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- Marcel Danesi

*Table of contents : PrefaceContentsAbout the EditorSection EditorsContributorsIntroduction Cognitive MathematicsSection 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 ReferencesSection 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 ReferencesSection 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 ReferencesSection 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 ReferencesSection 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 ReferencesSection 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 ReferencesSection 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 ReferencesSection 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 ReferencesSection 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 ReferencesSection 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 ReferencesIndex*

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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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

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

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

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

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Blending Theory and Mathematical Cognition Marcel Danesi

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

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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

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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

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

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

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

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

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

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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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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

469

501

539

543

20

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

573

21

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

595

22

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

617

Volume 2 Section VI Learning and Teaching Mathematics Dragana Martinovic 23

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

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

643

647 685

727

763

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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

805

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

843

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

869

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

799

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

xiii

1203

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.

xv

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

xviii

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

xix

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

xx

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

xxii

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

xxiii

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

xxiv

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

xxv

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

xxviii

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

xxix

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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Introduction

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.

Introduction

xxxi

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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Introduction

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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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.

References Baddeley, A. (2000). The episodic buffer: A new component of working memory? Trends in Cognitive Sciences, 4(11), 417–423. Baddeley, A. (2007). Working memory, thought, and action. Oxford University Press. Bagchi, R., & Davis, D. (2016). The role of numerosity in judgments and decision-making. Current Opinion in Psychology, 10, 89–93. Bellucci, F. (2018). Peirce’s speculative grammar: Logic as semiotics. Routledge. Boesch, C. (1996). The emergence of cultures among wild chimpanzees. Proceedings of the British Academy, 88, 251–268. Bortot, M., Agrillo, C., Avarguès-Weber, A., Bisazza, A., Miletto Petrazzini, M. E., & Giurfa, M. (2019). Honeybees use absolute rather than relative numerosity in number discrimination. Biology Letters, 15, 20190138. Butterworth, B. (1999). What counts: How every brain is hardwired for math. Free Press. Cooke, E. (2011). Peirce’s general theory of inquiry and the problem of mathematics. In M. E. Moore (Ed.), New essays on Peirce’s mathematical philosophy (pp. 169–202). Open Court Publishing. Coolidge, F. L., & Overmann, K. A. (2012). Numerosity, abstraction, and the emergence of symbolic thinking. Current Anthropology, 53(2), 204–225. Cowan, N. (2000). The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences, 24(1), 87–185. Danesi, M. (2019). Math puzzles as learning devices. In M. Danesi (Ed.), Interdisciplinary perspectives on math cognition (pp. 141–153). Springer Nature. Davis, H. (1993). Numerical competence in animals: Life beyond Clever Hans. In S. T. Boysen & E. J. Capaldi (Eds.), The development of numerical competence: Animal and human models (pp. 109–125). Lawrence Erlbaum Associates, Inc. Erlam, R. (2009). The elicited oral imitation test as a measure of implicit knowledge. In R. Ellis, S. Loewen, C. Elder, R. Erlam, J. Philp, & H. Reinders (Eds.), Implicit and explicit knowledge in second language learning, testing, and teaching (pp. 65–93). Multilingual Matters. Feigenson, L. S., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8, 307–314. Gallistel, C. R. (1990). The organization of learning. MIT Press. Gibson, J. J. (1979). The ecological approach to visual perception. Lawrence Erlbaum Associates. Harvey, B. M., Klein, B. P., Petridou, N., & Dumoulin, S. O. (2013). Topographical representation of numerosity in the human parietal cortex. Science, 341(6150), 1123–1126. Hummel, J. (1999). The binding problem. In R. A. W. F. C. Keil (Ed.), The MIT encyclopedia of cognitive sciences (pp. 85–86). MIT Press. Kaufman, E. L., Lord, M. W., Reese, T. W., & Volkmann, J. (1949). The discrimination of visual number. American Journal of Psychology, 62, 498–525.

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

116 117 118 121 123 125 126

Abstract

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

130 131 132 133 134 136 137 139 139 139 143 146 147 147 148 150 153 153 155 157 158

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

1

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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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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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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Lipka, J., Hogan, M. P., Webster, J. P., Yanez, E., Adams, B., Clark, S., & Lacy, D. (2005). Math in a cultural context: Two case studies of a successful culturally based math project. Anthropology in Education Quarterly, 36(4), 367–385. Loyd, S., Harding-DeKam, J. L., & Hamilton, B. (2014–2015). The power of multicultural mathematics picturebooks. Colorado Reading Journal, 25, 5–10. Maaka, M., Au, K. H., Lefcourt, Y. K., & Bogac, L. P. (2001). Raccoon? Wass dat?. Hawaiian preservice teachers reconceptualize culture, literacy, and schooling. In P. R. Schmidt & P. B. Mosenthal (Eds.), Reconceptualizing literacy in the new age of multiculturalism and pluralism (pp. 341–366). Honolulu Advertiser. Martin, D. B., & McGee, E. O. (2009). Mathematics literacy an liberation: Reframing mathematics education for African-American children. In B. Greer, S. Mukhopadhyay, A. B. Powell, & S. Nelson-Barber (Eds.), Culturally responsive mathematics education (pp. 207–238). New York: Routledge Taylor & Francis Group. Massarwe, K., Verner, I., Bshouty, D., & Verner, I. (2010). An ethnomathematics exercise in analyzing and constructing ornaments in a geometry class. Journal of Mathematics and Culture, 5(1), 1–20. Metcalfe, J., & Shimamura, A. P. (1994). Metacognition: Knowing about knowing. Cambridge, MA: MIT Press. Mogari, D. (2014). An in-service programme for introducing an ethno-mathematical approach to mathematics teachers. Africa Education Review, 11(3), 348–364. Moses, R., West, M. M., & Davis, F. E. (2009). Culturally responsive mathematics education in the algebra project. In B. Greer, S. Mukhopadhyay, A. B. Powell, & S. Nelson-Barber (Eds.), Culturally responsive mathematics education (pp. 239–256). New York: Routledge Taylor & Francis Group. Naresh, N. (2015). The role of a critical ethnomathematics curriculum in transforming and empowering learners. Revista Latinoamericana de Etnomatemática, 8(2), 450–471. Nasir, S. N. (2002). Identity, goals, and learning: Mathematics in cultural practice. Mathematical Thinking and Learning, 4(2&3), 213–247. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (2014). Principles to action: Ensuring mathematical success for all. Reston: National Council of Teachers of Mathematics. Nurdiansyah, I., Sarwi, & Haryani, S. (2021). Ethnomathematics contained of guided inquiry for elementary fourth grade students. Journal of Primary Education, 10(2), 160–165. Orey, D. C. (2017). The critical-reﬂective dimension of ethnomodelling. In M. Rosa, L. Shirley, M. E. Gavarrete, & W. F. Alangui (Eds.), Ethnomathematics and its diverse approaches for mathematics education (pp. 329–354). Cham: Springer International Publishing. Orey, D. C., & Rosa, M. (2015). Three approaches in the research ﬁeld of ethnomodeling: Emic (local), etic (global), and dialogical (glocal). Revista Latinoamericana de Etnomatemática, 8(2), 364–380. Owens, K. (2012). Identity and Ethnomathematics projects in Papua New Guinea. Adelaide: Mathematics Education Research Group of Australasia. Owens, K. (2017). The role of culture and ecology in visuospatial reasoning: The power of ethnomathematics. In M. Rosa, L. Shirley, M. E. Gavarrete, & W. F. Alangui (Eds.), Ethnomathematics and its diverse approaches for mathematics education (pp. 209–233). Cham: Springer. Owens, K., Paraides, P., Nutti, Y. J., Johansson, G., Bennet, M., Doolan, P., & Taylor, P. (2011). Cultural horizons for mathematics. Mathematics Education Research Journal, 23(2), 253. Pea, R. D. (2004). The social and technological dimensions of scaffolding and related theoretical concepts for learning, education, and human activity. The Journal of the Learning Sciences, 13(3), 423–451. https://doi.org/10.1207/s15327809jls1303_6. Powell, A. B., & Temple, O. L. (2001). Seeding ethnomathematics with oware: Sankofa. Teaching Children Mathematics, 7(6), 369–375. 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. (2011). Ethnomodeling: A pedagogical action for uncovering ethnomathematical practices. Journal of Mathematical Modelling and Application, 1(3), 58–67.

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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

10

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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Fig. 2 (a) Schematic illustration of the numerical ratio effect. It is easier to discriminate numerical quantities with a small ratio (e.g., 0.1) compared to numerical quantities with a large ratio (e.g., 0.9). (b) The approximate representation model of numerical quantities. Numerical quantities are represented as Gaussian distributions with scalar variability (i.e., Weber’s law). The increase in distributional variability results in larger uncertainty to represent large numerical quantities compared to small numerical quantities

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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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Abstract

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 m