpreted by the viewer. Therefore, their epistemological status in mathematics is debatable (see also discussion in section “Conclusion”). In any case, unfolding the information encoded by a picture is a good exercise in reasoning, even if while doing that, one also employs a nonvisual mode of thinking. In order to reason with figures, one needs to know (or be able to discover) the properties depicted in them. In particular, it is important to recognize some basic geometric configurations (BGCs) described as “simple but fundamental geometric facts expressed in drawings” (Kondratieva, 2011c). The drawings contain auxiliary elements and labels (e.g., equal angles, equal lengths, right angle, parallel lines) that allow recalling the statements along with the ideas of their proofs. BGCs may be given short names to enhance the memory. For example, the “bow-tie” and “StarTrek” BGCs are used for images showing relations between central and inscribed angles. The “isosceles triangle” BGC shows the line of symmetry, two equal angles and two equal sides. It may be supplemented by the construction tasks 1 and 2 from Example 4. BGCs are stepping-stones to proving or solving geometric problems. For example, from analysis of Fig. 4, one concludes that point M (the intersection of the angular bisector of ABC and the perpendicular bisector of AC) always lies on the circumcircle of the triangle ABC. This is because the perpendicular bisector of AC is also the angular bisector of the central angle AOC. The “Star-Trek” BGC relates the angle AOC with the inscribed angle ABC as well as the pairs of angles AOM ¼ 2ABM and MOC ¼ 2MBC produced by their angular bisectors, respectively. BGCs may support thinking, but only if learners grasp the mathematical structure, they represent (Arnheim, 1969). In the beginning, BGCs may be used to engage students in contemplations and articulation of what objects and relations they see and why the relations may hold. They may redraw BGCs on paper or implement them in dynamic geometry. Students should be trained to recognize BGCs as a part of more

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Fig. 11 A right triangle and its circumcircle

convoluted geometric figures. In advanced cases, students may need to draw auxiliary elements in a figure in order to use one of the BGCs. These experiences are needed because “the visual processes of gestalt recognition do not run in the same way as required and expected from a mathematical point of view” (Duval, 2006, p.112) and thus the viewer’s naïve intuition, without seeing many relations between the figure’s elements, may not be sufficient for their visual reasoning. The following two examples demonstrate the use of BGCs in visual proofs. Example 5 Investigating the right triangle and its circumcircle. Theorem A. For any right triangle, the hypotenuse is a diameter of its circumcircle. Proof: Let ABC be a right triangle with C ¼ 90 degrees. Rotate the triangle by 180 degrees around the midpoint O of the hypotenuse AB (Fig. 11). The image of A is B, the image of B is A, and the image of C is a new point D. Note that the segments AC and BD are parallel and of equal length, and so are segments BC and AD. So ACBD is a parallelogram. Since ACB ¼ 90 degrees, ACBD is a rectangle. The diagonal AB is then the diameter of the circumcircle. Theorem B: Let the side AB of a triangle ABC, inscribed in a circle, be a diameter of the circle. Then the opposite angle C is 90 degrees. Proof 1: Connect vertex C to the center O of the circle (Fig. 11). This way, two isosceles triangles AOC and BOC, are produced. Then the “isosceles triangle” BGC can be used to argue that in ABC angle C is the sum of angles A and B. By the “sum of 3 angles” BGC, C ¼ 180/2 ¼ 90 degrees. Proof 2: This theorem is a special case of the “Star-Trek” BGC, where the central angle AOB is 180 degrees, so the inscribed angle ACB is half of it.

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Proof 3: Rotate the triangle ABC by 180 degrees around the center O (Fig. 11). The original triangle, together with its rotated copy ABD, produces a quadrilateral ACBD with pairwise parallel sides. Since ACBD is inscribed in the circle, it is a rectangle. Thus, the triangle ABC is right-angled. Many BGCs allow “dynamical visual proofs, which are based on ‘drawing in movement’ that can be properly performed in a dynamical environment” (Gravina, 2008). The property depicted in Theorem B is one of them. Problems to prove can be treated as an interconnecting problem in a sense described in section “Geometrical Constructions as Solutions of an Interconnecting Problem.” The problem given in Example 6 below allows several approaches, each of which has visual elements as the basis of the solution. On several occasions, this problem was given to groups of in-service mathematics teachers with the assignment to produce at least three solutions, and these results were collected (Kondratieva, 2013b). The analysis of this collection clearly demonstrates the significance of the learner’s prior experiences and knowledge, as well as beliefs about what constitutes a proof. It also highlights that the most computationally efficient and insightful solutions may not be easy to arrive at. Why do learners overlook certain approaches that are within their capacity to produce and that the learners appreciate when they are exposed to them? This question is very relevant to visually based approaches but not specific to them as the same phenomena occur in other modes of thinking (Koichu, 2010). Example 6 Comparing the segments defined by a geometric construction. Problem: In a square ABCD with E at the mid-point of CD, join B to E and drop a perpendicular line from A to BE at F. Prove that the segments DF and AB are of equal length (Totten, 2007). Here we present a summary of various observed approaches, which demonstrates the variety of visual ideas relevant to the problem above. More details of each solution can be found in Kondratieva (2013b). The first approach consisted of direct measurement and comparison using various materials, including a ruler, string, Popsicle sticks, compass, or dynamic geometry software. The second approach was, in a sense, a symbolic implementation of the first one. Assuming that the side length of the square is a, we have the square with vertices at the points D(0,0); A(a,0); B(a, a); C(0,a). Then we find the equations of the pair of orthogonal lines AF and BE by determining their slopes and y-intercepts: y ¼ 2x + 2a and y ¼  x/2 + a/2. Coordinates of point F at which the two lines intersect are found by solving linear equations to get x ¼ 0.6a, y ¼ 0.8a. This pair of numbers satisfy the equation of the circle with center at D(0,0) and radius a: x2 + y2 ¼ a2. Thus, F lies on the circle and DF ¼ a ¼ DA. While we certainly see how the visual idea of using a compass to draw a circular arc AFC with center at D for comparing distances DA and DF (Fig. 12) inspired the

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Fig. 12 Several methods to show that DF¼DA

second solution, we also note that the rigor achieved by the second approach required more mathematical knowledge and technical skills. The third approach was based on the recognition of similar and congruent triangles. For that, an auxiliary line through vertex D and the midpoint X of segment AB was drawn (Fig. 12). Note that DX is parallel to BE. Denote by Y the intersection point of DX and AF. Then ABF and AXY are similar right triangles. This implies that Y is the midpoint of AF, and that DY is orthogonal to AF, so therefore, ADF is an isosceles triangle, and AD ¼ AF. It may be argued that to find this solution, one needs to recall BGC “isosceles triangle,” the image that contains the line of symmetry. Since we need to prove that AD ¼ AF, we draw the auxiliary line, the hypothetical line of symmetry, and then find the reasons for this being the case based on other details given in the figure. The fourth approach was based on recognizing that AFED is cyclic because the opposite angles ∠ADE and ∠AFE are both 90 degrees (Fig. 12). Now, note that angles CBE, BAF and DAE are equal; call this value y. Then ∠DAF ¼ 90o  ∠ BAF ¼ 90  y. From the “Star-trek” BGC, the relation for inscribed angle and central angle measured by arc, we have 2∠AFD ¼ arc (ADE)  arc(DE) ¼ 180  2y. Thus ∠AFD ¼ 90 - y. Since the two angles are equal, DAF is an isosceles triangle, DA ¼ DF. Observe that a property of isosceles triangles is employed here as well, but a bit differently. The decisive visual idea here was recognizing the cyclic quadrilateral, which triggered a chain of angle comparisons. However, note that the recognition of the cyclic quadrilateral can alternatively lead to another approach based on Ptolemy’s Theorem, a more advanced result in Euclidean geometry: In a cyclic quadrilateral, the product of its diagonal equals the sum of the products of opposite sides. The solution then reduces to the identification of required lengths by algebraic calculations. Finally, we present perhaps the most insightful solution. Extend lines AD and BE and call the intersection point G (Fig. 12). Note that DE is the midline in ABG, that is points, E and D are midpoints of sides BG and AG, respectively. Since AFG is a right

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triangle, then its vertices lie on a circle, and hypotenuse AG coincides with the diameter of the circle centered at D. Thus DF, DC, and DA are all equal to the radius of the circle. Note that drawing the circle with center at D and passing via point A, F, C was a natural idea presented in the first and second approaches. The right angle touching the circle is a part of the visual image of BGC emergent from Theorem A discussed in Example 5. The theorem was well known to most of the teachers who were asked to produce the solutions, but none of them arrived at this approach. However, when the solution was shown to them, the reaction was: WOW, how could I miss that? The BGCs that stand up for a learner when they examine a figure and the ones they finally choose to implement in their solution certainly depend on learners’ previous experience and knowledge, but the cognitive mechanism of insight is one of the greatest puzzles of human psychology.

Visualization in Non-Euclidean Settings In section “Visual Fallacies,” we touched upon visual fallacies in the ordinary setting, accessible to human sensation. Sections “Visual Constructions” and “Visual Reasoning” illustrated the role of formalism and rigor in interpretations, explanations, and support or dismissal of our visual perception of geometrical properties in the Euclidean plane. In this section, we discuss non-Euclidean geometries. They offer many properties and statements that are unusual, seemingly weird, or counterintuitive in comparison with the Euclidean case. Once again, we scrutinize the roles of both mathematical formalism and visual insight in such settings. “The Elements,” written by Euclid circa 300 BCE, introduced five postulates, nine axioms, and definitions/descriptions of objects by listing basic visual properties of the space, as they are perceived by humans. From them, Euclid deduced many other geometrical properties (theorems) of the space. Euclid’s postulates embrace the following observations: (1) one can draw a straight line (segment) between any two points; (2) any such segment could be extended to form an (infinite) straight line; (3) in a plane, one can draw a circle with any given center and radius; and (4) all right angles (defined as a half of an open angle formed by a straight line) are equal. A modern reformulation of the fifth postulate asserts: in a plane, given a straight line and a point not on it, a unique straight line can be drawn such that it does not intersect a given line and passes through the point. Many attempts of mathematicians over about two millennia to clarify whether or not the fifth postulate is a consequence of the first four postulates resulted in a realization that its alteration will lead to different geometrical spaces, regarded as non-Euclidean. The case most readily accessible for visualization is a sphere 2, the 2D surface of a 3D ball. Spherical trigonometry was of interest since antiquity in particular due to its importance for navigation and astronomy. However, a rigorous comparison between 2 and 2D Euclidean plane ℝ2, required such notions as geodesic, angle, distance, metrics, and space introduced within a more general mathematical theory.

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Fig. 13 Triangles on a sphere as an intersection of two-sided polygons

On 2 , a straight line (geodesic) is defined as an arc of a great circle (formed by the intersection of the sphere and a plane passing through its center). An intersection of the sphere with any other plane forms a circle. The angle between two lines can be measured as the angle between their tangents at the point of intersection. Here we rely on the extrinsic view of the 2D sphere as being placed in the 3D Euclidean space in order to imagine the surface, curves and tangents. However, in the intrinsic geometry of 2 , the distances between points on the sphere are measured along the surface rather than along a line segment connecting these points in 3D. With this realization, one can reinterpret the first four of Euclid’s postulates on 2. At the same time, every straight line passing through a point off a given straight line on 2 will intersect that line in exactly two points, making the fifth postulate invalid. Instead, two pairs of identical two-sided polygons will be produced by any two intersecting great circles. Two-sided polygons are clearly impossible in the Euclidean plane. Assuming the radius of the sphere to be 1 unit, its surface area is 4π square units. Then, the area of a two-sided polygon on the sphere (whose two angles are always A equal and the area is propositional to its angle), is SðAÞ ¼ 2π 4π ¼ 2A. Here A is the radian measure of the angle of the two-sided polygon. Further, one can view any geodesic triangle ABC on 2 as an intersection of three two-sided polygons with angles A, B, and C. As noted above, two-sided polygons occur in pairs. The three pairs of two-sided polygons (with angles A, B, and C, respectively) cover the entire sphere while overlapping over the original triangle ABC and its exact copy on the opposite side of the sphere (Fig. 13). From additive property of area, we have: 2S(A) + 2S(B) + 2S(C) ¼ 4π + 4S(ABC). Consequently, the area S(ABC) of a geodesic triangle ABC is the difference between the sum of its angles measured in radians and π: S(ABC) ¼ A + B + C  π. This is another striking distinction from the Euclidean case, where the difference given above is always zero. Further, on a sphere, one can imagine walking from a point on the equator to the North Pole, then turn by a quarter of a circle and move south down to another point on the equator and finally, walk along the equator to the starting point. The resulting

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path is a triangle with three right angles. In addition, if a person follows this very path without turning her head, if she starts facing north, she will come back facing another direction (east or west). This phenomenon occurs only in non-Euclidean spaces and is called holonomy. While a 2D Euclidean plane ℝ2 is flat, a sphere has curvature. Due to the symmetry, that curvature is the same at any point of a sphere. One can also observe that the bigger the radius, the smaller the curvature of a sphere. A sheet of paper can serve as a model of part of a Euclidean plane. This sheet could be bent into a portion of a cylinder or cone; on that reason, both the cylinder and cone are said to have a zero curvature, even though they may not look as flat as a plane. The curvature of a sphere differs from them, because a sphere cannot result from bending a paper sheet. These visual ideas, in particular the intuitive notion of curvature, are formalized by means of vector calculus and differential geometry, such as the parametric description of curves and surfaces, tangent and normal vectors, etc. Within this formal apparatus, precise notion of the Gaussian curvature is introduced, consistently with the idea that the unit sphere has curvature 1 at any point. The Gaussian curvature of an ellipsoid is positive but not constant (it differs from point to point, remaining positive). Some points on the torus have positive, some negative, and some zero curvature. The Gaussian curvature of a hyperbolic (one-sheet) hyperboloid is negative but not constant. Now, by analogy with the sphere, one can formally introduce a surface whose Gaussian curvature has the same negative value at all points, a 2D hyperbolic plane ℍ2 possessing Bolyai-Lobachevskian geometry. However, the question is, how to visualize this object? Since the visualization of ℍ2 is not so straightforward as in the case of positive constant curvature, mathematicians have developed several models – each of which has its advantages and limitations (Henderson & Taimina, 2005/2020). This includes Beltrami’s pseudosphere (tractricoid), Beltrami-Klein disc, Poincare disc, and Poincare half-plane. Neither of the models can truly depict all features of the entire hyperbolic plane, but different models can be employed in different situations to highlight a specific property or provide a basis for comparison with the Euclidean and spherical cases. Thus, define a square as a quadrilateral with four equal sides and four equal angles. In the Euclidean case, this leads to the property that each corner of a square is a right angle. A Euclidean plane can be tessellated (covered without gaps and overlaps) by squares, where four squares meet at one grid point. In contrast, there exists a square tiling on a sphere where only three squares meet at one grid point. This could be visualized by deforming the surface of a cube into a sphere and transforming faces into the geodesic square tiles on that sphere, as illustrated in CodeParade (2020). Note that each angle of such square is 120 degrees. An attempt to merge five squares at one grid point will lead to a 2D hyperbolic plane ℍ2 (Order 5 square tiling, 2021). Several properties of ℍ2are opposite to ones of the sphere 2. In particular, in ℍ2, there are at least two straight lines (geodesics) that pass through a given point and do not intersect a given line not passing through the point. The area of any triangle in ℍ2 is proportional to the difference of π and the sum of its angles (which is always less than π), and there exists triangles with all angles being zero. Two examples of the

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Fig. 14 Congruent triangles in the Poincare half-plane model

latter are given above (Fig. 14) in the Poincare half-plane model. The two triangles in Fig. 14 are congruent; their perimeter is infinite while the area is finite. To see these properties, one needs to enhance their visual perception with related theory and the model’s assumptions. The visual models mentioned above have been useful for mathematical studies, but they do not offer a direct experience of the hyperbolic plane, similar to one that is afforded by the sphere. There was a desire for a physical hand-made model that would capture the intrinsic nature of the hyperbolic plane and provide a good approximation to it. In 1970s, William Thurston proposed the following construction: “Cut out many identical annular (‘annulus’ is the region between two concentric circles) strips. Attach the strips together by taping the inner circle of one to the outer circle of the other. It is crucial that all the annular strips have the same inner radius and the same outer radius, but the lengths of the annular strips do not matter.” Since the radii of the two circles forming the strip are distinct, if one holds the middle strip flat, then the edges of neighboring strips will bend or riffle. “The resulting surface is of course only an approximation of the desired surface. The actual hyperbolic plane is obtained by letting the strips’ width approach zero while holding the radius fixed. [. . .] Daina [Taimina] discovered a process for crocheting the annular hyperbolic plane as described in Appendix A” (Henderson & Taimina, 2005/2020, p. 68). Interested readers may visually compare several crocheted hyperbolic spaces of various radii provided in the reference above, noticing the reciprocity between the radius and the apparent curvature of the space. Oftentimes, “the natural feeling of the surrounding Euclidean geometric world hinders the experience and exploration of other geometric settings” (Skrodzki, 2020, p. 6). In this case, one can avail of modern technology and devices such as shutter glasses. Indeed, “virtual reality can eliminate this coexistence between different geometrical setups by providing a fully immersive experience of the non-Euclidean space” (ibid, p. 6). Implementation of 3D hyperbolic space in virtual reality (see, e.g., Segerman, 2017) allows users an interactive way of experiencing such concepts as holonomy and divergence of trajectories. Thus, completing a closed path may change the orientation of the space around the viewer and objects’ size “grow exponentially as the user moves towards them” (Skrodzki, 2020, p. 9). Similar to tiling 2D planes with squares, cubes can tessellate the 3D space. In contrast with 3D Euclidean space, for a specific magnitude of negative curvature, the user will observe, “not four but six cubes fit around the edge [formed by walls]” (ibid, p. 11). These experiences make “the non-intuitive [ideas] accessible to users without a thorough understanding of mathematical formulae” (ibid, p. 10).

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New geometrical spaces deviate from their Euclidian sibling because they emerged from visual and embodied human practices that were theorized, generalized, and altered. The study of non-Euclidean geometries, while significantly relying on mathematical formalism, calls for supporting embodied experiences. Being produced from an entirely formal, axiomatic approach, non-Euclidean models nevertheless found many applications in astrophysics, engineering, architecture, and arts (Dunham, 2012). For example, they appear to be useful for depicting such phenomena as light bending in the vicinity of black holes. Thus, the formal geometrical models are acquiring new metaphorical, physical, and visual interpretations.

Conclusion Seeing is both detection and interpretation of visual signals. Processing of visual (and other) information by the brain leads to the emergence of mental images. “Descartes, in Discours de la method (1637), argued that the ability to create mental representational frameworks is innate” (Jackson, 2002). Remembering a walking route is considered to be an ordinary ability, which may, however, be seen as conjoint with other abilities such as perceiving spatial relations between objects’ positions and organizing them in mental maps. Studies of adults responding to tasks in which they were given verbal description of relative positions of objects placed on a table revealed that the subjects used visual mental models rather than a pure logical approach to produce their deductions (Byrne & Johnson-Laird, 1989). Students trained in performing mental arithmetic calculations with big numbers reported that they imagined working with an abacus as if they were using the physical tool, the phenomenon called mental abacus (Frank & Barner, 2012). While visual ability has the major impact for most people, all senses are involved in creation of mental images. This is confirmed by studies of visually impaired people, including blind mathematicians, who are accustomed to tactile information. For example, Jackson (2002) writes about geometer Bernard Morin, who “can, after manipulating a hand-held model for a couple of hours, retain the memory of its shape for years afterward.” Morin was blind since he was six. In the course of his mathematical research, he had made clay models occurring in his imaginative process of sphere inversion in order to “communicate to the sighted what he sees so clearly in his mind’s eye.” According to Morin’s own description, “Our spatial imagination is framed by manipulating objects [because] you act on objects with your hands, not with your eyes.” Sossinski (1999) observes, “it is not so surprising that many blind mathematicians work in geometry” because “the blind person (via his other senses) has an un-deformed, directly 3-dimensional intuition of space” (as quoted in Jackson, 2002, p. 1249). Mental spatial constructions based on analysis of information received through senses and actions allow “seeing the unseen” (Arcavi, 2003) for both blind and sighted learners of mathematics. People differ in their ability to understand and use visual materials. Moreover, “individual differences in types of imagery, quality and quantity, preference for and skill in using, persist through the school years and possibly through lifetimes,

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without evidence of general developmental trends in forms of imagery or in their personal use” (Presmeg, 2006). Visualization ability does not seem to develop by itself as individuals mature: “it is fallacious to assume that students’ knowledge of the properties of diagrams will increase substantially with age” (Diezmann, 2005). For this reason, visualization as a skill should be explicitly taught. “Students, especially if unaccustomed to a visual approach, may not find it helpful and the benefits of this approach cannot be taken for granted” (Nardi, 2014, p. 194). Examples presented in this chapter illustrate the importance of reasoning, analysis, and the sense of internal structure in working with visual images in mathematics, for both researchers and learners. Research on visualization has also an epistemological aspect (Giaquinto, 2020). What is the role of diagrams in developing mathematical knowledge and understanding? What is the role of visual aid when a proof is either constructed or followed by a scholar or learner? Can a figure constitute a proof by itself? Brown (1999) regarded some pictures as “windows to Plato’s heaven.” Indeed, there are diagrams presenting compelling proofs that basically “speak for themselves,” but this is rather an exception. Some great examples of proofs without words are collected in Nelson (1993). However, in contrast with verbal proofs, which outline the path from the premises to the conclusion, these figures are static. They present a wealth of information at once, so it becomes the viewer’s responsibility “to establish what is important (and what is not) and in what order the dependencies should be assessed” (Borwein & Jorgenson, 2001). Moreover, it could be the case that just by looking at these figures, “one cannot even understand how most of these ‘pictureproofs’ function as representations of mathematical objects, much less as valid mathematical arguments” (Hanna & Sidoli, 2007). Similarly, Davis (1993) proposes that formulas and explanation always must accompany figures, but he also believes that figures themselves encompass some information, which would be difficult to express otherwise, at least with the same explanatory effect on the reader. Oftentimes, however, figures are regarded as an optional addition to a rigorous symbolic-verbal proof; they aim to facilitate understanding of the proof by illustrating or summarizing certain aspects of it. This brings us to the following questions. In which cases are figures superfluous? And when are they perceived as an essential and reliably rigorous part of mathematical reasoning and proof? Examples of diagrammatic proofs that are neither superfluous nor replaceable exist in geometry and logic. Due to soundness of formal diagrammatic systems in these areas (Shin et al., 2013), a series of diagrams would constitute a derivation in the system if each next diagram were a permitted alteration of the previous diagram. Miller (2001) gives a pure diagrammatic version of Euclid’s proof of existence of an equilateral triangle on any segment. This proof consists of nine figures: the first shows a segment, the second adds a circle centered at one end and passing through another end of the segment, etc. until the last figure that shows the constructed triangle with all sides marked as equal. Casselman (2000) recommends an animation consisting of 16 consecutive figures showing instances in a possible

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proof of Pythagoras theorem. However, even in these step-by-step expositions, the viewer has to make some visual judgment about relations of the figures’ components, and in this respect, some verbal explanations could be helpful for orienting the viewer. In general, diagrams may be considered unreliable due to the following possibilities. They may be misleading in either deceiving our visual judgement or presenting verbally described construction incorrectly as it was demonstrated in section “Visual Fallacies.” As well, one may overlook a nontypical case by making certain visual assumptions. For example, visual inspection reveals that all rectangles have exactly two lines of symmetry. However, a square is a rectangle with more than two lines of symmetry. There is also a risk of unwarranted generalization because by using a figure, one may unwillingly rely on a detail not common to the whole class of objects under consideration. Thus, one may assume that an altitude dropped from a vertex on the opposite side of a triangle lies inside this triangle; however, this may not be true in the case of obtuse triangles. There are some common ways to ensure safety of implications, for instance, considering objects that are similar (e.g., circles, cubes) or proving by cases (e.g., treating separately the cases of obtuse, acute, and right triangles). The issue of reliability and rigorousness of diagrammatic proofs is further discussed in Barwise and Etchemendy (1996) and Jamnik (2001). Awareness of hidden assumptions and possible pitfalls in reasoning leads to a more general epistemological question: “what constitutes a proof, and for whom?” Such consideration naturally calls for a framework embracing different roles of proof in mathematics (Hanna, 2000; De Villiers, 1999). Mental and physical figures may be generated through bare human senses, as well as assisted with optical and other devises. Modern computer technology allows visual representation of data, modeling, and manipulation of mathematical objects that otherwise would not be possible (Palais, 1999). This includes implementation of hyperbolic spaces in virtual reality (cf. section “Visualization in Non-Euclidean Settings”). Some computer-generated images carry both scientific and artistic value (see, e.g., Peitgen & Richter, 1986). Even if a figure is not accepted as a proof, its mental counterpart may present a way to find a proof strategy and a way to negotiate its idea with others. From the cognitive perspective, visualization in many aspects aids in the process of truth discovery viewed as an individual act of establishing an internally consistent system of beliefs and belief-forming dispositions (Giaquinto, 1992, 2007). Visual images bring to mind relevant beliefs and trigger particular dispositions. An image reminds the viewer of a spatial property confirmed by all their senses and experiences. The viewer, in their role of either a learner or researcher of mathematics, possesses the image tied to a mathematical object. The links are produced by the viewer’s mind in the process of embedding the object in their entire mental universe by means of geometrical construction and reasoning while staying alert to possible inconsistencies. Thus, images formed in the process of conceptualization serve as a gateway to the mathematical objects that they represent within the viewer’s conceptual system. In this way, mathematical visualization becomes subjectively trusty.

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Numerical Abilities in Nonhumans: The Perspective of Comparative Studies

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Rosa Rugani and Lucia Regolin

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Background of Animal Numerical Abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proto-numerical and Numerical Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arithmetic Abilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial Numerical Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonhuman Animals Associate Numbers with Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number-Space Association in Magnitude-Estimation Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The history of the study of animal numerical cognition is characterized by an unfortunate start: the Clever Hans story, which caused a widespread skepticism across the scientific community. In the last decades, a growing body of evidence demonstrates numerical skills in nonhumans; nonetheless, studies have focused on adult subjects. Here, it will be discussed numerical comprehension in day-old chicks, a model species that allows an insight on the early development of cognitive abilities. Newborn chicks discriminate between diverse numbers, solve arithmetic calculations, and use ordinal information. This animal model allowed to also unveil another peculiar aspect of numbers: their association with space. This ordered representation of numbers in space is known in humans as the mental number line (MNL) and refers to an ascending mapping of numbers from left to right. Chicks associate smaller numbers with the left and larger numbers with the right space. The paradigm used to test spatial numerical association (SNA) in chicks has been proficiently used to also assess this association in R. Rugani (*) · L. Regolin Department of General Psychology, University of Padova, Padova, Italy e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_39

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human newborns, providing a suggestive example of animal research inspiring developmental studies. Overall findings from 1- or few-day-old chicks, virtually naïve or reared under strictly controlled conditions, constitute a strong case for the claim that numbers are a primitive and inherent information processed by animals. Keywords

Numerical cognition · Number sense · Numerical discrimination · Ordinal information · Arithmetic · Spatial numerical association · Mental number line · Domestic chick

Introduction Mathematics includes abstract and complex operations based on the use of specific symbols and rules, which only educated humans are able to master (Gallistel & Gelman, 1992; Dehaene, 2011). In spite of this, all adult humans are still able to master “non-symbolic” numerical tasks. This can be tested under specific experimental conditions. Syllable repetition prevents language use while solving nonsymbolic numerical tasks, which could consist of estimating which of the two arrays depicts a larger number of dots (Cordes et al., 2001). The extrapolation of numerical information from a visual or auditory stimulation, without any possible support from the symbolic system, is defined as a nonsymbolic ability (Feigenson et al., 2004; Rugani et al., 2017). Nonsymbolic processing is preserved in educated humans and can be compared to the performance of nonsymbolic individuals such as preverbal infants and nonhuman animals (Cordes et al., 2001; Cantlon & Brannon, 2007). Intriguingly, similarities have been found in the nonsymbolic numerical system across species. When the difference between two numbers increases, the task becomes easier (distance effect). For example, it is easier to decide that an array depicting eight dots is larger than another depicting two dots, and it is more difficult to assess that an array of eight dots is larger than one depicting seven dots. Another effect, the size effect, describes an increase of performance as the magnitude decreases. This means that, keeping the distance constant between two numbers, it is easier to deal with smaller magnitudes rather than larger ones. For example, it is easier to discriminate between 2 versus 3 dots than 80 versus 81 dots. The distance and side effect have been described in humans and nonhuman animals (Cantlon & Brannon, 2007; Scarf et al., 2011). As a result of combining the two effects, the ease of distinguishing two numbers depends on their ratio and not their absolute magnitudes. A numerical task becomes easier as the ratio between two numbers decreases. This is directly measured both as increased accuracy and as a reduction of response times. On the other side, the task becomes more difficult whenever the ratio increases: accuracy declines and response times increase (Gallistel & Gelman, 1992). Since similar effects have been shown in a variety of animals, humans

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included, the presence of an ancient nonsymbolic numerical system, shared across species and able to deal with numerical and quantity estimation, has been hypothesized (Cantlon & Brannon, 2007). Even if the link between the nonsymbolic and symbolic numerical system is still unclear, it seems that the first could serve as a building block onto which the abstract and purely symbolic knowledge may develop (Feigenson et al., 2004). In support of this view, it has been proven that nonsymbolic number sense in 6-month-old infants can predict mathematical ability in first graders (Starr et al., 2013) and that training the nonsymbolic number sense increases proficiency in mathematics (Park & Brannon, 2013). Nonsymbolic mathematics appears to be deeply connected to an ontogenetically precocious and evolutionarily ancient “number sense” (Dehaene, 2011), which arises in the first days of life (Butterworth, 2010). Nonsymbolic number sense would play a primary role in achieving abstract and symbolic mathematical thinking. A deeper knowledge of the nonsymbolic number system could allow for early identification of children who may develop an impaired mathematical comprehension and thus pave the way for educational interventions to increase their number sense, even before formal schooling.

Historical Background of Animal Numerical Abilities Animal numerical cognition attracted scientific interest in the twentieth century; nevertheless, a substantial growth in knowledge of comparative numerical abilities has occurred in the last three decades. The huge gap of interest in animal numerical studies was mainly caused by an unlucky beginning. In the early twentieth century, a horse, known as Clever Hans, was trained by his owner, Wilhelm von Osten, to solve several calculations, which comprised additions, subtractions, divisions, and square roots. In a typical examination, Hans stood in front of its owner, or another interviewer, who showed him a mathematical operation. Hans answered by tapping his hoof on the ground the exact number of times. Initially, the horse convinced a large majority of scientists of his arithmetic skills. A few years later, it turned out that Hans responded correctly only if he could see his interviewer while performing the task, so his answers were wrong when the interviewer was hidden. Moreover, if the interviewer knew a wrong answer for a specific mathematical question and Hans could observe his face while responding, he tapped his hoof on the ground the wrong number of times (Pfungst, 1907). Hans’s intelligence was not mathematical but social: even if he did not master mathematical reasoning, he was amazingly skilled at detecting and interpreting minimal behavioral gestures, subconsciously produced by his interviewer, upon reaching the correct number of taps. Such new evidence revolutionized Hans’ reputation, and he was no longer considered an intelligent animal. At the same time, an impressive skepticism towards animal numerical skills pervaded the scientific community, which was no longer predisposed to believe in animals’ numerical capabilities. Scientists aligned with the Aristotelian thought, which postulated that logic, represented in its higher form to numerical comprehension, was strictly intertwined with language. Thus, “logos” was considered a

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prerogative of human minds. Language and symbols were the prerequisites for cognition, and all creatures unable to grasp them, principally infants and animals, were considered incapable of any numerical comprehension. For several decades, only very few researchers were still persuaded that animals could possess basic but adaptive numerical understandings. They performed rigorous and elegant tests, avoiding face-to-face contacts, and controlled for several nonnumerical cues onto which animals could base their responses. The German zoologist Otto Koehler, around 1943, was the first to study numerical abilities in various bird species. He showed that a pigeon, Columba livia, could learn to select a set containing a certain number of grains to receive a food reward. For example, pigeons could choose the set containing four grains when it was paired with another set comprising either a smaller or larger number of grains. A jackdaw, Corvus monedula, succeeded in identifying a given number in a matching-to-sample task. The bird was initially presented with a card depicting a “sample” number of dots. Then the bird faced two dishes each covered by a lid. One lid illustrated the sample number of dots, and the other illustrated a different number. The jackdaw correctly matched the sample number with the corresponding one on the lid; in other words, the bird selected the lid depicting the number shown on the card, especially when the dimensions of the dots depicted on the card differed from those illustrated on the lid. Koehler also investigated action enumeration in pigeons and budgerigars, Melopsittacus undulatus. The birds learned to peck only a preestablished number of times; for example, if they were trained to peck only four times, they ate only four seeds when presented with 10. From his experiments, Koehler concluded that the birds had mastered simple abilities such as number discrimination and action enumeration. Moreover, he noted that various species exhibited incredible similarities in the upper numbers they could deal with: they mastered the tasks only when evaluations of about five or six items were required (Koehler, 1943). In all his studies, Koehler accurately avoided providing animals with any cue that may have triggered the infamous Clever Hans effect. To avoid direct contact between birds and the experimenter, he hid himself behind a panel and out of the birds’ sight. To prevent the experimenter from unconsciously biasing the birds’ behavior, he used mechanical devices to dispense rewards. Moreover, these were filmed to provide objective records of the experiments. However, despite his experimental rigor, Koehler’s evidence, as well as other researchers’, was criticized for its lack of controls on quantitative cues, among others: odor, brightness, size, color, shape, texture, and similar. It is worth mentioning that the numerical skills investigated by Koehler were much easier than the mathematical calculations presented to Clever Hans. The numerical magnitudes were smaller, and the required numerical computations were simpler; likewise, there was a discrimination of various numbers of dots in Koehler’s experiments versus symbolic arithmetic operation in Hans’ tasks. Koehler’s tasks involved more ecologically relevant numerical estimations, which can improve animals’ fitness in their natural environments. Different from Hans’ symbolic calculations, Koehler’s findings fall into the nonsymbolic category. Nevertheless, skepticism towards animal numerical competences persisted, and the general belief was that although animals could learn to deal with numerical

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problems, they would only use numbers as a last-resort strategy. Thus, animals were considered able to process numerical cues only when all other nonnumerical cues such as odor, brightness, size, color, shape, texture, and similar were prevented. Moreover, animals were considered incapable of a spontaneous use of numbers – instead, this was possible only after extensive trainings (Davis & Pérusse, 1988). In the last three decades, growing evidence has proven various kinds of nonsymbolic numerical competences in nonhuman species. Interestingly, some of these can also be shown in the absence of specific numerical training. Lions use sounds to decide whether to attack a potential troop of rivals. Hearing recorded sounds of potential rivals during controlled trials, lions decided to attack only if their own troop was double their rivals. Such behavior requires initial numerical estimation of their own and external troops and then a numerical discrimination between these to evaluate if the attack could be a convenient strategy (McComb et al., 1994). Breeding and nesting behaviors are other ecological contexts in which animals use numerical cues. American coots, Fulica americana, compute the number of their own eggs, neglecting the parasitic ones, to reach an optimal number of their nested eggs. This implies a certain kind of enumeration: they could “count” the number of their own eggs, separating them from the parasitic ones (Lyon, 2003). Also, foraging has been used to assess animal numerical comprehension and to compare across species. Facing diverse food options, animals choose the one ensuring the highest energetic gain. This spontaneous behavior has been exploited to explore numerical discrimination in the absence of any numerical trainings. When facing two foraging alternatives, each containing the same food but in differing quantities (e.g., one comprised of two and the other three bites of food), diverse species chose the bigger one (see Rugani (2017) for a review and the protonumerical and numerical discrimination of this chapter for a detailed description). These outcomes suggested that various species share the capability of discriminating two “numerousnesses.” These are specifically labelled as “numerousness” and not as “numbers,” since the number of food pieces covaries with several quantitative cues in these studies. Whenever a rigorous control for the possible use of quantity information was not performed, animals could equally use numerical and quantitative cues. Thus, it is appropriate to consider these abilities as proto-numerical. Nevertheless, their scientific relevance is high, since they proved that animals, in various contexts, spontaneously use proto-numerical cues to optimize their fitness in the absence of any numerical training. On the basis of these and other evidence, it has been proposed that numerousness estimation is one of the core knowledge systems that might have developed, under specific evolutionary pressures, to better interact with the natural environment. A current challenge in numerical cognition consists of disambiguating the relative role and importance of numerical versus quantitative cues and the way they interact in making numerical estimation. It is worth noticing that these are two sides of the same coin. The assessment of abstract numerical abilities requires controlling for all nonnumerical variables that can affect performance. On the flip side, the inner essence of nonsymbolic numbers is intrinsically related to their physical dimensions. A deeper understanding of this aspect requires reflection on what a nonsymbolic

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number is, namely the extraction of numerical information from a perceptual stimulus (Feigenson et al., 2004; Rugani et al., 2017). Being that perceptual entities, mainly objects and sounds, exist in the physical world in a specific time and space, they come together with volume, mass, surface area, and other quantitative measures. From this perspective, nonsymbolic numbers can be considered as a portion of a general system that estimates quantities, whether continuous, such as area and volume, or discrete, such as numbers (Gallistel, 2011). A large majority of researches have been conducted on altricial animals, which are almost impossible to test early in life, limiting the comprehension of developmental aspects of numerical cognition. Nevertheless, a peculiar animal model, the domestic chick, Gallus gallus, permits to disentangle “nature” from “nurture.” Such a precocious bird has shown a variety of numerical cognitions that animals can master from their first days of life, which will be described in the rest of this chapter.

Proto-numerical and Numerical Discrimination A very basic numerical ability displayed by animals is the capacity to compare two sets and judge their relative difference in magnitude – that is, which is “more” and which is “less” (Vallortigara, 2018). Such relative numerical discrimination leads animals to make a choice in favor (usually) of the larger amount of resources (Davis & Pérusse, 1988). In the laboratory, this ability is assessed by placing the animal in front of a choice among two stimuli featuring different numerousness of relevant objects (e.g., prey, conspecifics, and food items). In this paradigm, nonnumerical (quantitative) variables correlate with numerical cues, and choice of the larger number is confounded with choice of the larger amount. An interesting solution to control for nonnumerical cues has been that of presenting to the animals sets in which some of the elements were not fully visible, as they were partly hidden by occluders. This way the elements could still be perceived in full (through the perceptual process of amodal completion), but it was possible to manipulate and equalize at once the physical amount of their surface. For example, in the 2 versus 1 comparison, one object was fully visible on one side, whereas on the other side of the choice area, where the two objects should be presented, these were masked so that only half of each could actually be seen. Chicks, like humans and many other species, have the ability to complete the missing parts of objects when these are hidden behind other objects (occluders). As a result, the object that is only partly physically visible is still perceived as a whole object. This perceptual mechanism, deemed “amodal completion,” was first described in our own species: a circle with a missing sector, or even a missing half, is still perceived as a full circle, and we can effortlessly recognize the letters in a word/sentence even after they have been masked. The same mechanism is also well attested in nonhuman species, including the (very young) domestic chicken (Regolin & Vallortigara, 1995). By exploiting amodal completion to control for nonnumerical cues, it was possible to demonstrate that day-old chicks discriminate sets of objects by relying on nonquantitative cues to solve the task (Rugani et al., 2008). In this study, young chicks were trained to peck

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onto a small panel to retrieve some food reinforcement. Chicks had to discriminate between 1 versus 2 or between 2 versus 3 elements, and the elements were identical shapes depicted onto two different panels. For each comparison (e.g., 1 vs. 2), one group of chicks was reinforced for selecting one element and ignoring the two elements’ stimulus, and the other half of chicks was only reinforced when pecking onto the panel depicting two elements. During training, the birds could make use of numerical “as well as” quantitative (overall area or perimeter) cues to discriminate among the two stimuli. Thereafter, in a number of tests, chicks had to choose among the two stimuli experienced during training, but this time the quantitative cues were controlled. For example, when the elements presented during the test were displaced in various positions on the panel, this control for spatial distribution did not affect the chicks’ ability to select the stimulus that had been reinforced during training. In spite of the fact that most trials were unrewarded during the tests, this result was clear cut (i.e., they were “in extinction”). A key control consisted of equalizing at once both the contour length and the overall surface that could be seen in each of the two compared stimuli. In the stimulus depicting the smaller number of elements, these were entirely visible, whereas some of them could only be seen partially in the larger set of elements (but were visually and amodally completed by the observer). Basically, the exceeding area or perimeter that should have been present in the larger stimulus corresponded exactly to its occluded, hence missing, parts. For example, in the 1 versus 2 comparison, the occluder hid exactly half of each of the two dots of the stimulus representing 2 (the same occlude was also present in the other stimulus, but it did not overlap the single dot). Even in this case, chicks successfully identified the stimulus that had been reinforced during training (S+) in the 1 versus 2 and 2 versus 3 comparisons. However, with a larger set of comparisons (3 vs. 4 or 4 vs. 6), chicks failed to acquire the discrimination during training, in spite of the available possibility to use both numerical and nonnumerical cues at once (Rugani et al., 2008). Therefore, the ability of very young domestic chicks is attested to spontaneously encode the numerical cue during training (at least for sets up to three elements), in spite of the fact that quantitative cues were also available and could by themselves suffice to solve the discrimination learning task. In fact, chicks could still perform the discrimination successfully when, during the test, all other nonnumerical cues were controlled. This means that numerical cues are a natural and significant cue for animals: they are rather easily encoded and effectively used, even early in life, by the young chicks and possibly by other species. In this first set of experiments, a brief training of about 1 h was required to unveil chicks’ abilities to process numerical cues. However, would the young chicks be able to encode numbers truly spontaneously (i.e., in the absence of any training at all)? Animals do discriminate between various amounts of relevant objects, such as food patches, and usually prefer to go for the larger quantity. Several studies showed that animals prefer the more abundant source of food or energy according to what was predicted by an optimal foraging strategy. For example, salamanders prefer 2 to 1 and 3 to 2 mosquitos (Uller et al., 2003). Frogs prefer 2 to 1, 3 to 2, 6 to 3, and 8 to 4 prey (Stancher et al., 2015). Similar evidence was reported for birds (robins discriminating 6 vs. 8, 8 vs. 64, and 16 vs. 64; Garland et al., 2012). Some studies

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also assessed discrimination in nonhuman apes (orangutans, with comparisons of sets in a range from 1 to 6, Call, 2000) and human infants (1 vs. 2 and 2 vs. 3; Feigenson et al., 2002). Recently, numerical discrimination abilities have also been described in a variety of invertebrate species (Bortot et al., 2020), confirming how these abilities likely confer an evolutionary advantage to all living species. In this kind of study, differences in number covaried with differences in nonnumerical (also deemed “continuous” or “quantitative”) cues, such as volume, surface area, contour length, occupancy, density, and so on. This renders it impossible to prove that animals can process numerical information by itself. Some studies have attempted to run controls for continuous cues, but in all cases, at least some of the nonnumerical variables remained available to the subjects, offering an alternative explanation for the results. For example, animals may have relied on the perimeter when the total surface had been equated, and they may have relied on the surface when the overall perimeter had been equated. In an attempt to find a method to test the truly spontaneous abilities of animals in the absence of any extensive or supervised formal training and to convincingly control for the nonnumerical variables at the same time, the authors designed a new paradigm for newly hatched domestic chicks, exploiting the phenomenon of filial imprinting. Chicks in fact remember the fine features of the relevant social objects they have been reared together with soon after hatching (a few hours of exposure usually suffice to create a strong social bond and memory for those objects). These objects in the natural condition are real conspecifics (the mother hen or the siblings), but, in the laboratory, chicks can socially attach (imprint onto) artificial objects (e.g., 3D objects with surface, volume, etc., can be effectively controlled experimentally). Newly hatched chicks were reared for 2 days with various numbers of objects and then tested them on day 3 of their life for their spontaneous choice among the familiar and novel numerousness of those same objects. Chicks could freely choose which set to approach during the 6 min of the test. Chicks showed a preference not for the familiar but rather for the larger numerosity in the two sets presented. This means that chicks that had familiarized with the smaller set during the test chose the larger set. Interestingly, the choice was not based on the absolute numerosity of the test stimuli; rather, choice was based on the numerosity of the familiar object, which emerged when the number of novel objects was introduced so that the numerosity of the two possible alternative sets during the test was equalized (1 novel object was added to the group of 3 and 3 novel objects to the group of 1). The novel objects were identical in size and shape and only differed slightly for their color shade. Chicks obviously discriminated the familiar objects and chose the larger group of those objects. In this type of experiment, discrimination could of course be based either on numerical or nonnumerical cues because the overall surface area and volume covaried with the set size. To disentangle this issue, objects of varying size were used so that in each comparison (1 vs. 4, 1 vs. 6, and 1 vs. 3), either volume or surface area was equalized (it was unfortunately impossible to control for all continuous variables at once within the same condition). This also means that the

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size of the one object presented in the smaller stimulus was always larger than the size of the objects presented in the larger sets used as comparisons. In all these tests, chicks were shown to discriminate, but they approached the single larger object, possibly because this provided some sort of supernatural stimulus, resulting as more interesting or attractive than the other sets. An alternative strategy to control for nonnumerical cues (and avoid the presence of larger objects in one set) is to use sets made up of all heterogeneous objects (each differing from all others in terms of color, shape, and size). The first group of chicks was familiarized with two objects and another group with three objects. During the test, objects employed were completely different from those used during rearing in terms of their color, shape, and size. The overall surface and volume of the two groups presented during the test were also equalized. Under these conditions, both groups of chicks preferred the set comprising the familiar numerosity, and this also means that chicks reared with the smaller set (i.e., two objects) approached the set containing two objects being tested and not the set with the larger numerosity. Since no other cues are available, it can be concluded that the chicks do rely on numbers to discriminate between two sets (and choose the familiar numerosity). To summarize, chicks familiarized with a given number of objects then • Approached the larger set with objects all identical in size • Approached the set with the single larger object • Approached the familiar number of objects when the use of nonnumerical cues had been prevented through the use of all heterogeneous objects both at rearing and during the test In chicks, the information onto which their choice is based (number vs. quantity) seems to be induced by the rearing and testing conditions. This is not surprising. In fact, it echoes that which was reported in the developmental psychology literature. Infants in some studies were shown to discriminate between arrays of numbers when the overall perimeter is controlled (Xu et al., 2005), but in other studies, they showed higher sensitivity to the nonnumerical, rather than numerical, variables (e.g., perimeter and area, Feigenson et al., 2002). Based on all of the mentioned studies, it appears that young chicks can discriminate sets of elements on the basis of numerical information but only up to 3–4 elements. On the basis of such evidence, it was originally hypothesized that young chicks only possessed a small number system, in spite of the fact that discrimination of large numbers had not been investigated in this species. Two separate systems have been put forward to explain nonsymbolic number cognition: • A system capable of representing small numbers, deemed object file system (OFS) • A system to represent large numerical magnitudes, deemed analogue magnitude system (AMS)

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The first system, the OFS, seems to be an attentional-driven mechanism to represent and process the perceived objects as individual files (a new file is created for each object) within working memory. The original purpose of this system may serve to represent objects, not enumerate them. However, since there is a 1:1 correspondence between objects and files, the system also implicitly leads to representing the number of objects “individuated” in the visual scene. The signature of the OFS is its capacity: a maximum of (usually)
4 files/objects can be simultaneously held in working memory. Processing of numerousness larger than 4 therefore needs to be supported by the second system: the AMS. This second system does not appear to have an upper limit, but it operates according to Weber’s function: as the ratio between the two numbers to be discriminated increases, response time decreases and accuracy increases (Gallistel & Gelman, 1992). In spite of this clear-cut separation between the two systems, later evidence supported the idea that the AMS would actually also process numbers in the range of the small magnitudes (Brannon & Terrace, 1998; Cantlon & Brannon, 2007; Pepperberg, 2012; Rugani et al., 2014b). However, other studies maintained that small magnitudes are selectively processed by the OFS (Feigenson & Carey, 2005). The central issue in this debate is what kind of information prompts processing via either system: whether it is numerosity per se (OFS with less than 4 objects and AMS with numbers larger than 4) or another kind of contextual information. This second possibility was first elaborated by Hyde and Spelke (2011), who highlighted the role of attentional processes: attention to the whole group of elements (triggered, for example, by simultaneous presentation of the sets) would activate AMS processing; and attention focused onto individual objects (prompted by sequential/ one-by-one presentation of the elements) would activate OFS processing. The time was come to challenge the limits of number processing of day-old chicks. Their ability to succeed in the small and large numerical range by employing the AMS was assessed (Rugani et al., 2014b). Comparisons used during the test involved just small numbers (less than 3) or just large numbers (larger than 4), or a direct comparison between one small and one large numerousness. This last test was devised to investigate whether a processing gap existed when bridging across the two domains was required. Chicks had learned to associate food to numbers during spontaneous foraging and feeding (no supervised training required). During their regular rearing, for about 2 days, chicks learned that food was always located behind salient environmental stimuli (vertical panels depicting a specific numerosity). The panel hiding food depicted a given number of red squares. Other similar panels were there in the cage and depicted a different number of identical elements, which was not associated with food (nothing could be found behind such panels). The position of all panels was changed regularly during rearing for each chick so that food could not be located through association with a spatial location, and the only relevant cue to the food was the number of red elements on the panel. On day 3 of life, a 6-minute free-choice test was run, during which chicks could freely choose to approach a panel depicting the previously reinforced numerosity or a second panel depicting the familiar but never reinforced numerosity. Chicks discriminated and approached the number associated with food in all circumstances: when both numbers confronted were small (2 vs. 3) or large (6 vs. 9, 8 vs. 12, and 8 vs. 14) and even when one number was small and one was large (2 vs. 8). Chicks’

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performance was confirmed also when the nonnumerical variables (overall area and perimeter) had been controlled. The results of this study (Rugani et al., 2014b) offer additional support to the fact that animals can spontaneously encode numerical information and that at least some species can do so remarkably early in life. As already mentioned, no supervised training was required in this paradigm. The association among food and the landmark cue represented by the number of elements pictured on the vertical panel took place naturally during autonomous foraging behavior. Moreover, the fact that chicks were smoothly responding also in the 2 versus 8 comparison (1 small vs. 1 large numerousness) seems to favor the hypothesis that the same system, the AMS, operates along the whole numerical continuum. This would be coherent with previous evidence on adult humans engaged in nonsymbolic numerical tasks (Cordes et al., 2001), children (de Hevia & Spelke, 2009), infants (Cordes & Brannon, 2009), and nonhuman species such as gorilla (Hanus & Call, 2007), macaque monkey (Cantlon & Brannon, 2007; Brannon & Terrace, 1998), baboon and squirrel monkey (Smith et al., 2003), capuchin monkey (Judge et al., 2005), parrot (Pepperberg, 2012), and robin (Garland et al., 2012). In Rugani et al. (2008), why did the baby chicks appear to be capable of processing numerosities larger than 3? It has been hypothesized that a key element relied on the fact that, in previous studies, the pecking response had been actively shaped (during conditioning procedures) onto a specific stimulus. The birds, in order to coordinate the pecking response (which is some sort of grasping action) towards a specific target, likely needed to focus their attention on the features of that stimulus. However, when chicks are not formally trained to peck for food, but learn spontaneously to associate the food to a nearby environmental feature, they tend to perceive the stimulus more globally. Therefore, pecking onto the stimulus, required in some paradigms, triggered processing by the OFS, and as a result, the chicks missed the discrimination of sets larger than four elements. To prompt processing of the whole collection of elements, in subsequent experiments, the stimuli were located more distantly to make them clearly visible by presenting the sets onto vertical panels. Previous studies, in fact, used this paradigm to encourage configural processing – for example, when testing the discrimination between possible and impossible objects or chicks’ sensitivity to the Ebbinghaus illusions, also known as Titchener circles (review in Rugani (2017) and Vallortigara (2018)). In all, the findings presented are in line with the idea that attentional mechanisms can differentially activate the processing systems (Hyde & Spelke, 2011). Whenever attention is focused on the individual elements, processing would be supported by the OFS; whenever attention is directed to the overall set, processing would be undertaken by the AMS.

Arithmetic Abilities Besides representing numbers, adult humans can also manipulate numerical representations to perform arithmetic calculations. For a long time, this was considered a distinctive capacity of our own species, which had to be culturally framed and was

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only possible following a long and specific education (Gallistel & Gelman, 1992; Dehaene, 2011). In recent years, scientists started to wonder whether arithmetic could be performed at a nonsymbolic level. This issue has been tackled by studying nonhuman species and young infants. The aim of those studies was to ascertain whether nonsymbolic (animals) or pre-symbolic (infants) subjects can perform simple calculations such as adding/subtracting elements from a numerical representation to obtain a new mathematically correct representation (Dehaene, 2011). Not surprisingly, the first demonstrations that animals can add two or more numerousnesses came from studies that looked at our closest relatives: apes and other primates. Chimpanzees, for example, were able to choose the more advantageous (larger) combination of two smaller amounts. Each of the four subsets was constituted by a number of chocolates from 0 to 4, and to solve the task, the chimps had to sum the chocolates of two specific subsets and compare the result with the sum of the chocolates in the other two subsets. Subjects solved the task even when the correct (i.e., the overall larger) combination did not include the single larger subset as in 3 + 0 versus 2 + 2, ruling out the possibility that animals were using a simpler strategy (Rumbaugh et al., 1988). Another chimpanzee, named Sheba, confirmed these results in a different paradigm. To obtain the food she had previously seen during three different hidings (each concealing from 1 to 4 orange pieces), Sheba was required to choose the Arabic number corresponding to the summation of the three sets. Sheba solved the task even when food items were replaced with symbols (Arabic numbers). Thereafter, two macaque monkeys were trained (Washburn & Rumbaugh, 1991) to respond to symbols representing numbers. Animals were shown two combinations of two numbers and were allowed to eat an equivalent number of food pieces as the combination they selected. Monkeys chose the larger overall number even when confronted with novel combinations. Other paradigms involved training capuchin monkeys to associate tokens to numbers of food items. Capuchins were able to select the higher number among two combinations of tokens (Addessi et al., 2007). Some bird species also displayed remarkable mathematical skills. Pigeons learned to choose the combination of two visual symbols leading to the larger reward. However, they chose on the basis of the larger total amount of food and not on the basis of the larger number of food items (Olthof & Roberts, 2000). The gray parrot Alex, who was famously able to understand and respond to many vocal requests, could answer questions such as “How many blue objects plus yellow objects?” with sets of up to six objects. Alex extracted the two relevant numerical values from a complex scene containing other differently colored objects and correctly combined the two representations (Pepperberg, 2012). Besides the abovementioned studies on adult individuals of nonhuman species, convincing evidence of nonsymbolic arithmetic came from developmental studies. A seminal paper (Wynn, 1992) showed that 5-month-old infants mastered simple summations (1 + 1) and subtractions (2  1). For example, infants who saw two toys, one at a time, being positioned behind a single panel, expected to see exactly two

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objects once the panel was removed; in fact, infants stared significantly longer at the unexpected outcomes (which overtly violated their expectations) of either one or three objects. This result demonstrated that infants master the simple addition 1 + 1, expecting exactly 2 as the result. Similarly, after having seen two toys being placed behind the panel, and then one of them being removed (i.e., the 2  1 operation), infants looked longer at the incorrect outcome of two toys found behind the panel rather than at the correct outcome (one toy). This first study did not control for nonnumerical variables, such as volume or surface area; therefore, infants might have based their responses on the numerical result and on other continuous cues that covaried with the number. An interesting control was employed in a later study employing the same paradigm (Simon et al., 1995). In this study, the identity of the toys was manipulated so that the characters found after the panel was removed were novel, even if they corresponded in number to the arithmetical calculation. This study therefore assessed whether infants react to unexpected object identity or to numerically unexpected events. Infants showed surprise (i.e., they looked longer) only at the numerically unexpected outcome, whereas the object identity switch did not violate their expectations. This interestingly indicates that subjects represented the numerical information stripped from other nonnumerical features. The same paradigm was used to investigate arithmetic abilities in nonhumans, showing that monkey species looked longer at arithmetically incorrect outcomes (e.g., 1 + 1 ¼ 1 or 1 + 1 ¼ 3, or 2  1 ¼ 2). This led to the conclusion that monkeys were expecting exactly that 1 + 1 ¼ 2 and 2  1 ¼ 1 (Hauser et al., 1996). Summation of items over time (i.e., tracking and adding items one by one) was specifically demonstrated in chimpanzees and macaques (Beran & Beran, 2004), as well as in 12-month-old human infants. Infants chose the larger set in 1 versus 1 + 1, 1 versus 1 + 1 + 1, and 1 + 1 versus 1 + 1 + 1. Remarkably, they chose at random in the comparison of a small versus a large number (1 vs. 1 + 1 + 1 + 1; Feigenson et al., 2002). Although nonnumerical variables were not controlled in the studies just mentioned, it was demonstrated that infants as young as 9-months-old can add and subtract larger numbers (5 + 5 ¼ 10 rather than ¼ 5; and 10  5 ¼ 5 rather than ¼ 10) and that their performance is maintained even when the overall area and perimeter are carefully controlled by employing computer-presented stimuli (McCrink & Wynn, 2004). Computer-presented stimuli were also successfully employed to control for nonnumerical variables in adult monkeys. They were able to select, among two arrays of dots equalized for quantitative cues, the one array which corresponded to the numerical summation of two sets previously seen (Cantlon & Brannon, 2007). Nonsymbolic arithmetical abilities assessed in nonhuman adult individuals could not be directly compared with developmental studies in human infants. The only exception to the study of nonhuman adults was the work on domestic chicks. Arithmetic in baby chicks was studied by exploiting chicks’ tendency to approach the larger group of siblings or, in laboratory conditions, familiar objects (Rugani et al., 2010). The chicks had been reared, upon hatching, one per cage, together with five identical objects, and the test took place on day 3. During the test, each chick was presented with a series of events featuring their five familiar objects moving

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around one by one and hiding in either of the two identical locations (two vertical panels). If two objects went behind one panel and three behind the other one, chicks preferred to approach the panel hiding three objects in an attempt to rejoin the larger number of familiar objects, even in the case where the objects’ size had been manipulated so as to control for the overall area or overall perimeter of the two groups. Interestingly, when the overall perimeter was equalized, an inverse correlation was apparent between the number and overall area: the larger area belonging to the smaller number set. As a result, chicks’ choice for the larger set could not be based on the overall area or perimeter. These results showed for the first time that even extremely young organisms could combine a representation of multiple elements individually experienced. Dayold chicks created and updated two distinct representations, corresponding, respectively, to the number of objects present behind each of the two panels, and then compared these representations to find the larger group (Rugani et al., 2009). Chicks could also perform simple addition or subtraction. To this purpose, they were presented with two sequences of events, such as (4  1) versus (1 + 1) (or, in other conditions, 5  2 versus 0 + 2, 4  2 versus 1 + 2, or 5  3 versus 0 + 3). The result of the first series of object displacements (in bold) could be reversed by the second series of displacements so that the panel hiding the larger number of elements may or may not be the one that hid the smaller number at the end of the first part of the events. In all conditions at the end of all events, one panel hid two objects and the other hid three objects. Other possible cues were also controlled, such as the panel where the first or last object was seen to hide, the left-right position of either displacement, and where each final numerosity was to be found. The only tenable hypothesis to explain chicks’ behavior is that the subjects created and updated two mathematically correct representations of the object in each location (Rugani et al., 2009). Notably, representations of small numerousness would not suffice in this study because some of the comparisons involved numerousness higher than 4. These findings seem therefore to support that the AMS also processes small numbers or that the two systems may be integrated (Cordes & Brannon, 2009; Brannon & Terrace, 1998; Cantlon & Brannon, 2007; Judge et al., 2005; Cordes et al., 2001; Pepperberg, 2012). This would be in contrast with evidence already reported above from infants who could choose the larger set in a series of comparisons involving one, two, and three elements, but they chose at random when confronted with 1 versus 4 (retrieving one biscuit from one box vs. four biscuits from a second box). This surprising result has been considered as evidence of a distinct system dedicated to represent small numbers (Feigenson & Carey, 2005). Since infants are capable of adding a large number (5 + 5, McCrink & Wynn, 2004, see above), for infants, it generally seems that two separate and independent systems to represent numbers are there: one for values up to 4 and the other for values >4. Nevertheless, on this topic, scientific literature has reported contrasting evidence. On the one hand, some data suggest that small numbers can be treated solely via OFS (Starkey & Cooper, 1980).

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Domestic chicks offered to this purpose a good model for the comparative analysis of the 1 versus 1 + 1 + 1 + 1 dilemma. Would baby chicks also fail in the 1 versus 4 discrimination? It seems this was not the case. Chicks that presented with sequential events involving social objects or food objects succeeded in locating the larger set of the 1 versus 4 comparison. Chicks in the same study also succeeded in the 2 versus 4 discrimination, even when the nonnumerical cues had been controlled. Overall, the evidence obtained from domestic chicks in this task supports a continuity in the processing of small and large numbers (Rugani et al., 2014b). However, when presented with large numbers (sequential presentation of objects and final choice with 5 vs. 10 or 6 vs. 9), chicks were only successful when all objects had the same size and not whenever the nonnumerical cues had been controlled for (Rugani et al., 2011a). It was concluded that with more difficult tasks, animals need to rely on a convergence of multiple cues. This would be in line with that reported for human infants employing a different paradigm (Suanda et al., 2008) and other vertebrates (Stancher et al., 2015), thus introducing the need to investigate the coordinated role of numerical and quantitative cues in nonsymbolic numerical processing.

Spatial Numerical Association One of the most outstanding characteristics of number processing, which is apparently shared between humans and animals, is their tendency to associate numbers onto space. A peculiar characteristic of numbers is their intrinsic association with space: educated humans typically represent numbers along a left-right continuum, where small numbers are associated with their left and large numbers with their right space: the mental number line (MNL) (Galton, 1880; Dehaene et al., 1993). Traditionally, the MNL has been considered a by-product of culture and education, mainly writing and reading habits (Zebian, 2005). More recently, an association between numbers and space has been reported in preverbal infants and nonhuman animals (for a review, see Rugani & de Hevia, 2017; Vallortigara, 2018). The increasing quantity of evidence of a spatial numerical association (SNA), in nonverbal subjects, opens a scientific debate on its possible evolutionarily ancient and shared origin. The SNA and how it was first reported in animal species, specifically domestic chicks, will be discussed below, focusing on two central sources of evidence of the SNA, each based on peculiar numerical competences: ordinality and magnitude processing.

Nonhuman Animals Associate Numbers with Space Ordinality Ordinality refers to the capability to identify a target element in a series of identical elements on the basis of its ordinal position. In a seminal study aimed at testing if young animals could use ordinal information, an unexpected bias in counting

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directionality has been surprisingly found, which paved the way for a new perspective according to which nonhuman animals associate numbers and space. To be more precise, these studies indicated that young and inexperienced birds are perhaps also predisposed to “count from left to right.” Domestic chicks learned to select an item based on its ordinal position within a series of identical items (Rugani et al., 2007). During training, 5-day-old chicks gradually learned that a food reinforcement was always located in the container that occupied a target ordinal position (e.g., the fourth one), in a series of 10 identical and equidistant containers, which were sagittally displayed with respect to the chicks’ starting position. The ordinal location of the target varied for various groups of chicks: a group learned to find the food in the third, another in the fourth, and one more in the sixth container. Chicks took a few hours to learn such spatial/ordinal strategies to locate the food. During the test, they faced the same displacement of containers, and each group selected above chance only their target container. Nevertheless, given that the location of the containers was always identical, this evidence could not yet demonstrate that chicks processed serial/ordinal cues. In fact, chicks could have relied on spatial cues rather than ordinal ones. A variety of other usable cues were available. For example, chicks could have merely memorized either the distance separating the target container from the first container or from their starting position. Thus, a series of control experiments were fundamental to excluding these and other similar alternative explanations. One of these controls was designed to determine the distance of the target from the starting position, as well as to determine if its specific spatial location was indispensable to solving this task. Birds were initially trained to select the fourth container in a series of 10 sagittally oriented ones (Fig. 1a). During training, chicks always experienced the target in the same location, and, with the distance between the containers being constant, the target was always at the same distance from the starting point. During testing, the series was maintained as identical, but it was rotated by 90 , becoming fronto-parallel oriented with respect to the birds’ starting position (Fig. 1b). In this test displacement, chicks actually had two possible correct choices: the fourth left or the fourth right containers. Despite both options being in the same way consistent with the reinforcement contingencies learned during training, chicks significantly chose the fourth left element, neglecting the right one. This was the first proof of a preferential left-toright “counting” in nonhuman animals (Rugani et al., 2007). Such a left asymmetry is now considered a robust phenomenon, and it has been replicated in subsequent studies. Young domestic chicks and adult Clark’s nutcrackers, once trained to select either the fourth or sixth containers in a series of identical and equidistant containers, sagittally oriented with respect to the bird’s starting point, showed the same left asymmetry in the fronto-parallel test (Rugani et al., 2010b). Although intriguing, this left bias could be interpreted in several ways. A lateral preference for targets located on the left side could reflect a phenomenon similar to the human “pseudoneglect” (Albert, 1973). The pseudoneglect describes that humans preferentially allocate their attention on one side of the extracorporeal space, which typically is their left one. In a free foraging task, similar behavioral asymmetries had been reported in chicks and pigeons. Chicks and pigeons faced a

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Fig. 1 (a) Overhead view of the experimental apparatus and of the series of items, sagittally displaced with respect to the bird’s starting position. During training and the test, the items looked identical; here for illustrative purposes, the target item, the fourth one, has been filled. (b) The series of items as they were displaced during the fronto-parallel test. The filled circles show the two correct items

symmetrical grid of cells, each containing a grain of food, and were allowed to feed themselves for a limited amount of time. Both species ate the left grains while neglecting the right ones, showing a non-learned asymmetrical behavior in a simple spatial task (Diekamp et al., 2005). Similarly, chicks show a left bias also in a linebisection task. Birds were presented with a series of three aligned beads and learned to peck the central one for a food reward. When they faced larger numbers of beads during the test (i.e., 5, 7, or 9), they continued to peck the central one; nonetheless, their errors were mainly concentrated on their left side (Regolin, 2006). It has been proposed that all of these biases in spatial tasks as the researchers mentioned could reflect a right-hemispheric specialization for processing spatial information. In chicks, in fact, a right-hemispheric specialization for spatial processing has been well documented (review in Vallortigara (2018)). In the ordinal experiments described above, the distance between the items (hereafter inter-item distance) was kept constant; thus, the overall length of the series was also identical during the training and test. This way, chicks could rely on ordinal/numerical information as well as various kinds of metric cues. Between the other metric cues available in this version of the task, the one that more likely played a crucial role in determining the left bias was the distance of the target from the beginning of the series (in identifying the fourth containers – fourth left, fourth right – chicks anchor their counting to the fist container; see Rugani et al. (2007) for a broader discussion). This regular displacement could have prompted the birds to preferentially rely on spatial cues to resolve the task, eliciting a processing with the right hemisphere. The tendency to count from left to right can be considered a consequence of an allocation of attention to the left space. To test the influence of the metric cues in determining the left bias, a new study was designed. In various experiments, chicks were prevented from using metric cues

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Fig. 2 Overhead view of the experimental apparatus and of the items’ displacements with respect to the bird’s starting position. In all training and testing trials, the items looked alike. Here, to better illustrate the procedure, the target items are filled. (a) The displacement of the series during training. The distance (d) of the target item from the closest end of the series, which was identical in each training trial. (b) The items’ displacement in the fronto-parallel series in which the inter-item distances were increased. (c) The items’ displacement in the fronto-parallel series in which the inter-item distances were reduced

during training or during the test. In one experiment, chicks were trained to select the fourth container in a sagittal series of 10 identical containers, which were kept in a fixed location (Fig. 2a). At the fronto-parallel test, the inter-element distances were either increased for one group of animals or reduced for a second group. In both cases, the new inter-item distances were calculated to conflict between the spatial and ordinal cues; more precisely, whenever the distance was increased, the third container was in the location occupied during training by the fourth container. When the distance was reduced, the fifth container was in the location previously occupied by the fourth one (Fig. 2b, c). This test allowed the disentanglement of the use of the spatial versus numerical information. In these tests, chicks pecked at the fourth element and almost neglected those located at the correct distance from the beginning of the series. However, they did not show any left bias, choosing the left and right targets almost with the same frequency (Rugani et al., 2011b). This indicates that even if both cues were accessible and identically reliable during training, chicks used the numerical rather than the spatial one. The availability of the spatial cue seemed therefore to play a crucial role in the determination of the left-to-right counting direction. This hypothesis was also corroborated in another experiment. In this case, during training, the interelement distances were changed (i.e., either reduced or increased). This way, the numerical/ordinal cue was reliable, whereas the spatial one was not, thus forcing chicks to rely only on numerical information. Also, during the test, the inter-element distances were changed at each trial. Birds properly selected the target element. However, they did not display any spatial asymmetry (Rugani et al., 2011b). This demonstrates that chicks can solve an ordinal task by relying solely on numerical/ ordinal cues, supporting the idea that the integrated processing of numerical and spatial information is indispensable to show a preferential counting from left to right.

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An alternative explanation could be that this lack of left bias may be caused by a general novelty effect that can be triggered by some kind of modification of the test series with respect to the training one. Another experiment was designed to investigate this possibility. Chicks learned to find the reward in a sagittally oriented series of green containers, always maintained at static distances. During the fronto-parallel test, the metric features of the series remained identical, but the containers were red. In this case, chicks correctly selected the target item (the fourth one), and, interestingly enough, they showed a left bias (Rugani et al., 2011b). This shed light on the fact that it is not a generic novelty effect that affects the left bias – instead, it was clearer that a specific spatial change was the relevant aspect that could disrupt the tendency to count from left to right. Overall, these findings suggested that, while both cerebral hemispheres equally process numerical/ordinal information, spatial ones should be represented in the right hemisphere. To disentangle the engagement of each hemisphere in dealing with the ordinal task and determining the leftward bias, the monocular occlusion technique was used. This consists of restricting the visual input to one eye by placing a patch over it. Since the avian brain has no corpus callosum and displays a virtually complete decussation of fibers at the optic chiasm, by restricting the visual input to a single eye, it is possible to determine the functioning of the contralateral hemisphere (review in Daisley et al. (2009), Rogers et al. (2013), and Vallortigara (2018)). The chicks were binocularly trained to peck at the fourth container in a series of 10 identical and static containers, which were sagittally aligned with respect to the chick in its starting position. The fronto-parallel test was then conducted in three vision conditions: binocular, right monocular, and left monocular. In the right monocular condition, the chicks identified as correct the fourth container from the right. In both the left monocular condition and the binocular condition, the chicks solely selected the fourth left container. These results indicated that ordinal information is bilaterally represented in the cerebral hemispheres. Whenever both hemispheres process this information, the extra-activation of the right hemisphere takes place. This leads to the allocation of attention to the left hemispace and thus produces a bias to “count” selectively from left to right (Rugani et al., 2016). For fostering a better understanding of the involvement of the hemispheres in dealing with spatial versus numerical-ordinal cues, the monocular occlusion technique was used in a fronto-parallel test, characterized by a conflict between number and space. Again, the chicks learned to peck at the fourth container in a sagittal series of identical containers, maintained in static positions, to allow the birds to rely on both spatial and numerical cues. In the fronto-parallel tests, the inter-item distance was increased so that the third container was at the distance where the fourth one was training, thus compelling the chicks to use either a spatial cue or an ordinal cue. When seeing binocularly, the chicks selected the fourth left and right containers. In both monocular tests, the chicks equally chose the third and fourth containers selectively in their clear spaces, indicating that both hemispheres can process spatial as well as ordinal information. However, the hemispheres also interact to integrate both types of information (Rugani & Regolin, 2020). This may allow to speculate on a model –

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which assumes the two hemispheres’ differential encoding, processing, and integration of numerical and spatial information – to determine a specific SNA’s left-to-right orientation. The domestic chick and Clark’s nutcrackers (birds showing left-to-right-oriented counting strategies), together with the avian species characterized by laterally placed eyes, are recognized as unique models for investigating the effect of cerebral lateralization on behavioral functions. The neural substrates of these species are characterized by a complete decussation of visual fibers at the optic chiasm, as well as by a lack of any substantial interhemispheric connection tracts analogous to the mammalian corpus callosum, even though other smaller tracts allow for interhemispheric communication. As a consequence, in these species, behavioral lateralization is particularly noticeable, as information from each eye (and thus each spatial side) is mainly elaborated by the contralateral hemisphere (e.g., Daisley et al., 2009; Rogers et al., 2013). Such peculiarities of the avian brain anatomy could determine the left asymmetry observed in the fronto-parallel tests, making it difficult to generalize findings for chicks to humans (Drucker & Brannon, 2014). Nevertheless, rhesus macaques, which are characterized by a less asymmetrical brain, also showed a similar leftward lateralization in ordinal processing (Drucker & Brannon, 2014). Rhesus monkeys learned to select a target dot (the fourth from the bottom) in a series of identical vertically aligned dots. As in the fronto-parallel test used with the chicks, the monkeys faced a series rotated by 90 during their test. They selected the fourth dot from the left and not the fourth one from the right. Such a left bias was also maintained when the whole series was displaced on the screen in such a way that the fourth left dot was actually to their right (Drucker & Brannon, 2014). This indicates that left bias is not related to a tendency to respond to left stimuli but rather to count from left to right. A similar facilitation to order numerosities according to a left-to-right orientation was demonstrated in chimpanzees while they were performing a different task (Adachi, 2014). Chimpanzees were trained to touch Arabic numerals (in the interval of 1–9) in ascending order. During training, the numerals appeared in random locations; thus, no specific relation existed between each numeral and a spatial position. During the test, when they faced only the smallest (1) and the largest numeral (9) presented side by side, with one on the left and one on the right side of the screen, their speed was faster if 1 was on the left and 9 was on the right than when the spatial positions of the numerals were shifted (9 on the left and 1 on the right). This indicates that chimpanzees mapped the learned sequence in a left-to-rightoriented sequence (Adachi, 2014). These data on chimpanzees are parallel to those reported for monkeys in Drucker and Brannon (2014). Together, they suggest that the asymmetrical processing of ordinal information is not a prerogative of the highly lateralized bird brain. Instead, it is a more general phenomenon that potentially reflects the mechanism shared between distant species. The numerical biases reported for different animal species challenge the primary role of education in determining the left-to-right-oriented association between numbers and space. They also emphasize the nonlinguistic nature of this association.

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Number-Space Association in Magnitude-Estimation Tasks Associations between numbers and space have been reported in domestic chicks not only in ordinal tasks but also in tasks requiring magnitude estimations. A limitation of the above-described ordinal task is related to the constant left bias. Nevertheless, the human MNL is not only characterized by a tendency to count from left to right but also is reported to be in an association of larger numbers with the right space. The first study to show an association of a large numerical magnitude with the right space in animals, in the absence of any specific numerical training, is the work by Rugani, Rosa-Salva, and Regolin (2014a). To their aim, these authors exploited chicks’ tendency to rejoin a bigger group of artificial social companions (Rugani et al., 2009, 2010, 2011a, 2014a). All of these studies share a rearing procedure: chicks were exposed since their very first hours of life to a set of objects – for example, three identical red squares. After this type of brief exposure, birds will consider these objects to be their social companions and will be motivated to stay close to them. If they see two objects disappear behind a panel (P1) and only one object disappears behind another panel (P2), the birds will circumnavigate the panel, P1, to rejoin the larger group of social companions (Rugani et al., 2009). This experimental procedure has demonstrated that day-old birds can master simple arithmetic in a range of small numerousness (Rugani et al., 2009) and in a range of large numerousness (Rugani et al., 2011a). In the study by Rugani, Rosa-Salva, and Regolin (2014a), chicks were exposed for a few days to a set of identical objects. During the test, the birds faced two identical panels, one on their left and one on their right. They also observed some objects disappear behind the left panel, whereas other objects disappeared behind the right panel. For example, 5 objects disappeared behind the left panel and 10 behind the right panel. Then, the birds could freely approach either panel. The crucial manipulation involved the location where the larger group disappeared, which was randomized and balanced between trials: for half of the time, the larger group was hidden behind the left panel, and during the other half of the time, it was behind the right panel. The study’s rationale was that if larger numbers were preferentially associated with the right space, the chicks should perform better when the larger group was hidden in the right space. This is actually what happened: the chicks found the larger group of artificial social companions more easily when these were on the right side. Such a right advantage in remembering where the larger group was has been consistently reported in two numerical conditions (5 vs. 10 and 6 vs. 9). Nevertheless, an alternative explanation is still possible: the intrinsic motivation to join the larger group of social companions (a category that the left hemisphere usually processes) could have biased chicks’ behaviors towards the right space. With the goal of ruling out this alternative, in a control experiment, the same number of artificial companions (2 vs. 2) was hidden behind the two panels. Interestingly enough, in this case, the chicks did not show a directional bias, thus excluding any intrinsic motivational effect related to the social nature of the stimuli (Rugani et al., 2014a).

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The innovative aspect of the results of the latter study was to clarify the presence of a right spatial bias in responding to a large numerical magnitude. Nevertheless, so far, all of the studies have been able to solely demonstrate either an association of small numbers with the left or an association of bigger numbers with the right. Thus, the need still existed for a paradigm that would allow for the simultaneous demonstration of both small and large numerical associations, respectively, with the left and right spaces – an aspect that was intrinsically impossible in the design previously described. Moreover, the human MNL is characterized by another essential feature: relativity. In our species, the direction of the association of a given number with the left or with the right space is determined by the relation of a number with a reference number, which functions as an anchor value onto which all of the following numbers will be compared. As a consequence, the same number can be associated with the left space or the right space, with the anchor value simply being changed. For example, if our anchor value is 4, the number 6 will be on the right, but if the anchor value is 8, the number 6 will be associated with the left. In other words, the human MNL – and likewise, the number magnitude – is not absolute but rather relative (Dehaene, 2011). To summarize, for clearly proving that other species show a number-space association that is analogous to the human MNL, an experimental paradigm should simultaneously assess two main aspects: (i) a magnitude-related bidirectional bias: a left bias for small numbers and a right bias for large ones and (ii) the SNA “relativity”: changing the anchor value, with the bias switching sides. To the best of our knowledge, so far, only a few studies in nonverbal subjects have met these criteria. In a seminal study, 3-day-old chicks showed left responses when dealing with relatively smaller numerical magnitudes and right responses when dealing with relatively bigger numbers (Rugani et al., 2015a). The same numerical magnitude was associated with the left or with the right side depending on whether the anchor value was, respectively bigger or smaller, thus providing the first direct evidence on the SNA’s relativity in animals (Rugani et al., 2015a). Considering its scientific relevance, this study will be described in further detail. During the study, chicks were initially familiarized with a certain number, which became the anchor value. To this aim, the chicks underwent training during which they faced a panel, which was always located in the center of the apparatus, exactly in front of the chicks’ starting position. The panel always depicted the same numerical magnitude (5 or 20 squares for different groups of animals), but this was presented in different forms: the spatial organization of the squares was unique for each stimulus. In each trial, the panel hid a piece of food that the birds could reach by circumnavigating the panel. Once the chicks learned to promptly circumnavigate the central panel to reach the reward, they underwent two tests: a small number test and a large number test. In both tests, the chicks faced two identical panels located in two new locations: one on the chicks’ left and one on the chicks’ right. The panels depicted the same number of squares. In the small number test, the panels represented a number smaller than the one experienced during training, and in the

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Fig. 3 (a) Schematic illustration of the training setup: only one panel, depicting five items, was in the center of the experimental apparatus, in front of the chicks’ starting position. (b) Schematic illustration of the setup used for the small number test (2 vs. 2). Two panels were in novel positions, one on the left and one on the right side of the experimental apparatus. Chicks preferentially approached the left one. (c) The setup for the large number test (8 vs. 8). As in the small number test, the two identical panels were on the left and right sides. Chicks preferentially approached the right panel

large number test, the number was bigger than that experienced at training. For example, the chicks that experienced five squares during training (Fig. 3a) were presented with two squares (Fig. 3b) in the small number test and with eight squares in the large number test (Fig. 3c). Meanwhile, the chicks that experienced 20 squares during training (Fig. 4a) were presented with 8 squares in the small number test (Fig. 4b) and with 32 squares in the large number test (Fig. 4c). In this way, the same number of 8 corresponded to a large numerical magnitude for the first group of chicks and to a small one for the second group. As a consequence, if chicks do associate numerical magnitudes with space, the birds that had been formerly trained with five squares should have preferentially associated the exact same stimulus depicting eight squares with the right side. Likewise, the birds trained with 20 squares should have done this with the left side, thus allowing the relativity of the SNA to be tested. To explore the occurrence of such number-space associations, Rugani et al. (2015a) recorded whether birds circumnavigated the left or right panels in their

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Fig. 4 (a) Schematic illustration of the training setup. The chicks from their starting point faced a single central panel depicting 20 elements. (b) Schematic illustration of the setup for the small number test (8 vs. 8). Between the two identical lateral panels, chicks approached the left one. (c) The setup for the large number test (32 vs. 32). Between the two identical lateral panels, the chicks approached the left one

attempt to gain the reward that was never present during the test. The predictions were that during the small number test, the chicks would preferentially look for food behind the left panel, whereas during the large number test, they would prefer to direct their searches behind the right panel. Consistently with these predictions, the chicks trained with five squares circumnavigated the left panel when the stimuli depicted two squares (Fig. 3b), and they circumnavigated the right panel when the stimuli depicted eight squares (Fig. 3c). Remarkably, chicks trained with 20 squares preferentially circumnavigated the left panel when the stimuli represented 8 squares (Fig. 4b), and they circumnavigated the right panel when both stimuli represented 32 squares (Fig. 4c). Additional control experiments showed that such asymmetrical behavior stemmed from numerical information and not from other continuous physical cues, such as the overall perimeter, area, or density, which covaried with the number whenever the dimensions of the squares were identical in the stimuli’s set. For example, if we imagine that the squares in the stimuli were identical in size, the overall area and perimeter of the three-square stimuli would be smaller than those of

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the five-square stimuli. For the purpose of controlling for these two variables (overall area and perimeter), three control experiments were designed. In one experiment, the stimuli, instead of representing identical red squares, consisted of heterogeneous elements, each characterized by a peculiar shape, color, and size. In the second experiment, the elements were always red squares in each numerical stimulus. Nevertheless, their sizes varied in the stimuli so that they covered an identical overall red area. Finally, in the third experiment, the stimuli depicted red squares of different sizes. These stimuli were characterized by an identical overall perimeter, density (mean distance between the elements), and occupancy (overall surface covered by each square’s array). By maintaining an identical overall perimeter, in this last experiment, the stimuli also presented an inverse correlation between the overall area covered by the red squares and the number of squares; in other words, the total red area was smaller in the stimuli depicting eight squares than in a stimulus depicting two squares. See Rugani et al. (2017) for the mathematical formula that describes the variation of the overall area when the perimeter is constant. Consistently, in all of these control experiments, the chicks circumnavigated the left panel when facing smaller numbers and the right panel when facing larger numbers. This demonstrated that the asymmetrical number-space association was based on the numerical information and not on other continuous physical variables, such as the overall perimeter, area, color, light, occupancy, and density (Rugani et al., 2015a, 2015b). This evidence has literally unleashed a firestorm of controversy about the origin of the association between number and space among members of the scientific community. Psychologists, biologists, linguists, and cognitive scientists are divided into two opposite counterparties. On one side are those who believe that early SNA in chicks is unrelated to the human phenomenon and can simply be explained by individual bias or by preference for novelty. On the opposite side are those welcoming the new evidence, mainly coming from newborn humans and nonhuman species. In support of the latter hypothesis, in a more recent study, 3-day-old chicks were trained to circumnavigate a central panel that depicted five identical red squares. Then, the chicks underwent three tests: a small number test, in which the two stimuli on the lateral panels depicted two squares; a large number test, in which the two stimuli depicted eight squares; and a control test, in which the stimuli depicted five squares – the number they experienced during training. The left-sided choice (LC) was then computed on each chick’s performance in each test. Interestingly, the LC was explained by the number of squares and not by an individual lateral bias in the spatial task. Moreover, we found a linear trend with three points of reference (LC2 vs. 2 > LC5 vs. 5 > LC8 vs. 8), which provided for the first time convincing evidence in domestic chicks that the numerical magnitude of the stimuli and not individual bias drives the asymmetrical displacement of numbers into space. Overall, these studies concluded that chicks do associate small numbers with their left and larger ones with their right spaces. This evidence, together with that reported in human newborns, indicate that side biases similar to the ones observed in humans and driven by the MNL can be observed in illiterate and almost naïve subjects. This

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impacts the investigation on the origin of the SNA towards its neural representation (Brugger, 2015; Vallortigara, 2018). Using a similar paradigm, the SNA also has been shown in gorillas and orangutans (Gazes et al., 2017). Although present in most apes, the SNA was not equally oriented onto space in all individuals. Various animals showed either left-to-right- or right-to-left-oriented SNA. Distinctive experiences – for instance, the interactions with caregivers, rather than differences linked to species or handedness – have been identified as the central factors that might regulate the SNA’s orientation at an individual level (Gazes et al., 2017, but see Beran et al. (2019) for information on the failure of the replication of the SNA in rhesus monkeys). No SNA evidence has been found in a cleaner fish species (Triki & Bshary, 2018). This may indicate the presence of specific evolutionary factors in developing the SNA in only some vertebrate lineages. For example, the mechanisms underlying the spatial asymmetries in numerical processing perhaps evolved with amniotes, i.e., after the last common ancestor of the fish and reptiles. Nevertheless, any conclusions regarding this cannot be based on the idea of solitary evidence, especially considering the huge taxonomic and neuroanatomical diversity reported between fish species (Triki & Bshary, 2018). Follow-up studies on the SNA in various fish species may be dedicated to disentangling these open issues. More successful and intriguing findings were obtained in two studies that replicated Rugani and colleagues’ paradigm in newborns (Di Giorgio et al., 2019; de Hevia et al., 2017). These studies exploring human newborns under minimal to no exposure to adults’ scanning biases clearly assessed how precocious the association between number and space is in humans. Newborns were first habituated to a numerical value – for example, a group of 12 items – that was presented in two identical copies, one on their left and one on their right on a monitor. Immediately after the habituation, they were presented with two identical stimuli – again, one on the left and one on the right. These could represent either a smaller (4) or a larger (36) number of items. When they faced a smaller number, they looked longer at the left copy of the stimulus, but when facing a larger number, they looked longer at the right one. Moreover, the SNA was not absolute but rather relative: the same number “12” was associated with the left when newborns previously experienced a larger number – for example, “36” – but it was associated with the right side when they were habituated to a smaller number, like “4” (Di Giorgio et al., 2019). These findings demonstrate that a disposition to associate numbers on a left-to-rightoriented MNL exists independently of cultural factors and with little, if any, early exposure to directional cues. This supports the hypothesis that it is perhaps biologically predisposed in the brain.

Conclusions Some very early attempts to ascertain the presence of mathematical understanding in nonhumans were a clamorous failure (Pfungst, 1907). This was probably linked to the pervasive anthropocentric perspective of those times and approaches. It was also

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responsible for the subsequent trail of skepticism and mistrust towards the possibility of objectively investigating cognitive functions in nonhumans. Animals were thereafter regarded as being capable of acquiring only number-response-conditioned associations following intensive training. The key feature is that animals were considered to be incapable of responding to purely numerical information (Davis & Pérusse, 1988). According to this view, only adult humans could grasp the numbers. To them, these symbols were meaningful, and no calculations would be possible in the absence of language or other symbolic representations. This position is, of course, no longer tenable, as a huge number of demonstrations have been accumulating during the past few decades concerning the fact that many species can master numerical tasks. In some instances, animals successfully deal with purely numerical information. Notably, animals can encode a number even when other, nonnumerical cues are also available. This means that number processing is not employed as a last-resort strategy – for example, only when the nonnumerical cues are unavailable (Vallortigara, 2018). In the natural environment, though, usually numerical cues covary with other continuous information. Therefore, not surprisingly, when both types of cues are available, animals exploit them to solve more complex numerical tasks (Rugani et al., 2017). Parallel to the comparative evidence, a growing number of studies have demonstrated the presence of numerical skills in very young human infants as well, well before language and culture could be claimed as being responsible. The domestic chicken (Gallus gallus) has offered a unique model for bridging the comparative studies (carried out invariably in adult individuals) and the developmental evidence. Research on very young chicks, in fact, was able to assess early predispositions for number processing with strict controls over the role of experience. Chicks spontaneously discriminated the numerousness of sets of familiar objects and approached the larger set. Chicks even solved simple arithmetic computations with sequential presentations of objects, such as 1 + 1 + 1 judged as larger (hence preferred) than 1 + 1. Chicks seemingly operated based on representations of objects that were no longer visible in each set, and based on representations of their overall numerosities. These fine numerical abilities demonstrated that number processing is an inherent ability of animals that is crucial for survival, and these skills emerge even in the presence of an extremely reduced amount of experience. The sense of number (Dehaene, 2011) would be shared across species – obviously preceding school and cultural education – and it would constitute the core knowledge supporting successful interactions with the environment (Feigenson et al., 2004). Even in humans, the nonsymbolic numerical abilities are considered to be the foundations of more complex numerical reasoning (Starr et al., 2013; Park & Brannon, 2013). Another inherent property that appears (to the surprise of many) in animal species concerns the integration of two domains: numerical and spatial. A leftward bias has been described when animals locate a target by its ordinal position within a series of identical targets oriented from left to right in front of the subject. This bias was confirmed in three animal species: domestic chicks (Rugani et al., 2007), Clark’s nutcrackers (Rugani et al., 2010b), and rhesus monkeys (Drucker & Brannon, 2014).

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It remains unclear whether such bias depends on the same (homologous) mechanism shared across species or whether we are looking at similar but analogous mechanisms (developed separately in the different species) for number-spatial processing. The bias seems to be specific for the integrated use of numerical and spatial information: chicks manifest a left-bias when ordinal and spatial information are both available, but they do not do this when the numerical information is all that is available (Rugani & Regolin, 2020; Rugani et al., 2011b). This bias seems to be linked to the spatial representation of various (increasingly large from left to right) numerical quantities. Newborn chicks display a left-spatial bias when exploring a numerosity smaller than expected, and a right-spatial bias when exploring a numerosity larger than expected (Rugani et al., 2015a, 2020). Such space-number association was also confirmed in human infants (Bulf et al., 2015; Rugani & de Hevia, 2017) and in newborns (de Hevia et al., 2017; Di Giorgio et al., 2019). Clearly, the processing (and possibly the neurobiology) of numerical versus other types of dimensions are intertwined. The direction of the association is consistent across species: smaller numbers are associated with the left space, and larger numbers are associated with the right space. This fact is rather suggestive of a common origin of the underlying mechanisms at least in humans, other mammals, and other vertebrates (i.e., birds). The neural basis of nonsymbolic numerical cognition in general, and of number-space mapping mechanism in particular, is currently being studied in avian species. In humans and monkeys, the intraparietal sulcus seems to be involved in number processing, and topographically organized neurons responsive to small numerousness were found in the human parietal cortex (Harvey et al., 2013). Neurons that are responsive to numbers have been found in the nidopallium caudolaterale of crows, an associative area of the avian pallium engaged in higher cognitive functions. These neurons are thought to represent the analogues of the mammalian prefrontal-cortex (Ditz & Nieder, 2015). It has been proposed that this neural organization may determine the organization of magnitudes along a left-to-right-oriented space. A right hemisphere dominance in processing numerical and spatial information may prompt animals to start to count from left to right (Rugani et al., 2010b). Alternatively, the two hemispheres may be differently involved in processing small (right hemisphere) or large magnitudes (left hemisphere) (Vallortigara, 2018). Recent evidence in chicks showed that no one hemisphere preferentially deals with small or large numerical magnitudes (Rugani et al., 2020). Instead, the proficient use of numerical/ordinal information requires simultaneous processing by both hemispheres (Rugani & Regolin, 2020). How this integration occurs in the brain remains to be discovered. Overall, this pattern of results seems to suggest that educational factors could not be the only factor determining the development of the number-space associations. This does not mean that experience cannot affect the SNA. Recent data on newborn chicks demonstrated that experience can indeed modulate the strength of the SNA (Rugani et al., 2020). Nevertheless, the integration of brain numerical representation and experience with determining the direction of the SNA remains one of the greatest challenges.

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Acknowledgments This work has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie (grant agreement No. 795242 SNANeB) to R.R. and by a PRIN 2017 ERC-SH4–A grant (2017PSRHPZ) to L.R.

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Executive Dysfunction Among Children with ADHD: Contributions to Deficits in Mathematics

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Lauren M. Friedman, Gabrielle Fabrikant-Abzug, Sarah A. Orban, and Samuel J. Eckrich

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Is ADHD? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ADHD Diagnostic Criteria and Subtypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Impairments Associated with ADHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Findings from Structural and Functional Imaging Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ADHD as a Neurodevelopmental Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Executive Functioning in ADHD and Relations to Math Achievement . . . . . . . . . . . . . . . . . . . . . . . Working Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavioral Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Set-Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interventions for ADHD-Related Executive Function and Math Deficits . . . . . . . . . . . . . . . . . . . . . Behavioral Interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Psychostimulant Medication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cognitive Training Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Math Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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L. M. Friedman (*) · G. Fabrikant-Abzug Department of Psychology, Arizona State University, Tempe, AZ, USA e-mail: [email protected]; [email protected] S. A. Orban Department of Psychology, University of Tampa, Tampa, FL, USA e-mail: [email protected] S. J. Eckrich Department of Pediatric Neuropsychology, Kennedy Krieger Institute; Johns Hopkins University School of Medicine, Baltimore, MD, USA e-mail: [email protected] © Springer Nature Switzerland AG 2022 M. Danesi (ed.), Handbook of Cognitive Mathematics, https://doi.org/10.1007/978-3-031-03945-4_40

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Abstract

Math deficits among children with attention-deficit/hyperactivity disorder (ADHD) are well documented, yet little evidence supports a causal link between the ADHD core symptoms of inattention, hyperactivity, and impulsivity and poor math achievement. Rather, considerable evidence suggests that ADHD symptoms and cooccurring math difficulties are secondary to deficits in executive functioning. In the ensuing chapter, we discuss executive dysfunction as a potential explanatory mechanism for ADHD-related math challenges. First, ADHD diagnosis and related sequelae are discussed. Next, we review the supporting evidence that links ADHD-related math achievement deficits to challenges with executive functioning. Finally, we review the efficacy of gold-standard treatments (i.e., behavioral interventions, psychostimulant medications) and experimental interventions (i.e., computerized cognitive training programs, direct math instruction) for improving math deficits among children with ADHD. Keywords

ADHD (Attention-Deficit/Hyperactivity Disorder) · Math · Academic achievement · Executive functions · Working memory

Introduction Attention deficit/hyperactivity disorder (ADHD) is a neurodevelopmental disorder characterized by excessive and clinically impairing levels of inattention, hyperactivity, and/or impulsivity. ADHD is one of the most common mental health conditions of childhood and affects approximately 7% of school-aged children worldwide (Thomas et al., 2015). An estimated $124.5 billion USD is spent on the disorder within the United States annually (Zhao et al., 2019), with the majority of costs allocated towards treatment of the significant, adverse functional outcomes common across domains – including within the area of math achievement. Children with ADHD evince largemagnitude deficits on standardized assessments of math skills, are diagnosed with Specific Learning Disorder in Mathematics/Developmental Dyscalculia at disproportionate rates, receive lower report card grades in math performance, and demonstrate reduced academic productivity on math assignments relative to neurotypical peers. It may be hypothesized that the math difficulties associated with ADHD are the result of attention problems characteristic of the disorder. It is a reasonable assumption, as a child is not able to learn appropriately during classroom math instruction if he or she is daydreaming, attending to irrelevant peer conversations rather than his or her classroom teacher, or continuously getting out of his or her seat to sharpen a pencil during math class. Indeed, evidence suggests that mathematics deficits are more closely related to inattention symptoms relative to hyperactive and impulsive symptoms, and the phenotypic overlap between specific academic weaknesses in math and ADHD is driven largely by inattention symptoms in genetics studies.

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However, evidence for a causal link between inattention and children’s math problems appears weak. Consider the following thought experiment: if deficits in math performance among children with ADHD are entirely due to challenges with inattention, then normalization of attention problems through treatment should also cure children’s difficulties in math. This mental exercise is easily testable empirically, as several extant interventions are largely effective for improving inattention among children with ADHD. Disappointingly, several multiyear investigations fail to demonstrate improvement on math achievement despite significant improvement and normalization of attention following gold-standard treatments for the disorder (i.e., carefully titrated psychostimulant medication, multimodal behavioral interventions, and their combination). Thus, it is likely that inattentive symptomatology alone is insufficient for explaining the well-documented deficits in math achievement characteristic of ADHD, and consideration of alternative mechanisms is required to identify the putative factors underlying ADHD-related math difficulties. Over the last quarter-century, executive functions (EFs) have emerged as a promising etiological mechanism for explaining ADHD symptoms and related functional impairments – including difficulties in math performance. In the ensuing chapter, executive functions are reviewed as a potential explanatory factor for ADHD-related math deficits. First, the diagnosis and sequelae of ADHD, current etiological conceptualizations, and evidence from neuroimaging studies that support these causal theories are discussed. In the second section, support for executive functions as an underlying contributor to ADHD-related math deficits is reviewed. In the final section of the chapter, gold-standard and experimental treatments for the disorder are reviewed, the relative impotence of current interventions for treating both math challenges and executive dysfunction among children with ADHD are discussed, and future directions for the design of novel interventions based on the etiological science of ADHD are provided.

What Is ADHD? ADHD Diagnostic Criteria and Subtypes The Diagnostic and Statistical Manual of Mental Health Disorders (DSM) is the compendium of clinical guidelines and diagnostic criteria that mental health professionals use when determining psychiatric diagnoses, including ADHD. The disorder first appeared in the second edition of the DSM (1980) under the term “hyperkenetic reaction of childhood” and the diagnostic criteria have undergone extensive revision in each subsequent version of the DSM. The most recent iteration, DSM-5 (2013), recognizes ADHD as a neurodevelopmental disorder alongside other mental health conditions that occur early in the developmental process and are secondary to neurobiological dysfunction (e.g., autism spectrum disorder, intellectual disability, learning disorder). This conceptualization is largely the result of extant literature that documents the centrality, importance, and causal role of executive dysfunction as an underlying mechanism of ADHD symptoms and impairment

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– the focus of the present chapter. Two broad symptom dimensions are recognized in the DSM – inattention symptoms and hyperactive/impulsive symptoms – and children may exhibit problem behaviors in either one or both symptom clusters. Diagnostic criteria state that symptoms must be present before the age of 12 and cause significant impairment across multiple settings (e.g., school, home, occupational, social impairments). Like all mental health conditions, the symptoms that comprise ADHD exist on a continuum, and provision of an ADHD diagnosis artificially dichotomizes normally distributed rates of inattention, hyperactivity, and impulsivity. While consideration of the entire dimension of attentional abilities has several merits, the ensuing chapter exclusively discusses significant and impairing levels of symptoms that warrant an ADHD diagnosis. The decision to operationalize ADHD as a binary construct within this review reflects current practices within school systems and mental health communities wherein intervention services for ADHD and poor math achievement are rendered only following a diagnosis.

Functional Impairments Associated with ADHD While ADHD diagnostic status is determined based on the presence of inattentive and/or hyperactive symptoms, the adverse effects of the disorder are evident in areas far exceeding these narrow behavioral domains. Relative to neurotypical peers, children with ADHD evince significant deficits in social and emotional functioning such as reduced capacity for emotion regulation, poor parent/child relationships, and increased rates of peer rejection. Adults with a history of childhood ADHD face a heightened risk of both arrest and homelessness (24% vs 4%; García Murillo et al., 2016) compared to their non-ADHD peers. Children with ADHD also experience increased occupational problems as adults. These include being more likely to be fired, changing jobs more frequently, achieving a lower socioeconomic status, and earning a lower income (approximately $625,000 USD less over their lifetime) relative to peers (Barkley & Fischer, 2017). Many of these important long-term outcomes are secondary to deficits in early school performance and academic achievement. Children with ADHD exhibit academic achievement significantly below the level predicted by their age or IQ, and nearly half of all children with ADHD are diagnosed with a comorbid learning disability (DuPaul et al., 2013). Compared with neurotypical peers, students with ADHD complete fewer assignments, receive poorer grades, and fail more courses. Students with ADHD also have higher rates of grade retention and special education placement. Furthermore, children with ADHD incur more suspensions and expulsions than neurotypical peers and demonstrate high-school dropout rates as high as 32% (Barkley & Fischer, 2017). Mathematic deficits are especially crucial given math’s relations to important long-term outcomes. Deficits in math are related to lower high-school and college graduation rates, later delinquent behavior, and lower socioeconomic status in adulthood. These adverse outcomes are particularly salient for children with

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ADHD, who experience high rates of comorbidity with math learning disorders (DuPaul et al., 2013). Even in the absence of a diagnosable learning disorder, math deficits are still problematic for children with ADHD. One meta-analysis found that children with ADHD evince moderate to large magnitude (d ¼ 0.67) deficits in math skills relative to typically developing children and score in the bottom quartile on standardized tests of math achievement (Frazier et al., 2007). These disparities are consistent across measures of math-related behaviors (e.g., standardized achievement measures, report card grades, math productivity, and accuracy) and math skills (e.g., calculation, applied problem solving, math fact retrieval, and math fluency; Friedman et al., 2018b).

Findings from Structural and Functional Imaging Studies Neuroanatomical Brain Differences. Decades of magnetic resonance imaging (MRI) evidence converges to indicate that atypical brain structure is implicated in ADHD. By the time a child begins school, ADHD is associated with widespread structural brain abnormalities including reduced total cerebral cortex volume (3–8% volume reduction compared to typically developing individuals) to varying degrees across all cortices of the brain (Friedman & Rapoport, 2015). The most pronounced structural differences appear in the prefrontal cortex, which is implicated in executing complex cognitive processes such as executive functioning and other related abilities (e.g., mental arithmetic, sustaining attention, planning, organizing, evaluating rewards/risks, and inhibiting impulses). Shaw et al. (2007) provided some of the most groundbreaking insights into the development of cortical structures from childhood to young adulthood among those diagnosed with ADHD. Delayed cortical maturation was observed in children with ADHD across a large longitudinal sample and suggests that children with ADHD achieve peak cortical thickness in the prefrontal regions approximately 2.5–3 years behind typically developing peers. There are also structural atypicalities consistently found in the cerebellum and subcortical structures. A meta-analytic review examining subcortical region development in children with ADHD revealed reduced volume in multiple regions including the basal ganglia, nucleus accumbens, amygdala, hippocampus, and thalamus (Hoogman et al., 2017). These areas are integral to facilitating communication to the cortical regions related to emotion, threat, stress, reward/motivation, and memory. Interestingly, there also appears to be sex-differences among preschool-aged children suspected of ADHD. Boys tend to have more structural abnormalities in premotor brain areas, whereas girls tend to have significantly smaller volumes of prefrontal and subcortical structures (e.g., caudate, globus pallidus, thalamus; Rosch et al., 2018). These differences may account for heightened hyperactivity observed among boys with ADHD relative to girls. Functional Brain Activity. Electroencephalogram (EEG) is one of the oldest instruments for measuring brain activity among children with ADHD. EEG measures the electrical output of neuronal firing across the cerebral cortex. EEG

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investigations of children with “problem behaviors” have existed for over 80 years. The most robust EEG feature associated with ADHD is elevated power of slow waves (4–7 Hz “theta”) and/or decreased power of fast waves (14–30 Hz “beta”) typically recorded over the frontal lobe. Although EEGs are particularly useful for determining the timing of neuronal activity, they are somewhat imprecise at determining where in the brain the electrical signal originates. Functional magnetic resonance imaging (fMRI), magnetoencephalography (MEG), positron emission tomography (PET), single-photon emitted tomography (SPECT), and functional near-infrared spectroscopy (fNIRS) are functional imaging techniques that provide measures of spatial specificity and connectivity while offering a metabolic rationale for abnormal brain activity in children with ADHD. The most widely used tool is fMRI, which measures the amount of oxygen the brain recruits via blood vessels while performing cognitive tasks. In general, task-based fMRI studies reveal hypoactivation of the frontostriatal, frontoparietal, and mesocorticolimbic networks in children with ADHD relative to typically developing children (Rubia, 2018). That is, less oxygen is recruited to neurons that connect the frontal lobe (responsible for executive functioning, planning, organizing, integrating long-term and short-term memories, evaluating rewards) to the striatum (motor coordination), parietal lobe (language and mathematical operations), and limbic system (emotional regulation, memory formation, motivation/reward). More specifically, several studies have demonstrated disrupted connectivity within critically important networks among individuals with ADHD including the cognitive control network (e.g., anterior cingulate–premotor area– dorsolateral prefrontal–inferior frontal–anterior insula–posterior parietal) and the default mode network (e.g., medial prefrontal–posterior cingulate–lateral inferior parietal cortices.) A popular theory related to altered neuronal connectivity in children with ADHD involves the default mode network (DMN). In a typically developing person, the DMN switches off when focus is needed to complete a task and the cognitive control network (CCN) becomes activated. fMRI investigations reveal that the DMN does not switch off efficiently and the CCN does not adequately switch on among children with ADHD during tasks requiring high attentional demands. Typical activation of these networks integrates sensory information and helps children accomplish goals by appropriately orienting their attention, maintaining alertness, blocking distractions, and tolerating frustration while performing cognitively demanding tasks (e.g., tasks requiring executive function such as mental arithmetic and reading comprehension) – all areas of particular difficulty for individuals with ADHD. Investigation of functional neuroimaging may also help explain the trajectory of ADHD symptomology and associated deficits across the lifespan. For example, those with persistent symptoms of inattention into adulthood show abnormalities in the default mode network, whereas individuals with remission of inattention symptoms show similar DMN connectivity to individuals who have never been diagnosed with ADHD (Sudre et al., 2018). Additionally, the severity of abnormality within a particular structure of the cognitive control network (e.g., dorsolateral prefrontal cortex) was associated with the persistence of inattention symptoms in

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adulthood (Shaw et al., 2015). The dorsolateral prefrontal cortex is particularly implicated in executive functioning and complements neuropsychological evidence that deficits in executive functioning remain in the subset of individuals who exhibit symptom remittance in adulthood. Summary. Structural MRI studies have identified delayed development of cortical and subcortical brain tissue, and EEGs and functional neuroimaging investigations have identified underarousal of frontal/prefrontal areas and networks in children with ADHD – key areas implicated in executive functioning. Disruptions of these networks lead to a lack of integration among brain areas and are likely responsible for the behavioral symptoms of ADHD expressed as deficits in regulating attention, motivation, gross motor activity, and performing executive functioning tasks.

ADHD as a Neurodevelopmental Disorder Historically, ADHD was viewed as a disorder of volitional control and reflected contemporaneous thinking that symptoms of inattention, hyperactivity, and impulsivity were a result of children’s poor motivation to maintain attention or failure to manage behavioral impulses. This notion was reified in previous versions of the DSM, which classified ADHD as a “Behavioral (Overt) Disorder” along with disruptive behavior disorders such as Conduct Disorder and Oppositional Defiant Disorder. However, current conceptualizations and nosological classifications consider ADHD to be a neurodevelopmental disorder, and the symptoms and impairments characteristic of ADHD are hypothesized to result from underlying deficits in cognitive functioning rather than behavioral choices under the child’s control. This paradigm shift represents decades of empirical study documenting structural and functional neurobiological differences associated with an ADHD diagnosis, as reviewed above, coupled with neuropsychological evidence noting significant and serious deficits on measures of cognitive performance. Children with ADHD show substantial impairment in several areas of cognitive functioning, including speeded processing, fluid intelligence, reward processing and delay discounting, and executive functioning. Among the cognitive domains assessed, deficits in executive functioning are the most consistently implicated cognitive deficit (Kofler et al., 2019), associated with the largest performance differences relative to peers (Willcutt et al., 2005), and show strong persistence into adulthood (Karalunas et al., 2017). Deficits in executive functioning are found to underlie core ADHD symptoms (Rapport et al., 2013), and executive dysfunction predicts long-term academic and social functioning over and above ADHD diagnostic status alone (Miller & Hinshaw, 2010). These neuropsychological findings are complemented by neuroimaging studies that document significant delay and dysfunction in the prefrontal cortical areas that support executive functioning. Given the substantial and central role of executive functions in ADHD-related symptoms and impairment, most contemporary etiological models of ADHD implicate one or more executive functions as contributory to ADHD core symptoms and related functional impairments (see Table 1 for a brief review). Because of the well-

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Table 1 ADHD etiological theories implicating executive dysfunction Theory Behavioral Inhibition Model

Cognitive Neuroenergetic Model

Default Mode Network Model

Executive Function Model

Tripartite Pathway Model

Working Memory Model

Description Behavioral and cognitive difficulties in ADHD are attributable to deficient behavioral inhibition – a cognitive process thought to subserve behavioral regulation and includes the ability to withhold or stop an on-going response (Barkley, 1997) Neurocognitive deficits in ADHD are caused by a faulty neurometabolic process in which insufficient production of astrocytes across the brain leads to inadequate lactate production necessary to fuel neuronal activity (Russell et al., 2006) A brain network (e.g., cortico-striato-thalamo-cortical) implicated in nonfocused activities or wakeful rest is overactive in individuals with ADHD and contributes to behavioral and cognitive difficulties (Castellanos et al., 2005) Symptoms of ADHD arise from broad weaknesses in higher-order, neurocognitive processes involving executive control (i.e., maintenance of problem-solving skills to attain a future goal) (Willcutt et al., 2005) ADHD symptoms are caused by deficits in one or more cognitive processes including temporal processing, inhibitory control, and delay aversion (i.e., disrupted reward/motivation perception) (Sonuga-Barke et al., 2010) Delayed cortical maturation and underarousal in the prefrontal, temporal, and parietal lobes leads to deficient working memory performance, which is theorized to underlie ADHD core symptoms and related impairments (Rapport et al., 2008)

documented deficits in executive functioning among children with ADHD (Willcutt et al., 2005), considerable theoretical and empirical evidence supporting the importance of executive function deficits as causal ADHD processes, and complementary neuroimaging studies corroborating executive dysfunction among children with ADHD, we limit our review of the cognitive contributors to ADHD-related math difficulties to executive functions.

Executive Functioning in ADHD and Relations to Math Achievement Executive functions (EFs) are a series of interrelated cognitive processes that enable goal-directed thoughts and behaviors. The term has been applied to an array of fronto-parietal mediated tasks, yet considerable experimental, developmental, and neuroimaging evidence suggests the presence of three core EF abilities – working memory, behavioral inhibition, and set-shifting (see Friedman & Miyake, 2017 for a review). The three EF abilities represent separable cognitive functions, each associated with task-specific performance variance. However, EF abilities also display considerable relatedness. Neuroimaging studies indicate that EFs rely on the same neurological substrates (i.e., prefrontal cortical structures), and latent variable

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analyses reveal the presence of a higher-order, domain-general EF ability that predicts working memory-specific, inhibition-specific, and shifting-specific processes. This common EF factor represents domain-general goal attainment abilities that are common among the three EFs. It should be noted that several other EF abilities have been proposed in extant literature. These include organization, self-regulation, time management, problem solving, and planning skills. While these skills are critical for enabling goal attainment in daily life functioning, including math-related behaviors, it is likely inaccurate to classify these cognitive processes as core executive abilities. These skills fail to show expected patterns of relations following rigorous empirical scrutiny (Friedman & Miyake, 2017), and they are downstream of and incumbent upon a combination of core EF and other abilities. As illustrative of this, planning requires holding in mind all pertinent tasks while updating this list with newer, more relevant information (working memory), switching focus among the various tasks to be performed (set-shifting), and inhibiting irrelevant information from accessing the focus of attention while planning tasks (behavioral inhibition). Collectively, these oft-misattributed cognitive abilities are better conceptualized as downstream consequences of the three core EF abilities rather than executive processes themselves. The ensuing discussion on the neurocognitive predictors of ADHD-related math skills is therefore limited to the three core EF abilities.

Working Memory Working memory is a limited capacity system responsible for the temporary storage and processing of information. This ability is necessary and critically important for guiding behavior and executing complex cognitive tasks including learning, comprehension, planning, and reasoning. Contemporary models view working memory as a multicomponent system consisting of three anatomically distinct subdomains that work reciprocally to process internally held information (Baddeley, 2007). Two domain-specific subsidiary processes, the phonological and visuospatial short-term memory subsystems, are responsible for temporarily storing and rehearsing modality specific (verbal vs visuospatial) information. These two subcomponents represent the memory processes of working memory. A domain-general central executive (i.e., the working component of working memory) oversees the two storage and rehearsal subsystems and is responsible for reordering, updating, manipulating, and processing information while concomitantly focusing attention to task demands and interacting with long-term memory. The three working memory subcomponents show separable patterns of performance in individual difference studies and are associated with distinct areas of neural activation in fMRI studies – the central executive is associated with increased activity in the dorsolateral prefrontal cortex, while phonological and visuospatial short-term memory functions are supported by the left temporoparietal region/Broca’s area and posterior parietal/superior occipital regions, respectively (Baddeley, 2007). A fourth working memory component has been proposed in recent years. The episodic buffer is purported to bind verbal and

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Table 2 Common working and short-term memory task examples Task Working memory: Updating N-Back

Description A continuous stream of numbers or letters is presented. The participant must recall the digit presented N-times previously when prompted. For example, on a two-back task where T,P, S,H,C is presented, the correct response is “S,” as it appeared two times “back” A series of numbers is presented, and the participant is required to add each digit to the one that appeared previously

Paced Serial Auditory Serial Addition Test (PASAT) Working memory: Dual processing Operation Span Participants are presented with a series of math problems to solve, and a target word appears following each math fact (e.g., 7 + 13 ¼??; sheep). At the end of each span set, participants are required to recall the target words in the order presented Counting Span A display of target shapes (e.g., blue circles) and distractor shapes (e.g., red circles and blue squares) are presented, and the participant is required to count the number of target shapes that appear in each display. At the end of each span set, the participant must recall, in order, the number of target shapes that appeared on each display Working memory: Reordering Phonological Reordering Participants are given series of single-digit numbers and letters are instructed to recall the numbers in order from smallest to largest followed by the letters in alphabetical order. For example, 6HB2 is correctly recalled as 26BH Visuospatial Reordering A 3  3 offset grid appears on a computer screen. A series of black and red dots appear one at a time within the grid locations. Participants are required to recall the spatial location of the black dots in the order they appeared and recall the red dot in the last serial position Short-term memory Digit Span Participants are verbally presented a series of digits and are instructed to recall the stimuli in the same or reverse order presented Corsi Block A grid containing several target locations, or boxes, is presented to the participant. An experimenter taps locations in a predefined order. The participant is required to recreate the sequence of locations in either the same or reverse order List Learning Tasks In this task, participants immediately repeat words recalled from a long list (12 to 16 items) in any order

visuospatial information prior to information processing in the central executive; however, evidence for its existence and purpose is mixed (Baddeley & Hitch, 2019) and is particularly weak among school-aged children (Gray et al., 2017). Working memory is traditionally assessed using three broad categories of tasks (see Table 2), each designed to target three theoretically and neuroanatomically distinct functions of the central executive. During updating, information is actively

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monitored to assess task-relevancy while newly learned material is added and no-longer relevant data is removed from short-term memory. As an example, the N-back task requires participants to attend to a continuous stream of numbers and recall the digit presented N-times previously. To execute the task successfully, the information held in mind must be continually updated as new digits are presented. Reordering involves maintaining held information in active attention while concurrently rearranging the order of learned material for successful task execution. Reordering tasks typically require participants to reposition and recall a series of presented digits, letters, or words according to a prespecified rule (e.g., alphabetical order). During dual-processing, individuals must maintain target information in mind while simultaneously performing a second, unrelated yet cognitively demanding task. For example, complex span tasks interleave a primary task (e.g., judging the veracity of several presented statements) with a second memory task (e.g., remembering the last word of each statement). Children with better developed central executive abilities are more skilled at managing the dual task-requirements and recalling the necessary information when prompted. These central executive processes play important roles in children’s learning, abstract reasoning, planning, creativity, organization, reading and writing skills, and language use and comprehension, while significant yet more moderate relations are seen between these ecologically valid skills and short-term memory abilities (Baddeley, 2007). While the central executive serves a critical role in learning and information processing, it has no capacity for holding information on its own. Rather, the central executive works in tandem with the domain-specific short-term memory subsystems and acts upon the information stored within. The integration of working and memory components renders examination of component-specific effects challenging because no working memory task provides a pure estimate of central executive abilities. Instead, latent variable approaches (e.g., factor analytic and structural equation modeling) are employed to isolate statistical variance attributable to the central executive in order to assess its functioning. Rapport et al. (2008) developed a novel, regression-based approach for isolating performance of the central executive from the short-term memory components. Using this approach, children perform a series of separate, modality-specific (i.e., phonological and visuospatial) serial reordering tasks, and scores from each modality are regressed onto each other to isolate shared and unique task variance. Because each task requires domain-general central executive and domain-specific memory components for successful execution (see Fig. 1), shared variance among the two working memory tests is considered to represent central executive functioning, while unshared variance is used to assess modality-specific short-term memory abilities. Given that working memory components are separable, play unique roles in learning-related processes, and rely on distinct cortical areas for successful execution, examination of component-specific performance is critical for fully understanding working memory’s contributions to ADHD symptoms and deficits in math achievement. Working Memory Impairment in Children with ADHD. Among ADHD etiological theories, working memory abilities have received considerable attention as a potential core deficit (Rapport et al., 2008), related causal mechanism (Barkley,

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Fig. 1 Schematic depicting the regression-based approach used by Rapport et al. (2008) to isolate central executive and short-term memory components from phonological and visuospatial working memory tasks. Shared variance between the two tasks is used to provide a reliable estimate of central executive abilities, while unique task variance is considered to represent domain-specific (phonological vs visuospatial) short-term memory performance

1997; Willcutt et al., 2005), or possible endophenotype (Castellanos & Tannock, 2002) for explaining disorder-specific symptoms and impairments. Its prominence among causal models is largely owing to the significant, pervasive, large magnitude working memory deficits observed among children and adults with ADHD relative to neurotypical peers. Meta-analytic and experimental estimates indicate that approximately 67–98% of children with ADHD evince deficits in working memory performance (Fosco et al., 2020; Kasper et al., 2012). The observed impairments are of substantial magnitude (d ¼ 1.89–2.31; Rapport et al., 2008) and are stable over time (Karalunas et al., 2017). Importantly, ADHD-related working memory deficits appear unattributable to poor motivation (Dovis et al., 2013), inattention during task administration (Orban et al., 2018), insufficient inhibitory control (Alderson et al., 2010), or basic information processing difficulties (Raiker et al., 2019). Latent variable approaches dissociating working memory components find a consistent pattern of impairment among patients diagnosed with ADHD. Substantial, large-magnitude deficits are observed in the central executive component of working memory (d ¼ 2.76; Rapport et al., 2008), and follow-up studies examining separable central executive processes find that children with ADHD are most likely to evince deficits in reordering relative to updating and dual processing abilities (Fosco et al., 2020). Significant yet more moderate impairments are evident in the visuospatial (d ¼ 0.89) and phonological (d ¼ 0.55) storage and rehearsal subsidiary systems (Rapport et al., 2008), with deficits observed in both storage (d ¼ 1.15 to 1.98) and rehearsal (d ¼ 0.47 to 1.02) processes (Bolden et al., 2012). Collectively, converging evidence indicates that ADHD is associated with significant impairment in the working component of working memory but more moderate deficits in the memory components, leading to observations that children with ADHD are ‘okay holders but bad shufflers’ of information (Simone et al., 2016). Working memory also shows strong, potentially causal relations to core ADHD symptoms and related impairments. Deficits display reciprocal, longitudinal

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relations with ADHD symptom severity, and remittance of ADHD symptoms over time is limited to the subset of children who similarly evince improvements in working memory performance (Karalunas et al., 2017). Central executive performance has been found to covary with core ADHD symptoms, and experimental studies manipulating working memory demands show increased presence and severity of inattentive (Orban et al., 2018), hyperactive (Sarver et al., 2015), and impulsive (Raiker et al., 2012) symptoms as central executive demands increase. However, relations between short-term memory subsystems and ADHD symptoms are either more limited or nonsignificant. In their seminal study, Kofler et al. (2010) examined the causal link between working memory component processes and ADHD symptoms using the regression-based approach noted above to isolate working memory subsystem contributions. Children with ADHD evinced disproportionate decrements on objective measures of attentive behavior during low vs high working memory conditions relative to neurotypical peers. This relation was carried almost entirely by deficits in the central executive component of working memory, with moderate contributions attributed to phonological short-term memory. A series of follow-up regression analyses attempted to provide insight into the directionality of the working memory/inattention relation. Between group differences in attentive behaviors were no longer evident after controlling for working memory abilities; however, deficits in working memory remained after controlling for attention. These findings indicate that working memory deficits account for the totality of variance in children’s attention problems (but not the reverse) and add to the considerable evidence for the causal role of working memory deficits in ADHDrelated symptoms and impairments. Similar patterns of working memory involvement are observed across functional domains including social problems (Kofler et al., 2011), written expression deficits (Eckrich et al., 2019), and overall academic impairment (Calub et al., 2019). Friedman and colleagues recently completed a series of investigations examining working memory processes as potential underlying mechanisms for explaining reading comprehension deficits common among children with ADHD (Friedman et al., 2017, 2018b). The central executive component of working memory was a significant mediator of ADHD-related reading comprehension deficits, while phonological and visuospatial short-term memory failed to explain significant variance in the diagnostic status to reading comprehension relation. Follow-up analyses indicated that the central executive makes both direct contributions to children’s reading comprehension and indirect influences through its involvement with basic reading processes (i.e., word reading and decoding abilities). This model explained 61% of the variance in ADHD-related reading comprehension deficits and fully accounted for between group reading differences, suggesting that inefficient or unsuccessful central executive-mediated processes of updating, manipulation, and reordering of decoded text, combined with poor attentional control and activation of long-term memories, are responsible for ADHD-related reading comprehension deficits. Collectively, converging evidence indicates significant, sizable deficiencies in central executive functioning associated with a diagnosis of ADHD. These deficits

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display causal relations to ADHD diagnostic symptoms (inattention, hyperactivity, and impulsivity) as well as related impairments across functional domains, including academic achievement. More moderate deficits are observed in children’s phonological and visuospatial short-term memory abilities, and these memory functions are either limitedly or unrelated to ADHD symptoms and impairments. Given its importance for explaining disorder-related behaviors, the central executive component of working memory appears to be a promising mechanism for explicating the significant math achievement deficits observed among children with ADHD and a potential treatment target for improving ADHD symptoms and impairments. Evidence for Working Memory Involvement in Math. Working memory is an especially relevant domain of EF with regard to children’s mathematical functioning. The domain-general central executive and subsidiary short-term storage and rehearsal subsystems each provide unique contributions during execution of mathematical problemsolving tasks. Phonological short-term memory is responsible for temporarily holding numbers and mathematical rules, saving partial solutions during multistep problems, and storing the text of word problems while completing applied math tasks. Visuospatial short-term memory organizes visual information during calculations (e.g., lining up the tens place), aids in analyzing visual content (e.g., graphs), and serves as mental blackboard where a visual representation of the math problem may be created in the “mind’s eye.” The domain-general central executive manages and coordinates these two systems to accomplish several objectives. These goals include determining the taskrelevance of information given in a word problem, connecting information presented in the math problem with knowledge of math rules stored in long-term memory, sustaining attentional focus while simultaneously inhibiting irrelevant information, maintaining the overall goal of the mathematical problem, and updating, reordering, and manipulating the information contained within the short-term stores. Current experimental evidence indicates a prominent role for working memory in children’s mathematical performance. Early working memory abilities are significant predictors of applied problem-solving and calculation skills. Children with arithmetic difficulties also score lower on working memory tasks that require manipulation of information (Passolunghi & Cornoldi, 2008). Studies examining the relationship between EF and specific math skills, such as whole-number calculations and word-problem solving, find strong correlations between working memory and nearly all math skills, and this relation is particularly pronounced for complex math skills that require multiple steps to complete (Peng et al., 2016). Empirical research has further examined the separate effects of the central executive and phonological/visuospatial subsystems. These studies demonstrate significant, independent contributions of the central executive and phonological short-term memory to children’s mathematical problem-solving abilities. Results concerning visuospatial short-term memory are mixed with most, but not all, studies demonstrating a considerable relationship between visuospatial memory and mathematical abilities. Collectively, overwhelming evidence provides clear support for working memory as an important determinant for proficient mathematical ability and achievement, and particular importance is ascribed to the central executive and phonological short-term memory components of working memory.

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Working Memory as a Contributor to ADHD-Related Math Deficits. Because working memory is intricately involved in children’s mathematical skills and children with ADHD evince significant impairment in working memory, it stands to reason that working memory may explain the well-documented math achievement decrements observed among children with ADHD. This notion is supported by research demonstrating that performance on working memory tasks mediates ADHD-related deficits in math achievement, with one study finding that working memory explained nearly two-thirds of the variance in between-group (high vs low ADHD) math achievement differences (Rennie et al., 2014). The effects of working memory deficits have detrimental consequences beyond math performance as well. Children with ADHD who have more severe working memory deficits are more likely to repeat a grade or receive special education placement relative to those with less severe working memory impairments (Fried et al., 2016). To date, relatively few studies have examined the differential effects of working memory subcomponent processes on math deficits among children with ADHD. Most studies utilize measures of phonological and visuospatial working memory that assess domain general and specific component processes concomitantly. These investigations show significant support for the phonological working memory as an explanatory mechanism for ADHD-related math deficits. Most, but not all, studies also implicate visuospatial working memory. While informative, this approach does not allow for fractionation of central executive and subsidiary memory processes – a critical area of investigation given differential patterns of impairment among children with ADHD and the unique roles ascribed to subcomponent processes for executing math-related tasks. Among the few studies to investigate working memory component processes, Re et al. (2016) found that children with ADHD made more errors on word problems that required information updating, a central executive process, relative to word problems that did not require updating for successful execution. In contrast, typically developing controls did not show any difference in performance on math problems with low vs high executive demands, implicating central executive updating processes as a key contributor to ADHD-related applied problem-solving difficulties. To our knowledge, only one study has examined the effect of both domain general and specific working memory components on children’s math abilities (Friedman et al., 2018). Friedman and colleagues investigated the differential effect of central executive, phonological short-term memory, and visuospatial short-term memory on ADHD-related applied problem-solving deficits (i.e., word problems). The role of basic math skills (i.e., calculation) on applied problem-solving differences was also investigated, hypothesizing that working memory may make both direct and indirect contributions to problem solving difficulties through its influence on lower-level math processes. In other words, higher order math problem-solving abilities rely on more basic math processes, both of which rely on working memory. It may be the individual and collective effects of working memory and calculation that affect applied math problem-solving, and this notion was tested (see Fig. 2). Using the latent variable approach developed by Rapport (2008) to isolate working memory component processes, neither visuospatial nor phonological

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Fig. 2 Serial mediation model depicting the total effect, direct effect, and indirect effects presented in Friedman et al. (2018) (reprinted with authors permission). Both central executive abilities and math calculation skills, when modeled serially, fully mediated the diagnostic status to applied problem solving relation. * indicates significance based on 95% confidence intervals that do not include 0.0.; CE¼ Central Executive; ER¼Effect Ratio

short-term memory were significant mediators of ADHD-related applied problemsolving difficulties. In contrast, both central executive and calculation abilities when modeled separately were significant mediators of the diagnostic status to applied problem-solving relation. Using serial mediation analyses, the central executive and math calculation skills completely mitigated diagnostic group differences in applied problem-solving. This interactional finding may reflect deficiencies in multiple central executive-mediated processes that involve the recall of math-related information from long-term memory; updating, reordering, and manipulating information held in memory; and influence how this knowledge is connected with and applied to mathematical word problems. Another potential explanation for these results lies in recent findings that children with ADHD fail to develop appropriate skill automaticity relative to their peers. Even after considerable practice, children with ADHD are unable to achieve the same level of skill proficiency as their non-ADHD peers on various cognitive tasks, particularly on tasks that require high levels of working memory involvement (Huang-Pollock & Karalunas, 2010). Once automaticity is achieved, fewer executive resources are required for successful execution. Poor automaticity of basic math processes, coupled with the well-documented deficits in central executive performance, may therefore explain the significant applied problem-solving deficits among children with ADHD. When greater executive resources are dedicated to compensating for the lack of basic math automaticity, fewer resources are available for comprehending, updating, and processing mathematical word problems.

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Summary. Extensive evidence indicates significant, pervasive working memory deficits among children with ADHD. Empirical approaches fractionating working memory components find a consistent pattern of impairment wherein largemagnitude deficits are observed in the central executive “working” component of working memory, but smaller deficiencies are observed in the verbal and visuospatial “memory” components of working memory. Complementary evidence examining working memory as a potential explanatory mechanism for the severe math challenges typical of the disorder implicates the domain-general central executive as a key underlying contributor, but little support for the causal role of phonological and visuospatial short-term memory has been found. Collectively, the central executive component of working memory appears to be a key contributor to ADHDrelated math achievement deficits and a promising target for potential intervention.

Behavioral Inhibition Behavioral inhibition refers to the ability to suppress a prepotent or ongoing response and reduce interference from irrelevant thoughts and actions while executing goal directed behaviors. Behavioral inhibition is not a unitary construct but rather encompasses at least two distinct inhibitory abilities – motor response inhibition and interference control. Motor response inhibition reflects the ability to cancel or suppress a prepotent, automatic, or reflexive response. For example, response inhibition is necessary when a student refrains from leaving their seat when the bell rings because the teacher has not formally dismissed the class. Response inhibition is often measured using the Go/No-Go or Stop-Signal tasks (See Table 3 for a description of common behavioral inhibition tasks). Interference control reflects the ability to prevent irrelevant or previously relevant information from accessing one’s working memory while engaging in a cognitive task. For example, children use interference control abilities during a math test to ignore distracting, external, irrelevant stimuli in their immediate environment (e.g., people having a conversation in the hallway) or suppress internal, irrelevant thoughts that may interfere with completing a math test (e.g., thinking about afterschool activities). A number of empirical studies have examined the developmental continuity of behavioral inhibition skills in early and later childhood. These studies indicate that behavioral inhibition may be a necessary executive function in early development (i.e., preschool and early elementary school) but may play a less relevant role as children age (Brocki & Bohlin, 2006). This notion is supported by several studies across populations that find evidence of inhibition-specific variance in EF performance among younger children but fail to find evidentiary support for separable inhibitory abilities in older children and adults (see Friedman & Miyake, 2017). Explanations for these findings abound. It has been hypothesized that behavioral inhibition may be an early precursor to the development of other executive functions (i.e., set-shifting and working memory). It has also been postulated that inhibition abilities represent goal monitoring behaviors common among all EF tasks (i.e., behavioral inhibition is a part of common EF variance rather than a separable

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Table 3 Common behavioral inhibition task examples Task Response Inhibition Stop Signal

Continuous Performance Test (CPT) or Go/No-Go

Interference control Stroop

Flanker

Description Two different letters (e.g., “X” or “O”) appear randomly on a computer screen. The “X” and “O” appear in equal proportions throughout the task and are considered the “go” trials. The participant is told to press one button when they see an “X” and press a different button when they see an “O.” On a smaller proportion of trials (e.g., 25%), the participant will hear a stop signal (i.e., tone) in which they are told to not press either of the buttons when they see a letter. These trials are referred to as “no-go” trials. Poor inhibitory control is evident when the person fails to inhibit a response on a stop signal trial (i.e., hearing the tone, but still pressing the button when they see an “X”). In addition, stop signal reaction time (SSRT) or stop signal delay (SSD; in dynamic stop signal task paradigms) can be calculated to isolate inhibitory control processes from general response output Several versions of the CPT and Go/No-Go tasks exist, but they all share the following features: participants must respond to target stimuli but withhold responding to rare, nontarget stimuli. In one of the more commonly used CPTs, the Conners Continuous Performance Test presents a series of letters one at a time. The participant is required to respond to all letters aside from the letter “X.” The nontarget “X” appears in approximately 20% of the presented stimuli over the 14-minute task. Poor inhibitory control is reflected when a person commits an error by responding to a no-go trial (i.e., responding when an “X” appears) Traditional Stroop tasks consist of three blocks. In the first block (basic word reading), participants read color names (e.g., BLUE, RED, GREEN) printed in black ink. In the second block (basic color naming), participants name the color shown (e.g., “XXXXX” printed in red ink). In the third block (interference control), participants are presented with color names printed in different color ink (e.g., the word “RED” printed in blue ink). Participants are asked to name the ink color and refrain from the reflexive response of reading the word. Poor inhibitory control is reflected when participants read the words more slowly or commit more errors on the third block compared to the first two blocks This task measures selective attention and response competition. Arrows (e.g., > > > > >) appear on the center of a screen. The participant must press a button to determine which direction the central arrow is facing, either left (“”). On some of the trials, the flanking arrows are congruent with the central arrow (> > > > >) on other trials the flanking arrows are incongruent with the central arrow (< < >