Piaget’s Genetic Epistemology for Mathematics Education Research 9783031473852, 9783031473869

Table of contents :
Contents
About the Authors
Part I: Introduction to Piaget’s Genetic Epistemology and the Tradition of Use Featured in This Book
Chapter 1: Introduction to Piaget’s Genetic Epistemology
Introduction
Why Is Piaget’s Genetic Epistemology Useful?
Organization of the Book
References
Chapter 2: An Historical Reflection on Adapting Piaget’s Work for Ongoing Mathematics Education Research
Piaget and Modern Mathematics
Development vs. Learning
Developmental vs. Mathematical Structure
Preludes to IRON (Interdisciplinary Research on Number)
Piagetian Research
A Change in Research Program
A Fortunate Introduction
Interdisciplinary Research on Number (IRON)
Original Members of IRON
Ernst as Scientist
Piaget’s Epistemic Child
Conceptual Analysis
The First Type
Attentional Moments and the Unitizing Operation
The Second Type
Schemes
Systems of Schemes
The Initial Number Sequence as a Scheme
Operationalizing Reflective Abstraction
Constructing the Initial Number Sequence
Further Operationalizing Reflective Abstraction
Learning Stages
The Perceptual Stage
The Figurative Stage
Percent of First-Grade Children in the Perceptual Stage
Further Criteria of Learning Stages
Modifications in the Initial Number Sequence
Modifications in the Explicitly Nested Number Sequence
The Fractions Teaching Experiment
The Equipartitioning Scheme: A Functional Accommodation
Pre-fractional Children
The Partitive Fraction Scheme
The Iterative Fraction Scheme
Final Comments
References
Part II: Key Constructs from Genetic Epistemology Being Used in Ongoing Mathematics Education Research
Chapter 3: Schemes and Scheme Theory: Core Explanatory Constructs for Studying Mathematical Learning
Brief Overview of Glasersfeld’s Radical Constructivist Epistemology
Piaget’s Development and Glasersfeld’s Three-Part Definition of Schemes
Schemes from Reflexes
Sensory Motor and Conceptual Schemes
Empirical Example of Glasersfeld’s Three-Part Definition of Scheme: Michael Solves the Outfits Problem
Building on Glasersfeld’s Definition of Schemes: Steffe’s Tetrahedral Model
Nuances in Steffe’s Definition of a Scheme
Empirical Example of Nuances in Steffe’s Definition of Scheme: Nico’s Reversible Scheme
Using Schemes to Investigate Learning
Assimilation, Perturbation, and Accommodation
Empirical Example of Assimilation: Carlos’s Solution of the Flag Problem
Empirical Example of Perturbation: Carlos’s Solution of the Flag Problem
Empirical Example of Functional Accommodation: Carlos’s Solution of the Handshake Problem
Situating Investigations of Learning Within a Broader Framework: Stages
Glasersfeld’s Definition of Stage
Hackenberg’s and Norton’s Stages of Multiplicative Reasoning and a 2-slot MPS
Recursion in Thompson’s Definition of Scheme
Empirical Example of a Stage 2 Student Sequentially Using Her MPS: Mikayla Solves the Sandwich Problem
Empirical Example of a Stage 3 Student Recursively Inserting Operations into a Scheme: Tyrone Solves the Card Problem
Revisiting Theoretical Constructs Relative to the Data Examples
Investigating Learning of Stage 3 Students: Levels of Schemes
Empirical Example of Different Levels of a Scheme for Stage 3 Students: Armando Solves the Colored Digits Problem
Levels, Functional Metamorphic Accommodation, and Reflecting Abstraction
Conclusion
References
Chapter 4: Operationalizing Figurative and Operative Framings of Thought
Introduction
Some “Definitions”
Uses and Evolution of Figurative and Operative Thought
Transitioning the Constructs to Mathematics Education
Transitioning the Constructs to Higher Level Mathematics
Models of Students’ Graphical Thinking
Informing Generalized Models of Student Thinking
Adapting the Distinctions to Other Representations
Transitioning the Constructs Back to the Study of Meaning Construction
Implications for Methodology and Task-Design
Moving Forward
References
Chapter 5: Figurative and Operative Imagery: Essential Aspects of Reflection in the Development of Schemes and Meanings
Imagery
Imagery, Schemes, and Meanings
Images and Schemes
First-Level Imagery (Deferred Imitation)
Second-Level (Figurative) Imagery
Third-Level (Operative) Imagery
Summary
Imagery, Schemes, and Reflective Abstraction
Case Studies
Imagery in the Construction of a Nim Scheme
Session 1: June 17, 2020
21 and 3
38 and 8
33 and 7
Session 2: July 28, 2020
21 and 3
36 and 5
General Nim
Discussion
Nim Scheme
General Nim Scheme
Implications for Math Education
Implications for Mathematics Teaching
Implications for Mathematics Education Research
Imagery in the Projection from Figurative to Reflected Thought
Implications for Mathematics Education
Implications for Mathematics Teaching and Mathematics Education Research
Discussion
References
Chapter 6: Empirical and Reflective Abstraction
The Enduring Attention to Abstraction
Considering Abstraction When Making Sense of Student Reasoning
Angelo
Willow
The Basis for Angelo and Willow’s Abstractions
Empirical Abstraction
Reflective Abstraction
Two Phases of Reflective Abstraction
Pseudo-empirical Abstraction
Reflecting Abstraction
Reflected Abstraction
Data Episodes
The Faucet Task
Mario Engages in Empirical and Reflective Abstractions
Kendis and Camila Engage in Pseudo-empirical, Reflecting, and Reflected Abstractions
The Passwords Activity
Tyler Engages in Pseudo-empirical and Reflecting Abstraction
A Group of Students Engage in Pseudo-empirical, Reflecting, and Reflected Abstraction
Discussion
Standards of Evidence
The Cyclical Nature of Abstraction
The Value of Abstraction as a Construct
References
Chapter 7: Groups and Group-Like Structures
Two Kinds of Structure
Groups
Closure
Identity and Reversibility
Associativity
Group-Like Structures
Properties of Groupings
A Critical Analysis of Groupings
The Splitting Loope and the Splitting Group
Genetic Roots in Mathematics
The Erlangen Program
INRC and the Bourbaki
Applying Mathematical Structures to Mathematics Education Research
Summary
References
Chapter 8: Reflected Abstraction
Introduction
Piagetian Abstraction
Empirical Abstraction
Pseudo-Empirical Abstraction
Reflecting Abstraction
Reflected Abstraction
What Is Reflected Abstraction?
Supporting Retroactive Thematization
Example of Applying a Reflected Scheme
Relation of Reflected Abstraction to Other Piagetian Constructs
The Semiotic Function and Representational Thought
Schemes and Equilibration
Imagery
Figurative and Operative Modes of Thought
Relation of Reflected Abstraction to Theoretical Constructs within Mathematics Education Research
APOS Theory
Harel’s Duality Principle
Quantitative and Covariational Reasoning
Lobato’s Actor-Oriented Transfer
Implications of Reflected Abstraction
Implications for Mathematics Education Research on Student Learning
Implications for Supporting Students’ Learning in Teaching Contexts
Implications for Researching and Supporting Teachers’ Pedagogical Content Knowledge
Final Comments on Genetic Epistemology’s Place in Educational Psychology and Mathematics Education
References
Chapter 9: The Construct of Decentering in Research on Mathematics Learning and Teaching
Theoretical Background, Framing, and Connections
Decentering’s Origins and Adaptation for Use in Mathematics Education Research
The Origins of Decentering in Piaget’s Genetic Epistemology
Adapting Decentering for Use in Mathematics Education Research
Connections Between Decentering and Other Theoretical Constructs Used in Mathematics Education
Reflecting and Reflected Abstraction and Their Connection to Decentering in the Context of Teaching
First- and Second-Order Models and Their Connection to Decentering
Mathematical Knowledge for Teaching and Its Connection to Decentering
Key Developmental Understanding (KDU) and Pedagogical Understanding
Epistemic Students Emerge from Conceptual Analysis and Second-Order Models
Clinical Interview Methodology
The Role of Decentering in Clinical Interview Data Collection
Examples of Decentering in Mathematics Education Research
Example 1: How a Researcher’s Meaning for Rate of Change Informed Data Collection and Data Analysis
Example 2: Researcher Decentering in a Teaching Experiment on Logarithms
Iterative Models of a Student’s Thinking Informs Teaching Experiment Design
Exploratory Teaching Interviews Lead to Advancements in a Researcher’s First-Order Model
First-Order Models, Decentering, Second-Order Models, and Conceptual Analysis Inform Task Design
Modeling Student Thinking in the Context of a Teaching Experiment
Comments on Examples 1 and 2
Uses of Decentering When Studying Teachers and Teaching
Elaborating Our Meaning for MMT and Its Symbiotic Relationship with Decentering
The Symbiotic Relationship Between a Teacher’s Meaning for an Idea and Her Decentering Actions
Advances in a Teacher’s Ways of Thinking About Teaching an Idea
Shifts in a Teacher’s Meaning for the Idea of Average Rate of Change (AROC) and Her Ways of Thinking About Teaching the Idea of AROC
Decentering Actions Lead to Advances in a Teacher’s MMT for Teaching AROC
Characterizing Teacher Decentering
Decentering as a Construct for Studying Teachers
A Decentering Framework for Studying Teaching
Concluding Remarks
References
Chapter 10: Logic in Genetic Epistemology
Introduction and Goals
The Nature of Logic and Its Possible Relations to Human Reasoning
Logic and Psychology: Respectfully Disjoint
Logic and Psychology: A Recurring Methodological Conundrum
Extension and Intension
Implication and Inference
The Tasks
Modeling Stages and Stages of Modeling
Varying Use of Propositional Variable Expressions
Syntactic Transformations as Researcher Inferences
Elaborating Possibilities and “All Other Things Being Equal” Reasoning
The Child and Researcher’s Constructions of the Truth Table of 16 Possibilities
Logic of Meanings: Assimilation as Extension
Concluding Lessons
References
Chapter 11: Students’ Units Coordinations
Introduction
What Is Units Coordination?
Definitions and Characterizations, with Examples
Students at Stage 1
Students at Stage 2
Students at Stage 3
How Have Researchers Used Units Coordination in Research?
Units Coordination and Fractions Knowledge: An Overview
Units Coordination and Fraction Knowledge: Examples
Stage 1
Stage 2
Stage 3
Units Coordination and Algebraic Reasoning: Examples
Stages 1, 2, and 3: Quantitative Unknowns and Conjectures About Them
Stages 2 and 3: Drawings of Quantitative Unknowns
Stages 2 and 3: Equation Writing
Data Excerpt 1: Elliot and the Teacher Converse About His Ideas About Equations
Stage 3: Reciprocal Reasoning
Standards of Evidence for Making Claims about Units Coordination
Preliminary Requirements
Good Tasks
Good Probing of Students
Good Records of Students’ Interactions and Work
Data Analysis and Claims: An Example
Working Model of Emily’s Mathematics in September
Retrospective Model of Emily’s Mathematics in September
Data Except 2: Emily’s Work on the Tiles Problem
Data Excerpt 3: Determining the Number of Cans of Juice in the Crate
Continuation of Data Except 3: Continuing to Determine the Number of Cans in the Crate
Second Continuation of Data Except 3: What Does the 32 Mean?
Population Estimates and Stage Changes
Conclusion
References
Chapter 12: Modeling Quantitative and Covariational Reasoning
Modeling Quantitative and Covariational Reasoning
Meanings for “Quantitative” and “Covariational”
Gross, Intensive, and Extensive Quantities
The Meaning of Quantity
Arithmetic Reasoning, Quantitative Reasoning, and Reasoning Quantitatively
Reasoning Quantitatively
Modeling Students’ Images of Speed
Variational Reasoning
Models of Continuous Variation
Chunky, Smooth, and Scaling Continuous Variational Reasoning
Quantitative Variational Reasoning
Covariational Reasoning as Correspondence Between Variations
Methodological Considerations for Investigating Quantitative and Covariational Reasoning
The Role of Technology in Quantitative and Covariational Reasoning
Role of Modeling Education in Promoting Quantitative and Covariational Reasoning
References
Part III: Commentaries on Genetic Epistemology and Its Use in Ongoing Research
Chapter 13: Genetic Epistemology as a Complex and Unified Theory of Knowing
Genetic Epistemology
Historicocritical and Psychogenetic Methods
How Is Mathematical Knowledge Possible?
Connections to Prior Chapters and Critical Constructs
Chapter 3: Schemes and Operations
Chapter 4: Figurative and Operative Thought
Chapter 5: Images
Chapter 6: Empirical, Pseudoempirical, and Reflective Abstraction
Chapter 7: Groups and Groupings
Chapter 8: Reflected Abstraction
Chapter 9: Decentering
Chapter 10: Logic
Chapter 11: Units Coordination
Chapter 12: Quantitative and Covariational Reasoning
Future Directions for Genetic Research
What Are the Relationships Between Units Coordination and Covariational Reasoning?
How Can We Support Students’ Stagewise Development of Units Coordination?
Can We Specify the Coordination of Actions That Constitute Reflective Abstraction Across Various Mathematical Domains?
What Is the Appropriate Role of Formalization in Mathematics Education?
How Might Teachers Assess Students’ Available Mental Actions and Model Their Coordination as Reversible and Composable Operations?
Conclusion
References
Chapter 14: Second-Order Models as Acts of Equity
Defining Acts of Equity
How Are Making and Using Second-Order Models Acts of Equity?
Making Second-Order Models
What Is a Second-Order Model?
Why Is Making a Second-Order Model an Act of Equity?
Establishing Epistemic Students
What Is an Epistemic Student?
Why Is Establishing an Epistemic Student an Act of Equity?
Using Second-Order Models and Epistemic Students
What Does It Mean to Use Second-Order Models and Epistemic Students?
How Is Using Epistemic Students an Act of Equity?
How Can We Enhance Current Second-Order Models?
Social Identity Categories and Social Identities
Whose Reasoning Is Represented in Our Models? Participants’ Social Identity Categories
Who Are the Model Builders? Researchers’ Social Identity Categories
Theorizing About Social Identity Categories and Social Identities
Considerations for Making Second-Order Models That Account for Social Identities
Example 1: Addressing gender equity in interactions
Acts of Equity in an Interaction
Making Second-Order Models That Include Acts of Equity in Interactions
Example 2: Designing to address equity
Designing Interactions to Address an Equity Issue
Making Second-Order Models from the Study
Looking Ahead
References
Chapter 15: Reflections on the Power of Genetic Epistemology by the Modern Cognitive Psychologist
Reflections of Respectful Tourists
What a GE Approach to Mathematical Cognition Offers for Psychology
A Paean to Construct Validity
GE’s Potential Contribution to Psychology, Construct #1: Fraction Schemes
GE’s Potential Contribution to Psychology, Construct #2: The Figurative/Operative Distinction
What Can Psychology Offer to the Genetic Epistemologist?
The GE Approach Would Be Much more Powerful If Updated to Feature a Theory of Memory
The GE Approach Would Be Much more Powerful If Updated with a Probabilistic, Emergentist Conception of Cognition
A Major “What If”
References
Chapter 16: Skepticism and Constructivism
Key Lessons I Glean from Piagetian Constructivism
A Useful Example of a Piagetian Experiment
The Principle of Subjective Rationality
Rationality and Normativity
Knowledge Assumes a Knower
Skepticism in Modeling
Scientific Model Building and Ontological Correspondence
Wittgenstein’s on Certainty (1969)
Considering Constructivist Skepticism
The Contradiction of Radical Constructivism and Philosophical Skepticism
Improving Communication
Summary and Conclusions
References
Part IV: Using Constructs from Genetic Epistemology to Develop Agendas of Research
Chapter 17: Researching Special Education: Using and Expanding Upon Genetic Epistemology Constructs
Mathematics Interventions and Students with Disabilities
Turning Around
Genetic Epistemology as Part of a Framework for Equity and Inclusion
Expanding Theory and Building a New Evidence Base
References
Chapter 18: Research in Subitizing to Examine Early Number Construction
Questions Regarding Children’s Number Construction
Units Construction and Subitizing Activity
Genetic Epistemology as a Pathway to Early Childhood Mathematics Education Scholarship
Next Steps in Early Childhood Mathematics Education
Equitable Access to Opportunities for Number Construction
Conclusion and Final Thoughts
References
Chapter 19: Researching Coordinate Systems Using Genetic Epistemology Constructs
Piaget’s Distinctions in Children’s Organizations of Space
Piaget’s Logical Multiplication of Measurements and Units Coordination
References
Chapter 20: Researching Quantifications of Angularity Using Genetic Epistemology Constructs
Distinguishing Quantifications of Angularity Using Piagetian Constructs
Thinking About One-Degree Angles in the Content Course
Establishing Orienting Constructs in the Doctoral Seminar and Elsewhere
Consulting and Organizing Prior Literature
Formulating Hypotheses
Designing Some Initial Interview Tasks
Concluding Remarks
References
Chapter 21: Using Constructivism to Develop an Agenda of Research in Stochastics Education Research
Researching People’s Meanings
Researching Teaching
A Radical Constructivist Statistician
Final Thoughts
References
Index

Citation preview

Research in Mathematics Education Series Editors: Jinfa Cai · James A. Middleton

Paul Christian Dawkins Amy J. Hackenberg Anderson Norton   Editors

Piaget’s Genetic Epistemology for Mathematics Education Research

Research in Mathematics Education Series Editors Jinfa Cai, Newark, DE, USA James A. Middleton, Tempe, AZ, USA

This series is designed to produce thematic volumes, allowing researchers to access numerous studies on a theme in a single, peer-reviewed source. Our intent for this series is to publish the latest research in the field in a timely fashion. This design is particularly geared toward highlighting the work of promising graduate students and junior faculty working in conjunction with senior scholars. The audience for this monograph series consists of those in the intersection between researchers and mathematics education leaders—people who need the highest quality research, methodological rigor, and potentially transformative implications ready at hand to help them make decisions regarding the improvement of teaching, learning, policy, and practice. With this vision, our mission of this book series is: (1) To support the sharing of critical research findings among members of the mathematics education community; (2) To support graduate students and junior faculty and induct them into the research community by pairing them with senior faculty in the production of the highest quality peer-reviewed research papers; and (3) To support the usefulness and widespread adoption of research-based innovation.

Paul Christian Dawkins Amy J. Hackenberg  •  Anderson Norton Editors

Piaget’s Genetic Epistemology for Mathematics Education Research

Editors Paul Christian Dawkins Department of Mathematics Texas State University San Marcos, TX, USA

Amy J. Hackenberg Department of Curriculum and Instruction Indiana University Bloomington, IN, USA

Anderson Norton Department of Mathematics Virginia Tech Blacksburg, VA, USA

ISSN 2570-4729     ISSN 2570-4737 (electronic) Research in Mathematics Education ISBN 978-3-031-47385-2    ISBN 978-3-031-47386-9 (eBook) https://doi.org/10.1007/978-3-031-47386-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed 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 Paper in this product is recyclable.

To our teachers, most prominently the children who teach us their mathematics as well as Les Steffe and Pat Thompson who paved the way for this flourishing work.

Contents

Part I Introduction to Piaget’s Genetic Epistemology and the Tradition of Use Featured in This Book 1

 Introduction to Piaget’s Genetic Epistemology������������������������������������    3 Paul Christian Dawkins, Amy J. Hackenberg, and Andy Norton

2

 Historical Reflection on Adapting Piaget’s Work for Ongoing An Mathematics Education Research����������������������������������������������������������   11 Leslie P. Steffe

Part II Key Constructs from Genetic Epistemology Being Used in Ongoing Mathematics Education Research 3

Schemes and Scheme Theory: Core Explanatory Constructs for Studying Mathematical Learning����������������������������������������������������   47 Erik S. Tillema and Andrew M. Gatza

4

 Operationalizing Figurative and Operative Framings of Thought ����   89 Kevin C. Moore, Irma E. Stevens, Halil I. Tasova, and Biyao Liang

5

Figurative and Operative Imagery: Essential Aspects of Reflection in the Development of Schemes and Meanings ������������������������������������  129 Patrick W. Thompson, Cameron Byerley, and Alan O’Bryan

6

Empirical and Reflective Abstraction����������������������������������������������������  169 Amy Ellis, Teo Paoletti, and Elise Lockwood

7

 Groups and Group-Like Structures ������������������������������������������������������  209 Anderson Norton

8

Reflected Abstraction������������������������������������������������������������������������������  239 Michael A. Tallman and Alan E. O’Bryan

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Contents

The Construct of Decentering in Research on Mathematics Learning and Teaching����������������������������������������������������  289 Marilyn P. Carlson, Sinem Bas-Ader, Alan E. O’Bryan, and Abby Rocha

10 Logic  in Genetic Epistemology ��������������������������������������������������������������  339 Paul Christian Dawkins 11 Students’ Units Coordinations����������������������������������������������������������������  371 Amy J. Hackenberg and Serife Sevinc 12 Modeling  Quantitative and Covariational Reasoning��������������������������  413 Steven Boyce Part III Commentaries on Genetic Epistemology and Its Use in Ongoing Research 13 Genetic  Epistemology as a Complex and Unified Theory of Knowing������������������������������������������������������������������������������������������������  447 Anderson Norton 14 Second-Order  Models as Acts of Equity������������������������������������������������  475 Amy J. Hackenberg, Erik S. Tillema, and Andrew M. Gatza 15 Reflections  on the Power of Genetic Epistemology by the Modern Cognitive Psychologist ����������������������������������������������������������������������������  511 Percival Matthews and Alexandria Viegut 16 Skepticism and Constructivism��������������������������������������������������������������  541 Paul Christian Dawkins Part IV Using Constructs from Genetic Epistemology to Develop Agendas of Research 17 Researching  Special Education: Using and Expanding Upon Genetic Epistemology Constructs������������������������������������������������������������������������  565 Jessica H. Hunt 18 Research  in Subitizing to Examine Early Number Construction�������  573 Beth L. MacDonald 19 Researching  Coordinate Systems Using Genetic Epistemology Constructs������������������������������������������������������������������������������������������������  585 Hwa Young Lee 20 Researching  Quantifications of Angularity Using Genetic Epistemology Constructs����������������������������������������������������������  595 Hamilton L. Hardison

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21 Using  Constructivism to Develop an Agenda of Research in Stochastics Education Research ��������������������������������������������������������  605 Neil Hatfield Index������������������������������������������������������������������������������������������������������������������  613

About the Authors

Sinem Bas-Ader  is an assistant professor in the Department of Mathematics and Science Education at Istanbul Aydin University. She received her PhD from Middle East Technical University. She is studying teacher noticing of students’ mathematical thinking and she is working with both pre-service and in-service mathematics teachers in various professional development contexts. In her postdoctoral study as a visiting scholar at Arizona State University, she joined an NSF-funded Pathways Project team directed by Professor Marilyn P. Carlson. She developed an interest for constructivism in mathematics teaching and particularly focused on Piaget’s construct of decentering as a key competence for responsive teaching. Steven  Boyce  is an associate professor of Mathematics Education at Portland State University in Portland, Oregon. His passion is modeling students’ mathematical thinking, in particular connections between adolescents’ and adults’ coordinations of numerical and quantitative units across contexts. Dr. Boyce is the author of dozens of articles and conference proceedings pertaining to units coordination, fractions, pre-calculus, calculus, and teacher education. He is currently serving as guest editor of a special issue on Units Construction and Coordination across the Curriculum, PK-20, to appear in the journal Investigations in Mathematics Learning in Fall 2024. Cameron  Byerley  is an assistant professor in the Department of Mathematics, Statistics, and Social Studies Education at the University of Georgia. She is interested in using models of mathematical thinking to improve communication of critical real-world quantitative information to the public. Marilyn P. Carlson  is a professor of Mathematics Education studying the teaching and learning of ideas of precalculus and beginning calculus. Her current research is studying mechanisms for supporting precalculus level instructors in transitioning their instruction to be more coherent, engaging, and meaningful for students. She is exploring the interaction between a teacher’s mathematical meanings for teaching an idea, their decentering actions, and their effectiveness in modeling students’ xi

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About the Authors

thinking. She and her colleagues are working to design interventions to support precalculus instructors in modeling students’ thinking and using these models when planning their lessons, interacting with students, and assessing student learning. Paul  Christian  Dawkins  is a professor of Mathematics Education in the Department of Mathematics at Texas State University. He studies the teaching and learning of proof-based mathematics at the undergraduate level. He focuses particularly on real analysis, geometry, and logic. His teaching experiments focus on modeling student thinking about mathematical proofs and developing teaching sequences and learning trajectories by which students can learn the epistemology of mathematical proving. He received the Selden Award for research in undergraduate mathematics education from the Mathematics Association of America. Amy Ellis  is a professor of Mathematics Education at the University of Georgia. She received her Ph.D. in Mathematics Education from the University of California, San Diego and San Diego State University. Amy’s research agenda addresses students’ algebraic reasoning, teachers’ pedagogical practices for fostering meaningful student engagement, and playful math, which is the investigation of students’ playful engagement with mathematical ideas. Amy has received 15 grants from national and state organizations, including the National Science Foundation and the Institute of Education Sciences, and has published three books for the National Council of Teachers of Mathematics Essential Understanding Series. Andrew M. Gatza  is a former middle school mathematics teacher and completed doctoral programs in Mathematics Education (Indiana University, Bloomington) and Urban Education Studies (Indiana University, Indianapolis). He is currently an assistant professor of Mathematics Education in the Department of Mathematical Sciences at Ball State University. His work focuses on bringing together equity and justice issues with rich mathematical problem sequences to investigate the kinds of reasoning in which students engage. Additionally, he is focused on cultivating mutually supportive school partnerships and developing socially conscious mathematics educators who foster positive mathematics identities and understand and support students’ ways of reasoning. Amy J. Hackenberg  taught middle and high school students for 9 years in Los Angeles and the Chicago area. She earned her MAT degree in Mathematics Education from the University of Chicago and her PhD from the University of Georgia. Amy is a professor of Mathematics Education at Indiana University, Bloomington. She studies how middle and high school students construct fractions, rational number knowledge, and algebraic reasoning; how to differentiate instruction for students with diverse ways of thinking; and how to support teachers to support their students to feel a stronger sense of belonging in math class.

About the Authors

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Hamilton L. Hardison  is an assistant professor of Mathematics at Texas State University. He earned his PhD in Mathematics Education at the University of Georgia, and his primary research interests lie in investigating students’ mathematical thinking. His current research focuses on modeling how students quantify angularity, how these quantifications change over time, and how they vary across contexts. His additional interests include studying mathematical classroom discourse and discussing radical constructivism. Neil Hatfield  is an assistant research professor in the Department of Statistics at the Pennsylvania State University (USA). His research foci include cognition around the concept of distribution and related stochastic concepts, the teaching of stochastics (statistics, probability, and data science), and on diversity, equity, and inclusion issues in STEM. Jessica H. Hunt  began her career in education as a middle school mathematics teacher in a technology demonstration school in Florida. From that work, she grew to love teaching students at risk for mathematics difficulties or disabilities. Hunt argues that mathematics instruction for these students should (a) uncover strengths, (b) give access to their mathematical reasoning, and (c) support the advance of that reasoning. Hwa Young Lee  received her PhD in Mathematics Education from the University of Georgia and is an associate professor in the Department of Mathematics at Texas State University. Her main research interest is in investigating students’ mathematical thinking—specifically, students’ constructions of frames of reference, coordinate systems, and graphs—and in learning how teachers can facilitate and support their students’ mathematical thinking. Biyao Liang  obtained her PhD degree in Mathematics Education at the University of Georgia. She is currently an assistant professor at The University of Hong Kong. Her research program covers the areas of mathematical cognition, social interactions, and teacher education. She is interested in students’ and teachers’ mathematical thinking in a variety of contexts. Her current projects investigate the affordances of computer programming in engendering and supporting uncanonical and productive ways of reasoning with mathematics. She is also interested in teachers’ mathematical learning through social interactions with students and designing educational opportunities, tools, and materials that can support teachers’ learning about students. Elise Lockwood  is a professor in the Department of Mathematics at Oregon State University. She received her PhD in Mathematics Education from Portland State University and was a postdoctoral scholar at the University of Wisconsin–Madison. Her primary research interests focus on undergraduate students’ reasoning about combinatorics, and she is passionate about improving the teaching and learning of discrete mathematics. Her work has been funded by the National Science Foundation and Google, and her current work explores ways in which computation and

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programming can be leveraged to support students’ combinatorial thinking and activity. Elise currently serves as co-editor in chief of the International Journal of Research in Undergraduate Mathematics Education. In 2019, she was a Fulbright Scholar to Norway, working with researchers at the University of Oslo. Beth L. MacDonald  is an associate professor of Early Childhood Mathematics Education in the School of Teaching and Learning at Illinois State University. Beth taught elementary school for 15 years and served as a PreK-5 Instructional Specialist for 2 years in Virginia. While teaching, Beth completed her Master’s degree and her Doctor of Philosophy degree, each with a Curriculum and Instruction concentration and focus on Mathematics Education, both from Virginia Tech. Her research broadly focuses on young students’ development of number and subitizing activity. Beth collaborates with colleagues examining teachers’ specialized content knowledge development and marginalized students’ number understanding development. Percival  Matthews  is an associate professor of Educational Psychology at the University of Wisconsin–Madison. He earned is PhD in Psychology from Vanderbilt University and completed his postdoctoral research at the University of Notre Dame. The bulk of his research is organized around two primary goals: (1) understanding the basic underpinnings of human mathematical cognition and (2) finding ways to leverage this understanding to create effective pedagogical techniques that can be used to impact the life chances of everyday students. Kevin  C.  Moore  Born and raised in Ohio, I attended The University of Akron from 2001 to 2006. I worked as a Graduate Assistant in the Department of Mathematics and grew curious about my students’ mathematical thinking when teaching as part of the assistantship duties. This curiosity landed me at Arizona State University under the guidance of Professor Marilyn P. Carlson. I immediately grew interested in the constructivist movement in mathematics education, and specifically the ability to take a scientific-inquiry approach to modeling students’ mathematical thinking. Since this initial interest, I have rooted myself with other researchers who participate in this progressive research program in the hopes of better understanding students’ mathematical thinking, improving the teaching and learning of mathematics, and opposing outcome-based forces in education. Anderson Norton  is a professor of Mathematics Education in the Department of Mathematics at Virginia Tech. His research focuses on building psychological models of students’ mathematical development, particularly in the domain of fractions knowledge, and the epistemology of mathematics. He has served as chair for the editorial panel of the Journal for Research in Mathematics Education, chair of the steering committee for the North American Chapter of the International Group for the Psychology of Mathematics Education, and lead editor for the Springer book, Constructing Number. In 2013, in recognition of his outreach efforts, he received the Early Career Award from the Association of Mathematics Teacher Educators.

About the Authors

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Alan  O’Bryan  has worked for almost 20 years as a mathematics education researcher, curriculum developer, professional development leader, and teacher trainer. His research focuses on examining student learning of mathematical ideas and teacher change through the lens of cognitive psychology. He is the co-author of six secondary and post-secondary mathematics textbooks and has delivered nearly 100 professional development workshops across the United States for teachers of all levels from elementary to post-secondary instructors. His presentations and workshops are characterized by the way they make mathematics education research findings accessible and practical for classroom teachers. Teo Paoletti  is an associate professor specializing in Mathematics Education in the School of Education at the University of Delaware. The primary goal of his research agenda is to explore student understanding of mathematical ideas at various levels (e.g., middle school through post-­secondary). He leverages design-based methods to explore ways in which students can leverage reasoning about relationships between quantities to construct and reason about critical mathematics concepts. His recent work entails designing task sequences that leverage various dynamic mathematical software (e.g., GeoGebra, Desmos, GSP) to support middleschool students developing meanings for various function classes, graphs, and inequalities. Abby  Rocha  completed her PhD in Mathematics Education at Arizona State University, where she was recognized with multiple awards for her research and teaching excellence. Abby’s research investigates relationships between teachers’ mathematical meanings for teaching and their instructional practices, including their actions to decenter. Currently, Abby is a postdoctoral fellow in the STEM Learning Center at the University of Arizona. During her fellowship, Abby’s research will focus on supporting STEM faculty in engaging and integrating culturally responsive pedagogical and curricular practices through professional development for faculty teaching gateway lecture and lab courses. Serife Sevinc (Şerife Sevinç; in native language) is an associate professor at the Department of Mathematics and Science Education at Middle East Technical University, Turkiye. She received her doctoral degree in Curriculum and Instruction with Mathematics Education specialization at Indiana University and a minor PhD degree in Inquiry Methodology. Her main research interests involve investigating students’ and preservice teachers’ mathematical thinking in problem-solving and problem-posing processes, mathematical modeling process that enhances students’ and teachers’ conceptual understanding in mathematics, and the nature of pre-service teachers’ knowledge in designing realistic mathematics problems. Leslie  P.  Steffe  is a research professor emeritus of Mathematics Education, University of Georgia. He received his Ph.D. from the University of Wisconsin in 1966 and joined the Department of Mathematics Education at the University of Georgia in 1967 from which he retired in 2017. He developed the research program

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Interdisciplinary Research on Number (IRON) with the epistemologist, Ernst von Glasersfeld, and the philosopher, John Richards, starting circa 1977 and lasting until circa 2000. From that point on, he formed the research program Ontogenetic Analysis of Algebraic Knowing (OAK) which lasted until he retired in 2017. Irma E. Stevens  is an assistant professor in the Department of Mathematics and Applied Mathematical Sciences at the University of Rhode Island. She earned her PhD in Mathematics Education and MA in Mathematics at the University of Georgia and her BS in Mathematics at the University of North Carolina at Charlotte. Her research interests involve teacher decision making and student learning at the secondary and postsecondary level. Her current projects include understanding undergraduate students’ reasoning with dynamic quantities and formulas. Her work uses technology that has resulted in the construction of materials used for research, professional development, and classroom instruction. Michael  A.  Tallman  received his BSc and MA in Mathematics from the University of Northern Colorado and his PhD in Mathematics Education from Arizona State University. Dr. Tallman has taught mathematics at the secondary and post-secondary levels. His primary research is in the area of mathematical knowledge for teaching secondary and post-secondary mathematics. Dr. Tallman’s work informs the design of teacher preparation programs and professional development initiatives through an investigation of the factors that affect the nature and quality of the mathematical knowledge teachers leverage during instruction. In particular, his research examines how various factors like curricula, emotional regulation, identity, and teachers’ construction and appraisal of instructional constraints mediate the enactment of their subject matter knowledge. Halil I. Tasova  is an assistant professor of Mathematics Education at California State University, San Bernardino. He earned his PhD in Mathematics Education from the University of Georgia and holds an MS and BS in Secondary Mathematics Education from Marmara University in Türkiye. With prior experience as a high school math teacher, Dr. Tasova brings valuable classroom insights to his current role. His research focuses on students’ mathematical thinking and learning, particularly their reasoning about quantities. He explores the construction and interpretation of graphs from the perspective of quantitative and covariational reasoning. Patrick  W.  Thompson  is professor emeritus of Mathematics Education in the School of Mathematical and Statistical Sciences at Arizona State University. He spent his career researching the learning and teaching of mathematics at elementary school, high school, and university levels through a lens of quantitative reasoning. He has contributed to the understandings of learning and teaching of arithmetic, algebra, statistics, and calculus, and to areas of research methodology in mathematics education.

About the Authors

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Erik S. Tillema  is a former middle and high school mathematics teacher and current associate professor of Mathematics Education at Indiana University. His work focuses on making second-­order models of middle grades and secondary students’ reasoning with the aim of having such models be the basis for curricular design and instruction. He is particularly interested in second-­order models of students’ combinatorial and quantitative reasoning and how generalizations of this reasoning can support students’ algebraic reasoning. His recent work involves collaborating with secondary teachers in the endeavor of using combinatorics problems to support their students to generalize algebraic structure. Alexandria  Viegut  is an assistant professor of Psychology at University of Wisconsin-Eau Claire. She earned a PhD in Educational Psychology from UW– Madison. Alex’s current research investigates students’ understanding of fractions, the connections between fractions knowledge and algebra knowledge, and the role of visual representations in mathematical cognition.

Part I

Introduction to Piaget’s Genetic Epistemology and the Tradition of Use Featured in This Book

Chapter 1

Introduction to Piaget’s Genetic Epistemology Paul Christian Dawkins, Amy J. Hackenberg, and Andy Norton

Introduction Piaget is known for his work in developmental psychology, but he began his career as a biologist whose primary interests evolved into epistemology; that is, theories of knowledge and knowing. While studying snails, he was introduced to Bergson’s (1998) idea of creative evolution, in response to which he later said, The problem of knowing (properly called the epistemological problem) suddenly appeared to me in an entirely new perspective and as an absorbing topic of study. It made me decide to consecrate my life to the biological explanation of knowledge. (Vidal, 1994, p. 52)

Piaget began to study children’s psychological development as a means of investigating the biological origins of logic and mathematics. In this pursuit, he followed the biogenetic law that “ontology recapitulates phylogeny”: the development of the individual follows a similar trajectory as the development of humankind.

P. C. Dawkins (*) Department of Mathematics, Texas State University, San Marcos, TX, USA e-mail: [email protected] A. J. Hackenberg Department of Curriculum & Instruction, Indiana University, Bloomington, IN, USA e-mail: [email protected] A. Norton Department of Mathematics, Virginia Tech, Blacksburg, VA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 P. C. Dawkins et al. (eds.), Piaget’s Genetic Epistemology for Mathematics Education Research, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-47386-9_1

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We can sum up the constructivist epistemology1 with the mantra, “All knowledge is constructed” (Glasersfeld, 1995, p. 160). Here, we focus on Piaget’s genetic epistemology, which implies more than its constructivist tenets. We frame Piaget’s genetic epistemology in terms of his lifelong pursuit to understand the power and origins of logico-mathematical operations. It directs us back to the biological origins of our logico-mathematical operations and to the structures that organize them so that they, in turn, might organize the world. To say “all knowledge is constructed” is to challenge the notion that knowledge comes in from outside a person in some simple or reliable way. Certainly, humans build knowledge based on their experiences in their environments, but those experiences are unique to each individual. In the same way, the physical form that an organism takes is largely driven by its internal structure (e.g., DNA), humans’ construction of knowledge operates through internal structures in a complex interplay with (their sensorimotor experience of) their environment. This is primarily an epistemological claim about the construction of knowledge, not an ontological claim about the extent to which the knowledge so constructed is, in some sense, an “accurate” depiction of the world. A key feature of this epistemology is that it can explain learning without a strong notion of correspondence to reality. Modern biology shows that different kinds of animals experience the world very differently from humans. By framing human psychology as fundamentally biological, we gain at least two important perspectives. First, we learn to respect how much our experience of the world is structured by our bodies and by our cognitive processes. Second, we may consider how much those experiences have changed over the course of our lifetimes. Children at different stages of development may experience different worlds and reason in markedly different ways from us as adults. We may question how much of our current experience depends intrinsically on earlier constructions that we cannot recall ever having done without. In his genetic epistemology, Piaget adopted a Kantian perspective, but empirical results from his research with children challenged core assumptions in Kant’s philosophy. Kant’s (1781) Critique of Pure Reason blended empiricism with rationalism by accepting a few principal cognitive structures as innate. These innate structures included space, time, and number, which enable us to organize our experiences in the world. They also explain how we might take experiences as shared. If we all construct reality within the same God-given space–time framework, we can expect some commonalities among those realities. However, Piaget demonstrated that children construct these foundational structures too, during their first few years of life. In other words, Piaget’s research countermanded Kant’s assumptions. Few, if any of us, can recall the years we spent playfully organizing our worlds, so we take those constructions (e.g., space, time) for granted.

  We have found Noddings (1990) explanation of constructivist epistemology, especially illuminating. 1

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Consider Kant’s (1781) famous statement: “The concept of Euclidean space is by no means of empirical origin, but is the inevitable necessity of thought.” From our perspective now, this statement is an error. The mere possibility of non-Euclidean geometries, such as the one Gauss invented during the nineteenth century, refutes Kant’s claim. Another century later, Piaget and Inhelder (1967) demonstrated that children construct space during their first years of life, on the basis of their own sensorimotor activity in the world. Thereafter, the objects children experience have a home to persist in even when they are out of sight (i.e., object permanence). Piaget demonstrated that children construct number, too. Steffe has elaborated on this construction through learning levels that he referred to as children’s number sequences. At every stage, development depends upon the coordination of actions— first sensorimotor, then internalized as mental actions, and finally organized within structures that render them logico-mathematical operations. These structures, both spatial and numerical, serve the role Kant envisioned for them, but they are the result of years of labor. Once we have constructed them, it becomes difficult to imagine a world without them. What then allows us to build up concepts like space, time, and number if we do not, as Kant claimed, begin with certain structures already in place? The heart of Piaget’s answer to this is the organization of our own activity. Children act in their experiential worlds, and their organization of those actions provides the basis for the organization of their experiential worlds. Piaget’s genetic epistemology emphasizes the unique status of logico-­ mathematical operations within human knowledge. It also affords a different account of mathematical objects themselves. If knowledge corresponds to reality in a strong sense, then it raises questions about the nature and source of abstract concepts such as number, line, function, and set. The philosophical stance known as Platonism classically solves this problem by asserting the real existence of abstract entities (an ontological claim). This allows us to somehow learn abstract concepts under the assumption that they come in from the outside world (an epistemological claim). Since constructivism provides an alternative account of how concepts form, it provides a resolution of this epistemological issue that can remain ontologically neutral. It thus provides an alternative to Platonism in explaining the power of logic and mathematics. This power owes to the structures that we construct through the coordination of our own mental actions rather than structures imposed upon us by the worlds we ourselves organize through those very same structures. In all, the chief apparent advantage of Platonism, which is to account for the objective robustness of logico-mathematical entities and structures, is guaranteed in the same way by the concept of the general and internal co-ordinations of actions and operations. That hypothesis that ideal entities are external is thus unnecessary to guarantee the independence of structures in view of the free will of individual subjects. (Beth & Piaget, 1966, p. 294)

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Why Is Piaget’s Genetic Epistemology Useful? For Glasersfeld (1995), a way of knowing is valuable to the extent it is useful. Another way to say this is that people construct ways of knowing to serve purposes. We apply this orientation repeatedly in this book to articulate why Piaget’s genetic epistemology—and the research tradition in Mathematics Education that has been built from it—is useful. One reason it is useful is that it acknowledges that people build up knowledge to organize their experiential worlds and pursue goals within those worlds, not to describe an observer-independent world. As Glasersfeld (1995) pointed out, Piaget was not the first to take this position on knowledge, but he was the first to take a developmental approach (p. 13). As we have introduced above, Piaget viewed the construction of knowing in an individual (1) to be a process of construction over a lifetime and (2) to reflect the construction of knowing in humans as a species. The first point means that no person’s ways of knowing are ever complete—they are always evolving. The second point means that understanding the nature of knowing requires studying its ontogenesis—its development in humans across their lives. In our experience, as researchers and teachers in mathematics education, Piaget’s views on knowing provide the basis for generating rich tools for describing and accounting for students’ mathematics. Piaget’s views also enlarge what is considered mathematical—and, therefore, who is considered a mathematical thinker. Piaget’s views on knowing imply that a great variety of ways of knowing and thinking can be admitted into mathematical knowledge, including children’s mathematics (Steffe & Olive, 2010). So, researchers can co-construct with participants, including children, ways of thinking that can be understood as mathematical beyond what has traditionally been viewed as mathematical. These ways of thinking are models of participants’ mathematical knowledge that researchers can use to support future interactions with other participants (see Chaps. 9 and 14). As a consequence, students whose ways of thinking differ from what has traditionally been considered standard mathematical ideas can be legitimated, and these students can be seen more fully as mathematical thinkers (e.g., Hackenberg, 2013; Hackenberg & Sevinc, 2021; Norton & Boyce, 2015). Piaget is known for developmental stages (e.g., concrete operations, formal operations) that have been critiqued as being rigid and nonrepresentative of all people. We would like to address that directly using children’s number sequences (Steffe et al., 1983; Steffe & Cobb, 1988) as an example. In this research, Steffe and colleagues studied how young children construct whole numbers by studying how they count and how the nature of counting changes with successive constructions. They found that children constructed approximately four number sequences, and these occur in order because later number sequences involve more complex organizations of units. Such descriptions can help teachers and researchers organize instructional interactions with a range of elementary school students. Yet, the descriptions of number sequences of children do not follow a lockstep set of stages at the same ages—that is not what the developmental aspect of genetic epistemology means. So,

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the developmental aspect of genetic epistemology means that children tend to construct number sequences in a certain order, but there is great variation in how students at a particular age conceive of number. And yet, the usefulness of genetic epistemology does not stop with description. The use and development of other Piagetian tools, such as accommodation and reflective abstraction (discussed in Chaps. 3 and 6), provide the means for explaining students’ mathematical thinking and learning.2 Together, descriptions of mathematical thinking with explanations of how learning proceeded—or did not proceed—are key components of making models of students’ mathematical thinking and learning (Steffe & Thompson, 2000; Ulrich et al., 2014). Consistent with Piaget’s overall insight that knowledge need not be explained in terms of correspondence with reality, researchers using genetic epistemology recognize research as their process of knowledge construction. As a result, we must be careful to distinguish what we try to learn about student knowledge (our models of their knowing) from student knowledge itself. We cannot know whether these models, co-constructed with students, are what we would experience if we somehow became these students or otherwise fully adopted their ways of knowing. Students’ ways of knowing are not directly accessible to us. Instead, the models are our ways of knowing that fit with our interactions with the students—they are what Steffe refers to as second-order knowledge (2010), or the mathematics of students.3 Such models take extensive work for researchers to construct and refine (see Chap. 14). Robustly developed models can be regarded as legitimate mathematical ways of knowing—and thus, what gets considered to be mathematics gets expanded. For example, when working to build fractions knowledge, students who are trying to draw 3/5 of a bar can learn to partition the bar into five equal parts, take out one part, and repeat the part to make three parts (Fig. 1.1). In other words, they can create 3/5 of a bar as 1/5 of the bar, another 1/5, and another 1/5. To observers, it might look like the student thinks of 3/5 as 3 times 1/5. However, that may not be the case. Third- through fifth-grade students taught Steffe and Olive (2010) that they may not think of 3/5 in this way. Rather, they may rely on part–whole meanings for the result, thinking of 3/5 as three parts out of five, despite the actions they took to make the 3/5. Because this way of thinking was a regularity in how students operated in a longitudinal teaching experiment (Steffe & Olive, 2010), Steffe and Olive formulated a scheme (see Chaps. 2 and 3 of this volume) to describe these students’ way of thinking about fractions, the partitive fraction scheme (Steffe & Olive, 2010). This way of thinking about fractions is challenging to understand for those who conceive of fractions as multiples of unit fractions; it is hard to see that the students’ meaning could be non-multiplicative when the actions look like what a person would do who thinks of 3/5 as 3 times 1/5. Indeed, a person with multiplicative meanings could engage in the same physical behavior for 3/5. However,  We will use the term “students” rather than “children,” since not all research has been with young children. 3  First-order mathematical knowledge is the mathematical ways of knowing we have built to organize our own experiential worlds. 2

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Fig. 1.1  3/5 as three of 1/5

differences quickly arise for students when fractions exceed the whole (e.g., Hackenberg, 2007; Norton & Wilkins, 2012; Steffe & Olive, 2010). Students who have constructed a partitive fraction scheme find drawing, for example, 7/5 of a bar, very mysterious. How can a person draw 7 parts out of 5? This example shows both aspects of the usefulness of Piaget’s views: The partitive fraction is an example of the use of scheme as a powerful tool for modeling student thinking, and this scheme is an expansion of ways of thinking with fractions that can be considered legitimately mathematical. Second-order models of particular students can be very satisfying to make: When they are developed, they represent to the researcher an understanding of the ways of thinking of the students, and they can show why it makes sense that a student solved a problem or thought about a topic in a particular way. Thus, models can provide a researcher with a great sense of fit. And yet, the models are actually instruments of interaction (Steffe & Olive, 2010): They allow researchers to better interact with these particular students because the researchers can base problems and questions on the ways of thinking in the model. Doing so can facilitate communication about mathematical ideas with particular students. This aspect of models can also feel satisfying because it can engender a sense of connection between the researcher and students (see Chaps. 9 and 14). And yet, if the models were only useful for the particular students with whom researchers were working in particular studies, that would be quite limiting as research. Fortunately, experience shows that is not at all the case. Another reason genetic epistemology is useful is that the models developed with a few students usually allow researchers to interact more broadly with other students who have similarities to the students from whom the models were made (see Chap. 14). So, as researchers build models for particular students, they are usually building models that are useful with a wide range of students.

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Organization of the Book We mention these models of students’ mathematics because they portray how Piaget’s genetic epistemology has contributed to mathematics education research. More importantly, Piaget provided a rich set of theoretical tools for pursuing this kind of research. The goal of this book is not to describe particular models of students’ mathematics that have been developed but rather to describe the tools that mathematics educators use to construct such models. We have thus organized the main body of the book—Part 2, Chaps. 3, 4, 5, 6, 7, 8, 9, 10, and 11—around clusters of related constructs. To be precise, the first seven of those chapters describe constructs directly descended from Piaget’s research, and the last two describe constructs developed later on but whose importance to mathematics education research warranted their inclusion in this volume. The rest of the book—Parts 3 and 4, Chaps. 12, 13, 14, 15, 16, 17, 18, 19, 20, and 21—contains two parts corresponding to two different ways of building on the construct chapters in Part 2. The chapters in Part 3 each contain commentaries on the first part and on genetic epistemology more broadly. The chapters in Part 4 each summarize the research agenda of a younger mathematics education scholar who draws upon genetic epistemology in their work. These final chapters provide further examples of the utility and fecundity of this body of theory. We could have adopted other organizational approaches such as a historical account of how ideas developed, by focusing on the various scholars who drew upon Piaget in their research, or by surveying key findings and contributions developed in this tradition. We adopted the current organization because we anticipated it would be most useful to scholars who want to learn about these tools to engage in mathematics education research. In other words, we organized the book looking forward to future research rather than trying to survey or summarize previous research. As a result, the contributions of many important mathematics educators who draw heavily upon Piaget’s work may be underrepresented or omitted in these pages. We have included Chap. 2 as an acknowledgment of the history of research and the intellectual heritage by which this body of theory has come to us. Dr. Les Steffe is one of the central scholars who draws upon Piaget’s work to study children’s mathematics and who trained many of the other authors to do the same. Chapter 2 presents Steffe’s historical reflection on Piaget’s influence on mathematics education. This book was formulated to serve as a graduate textbook for those studying to become researchers in mathematics education. We sense that our field needs more such texts, especially regarding rich and complex bodies of theory such as genetic epistemology. We sincerely hope that this book provides a helpful starting point for those newer to these ideas and a productive resource for those more experienced. We have learned much from the chapters our excellent coauthors contributed, which makes us confident that what follows will be of value to the field. It is a joy to be continually engaged as learners: learners of mathematics, especially students’ mathematics, and learners among the community of researchers trying to support quality mathematics education.

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References Bergson, H. (1998). Creative evolution (A.  Mitchell, Trans.). Dover. (Original work published in 1911). Beth, E. W., & Piaget, J. (1966). Mathematical epistemology and psychology. D. Reidel Publishing Company. Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior, 26(1), 27–47. https://doi. org/10.1016/j.jmathb.2007.03.002 Hackenberg, A. J. (2013). The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior, 32(3), 538–563. Hackenberg, A.  J., & Sevinc, S. (2021). A boundary of the second multiplicative concept: The case of Milo. Educational Studies in Mathematics., 109, 177. https://doi.org/10.1007/ s10649-­021-­10083-­8 Kant, I. (1781). Critique of pure reason (N.K. Smith, Trans.). Macmillan. Noddings, N. (1990). Constructivism in mathematics education. Journal for Research in Mathematics Education (Monograph): Constructivist Views on the Teaching and Learning of Mathematics, 4, 7–18. Norton, A., & Boyce, S. (2015). Provoking the construction of a structure for coordinating n+ 1 levels of units. The Journal of Mathematical Behavior, 40, 211–232. Norton, A., & Wilkins, J. L. M. (2012). The splitting group. Journal for Research in Mathematics Education, 43(5), 557–583. https://doi.org/10.5951/jresematheduc.43.5.0557 Piaget, J., & Inhelder, B. (1967). The child’s conception of space (F.J. Langdon & J.L. Lunzer, Trans.). Norton.. (Original work published in 1948). Steffe, L.  P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies. Springer-Verlag. Steffe, L.  P., & Olive, J. (2010). Children’s fractional knowledge. Springer Science & Business Media. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Erlbaum. Steffe, L.  P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children’s counting types: Philosophy, theory, and application. Praeger Scientific. Ulrich, C., Tillema, E. S., Hackenberg, A. J., & Norton, A. (2014). Constructivist model building: Empirical examples from mathematics education. Constructivist Foundations, 9(3), 328–359. Vidal, F. (1994). Piaget before Piaget. Harvard University Press. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Falmer.

Chapter 2

An Historical Reflection on Adapting Piaget’s Work for Ongoing Mathematics Education Research Leslie P. Steffe

Piaget and Modern Mathematics Piaget was “rediscovered” (Ripple & Rockcastle, 1964) during the 1960s by mathematicians and mathematics educators whose goal was to reform mathematics curricula based on modern mathematics (e.g., Allendoerfer & Oakley, 1959; School Mathematics Study Group, 1965). Logical–mathematical structure served as the basic rationale for the new math programs that, in many cases, resembled collegiate mathematics. Although classical idealism, the doctrine that reality, or reality as we know it, is fundamentally mental, served operationally as the epistemological position of the reformers, empiricism and realism were still the more general positions in the United States as indicated by a return to behaviorism in the decade following the modernist programs. Problem solving, along with learning by discovery, was the major psychological emphases among the reformers (Pólya, 1945, 1981) for which Wertheimer’s1 (1945) work on productive thinking served as a basic rationale. Piaget’s genetic epistemology (Piaget, 1970) did not serve as an epistemological basis for the modern programs, nor was it explicitly emphasized at a conference held at Cornell University and the University of California to investigate implications of Piaget’s work for mathematics education (Ripple & Rockcastle, 1964). The interest of the conference organizers was in exploring the implications of Piaget’s  Wertheimer was one of the three founders of Gestalt psychology along with Kurt Koffka and Wolfgang Köhler. 1

L. P. Steffe (*) Department of Mathematics, Statistics, and Social Studies Education, University of Georgia, Athens, GA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 P. C. Dawkins et al. (eds.), Piaget’s Genetic Epistemology for Mathematics Education Research, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-47386-9_2

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stages of cognitive development as a rationale for the elementary programs because those programs were left without a psychological rationale (Ripple & Rockcastle, 1964). Piaget (1964) was invited to present four papers at the conference that he titled, “Development and learning,” “The development of mental imagery,” “Mother structures and the notion of number,” and “Relations between the notions of time and speed in children.” Although he made no reference to genetic epistemology in these papers, by presenting them, he did implicitly explain the concept of genetic epistemology that he presented at the Woodbridge Lectures at Columbia University in 1968 (Piaget, 1970). Genetic epistemology attempts to explain knowledge, and in particular scientific knowledge, on the basis of its history, its sociogenesis, and especially the psychological origins of the notions and operations upon which it is based. (p. 1)

Development vs. Learning Although he could have oriented his papers as elaborations of genetic epistemology, his emphasis at the conferences was on explaining the cognitive development of number, space, and time as opposed to teaching such concepts and expecting them to be learned. He made a sharp distinction between development and learning in that development is a spontaneous process tied to the whole process of embryogenesis. Embryogenesis concerns the development of the body but it concerns as well the development of the mental functions. In the case of the development of knowledge in children, embryogenesis ends only in adulthood. … In other words, development is a process— which concerns the totality of the structures of knowledge. (Piaget, 1964, p. 8)

In contrast, he explained learning as presenting the opposite case. In general, learning is provoked by situations. … It is provoked in general as opposed to spontaneous. In addition, it is a limited process—limited to a single problem or to a single structure. (Piaget, 1964, p. 8)

Although I do not regard learning as such a limited process, the developers of the modern programs firmly believed that their programs could be learned and would either accelerate Piaget’s account of the cognitive development of basic mathematical notions or essentially replace developmental processes. Most, if not all, of the major curricular reform projects have taken the logic (or structures) of the subject matter as a point of departure rather than psychological learning theory or studies of cognitive development. This point is made abundantly clear, for example, in Jeremy Kilpatrick’s paper on the SMSG Program included in this report. (Ripple & Rockcastle, 1964, p. iii)

The curriculum developers considered Piaget to be an observer rather than a teacher, and the elasticity of the limits of children’s minds was not considered as having been established.

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These reformers (and I speak now not only of SMSG) have been so successful in teaching relatively complex ideas to young children, and thus doing considerable violence to some old notions of readiness, that they have become highly optimistic about what mathematics can and should be taught in the early grades. (Kilpatrick, 1964, p. 129)

The lack of appreciation for genetic epistemology was addressed by Eleanor Duckworth, a former student of Piaget, who served as an intermediary between Piaget and the conference attendees. She addressed the teaching the “structure” of a subject matter. The pedagogical idea is that children should be taught the unifying themes of a subject matter area, after which they will be able to relate individual items to this general structure. (This seems to be what Bruner often means by ‘teaching the structure’ in the Process of Education). (Duckworth, 1964, p. 3)

Developmental vs. Mathematical Structure The structural emphasis of the modern programs was not compatible with Piaget’s emphasis on the structure of operational thought. An operation is an interiorized action. … Above all, an operation is never isolated. It is always linked to other operations and as a result it is always a part of a total structure. (Piaget, 1964. p. 7)

A major difficulty was that “structure” had very different meanings for Piaget and for the curriculum developers. Piaget’s structures were second-order models, “the hypothetical models observers may construct of the subject’s knowledge in order to explain their observations (i.e., their experience) of the subject’s states and activities” (Steffe et al., 1983, p. xvi). The mathematical structures of the modern programs were first-order models, or renditions of the mathematical knowledge of the curricular developers (cf. Steffe et al., 1983). This distinction between the mathematical thought of the child from the point of view of the adult and the adult’s own mathematical knowledge that he or she would not attribute to the child has been and remains a major issue in the mathematics education of children. Even though it is assumed in genetic epistemology that the mathematical thinking of children as it evolves over time has something to do with mature mathematical thinking, it takes major decentering for an adult mathematical thinker to think as if he or she is a child (Thompson & Thompson, 1994). Further, in those cases where the adult does learn to think as if he or she is a child, developing models of how such an evolution might occur is quite exacting. Hermine Sinclair succinctly pointed out such difficulties at the level of child thought in attempts by mathematics educators to use Piaget’s genetic epistemology at a conference held at Columbia University in 1970. At first sight it would seem that a psychological theory that is regarded by its author as a “by-product” of his epistemological research and is therefore principally directed toward the investigation of knowledge and its changes in the history of mankind, as well as in the growing child, is ideally suited to educational applications. (Sinclair, 1971, p. 1)

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Sinclair used a metaphor to explain what she regarded as difficulties in trying to apply Piaget’s stage theory of cognitive development in an attempt to provoke nonoperational children to become operational. Piaget’s tasks are like the core samples a geologist takes from a fertile area and from which he can infer the general structure of a fertile soil; but it is absurd to hope that transplanting these samples to a field of nonfertile soil will make the whole area fertile. (Sinclair, 1971, p. 1)

Preludes to IRON (Interdisciplinary Research on Number) Piagetian Research Professor Henry Van Engen introduced me to Piaget’s work while I was a doctoral student at the University of Wisconsin, working as a research associate in the Research and Development Center for Cognitive Learning. Following Bridgeman (1927), mental operations were at the heart of Van Engen’s meaning theory of arithmetic, where he defined the meaning of a symbol as an intention to act (Van Engen, 1949a, b). Piaget’s emphasis on mental operations and operational thought was a major point of convergence and served as the basis of Van Engen’s interest in Piaget (Van Engen, 1971). As a research associate, it was my wont to apply Piaget’s theory of the development of number in the mathematics education of young children using scientific methods, which in part translated to investigating the importance of conservation of numerosity of first-grade children on arithmetical tasks using research design (Stanley & Campbell, 1963) and statistical methods (Steffe, 1966). I continued on with this program of research, which became known as “Piagetian Research” (Steffe & Kieren, 1994), for 7 years after joining the faculty of mathematics education at the University of Georgia, a time during which I directed ten doctoral students in applying Piaget’s research in the mathematics education of children. Professor Charles Smock of the Psychology Department, who was a Piagetian, served on the committees of most of my doctoral students during that time, which was quite important because he was the one who eventually introduced me to Ernst von Glasersfeld.

A Change in Research Program I was working as a realist and an empiricist in my attempts to apply Piaget’s developmental theory in the mathematics education of children, and I was making only accretional rather than recursive progress. As a consequence of making only minimal progress, I abandoned my attempts to apply Piaget’s research on number and quantity as well as my statistical method of application and taught a group of first graders with the help of two of my advanced doctoral students for an academic year

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in an attempt to let children teach me what was important in their numerical ways of operating (Steffe et  al., 1976a, b). The importance of this work was two-fold. First, it was the impetus for the development and use of the constructivist teaching experiment as a legitimate scientific method of doing research (Steffe, 1983, 1991b; Steffe & Cobb, 1983; Steffe & Thompson, 2000; Steffe & Ulrich, 2013). Second, the finding that counting was the children’s basic method of solving arithmetical tasks led to abandoning attempts to apply Piaget’s idea that number is constructed as a synthesis of classification and seriation operations concurrently with the introduction of the arithmetical unit and to investigating children’s construction of number sequences in the context of their spontaneous use of counting in solution of arithmetical tasks in teaching experiments (Steffe, 1994, 1996; Steffe et al., 1983; Steffe & Cobb, 1988).

A Fortunate Introduction It was extremely fortunate, not only to me at the time but also to my doctoral students and to the field of mathematics education at large, that Professor Smock invited me to a seminar given by Ernst von Glasersfeld (Steffe, 2013). The seminar event arranged by Smock occurred around 1974, shortly after the demise of the modern mathematics movement and during the move back to behaviorism that occurred in the 1970s. At the time, the question concerning whether mathematics was invented or discovered held little sway with me even though I had read Piagetian basic books such as The Child’s Conception of Number (Piaget & Szeminska, 1952), The Child’s Conception of Geometry (Piaget et  al., 1960), The Child’s Conception of Space (Piaget & Inhelder, 1967), and The Growth of Logical Thinking from Childhood to Adolescence (Inhelder & Piaget, 1958). I understood that children developed mathematical knowledge, but what developed I regarded as a prelude to what was “out there” in some mathematical reality. My conception of mathematics was, and still is, widely shared by mathematics educators as well as mathematicians. According to Stolzenberg (1984), it is indisputable that the contemporary mathematician operates within a belief system whose core belief is that mathematics is discovered rather than created or invented by human beings. My belief in the objective existence of mathematics was seriously questioned by a story Glasersfeld recounted in the seminar. The story, taken from Letvin et  al. (1959), clarified that the only contact we have with what is “out there” is through our senses. When talking about a frog as a fly-catcher, he commented that: The system [the frog’s visual system] as a whole makes the frog an efficient fly-catcher, because it is tuned for small dark “objects” that move about in an abrupt fly-like way. In the frog’s natural habitat, as we, who observe the frog see it, every item that possesses the characteristics necessary to trigger the frog’s detectors in the proper sequence is a fly or bug or other morsel of food for the frog. But if the frog is presented with a black bead, an air-gun pellet, or any other small dark moving item, it will snap it up as though it were a fly. In fact,

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L. P. Steffe to the normal frog’s visual apparatus, anything that triggers the detectors in the right way, is a “fly.” (Glasersfeld, 1974, p. 15; reprinted in Glasersfeld, 1987, pp. 106–07)

The point of the story, of course, is that whatever is perceived is basically a composition of sensory signals generated in our various sensory channels. We are, of course, free to consider these original signals the effect of some outside causes. But since there is no way of approaching or “observing” these hypothetical causes, except through their effects, we are in the same relation to that “outside” in which the first cyberneticists found themselves with regard to living organisms—that is to say, we are facing a “black box.” (Glasersfeld, 1974, p. 16; reprinted in Glasersfeld, 1987, p. 107)

Soon after the colloquium, Ernst presented his seminal paper on radical constructivism at a conference sponsored by Charles Smock’s Mathemagenic Activities Program (von Glasersfeld, 1974). In the paper, Glasersfeld argued that radical constructivism constituted a legitimate interpretation of Piaget’s genetic epistemology. The radical constructivist’s interpretation of Piaget’s Genetic Epistemology then, consists in this: The organism’s representation of his environment, his knowledge of the world, is under all circumstances the result of his own cognitive activity. The raw material of his construction is “sense data,” but by this the constructivist intends “particles of experience”: that is to say, items which do not entail any specific “interaction” or causation on the part of an already constructed reality that lies beyond the organism’s experiential interface. (p. 22)

Ernst once told me that he had discussed his interpretation of Piaget’s genetic epistemology with Barbel Inhelder, Piaget’s close collaborator, and she agreed with it.

Interdisciplinary Research on Number (IRON) Original Members of IRON After becoming acquainted with Glasersfeld, along with the philosopher John Richards and Patrick Thompson (who was then a doctoral student), we organized the radical constructivist research program, Interdisciplinary Research on Number (IRON). Although our general goal was to start a constructivist revolution in mathematics education of no less magnitude than the modern mathematics movement, we each had our own goals. Glasersfeld (1995, p.  15) later commented that his motivation for joining IRON was to develop the empirical data to countermand empiricism and, further, to engage in work in a field where he would be taken seriously as an epistemologist and philosopher. His particular goal was to develop his attentional moments model (cf. Glasersfeld, 1981) and use it in an analysis of how children construct numerical units in counting. Richard’s goal was to reconstitute the foundations of mathematics based on how mathematics is constructed by human beings. I had a videotape library of teaching episodes with first-grade children in a year-long teaching experiment that I had conducted after my first encounter teaching first-grade children (Steffe et al., 1976a, b). My initial purpose for engaging in

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IRON was to construct explanations of the unit items children established and counted in the previous teaching experiment.2

Ernst as Scientist During the time we engaged in IRON, I knew Glasersfeld best as a coinvestigator, as a scientist conducing conceptual analysis of children shown on the video-tapes when they engaged in solving numerical situations (Steffe, 2011). Of these analyses, he commented that, He [Les Steffe], a graduate student of his [Pat Thompson], the philosopher John Richards, and myself would spend countless hours viewing these tapes and trying to agree on what we gathered from them. We had heated arguments and for all of us it was a powerful lesson, hammering in the fundamental fact that what one observer sees is not what another may see and that a common view can be achieved only by a strenuous effort of mutual adaptation. (Glasersfeld, 2005, p. 10)

Ernst’s comment is testimony to how we, as an interdisciplinary team, proceeded in interdisciplinary research in the context of analyzing children’s counting. It was through these intensive social interactions that we learned the language and thinking of the other members of the team, and I carried this lesson on throughout the IRON research program. Analyzing children’s counting may have seemed inconsistent with our identification as “Piagetians” because Piaget and Szeminska (1952) excluded children’s counting in their account of children’s construction of number. At a meeting of the International Group for the Psychology of Mathematics Education at Grenoble, France, in 1981, that question was put to Ernst in no uncertain terms. His answer was incisive, and it at once established the originality of the research we were doing in IRON. Piaget’s account of children’s construction of number should not be regarded as an objective mental reality independent of the observer, a point on which he later elaborated. (Steffe, 2011, p. 173)

By making this comment, Glasersfeld made clear that Piaget’s models of the epistemic child were Piaget’s explanations of his observations. Our models differed from Piaget’s because we used teaching experiments rather than clinical interviews as our method of observation as well as different analytical constructs.

 For example, I thought that when children put up a finger synchronous with uttering a number word in an act of counting, putting up a finger was necessarily an arithmetical unit item. 2

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Piaget’s Epistemic Child There was also widespread interest in Piaget’s account of the spontaneous development of proportional reasoning during the era of modern mathematics and on into the next decade (e.g., Lovell, 1971; Steffe & Parr, 1968). He used the operations that he called inversion, negation, reciprocal, and correlative, which, when working in consort, formed a model of proportional reasoning that he called the INRC group. Piaget thought that proportional reasoning emerged at approximately 12 years of age. Similar to Piaget’s models for children’s construction of number, the INRC group was not of interest in later research of IRON that was done with fractions and algebraic reasoning primarily because we constructed our own models of children’s reasoning in these areas based on teaching experiments. A reader might infer that we abandoned Piaget’s analysis of situations that he used to analyze preoperational, concrete operational, and formal operational thought as contained in the basic books cited above. That is, a reader might infer that we abandoned Piaget’s analysis of his observations. But nothing could be further from the case. Piaget’s grouping structures that he used to model concrete operational thought and his INRC group that he used to model formal operational thought were models of the epistemic child, which Piaget (1966) explained as “that which is common to all subjects at the same level of development, whose cognitive structures derive from the most general mechanisms of the co-ordination of actions” (p. 308). Although we had abandoned Piaget’s accounts of his epistemic child, we had not abandoned his analyses of his copious observations and, at times, used them in our analyses of our observations (cf. Steffe & Olive, 2010). We had mounted what Lakatos (1970) referred to as a progressive research program in which we were interested in exploring children’s construction of mathematics in teaching experiments throughout the elementary, middle school, and early secondary school years. At the outset of IRON, our immediate concern was with young children’s construction of units and number. The explanatory models that were later constructed were simply not available at the outset of our work in IRON.

Conceptual Analysis The First Type Before Glasersfeld came to the United States, he had worked with Silvio Ceccato (1974) in the Italian Operational School (Glasersfeld, 2005) and brought with him the idea of conceptual analysis (Glasersfeld, 1981). Much of our initial effort that led to Glasersfeld’s above comment was devoted to engaging in conceptual analysis using the concept of attention in explaining the units children create and count. In explaining word meaning, the goal of Silvio Ceccato’s Italian Operational School was to “reduce all linguistic meaning, not to other words, but to ‘mental

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operations’” (Glasersfeld, 1995, p.  78). This goal is defined by a question from Ceccato’s group: “What mental operations must be carried out to see the presented situation in the particular way one is seeing it?” (Glasersfeld, 1995, p. 78). We carried out two major types of conceptual analysis (Thompson, 2000). The first one is first-order conceptual analysis, or an analysis of one’s own conceptions, of the kind that Glasersfeld carried out using the basic idea of attentional moments. The firstorder conceptual analysis extends, of course, to analysis of one’s own mathematical conceptions. For example, Thompson (2008, p.  43) used conceptual analysis to “devise ways of understanding an idea that, if students had them, might be propitious for building more powerful ways to deal mathematically with their environments than they would build otherwise.”3 An interesting twist in Thompson’s conceptual analysis is that he often referenced students’ thinking.

Attentional Moments and the Unitizing Operation Glasersfeld explained attention as a mechanism that compounds sensory material from various sensory channels into a “whole” or “thing.” Attention is not to be understood as a state that can be extended over longish periods. Instead, I intend a pulselike succession of moments of attention, each one of which may or may not be “focused” on some neural event in the organism. (Glasersfeld, 1983, p. 126)

Focused moments of attention in between unfocused moments constitute the operation that is used to establish a unitary item—the unitizing operation. According to our analysis, the attentional operations that create unitary items in all areas of experience are essentially the same as those that generate units of units that have been called “number” since Euclid. If that is so, it follows that these “unitizing” operations play a crucial role in all activities that involve numbers. (Glasersfeld & Richards, 1983, p. 4)

Glasersfeld (1981, 1983) used his attentional model to develop a hypothetical, first-­ order model of the operations that produce number starting with the infant’s earliest sensory items. At the end of his presentation, he commented, To conclude, I should like to stress two points that, to me, are the central ideas explicated by this model. First, when we speak of “things,” “wholes,” “units,” and “singulars,” on the one hand and “plurals,” “pluralities,” “units,” “collections,” and “lots,” on the other, we refer to conceptual structures that are dependent on material supplied by sensory experience. Insofar as these concepts involve sensory-motor signals, they do not belong to the realm of number. They enter that rarified realm through the process of reflective abstraction, which extricates attentional patterns from instantiations in sensory-motor experience. (Glasersfeld, 1983, p. 134)

I recall Glasersfeld commenting that extracting attentional patterns from sensory-­ motor items and recording them is the simplest form of reflective abstraction. Glasersfeld’s (1981) attentional model of the conceptual construction of units and  See Thompson and Saldanha (2003) for an example.

3

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number has not itself been widely used in mathematics education research. But what follows from the model, units and their systems, have been widely used. The model doesn’t specify the material of construction, but it does provide a way of thinking about the unitizing operation and its products.

The Second Type The second type of conceptual analysis is found in the first publication of IRON, “Children’s Counting Types” (Steffe et al., 1983), which included our analysis of the video tapes that I had stored in my tape library at the outset of IRON using Glasersfeld’s attentional model (Glasersfeld, 1981). This type of conceptual analysis is a second-order conceptual analysis in that it is an analysis of the conceptions of the other. The goal of this analysis is to produce second-order models of the “black boxes” constituted by children’s minds. To account for our observations, we found it necessary to distinguish counters of perceptual unit items, figurative unit items, and abstract unit items. The importance of making these distinctions among children’s counting types was that they were later interpreted in terms of distinct stages in children’s construction of counting schemes, multiplicative schemes, and fraction schemes (cf. Steffe, 1992; Steffe & Cobb, 1988; Steffe & Olive, 2010).

Schemes After the original publication of IRON (Steffe et al., 1983), I mounted a new 2-year teaching experiment starting in 1980 along with a doctoral student, Paul Cobb, with six first-grade children in order to replicate the results of my original experiment and to generate new empirical content not previously available in the original experiment. Glasersfeld’s (1980) paper, “The Concept of Equilibration in a Constructivist Theory of Knowledge,” was his most important paper for our work in this teaching experiment as well as those that followed. In the paper, Glasersfeld interpreted Piaget’s (1980) concept of scheme, which Piaget defined as “action that is repeatable or generalized through application to new objects” (p. 24).4 In harmony with Piaget, Glasersfeld (1980) explained the concept as an instrument of action or interaction and elaborated it in a way that opened the possibility of focusing on assimilation in the construction or use of a scheme as well as what might go on subsequent to action or interaction. Schemes are basic sequences of events that consist of three parts. An initial part that serves as trigger or occasion.… The second part, that follows upon it, is an action (“response”) or

 Cf. Thompson and Saldanha (2003) for an example

4

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an operation (conceptual or internalized activity).… The third part is… what I call the results or sequel of the activity.

… This feature of the Piagetian model, as I see it, constitutes its main basis as a constructivist theory of cognition in which “knowledge” is no longer a true or false representation of reality but simply the schemes of action and the schemes of operating that are functioning reliably and effectively. (pp. 80, 83)

He illustrated his concept of scheme using the sucking scheme, a scheme that Piaget considered a fundamental scheme babies use in their construction of object concepts. I have no access to the schemes that Glasersfeld might have constructed regarding the second teaching experiment because he did not participate in the conceptual analyses of the video tapes. The schemes that did emerge from the children’s behavior were novel and not known to me prior to the teaching experiment, although I had experienced children’s counting in my first teaching experiment that served in the original publication of IRON. To make a claim that a child has constructed a scheme follows from constructing the scheme for oneself as a result of intensive as well as extensive interaction with one or more children and conceptual analysis of their mathematical behavior. It is a major claim based on original contributions children make to task solutions as well as on repeated observations over a period of time long enough so that one is convinced their behavior is characteristic and stable (Steffe, 1994, 1996; Steffe & Cobb, 1988; Steffe & Olive, 2010).

Systems of Schemes Thompson (2013) critiqued Glasersfeld’s concept of scheme in his work on quantitative reasoning. My understanding of what Piaget meant by “scheme” differs from that proposed by Cobb and Glasersfeld (1983) and by Glasersfeld. (1995, 1998)

… I believe what Cobb and Glasersfeld described fits better with what Piaget called a schema of action (Piaget, 1968, p. 11; Piaget & Inhelder, 1969, p. 4). Piaget spoke of a child’s sucking schema, for example. I believe Piaget had larger organizations in mind when he spoke of schemes—organizations of operations, images, schemata, and schemes—that did not have easily identified entry points that might trigger action. (p. 60)

Glasersfeld did illustrate the concept of scheme as well as assimilation and accommodation using the sucking scheme, as I noted above. Because of his interest in using his attentional model for the conceptual construction of units and number, however, I never felt constrained to using his interpretation of scheme in the case of only his most elementary composite wholes that he called “collection” or “lot” as the first part of a scheme (Glasersfeld, 1981, p. 90). When teaching the children in the teaching experiment, it became evident that the nature of counting varied within

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a particular child over time as well as across the children. As a consequence, five different counting schemes using Glasersfeld’s characterization were isolated and explained: the perceptual counting scheme, the figurative counting scheme, the initial number sequence, the tacitly nested number sequence, and the explicitly nested number sequence (Tzur, 1999). The latter three schemes were operative schemes5 whose assimilative structures were numerical structures. In two later teaching experiments (Steffe, 1992; Steffe & Olive, 2010), the explicitly nested number sequence was reorganized in such a way that the reorganizing children constructed, or were capable of constructing, systems of schemes that I find compatible with Thompson’s conception of scheme. In fact, beyond the most elementary mathematical schemes, it has always been necessary to construct systems of schemes to explain children’s mathematical behavior. But it is not identical because of the kinds of conceptual analyses in which we engaged.

The Initial Number Sequence as a Scheme To illustrate the concept of scheme, I use the initial number sequence for which counting-on is the behavioral indicator (Steffe, 1988a; Steffe & Olive, 2010, pp.  35ff). The first part of the scheme, the structure that is used in assimilation, consists of a pattern of numerical unit items that contain records of counting.6 Consider this question: “If four jacks are placed in a bag holding six jacks, how many jacks are in the bag?” Upon assimilation of the question, as heard by the student, it may constitute a perturbation that I understand as “not just inputs but only such inputs as upsets the organism’s equilibrium” (Glasersfeld, 1980, p. 78). Since the goal is to restore equilibrium, it can be thought of cognitively7 as the student generating a goal to find how many jacks are in the bag in order to restore equilibrium. The goal activates the activity of the scheme, which in this case is to count four more times beyond six. As I have already pointed out, the activity of the scheme is “contained in” the first part of the scheme, as the scheme is an operative numerical scheme.8 The counting activity feeds back into the goal of the scheme as the activity is in progress, and vice versa. The third part of the scheme consists of a numerical composite of definite numerosity produced by counting that completes the feedback into the goal and closes the scheme.9 The feedback system operates as a monitor of the activity and is essential in knowing when to stop counting (Steffe & Olive, 2010, p. 23) (Fig. 2.1).  A counting scheme is judged as operative if a child can at least count-on.  When the unitizing operation operates on counted items, records of the acts of counting as well as of the involved sensory items are part of the abstracted unit items. 7  The disequilibria can also have emotional or sensory overtones. 8  “A stimulus is really a stimulus only when it is assimilated into a structure and it is this structure which sets off the response” (Piaget, 1964, p. 15). 9  The perturbation has been eliminated and the scheme is in a state of equilibrium. 5 6

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Generated Goal

•

Find how many jacks are in the bag

Situation

• • •

Result

Heard question Activated numerical composite Disequilibrium

• •

A numerical composite, ten Equilibrium

Generated Activity

•

Counting four more times beyond six

Fig. 2.1  The concept of scheme

Operationalizing Reflective Abstraction While Paul Cobb and I were involved in the teaching experiment with the six children, Glasersfeld continued on with his analysis of Piaget’s two volumes on reflective abstraction, which were available only in French (Piaget & collaborators, 1977). Glasersfeld’s (1991) analysis was particularly important for our work in the teaching experiments.10 Of reflective abstraction, Glasersfeld (1991) quoted Piaget as follows11: Reflective abstraction always involves two inseparable features: a “réfléchissement” in the sense of the projection of something borrowed from a preceding level onto a higher one, and a “réflexion” in the sense of a (more or less conscious) cognitive reconstruction or reorganization of what has been transferred. (Piaget, 1975, p. 41) (p. 58)

Reflective abstraction provided a general explanatory construct that Piaget used to explain the transition from the preoperational stage of representation to the stage of concrete operations. In the previous stage, children do not reason logically whereas in the latter stage, children can reason logically. For example, given a collection of sticks all of different length, children in the stage of concrete operations order the sticks according to their length, whereas preoperational children would most likely order them pairwise if they ordered them at all. Although Piaget thought that four factors contribute to development—social interaction, experience, maturation, and equilibration (Duckworth, 1964), to explain the general transition, he used reflective  I have a draft of Glasersfeld’s paper written in 1982 which he first published in 1991.  Glasersfeld was fluent in four languages of which French was one. So, this was an original translation that he made. 10 11

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abstraction. One of the major goals of the teaching experiment was to operationalize reflective abstraction, so it could be useful to mathematics teachers. Operationalizing reflective abstraction was similar to operationalizing Glasersfeld’s scheme theory in constructing second-order models of children’s mathematics.

Constructing the Initial Number Sequence Tyrone I found occasions that could be interpreted as Piaget’s reflective abstraction if one does not expect both aspects to occur in close proximity. One occasion occurred in my work in the teaching experiment when I was working with a child, Tyrone, in his construction of the initial number sequence (Steffe, 1988b, pp. 284–322). Tyrone had constructed the figurative counting scheme, which observationally means that he could count items that were not in his immediate visual field, but he was yet to count on. He was presented with a collection of items, seven of which were covered by one cloth and five by another cloth, and was asked to count the items.12 He pointed to places on the first cloth he took as covering an item as he subvocally uttered number words “1, 2, 3, 4, 5, 6, 7.” He then proceeded to touch the second cloth six times in a row, whispering, “8, 9, 10, 11, 12, 13,” which indicates that he did not have a five pattern that he could use to keep track of counting five more times. Failing to recognize a pattern that would close his scheme, he started over twice. On his last attempt, he again started counting from “one” and stopped at “twelve” with conviction. When counting five more times past “seven,” he stared into space while touching the cloth synchronously with uttering number words. I interpreted this final count over the second cloth as Tyrone creating a linear pattern for “five” by intentionally monitoring his counting acts in the following way. After saying “8, 9,” say, and making two pointing acts, he must have re-presented13 the results of these counting acts and “held the results at a distance,” reflecting on them as indicated by staring into space. Distancing himself in this way requires an operation not provided by re-presentation. I inferred that he took each visualized counting act as a unit using the unitizing operation, creating an arithmetical unit item containing records of the counting act from which it was abstracted. This is the process that creates interiorized counting acts14 as distinct from the internalized counting acts from which they were abstracted. The process continued on until Tyrone established a pattern of five interiorized arithmetical unit items or a numerical composite (cf. Steffe & Cobb, 1988).

 This task was designed so that Tyrone would at least need to visualize the items he counted as well as eliminate visual cues for when to stop counting. 13  A re-presentation is a re-generation of a past experience (Glasersfeld, 1991, p. 49). 14  An interiorized counting act is purged of its sensory-motor material. I think of it as a slot that contains records of the sensory-motor material. 12

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The establishment of a numerical composite is a projection of the five counting acts, “8, 9, 10, 11, 12,” to the interiorized level, which is one level beyond his figurative counting acts, which were at the level of his figurative counting scheme. I called the accommodation he made in his figurative counting scheme an engendering accommodation because he came back from Christmas vacation able to count-on. Counting-on indicated that a metamorphosis had occurred in his counting scheme over Christmas vacation. To account for the observed metamorphosis, the numerical pattern “8, 9, 10, 11, 12” that Tyrone established was an interiorized pattern of arithmetical unit items, whereas his figurative counting scheme consisted of internalized counting acts. That is, the numerical pattern “8, 9, 10, 11, 12” had been disembedded from his internalized counting scheme as an abstracted numerical concept and was established at a level “above” his internalized counting scheme of which the figurative counting acts were still a part. However, I wouldn’t say that the sequence of interiorized counting acts is applied to the higher level. Rather, monitoring counting is that process that creates the level of interiorization. The presence of the numerical pattern would have the effect of setting up a systemic perturbation that would drive the interiorization of “close by” counting acts that would continue until the perturbation dissipated. I don’t know how far his interiorized number sequence extended forward from “one,” but it was far enough for him to engage numerically in solving the arithmetical tasks that were presented to him after Christmas. I referred to this second accommodation as a metamorphic accommodation that followed on from the accommodation that engendered it and interpreted it as the “réflexion” part of Piaget’s reflective abstraction. Jason The primary reason that three children were selected for the teaching experiment whose counting schemes were judged as figurative schemes at the outset of the experiment was to explore if the accommodations the three children made in the schemes in the solution of presented tasks could legitimately be judged as equivalent. So, the integration operation emerged for Tyrone’s two cohorts in the teaching experiment in the context of patterns as well confirmed the crucial role of patterns in the construction of numerical composites. In what follows, I focus on the task where Jason was observed establishing a numerical pattern. The teacher/researcher (Paul Cobb) pretended to place eight cookies under a cloth and then some more and told Jason that there were ten cookies under the cloth (Steffe & Cobb, 1988). Jason interpreted the task as the teacher/researcher placing eight cookies under the cloth and then placing ten more with them. Like Tyrone, Jason concentrated intensely on keeping track of extending his first eight counting acts ten more times when the “breakthrough” occurred; that is, he kept track of counting ten more times and stopped after his counting act, “eighteen.” This counting episode is taken as indication of Jason projecting the involved internalized counting acts to the interiorized level. After this observation on March 18, 1981, Jason’s counting scheme remained figurative (he always started counting from “one”) until May 21, 1981 of his first grade when he applied the integration

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operation sequentially and counted on. So, like Tyrone, the projective aspect of reflective abstraction occurred in the context of solving tasks on March 18, but it wasn’t until approximately 2  months later that his figurative counting scheme emerged as an interiorized number sequence, which was indeed a metamorphosis of the figurative counting scheme. So, similar to Tyrone, the “réflexion” aspect of Piaget’s reflective abstraction, which he characterized as “a (more or less conscious) cognitive reconstruction or reorganization of what has been transferred,” did not occur in close temporal proximity to the observation of the “projective” aspect. It is important to emphasize that operationalizing reflective abstraction was accomplished in the context of intensive communicative interaction with the two children in a teaching experiment. It is also important to emphasize that it shouldn’t be inferred that the initial reflective abstraction that I have explained, whose result was the construction of an interiorized counting scheme, was sufficient for the children to construct, for example, subtraction as the inversion of addition nor to strategically find the pairs of numbers whose sum is ten. In general, children’s schemes and their affordances and constraints are, in part, constructed by a teacher/researcher in the context of conceptual analysis of recorded teaching episodes. More importantly, they are also constructed by means of actually teaching children in teaching episodes. I have been emphasizing the former without emphasizing the creativity of the researcher as a teacher in constructing what I call experiential models of the participating children’s mathematics. It is these experiential models that ground the conceptual analyses of the corpus of video material. In short, the basic premise of teaching experiments is that the teacher/researcher must construct the mathematics of children, which are second-­ order models of children’s mathematics.

Further Operationalizing Reflective Abstraction Three of the six children in the teaching experiment, one of whom was a child named Brenda, entered the experiment as counters of perceptual unit items, meaning they could not count unless the countable items were in their perceptual field. To illustrate, an interviewer covered six of nine marbles with his hand and asked Brenda to count all the marbles. Brenda first counted the interviewer’s five fingers and then counted the three visible marbles pointing to each in turn. The interviewer pointed out that he had six marbles beneath his hand, and Brenda replied, “I don’t see no six!” (Steffe & Cobb, 1988, p. 23). I inferred that Brenda was aware that marbles were hidden by the interviewer’s hand as indicated by her initializing counting the interviewer’s fingers as substitute15 countable items. Had she pointed to the interviewer’s hand at places she thought that marbles were hidden rather than point to the

 The interviewer’s fingers were not representatives of the hidden marbles. Rather, they were perceptual replacements. 15

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interviewer’s fingers, I would have inferred that she re-presented more than one, or a plurality, of marbles and took these images as countable items in a way similar to Tyrone when he counted to “seven.” I taught these three children approximately 60 times over the course of the experiment, and they also participated in their regular mathematics classrooms. Unfortunately, counting-on did not emerge in these three children over the 2 years that they spent in the teaching experiment in spite of my best efforts to provoke it and, presumably, the best efforts of their teachers. Rather than re-present collections of perceptual unit items and count figurative unit items as Tyrone and Jason did, the three children re-presented finger patterns during their first grade. The re-presented finger patterns served as substitutes for, but not representatives of, spatial patterns. These children’s re-presentation of finger patterns is an elementary example of projection in Piaget’s account of reflective abstraction in its functional forms. Although the three children were not without a semblance of the projective aspect of reflective abstraction, their counting scheme remained as a perceptual scheme throughout their first grade.

Learning Stages A major consequence of Piaget’s reflective abstraction that is still not taken seriously in mathematics education is that it produces developmental stages in children’s spontaneous constructive activity. Although I had abandoned Piaget’s epistemic child and his cognitive models of the preoperational and concrete operational child, as I have already commented, I had not abandoned the essence of his observations. However, rather than speak of what preoperational children could not do, which was to conserve numerosity or to solve the class inclusion problem, I now spoke more positively of what children taught me that they could do, which was, in the case of the three children, to engage in the goal-directed activity of counting perceptual unit items to specify the plurality of perceptual collections. My hypothesis at the outset of the teaching experiment was that the three children’s counting schemes would remain essentially equivalent throughout the time that I taught them, but it was my goal to provoke the maximal progress that I could in their constructive activity.

The Perceptual Stage The Period Criterion I didn’t start the teaching experiment with the assumption that the children’s constructive activity would occur in learning stages (cf. Ulrich, 2015 for an elaboration

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of stages).16 The children imposed that observation upon me. I was compelled to infer that being a counter of perceptual unit items constituted a stage according to Glasersfeld and Kelly’s (1983) criteria for stages. The first of these is that an item must remain constant throughout a period of time, which was counting perceptual unit items. I don’t know when these three children learned how to count perceptual unit items, but it preceded their entry into their first grade. So, there must have been a period of time when they were 5 years of age or younger when they first learned to count perceptual unit items. They remained as counters of perceptual unit items throughout their first grade in school, so I regarded the period criterion as satisfied for these three children. The Incorporation Criterion Piaget required that an earlier developmental stage be “incorporated” into a succeeding stage, a criterion that was adopted by Glasersfeld and Kelly. To discuss this criterion for these three children, I appeal to their construction of collections of perceptual unit items for which our use of Glasersfeld’s (1981) attentional moments model provided a second-order model. But rather than say that collections of perceptual unit items were incorporated in the perceptual counting scheme, I would rather say that the perceptual counting scheme is based on children having already constructed collections of perceptual unit items. Further, it is based on children’s awareness of a plurality, or more than one, of such items, an awareness that serves to establish a goal to make this quantitative property of the collection definite. That is, being a counter of perceptual items means a lot more than knowing how to count. As I have already commented, the goal of children counting collections of perceptual items is to find how many items are in the collections. The Invariant Sequence Criterion Glasersfeld and Kelly (1983) also required that stages appear in an invariant sequence. It was very alarming that Brenda, Tarus, and James did not construct figurative collections and count their elements to specify their plurality during their first grade, as did Tyrone and Jason. So, I had to wait until the children’s second grade to evaluate this criterion. As I explained above, the three children did re-present finger patterns, but not being able to produce a figurative collection and count the elements of the collection was a major constraint for the children throughout their first grade. They could recognize perceptual patterns—finger patterns, spatial patterns, spatio-­ motor patterns, and auditory patterns—as meaning of number words, but they did not re-present such patterns and count the re-presented elements of the patterns during their first grade, nor did they engage in monitoring counting to produce a numerical pattern, as did Tyrone or his two counterparts in spite of my best attempts.

 A major distinction between developmental stages and learning stages is that the former are the result of spontaneous development whereas the latter are the result of interacting socially or otherwise in teaching episodes as well. 16

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The Figurative Stage It wasn’t until the second grade that the three children made progress and constructed figurative collections. To illustrate this point, consider Tarus’ attempt to solve the following situation at the end of January of his second grade. I presented him with a cylindrical tube open at one end where a marble would just fit into the tube. After Tarus put 11 marbles into the tube, I poured three marbles out of the tube into his hand and asked him to find how many marbles were left in the tube. Tarus buried his head in his arms and played with a marble and then said, “ten.” In explanation, he said, “I count.” Based on Tarus burying his head in his arms and playing with a marble, I inferred that he was aware of marbles in the tube (an awareness of a figurative collection) and “ten” referred to marbles. But I also inferred that his figurative counting scheme did not contain the numerical operations necessary to find how many marbles were left. The internal constraints that Tarus experienced because his counting scheme was not an operative scheme were striking throughout the 2 years that I taught him (cf. Steffe & Cobb, 1988). Still, Tarus made progress over the 2-year period in that his counting scheme became a figurative scheme during that period. Tarus could count-on while he was in his third grade, but I have no information on the two other children. The best scenario is that all three children constructed their initial number sequences by that time. In any event, I inferred that the invariant sequence criterion for stages was satisfied in the case of the perceptual and figurative stages in the construction of the number sequence for these three children.

Percent of First-Grade Children in the Perceptual Stage Children who are counters of perceptual unit items at the beginning of their first grade in school are among those children who will essentially fail mathematics all the way through school because their schemes and operations are out of synchrony with school mathematics curricula. It is not just a few children who enter their first grade in school as counters of perceptual unit items. Based on my experience teaching the three children, on my experience in teacher education at UGA, and on data that was supplied to me by Professor Bob Wright of Southern Cross University, Australia, who started the Mathematical Recovery Program (Wright et al., 1998; Wright et al., 2000) and who was associated with IRON as a doctoral student, I estimated that 40% of entering first graders in the United States are counters of perceptual unit items. Of this estimate, Professor Wright commented that “I think that is a good estimate for the number in the perceptual stage or lower, that is, the children who can’t yet count perceptual items. I think the percentage would be lower in Australia and New Zealand, say about 30%” (Personal Communication, 2010).

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Further Criteria of Learning Stages I conducted a 2-year teaching experiment starting in September of 1985 with six third-grade children to investigate their construction of multiplicative schemes. My immediate goal here is to recount the progress of two children who had constructed only the initial number sequence. Zachary was the more advanced of the two children, so I concentrated on him in my conceptual analysis (Steffe, 1992).17 In what follows, I recount my efforts to explore criteria for learning stages that are not relevant to Piaget’s developmental stages in the context of Zachary’s productive attempts to solve tasks. If a productive attempt constituted a functional accommodation18 of Zachary’s initial number sequence, the results of the revised scheme (a revision of the initial number sequence) would need to close the scheme, and the change would need to be permanent. If a functional accommodation led to a metamorphic accommodation like Tyrone’s and Jason’s, it would be necessary to be able to legitimately infer that a change in the assimilating structure (or operations) of the initial number sequence occurred. So, of central concern was whether it could be judged that a functional accommodation left the operations of the initial number sequence invariant, which is a criterion for learning stages.

Modifications in the Initial Number Sequence In an exploration of Zachary’s possible functional accommodations, I presented him with a situation where four rows of blocks with three per row were hidden, some more rows with three blocks per row were hidden with the four and told Zachary that seven rows were now hidden. He was to find how many more rows were hidden. This task was designed to explore if Zachary could take the three blocks in a row as a composite unit and operate as if the rows were units of one. That is, would Zachary simply count from “four” up to “seven” to find how many more rows were hidden? Zachary initially formed the goal of finding how many blocks rather than rows were hidden. After counting the blocks in the first two rows, “3, 6,” he proceeded as follows. He sequentially put down his thumb, index finger, and middle finger repeatedly as he whispered number words, “7, 8, 9; 10, 11, 12; 13, 14, 15; 16, 17, 18; 19, 20, 21; 22, 23, 24.”19 Apparently realizing that he didn’t know when to stop counting, he started over, this time sweeping a finger over each of the

 I had no information on Zachary while he was in his second grade. But based on how he used his initial number sequence at the outset, I would say that he constructed it at some point in his second grade. 18  A functional accommodation of a scheme occurs in the use of the scheme. 19  The semicolons represent breaks between completing three counting acts and starting three more counting acts. 17

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hidden rows and then sweeping it once more as if counting one more row. He then stopped and said, “I can’t do that one.” When counting, Zachary used his finger pattern for “three” by sequentially putting down three fingers repeatedly. Apparently, his intention was to use his finger pattern seven times. But he apparently didn’t know when to stop counting, which indicates that he didn’t keep track of how many times he used it. Had he kept track, I would have inferred that he took each finger pattern as a composite unit—as one thing—and counted those composite units. That is, he would have monitored repeatedly using his finger pattern. His comment, “I can’t do that one,” confirms a lack of monitoring activity and a lack of making composite units of three as he proceeded. His behavior is consistent with the interpretation that the assimilating structure of his initial number sequence was numerical composites rather than composite units containing them. It is also consistent with the interpretation that he was aware of a plurality20 of rows with a trio of blocks in each row prior to activity. Because Zachary’s modification in how he used his initial number sequence didn’t close the scheme, the modification in how he counted couldn’t be said to constitute a functional accommodation. I interpret Zachary’s modification in how he counted as producing a pseudo-empirical abstraction, which is any abstraction concerning an activity that a child has produced but whose results cannot yet be obtained without actually carrying out the activity (Steffe & Cobb, 1988). Corroboration that he made a pseudo-empirical abstraction is based on how he later tried to find how many piles of three blocks he could make from a pile of 12 blocks (Steffe, 1992). He squeezed his right thumb, index finger, and middle finger together and placed them on the table in front of him and said, “three.” He continued on in this way four times until he reached “twelve.” He then looked at the nine fingers that he used and said, “Nine times.” Upon saying “Nine times,” he apparently experienced a conflict because he started over, but he was again unsuccessful. The above modification that Zachary made in his initial number sequence did lead to a functional accommodation in that I was able to infer that he constructed experiential composite units in activity in the context of engaging in enactive units-­ coordination. After Zachary made four rows of blocks with three per row, I covered them and asked him how many blocks were in the rows. He looked straight ahead as he sequentially put up his right thumb, index finger, and middle finger four times and then said, “Twelve.” He only used one finger pattern for three and clearly kept track of how many times he used it as he was counting to “Twelve” in modules of three, which indicates that he took his finger pattern as one thing. Taking his finger pattern for three as one thing in activity constitutes making experiential composite units, and keeping track of how many finger patterns he made constitutes enactive units-coordination of the composites four and three, or a units-coordination in activity. Zachary made still further progress, but there was never an occasion where I could infer that he made a functional accommodation in his number sequence that I could judge as metamorphic. So, he remained in the stage of the initial number

20

 The term “plurality” means more than one unitary item.

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sequence for his entire third grade in school with the proviso that he progressed to making experiential composite units in activity during the latter part of the school year. Modifications in the Tacitly Nested Number Sequence The child Maya had constructed the tacitly nested number sequence at the outset of the multiplication teaching experiment, which is characterized in part by the construction of composite units (Ulrich, 2016a, p.  34). To illustrate Maya’s number sequence, I analyzed her solution to a task where I asked her to place eight marbles in a cup and seven in another. To find how many were in both cups, I inferred that she double counted seven onto eight, which means that she counted “Eight. Nine is one; ten is two, …, fifteen is seven. Fifteen.” She then placed three more marbles with the seven upon my request and I asked her how many marbles were in the cups now. She immediately said, “Eighteen” and explained, “Because there were eight and seven, and you know that eight and seven is fifteen, and then you put the other three, and there’s fifteen, 16, 17, 18.” Her solution to this task was quite beyond any solution that I observed Zachary produce. To explain how she operated as she did, I appealed to the recursive property of her number sequence, which means that she could take the number sequence as its own input to create countable items, as illustrated by double counting (Steffe, 1992). My Construction of Units-Coordinating Operations As I was analyzing the teaching episodes after the teaching experiment with the children was completed, the way in which Maya solved the following task had me “staring at the wall” for the better part of one summer in an attempt to viably interpret her solution. I recount it because it was the occasion for constructing units-­ coordinating operations as constitutive multiplicative operations. I also recount it to indicate the challenges of engaging in a conceptual analysis of the video records of a teaching experiment that I have referred to as a retrospective conceptual analysis. In the presentation of the task, I placed a red piece of construction paper in front of Maya and several congruent rectangular blue pieces cut so that six blue pieces fit on the red piece. After she said, “Six,” I removed three of them and covered one of them with two orange pieces that fit exactly and asked her how many orange pieces go on the blue piece. After looking straight ahead, she subvocally uttered number words and said, “Twelve.” In explanation, she tapped the table twice with each of six fingers, “1, 2; 3, 4; 5, 6; 7, 8; 9, 10; 11, 12.” Tapping each of her fingers twice indicated that she was counting the orange pieces on each red piece. My issue, which led me to “staring at the wall,” was whether the solution was based on simply re-presenting placing an orange piece on each blue piece or whether she inserted a unit of two into each unit of six, which would be an operative rather than simply a figurative solution. How she organized her explanation into modules of two counting acts eventually was the determining element that eliminated the purely figurative interpretation because she was definitely counting six twos. So, the interpretation that she coordinated inserting a unit of two into each unit of a unit of six either before or in operating was the more viable interpretation. As indicated by other

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tasks that she solved, Maya was explicitly aware of units of two or three as well as the items they contained. I interpreted Maya inserting a unit of two into each unit of her composite unit of six units as an accommodation. The results of the scheme closed the scheme, and she was observed using the scheme in other cases that, from my perspective, were multiplicative situations. This scheme was quite distinct from regarding multiplication in elementary school as repeated addition, which is essentially what Zachary had constructed when he found how many blocks were in four covered rows with three blocks in each row that I explained above. In that case, he made an enactive units-coordination (or a units-coordination in activity) of the four rows and the three blocks in each row. In Maya’s case, she made a units-coordination prior to initiating units-coordinating activity. As the issue was whether Maya’s units-coordination constituted a metamorphic accommodation on a par with Tyrone’s and Jason’s metamorphic accommodation or whether it was a functional accommodation that produced a scheme, her units-coordinating scheme that was at the same level of reflective abstraction as her tacitly nested number sequence. To make that decision, I looked to other tasks that involved units-coordination to solve. Constraints in Provoking Reflective Abstraction I presented a task to Maya that children who have constructed the explicitly nested number sequence could solve in an attempt to analyze if Maya’s units-coordinating scheme was a stage beyond her tacitly nested number sequence. Had Maya spontaneously solved the tasks, I would have inferred that her construction of her units-­ coordinating scheme involved a reflective abstraction. Although Maya solved the tasks upon my urging, her solution attempts were not spontaneously initiated, which prohibited inferring that her accommodation was produced by a reflective abstraction (Steffe, 1992). In spite of my best attempts, I should be clear that I did not experience Zachary nor Maya solve a task I presented to them that could be analyzed as entailing either a reflective abstraction or the results of a reflective abstraction throughout their third grade in school. They remained in their respective stages throughout that school year. I did document solutions of tasks that involved important functional accommodations that I have referred to as “lateral learning” rather than “vertical learning” (Steffe, 1991a). The question concerning whether reflective abstraction or vertical learning can be provoked in children like Zachary or Maya in their third grade remains a crucial question because both children were out of synchrony with what is expected of a third-grade child in the mathematics curricula.

Modifications in the Explicitly Nested Number Sequence Numerical part-to-whole reasoning is the identifying characteristic of the explicitly nested number sequence (Steffe, 1992, p.  290). That is, children who have constructed the explicitly nested number sequence can disembed a composite unit from

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a given composite unit and use the disembedded unit in further reasoning.21 To illustrate, in a “missing rows” situation, Johanna, a child in the multiplication teaching experiment, made four rows of blocks with three per row. I then covered some more rows of three and told her that there were now seven rows of three and asked her to find how many rows were hidden. She said, “three” almost immediately, treating the rows as if they were singleton units in contrast to Maya’s attempted solution of a similar task (Steffe, 1992). To again compare Johanna’s and Maya’s task solutions, I presented the same task to Johanna that I spent most of one summer “staring at the wall” when analyzing it in Maya’s case. In solution, Johanna said “twelve” after about 15  seconds and explained, “Well, six plus six is twelve, and each two blocks fit on one big block, and that makes twelve” (Steffe, 1992, p. 292). I inferred that Johanna was reflective aware of six units of two and decomposed them into two units of six, whereas Maya took six units of one as a given and then substituted a unit of two for each unit of one. Based on Johanna’s spontaneous reasoning, I inferred that her units-­coordinating scheme constituted a functional accommodation in her explicitly nested number sequence, an inference that was corroborated throughout the teaching experiment. Johanna continued making progress and used her units-coordinating scheme in the construction of three levels of units, or a unit of units of units. For example, after hiding five rows of blocks with four per row that she had made, I asked her how many blocks were in the first three rows. After about 25 seconds, when she was in deep concentration, she said, “Twelve.” I then asked her how many blocks were in all five rows after she found that eight blocks were in the remaining two rows. After sitting silently for about 15 seconds, she replied “twenty” and explained, “Because I added up. Twelve plus four is 16, and 16 plus four is 20!” (Steffe, 1992, p. 292). I inferred that Johanna united the first three units of four into another composite unit containing them whose numerosity she found as “twelve.” My inference that she operated in this way is also based on that it took her 25 seconds to answer “twelve” (Steffe, 1992). My inference that making units of units of units was an accommodation in her units-coordinating scheme would not have been viable unless there were other composite unit tasks where she exhibited similar productive mathematical activity in solution. Corroboration is an essential element of the process of constructing second-order models, particularly when inferring the occurrence of accommodation. Corroboration is but one aspect of the teacher/researcher constructing interiorized experiential models of the children’s mathematics that the teacher/researcher uses to predict how a particular child will operate in other tasks. That is, the teacher/ researcher becomes the child and attempts to think as if he or she is the child (Steffe & Thompson, 2000). I have already commented that such experiential models are critical in the retrospective second-order analysis of the video-recorded material at the end of the teaching episodes in constructing explanations of children’s mathematics. In constructing second-order models like the number sequences, it is

21

 An analogy is conceiving of a subset of a set.

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important to stress that these models are explanations the researcher makes that are grounded in the activity of children that the teacher/researchers are involved in provoking through experimental teaching. So, the researcher is inextricably involved in creating the observations for which the researcher creates explanations, which is compatible with thinking in second-order cybernetics.22

The Fractions Teaching Experiment After the completion of the multiplication teaching experiment, my interest turned to exploring children’s construction of fractions. My hypothesis was that fraction knowing can emerge as accommodations in number sequences (Steffe & Olive, 2010). At the time, there was a general belief that whole number knowing interferes with the learning of fractions, a belief that is still pervasive. It translates in school mathematics curricula as fractions being presented independently of work with whole numbers. To test the reorganization hypothesis, John Olive and I, along with Ron Tzur and Barry Biddlecomb, who were then graduate students, started a 3-year teaching experiment that we referred to as the fractions teaching experiment in 199123 with third-grade children. This project continued on until 2002, and the book Children’s Fractional Knowledge was printed in 2010 (Steffe & Olive, 2010).24

The Equipartitioning Scheme: A Functional Accommodation The reorganization hypothesis was unexpectedly partially confirmed by two children, Jason and Patricia, both of whom had constructed the explicitly nested number sequence. The children partitioned a computer-generated drawing of a segment that we referred to as a “stick” in the experiment (Steffe & Olive, 2010). To make a share of a stick for one of four people, Jason used his concept, four, as a partitioning template to mark off one of four equal parts. He then reassembled the parts into a unit of four equal parts of the stick and compared the four parts with the original stick to find if four of them together were the same length as the original. In doing so, he constructed a partitioning scheme that I called the equipartitioning scheme that satisfied six of the seven criteria for Piaget’s operational subdivision25 (Piaget et al., 1960). The criterion it didn’t satisfy was the criterion that the units of the subdivision can be taken as units to be subdivided further. Patricia, too, demonstrated that

 See Steier (1995) for an introduction to second-order cybernetics.  Dr. Heide Wiegel also worked on the teaching experiment organizing, cataloging, and analyzing data. 24  See Hackenberg et al. (2016) for an interpretation for teachers. 25  Partitioning is fundamental in constructing fraction schemes. 22 23

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she could use her number concepts as partitioning templates in the establishment of connected numbers.

Pre-fractional Children Partitioning continuous units using numerical concepts like Johanna’s was found to be one of the key elements in corroboration of the reorganization hypothesis. However, for children like Zachary, who are in the stage of the initial number sequence, fragmenting a continuous unit proved to be problematic due to a general lack of coordinating the number and size of the parts with exhausting the whole, which meant that they are yet pre-fractional (Biddlecomb, 2002; Steffe & Olive, 2010). These limitations in fragmenting were in part removed by children like Maya, who had constructed the tacitly nested number sequence because number words now symbolized composite units prior to action, which enables them to partition a stick into four equal parts should that be their goal. What would be still lacking is an understanding that any of the unit parts could be used to reconstitute a continuous but segmented unit equivalent to the original unit—that is, the unit parts are not iterable units like they were for Jason because the units of the composite unit used in fragmenting are not iterable (Steffe & Olive, 2010; Ulrich, 2016b). In fact, after working with a child who had constructed the tacitly nested number sequence throughout the child’s fourth grade, the child did not construct any fraction schemes (Steffe & Olive, 2010).

The Partitive Fraction Scheme A burning question of the fraction teaching experiment was whether children who had constructed the explicitly nested number sequence could also construct the fundamental elements of fraction knowing. Jason and Patricia did use their operative equipartitioning scheme to construct what Tzur (1999) called partitive fractions, or “proper” fractions of the fractional whole (cf. Steffe & Olive, 2010, pp.  323) by means of a functional accommodation. At the time it was startling that the partitive fraction scheme was not sufficient to construct fractions beyond the fractional whole. Other than being restricted to making proper fractions, children who had constructed the partitive fraction scheme were not able to, say, given a segment that is said to be three-fifths of some other segment, construct the other segment by partitioning the given segment into three parts and use those parts in making the other segment (Steffe & Olive, 2010).

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The Iterative Fraction Scheme It was a major finding that three levels of units were necessary to construct fractions greater than the fractional whole. For example, after Patricia constructed three levels of units, she explained why a stick was eight-sevenths in length was because the eight parts of the stick were the same as the little pieces in the seventh stick (the fractional whole), so its eight-sevenths (Steffe & Olive, 2010). I interpreted her comments as indicating that one-seventh was freed from the fractional whole of which it was a part and was constituted as a unit fraction without reference to the fractional whole. That is, “one-seventh” referred to any one of the seven parts and, in that sense, it was a hypothetical part of seven-sevenths that could be iterated eight times to produce eight-sevenths. So, the eight units of the eight-sevenths were conceived of as one-seventh eight times. To illustrate, using the available computer operations, Patricia used the calculator to multiply a 14/99 stick by ten and explained why the stick she made would be a 140/99 stick was because “you always have the same little stick you started off with” (Steffe & Olive, 2010, p. 334). That is, 140/99 was constituted as a fractional number because it was 140 times the fractional unit that was used in producing it, analogous to one iterated 140 times to produce the number 140. This was a critical conception of fractions in that the entirety of the composite unit comprised by 140/99 doesn’t need to be produced to give meaning to the fraction, so the numeral can be a symbol for those operations. In fact, Patricia and her cohort produced a fraction number sequence that Patricia thought would continue on to infinity quite similar to how she could use her unit of one to produce indefinitely large whole numbers. We referred to Patricia’s fraction scheme as the iterative fraction scheme. It was one stage beyond her partitive fraction scheme, a claim that was based not only on the fractional numbers that she produced but also on the level of units on which it was based. We never observed what could be considered a metamorphic accommodation that produced the three levels of units. But I believe that the third level was constructed by means of a metamorphic accommodation in the context of the teaching experiment. In Jason’s case, he returned from summer vacation able to operate in ways that were beyond him when he left for vacation after his fourth grade. When he returned from summer vacation, he reasoned in such a way that implied that he had constructed three levels of units (Steffe & Olive, 2010). There were no observations that could be used to operationalize reflective abstraction during his fourth grade, so if he made a reflective abstraction, it occurred over the summer months.

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Final Comments To provide perspective on the learning stages in the school population, I present my estimates of the percentage of children in the first, second, third, fifth, and sixth grades who have constructed the various counting schemes. For the 40% of the school population who enter the first grade in the perceptual stage, I have argued that it is reasonable to expect that the majority of them will construct the initial number sequence during their third grade. Wright estimated that from 5 to 8% might not be counting on by the third grade (Personal Communication). From that point on, Zachary, notwithstanding, my expectation is that the majority of them will remain in that stage throughout the fifth grade. The relative percentages are not certain, but my best estimate is that approximately 30% of the children entering the sixth grade will be in the stage of the initial number sequence. Wright’s estimate was “that about thirty percent of kids entering the sixth grade in the US will only be able to count on” (Personal Communication). And those who are in that stage will remain there until their eighth grade.26 I also estimate that at most one child in ten in the first grade will have constructed the explicitly nested number sequence. Based on this estimate, no more than approximately 50% of children entering the first grade will have constructed the initial number sequence sometime during their first grade. The 10% estimate is based on Piaget et al.’s (1960) finding that only one child in ten of those from 6 to 7 years of age (first graders) attain the iterable length unit, which I take as analogous to the explicitly nested number sequence. Because of the homogeneity of the Swiss population, I consider the 10% as an upper bound on the percent of first-grade children who have constructed or will construct the explicitly nested number sequence. In the second grade, my estimate is that approximately 40% of the children are in the figurative stage because these are the children who are in the perceptual stage in the first grade. Piaget et al. (1960) estimate that one-half of Swiss children from 7 to 7 years 6 months (second graders) attain the iterable length unit. Because of the homogeneity of the Swiss population, I consider it implausible that 50% of second graders have constructed the explicitly nested number sequence and estimate that no more than 30% do so. On that basis, 30% of second graders remain in the stage of the initial number sequence. Whatever these two percentages may be, approximately 70% of the second-grade population is yet to construct the explicitly nested number sequence, which is the number sequence I consider is assumed in normal second-grade mathematics classrooms given the curriculum. And 40% are yet pre-­ numerical (figurative), which means that they are seriously at risk in school mathematics not only in their first and second grades but also throughout their mathematics education (Table 2.1).

 In a 3-year teaching experiment with sixth, seventh, and eighth graders, three children who had constructed the initial number sequence remained in that stage for the first 2 years of the experiment (Hackenberg, 2005; Tillema, 2007). 26

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Table 2.1  Estimated percent of children at each level of units by grade Grade/ stage First Second Third Fifth Sixth

Perceptual/ figurative 40% P 40% F – – –

Initial/ tacitly nested (%) 50 30 60 35 30

Explicitly nested (%) 10 30 40 40 30

Three levels of units (%) – – – 25 40

Iterative fraction

No estimate No estimate

Throughout the third grade, I assume that a majority of the children in the figurative stage will construct the initial number sequence (approximately 35%). Of the remaining 65%, my best estimate is that 40% of the total population will construct the explicitly nested number sequence, and the remaining 25% will remain in the stages of the initial or the tacitly nested number sequence.27 Based on the fraction teaching experiment, children who have constructed the explicitly nested number sequence in the third grade can construct three levels of units by the time they enter their fifth grade under the influence of intensive teaching. So, optimistically, I expect that 25% of the population will have constructed three levels of units by the time they enter the fifth grade, and 40% will have constructed the explicitly nested number sequence. I estimated that 25 and 40% of the fifth- and sixth-grade populations have constructed three levels of units. Data presented by Norton and Wilkins (2009) in Table 2.2 confirm my estimates in broad outline. These are the children who should have constructed the iterative fraction scheme and fractional numbers. However, Norton and Wilkins found that only 14% of the fifth graders and 18% of the sixth graders have constructed the iterative fraction scheme and that these percentages do not improve for the seventh and eighth grades. This is quite concerning because the children in our fractions teaching experiment who had constructed three levels of units did construct that scheme and its use in producing fractional numbers, which confirms that researchers should be involved in teaching the children in their teaching experiments. Even more surprising is the finding by these two authors that fewer than 20% of students throughout the middle school grades can produce a fractional whole given a proper fraction of the whole (Wilkins & Norton, 2018). Clearly, the presence of learning stages in children’s constructive activity presents major challenges for constructivist researchers in mathematics education in a way similar to the challenges that spontaneous development presented for curriculum developers during the era of modern mathematics.

27

 Zachary and Maya were among this 25%.

40 Table 2.2  Norton and Wilkins data

L. P. Steffe Grade/level Three levels of units (%) Fifth 34 Sixth 35 Seventh Eighth

Iterative fraction (%) 14 18 13 18

References Allendoerfer, C. B., & Oakley, C. O. (1959). Fundamentals of freshman mathematics. McGraw Hill Book Company, Inc. Biddlecomb, B. (2002). Numerical knowledge as enabling and constraining fraction knowledge: An example of the reorganization hypothesis. Journal of Mathematical Behavior, 21, 167–190. Bridgman, P. W. (1927). The logic of modern physics. Macmillan. Duckworth, E. (1964). Piaget rediscovered. In R.  E. Ripple & V.  N. Rockcastle (Eds.), Piaget rediscovered: A report of the conference on cognitive studies and curriculum development (pp. 1–5). School of Education. Cornell University. Hackenberg, A. J. (2005). Construction of algebraic reasoning and mathematical caring relations. Unpublished doctoral dissertation, University of Georgia. Hackenberg, A. J., Norton, A., & Wright, R. J. (2016). Developing fractions knowledge. Sage. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. Routledge & Kegan Paul. Kilpatrick, J. (1964). Cognitive theory and the SMSG program. In R. E. Ripple & V. N. Rockcastle (Eds.), Piaget rediscovered: A report of the conference on cognitive studies and curriculum development (pp. 128–133). School of Education. Cornell University. Lakatos, I. (1970). Falsification and the methodology of scientific research programs. In I. Lakatos & A.  Musgrave (Eds.), Criticism and the growth of knowledge (pp.  91–195). Cambridge University Press. Letvin, J. Y., Maturana, H. R., McCulloch, W. S., & Pitts, W. H. (1959). What the frog’s eye tells the frog’s brain. Proceedings of the Institute of Radio Engineers, 47, 1949–1951. Reprinted in W. S. McCulloch (Ed.) (1965). In Embodiments of mind. MIT Press. Lovell, K. (1971). Proportionality and probability. In M.  Rosskopf, L.  P. Steffe, & S.  Taback (Eds.), Piagetian cognitive development and mathematical education (pp. 136–148). National Council of Teachers of Mathematics. Norton, A., & Wilkins, J. L. M. (2009). A quantitative analysis of children’s splitting operations and fraction schemes. Journal of Mathematical Behavior, 28(2/3), 150–161. Piaget, J. (1964). The Piaget papers. In R.  E. Ripple & V.  N. Rockcastle (Eds.), Piaget rediscovered: Report of a conference on cognitive studies and curriculum development. Cornell University Press. Piaget, J. (1966). General conclusions. In E. W. Beth & J. Piaget (Eds.), Mathematical epistemology and psychology (pp. 305–312). Dordrecht. Piaget, J. (1968). Six psychological studies. Vintage Books. Piaget, J. (1970). Genetic epistemology. Columbia University Press. Piaget, J. (1975). L’equilibration des structures cognitives. Presses Universitaires de France. Piaget, J. (1980). The psychogenesis of knowledge and its epistemological significance. In M.  Piattelli-Palmarini (Ed.), Language and learning: The debate between Jean Piaget and Noam Chomsky (pp. 23–34). Harvard University Press. Piaget, J., & collaborators. (1977). Recherches Sur l’abstraction reflechissante (Vol. I & II). Presses Universitaires de France. Piaget, J., & Inhelder, B. (1967). The child’s conception of space. W. W. Norton & Company, Inc (First published 1956).

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Piaget, J., & Inhelder, B. (1969). The psychology of the child. Basic Books. Piaget, J., & Szeminska, A. (1952). The child’s conceptions of number. Routledge & Kegan Paul. Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry. Basic Books. Pólya, G. (1945). How to solve it (2nd ed.). Princeton University Press (1957). Pólya, G. (1981). Mathematical discovery. (Volumes 1 and 2, Combined paperback edition). Wiley. Ripple, R. E., & Rockcastle, V. N. (Eds.). (1964). Piaget rediscovered: A report of the conference on cognitive studies and curriculum development. School of education. Cornell University Press. School Mathematics Study Group. (1965). Mathematics for the elementary school: Grade 6, Parts I and II. A. C. Vroman, Inc. Sinclair, H. (1971). Piaget’s theory of development: The main stages. In M. Rosskopf, L. P. Steffe, & S. Taback (Eds.), Piagetian cognitive development and mathematical education (pp. 1–11). National Council of Teachers of Mathematics. Stanley, J.  C., & Campbell, D.  T. (1963). Experimental and quasi-experimental designs for research on teaching. In N. L. Gage (Ed.), Handbook for research on teaching (pp. 171–246). Rand McNally & Company. Steffe, L.  P. (1966). The performance of first grade children in four levels of conservation of numerousness and three IQ groups when solving arithmetic addition problems. Technical Report No. 14, Research and Development Center for Cognitive Learning and Re-Education. University of Wisconsin. Steffe, L. P. (1983). The teaching experiment in a constructivist research program. In M. Zweng, T. Green, J. Kilpatrick, H. Pollack, & M. Suydam (Eds.), Proceedings of the fourth international congress on mathematical education (pp. 469–471). Birkhauser. Steffe, L.  P. (1988a). Children's construction of number sequences and multiplying schemes. In J.  Hiebert & M.  Behr (Eds.), Number concepts and operations in the middle grades (pp. 119–140). Lawrence Erlbaum Associates. Steffe, L. P. (1988b). Modifications of the counting scheme. In I. L. P. Steffe & P. Cobb (Eds.), Construction of arithmetical meanings and strategies (pp. 284–322). Springer Verlag. Steffe, L. P. (Ed.). (1991a). Epistemological foundations of mathematical experience (pp. 45–67). Springer. http://www.vonglasersfeld.com/130 Steffe, L.  P. (1991b). The constructivist teaching experiment: Illustrations and implications. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics education (pp.  177–194). D. Reidel Company. Steffe, L.  P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309. Steffe, L. P. (1994). Children’s construction of meanings for arithmetical words. In D. Tirosh (Ed.), Implicit and explicit knowledge: An educational approach (pp. 131–169). Ablex. Steffe, L. P. (1996). Stages in the construction of the number sequence. In J. Bideaud, C. Meljac, & J.-P. Fischer (Eds.), Pathways to number (pp. 83–98). Routledge. Steffe, L.  P. (2011). The honor of working with Ernst von Glasersfeld: Partial recollections. Constructivist Foundations, 6(2), 172–176. Steffe, L. P. (2013). Establishing mathematics education as an academic field. Journal for Research in Mathematics Education, 44(2), 353–370. Steffe, L. P., & Cobb, P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2), 83–94. Steffe, L.  P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies. Springer Verlag. Steffe, L. P., & Kieren, T. (1994). Radical constructivism and mathematics education. Journal for Research in Mathematics Education, 25(6), 711–733. Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. Springer. Steffe, L. P., & Parr, R. B. (1968). The development of the concepts of ratio and fraction in the fourth, fifth, and sixth years of the elementary school. Technical Report No. 49, Research and Development Center for Cognitive Learning and Re-Education, C-03, OE 5-10-154. University of Wisconsin.

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Steffe, L. P., & Thompson, P. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. Kelly & R. Lesh (Eds.), Research design in mathematics and science education (pp. 267–307). Lawrence Erlbaum. Steffe, L.  P., & Ulrich, C. (2013). The constructivist teaching experiment. In S.  Lerman (Ed.), Encyclopedia of mathematics education (pp. 102–109). Springer. Steffe, L. P., Hirstein, J., & Spikes, C. (1976a). Quantitative comparison and class inclusion as readiness variables for learning first grade arithmetic content. Technical Report No. 9. ERIC Document Reproduction Service No. ED144808. Project for Mathematical Development of Children. Steffe, L. P., Hirstein, J., & Spikes, C. (1976b). Quantitative comparison and class inclusion as readiness variables for learning first grade arithmetic content. Technical Report No. 9, Project for Mathematical Development of Children, (ERIC Document Reproduction Service No. ED144808). Steffe, L.  P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children’s counting types: Philosophy, theory, and application. Praeger. Steier, F. (1995). From universing to conversing: An ecological constructionist approach to learning and multiple description. In L.  P. Steffe & J.  Gale (Eds.), Constructivism in education (pp. 67–84). Lawrence Erlbaum Associates. Stolzenberg, G. (1984). Can an inquiry into the foundations of mathematics tell us anything interesting about mind? In P. Watzlawick (Ed.), The invented reality (pp. 257–308). W.W. Norton & Company. Thompson, P.  W. (2000). Radical constructivism: Reflections and directions. In L.  P. Steffe & P. W. Thompson (Eds.), Radical constructivism in action: Building on the pioneering work of Ernst von Glasersfeld (pp. 412–448). Falmer Press. Thompson, P.  W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education. In O.  Figueras, J.  L. Cortina, S.  Alatorre, T.  Rojano, & A. Sèpulveda (Eds.), Proceedings of the annual meeting of the International Group for the Psychology of mathematics education (Vol. 1, pp. 45–64). PME. Thompson, P.  W. (2013). In the absence of meaning. In K.  Leatham (Ed.), Vital directions for research in mathematics education (pp. 57–93). Springer. Thompson, P. W., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), Research companion to the principles and standards for school mathematics (pp. 95–114). National Council of Teachers of Mathematics. Thompson, P. W., & Thompson, A. G. (1994). Talking about rates conceptually, Part I: A teacher’s struggle. Journal for Research in Mathematics Education, 25(3), 279–303. Tillema, E. (2007). Students’ construction of algebraic symbol systems. Unpublished doctoral dissertation, The University of Georgia. Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for Research in Mathematics Education, 30, 390–416. Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (part 1). For the Learning of Mathematics, 35(3), 2–7. Ulrich, C. (2016a). Stages in constructing and coordinating units additively and multiplicatively (part 2). For the Learning of Mathematics, 36(1), 34–39. Ulrich, C. (2016b). The tacitly nested number sequence in sixth grade: The case of Adam. The Journal of Mathematical Behavior, 43, 1–19. Van Engen, H. (1949a). An analysis of meaning in arithmetic. I. The Elementary School Journal, 49(6), 321–329. Van Engen, H. (1949b). An analysis of meaning in arithmetic. II. The Elementary School Journal, 49(7), 395–400. Van Engen, H. (1971). Epistemology, research, and instruction. In M. F. Rosskopf, L. P. Steffe, & S.  Taback (Eds.), Piagetian cognitive-development research and mathematical education (pp. 34–52). National Council of Teachers of Mathematics.

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von Glasersfeld, E. (1974). Piaget and the radical constructivist epistemology. In C.  Smock & E. von Glasersfeld (Eds.), Epistemology and education: The implications of radical constructivism for knowledge acquisition (pp. 1–26). University of Georgia. von Glasersfeld, E. (1980). The concept of equilibration in a constructivist theory of knowledge. In F.  Benseler, P.  M. Hejl, & W.  K. Kock (Eds.), Autopoisis, communication, and society (pp. 75–85). Campus Verlag. von Glasersfeld, E. (1981). An attentional model for the conceptual construction of units and number. Journal for Research in Mathematics Education, 12, 83–94. von Glasersfeld, E. (1983). An attentional model for the conceptual construction of units and number. In L.  P. Steffe, E. von Glasersfeld, J.  Richards, & P.  Cobb (Eds.), Children’s counting types: Philosophy, theory, and application (pp. 124–136). Praeger. von Glasersfeld, E. (1987). The construction of knowledge: Contributions to conceptual semantics. Intersystems Publications. von Glasersfeld, E. (1991). Abstraction, re-presentation, and reflection: An interpretation of experience and of Piaget’s approach. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 45–67). Springer. http://www.vonglasersfeld.com/130 von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Falmer Press. von Glasersfeld, E. (1998). Scheme theory as a key to the learning paradox. Paper presented at the 15th Advanced Course, Archives Jean Piaget. von Glasersfeld, E. (2005). Thirty years constructivism. Constructivist Foundations, 1(1), 9–12. http://constructivist.info/1/1/009 von Glasersfeld, E., & Kelly, M. (1983). On the concepts of period, phase, stage, and level. Human Development, 25, 152–160. von Glasersfeld, E., & Richards, J. (1983). The creation of units as a prerequisite for number: A philosophical review. In L.  P. Steffe, E. von Glasersfeld, J.  Richards, & P.  Cobb (Eds.), Children’s counting types: Philosophy, theory, and application (pp. 1–20). Praeger. Wertheimer, M. (1945). Productive thinking. Harper. Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. Journal of Stem Education, 5, 27. Wright, R.  J., Stewart, R., Stafford, A., & Cain, R. (1998). In K.  Norwood & L.  Stiff (Eds.), Assessing and documenting student knowledge and progress in early mathematics (pp. 211–216). Proceedings of the Twentieth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Wright, R. J., Martland, J., & Stafford, A. K. (2000). Early numeracy: Assessment for teaching and intervention. Paul Chapman Publishing.

Part II

Key Constructs from Genetic Epistemology Being Used in Ongoing Mathematics Education Research

Chapter 3

Schemes and Scheme Theory: Core Explanatory Constructs for Studying Mathematical Learning Erik S. Tillema and Andrew M. Gatza

The notion of scheme has given and is still giving rise to different interpretations. (Inhelder & de Caprona, 1992, p. 41)

As part of writing this chapter, we have had the enjoyable experience of re-­ reading many of Glasersfeld’s articles related to schemes. Re-reading Glasersfeld’s articles gave us an appreciation anew for the contributions he made in interpreting Piaget’s construct of scheme. In his later writings, Glasersfeld (e.g., 2001) came to call the network of ideas—schemes, adaptation, viability, assimilation, accommodation, perturbation, and equilibration—“scheme theory.” He points out that this network of ideas was not laid out for readers of Piaget in a single place as a theory. Rather Piaget created and refined these ideas over a significant period of time (Glasersfeld, 1997). One contribution, then, of Glasersfeld’s work is a coherent interpretation of Piaget’s writing, including the ideas that make up scheme theory. A second contribution of his work is an explicit articulation of an epistemological position, radical constructivism, and its role in scheme theory. As we highlight in this chapter, Glasersfeld’s (1995) epistemological position has a significant impact on the way that he interprets Piaget’s constructs. Therefore, we begin this chapter with some epistemological considerations related to interpreting scheme theory. We see these epistemological considerations as particularly helpful to interpreting how Piaget developed and used the construct of scheme in his own work and how these constructs have been subsequently taken up in mathematics education research. To this end, we provide a brief history of the development of schemes in Piaget’s work with discussion of how mathematics educators have E. S. Tillema (*) Department of Curriculum & Instruction, Indiana University, Bloomington, IN, USA e-mail: [email protected] A. M. Gatza Department of Mathematical Sciences, Ball State University, Muncie, IN, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 P. C. Dawkins et al. (eds.), Piaget’s Genetic Epistemology for Mathematics Education Research, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-47386-9_3

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interpreted anew a few key ideas from Piaget for their work in mathematics education (e.g., Steffe et  al., 1983; Thompson, 1985). Once these initial pieces are in place, throughout the rest of the chapter, we focus on illustrating constructs related to scheme theory relative to what we call a multiplicative pairing scheme (MPS), a scheme students can construct in the context of solving combinatorics problems. Our discussion of an MPS begins with illustrations of two definitions of the term scheme, Glasersfeld’s (1995) and Steffe’s (2010a) definitions. These initial examples are intended to show “schemes in action” and to highlight important features related to using the respective definitions of scheme. We then situate our discussion of schemes within a framework for studying learning, introducing and illustrating the constructs of assimilation, perturbation, and functional accommodation. We conclude the chapter with a discussion of an MPS relative to stages of units coordination and levels within a stage of units coordination where we see stages and levels as a way to situate schemes within a broader framework. We have chosen to discuss the same scheme throughout the chapter to help the reader focus on the different constructs related to scheme theory that we develop in the chapter. That is, we intend for the discussion to highlight issues that are of importance in general to using schemes and scheme theory in mathematics education research. Schemes are living constructs, and as such, researchers continue to make interpretations of key epistemological issues related to scheme theory (e.g., Steffe & Thompson, 2000a; Thompson, 2008), including interpretations of the term scheme itself (e.g., Steffe, 2010a; Thompson, 2013; Thompson et al., 2014). We make this point up front to highlight that Inhelder and de Caprona’s (1992, p.  41) quote is relevant to this chapter in two ways. The first is that what we are presenting is an interpretation of schemes and scheme theory situated within a radical constructivist epistemology. The second is that there is continued space for discussion and development of scheme theory in mathematics education research. Thus, we hope this chapter both informs readers about how schemes and scheme theory have evolved and inspires them to contribute to the continuing discussion and development of these constructs—constructs that we see as still possessing significant power for studying mathematical learning.

 rief Overview of Glasersfeld’s Radical B Constructivist Epistemology Throughout Piaget’s work, he insisted that knowledge always requires an active agent: To know an object, to know an event, is not simply to look at it and make a mental copy or image of it. To know an object is to act on it. To know is to modify, to transform the object, to understand the process of this transformation, and as a consequence, to understand the way the object is constructed. (Piaget, 1964, p. 176)

This orientation to knowledge broke with some of the foundational relationships between knowledge and reality as expressed in Western philosophy (Glasersfeld,

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1974). That is, Western philosophers often position knowledge as true when it represents “a state or feature of an experiencer-independent world” (Glasersfeld, 1990, p. 127) where the “experiencer-independent world” is considered to be ontological reality. Thus, in Western philosophy, truth is a central way of relating knowledge to reality—something is true if it corresponds to a reality that is independent of a knower (Glasersfeld, 1983). Piaget’s (1970) genetic epistemology breaks with this relationship precisely because an active agent is required in the production of all knowledge. Put another way, a person only has access to their own ways of perceiving and conceiving and thus has no way to step outside of themselves to determine whether their knowledge matches an ontological reality (Piaget, 1970, p.  15; Glasersfeld, 1983). For this reason, Glasersfeld (1990) proposed speaking of a person’s experiential reality rather than speaking about a person coming to know an ontological reality, where a person’s experiential reality is the reality they construct in the context of interacting in their physical and social world. These considerations led Glasersfeld (1980a, 1990) to replace the notion of truth as the fundamental relation between knowledge and reality with the notion of viability—a person’s knowledge is viable within their experiential reality. Within the physical world, viability means that a person is able to survive environmental constraints, a link to biology. Within the conceptual world, viability means that a person’s knowledge is noncontradictory, where noncontradictory means that a person can attain a state of equilibrium, meaning a person is not in a constant state of perturbation (Glasersfeld, 1981a). We point out that this definition of noncontradictory is a statement from an actor’s perspective, not an observer’s perspective; it is a statement of what noncontradictory means for an actor, not a statement about whether or not an observer deems another person’s knowledge to be noncontradictory. Philosophically, we see two important questions that arise from the shift to an experiential reality: (1) Is this shift a denial of the existence of any form of what people call ontological reality? and (2) Why is it common for people to act as if there is an ontological reality? To respond to the first question, Glasersfeld (1984, 1995) is clear that shifting from an ontological to an experiential reality is not a denial of an ontological reality, simply a statement that a person cannot come to know ontological reality. Moreover, this position is not a statement that a person’s constructions can be made up freely. Instead, it is a statement that a person experiences constraints within their experiential reality of both the physical and conceptual variety; it is not possible for me to walk through my desk (physical constraint), and I cannot call my concept of one-fourth both one-fourth and one-fifth and maintain a state of equilibrium (conceptual constraint). In this sense, our experiential reality limits what is possible, but it does not determine or cause how a person responds to these constraints (Glasersfeld, 1983, p. 4).1  We note that conceptual constraints are always relative to a person’s current schemes. Therefore, we are not suggesting here that, for example, elementary grade students would necessarily experience a perturbation (i.e., a loss of equilibrium) when talking about fractions like thirds, fourths, or fifths using the language “halving it up.” 1

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A response to the second question can be found in Elkind’s introduction to Six Psychological Studies when he says: Once a concept is constructed, it is immediately externalized so that it appears to the subject as a perceptually given property of the object and independent of the subject’s own mental activity. (Elkind in Piaget, 1967, p. xii, italicized in the original)

That is, for Piaget, knowing entails an active agent introducing actions or operations into their experiential world and then subsequently, once a concept is constructed, treating these actions or operations as if they are external to and independent of the acting agent. Once the process of externalization takes place, a person’s sense of the “existence of their concepts in an independent world” is strengthened in two ways. First, when a person’s constructions prove to be viable in future experience, a person’s sense of “their existence” is strengthened. As Thompson (2013, p. 61) puts it, “we construct stable understandings by repeatedly constructing them anew.” Second, when a person attributes to other people compatible constructions, “their existence” is also strengthened. The process of two or more people attributing to each other compatible constructions is what Glasersfeld (1995) calls establishing intersubjectivity, which he identifies as the highest form of reality. We start with these epistemological considerations because schemes are one of the primary tools that Piaget used to account for the construction of what Glasersfeld calls a person’s experiential reality. This constructive process has a definite beginning—at birth. Therefore, it is not surprising that Piaget developed schemes from his adjustment of reflexes, which evolutionary biologists consider to be present at birth. We discuss Piaget’s adjustment of reflexes to create schemes next.

 iaget’s Development and Glasersfeld’s Three-Part Definition P of Schemes Schemes from Reflexes Glasersfeld (1982, 1993) identifies that evolutionary biologists typically conceive of reflexes as involving a “stimulus-response” mechanism where the response is a “fixed action” (Fig. 3.1, Glasersfeld, 1995, p. 64). An example of a reflex in humans is the rooting reflex, where the “stimulus” is a baby’s experience of a brushing sensation on his or her cheek, and the baby’s response is to turn its head and root. Evolutionary biologists consider reflexes to be due to random genetic variation, where a reflex provides a critical advantage to members of a species who possess

Fig. 3.1  Glasersfeld’s depiction of a reflex

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them because they increase the likelihood of survival. The advantage that the rooting reflex offers is a higher likelihood of providing nutrition to a baby. Piaget took the idea of a reflex as a starting point for creating schemes. He conceptualized schemes as having three parts rather than two (Glasersfeld, 1993): a perceived or conceived situation, an activity, and a result (Fig.  3.2, Glasersfeld, 1995, p. 65). To make sense of Piaget’s replacement of a stimulus with a perceived or conceived situation involves understanding his intent. He intended to make explicit that stimuli neither exist in an observer-independent environment nor do they carry information (Glasersfeld, 1980b). Rather an active agent has to select and coordinate sensory signals or reproduce them in their absence to establish a perceived or conceived situation. Piaget also adjusted the notion of a “fixed action” in a reflex, removing the notion that a response is fixed (Glasersfeld, 1993). For example, as methods of feeding a baby change so too does the baby’s response to having its cheek brushed. Removing the fixedness between stimulus and response removes the idea that a stimulus and response are in one-to-one correspondence with one another. That is, multiple perceived or conceived situations may activate the same scheme (Thompson, 2013). The final adjustment Piaget made explicit in creating schemes was the inclusion of a result, which is a component that is not typically explicitly identified as part of a reflex. In the case of the rooting reflex, the result would be that the baby’s hunger is sated. Adding the notion of a result to reflexes is important to scheme theory because it introduces the idea that a person can develop an expected result. An expected result is an anticipation of what the result of a scheme will be based on prior experience. Thus, once a baby has gained sufficient experience, the rooting reflex can be considered a sucking scheme (sufficient experience for Piaget included the first 6 weeks of life); a baby may perceive a situation (e.g., brushing against its cheek) as an occasion for the activity of rooting and come to expect that this activity will satisfy its sensation of hunger. To sum up, Glasersfeld proposes that Piaget made these adjustments in creating schemes from reflexes and proposes a three-part definition of Piaget’s term scheme, a scheme entails a perceived or conceived situation, an activity, and a result (shown in Fig. 3.2).

Sensory Motor and Conceptual Schemes For researchers to effectively use Glasersfeld’s three-part definition of scheme involves understanding more about two issues in Piaget’s framing of the construct: (1) the genesis of mental operations and their role in schemes, and (2) the distinction Fig. 3.2  Von Glasersfeld’s depiction of a scheme

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between figurative and operative. To discuss the first issue, we note that Piaget (1971) used schemes as an explanatory construct for knowing and learning in both the sensory motor and conceptual domains (i.e., both the physical world and world of ideas). As he states: All knowledge is tied to action and knowing an object or an event is to use it by assimilating it to an action scheme … This is true on the most elementary sensory-motor level and all the way up to the highest logical-mathematical operations. (Piaget, 1971, pp. 14–15, 17)

For Piaget, schemes in the sensory motor domain involve the coordination of sensory motor actions—e.g., walking and opening a jar—and within the conceptual domain the coordination of mental operations—e.g., partitioning a linear whole into five equal parts (Glasersfeld, 1995). Piaget (1964) considered these two domains to be linked in the following way: An operation is an interiorized action. But, in addition, it is a reversible action; that is, it can take place in both directions, for instance, adding or subtracting, joining or separating. (p. 176)

For Piaget, the genesis of mental operations was sensory motor actions (i.e., kinesthetic activity, composition of sensory motor data in perception). So, for example, a partitioning operation has its origins in sensory motor situations that involve sharing a continuous whole, like breaking a cake into two parts or marking a line segment into four parts (Piaget et al., 1960; Steffe, 2010b).2 These sensory-motor actions get internalized, which means a child can carry them out in visualized imagination without carrying out any kinesthetic actions on perceptually available material like a cake or a line segment (Norton et al., 2018). However, internalized actions are not yet reversible, in that a child cannot move backward and forward through an experience that they carry out in visualized imagination (Steffe, personal communication, 12/08/21). So, for example, a child cannot move from a partitioned cake back to a whole cake. Once an internalized action becomes reversible, it is an interiorized operation, which means, for example, that a child can in visualized imagination move back and forth from whole cake to partitioned cake, and back to whole cake again. This account of the genesis of operations from sensory motor actions that get internalized and then interiorized is a reasonable starting point for understanding Piaget’s perspective on mental operations. However, there is a missing subtlety that is not addressed. To address this subtlety, we consider Steffe et al.’s (1983) work on counting that is rooted in Glasersfeld’s (1981b) account of the genesis of the unitizing operation. Glasersfeld proposes that the origins of the unitizing operation are in a person’s focused and unfocused moments of attention. This account differs subtly from an account of operations as internalized and interiorized sensory motor actions in that sensory motor actions are often also characterized as visible to an

 Steffe (2010a, b, Chap. 4) offers a much more nuanced account of the origins of partitioning that include five levels of fragmenting where at the fourth level of fragmenting there is the potential for a student to engage in what he calls equipartitioning. 2

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observer. With such a characterization of sensory motor actions, the distinction between making an account of visible behavior and making inferences about operations can easily be conflated. Glasersfeld’s account of the genesis of the unitizing operation based on focused and unfocused moments of attention highlights that visible behaviors are only a marker for making inferences about even the most basic operations; that is, moments of attention are not visible to an observer, and must be introduced by an active agent. These observations are critical to making sense of Piaget’s notion of an operation, and they are not captured solely from the first account we gave of operations as interiorized sensory motor actions. Another subtle difference that Steffe (personal communication, 12/08/21) made in transitioning these constructs to mathematics education research is that he relaxed the criteria that operations necessarily had to be reversible. For our work, we have followed Steffe’s definition of operations and we have construed Piaget’s (1964) criteria that—“Above all, an operation is never isolated. It is always linked to other operations, and as a result, it is always a part of a total structure (p. 176)”—as a way to relate operations to conceptual schemes. We see operations as the building blocks for conceptual schemes where operations make up the activity of a conceptual scheme.3 A significant part of the work, then, of using Glasersfeld’s three-part definition of scheme is to specify the operations that constitute it. The second issue, the distinction between figurative and operative, is also an important distinction for characterizing students’ conceptual schemes. We begin our discussion of this distinction with Glasersfeld’s (1995) quote about Piaget’s use of these terms: Throughout Piaget’s work, the distinction he makes between “figurative” and “operative”, and the concomitant distinction between (physical) “acting” and mental “operating” are indispensable for an understanding of his theoretical position. “Figurative” refers to the domain of sensation and includes sensations generated by motion (kinaesthesia), by metabolism of the organism, and the composition of specific sensory data in perception. “Acting” refers to actions on that sensorimotor level, and it is observable because it involves sensory objects and physical motion. Any abstraction of patterns composed of specific sensory and/ or motor signals is what Piaget calls “empirical.”…In contrast, any result of conceptual construction that does not depend on specific sensory material but is determined by what the subject does, is “operative” in Piaget’s terminology. “Operations,” therefore, are always operations of the mind and, as such, not observable. Whatever results reflection upon these mental processes produces are then called “reflective abstraction.” (p. 69)

As Glasersfeld frames it here, Piaget’s distinction between figurative and operative focuses on the difference between physical acting and mental operating, and in this framing the figurative domain is tied to actions in the sensory motor domain. So, if we return to our example of partitioning a cake, when a young child, 3 years of age, say, cuts a perceptually present cake into parts, that would fall into Piaget’s figurative domain if the child is constrained to carrying this activity out on a perceptually present cake. However, this seemingly sharp distinction can easily get

 Piaget does not discuss what he means by “total structure” in the immediate context of this quotation. He could mean either a scheme or a grouping structure. We have chosen to interpret his meaning to be a scheme. 3

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muddied: how should a researcher characterize a sixth-grade student who seemingly needs to carry out partitioning activity on a fraction bar to determine the result of multiplying a fraction by a fraction? Here a sixth-grade student is not, generally, constrained to actions in the sensory motor domain. To sharpen our distinction, we return to our more basic example of a 3-year-old; in this example, it is possible to differentiate between the cake and the act of cutting the cake. In Piagetian terms, the cake is an object concept, and the act of partitioning the cake is either a sensory motor or internalized action. This distinction between the material that gets operated on (i.e., the cake or an object concept) and the actions or operations that get performed on it (i.e., the partitioning of the cake) helps to sharpen the distinction between figurative and operative. The material that gets operated on is figurative and the actions or operations performed on it can be considered operative (Glasersfeld, 1991). This distinction is the basis for the way Thompson (1985) generalized Piaget’s distinction between figurative and operative. Thompson proposed that the differentiation between figurative and operative could be seen as a distinction between a controlling scheme and a subordinate scheme where both schemes had the potential to be operative in the sense that both could involve mental operations. With Thompson’s generalization, the distinction, then, between figurative and operative is about a relationship between schemes; a current operative scheme needs material to operate on, and this material is a previously constructed scheme, which serves as figurative material for the operative scheme. This observation means that figurative material is always relative to the current cycle of operations or what is currently operative for a person. This generalization makes the distinction between figurative and operative useful even for studying students’ schemes related to advanced mathematical concepts (Chap. 4; Thompson, 1985).

 mpirical Example of Glasersfeld’s Three-Part Definition E of Scheme: Michael Solves the Outfits Problem With the above distinctions in mind, we now use an empirical example to illustrate Glasersfeld’s three-part definition of scheme. To do so, we use Michael’s solution of the Outfits Problem. Outfits Problem. You have 3 shirts and 4 pants. An outfit is 1 shirt and 1 pants. How many outfits can you make?

Michael was an eighth grader who participated in a 3-year teaching experiment. He solved the Outfits Problem at the beginning of his third year in the project. We use this example for two reasons. First, it is an example of a student’s solution to a

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problem that is not particularly complex, especially for an eighth-grade student, which allows us to make a clear exposition of the mental operations that we have proposed were involved in Michael’s scheme. Second, Michael had developed complex multiplicative reasoning with whole numbers and fractions during the first 2 years of the teaching experiment (Hackenberg, 2010). However, he did not initially solve the Outfits Problem in the way that we anticipated (i.e., the situation did not initially activate his powerful multiplicative reasoning with whole numbers). Therefore, his solution to the problem was an occasion for us to consider that he used novel mental operations. With this context, we now present Michael’s solution to the problem. Data Excerpt 1: Illustrating a Conceptual Scheme M: Okay, you said four pairs of pants. [Begins to draw a picture of one pair of pants and one shirt and writes “x 4” next to the pants and “x 3” next to the shirts.] You could take one (shirt and one pants) and make one outfit. Take another one (shirt and pants to make a second outfit) and take another one (shirt and pants to make a third outfit) and then you have one pants (leftover). Wait do we do every possibility? T: Every possibility you could possibly get. M: [draws three tally marks followed by four tally marks, making two rows of tally marks (Fig. 3.3).] You could do that, that, that [connects each tally mark in the first row with the first tally mark in the second row.] You could do a lot of possibilities of this one. [M draws similar connecting lines between the rows of tally marks that end up in a jumble.] There is three (outfits) for every pants so basically twelve.

We interpret Michael’s initial response of three outfits with one pair of pants leftover (Lines 1–5) as part of his establishing a situation. We note that from the researcher’s (and likely the reader’s) perspective, the statement of the problem is sufficient to establish a well-defined situation. However, that is often not the case for the learner—the learner may make initial interpretations of a situation that get revised as part of establishing, for themselves, a definite problem situation. Michael’s question at the end of Line 5 indicates that he established the Outfits Problem as a situation about creating possibilities, which contrasted with his initial interpretation, where he considered himself constrained to using a shirt or pants only once. The activity of Michael’s scheme involved two mental operations, ordering and pairing, that he carried out on two composite units,4 four and three, where the composite units were the figurative material for his scheme (Lines 7–11). Elsewhere we Fig. 3.3  Michael’s tally marks

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have defined an ordering operation as creating a first, second, third, and so on unit in a composite unit, and pairing as uniting a unit from each of two composite units to create a single unit, a pair, that contains two units (Tillema, 2013, 2014). To translate the abstract language of operations back into context, he created a first, a second, and a third shirt; created a first, a second, a third, and fourth pants; and paired a shirt with a pants to create an outfit (i.e., a pair) 12 times.5 He organized creating pairs in a lexicographic order, fixing the first pants and pairing it with the first, second, and third shirt before creating any outfits with the second pants where he again repeated pairing it with the first, second, and third shirt (see English, 1991, odometer method). Michael did not anticipate a definite number of possibilities before he solved the problem. Partway through his solution, he had an expectation that there would be “a lot of possibilities (Line 9),” but even after he stated this expectation, it did not seem to be accompanied by him knowing a definite number of possibilities. When he finished operating on the tally marks, we interpret his statement, “There is three for every pants so basically twelve (Line 11),” as him interpreting the result of his scheme in terms of his composite units being iterable units (i.e., I can make three with the first pants and so could do that with every pants), and his scheme closed. However, we claim that the four threes he produced in this situation were a different kind of composite unit; he produced a composite unit of pairs, three pairs, as opposed to a composite unit of ones, three ones. We have used this distinction as a way to make claims about the multiplicative relationships students establish between one- and two-dimensional discrete units where a pair is a discrete two-dimensional unit. Later in this chapter, we discuss these differences in more detail to differentiate among students at different stages of units coordination. Here, we use this data excerpt to focus on moving from a data excerpt to making an account of the data excerpt using the three-part definition of schemes. We call Michael’s scheme a multiplicative pairing scheme (MPS), where he established a situation, engaged in mental operations on figurative material, and produced a result. He established the situation using two composite units where the units of these composite units were iterable units of one. He operated on these two composite units, using ordering and pairing operations, and interpreted the result he produced in terms of his iterable composite units (i.e., each three I made is the same). We have claimed there were differences between his iterable composite units and the result he produced in this situation, a composite unit of pairs. However, Michael

 Michael’s units of one and composite units were both iterable. Steffe (2010b) considers a unit of one to be iterable when a student can take a number word like three as implying three iterations of a unit of one without having to actually make these iterations. He considers composite units to be iterable when a student can take one composite unit to stand in for any number of iterations of the composite unit. 5  From our perspective, the contextualized language helps make some sense of the language of operations, but it does not help identify key differences among students’ reasoning because the same contextualized language can be used to describe students who establish qualitatively different multiplicative relationships in these situations. 4

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did not make any differentiation in this situation, and so his scheme closed. We note that Thompson’s (1985) reformulation of figurative material is essential in making sense of our interpretation of Michael, in that Michael’s composite units, which we have called figurative material for Michael, were the result of a prior conceptual scheme (i.e., a scheme based on mental operations) that made the composite units available to him as material to operate on in this situation. Michael’s data brings up one last interesting question: When should a researcher claim that a student has constructed a scheme? Piaget (1936/1977), as quoted in Thompson et al. (2014, p. 10), characterizes a scheme as “organized totalities whose internal elements are mutually implied.” Based on this characterization, we would claim that Michael was in the process of constructing a new scheme rather than that he had already constructed one (Thompson et  al.’s, 2014, understanding in the moment versus stable understanding, pp. 12–14). This characterization is aligned with our claim that Michael produced a new type of unit in this situation, a composite unit of pairs, but that he himself did not differentiate a composite unit of pairs from a composite unit of ones. Therefore, this data excerpt represents a student whose scheme was under construction. In such cases, there are two important questions for a researcher to consider: (1) is it necessary to define a new scheme? and (2) if so, what relationship is there between a new and prior scheme? One potential relationship is that a new scheme supersedes a prior scheme, in which case a person can use the new scheme to solve all the situations that they used the prior scheme to solve, and they can also use it to solve additional situations. In these cases, the new scheme, in essence, replaces the old scheme. In our example, we did not see the new scheme as superseding Michael’s old scheme for whole number multiplication because this new scheme involved him in establishing a novel relationship between one- and two-dimensional discrete units. Therefore, we chose to define a new scheme, an MPS, that was related to his scheme for whole number multiplicative reasoning but distinct from it.

 uilding on Glasersfeld’s Definition of Schemes: Steffe’s B Tetrahedral Model Nuances in Steffe’s Definition of a Scheme Steffe (2010a) presents a tetrahedral model for schemes that builds on Glasersfeld’s three-part definition in order to make several additional features of schemes explicit. One aspect the tetrahedral model makes explicit is that schemes function in relation to a goal or goals. He places goals at the apex of a tetrahedron because a person’s goals guide all three parts of a scheme (Fig. 3.4a); this relationship between a person’s goals and their scheme is portrayed by the bi-directional arrows between the goals and the three parts of a scheme, perceived or conceived situation, activity, and result, where the three parts of a scheme are represented on the vertices of a tetrahedron (Fig.  3.4b). Throughout his work, Steffe has emphasized the importance of

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Fig. 3.4  Steffe’s tetrahedral model of a scheme (a, top left; b, top right; c, bottom)

internal goals rather than external stimuli in guiding schemes. He uses the sucking scheme to illustrate this issue where he emphasizes that upon experiencing a sensation of hunger a baby may establish a goal to satiate hunger (Steffe, 2010a, pp. 21–22). His discussion highlights that an internal goal, the goal to satiate hunger, not external stimuli, brushing the cheek, is what guides even very basic schemes like the sucking scheme. This example indicates the depth of re-formulation that took place in developing schemes from reflexes.6 Steffe (2010a) also includes the bi-directional arrows from the goal to the parts of a scheme (Fig. 3.4b) to make explicit that any one of the three parts of a scheme coupled with a goal can activate the other parts of a scheme. The potential for any one of the parts of a scheme to activate the other parts is represented by the bi-­ directional arrows between the three parts of the scheme itself (Fig.  3.4c). This aspect of Steffe’s model highlights that schemes are not linear, i.e., they do not necessarily flow from perceived situation to activity to expected result (compare Fig. 3.4c with Fig. 3.2). Rather there is a dynamic relationship among the parts of a scheme where any one of the parts can activate the others. We give a hypothetical example of the nonlinearity of a scheme based on extending the example we started with Michael. It is possible to imagine Michael getting dressed in the morning and creating an outfit by pairing a shirt with pants that he has in his dresser. This activity might support him in establishing a goal of determining the total number of possible outfits he could create from the shirts and pants in his

 We warn the reader against thinking of Fig. 3.4a or b as indicating that a person’s goals are a part of the scheme itself. Rather once a person establishes a goal or goals, the goal or goals mediate all three parts of the scheme. 6

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dresser. This hypothetical example, then, illustrates how the activity of pairing a shirt with a pants and his goal to determine the total number of possibilities might activate his MPS. Once activated, it could be an occasion for him to establish a situation, similar to the Outfits Problem, based on the number of shirts and pants he has in his dresser and to determine the total number of possible outfits he could make, the result of his scheme. Thus, Michael could use an activity and goal to establish a situation and a result. The main point is that any of the three parts of a scheme coupled with a goal can activate the other parts; a scheme does not necessarily proceed in a linear fashion.A final aspect of Steffe’s tetrahedral model is that it explicitly portrays a person’s expectation, which Steffe shows with bi-directional arrows between a conceived situation and result (see Fig. 3.4c and compare to Fig. 3.2). The arrow in Steffe’s model that points from establishing a situation to a result is intended to convey that as part of establishing a situation a person may also establish an expectation of what the result of their scheme will be. The arrow from the result back to establishing a situation is intended to convey that once a person produces an actual result, they may compare the actual result to their expected result (cf. Glasersfeld, 1980b, p. 75). We would add that a person can develop expectations for any of the three parts of a scheme, as well as how these parts are related to each other.

 mpirical Example of Nuances in Steffe’s Definition of Scheme: E Nico’s Reversible Scheme This observation of the dynamic relatedness of the parts of a scheme is of particular interest to mathematics education researchers when a person can start from the result of a scheme and return to the situation that produced it. In these instances, we consider a scheme to be reversible (Hackenberg, 2010; Inhelder & Piaget, 1958). The reason this kind of reasoning is of particular interest to mathematics educators is because reversibility is the basis for establishing important relationships, like subtraction as the inverse of addition, division as the inverse of multiplication, or square rooting as the inverse of squaring. We illustrate the reversibility of a scheme, using Steffe’s tetrahedral model, where Nico, a seventh-grade student, starts from a result and a goal to establish a situation and activity that produced that result. Nico was solving the following version of the Flag Problem, where he agreed to respond to this single question outside of the context of working on any other related problems. The interviewer had the following interaction with Nico. Flag Problem. You create a two-striped flag using one color in the top stripe and one color in the bottom stripe. The color in the top and bottom stripe can be the same (e.g., red-red flag is allowed), and you count as different, for example, a red-orange and an orange-red flag. How many colors would you need to make 225 two-striped flags?

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Fig. 3.5  A two striped flag and Nico’s notation for the Flag problem (a) Left and (b) right

Data Excerpt 2: Illustrating Reversibility in Steffe’s Tetrahedral Model of Schemes T [shows Nico a flag that has two horizontal stripes (Fig. 3.5a).]: If I told you that there were enough colors to make 225 (two-striped) flags, can you tell me how many colors there were to start out with? Where a flag would be like red and blue or yellow and green or red and green. N: Let’s see…ah, no [indicating initial uncertainty]. [He continues thinking] Um, 15. T: Okay. Do you want to say why? [Nico says he took the square root of 225. He has some difficulty articulating why he took the square root of 225; he says that he does not want to write down all 225 possible flags. To avoid doing so, he initiates showing the interviewer a smaller case of the problem. He proposes a case where there are 9 two-striped flags and makes a list to show that 3 colors would produce 9 flags (Fig. 3.5b). He then explains why 15 colors would produce 225 flags.] N: So you have 15 colors, and then what you could do is like match, if you had like, red, and then you had just a bunch of other colors you could match red with all of the other colors. [He momentarily wanders in his explanation and then continues] …. So you match red with all the colors, then you could move on to blue and you would match that with all the colors. And then move onto the next one and match that with all the colors. And eventually you’d get 225.

We interpret this data excerpt as Nico starting from what he considered to be a result of his MPS (225 possible flags) coupled with a goal of determining how many colors he would need to produce that number of flags. Doing so activated his expectation that taking the square root would produce the correct number of possible colors (Lines 6–7); he expected that 225 was produced from a situation where he would square some number of possible colors.7 He experienced difficulty explaining what activity he would engage in to connect a situation involving 15 colors to the 225 possible flags, where his difficulty largely seemed to be about his not wanting to write down all 225 possible flags (Lines 7–8). To address his difficulty, he created an analogous situation that involved nine flags where he could more easily instantiate the activity of his scheme. He ordered the three colors and paired the first

 We largely use contextualized language for this example because the main purpose is to show the dynamic relatedness of the parts of a scheme. In the language of unit structures, we consider Nico’s expectation to have arisen from his establishing a multiplicative relationship among two composite units and a unit of units of pairs. We discuss this multiplicative relationship later in this chapter. For this particular situation, the numerosity of the two composite units was unknown, and their numerosity had to be equal. 7

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color with each of the other colors (red with red; red with orange; red with yellow), continuing in this way until he had produced nine ordered pairs (Lines 8–11). Once he had done so, he was able to explain what activity he would use to move from 15 colors (an initial situation) to 225 flags (a result) without producing all 225 flags (Lines 12–18). This example, like the hypothetical example with Michael, shows how Steffe’s formulation of schemes posits a dynamic relatedness among the parts of a scheme; in this case, Nico used a result and goal which activated the other parts of his scheme, a situation and activity. A careful reader might object to this characterization in that Nico clearly must have established a situation involving 225 flags (i.e., he started from a situation he established). The point, though, is that he considered that situation to be the result of his MPS. It is in this way that he used a result to regenerate a situation and activity. This observation indicates the importance of seeing schemes as relative to one’s current model of a learner and situated within a broader goal of making meaningful distinctions in qualities of student reasoning. In this case, one quality of student reasoning that is important is that Nico’s scheme was reversible. When a scheme is reversible, the result and the initial situation imply one another (two composite units of 15 implies 225 ordered pairs, and 225 ordered pairs implies two composite units of 15). For Nico, this implication enabled him to link taking a square root and squaring a number.

Using Schemes to Investigate Learning Assimilation, Perturbation, and Accommodation One difference between Nico’s data excerpt (Data Excerpt 2) and Michael’s data excerpt (Data Excerpt 1) is that we do not think Nico’s data excerpt was a situation of learning, whereas we do think Michael’s data excerpt had the possibility to be a situation of learning. We consider the potential for learning to occur when a person experiences a perturbation. Understanding perturbations requires an understanding of Piaget’s constructs of assimilation and accommodation. Piaget (1971, pp. 4–6) considered assimilation to be the process that an organism uses to integrate some material (whether that be physical or conceptual) into its structures. He gives many biological examples at the beginning of Biology and Cognition of assimilation where, for example, a plant assimilates light energy into its structures, and in the process of integration, transforms the light energy into chemical energy. Piaget emphasized that assimilation always involves a transformation. We find Glasersfeld’s (1995) characterization that assimilation involves a person treating a current experience (current problem situation) as an instance of a prior experience (prior problem situation) to capture the transformative component of assimilation, but perhaps in an unexpected way. That is, a person transforms a current experience vis-à-vis assimilation into one that is already known. An observer

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may see differences between a current and prior experience that the assimilating person (i.e., the actor) either ignores or simply does not consider. It is in this sense that Steffe and Thompson (2000a) have described assimilation as constitutive for an actor: an observer may be able to point to differences between current and past experiences that have activated a person’s scheme, but the person themselves may not see or does not act on these differences. Glasersfeld (1995) maintains that assimilation is active both when a person establishes a problem situation (first part of a scheme) and in comparing an expected result to an actual result (last part of a scheme). When a person assimilates their actual result to their expectation, their scheme closes—the scheme has functioned as intended, producing the expected result. On the other hand, a person may experience a perturbation when they establish a difference between what they expect—where their expectations have been developed in prior uses of their scheme—and a current instantiation of a scheme (Glasersfeld, 1980b, p. 75). Two common examples of when perturbations occur are: (1) when a current situation activates a scheme but does not meet the same conditions as prior situations that have activated it, or (2) when a result of a scheme differs from the result the person expects (Glasersfeld, 1995). We note that perturbations often occur outside of a person’s conscious awareness, and in such cases, “establishing a difference” may be entirely implicit for the person. In response to a perturbation, a person may review any of the three parts of their scheme and make changes to it. Following Steffe (1991), we call changes a person makes in their scheme, while the scheme is in use, functional accommodations, and we consider functional accommodations to be one kind of learning. We note that making an inference that a person has experienced a perturbation is not the same as making an inference that a person has made a functional accommodation. To infer that a person has made a functional accommodation requires evidence that the perturbation fed back into the scheme in a way that led to changes in one or more parts of the scheme. Moreover, we consider an inference of a functional accommodation to involve establishing that the changes were more or less permanent, similar to Thompson et al.’s (2014, p. 13) distinction between in the moment and stable. Here we use the terminology more or less permanent to mean that future situations activate a scheme in a way that indicates the person integrates the novel ways of operating into their future uses of the scheme. To illustrate assimilation, perturbation, and accommodation, we now present an extended empirical example.

 mpirical Example of Assimilation: Carlos’s Solution E of the Flag Problem To illustrate assimilation, we use a data excerpt of Carlos, an eighth-grade student, who worked with Michael as a partner. Carlos also participated in the first 2 years of a teaching experiment, and his eighth-grade year was his third year in the project. Carlos spent the first two teaching episodes of his eighth grade solving problems

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Fig. 3.6  Carlos’s list for the Flag Problem

like the Outfits Problem. Unlike Michael, we could attribute to Carlos his construction of an MPS from the start of the teaching episodes. He could represent these problems using lists, tree diagrams, and arrays and was able to anticipate correct multiplication problems for these situations. At the beginning of the third teaching episode, he solved a version of the Flag Problem that did not involve reversibility. Flag Problem. You have 15 colors with which to make a two-stripe flag. One color in the top stripe and one color in the bottom stripe. How many two stripe flags could you make?8

To set up the Flag Problem, the teacher-researcher (first author) asked Carlos and his partner to make some example flags. Together they made some flags with red in the top stripe and some flags with red in the bottom stripe, but neither made a flag with the same color in both stripes nor did they make two flags that had the colors switched (e.g., the red-blue and blue-red flag). Once the boys had made some example flags, the teacher-researcher asked them to make a list of all the possible flags,

 When I say problems “like the Outfits Problem” or problems “like the Flag Problem,” I mean that the statement of the Outfits Problem refers to two composite units (4 pants and 3 shirts) whereas the statement of the Flag Problem refers to only a single composite unit (15 colors). This differentiation is ours; a student may assimilate problems like the Flag Problem using two composite units, depending on the state of their 2-slot MPS. 8

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leaving open whether they should count flags that had the same color in each stripe, and whether they should count flags that had the colors switched (i.e., count as different the red-blue and blue-red flag). The following exchange between Carlos, Michael, and the teacher-researcher took place over an 11-minute period. Carlos’s response during this time illustrates well features of how assimilation functions. Data Excerpt 3: Assimilation as Constitutive C [in response to the Flag Problem writes “15 × 15” on his paper and uses his multiplication algorithm to find the result should be 275 (an error in carrying out the algorithm). He begins to write possible colors, associating a number with each. He writes out the first column of notation by writing “1,1”, “1,2”, etc. (Fig.  3.6). He writes out the second column the same as the first. When he gets to the third column, he writes, “3”, thirteen times, and then fills in the appropriate number next to each “3”. He writes the fourth and fifth column of notation in a similar manner to the third column. He is writing down one less flag in each column of Fig. 3.6 so the total number of flags in the first five columns is 15 + 14 + 13 + 12 + 11] [Six minutes have passed. Carlos is finishing his ninth column of notation] W: You guys are going to need more paper. C: I already have the answer. That is [points to where he has written “15 × 15”] how many pairs you could make. [Carlos has incorrectly identified that his list will contain “15 × 15” flags when he is done. Michael tells Carlos not to give away the answer. Both continue to make their list where Michael’s list is similar to Carlos’s list.] M [indicating that the number of flags in each column of his list is one less than the number in the previous column]: It just decreases by one every time. T [To Michael]: Yeah, it just decreases by one every time. C: [Carlos starts the tenth column writing “10” four times. He counts the number of flags he has symbolized in the ninth column, finds he has symbolized seven flags, and writes “10” twice more for a total of six times. He then writes “10”; “11”, etc. next to each “10” he has written to complete the tenth column. He repeats this process for the remaining columns.] M [finishes writing a similar list of notation]: See there is like fifteen and it decreases by one every time so it’s fifteen plus fourteen plus thirteen plus twelve until you get to one. T: Go ahead and write that down and see if you can figure out how many you got that way. [Carlos finishes his list of notation. 10 minutes have passed since he began. Carlos sits as if satisfied that he has solved the problem.] Carlos, I wonder if you could figure out how many pairs you got here [pointing to his list in Fig. 3.6]? C: Two hundred and seventy-five [points to where he has computed 15 × 15.] T: Why don’t you check that by figuring out what this [points to Fig. 3.6] is? C: Okay [writes out the sum 15 + 14 + 13 + … + 1 vertically.] T [To Michael & Carlos]: I wonder if you could figure out a quick way to add those up. C: Fifteen times fifteen.

We give an interpretation of this data excerpt and then relate it back to the idea that assimilation is constitutive. From this data excerpt, we infer that for Carlos, the Flag Problem activated his MPS just as the Outfits Problem had. We note that his response of 15 × 15 yields a correct total number of outcomes for the Flag Problem had he counted ordered pairs (e.g., counted the red-blue and the blue-red flag). However, as his solution went on, it became apparent that he was not intending to count ordered pairs—he did not include them in his list. Therefore, we do not interpret his multiplication statement as indicating his intent to count ordered pairs. Rather, we see his response as him treating the Flag Problem as if it were the same

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as the Outfits Problem, where, for him, the Outfits Problem did not involve ordered pairs and did involve multiplication. We infer he assumed the same to be the case for the Flag Problem. We think it is noteworthy that the way Carlos created his list indicated that he anticipated the number of flags he would write in each column before writing them; that is, he made the third column of his list by writing 13 “3s,” and then filling in the numbers 3 through 15 afterward, indicating that he anticipated he would make 13 flags before he actually made them. This way of creating his list is noteworthy because it shows that he anticipated he would write down one less flag each time he started a new column. It is not surprising, then, that he wrote a sum, 15 + 14 + 13 + … + 1, for the total number of flags in Fig. 3.6. However, he thought the sum would produce the same total number of flags as his multiplication problem, 15 × 15. We take that as indicating he did not have a way to compare the result he produced, 15 + 14 + 13 + … + 1, to his expected result, 15 × 15, except through a computation. Here, we do not make a full account of why that occurred or why we think it is important that it did occur (see Tillema, 2014). Instead, we use this data excerpt to illustrate that assimilation involves treating a new situation (i.e., the Flag Problem) as an instance of an already known situation (i.e., problems like the Outfits Problem). Moreover, doing so was constitutive for Carlos in that he gave no indication that the Flag Problem was in some way different from problems like the Outfits Problem. In fact, his response illustrates how robust a previously constructed scheme can be; he had multiple opportunities during the time he was making his list (an 11-minute period) to identify differences in his solution of the Flag Problem relative to his solution of problems like the Outfits Problem. Nonetheless, at the end of that time, he still asserted that his sum and the multiplication problem would produce the same number of flags. We are confident that Carlos knew that the sum was not the multiplication problem, so we infer his sense of their sameness came from assuming each would produce the same total number of flags. Doing so is another instance of assimilation; Carlos treated the actual result (the sum) as an instance of his expected result (the multiplication problem) without differentiating between them. We infer he made no differentiation between them because he did not, at that moment, have a way to make a comparison between the two, and without a way to compare them he simply considered them to be the same.

 mpirical Example of Perturbation: Carlos’s Solution E of the Flag Problem We continue with Carlos’s solution of the Flag Problem to illustrate a perturbation that arose from comparing his actual result (the sum) to his expected result (the multiplication problem). We note that when a person assimilates their actual result to their expected result, we would assert that their scheme closes; the scheme has functioned in the way that the person has expected and no differentiation is made between the current and prior uses of the scheme. In Carlos’s case, the

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teacher-­researcher wanted to provide a further opportunity for Carlos to differentiate his actual and expected result. So, he requested that Carlos continue his solution of the Flag Problem by adding up the numbers in the sum. The following interaction took place. Data Excerpt 4: Comparing an Actual and Expected Result C [evaluates his sum by adding 15 and 5, then adding 9 and 1, 8 and 2, 7 and 3, and 6 and 4. He then adds 14, 13, 12, 11, and 10 in his head. The teacher asks both boys what they got.] M: I got one hundred and ten possibilities. C: I got one hundred and thirty. T [Neither of the boys’ responses is correct. The total number of flags should be 120]: Uh-oh. We got to figure out some way to add those numbers up and actually check. [two and a half minutes have passed] W: What’s Carlos doing over here? Are you doing the same way (evaluating the sum in the same way as Michael)? C [with excitement]: No, I’m not subtracting. I just figured out something! That once you got that one [C points to the “15” in his sum], then you got that one and that one [Carlos points to “14” and “1”], then you make another fifteen, that one and that one [Carlos points to “13” and “2”] and make another fifteen. You keep on going all the way down until you get to seven and eight which (means) you have eight fifteens so you should have just timesed fifteen times eight and you would have got the answer [Carlos crosses out where he has written “15 × 15”.]

Carlos initially evaluated the sum and found that there were 130 flags. Neither 275 from the multiplication problem nor 130 flags from evaluating the sum was a correct calculation. However, we infer it allowed him to numerically compare his actual and expected result, and that in doing so he experienced a perturbation. Carlos never stated that he made a numerical comparison—we could only infer this comparison and his perturbation from what he did next. He related his actual result, the sum, to his expected result, the multiplication problem, by transforming the sum into 15s. We note that perturbations are sometimes construed as consciously conflictive, but in this case, the evidence that Carlos experienced a perturbation did not come from an explicit statement that he was experiencing a conflict. Rather the inference that he experienced a perturbation came from his expression that he “just figured out something,” indicating that he was searching for a way to make a comparison between his actual and expected result. Thus, in this case, it was the fact that Fig. 3.7  Carlos’s list for the Handshake Problem

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he was searching for a way to compare the two that led to making the inference that he had experienced a perturbation. We highlight that an important component of a researcher making an inference of a person experiencing a perturbation is the researcher’s current model of a student’s schemes. We make this statement because the less consciously conflictive or obviously different an actual result is from an expected result, the less likely a researcher is to identify opportunities to occasion perturbations for students. To highlight this point, we return to the example of Michael where we claimed that he produced a composite unit of pairs in solving the Outfits Problem, but that he assimilated this result using his composite units where his composite units were composite units of one. Doing so meant that he did not experience a perturbation in the Outfits Problem; he treated the actual result as if it were the same as his expectation of what the result would be. It is possible for a researcher himself or herself not to make this differentiation, in which case they might reduce the likelihood of responding in ways that could occasion perturbations for students. We consider the strength of the models a researcher makes, then, to be connected, though not causally (Steffe, 1996), to the learning opportunities a student might have.

 mpirical Example of Functional Accommodation: Carlos’s E Solution of the Handshake Problem Carlos’s solution of the Flag Problem does not provide evidence that his perturbation fed back into his MPS—it simply provides evidence that he found a way to compare his actual and expected result. Therefore, we look at the next teaching episode to illustrate data that we would take as sufficient evidence for inferring he had at least made temporary changes to his MPS, which we frame as necessary, but not sufficient evidence of a functional accommodation. Carlos began the next teaching episode solving the Handshake Problem. His solution illustrates that his perturbation in the Flag Problem did indeed feed back into his scheme, which is shown in the following data excerpt. Handshake Problem. There are 10 people in a room. Each person wants to shake every other person’s hand. How many handshakes will there be? Data Excerpt 5: Initial Signs of a Functional Accommodation C [begins by recording the handshakes shown in Fig. 3.7. Later he writes A = 1, B = 2, etc. to show that A and 1 represent the same person. In his list in Fig. 3.7, he is eliminating self-handshakes, for example, he has not recorded 1A, and also duplicate handshakes, for example, he has not recorded 2A because he had already recorded that handshake as 1B. Michael is also making a list for the handshakes.]

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We take this excerpt as sufficient evidence to claim that Carlos’s perturbation from the Flag Problem fed back into his scheme. That is, we interpret Carlos’s comment, “minus two actually,” as an indication that he was comparing the total number of possible handshakes that the second person could have had (10 possible handshakes) to the total number of handshakes that he counted the second person actually making (8 handshakes). This statement indicates that he was comparing how the activity of his scheme in the Handshake Problem differed from the activity he produced in solving problems like the Outfits Problem. In addition, he insisted that the Handshake and Flag Problems were different—in one, he counted flags with the same color stripes, and in the other, he did not count self-handshakes. We take this statement as further evidence that he was comparing his activity in this problem to his activity in the Flag Problem. These statements are sufficient evidence that the perturbation he experienced in the Flag Problem fed back into his MPS and that he had at least temporarily made a change in the activity of his scheme.9 We would require further evidence that this change was more or less permanent to make a claim that Carlos made a functional accommodation in his MPS. By permanent, we mean that should Carlos’s scheme get activated in future situations, he would, with relative ease, establish similarity between the future situations and the distinctions he made here. We also suggest that making a claim that a student has made a functional accommodation is different from characterizing what the functional accommodation was. To make an account of a functional accommodation requires a characterization of changes the person made to their scheme, which involves detailing how the operations in their scheme or coordination among schemes changed. As we discussed with Michael, it also involves determining whether the functional accommodation produces a new scheme; unlike in Michael’s case, here we do not assert that Carlos’s functional accommodation entailed the construction of a new scheme. Instead, we frame the initial signs of a functional accommodation as changes he made to his MPS that allowed him to compare differences among the situations that activated his scheme. We do not provide details here about what changes we propose occurred in the parts of his scheme; we simply state that it is important to detail such changes in making a claim of a functional accommodation (see Tillema, 2014, for details about the functional accommodation).

 Michael’s statement, “same problem as last time, minus one and keep going and going and going,” indicates he was comparing his anticipation of what the result of his scheme would be in this situation to the result he had produced in his solution of the Flag Problem, the sum. 9

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 ituating Investigations of Learning Within a Broader S Framework: Stages Glasersfeld’s Definition of Stage We find investigations of learning using scheme theory to be most powerful when they are situated within a broader framework for interpreting how students might progress within a given mathematical domain and what possibilities may be open to them across distinct mathematical domains. For us, the broader framework that has informed our work on combinatorial schemes builds from Steffe’s (1992, 1994, 2010b) work on number sequences that Hackenberg et al. (2016) have subsequently used to identify three distinct stages of units coordination: stage 1, stage 2, and stage 3 (Chap. 11; Ulrich, 2015, 2016).10 Glasersfeld and Kelley (1982) define a stage as: Designat[ing] a stretch of time that is characterized by a qualitative change that differentiates it from adjacent periods and constitutes one step in a progression…the concept of qualitative change and, consequently, the concept of stage, cannot be derived from a quantitative measurement but requires a binary judgment of presence versus absence. (Glasersfeld & Kelley, 1982, p. 155)

We interpret Glasersfeld’s definition of stage as indicating that students at different stages construct qualitatively distinct schemes. Moreover, we see a claim that two students are at different stages as a claim that the difference in stage is unlikely to change over the course of multiple interactions with a student.11 As Steffe (personal communication, 4/20/22) puts it, the role that experience plays in stage change is ambiguous where we interpret Steffe as meaning that the relationship between experience and stage change is unclear or inexact not that experience is unimportant in promoting such changes. Thus, a claim that two students are at different stages helps to provide a boundary for the schemes a researcher might expect a student to construct in a particular domain of reasoning. With this purpose in mind, we illustrate the relationship between stages and schemes by: (1) outlining qualitative differences in students’ MPS at each of Hackenberg et  al.’s (2016) stages of units coordination, and then (2) illustrating a qualitative difference in stage 2 and stage 3 students’ schemes for solving 3-D combinatorics problems.12

10  Hackenberg (2007, 2010) and colleagues (e.g., Hackenberg & Sevinc, 2022) have referred to these as multiplicative concepts rather than as stages in some research publications. However, she and Norton have both used stages as well (e.g., Hackenberg et  al., 2016; Norton et  al., 2015) because they fit Glasersfeld’s definition of stage. 11  This assertion is supported in the research literature in that there are only a few examples of researchers working in longitudinal studies reporting that a student has made a stage change (e.g., Steffe, 1991, 1994; Hackenberg et al., 2021). 12  Characterizing combinatorics problems as 3-D is our characterization of the problems.

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 ackenberg’s and Norton’s Stages of Multiplicative Reasoning H and a 2-slot MPS We return to the Outfits Problem to characterize qualitative differences in students’ MPS across the three stages of multiplicative reasoning. As our earlier analysis of Michael’s solution to the Outfits Problem indicated, ordering and pairing operations are central to students’ construction of an MPS. The way students use these two operations, along with their disembedding operation to produce distinct multiplicative relationships, are the primary qualitative differences in students’ MPS across students at different stages of units coordination. A disembedding operation allows a student to treat a part of a composite unit as independent from, but related to, the composite unit without destroying the composite unit (e.g., 2 can be treated as independent of 6 in the process of, for example, adding 6 and 8 by adding 2 to 8 to make 10 and then adding the remainder of 4). Stage 1 students’ MPS involves them in operating on two composite units13 by ordering the units of each composite unit, disembedding a single unit from each composite unit, and pairing the unit of each composite unit to create a unit that contains two units, but is counted as a single unit, a pair. Stage 1 students’ use of an ordering operation can support them in producing the pairs in a lexicographic order. However, stage 1 students have yet to construct a disembedding operation, so they need the support of perceptual material (e.g., a list or array) to treat the pairs they produce as independent from the units of one that they use to create pairs. We have identified that stage 1 students (Tillema, 2018) produce a multiplicative relationship among a unit of one, a unit of one, and a pair (Fig. 3.8) as the result of their scheme; it is not a relationship that they can take as a given prior to their activity. Two common consequences of establishing this relationship as the result of their scheme are that: (1) they have to create pairs before they can take them as countable and (2) they may not see points in an array as pairs unless they have created these pairs in immediate past experience. Establishing the multiplicative relationship shown in Fig. 3.8 involves establishing the basic multiplicative relationship between two discrete one-­ dimensional units and one discrete two-dimensional unit (one unit times one unit produces one pair). Stage 2 students differ from stage 1 students in that they can interiorize the multiplicative relationship between a unit of one, a unit of one, and a pair (Fig. 3.8). The interiorization of this multiplicative relationship means that stage 2 students no longer need to create a pair in immediate past experience to (1) take them as countable and (2) treat the points in an array as pairs. Similar to stage 1 students, stage 2 students use ordering, disembedding, and pairing operations in their solution of problems like the Outfits Problem and can learn to produce pairs in lexicographic  We note that stage 1 students have not yet constructed iterable units of one so their composite units are qualitatively distinct from stage 2 students. Stage 2 students have constructed iterable units of one, but have not constructed iterable composite units so their composite units are qualitatively distinct from stage 3 students whose composite units are iterable (Chap. 11; Steffe, 2010b). 13

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Fig. 3.8  A multiplicative relationship created by stage 1 students using their MPS

Fig. 3.9  A multiplicative relationship created by stage 2 students using their 2-slot MPS

order. However, they have constructed a disembedding operation, and so can disembed a unit of one from one composite unit and the entire composite unit from the other composite unit. In doing so, these students can produce a multiplicative relationship that stage 1 students do not—a multiplicative relationship between a unit of one, a unit of units, and a unit of pairs (Fig. 3.9). They produce this multiplicative relationship as the result of their scheme; it is not a relationship that they can take as a given prior to their activity. We have considered producing this multiplicative relationship as akin to establishing a row or column of a 2-D array multiplicatively where a student differentiates the one-dimensional units from the two-dimensional units that make up the row or column (e.g., one unit times a unit of four units produces a unit of four pairs) (Tillema, 2018). Stage 3 students differ from stage 2 students in that they can interiorize the multiplicative relationship in Fig. 3.9. The interiorization of this multiplicative relationship means that stage 3 students can take a row or column of an array as multiplicatively created in the absence of creating it in immediate past experience. Similar to stage 1 and stage 2 students, stage 3 students use ordering, disembedding, and pairing operations in their solution of problems like the Outfits Problem, and they can learn to produce pairs in lexicographic order. However, they use their disembedding operation differently than stage 2 students in that they can disembed the entire composite unit from each of the composite units. Doing so, coupled with the unit structure they establish on the pairs, means that they can also interiorize a more

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Fig. 3.10  Multiplicative relationship interiorized by stage 3 students using their 2-slot MPS

advanced multiplicative relationship between a unit of units, a units of units, and a unit of units of pairs (Fig. 3.10). We have considered establishing the multiplicative relationship shown in Fig. 3.10 as akin to establishing a 2-D array multiplicatively, where a student differentiates between the one- and two-dimensional units in a 2-D array (e.g., a unit of three units times a unit of four units produces a unit of four units of three pairs or a unit of three units of four pairs) (Tillema, 2013). We also refer to stage 3 students’ scheme as a 2-slot MPS (as opposed to just an MPS) because the interiorized multiplicative relationship involves two composite units. The qualitative differences in students’ MPSs at each stage involve differences in the way they use operations and in the multiplicative relationships that they have interiorized between one- and two-dimensional units. It is within the framework of these differences that we have investigated subsequent functional accommodations that students make to their MPSs—for example, the functional accommodations Michael and Carlos made, as well as those that lead to the construction of ordered pairs rather than just pairs (e.g., Tillema, 2021). Now that we have introduced stages, we discuss an important feature of schemes that Thompson (2013; Thompson et al., 2014) captures in his formulation that helps us to identify a qualitative difference in the functional accommodations stage 2 and stage 3 students establish in the context of solving 3-D combinatorics problems.

Recursion in Thompson’s Definition of Scheme Thompson et al. (2014) elaborate on Glasersfeld’s (1995) three-part definition and Steffe’s (2010a) tetrahedral model of schemes to “make [their] explanatory power more evident (p.  10).” He defines a scheme as “an organization of actions,

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Fig. 3.11  Mikayla’s representation for the Sandwich Problem (a, upper left; b, upper right; c, bottom)

operations, images, or schemes—which can have many entry points that trigger action—and anticipations of outcomes of the organization’s activity (p. 11).” One feature of Thompson’s definition of scheme that helps to make the explanatory power more evident is that his definition highlights the recursive nature of schemes: a current scheme can, and usually does, contain prior schemes and/or operations.14 Given the recursive nature of schemes, it is important to determine what a researcher considers to be a single scheme versus what is considered an instantiation of multiple schemes. To capture this difference, we have contrasted a sequential use of two or more schemes with the insertion of one scheme inside another scheme where this insertion creates a new single scheme from what was previously two distinct schemes (Tillema & Burch, 2022). By a sequential use of schemes, we mean that one scheme closes (i.e., the goal of the scheme is satisfied) before the second scheme opens and the two (or more) schemes remain distinct from one another. In contrast, an insertion of a scheme or operations into another scheme entails creating a new singular scheme composed of multiple prior schemes  or a new singular scheme that contains novel operations where we frame this insertion as a recursive insertion. We use these ideas to illustrate a qualitative difference that can occur in the schemes of  stage 2 and 3 students in their solution of 3-D combinatorics

 There are differences between Thompson’s, Steffe’s, and Von Glasersfeld’s interpretation of scheme. Thompson’s definition of scheme, however, explicitly captures the recursive nature of schemes, which we see as a powerful addition in the context of using either Von Glasersfeld’s three-part definition or Steffe’s tetrahedral model. 14

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problems: in our examples, the stage 2 student sequentially uses her MPS, while the stage 3 student recursively inserts the operations of his  2-slot MPS inside of the scheme. 

 mpirical Example of a Stage 2 Student Sequentially Using Her E MPS: Mikayla Solves the Sandwich Problem We illustrate a sequential use of schemes with data from Mikayla’s solution to the Sandwich Problem. Mikayla, a stage 2 student, was a preservice secondary teacher in her senior year of college. She and a partner participated as students in 12 60–90-minute teaching episodes whose goal was to use combinatorics problems as a launch point for producing algebraic identities (Tillema & Burch, 2020). During the first teaching episode, she solved the Outfits and Flag Problems (2-D combinatorics problems), and then was given the Sandwich Problem (3-D combinatorics problem). Her solution to the Sandwich Problem is as follows. Sandwich Problem. Subway has 4 kinds of bread, 3 kinds of cheeses, and 5 kinds of meats. A sandwich is 1 bread, 1 cheese, and 1 meat. How many possible sandwiches can Subway make? Data Excerpt 6: Mikayla’s Sequential Use of Her Scheme M [writes Fig. 3.11a, then writes Fig. 3.11b, and next writes Fig. 3.11c.] T: Why don’t you go ahead and just say what you did? M: So, I started with one combination between the two, first. I started with the bread and the cheese [referring to Fig. 3.11b]. So, I did what I did with the colors [reference to the Flag Problem]. I did bread in the x-column (first position) or the first part of the stem and leaf plot, and then the cheese in the other one (in the second position). And got three combinations for all of those (each bread), so a total of twelve combinations just for bread and cheese. And then I numbered each one of those combinations, b1c1 was one, b1c2 was two, and so on all the way up to b4c3 is twelve [referring to the numbers in the bread and cheese column in Fig. 3.11c]. And then I made another one the same way [referring to the fact that she considers Fig. 3.11c to be similar to Fig. 3.11b] using the numbers one through twelve to not write those all over again. (The numbers) represented the bread and cheese combinations, and then the possibility that they have for the meat combinations. So like b1c1, or choice one, could have then five meats next to it. And then I did that all the way down. So, there were twelve possible bread and cheese combinations times the, five, for each one of those for the meat combinations to get sixty total sandwiches.

We take Mikayla’s solution of the Sandwich Problem as indication that she created the ordered triples through a sequential use of her MPS. That is, she ordered the units of all three composite units (Fig. 3.11a), but then paired only the units of two of the three composite units to make 12 ordered pairs. We infer from her creation of Fig.  3.11b that her goal was to produce all of the ordered pairs (i.e.,

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bread–cheese combinations) and that her scheme closed once that goal was satisfied. We make the inference that her scheme closed because she made no attempt to incorporate the units of the third composite unit into her creation of ordered pairs until after she had completed making all of the ordered pairs. Stage 2 students have interiorized a relationship between a unit of one, a unit of one, and an ordered pair, which is what we infer enabled her to take the result of her MPS, the 12 ordered pairs, as input for a second use of this same scheme. We note that her language (“So like b1c1, or choice one”) indicated these retained their status as ordered pairs. She explicitly ordered these pairs with the numbers 1 through 12 (Fig. 3.11c) and paired each one with the units of the third composite unit to make 60 ordered triples. Using language back in the context of the problem, she ordered the bread–cheese combinations and paired each bread–cheese combination with the five meats to create a sandwich. We interpret Mikayla’s solution as involving a sequential use of her MPS. She used this scheme first to make ordered pairs and then a second time to make ordered triples, where we consider her to have interiorized the relationship between a unit of one, a unit of one, a unit of one, and an ordered triple. Moreover, in her sequential use of her MPS, we consider her to have established a multiplicative relationship between a unit of one, a unit of one, a unit of five units, and a unit of five ordered triples in activity (akin to multiplicatively producing a row or column in a 3-D array). We illustrate one consequence of Mikayla’s sequential use of her MPS from the ninth teaching episode when we presented her with the Passwords Problem. Passwords Problem. You have the letters A, B, C, and D.  How many four-­ character passwords can you make if the letters in a password cannot be repeated (i.e., do not count AABD) and order matters (i.e., ABCD and ABDC are different passwords)?

From our perspective, the Passwords Problem involves determining the number of permutations of four letters. At this point in the teaching episodes, Mikayla had solved a variety of different combinatorics problems, including other problems that we would consider permutation problems. Data Excerpt 7: Mikayla’s Initial Solution of the Passwords Problem TR: Do you know how many total, of these passwords, there would be? M: Sixteen? TR: Where are we getting sixteen? M: Because there is four (letters) and we have four choices.

We interpret Mikayla’s response as indication that the Password Problem activated her MPS, and that she interpreted the solution of the Password Problem as involving two composite units, four and four, the number of letters, and the length of the passwords, respectively. Because she was constrained to sequentially using her MPS, it precluded her from anticipating, for example, a four-factor product like

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4 × 3 × 2 × 1. She could determine such a product through the sequential use of her scheme, but she did not anticipate a multifactor product. Her lack of anticipation of multifactor products was one constraint that we experienced and attributed to her sequential use of her MPS.

 mpirical Example of a Stage 3 Student Recursively Inserting E Operations into a Scheme: Tyrone Solves the Card Problem We contrast Mikayla’s sequential use of her MPS with Tyrone’s recursive insertion of operations inside his 2-slot MPS. Tyrone, a stage 3 student, was a tenth grader who participated in an interview study that consisted of 2-hour-long interviews. During the first interview, he solved the Sandwich Problem like Mikayla, and then he solved the Card Problem. We focus on his solution to the Card Problem because, during his solution to this problem, he inserted his pairing and ordering operations into his 2-slot MPS, a recursive insertion of operations, which entailed an extension of his 2-slot MPS. Card Problem. You have the 2, 3, and King of Spades, a friend has the 2, 3, and King of Hearts, and another friend has the 2, 3, and King of Diamonds. A three-card hand consists of one card from each person’s hand (order does not matter). How many different three-card hands are possible to make?

Tyrone began his solution to the Card Problem by saying he thought there would be “eighteen hands” because he thought his “three spades” could be paired with “six combos.” We interpret this statement as evidence that his 2-slot MPS was activated, and that he combined the heart and diamond cards to get six non-spade cards that he anticipated he could pair with the three spade cards. At the interviewer’s request, he attempted to show the 18 three-card hands using actual cards. He did so by fixing the king of spades and simultaneously moving a heart and diamond card next to the king of spades, and then simultaneously moving the heart and diamond card away from the king of spades. He was unable to keep track of which three-card hands he had made until he fixed a card in both the first and second position, which was a novelty he introduced into his solution. This novelty is illustrated in the following excerpt. Data Excerpt 8: Tyrone Recursively Inserts Operations into His Scheme T [puts the king of diamonds next to the king of spades. He leaves those two cards next to each other, and places the two of hearts next to them to create one three-card hand.] One. [He moves only the two of hearts away from the king of spades and king of diamonds and places the three of hearts next to the king of spades and king of diamonds to create a second three-card hand.] Two.

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[He moves the two of hearts away from the king of spades and king of diamonds and moves the king of hearts next to the king of spades and king of diamonds to create a third three-­ card hand.] Three [He has made the first three three-card hands that appear in Fig. 3.12a. He has made them with actual cards and not written them down like in Fig. 3.12a]. [He leaves the king of spades out and removes the king of diamonds and king of hearts. He places the three of diamonds next to the king of spades and one at a time moves each of the heart cards next to the king of spades and three of diamonds to create the next three three-card hands shown in Fig. 3.12a, saying]. Four, five, six. [He leaves the king of spades out and removes the three of diamonds and three of hearts cards from next to it. He puts the two of diamonds next to the king of spades and one at a time moves each of the heart cards next to the king of spades and two of diamonds, the last three three-card hands shown in Fig. 3.12a. He says,] Seven, eight, nine. [He leaves the king of spades out and moves the two of diamonds and the king of hearts card away. He moves the three of hearts next to the king of spades and puts the two of spades next to those two cards.] Ten. [His tenth three-card hand includes two spade cards, which is not a three-card hand that should be counted.] Hold on, hold on. See I’m getting lost. Alright I gotta re-start. I got lost. I got ahead of myself. [Over the course of eight minutes he makes four distinct attempts to produce all three-card hands using actual cards. On his final attempt he is successful. The interviewer then asks him to create a list for the three-card hands. He produces Fig. 3.12b, which includes all of the three-card hands produced in a lexicographic order]

We contrast our interpretation of this excerpt to the one of Mikayla. Across all of Tyrone’s attempts, he worked with the units of all three composite units15 at the same time as he produced triples. That is, we infer that he paired the first unit of one composite unit with the first unit of the other composite unit, and then recursively inserted his pairing operation into his 2-slot MPS, pairing the first, second, and third units of the last composite unit with the pair he had created. Using contextualized language, he paired the king of spades with the king of diamonds to create a two-­ card hand and then paired this two-card hand with the two, three, and king of hearts (see the first three three-card hands in Fig. 3.12a). We make the inference that he used his pairing operation recursively based on his fixing the cards in the first and second position, and then cycling through all possible cards in the third position. One reason it took Tyrone multiple attempts before he finally produced all three-­ card hands was that he did not initially organize his production of triples using his ordering operation. This facet of his reasoning can be seen in Fig. 3.12a, where he did not use a consistent order for the heart cards; for the first three three-card hands, he ordered the hearts as 2, 3, K and for the second three three-card hands he ordered the hearts as K, 2, 3 (Fig. 3.12a, yellow circles). He also ordered the diamonds as K, 3, 2 (Fig. 3.12a, red circle), which was different from any of the orders he used for the heart cards (Fig. 3.12a, yellow circles). On his final attempt with the cards, and in his list (Fig. 3.12b), he ordered the cards in each suit as 2, 3, K (Fig. 3.12b, yellow circle, red circle, and blue circles). His list indicated that he eventually recursively used his ordering operation; he ordered the units of all three composite units before he began creating triples.16 This recursive insertion of his ordering and pairing operations into his 2-slot MPS is evidence of extending his 2-slot MPS through a recursive insertion of his ordering and pairing operations into his 2-slot MPS. 15

 Tyrone’s units of one and composite units were iterable units as described earlier.

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Fig. 3.12  Three-card hands Tyrone made on his first attempt (a) and in his final list (b)

Revisiting Theoretical Constructs Relative to the Data Examples The difference between Mikayla (sequential use of MPS) and Tyrone (recursive insertion of operations in the extension of his 2-slot MPS) is an instance of a qualitative difference in schemes that students at different stages of units coordination construct. Our claim, then, about Mikayla was that she was constrained to (Steffe & Thompson, 2000b) this sequential use of her scheme over the course of the 12 teaching episodes. The claim of being constrained to sequentially using her MPS means that the functional accommodations she made during the teaching episodes were relative to her sequential use of her MPS. In contrast, the functional accommodations Tyrone he during the interviews were relative to the recursive insertion of his ordering and pairing operations in his 2-slot MPS. This comparison fits the two criteria that Glasersfeld provides for differences in stages: inferring a difference in stage is a judgment of presence versus absence, and there is a qualitative difference in the schemes of students at different stages. For us, this qualitative difference in schemes helps us to situate future observations of the functional accommodations Mikayla and Tyrone made during the teaching episodes

 We do not claim that Tyrone produced ordered triples just that he produced triples. Producing ordered triples entails ordering the composite units themselves, which in contextualized language would involve Tyrone treating the spades as the first suit, the diamonds as the second suit, and the hearts as the third suit. The fact that Tyrone did not establish ordered triples contrasts with our inference about Mikalya who did establish ordered triples in the context of the Sandwich Problem. We attribute this particular difference to Tyrone’s lack of experience with combinatorics problems (i.e., constructing ordered triples was a functional accommodation he could make, but one he had not yet made). Generally speaking, stage 2 and stage 3 students conceive of ordered triples in qualitatively distinct ways. 16

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within a broader framework. This broader framework contributes to the creation of a constructive itinerary (i.e., what is possible) for stage 2 and stage 3 students in terms of their combinatorial schemes. It is in this way that qualitative differences in schemes can serve as a boundary for the claims a researcher makes about student learning. With that said, we do not interpret stages to be deterministic for two reasons: (1) within a given stage there is a broad range of schemes students construct, and they make functional accommodations relative to these schemes and (2) a researcher always remains open to the possibility that a student makes a stage change in the context of interactions with a student.

Investigating Learning of Stage 3 Students: Levels of Schemes Researchers have not proposed stages of units coordination beyond stage 3. However, they have begun to identify significant qualitative differences in the reasoning of stage 3 students in empirical studies (e.g., Hackenberg & Sevinc, 2022; Shin et al., 2020; Tillema & Burch, 2022) or proposed what would be significant differences in students’ schemes through conceptual analyses (e.g., Thompson et al., 2014). We propose that it may be fruitful to use Glasersfeld’s definition of level as a way of interpreting at least some of these differences. Glasersfeld and Kelley (1982) contrast stages with levels where they define a level as: not refer[ring] to a stretch of time. Level is a relative concept interpretable only with regard to the kind of scale employed….[Levels refer to] hierarchical divisions within a system of classification or measurement and refers to that system and not to any structural or qualitative feature per se. (Glasersfeld & Kelley, 1982, p. 158)

Glasersfeld does not tie his definition of level to a stretch of time, or to making a qualitative judgment of presence or absence. In this way, we see Glasersfeld’s definition of level as a tool to identify differences and similarities among students who are operating within the same stage, and specifically stage 3 students because no further stages have, as yet, been proposed. Therefore, we consider levels to be a useful tool for creating a more nuanced landscape of schemes within a given stage. We illustrate one instance of using levels with empirical data for a stage 3 student where we use the recursive aspect of Thompson et al.’s (2014) definition of scheme to differentiate among different levels of a scheme.

 mpirical Example of Different Levels of a Scheme for Stage 3 E Students: Armando Solves the Colored Digits Problem Armando, a stage 3 eighth-grade student, participated in 2-hour-long interviews. Overall, the goal of the interviews was to investigate students’ development of the quadratic identity that (nx)2 = n2x2 using cases of the Colored Digits Problem (two

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cases are given). Figure  3.13 shows  Armando’s 2-D array for the 4-case of the Colored Digits Problem, part b. The Colored Digits Problem. You have a deck of number cards. You draw a card, replace it, and draw a second card in order to make coordinate points, such as (1,1). (a) Suppose the deck of number cards includes the numbers 1 through 7. The numbers are colored yellow. What color combination(s) are the coordinate points (e.g., first digit yellow and second digit yellow gives a yellow–yellow color combination)? How many coordinate points could you make? (1-Case: 1 color and 7 digits) (b) Suppose the deck of number cards has the numbers 1 through 28 on them. Every 7 digits is a different color (digits 1 through 7 are yellow, 8 through 14 are red, 15 through 21 are green, and 22 through 28 are pink). How many color combinations could you make (e.g., first digit yellow and second digit red gives a yellow–red color combination)? How many coordinate points are in each color combination? How many total coordinate points could you make? (4-Case: 4 colors and 28 digits) For the 4-case of the problem, the goal was for him to establish the equivalence that 282 = (4 × 7)2 = 42 × 72 where 282 refers to one way to count the total number of coordinate points; (4 × 7)2 is a way to re-express 282 to show the number of colors and number of digits per color; and 42 × 72 is a way to express the total number of coordinate points where 42 expresses the number of color combinations and 72 the number of coordinate points in each color combination. Upon entering the interviews, Armando had not constructed a 2-slot MPS. However, he constructed a 2-slot MPS during the first 30  minutes of the first interview. Once

Fig. 3.13  Armando’s array

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Armando had constructed a 2-slot MPS, he solved cases of the Colored Digits Problem where part a was an initial case and part b was a later case. We show data of Armando explaining part a of the problem to illustrate his construction of a 2-slot MPS, and then show him explaining part b of the problem in order to discuss distinct levels of a 2-slot MPS. Data Excerpt 9: Identifying Levels of a Scheme A [Provides the following explanation for part a of the problem]: It’s 49…the first 7 (coordinate points) would be like one-one, one-two, one-three, one-four, one-five, one-six, and one-seven. And it would go on like that. Just switching the one to a two, then a three, then a four, five, six, and seven. [Later he says that 49 is 7 times 7 or 7 squared.] …. [A has represented part b of the problem using an array (Fig. 3.13) where he has shown the different color combinations and drawn all coordinate points in the yellow–yellow color combination. He has just made a box to show the coordinate points in the other color combinations. He has recorded the following notation for his picture: “784 = 42 × 72 = 282 = (4 × 7)(4 × 7)”] TR: Do you want to talk about your notation and just relate it to your picture? A: Alright, um, in all it would be 28 squared (for the total number of coordinate points), since there’s 28 (digits), and so it would be 28 times 28 or 28 squared to put it simple. And digit and, what’s the word for it? Digit and plotted (color) wise, it would be 7 times 4, I forgot to add something in this [adds the dot multiplication sign in his notation “784  = 42 × 72 = 282 = (4 × 7) ∙ (4 × 7)”], 7 times 4 times 7 times 4. TR: Do you just want to show on your picture where you see 7 times 4 times 7 times 4? A: Well since there are all these 7s, like 7 here [spanning the yellow digits on the horizontal axis with his thumb and index finger], 7 there [spanning the red digits on the horizontal axis with his thumb and index finger], there [spanning the green digits on the horizontal axis with his thumb and index finger], and up to there [spanning the pink digits on the horizontal axis with his thumb and index finger]. Those all work out to 7 times 4. And then the same thing up here 7, 7, 7, 7 [each time he says seven he spans his fingers on seven digits on the vertical axis]. Those two (the 4 sevens times the 4 sevens) multiplied together would give me the same answer as 28 squared would give me. And then over here would be 7 squared [points to the coordinate points in the yellow–yellow color combination] meaning 7 (squared) in (each of) the colors (combinations), I want to say. And 4 squared meaning all the colors, like 4 there [uses his thumb and middle finger of his right hand to span the colors on the horizontal axis] and 4 here [uses his thumb and middle finger of his left hand to span the colors on the vertical axis], multiplied together [moves his right and left hand into the interior of the array touching his right and left thumb and his right and left middle finger].

As mentioned, his explanation of part a of the Colored Digits Problem is indication that the problem activated his 2-slot MPS, and that he considered there to be a multiplicative relationship among two composite units, a unit of seven units, a unit of seven units, and a unit of seven units of seven pairs (Lines 1–5). We consider his solution of part b as indication that he inserted his 2-slot MPS inside itself twice. That is, we infer that part b of the problem activated his 2-slot MPS, and that his multiplication statement 282 was a result of reasoning that was similar to his reasoning in part a (Lines 13–14), where he established a multiplicative relationship among a unit of 28 units, a unit of 28 units, and a unit of 28 units of 28 pairs.

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He also structured the 28 digits along each axis as a unit of 4 units of 7 units (Lines 20–26, shown with the four colors along each axis in Fig.  3.13). He then envisioned the total number of coordinate points as the number of coordinate points, 7 squared, within each color combination (Lines 28–30), where there were a total of 4 squared color combinations (Lines 30–35). The 4 squared color combinations are shown in his array with the colored pairings inside the boxes. The 7 squared coordinate points are shown in the bottom left of his array, after which Armando said, “It’d take a really long time to make all of those points so I boxed off where they are for the others.” We see his array and explanation of notation as evidence of him inserting his 2-slot MPS inside itself twice; that is, contained in 28 squared (i.e., all the coordinate points, which he produced using his 2-slot MPS) was 4 squared (i.e., the color combinations, which he produced using his 2-slot MPS) where each color combination contained 7 squared (i.e., the coordinate points within each color combination, which he produced using his 2-slot MPS). Thus, we see his solution of part b of the problem as a recursive insertion of his scheme inside itself twice; he inserted his 2-slot MPS inside his 2-slot MPS inside his 2-slot MPS. We use this data excerpt to propose three levels of a 2-slot MPS in order to differentiate among stage 3 student reasoning. The first level is simply the one that Armando used to solve part a of the problem. It involves establishing a multiplicative relationship among a unit of units, a unit of units, and a unit of units of pairs, as Armando did when he determined the total number of coordinate points for part a of the problem. The second level of a 2-slot MPS entails creating an equivalence like 282 = 42 × 72, where the equivalence is generated because 282 and 42 × 72 are two distinct ways of counting the total number of coordinate points (Gatza, 2021). A student might then use a single instantiation of their 2-slot MPS to determine that the coordinate points can be counted as 282. Then their scheme closes (i.e., they satisfy their goal of counting the total number of coordinate points) before they find a second way of counting the total number of coordinate points. The second way of counting the coordinate points involves recursively inserting their 2-slot MPS inside itself; the student recursively inserts the total number of coordinate points per color combination (one instantiation of their 2-slot MPS) inside each color combination (a second instantiation of their 2-slot MPS). The third level is what we proposed Armando did, in which he inserted the insertion of his 2-slot MPS inside itself (three levels of recursive insertion). That is, he inserted the coordinate points per color combination into the color combinations (symbolized as 42 × 72), all of which were contained in the total number of coordinate points (symbolized as 282). We consider his doing so as a crucial part of producing the (4 × 7)2 in the equivalence that 282 = (4 × 7)2 = 42 × 72, because it made available to him each of the 28s that he could restructure into a unit of 4 units of 7 units.

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 evels, Functional Metamorphic Accommodation, L and Reflecting Abstraction This example fits Glasersfeld’s definition of level in that we do not attribute a period of time to the distinct levels of the scheme we have outlined. Moreover, the levels are a way to capture what we see as discernably distinct ways of reasoning within a particular stage that are hierarchical in nature. The differences in this example that we outlined are hierarchical in nature precisely because they entailed recursive insertion of a scheme inside itself; thus, the third level could only be attained after the second, and the second only after the first. We consider there to be several important reasons to define distinct levels of reasoning, especially for stage 3 students. The first is that it provides the opportunity to differentiate between two kinds of learning, a functional accommodation and a functional metamorphic accommodation (cf. Steffe, 1991). Our earlier example of Carlos’s learning in his solution of the Flag Problem we considered to be a functional accommodation, where we have documented the changes he made to his 2-slot MPS. His functional accommodation, however, did not involve us in proposing distinct levels of learning. As we documented elsewhere (Tillema, 2014), Carlos changed the criteria he had for situations that activated his 2-slot MPS, used operations in a novel way that restructured the activity of his 2-slot MPS, and made changes to  the expected result of  his 2-slot MPS.  None of these changes to his scheme involved defining distinct levels. On the other hand, we consider Armando’s learning over the course of the two interviews to be in the province of a functional metamorphic accommodation. We consider his accommodation to be metamorphic precisely because we could define distinct levels of his scheme based on the recursive insertion of his 2-slot MPS inside of itself. Further, we consider it to be a way to operationalize the projective aspect of Piaget’s (1977/2001) reflecting abstraction. That is, two key components of Piaget’s reflecting abstraction are that a coordination of schemes gets projected from a lower to higher plane of learning and that this coordination gets reorganized at the higher plane of learning (cf. Chap. 6). We have considered the insertion of operations or schemes as a way to account for the projective aspect of Piaget’s (1977/2001) reflecting abstraction (Tillema & Burch, 2022). When a person recursively inserts operations or schemes into a scheme to create a new single scheme, we have considered the new single scheme to be at a higher plane of learning precisely because the new single scheme incorporates multiple prior schemes or operations into a new singular scheme. Thus, we consider levels, and identifying functional metamorphic accommodations, that produce changes between levels to be one way to operationalize the projective aspect of reflecting abstraction. More broadly, we see these constructs as an important way to make distinctions about different qualities in the kind of learning that a researcher is documenting.

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Conclusion Schemes and the corresponding constructs that we have outlined in this chapter provide researchers with a set of tools to make accounts of another person’s mathematical reasoning. Steffe et  al. (1983) have called these accounts second-order models. We often begin the work of creating a second-order model with questions like the following in mind: • Can I clearly characterize the three parts and goal of a person’s scheme? (Michael’s example in Data Excerpt 1) • What is figurative and what is operative relative to a person’s current scheme? (Michael’s example in Data Excerpt 1) • How are the parts of a scheme dynamically related to each other? (Steffe’s model of scheme, Michael’s hypothetical example) • Is a person’s scheme reversible? What affordances does reversibility have in the students’ reasoning? (Nico’s example in Data Excerpt 2) These questions serve as a basis for examining data where our initial goal is to create a scheme or schemes that are a local account of student reasoning, including qualities of a scheme, like whether it is reversible or not. We often find responding to the second question to be the most difficult; that is, since figurative material is always relative to a person’s current operative schemes (Thompson, 1985; Glasersfeld, 1991), identifying the relation between the operations and figurative material of a scheme requires situating the person within a broader framework of mathematical reasoning. We have used the stages of units coordination (Hackenberg et al., 2016) to accomplish this goal in our work where the stages of units coordination offer a way to understand key differences in the operations available to a student at a certain stage, and differences in the nature of, for example, students’ composite units (i.e., the figurative material for the MPS) at each stage. Based on this observation, we consider a broader framework to be helpful in scheme creation because it supports a researcher to consider the nature of the operations available to a student and the nature of the figurative material they are operating on, both of which are critical to scheme creation. We consider a researcher’s creation of a scheme to be a baseline from which to make further observations about student reasoning. These further observations include efforts to investigate learning—where one kind of learning is what we have called functional accommodations, which are accommodations that occur while a scheme is in use and do not involve what a researcher deems to be distinct levels of learning (cf. Steffe, 1991). As we engage in this work, we are often considering questions like the following: • What situations activate a scheme? How do the activating situations reveal information about a person’s expectations? (Carlos’s assimilation in Data Excerpt 3) • What markers are there that a person is experiencing a perturbation? (Carlos’s perturbation in Data Excerpt 4)

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• What evidence is there that a perturbation feeds back into a scheme, leading to a functional accommodation? (Carlos’s initial functional accommodation in Data Excerpt 5) • What evidence is there that the changes to a scheme are more or less permanent? (Criteria for claiming a functional accommodation) We use these questions as a way to support making accounts of learning. As part of doing so, we find it helpful to think of assimilation as the mechanism by which a person transforms a given situation into an already known situation (e.g., for Carlos, the Flag Problem became like the Outfits Problem). From an explanatory standpoint, considering how assimilation is functioning in a situation can support making an account of a person’s expectations, their expectations both in terms of situations that activate a scheme and expectations they have for the result of their scheme. Making an account of a person’s expectations can then support seeing when and why a person experiences a perturbation. As we suggested, evidence for a perturbation may be subtle in that it may not be consciously conflictive for a person. Once a researcher has identified a perturbation, it is important to determine whether and if the perturbation has fed back into a person’s scheme. Doing so returns a researcher to the baseline characterization of a scheme with the aim of making an account of what changes a person made to their scheme and whether there is evidence that the changes are more or less permanent. These criteria support making an inference of a functional accommodation. The tools we have described so far help a researcher in the initial creation and subsequent use of schemes to model learning. We consider stages and levels to add two additional layers to a researcher’s creation of schemes. Stages can help a researcher make an account of how students’ schemes at different stages are qualitatively distinct. Levels can support a researcher in making hierarchical distinctions among students’ schemes within the same stage. When considering stages and levels in scheme creation, we often have the following kinds of questions in mind: • What schemes do students at different stages of units coordination construct? How are these schemes qualitatively distinct from one another? (Mikayla and Tyrone in Data Excerpts 6 and 7) • How can differentiating levels help to situate observations about students’ schemes who are at the same stage? (Armando in Data Excerpt 8) • How do levels support differentiation in the kind of learning that a researcher is observing? (Discussion of functional versus functional metamorphic accommodation) Consideration of the first question helps us to capture important differences in students’ schemes that are at different stages. These differences go beyond ones that we would expect could be explained only through functional accommodations. That is, we view functional accommodations relative to differences in the schemes students at different stages construct where our expectation is that, without a change in stage, the schemes of students at different stages would remain qualitatively distinct. We do not think of stages as deterministic but rather orienting when we are

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making second-order models; we remain open to the possibility that a student makes a stage change while at the same time valuing the learning of a learner at a given stage. On the other hand, levels support us to differentiate what we see as discernably distinct ways of reasoning within a particular stage that are hierarchical in nature. One of the primary reasons for introducing levels is to signal potential differences in qualities of learning where we have considered changes in levels to entail a functional metamorphic accommodation as opposed to just a functional accommodation. In the above example, we used the recursion of a scheme to capture the hierarchical nature of levels. However, characterizations of different levels of a scheme are one area for research that is relatively underdeveloped, and we specifically consider there to be a need to develop such accounts for stage 3 students. These accounts, accompanied with the use of functional metamorphic accommodation, would contribute to mathematics educators’ work to operationalize Piaget’s construct of reflective abstraction (see Chaps. 6 and 8 for other ways to do the same). These tools, as well as the accompanying questions, have helped us to see nuance in student reasoning and to organize this nuance into connected accounts of learning. This work, in turn, has supported us in having productive interactions with students and served as a basis for our interactions with pre- and in-service teachers. It is in this way that we consider scheme theory to be a living tool that dynamically supports us in our work in research and teaching. Acknowledgments  This research was supported by the National Science Foundation under Grants No. DRL-1920538 and No. DRL-1419973. The views expressed do not necessarily reflect official positions of the foundation. The authors would like to thank the following people for their helpful comments on this chapter: Les Steffe, Pat Thompson, Paul Dawkins, Derek Eckman, and Amy J. Hackenberg.

References English, L. (1991). Young children’s combinatoric strategies. Educational Studies in Mathematics, 22(5), 451–474. Gatza, A. M. (2021). Not just mathematics, “just” mathematics: Investigating mathematical learning and critical race consciousness [Doctoral dissertation, Indiana University, Bloomington]. ProQuest. Glasersfeld, E.  V. (1974). Piaget and the radical constructivist epistemology. In C.  D. Smock & E. von Glasersfeld (Eds.), Epistemology and education (pp.  1–24). Follow Through Publications. Glasersfeld, E. V. (1980a). Viability and the concept of selection. American Psychologist, 35(11), 970–974. Glasersfeld, E. V. (1980b). The concept of equilibration in a constructivist theory of knowledge. In F. Benseler, P. M. Hejl, & W. K. Koeck (Eds.), Autopoiesis, communication, and society (pp. 75–85). Campus. Glasersfeld, E. V. (1981a). The concepts of adaptation and viability in a constructivist theory of knowledge. In I. E. Sigel, D. M. Brodzinsky, & R. M. Golinkoff (Eds.), Piagetian theory and research (pp. 87–95). Hillsdale.

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Glasersfeld, E. V. (1981b). An attentional model for the conceptual construction of units and number. Journal for Research in Mathematics Education, 12(2), 83–94. Glasersfeld, E. V. (1982). An interpretation of Piaget’s constructivism. Revue Internationale de Philosophie, 36(4), 612–635. Glasersfeld, E. V. (1983). Learning as constructive activity. In J. C. Bergeron & N. Herscovics (Eds.), Proceedings of the 5th annual meeting of the North American Group of Psychology in Mathematics Education (Vol. 1, pp. 41–101). PME-NA. Glasersfeld, E. V. (1984). An introduction to radical constructivism. In P. Watzlawick (Ed.), The invented reality (pp. 17–40). Norton. Glasersfeld, E. V. (1990). An exposition of constructivism: Why some like it radical. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 19–29). National Council of Teachers of Mathematics. Glasersfeld, E.  V. (1991). Abstraction, re-presentation, and reflection. In L.  P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 45–67). Springer. Glasersfeld, E. V. (1993). Learning and adaptation in the theory of constructivism. Communication and Cognition, 26(3/4), 393–402. Glasersfeld, E. V. (1995). Radical constructivism: A way of knowing and learning. Falmer Press. Glasersfeld, E. V. (1997). Homage to Jean Piaget. The Irish Journal of Psychology, 18(3), 293–306. Glasersfeld, E.  V. (2001). Scheme theory as a key to the learning paradox. In A.  Philipp & J. Vonèche (Eds.), Working with Piaget: Essays in honour of Bärbel Inhelder (pp. 139–146). Psychology Press. Glasersfeld, E., & Kelley, M. F. (1982). On the concepts of period, phase, stage, and level. Human Development, 25(2), 152–160. Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. The Journal of Mathematical Behavior, 26(1), 27–47. Hackenberg, A.  J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383–432. Hackenberg, A. J., & Sevinc, S. (2022). A boundary of the second multiplicative concept: The case of Milo. Educational Studies in Mathematics, 109(1), 177–193. Hackenberg, A. J., Norton, A., & Wright, R. J. (2016). Developing fractions knowledge. Sage. Hackenberg, A. J., Walsh, P. A., & Valero, J. R. (2021). A case of units coordination stage change in middle school. In Brief research report at the forty third annual meeting of the International Group for Psychology of Mathematics Education in North America (pp. 1281–1286). Towson University. Inhelder, B., & de Caprona, D. (1992). Vers le constructivisme psychologique: Structures? Procédures? Les deux indissociables. In B. Inhelder & G. Cellérier (Eds.), Le cheminement des déscouvertes de l’enfant (pp. 19–50). Delachaux et Niestlé. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence: An essay on the construction of formal operations (Vol. 22). Psychology Press. Norton, A., Boyce, S., Phillips, N., Anwyll, T., Ulrich, C., & Wilkins, J. (2015). A written instrument for assessing students’ units coordination structures. International Electronic Journal of Mathematics Education, 10(2), 111–136. https://doi.org/10.12973/mathedu.2015.108a Norton, A., Ulrich, C., Bell, M. A., & Cate, A. (2018). Mathematics at hand. The mathematics educator, 27(1), 33–59. Piaget, J. (1964). Part I: Cognitive development in children: Piaget development and learning. Journal of Research in Science Teaching, 2(3), 176–186. Piaget, J. (1967). Six psychological studies (D. Elkind, Trans.). Random House. Piaget, J. (1970). Genetic epistemology. Columbia University Press. Piaget, J. (1971). Biology and knowledge: An essay on the relations between organic regulations and cognitive processes. University of Chicago Press. Piaget, J. (1936/1977). The Origin of Intelligence in the Child. United Kingdom: Penguin. Piaget, J. (1977/2001). Studies in reflecting abstraction (R. Campbell, Trans.). Psychology Press. Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry. Basic Books.

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Shin, J., Lee, S. J., & Steffe, L. P. (2020). Problem solving activities of two middle school students with distinct levels of units coordination. The Journal of Mathematical Behavior, 59, 1–19. Steffe, L. P. (1991). The learning paradox: A plausible counterexample. In Epistemological foundations of mathematical experience (pp. 26–44). Springer. Steffe, L.  P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309. Steffe, L. P. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 1–41). SUNY Press. Steffe, L. P. (1996). Social-cultural approaches in early childhood mathematics education: A discussion. In Mathematics for tomorrow’s young children (pp. 79–99). Springer. Steffe, L.  P. (2010a). Perspectives on children’s fraction knowledge. In L.  P. Steffe & J.  Olive (Eds.), Children’s fractional knowledge (pp. 13–25). Springer. Steffe, L. P. (2010b). Articulation of the reorganization hypothesis. In L. P. Steffe & J. Olive (Eds.), Children’s fractional knowledge (pp. 49–74). Springer. Steffe, L.  P., & Thompson, P.  W. (2000a). Interaction or intersubjectivity? A reply to Lerman. Journal for Research in Mathematics Education, 31(2), 191–209. Steffe, L. P., & Thompson, P. W. (2000b). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education. Kluwer. Steffe, L.  P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children’s counting types: Philosophy, theory, and application. Praeger Publishers. Thompson, P. W. (1985). Experience, problem solving, and learning mathematics: Considerations in developing mathematics curricula. In E. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 189–243). Erlbaum. Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education. Plenary paper delivered at the 32nd Annual Meeting of the International Group for the Psychology of Mathematics Education. In O. Figueras, J. L. Cortina, S.  Alatorre, T.  Rojano, & A.  SÈpulveda (Eds.), Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 45–64). PME. Thompson, P.  W. (2013). In the absence of meaning. In K.  Leatham (Ed.), Vital directions for research in mathematics education (pp. 57–93). Springer. Thompson, P. W., Carlson, M. P., Byerley, C., & Hatfield, N. (2014). Schemes for thinking with magnitudes: A hypothesis about foundational reasoning abilities in algebra. In K. C. Moore, L. P. Steffe, & L. L. Hatfield (Eds.), Epistemic algebra students: Emerging models of students’ algebraic knowing (WISDOMe Monographs) (Vol. 4, pp. 1–24). University of Wyoming. Tillema, E.  S. (2013). A power meaning of multiplication: Three eighth graders’ solutions of Cartesian product problems. The Journal of Mathematical Behavior, 32(3), 331–352. Tillema, E. S. (2014). Students’ coordination of lower and higher dimensional units in the context of constructing and evaluating sums of consecutive whole numbers. The Journal of Mathematical Behavior, 36, 51–72. Tillema, E. S. (2018). An investigation of 6th graders’ solutions of combinatorics problems and representation of these problems using arrays. The Journal of Mathematical Behavior, 52, 1–20. Tillema, E. S. (2021). Students’ solution of arrangement problems and their connection to Cartesian product problems. Mathematical Thinking and Learning, 22, 23–55. Tillema, E.  S., & Burch L.  J. (2020). Leveraging combinatorial and quantitative reasoning to support the generalization of advanced algebraic identities. Invited paper presentation at the International Congress on Mathematical Education to the Topic Study Group on the Teaching and Learning of Discrete Mathematics in Shanghai, China. Tillema, E. S., & Burch, L. J. (2022). Using combinatorics problems to support secondary teachers understanding of algebraic structure. Zentralblatt für Didaktik der Mathematik, 54, 777. https://doi.org/10.1007/s11858-­022-­01359-­1 Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (Part 1). For the Learning of Mathematics, 35(7), 2–7. Ulrich, C. (2016). Stages in constructing and coordinating units additively and multiplicatively (Part 2). For the Learning of Mathematics, 36(1), 34–39.

Chapter 4

Operationalizing Figurative and Operative Framings of Thought Kevin C. Moore, Irma E. Stevens, Halil I. Tasova, and Biyao Liang

Introduction Theory is in the end, as has been well said, the most practical of all things, because this widening of the range of attention beyond nearby purpose and desire eventually results in the creation of wider and farther-reaching purposes and enables us to use a much wider and deeper range of conditions and means than were expressed in the observation of primitive practical purposes. For the time being, however, the formation of theories demands a resolute turning aside from the needs of practical operations previously performed. (Dewey, 1929) Theory is the stuff by which we act with anticipation of our actions’ outcomes and it is the stuff by which we formulate problems and plan solutions to them. (Thompson, 1994b, p. 229, in reference to Dewey’s quote)

K. C. Moore (*) Department of Mathematics, Science, and Social Studies Education, University of Georgia, Athens, GA, USA e-mail: [email protected] I. E. Stevens Department of Mathematics and Applied Mathematical Sciences, University of Rhode Island, Kingston, RI, USA e-mail: [email protected] H. I. Tasova Department of Teacher Education and Foundations, California State University, San Bernardino, San Bernardino, CA, USA e-mail: [email protected] B. Liang Academic Unit of Teacher Education and Learning Leadership, The University of Hong Kong, Hong Kong, SAR, China e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 P. C. Dawkins et al. (eds.), Piaget’s Genetic Epistemology for Mathematics Education Research, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-47386-9_4

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One framing of theory is that it provides ready-made tools to be applied as ways to understand, organize, and explain phenomena. In such a framing, theoretical constructs are mostly static in their definitions, with researchers using them as defined. For instance, Moore (2014a) used Carlson et al.’ (2002) covariation framework to model a student’s construction of the sine relationship in graphical and unit circle contexts. In adopting the mental actions introduced by Carlson et al. (2002), Moore did not make modifications to their definitions, instead applying them as defined in order to provide a hypothetical account of the cognitive actions driving the student’s activity. Another framing of theory is that it provides malleable tools for inquiry and research pursuits. Rather than providing ready-made tools to be applied, theory and theoretical constructs provide a general roadmap or orientation. Theoretical constructs are dynamic in their definitions, with researchers adapting constructs as they carry out empirical work so that the researcher can operationalize theoretical constructs in the context of novel settings and data. The adaptations result in an evolution of the theoretical constructs that account for novel settings while maintaining some central tenets of those constructs. For example, in adopting Ellis’s generalization framework (Ellis, 2007b), Ellis and colleagues (2022) found it necessary to adapt that framework to model the reasoning of students across different grade bands and content areas. We write this chapter in the spirit of this latter framing. The notions of figurative and operative thought were introduced over a half-­ century ago by Piaget (1969; Piaget & Inhelder, 1971, 1973), and since then, mathematics education researchers have used, extended, and shaped them in several ways. Their uses have been varied, and no singular definition of either sufficiently captures its development and use. Thus, in this chapter, we present various uses of the two notions of thought with the intention of highlighting their value in modeling student thinking. In doing so, we situate evolutions in the two constructs, while underscoring common tenets that span across these evolutions. We discuss methodological concerns regarding notions of figurative and operative thought, including how the constructs can be used in task design of empirical studies. We also discuss the constructs’ implications for researchers’ claims regarding students’ meanings. Because theoretical constructs should be continuously tested in new areas to assess their viability and expand their generalizability, we end the chapter with various ideas for future research that can contribute to continued evolutions in the notions of figurative and operative thought.

Some “Definitions” As mentioned in the introduction, we do not intend to provide definitive definitions of figurative and operative thought. This choice is not to meant to imply that definitions are not important. Definitions help paint a picture of the constructs’ central tenets when combined with researchers’ uses of the constructs, but what ultimately

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matters is the use of the constructs. As a resource to the reader, we provide a non-­ exhaustive collection of researcher quotes related to defining figurative and operative thought (Table 4.1).

Uses and Evolution of Figurative and Operative Thought As can be inferred from Table  4.1, one of the more difficult aspects of reading Piaget is that he did not always provide concise definitions, and when he did, his definitions evolved to capture changes in his conception of knowledge and knowing. Piaget’s approach to his writing and work mirrors his genetic epistemology and his conception of objects. To know an object is to act on it or engage in its use, as an object is comprised of operations of thought. Or, “To my way of thinking, knowing an object does not mean copying it—it means acting upon it” (Piaget, 1970, p. 15). It is through acting on an object (e.g., operationalizing a construct) that an individual furthers their understanding of the object via the mental operations imbued to the object. Fortunate to our role as authors of this chapter, the figurative and operative distinction did not see significant evolutions in Piaget’s writing and use. We hypothesize that this is because the distinction formed a general categorization of different knowledge forms, as opposed to constructs that emerged through nuanced characterizations of knowledge development (e.g., his various forms of abstraction). In fact, references to figurative or operative forms of thought occur in the context of Piaget describing forms of abstraction, in which he situates operative forms of thought with more sophisticated forms of abstraction. As Montangero and Maurice-Naville (1997) reported, the figurative and operative distinction emerged during writings by Piaget and his colleagues on perception, representation, mental imagery, and memory (Piaget, 1969; Piaget & Inhelder, 1971, 1973). Defined generally, figurative aspects of thought include those things that involve perception, sensation, sensory objects, and physical motion. Figurative aspects of thought deal with configurations and states, of which an individual looks to mimic in production. Operative aspects of thought include the coordination of mental actions and their transformations, and are those mental operations that define objects and allow individuals “to free themselves from the illusions and deformations aroused by perception” (Montangero & Maurice-Naville, 1997, p.  144). Operative forms of thought enable a student to isolate numerosity or make compensations of dimensions when engaged in Piaget’s famous pebble counting or glass and volume tasks. Figurative forms of thought lead to children relying on perceptual properties (including spatial positioning of endpoints) to draw conclusions about numerosity or volume. Before transitioning to the ways in which the figurative and operative distinction has been used in depth by mathematics educators, we perceive Piaget’s distinction to be related to (and have likely informed) his work on children’s conception of

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Table 4.1  Figurative and operative quotes Figurative and operative thought quotes Figurative thought quotes Piaget did admit an inferior species of knowledge that merely pictured static configurations; he called the lesser kind figurative knowledge. For Piaget perception and—usually—language are figurative. Figurative knowledge is knowledge of observables… ‘Figurative’ refers to the domain of sensation and includes sensations generated by motion (kinaesthesia), by the metabolism of the organism (proprioception), and the composition of specific sensory data in perception. [the figurative aspect of thought] tends to include the figural character of reality, i.e., configurations as such. With it can be grouped: (a) perception, which functions only when an object is present and through the intermediary of a sensorial field; (b) imitation in the broad sense of the term (gestural, phonic, graphic, etc.) which functions either with or without the presence of the object, but in any case through overt or covert motor reproduction; (c) the mental image, which functions only when there is no object present. The figurative functions have no tendency to transform objects, but tend to supply imitations of them…static configurations, which are relatively easy to translate into images; even when they concern movements or transformations, they do so in order to produce the appropriate configurations, not the changes of state. When a person’s actions of thought remain predominantly within schemata associated with a given level (of control), his or her thinking can be said to be figurative in relation to that level. Adopting Thompson’s framing, a researcher drawing distinctions between figurative and operative thought is thus an issue of characterizing whether an individual’s meanings are tied to carrying out particular actions and their results… Operative thought quotes For Piaget, knowledge is not a matter of pictures or sentences or symbolic data structures… knowledge is fundamentally operative; it is knowledge of what to do with something under certain possible conditions. Or it is knowledge of what that thing will do under different conditions… operative knowledge consists of cognitive structures. …whereas operative knowledge, which pertains to actions and operations, is knowledge of transformations. Now, logico-arithmetic operations cannot be rendered figuratively, unless this is done symbolically. The operative aspect of thought deals not with states but with transformations from one state to another. In contrast, any result of conceptual construction that does not depend on specific sensory material but is determined by what the subject does, is ‘operative’ in Piaget’s terminology.

Source Piaget (2001, p. 2) Piaget (2001, p. 298) von Glasersfeld (1995, p. 69) Piaget and Inhelder (1971, pp. 11–12)

Piaget and Inhelder (1973, p. 9)

Thompson (1985, p. 195) Moore et al. (2019b, p. 2)

Piaget (2001, pp. 2–3)

Piaget (2001, pp. 298–299) Piaget (1970, p. 14) von Glasersfeld (1995, p. 69) (continued)

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4  Operationalizing Figurative and Operative Framings of Thought Table 4.1 (continued) Figurative and operative thought quotes Figurative thought quotes The operative aspect on the other hand takes in those forms of cognitive experience or of deduction whose function consists of modifying the object in such a way as to apprehend transformations as such. This includes: (a) sensorimotor actions (with the exception of imitation), the sole instruments of sensorimotor intelligence to be organized before language; (b) internalized actions that prolong previous ones right from a preoperational level; (c) the operations proper of the representational intelligence, or reversible internalized actions which organize themselves as a set of structures or as transformation systems. When the actions of thought move to the level of controlling schemata, then the thinking can be said to be operative in relation to the level of the figurative schemata. …[operative thought] foregrounds the coordination of internalized mental actions so that figurative aspects of thought are subordinate to this coordination… the individual can call forth and control a scheme and its results.

Source Piaget and Inhelder (1971, pp. 11–12)

Thompson (1985, p. 195) Moore et al. (2019b, p. 2)

geometry (Piaget et al., 1981). Recall that to Piaget, objects are defined by operations of thought and thus the understanding of geometric objects involves moving beyond the observable. Despite the ease at which the study of geometric objects can be framed as the study of the observable, Piaget’s work underscores that the construction of geometric objects involves dissociating operative forms of thought from their figurative content so that the operative takes precedence over its figurative content, forms, and states (Norton, 2022; Piaget et al., 1981). Take, for instance, a circle. While the reader can likely evoke an image of a circle with ease, the mental coordination of rotating a fixed length about a fixed point and anticipating the trace of the rotating end-point defines the circle. We pull this tenet—one that involves identifying whether it is figurative or operative aspects of thought that take precedence in one’s meanings—through the rest of the chapter.

Transitioning the Constructs to Mathematics Education Piaget and his colleagues’ work has influenced many mathematics educators’ research programs. Whether it be aspects of his genetic epistemology, his discussion of schemes and operations, his identified stages of development, or his framing of abstraction, Piaget’s body of work has proved fruitful in providing mathematics educators with tools to research, model, and promote individuals’ mathematical development. With respect to the figurative and operative distinctions, von Glasersfeld (1987, 1995) and Steffe (1991a, b; Steffe & Olive, 2010), as close collaborators, were early adopters who did not make significant adaptations to the distinctions.

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von Glasersfeld (1995) incorporated Piaget’s distinction in his exposition on the nature of knowing and learning in order to distinguish between two forms of thought with categorical differences. Consistent with Piaget, he explained that figurative thought involves sensation, perception, sensory objects, and physical motion, and is thus directly related to any abstractions based in specific sensory material, including motor signals. He contrasted this by explaining that operative thought involves conceptual operations and their coordination. He explained that the enactment of such operations occurs in the context of specific figurative material, their abstraction is such that they are not dependent on specific sensory material, and these abstractions can provide the figurative material for subsequent abstractions. At a more specified level, one of von Glasersfeld’s (1987, 1995) primary interests was with representation, whether in terms of language, icons, or symbols. In reference to the figurative and operative distinction, von Glasersfeld (1987) explained, With regard to icons, Piaget’s distinction between the “figurative” and the “operative” would seem to be of some importance. Number is not a perceptual but a conceptual construct; thus, it is operative and not figurative. Yet, perceptual arrangements can be used to “represent” a number figuratively. Three scratches on a prehistoric figurine, for instance, can be interpreted as a record of three events. In that sense, they may be said to be “iconic”— but their iconicity is indirect. They do not depict “threeness”, they merely provide the beholder with an occasion to carry out the conceptual operations that constitute threeness (Glasersfeld, 1981, 1982). Carrying out these operations does not involve reference to some prior sensorimotor item or elements of such items—it is the operating itself that each time constitutes the abstract conception of threeness… An analogous distinction must be made in the case of symbols. On the one hand, there are symbols that refer to figurative items or sensorimotor situations, such as the King of France or the act of smoking; on the other, there are symbols that do not refer to sensorimotor experience at all but are merely indicators that a certain conceptual operation is to be performed. I would call this second category operative symbols and would list among them not only number words, numerals, and mathematical signs, such as “+”, “—”, and “=”, but also prepositions, conjunctions, and certain other words whose interpretation does not depend on the recall of sensorimotor experiences but requires the construction of some operative conceptual relation. (Glasersfeld, 1987, pp. 7–8)

von Glasersfeld (1987) described that an icon refers to another item or experience via sensorimotor similarity, whereas a symbol refers to other items or experiences more arbitrarily. With respect to both icons and symbols, von Glasersfeld drew on Piaget’s figurative and operative distinction to categorize those icons and symbols that form or refer to figurative items and sensorimotor situations, and those icons and symbols that (possibly in addition to forming figurative items) occasion or point to the coordination of mental operations.1 In von Glasersfeld (1995), he echoed this distinction in reference to words. He explained that an individual’s meaning for a word can be abstracted from sensorimotor experience and thus be figurative, or it can point to a coordination of mental operations and thus be operative. In the context of words, he added that a symbol or icon might merely cause a response in the  It is important to note that von Glasersfeld used symbol differently than Piaget as he described in his writings on radical constructivism (von Glasersfeld, 1995). 1

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form of a carried-out action, a case in which the symbol or icon acts as a signal as opposed to an abstracted coordination of mental operations (Moore, 2014b; Glasersfeld, 1995). von  Glasersfeld’s application of Piaget’s distinction to icons, symbols, and words is reflected in graphical associations related to static and emergent shape thinking discussed in section “Informing generalized models of student thinking”. With respect to Steffe and his colleagues, they adopted the figurative and operative distinctions to characterize children’s development of schemes in the context of counting and quantities, including length. With respect to the example of length, Steffe and Olive (2010) identified that figurative length entails a visualized path, motion producing the path, and some sense of the duration of the motion. Steffe and Olive (2010) further identify that figurative length provides the foundation for an individual to construct quantitative properties through the individual’s actions and reflection upon those actions in order to construct a system of coordinated operations (i.e., an operative meaning). Steffe and Olive’s characterization of length underscores that characterizing a meaning as figurative need not carry negative connotations. Instead, a figurative meaning may be a natural part of conceptual development as an individual transitions from meanings based in perception, sensation, and motor actions to meanings based in mental actions introduced and eventually coordinated by the individual. With respect to counting, and situated within children’s construction of number, Steffe and colleagues (1991b; Steffe & Olive, 2010) distinguished between figurative and operative counting schemes. Whereas figurative length primarily foregrounded notions of motion and duration, a figurative counting scheme requires re-presenting perceptual material or engaging in some form of sensorimotor action when counting.2 In order to support their counting of objects, especially if hidden, a child with a figurative counting scheme will produce or imagine material that is visual, such as an array of objects, or that is sensory, such as pointing in space or tapping their finger (Steffe, 1991b). Steffe and Olive (2010) introduced the term figural unit items for these re-presentational objects due to their basis in figurative material and sensorimotor action. An operative counting scheme is one in which the individual has constructed “a sequence of abstract unit items that contain the records of the sensory-motor material” (Steffe, 1991a, p. 35). Such a construction occurs through the individual coordinating the mental operations involved in counting so that they no longer need to produce or imagine figurative material or sensorimotor actions. The individual thus constructs unitized records of counting that they can then operate on to construct more sophisticated notions of quantity. Important to the figurative and operative distinction, including Thompson’s extension described below, these unitized records or abstract unit items are not tied to specific  We use “re-present” (or “re-presentation”) and “representation” in distinct ways. Drawing on Piaget and von Glasersfeld (von Glasersfeld, 1995), as well as Liang and Moore (2021), we use “re-presentation” to refer to the enactment and regeneration of schemes and operations. We use “representation” in the canonical sense to refer to the modes of display and symbolization associated with the field of mathematics (e.g., graphs, inscriptions, and verbal statements). 2

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re-presentational actions and associated figurative material. The regeneration of figurative material is necessary to re-present the records of counting, but it is the abstract unit items and unitized records of counting that drive the regeneration of that material.

Transitioning the Constructs to Higher Level Mathematics For Steffe and colleagues, the figurative and operative distinctions proved fruitful in describing number meanings and the extent figurative material or sensorimotor activity were a necessary aspect of counting actions. Their distinction provided a productive avenue to expanding Piaget’s discussion of concrete-operational intelligence (e.g., thought involving logical operations but limited to concrete or physical situations), which was sensible given that the development of number and counting schemes was their content area of study. This distinction is not as useful when discussing students’ meanings for more advanced mathematics topics built on complex, intertwined systems of meanings, representational activity, and symbolization. Researchers have thus reframed the figurative and operative thought distinction as illustrated by the work of Thompson, Moore, and colleagues (Liang & Moore, 2021; Moore, 2021; Moore et al., 2019a, b; Thompson, 1985). Thompson (1985) described Piaget’s distinction as one of “the most significant that I know of for mathematics education” (p.  194). He proposed generalizing notions of figurative and operative thought to any level of meanings in order to account for the developmental nature of mathematical meanings, and the fact that a system of meanings at one developmental level can become the source material (i.e., the figurative ground) for a system of meanings at a subsequent developmental level. Thompson explained, When a person’s actions of thought remain predominantly within schemata associated with a given level (of control), his or her thinking can be said to be figurative in relation to that level. When the actions of thought move to the level of controlling schemata, then the thinking can be said to be operative in relation to the level of the figurative schemata. That is to say, the relationship between figurative and operative thought is one of figure to ground. Any set of schemata can be characterized as figurative or operative, depending upon whether one is portraying it as background for its controlling schemata or as foreground for the schemata that it controls. For instance, the thinking of a college mathematics major in an advanced calculus course, which certainly would be classified as being formal operational in Piaget’s fixed sequence of cognitive development, could nevertheless be classified as figurative regarding the kind of thinking required in a graduate course in real analysis. Of course, we would have to make apparent to ourselves the possibility that the “objects” of such a student’s thinking are things like functions, classes of functions, and associated operations. (Thompson, 1985, p. 195)3

 Here, we believe Thompson’s use of schemata is consistent with scheme or meanings than his use of schema (see Thompson, 2013b). For a more extensive discussion of schemes, see Tillema and Gatza (Chap. 3). 3

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Thompson’s discussion of operative and figurative thought reflects his interest in characterizing complex systems of knowing that entail a web of meanings that build on themselves during a student’s development. Importantly, this development can occur both in the short-term and over a long period. For example, in the short-term, a student could construct a directional relationship (see Carlson et al., 2002) between two covarying quantities, which then can become the material on which they operate to construct an amounts of change relationship. Alternatively, over a longer period, a student could construct particular additive partitioning schemes that then become the source material for making multiplicative comparisons and constructing rate of change meanings (Byerley, 2019; Thompson, 1994a). Importantly, by identifying that what is operative at one level can become figurative at the next level as the individual attempts to operate on prior abstracted mental actions, Thompson provides researchers a way to characterize students’ actions or meanings as figurative or operative no matter the mathematical content or “object.” Also expanding the applicability of figurative and operative thought, Thompson’s reframing breaks free from a distinction resting on the availability or production of figurative material and sensorimotor actions. Thompson instead foregrounds the extent an individual’s meanings are tied to carrying out particular actions and obtaining particular results, as opposed to their being able to transform their actions to account for novel experiences and associated figurative material. Said another way, Thompson’s framing differentiates the focus of reasoning being properties of figurative material itself (including the results of activity) from the focus of reasoning being the mental actions and their coordination that generate associated figurative material and results.4 Thompson broadened the figurative and operative distinction to allow for the presence of figurative material (a point upon which we subsequently expand upon as it relates to task design), but it is not incompatible with Steffe and Olive’s description of figurative counting schemes. A child with a figurative counting scheme is constrained to carrying out particular actions, and specifically that of producing sensorimotor actions or using available perceptual material. On the other hand, a child having constructed a sequence of abstract unit items is in a position to use those items in the presence of novel experiences and associated figurative material. Thompson’s reframing is also faithful to tenets of Piaget’s use of figurative and operative thought. Piaget included imitation in his characterization of figurative aspects of thought but with a focus on sensorimotor reproduction. Piaget also noted that figurative aspects of thought involve imitations of objects and the production of configurations and observables. Thompson extends Piaget’s notions of imitation and reproduction to include those meanings or actions that foreground imitating or habitually reproducing previously enacted actions, whether mental or sensorimotor, for the purpose of obtaining particular results and states. As Ellis et al. (Chap. 6) explain, such meanings are often tied to abstractions based on the results or products

 We note that this framing from Thompson is related to various forms of abstraction. For more information on those forms, see chapters by Ellis et al. (Chap. 6) and Tallman (Chap. 8). 4

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of mental actions, as opposed to the coordination of the mental actions that produced particular results and objects (Moore, 2014b; Moore et al., 2019a, b). With respect to operative thought, Piaget’s descriptions foregrounded internalized actions, organized and logico-mathematical structures, and an individual’s ability to reason about transformations. This is consistent with Thompson’s use of operative, with Thompson adding that what is operative on one level can become figurative in nature during subsequent experiences and development. Importantly, Thompson’s addition is consistent with the goal-directed foundation of Piaget’s genetic epistemology (Piaget, 1970; Steffe, 1991a; Glasersfeld, 1995). Thompson’s reframing has enabled researchers to characterize marked differences in students’ meanings for the representations and symbolizations they construct, including the extent those meanings are tied to carrying out particular actions and their results. To illustrate the usefulness of Thompson’s framing of figurative and operative thought in this regard, we first draw on results from investigations of students’ graphical meanings and activity. This line of research has been prevalent because graphing necessarily entails aspects of both figurative and operative thought (Moore et al., 2019b). Before we report on data and empirical studies that draw on Thompson’s framing, we provide a few concise definitions and resources (Table 4.2) that will be of use to the reader. We include a list of relevant citations with each definition for the reader interested in reading more detailed works that use the associated term or construct. The connection between notions of figurative thought, operative thought, quantitative reasoning, and covariational reasoning is not by chance. Quantitative and covariational reasoning are apropos examples of operative thought, and specifically the enactment of logico-mathematical operations and associated meanings (Moore, 2019b; Norton, 2014; Thompson, 1985). We attempt to provide examples in the following sections so that they do not require a significant background in quantitative and covariational reasoning, but familiarity with research in those areas will undoubtedly assist the reader in situating our illustrations of figurative and operative notions of thought.

Models of Students’ Graphical Thinking Moore, Stevens, et al. (2019b) adopted Thompson’s (1985) reframing of figurative and operative thought to characterize prospective secondary teachers’ (PSTs’) graphical meanings. The authors did so in order to highlight a marked difference in the PSTs’ actions as they engaged in solving particular problems involving the interpretation or construction of graphical representations. The authors explained that some PSTs’ actions suggested their enacting meanings were dominated by fragments of sensorimotor experience and perceptual properties of shape, while other PSTs’ actions suggested their enacting meanings were dominated by the coordination of mental operations in the form of quantitative reasoning.

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Table 4.2  Useful definitions Construct Quantity

Definition “A quantity is a quality of something that one has conceived as admitting some measurement process” (Thompson, 1990, p. 5).

Magnitude

“The idea of magnitude, at all levels, is grounded in the idea of a quantity’s size” (Thompson et al., 2014, p. 1). “Magnitude refers to the size or amount-ness of a quantity that remains invariant with respect to changes in the unit used to measure the quantity” (Moore et al., 2019b, p. 3). “The numerical result of a quantification process applied Moore et al. to [a quantity]” (Thompson, 1990, p. 6). (2013), Smith III and Thompson (2007), and Thompson (2011) Smith III and We use “number” to refer to when a student is referencing a numerical signifier that is not the result of Thompson (2007) a quantification process applied to a quantity to obtain a and Thompson et al. (1994) measure. Liang and Moore We use quantitative operations to refer to both “the (2021), Steffe and conception of two quantities being taken to produce a Olive (2010) and new quantity” (Thompson, 1990, p. 11) as well as the Thompson (1994a) operations that generate quantity or are involved in measuring a quantity, such as partitioning, unitizing, and iterating (Steffe, 1991b). “[Operations] used to calculate a quantity’s value; there Ellis (2007a), is no direct correspondence, except in a canonical sense, Smith III and Thompson (2007) between quantitative operations and the arithmetic and Thompson operations actually used to calculate a quantity’s value et al. (1994) in a given situation” (Thompson, 1990, p. 12). Moore et al. (2013) “A quantitative relationship is the conception of three and Thompson and quantities, two of which determine the third by a Thompson (1996) quantitative operation” (Thompson, 1990, p. 13). “Quantitative reasoning is the analysis of a situation into Ellis (2007a), Moore (2014a) and a quantitative structure—a network of quantities and Smith III and quantitative relationships” (Thompson, 1990, p. 13). Thompson (2007) Carlson et al. “…someone holding in mind a sustained image of two (2002), Johnson quantities values (magnitudes) simultaneously…one (2012) and Paoletti tracks either quantity’s value with the immediate, explicit, and persistent realization that, at every moment, and Moore (2018) the other quantity also has a value” (Saldanha & Thompson, 1998, p. 298). Moore (2014a), and “Coordinating the direction of change of one variable with changes in the other variable” (Carlson et al., 2002, Thompson and Carlson (2017) p. 357).

Value (or measure)

Number

Quantitative operations

Arithmetic or numerical operations

Quantitative relationship Quantitative reasoning

Covariational reasoning

Directional change

Other references Smith III and Thompson (2007), Steffe, (1991b) and Thompson (1993) Liang and Moore (2021)

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100 Table 4.2 (continued) Construct Amounts of change

Definition “Coordinating the amount of change of one variable with changes in the other variable” (Carlson et al., 2002, p. 357).

Other references Ellis et al. (2015), Johnson (2015) and Liang and Moore (2021)

Moore, Stevens, et  al. (2019b) provide an apropos example of Thompson’s (1985) reframing because all PSTs in the study undoubtedly enacted mental operations when problem solving. Furthermore, they did so in the presence of available or produced figurative material, thus not having to regenerate material entirely from scratch. Accordingly, early distinctions of figurative and operative thought would not have been productive in providing differentiated accounts of their actions. By adopting Thompson’s framing, Moore, Stevens, et al. (2019b) differentiated whether the PSTs’ actions foregrounded properties of figurative material itself, including that which resulted from their activity, or whether their actions foregrounded the coordination of mental operations not necessarily tied to particular properties of figurative material. In all, they identified that some PSTs’ actions foregrounded figurative aspects of thought, including requiring starting a graph on the vertical axis, drawing or interpreting a graph left-to-right, drawing a graph that passes the “vertical line test,” and drawing a graph that maintains some template or recalled shape, to mention a few. On the other hand, they identified some PSTs’ actions that foreground operative aspects of thought in the form of quantitative and covariational reasoning so that their construction and interpretation of graphs were dominated by those operations no matter the resulting shape of their graph; the shape was entirely a consequence of their coordination and tracking of quantities’ variation. As an illustrative example, Moore, Stevens, et al. (2019b) identified some PSTs’ meanings that involved their perceiving “slope” in terms of left, right, up, and down physical movements, and then associating paired movements with numerical properties for slope. For instance, one student had constructed the following (movement, slope) pairs: (left-up, negative), (right-down, negative), (left-down, positive), and (right-up, positive). Importantly, these (movement, slope) pairs were not dependent on axes’ orientations or labeling, but instead properties of a line qua line. Moore, Stevens, et  al. (2019b) illustrated that such a meaning is viable when particular Cartesian conventions are maintained, but its foregrounding of particular sensorimotor movement results in significant perturbation when those conventions are not maintained.5 For instance, a PST in their study perceived the two displayed graphs in Fig.  4.1 to have a positive and negative “slope,” respectively. She was unable to reconcile to her satisfaction the perturbation that stemmed from each displayed graph having different “slopes,” yet each being associated with the same formula that implied a positive “slope” of 3.  As Moore, Silverman, et al. (2019a) described, our use of “convention” here is not precise. What the researchers perceived as a convention was certainly not a convention to the referenced participants of the study.

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Fig. 4.1  Two displayed graphs of y = 3x, each with noncanonical axes orientations

Fig. 4.2  A displayed graph of y = 3x when scales are homogeneous; and y is oriented with positive values horizontally to the right and x is oriented with positive values vertically downward, or y is oriented with positive values horizontally to the left and x is oriented with positive values vertically upward

On the other hand, Moore, Stevens, et al. (2019b) illustrated other PSTs’ meanings involved their perceiving properties of “slope” as a consequence of rate of change and its representation under the constraints of a coordinate system. These PSTs thus associated slope with the coordination of quantities’ values and a comparison of amounts of change. Reflecting the generativity of operative forms of thought, such a meaning proved viable for the PSTs no matter the coordinate orientation, as their reasoning persistently foregrounded the coordination of quantitative operations so that perceptual properties of displayed graphs were always a consequence of those coordinations. For instance, a PST in their study conceived numerous axes-orientations and labeling to achieve a positive “slope” with the displayed

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graph in Fig. 4.2. The PST’s comment on a student who claims that the line has “a negative slope” further illustrates the foregrounding of quantitative and covariational reasoning (Excerpt 4.1). Excerpt 4.1 Annika Coordinating Two Quantities’ Values to Address a Student Claim  Annika: You’d have to notice that even though it looks like a negative slope [making a hand motion down and to the right] because we call it slope because it’s visual and it’s easy to visualize a negative and positive slope [making hand motions to indicate different slopes]. But that’s only visual on our conventions of how we set it up. Um, but like [pointing to the graph] if slope is rate of change we can still see that for like equal increases of x [making hand motions to indicate equal magnitude increases] we have an equal increase of y [making hand motions to indicate equal magnitude increases] of three. And so for equal positive increase of one [sweeping fingers vertically downward to indicate an increase of one], we have an equal positive increase of three [sweeping fingers horizontally rightward to indicate an increase of three]. And so it is a positive slope. (Moore et al., 2019b, p. 12)

Informing Generalized Models of Student Thinking Whereas Moore, Stevens, et al. (2019b) focused on students’ graphical meanings enacted in-the-moment of problem-solving, the distinctions between figurative and operative thought are also useful in articulating generalized models of students’ graphical meanings. These generalized models, referred to as epistemic subjects, are conceptual models that specify categorical differences among students’ meanings and ways of thinking (Steffe & Norton, 2014; Thompson, 2013a). Specifically, the distinctions between figurative and operative aspects of thought informed Moore and Thompson’s constructs of static (graphical) shape thinking and emergent (graphical) shape thinking, respectively. Moore and Thompson (Moore, 2021; Moore & Thompson, 2015) introduced static shape thinking as a way to refer to meanings that involve conceiving a graph as if it is essentially a malleable piece of wire (graph-as-wire). They introduced emergent shape thinking as a way to refer to meanings that involve conceiving a graph as a trace entailing corresponding values (or magnitudes) that are produced through the covariation of quantities. Examples of static shape thinking include iconic translations—associations between the visual features of a situation and those of a graph (Monk, 1992)—and thematic associations—associating properties of event phenomena with shape properties of a graph that are superfluous to the stated quantitative referents (Thompson, 2016). An example of an iconic translation is a student perceiving a graph as containing a circle because the situation entails a circle. An example of a thematic association is a student perceiving an object traveling at a constant speed as necessarily implying a line for a graph or an object traveling at a varying speed necessarily implying a curve for a graph, regardless of the quantities being graphed. Other examples of static shape thinking include facts of shape in which figurative

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properties of a graph (or perceptual comparisons between graphs) imply particular equations, analytic rules, names, mathematical properties, and so on. For instance, an individual might associate a graph that curves up with an exponential relationship, or an individual might associate a line with a linear relationship regardless of the coordinate system. As another example, a student might associate the shape of a graph with only one analytic rule, which constrains their ability to accommodate novel graphing experiences (Moore, 2021). In each case (Fig. 4.3), the associations foreground perceptual properties of shape and associations based on those properties. Examples of emergent shape thinking include any instance in which an individual conceives a displayed graph as the product of coordinating and tracking two or more varying quantities. This could occur in the context of a student conceiving a displayed graph as having emerged through tracking quantities’ covariation, or it could occur in the context of a student producing a displayed graph as a trace that captures some conceived covariational relationship. The latter case is often for the purpose of producing a displayed graph that is a quantitative model of some situation or relationship held in mind. Producing such a displayed graph involves the student re-constructing the coordination of covarying quantities to produce a trace they perceive as capturing a covariational relationship equivalent to the relationship conceived in the situation or relationship held in mind (Fig. 4.4). Relatedly, a student can reason emergently to compare and contrast displayed graphs in terms of the covariational properties involved in producing their trace. For instance, a student can reason emergently to conceive two displayed graphs as representing an invariant covariational relationship despite perceptual differences between the graphs due to their being represented under different scales (Fig. 4.5), orientations (Fig. 4.1), or coordinate systems (Fig. 4.6) (Moore et al., 2013). Similarly, emergent shape thinking could involve a student conceiving a displayed graph as being produced by two different covariational relationships despite the result of each trace being perceptually equivalent (Fig. 4.7, left and right, each illustrating a progressive trace). In each case, any association a student makes with a graph is an implication of the covariational relationship they understand the displayed graph to have emerged as a trace of. The relationships between Thompson’s framing of figurative and operative thought and Moore and Thompson’s graphical shape thinking constructs are apparent by the object of reasoning—a displayed graph—and its associations. With respect to static shape thinking, such a way of thinking foregrounds figurative aspects of thought; static shape thinking entails actions based in perceptual cues and properties of shape. A displayed graph is essentially an object in and of itself, and associations with the displayed graph are learned facts of the shape qua shape. Such associations are consistent with von Glasersfeld’s (1987, 1995) discussion of figurative icons, symbols, or words. In the case of constructing a displayed graph, a student thinking statically carries out activity for the purpose of drawing a shape consistent with learned or memorized associations. Those learned or memorized associations are not conceived as organic or inherent to the production of the graph, but rather they are simply taken as assigned to a produced shape that is perceptually consistent with the association. The learned or memorized associations are abstractions based on the products of activity and particular states. Furthermore, because a

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Fig. 4.4  Constructing a covariationally equivalent emergent trace

Fig. 4.5  Two displayed graphs of y = 45 sin(2πx)

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Fig. 4.6  Two displayed graphs equivalent to the relationship defined by z = 2 k+1

student thinking statically foregrounds perceived properties of a produced or anticipated shape, they can reject a displayed graph generated via reasoning emergently when the resulting properties of the produced shape are incompatible with the student’s static shape thinking (see Polly in Moore et al., 2019b). With respect to emergent shape thinking, such a way of thinking foregrounds operative aspects of thought; emergent shape thinking entails the coordination of mental operations in the form of quantitative and covariational reasoning. A displayed graph is understood as reproducible through the coordination of the mental operations that resulted in its trace, and associations with the graph are anticipated as the mathematical properties organic to the operations that produce it. Any learned or memorized associations are understood as properties of or symbolizing the coordination of operations represented by the displayed graph. Such associations are consistent with von Glasersfeld’s (1987, 1995) discussion of operative icons, symbols, or words. Furthermore, because a student’s thinking emergently foregrounds the coordination of mental operations, their meanings are positioned to accommodate novel graphing experiences. The coordination of those mental operations and their perceived mathematical properties provide actions of thought that can be transformed into novel figurative material, all the while maintaining invariance in the logico-mathematical form. That is, emergent shape thinking enables the enactment of operations that can differ in terms of their figurative entailments yet be compared on the basis of similarities and differences in the mathematical properties of those operations.

Adapting the Distinctions to Other Representations Thus far, we have illustrated examples of figurative and operative thought through graphical representations and phenomena. However, definitions of figurative and operative thought have been adapted in ways to address other representations. In

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Fig. 4.7  One completed displayed graph, two covariational relationships (left, r = sin(2θ); right, r = |sin(2θ)|)

this section, we consider these adaptations when discussing students’ reasoning with formulas. As a reminder, Thompson (1985, p. 195) said, “Any set of schemata can be characterized as figurative or operative, depending upon whether one is portraying it as background for its controlling schemata or as foreground for the schemata that it controls.” In considering how to adapt the constructs of figurative and operative thought in a new representation, one needs to consider what mental operations are involved in students’ acts of re-presentation. With formulas, a researcher characterizing an enacted meaning does not rely on perceptual actions taken on material,

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such as tracing right to left as Moore, Stevens, et al. (2019b) described, or repeating partitioning activities across contexts as Liang and Moore (2021) described. Whereas graphical representations and phenomena provide material to engage operatively or figuratively, formulas merely provide symbols in the form of inscriptions or glyphs. Thus, a researcher characterizing a student’s formula meanings as figurative or operative relies on the associations evoked by a symbol (or collection of symbols) in a formula. In a teaching experiment in which Stevens (2019) attempted to understand and support students’ meanings for formulas through reasoning with dynamic situations, she built upon research involving the Piagetian constructs of multiplicative objects and figurative and operative thought to model students’ ways of thinking about graphs. Her strategy of the experiment focused on designing tasks that would enable her to construct models of students’ mathematics by identifying opportunities for students to reason quantitatively and covariationally with formulas, and her analysis focused on characterizing students’ mental actions in terms of the nature of associations evoked by given or constructed formulas. In particular, she adopted the constructs of figurative and operative thought to differentiate between associations that stemmed from perceptual similarities or learned facts and those that stemmed from enacting quantitative operations and abstracting their mathematical properties. To discuss figurative thought in more detail, consider how students might answer the following prompt: “Describe a situation in which the formula A=2πrh describes a relationship between quantities. How does your situation describe that relationship?” The meanings students have for the formula A = 2πrh dictate the associations they will draw between a situation and the formula. Consistent with static shape thinking examples above, a meaning foregrounding figurative aspects of thought would foreground associations between attributes or shapes (e.g., area of circle, or surface area of a cylinder or spherical cap) and aspects of the symbols constituting the formula. These associations may or may not have quantitative entailments, but importantly those entailments are not inherent to the association. For example, a student may conceive the formula A = 2πrh as the normative formula for the surface area of an open cylinder, and only an open cylinder (see Stevens (in press) for this example). As a second example, a student might also identify “2π” as necessarily pointing to a feature of a circle and, thus, that the relevant situation must include a circle. As Stevens (in press) illustrated, a student stated that A  =  2πrh could not represent a formula associated with a parallelogram. They explained, “Because I don’t know what it [2π] would represent in a parallelogram. Like, in a circle, it’s because you can like divide a circle into two pi radii, but you don’t have anything even here that you could do that with.” In both examples, the reasoning foregrounds associations between the arrangement or presence of the glyphs in the formula and particular shapes. Thus, in the event that the associations do have quantitative entailments, those entailments are constrained to the objects of association. Alternatively, operative thought would entail meanings that focus on the formula as representing quantitative relationships that could be re-presented in a variety of contexts. As an example, A = 2πrh represents a linear relationship between height

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and area that could be relevant to a rectangle, cylinder, or spherical cap, depending on what quantities the symbols represent. A student could thus generalize that A = 2πrh means that for equal changes in height, the surface area of the relevant shape, whatever it is, increases by an amount that is always 2πr times as large as that change in height. Stevens (2019) described a student who reasoned that such a relationship was implied by the formula, and the student then generated that relationship using cylinder and spherical cap contexts. Figure 4.8 provides an illustration of this way of reasoning; the corresponding strips of the cylinder and spherical cap have equal surface area. Consistent with the notion of operative thought, the student’s reasoning foregrounds quantitative relationships, and the coordination of quantitative operations drives their activity and products. It is important to note that figurative associations, such as associating πr2 with the area of a circle, can be useful in constructing a quantitative structure in a context. Similar to the notion of expert shape thinking (Moore & Thompson, 2015), as long as a student can unpack those associations quantitatively (similar to the reasoning illustrated in Fig. 4.8), while also anticipating that there can be other potential associations. Such associations reduce cognitive effort; it is inefficient to always enact quantitative operations. It is also important to note the effects of foregrounding figurative associations in constructing formulas. A student who constructs figurative associations between geometric shapes and formulas in a way that dominates any quantitative entailments may attempt to identify perceptual features in the context and then attempt to incorporate formulas associated with those figurative elements in their construction of a formula, resulting in a nonquantitative formula. For instance, in Stevens (2019), a student, Kimberley, identified both a dynamic cone and cylinder as including circles and varying height. She concluded that only one should be able to be associated with the formula A  =  2πrh. She eliminated the cone by foregrounding figurative aspects of thought and breaking the cone into particular shapes that she could then associate with area formulas. As a result, Kimberley’s formula for the cone combined her formulas for the area of the circle with the area of a triangle (Fig. 4.9). In summary, although mathematics education researchers have primarily focused the figurative and operative distinctions on representations or phenomena in which students can enact quantitative operations, the distinctions are also applicable to the

Fig. 4.8  An example of operative thought with A = 2πrh by considering A and h as variables to conclude that for equal changes in height, there are equal changes in surface area for cylinder and spherical cap (A = surface area, r = radius length of circle/sphere, h = vertical height)

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Fig. 4.9  Example of figurative thought used to construct formula for surface area of a cone by multiplying together shapes in geometric figure

associations students construct for symbols, groups of symbols, or formulas. Furthermore, such associations are consistent with von Glasersfeld’s (1987, 1995) broader discussion of icons or symbols with attention to the extent a formula evokes a learned association or a set of anticipated quantitative operations. Incorporating a focus on formulas also helps illustrate that although associations that are figurative in nature (e.g., A = πr2 provides a way to calculate the area of a circle) are beneficial for reducing cognitive load, it is important that such associations are not constructed at the loss of formulas representing a quantitative relationship specific to a context or particular object (e.g., A = πr2 representing that the area of a circle increases by increasing amounts as the radius increases by a constant, successive amount). Furthermore, it is important that such operative associations are not constructed at the loss of a formula potentially representing a variety of quantitative relationships (e.g., A = πr2 is the area of a rectangle with a side π times as large as the other side). In summary, operative meanings that foreground symbols and groups of symbols in formulas as pointers to quantities, quantitative operations, and covariational operations are generative in that they enable a student to conceive formulas as relevant to both experienced and unexperienced situations, all the while understanding the formula captures some invariance property across those situations.

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 ransitioning the Constructs Back to the Study T of Meaning Construction The examples in the prior section used figurative and operative distinctions when discussing undergraduate students’ reasoning with representations (e.g., graphs and formulas). The aforementioned work occurred after students had constructed particular meanings for those representations through schooling; the studies did not address students’ initial construction of graphical meanings. In this section, we illustrate the extension of these constructs to the construction of graphs, which was led by Tasova (2021). Tasova (2021) adopted figurative and operative thought to characterize middle-school students’ graphing meanings as they engaged in solving problems involving the construction of graphical representations of varying quantities for what appeared to be the first time (at least within formal settings). In doing so, he illustrated the viability of explaining middle-school students’ reasoning with the figurative and operative distinction, while also adapting those distinctions to explain his data. Ultimately, he identified meanings compatible with figurative or operative aspects of thought. Underscoring von Glasersfeld’s (1995) framing that operations always have to operate on something, Tasova also identified important meanings that entailed aspects of both figurative and operative thought in the same activity. We summarize those findings here. Before supporting middle schoolers in developing meanings for graphs in two-­ dimensional space, Tasova (2021) provided them opportunities to engage with quantities’ magnitudes represented by varying lengths of directed bars placed on empty number lines (also called magnitude lines, see Fig. 4.10, right). Students had opportunities to use these dynamic segments to represent individual quantities in one-dimensional space. As one example, students played the bike animation (see Fig. 4.10, left) and pulled the blue segment in ways that could represent how the bike’s distance from Arch (DfA) varied in the situation. Tasova (2021) identified two meanings for the dynamic segments that align with figurative and operative aspects of thought. An individual who conceives the bike’s DfA in the situation as decreasing and increasing might vary the length of the segment on the magnitude line accordingly. This is an example of an operative aspect of thought as the individual disembeds the conceived quantity’s magnitude from the situation, re-presents it on the magnitude line, and then coordinates the length of the

Fig. 4.10  Downtown Athens Bike Task and magnitude line

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blue bar, ensuring to preserve its length as the animation played. As an alternative conception, an individual can simulate the physical bike on the magnitude line. For example, they might coordinate the direction they “pull” one end of the blue segment with the bike’s right and left movements, thus associating the movement of the bike with the movement of the segment endpoint. Tasova identified this as an example of a figurative aspect of thought as the student’s thought is dominated by imitating the moving object in the situation. As a second example of using the figurative and operative distinction to characterize a nuanced difference in middle-school students’ actions, we use another case that involves re-presenting a quantity on the magnitude line. In this case, the bike in the animation and the dynamic segment on the magnitude line are synchronized— the length of the blue bar on the magnitude line varies according to the bike’s movement on the map by design (see the animation at the following link: https://youtu. be/6kdbDeVEF9w). For example, while moving the bike to the right from its position (Fig. 4.10, left), the right end side of the blue bar on the magnitude line moves to the left (indicating the bike’s DfA is decreasing). This is different from the previous task in that the endpoint moves in the opposite direction of the bike. Explaining how this could happen, a seventh-grade student referred to the map and claimed, “The bike is getting closer to Arch.” Then, by pointing to the blue bar on the magnitude line and tracing the pen over the line from right to left, the student said it “is gonna get closer to right here [pointing to the zero point on the magnitude line], which is Arch.” Tasova argued that the student was not conceiving the bar as a varying length, but instead that the student conceived of the Arch and bike as physically placed on the left and right endpoints of the blue bar, respectively. He thus perceived the blue bar’s change to be a product of the bike moving closer in proximity to the Arch along the blue bar. Tasova contrasted this with a response from another student who first determined that the bike’s DfA is decreasing, while moving the bike to the right on the map. The student then conceived the length of the blue bar on the magnitude line as a re-presentation of the bike’s DfA, thus requiring that the length decrease to remain equivalent to the length in the situation. She explained, “it [pointing to the blue bar] is gonna get smaller because distance is smaller on the number line too.” Moreover, she labeled the starting point as “zero,” whereas the previous student conceived the same point on the magnitude line as “Arch.” Reflecting on the two students’ actions, they are compatible in that they each had the same result from an observer’s perspective: the students moved the blue bar endpoint so that it decreased in length. But their meanings were notably different. In the first case, and consistent with reasoning that foregrounds figurative notions of thought, the meaning foregrounded the spatial proximity of two objects and moving them accordingly. Any change in the blue bar came as a consequence of changing their spatial proximity. In the second case, and consistent with reasoning that foregrounds operative notions of thought, the meaning foregrounded the distance between two objects. The length of the blue bar was persistently in mind, and the student’s goal was to maintain a length equivalent to that in the situation. Although these differences are subtle, Tasova (2021) illustrated they have significant

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Fig. 4.11 (a) DABT, (b) Zane’s graph, and (c) Melvin’s graph

consequences for students’ graphing actions. To illustrate, we use two middle-­ school students’ activity during the Downtown Athens Bike Task (DABT). DABT prompts students to use a Cartesian coordinate system to draw a sketch of the relationship between the bike’s DfA and the bike’s distance from Cannon (DfC) as the bike moves on Clayton St. in Downtown Athens (see Fig. 4.11a). The bike starts from the West side of the street and moves at a constant speed. To illustrate an example of figurative thought, we draw on Zane’s (a seventh grader) activity in which his graphical meanings representing literal pictures of a particular situation. Zane conceived the Arch and Cannon placed at physical locations on the vertical and horizontal axes, respectively, as implied by the labels (see blue and purple dots on each axis for Arch and Cannon, respectively, in Fig. 4.11b). Because Arch is at the top and Cannon is at the bottom of the plane, Zane then imagined rotating the plane in a way that Arch is at the bottom and Cannon is at the top, just as with the map. He then drew a segment on the plane, identifying it as the path of the bike. He also added dots on the plane (i.e., one that he labeled “Bike” and the other at the origin), which he placed to match the dots at each end side of the bike’s path on the map (see Fig. 4.11a). Zane went to the extent of using the same coloring for the segment on the plane as that on the map (i.e., light blue). Drawing on the notion of iconic translation (Clement, 1989; Monk, 1992), Tasova (2021) characterized

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Zane’s meaning as transformed iconic translation. Zane translated a transformed version of perceptual features of the situation to the plane as he rotated the plane and overlaid it into the map in order to perceptually match the graph that he drew on the plane with the bike’s path on the map. Whereas Zane’s example is a rather typical example of a graphical meaning that foregrounds figurative aspects of thought, Melvin’s activity in DABT provides a novel illustration of how an individual’s meaning might involve both figurative and operative aspects of thought. Melvin conceived the entire vertical and horizontal axes of the plane as the physical Cannon and Arch, respectively. The axes did not represent magnitude or number lines, but instead were referents to the objects themselves. Melvin then drew a line upward from left to right to represent “where the bike travels.” He added tick marks and dots on his line graph “to represent like where the bike could be.” For example, when the bike is at location 2 in the map (i.e., DfA and DfC are each at their minimum), he said, “It [the bike] would go right here [pointing to the tick mark on his graph near 2 in Fig. 4.11c].” Similar to Zane, and consistent with reasoning that foregrounds figurative aspects of thought, Melvin conceived each point on his graph as the physical bike moving on its path on the plane. Differing from Zane, Melvin also imagined moving the bike on the plane according to the variation of its DfA and DfC. He conceived DfC and DfA as being re-­ presented by vertical and horizontal segments drawn on the plane, respectively. The vertical segments on the plane represented the bike’s DfC because Melvin assimilated the horizontal axis as a reference ray that he measured the bike’s DfC from. Similarly, the horizontal segments on the plane represented the bike’s DfA because he assimilated the vertical axis as a refence ray he measured the bike’s DfA from. Increasing length of the horizontal and vertical segments on the plane indicated an increase in the bike’s DfA and DfC. Moreover, Melvin understood that the length of the horizontal segment is shorter than the length of the vertical segment for each point on the graph as the bike’s DfA is always less than the bike’s DfC on the map. Thus, although Melvin assimilated his graph as “where the bike travels,” Melvin’s graph was not the bike’s path as it is seen on the map; it was not an iconic translation. Melvin’s meaning of the points included determining quantitative features of the bike in the situation (i.e., its DfA and its DfC), and he ensured they preserved those quantitative properties on the plane. We consider Melvin’s graphing activity as a different way of graphing relationships because his initial activity foregrounded figurative aspects of thought, but he then engaged in quantitative operations to make sense of the results of that initial activity and produce a graph to his satisfaction. Specifically, he represented the quantities’ magnitudes in the space by committing to two frames of reference (i.e., the axes), and then placing a point where those magnitudes meet on the path he produced. Tasova and Moore (2020) termed this meaning of points, which has both operative and figurative entailments, as a spatial-quantitative multiplicative object, which involves an individual envisioning a point on the plane as a location/object that also entails quantitative properties. This example underscores that the figurative and operative distinction is not merely binary with respect to students’ meanings.

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As we discuss below, we believe this meaning, along with other meanings Tasova identified, illustrates the need to better understand how figurative and operative notions of thought play a role in students’ construction of meanings that serve as both immediately and longitudinally productive.

Implications for Methodology and Task-Design In addition to providing tools to construct second-order models of students’ mathematics, as illustrated above, we have, in turn, found the figurative and operative distinctions useful in designing empirical studies. Specifically, we have found the distinctions useful in designing tasks and framing evidence for students’ quantitative and covariational reasoning. Their usefulness in this regard can be organized by Moore et al. (2022) notion of an abstracted quantitative structure (AQS), which the authors introduced as criteria for concept construction from a quantitative reasoning perspective. They described an AQS as a system of quantitative operations that an individual has interiorized so that it: C1. is recurrently usable beyond its initial experiential construction; C2. can be re-presented in the absence of available figurative material including that in which it was initially constructed; C3. can be transformed to accommodate to novel contexts permitting the associated quantitative operations; C4. is anticipated as re-presentable in any figurative material that permits the associated quantitative operations. (Moore et al., 2022).

The criteria were directly informed by the figurative and operative distinctions (Moore et  al., 2022). C2 reflects the distinction between figurative and operative counting schemes as discussed by Steffe and colleagues (Steffe & Olive, 2010), as well as Glasersfeld’s (1995) distinction between recognition and re-presentation. C3 captures the distinction introduced by Thompson (1985), allowing for the presence of figurative material and resting on the extent an individual is able to transform their actions to account for novel experiences and associated figurative material. C4 further extends C2 and C3  in a way that combines aspects of both Steffe’s and Thompson’s framings. Reflecting Steffe’s framing, C4 involves an individual anticipating particular quantitative operations and their mathematical properties in the absence of figurative material. Reflecting Thompson’s framing, C4 also involves the individual understanding that those operations and their mathematical properties are generalizable to any instance in which those operations are viable, thus allowing the individual to anticipate their enactment in the presence of novel figurative material. Such anticipation enables us to attribute particular quantitative and covariational relationships to quantities and figurative material too complex for us to enact those operations within. For instance, temperature is far too quantitatively complex for most individuals to enact particular quantitative operations, but we can anticipate variations in temperature and use proxies (e.g., values, number

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lines, and coordinate systems) to represent those variations in a way that enable us to enact quantitative operations as if we are reasoning about temperature.6 As mentioned above, Moore et  al.’s (2022) main intention of introducing the construct of an AQS was to provide guiding criteria for concept construction, but it also provides a framing for designing empirical studies and situating researcher claims regarding students’ quantitative and covariational reasoning. Said succinctly, a researcher’s claims regarding a student’s quantitative and covariational reasoning are necessarily constrained by the nature of the tasks they use with a student.7 To illustrate this, we discuss each framing criteria above with respect to task design and potential claims regarding students’ quantitative and covariational reasoning. C1 refers to a quantitative structure that is recurrently usable beyond its initial experiential construction. From a researcher’s perspective, and when considered independently of C2–C4, C1 is the most trivial of the criteria because it refers to an individual re-enacting a previously constructed quantitative structure in the presence of previously experienced context or figurative material. Thus, from a task-­ design perspective, this involves engaging the student in a task the student has already experienced and enacted the relevant quantitative operations. Here, “the same task” need not mean a carbon-copy of the previous task. Rather, “the same task” encompasses the figurative material provided and the quantitative operations needed to solve the task in the way the researcher hypothesizes. The researcher might change aspects of the tasks they consider to be surface-level features like values (e.g., the numbers being summed in a counting task) based on their model of the student’s mathematics and what might be perceived as the same from the student’s perspective. A contraindication of C1—the student not being able to re-enact particular operations in the presence of an identical task—is evidence of the absence of quantitative or covariational reasoning. We do not consider indications of students having constructed an AQS consistent with C1 to enable strong claims regarding a student reasoning quantitatively or covariationally. Indications of C1 can be produced by a student mimicking or reproducing memorized actions from previous experiences. In such cases, a student is able to rely on the available figurative material and recall traces of activity from previous experience to reproduce previous results. However, to the researcher, the observable actions of such a student might be indistinguishable from those of a student enacting quantitative operations. It is thus necessary to design tasks that enable differentiating from a student mimicking or reproducing  This underscores one of the more critical roles graphing plays in mathematics. Coordinate systems and their graphs provide quantities (e.g., oriented lengths or angle-openness) of which individuals can enact quantitative operations, thus serving as useful proxies for those quantities on which we cannot enact quantitative operations. 7  It is more accurate and compatible with our theoretical perspective to say that a researcher’s claims regarding a student’s quantitative and covariational reasoning is necessarily constrained by the observable actions of the student’s engagement. For the purposes of this chapter, it is more straightforward to discuss tasks as if they have some inherent or objective design feature, but we remind the reader that no such features exist and a task is always defined by the student’s engagement. 6

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memorized actions from a student whose reasoning foregrounds quantitative operations, which leads us to C2. Much like C1, C2 is based on a student being able to re-enact particular quantitative operations. Unlike C1, the re-enactment comes in the absence of all or some subset of figurative material. With respect to a counting situation that involves an individual summing two collections of stones, the researcher might adjust the task so that one collection or both collections of stones may be hidden from the student with them only being told the numerosity of the hidden collections. As another example drawn from our own work, after a student constructs a particular relationship (e.g., the sine relationship) in the context of a Ferris wheel rider (Fig. 4.12), the researcher might prompt the student to re-construct that relationship with no figurative material provided (Fig. 4.13). Or, the researcher might provide a subset of figurative material as we did with the Which One? task in which a participant is asked to choose which red segments, if any, appropriately represent the height above the center (Liang & Moore, 2021, p. 300) (Fig. 4.14, also see the following link: https:// youtu.be/2pVVGl8eEr0). Note that these collections of tasks vary with respect to what is perceptually available to the student and what the student might be asked to mentally (or physically) generate, thus supporting a researcher in characterizing a student’s re-presentation capacity. In the Which One?  task,  we removed the Ferris wheel spokes because those spokes provide figurative material that can assist with partitioning the rider’s trip

Fig. 4.12  Amounts of change and the sine relationship for π/2 radians of rotated arc

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Fig. 4.13  A task with no provided figurative material

Fig. 4.14 The Which One? task environment as presented to students (Liang & Moore, 2021, p. 300), in which they are prompted to determine which red segments, if any, represent the point’s height above the center of the Ferris wheel in relation to the arc length traversed by the point

into equal increases in arc length and corresponding variations in height. We also designed the task to include a digital environment in which they did have the availability of red segments, and these red segments could be re-oriented and embedded in the circle if the participant chose to do so. However, we designed the task so that the participant could not physically draw within the environment. Thus, as they moved the point around the circle, the chosen red segment varied accordingly, either in its given position or in a re-oriented position if they chose to do so. This required that the participant be able to hold in mind different states of the red segment and mentally produce amounts of change in that segment without the assistance of constantly available figurative material (Fig. 4.15).8 A contraindication of C2—the student not being able to re-enact particular operations in the absence of figurative material—is evidence that their meanings are tied to particular figurative material and carrying out activity tied to perceptual properties of that material. On the other hand, indications of students having constructed an AQS consistent with C2 enables a researcher to make relatively stronger claims

 The reader might note that the correct red segment can be determined solely through identifying that it is the correct length at each state. If a student bases their choice by checking states, we ask “Show us how to see the amounts of change in the red segment?” in order to prompt them to describe or generate partitions illustrating appropriate amounts of change. 8

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Fig. 4.15 The Which One? task  and re-positioning a red segment as the object moves around the circle

regarding a student reasoning quantitatively or covariationally. An indication of C2 provides some evidence that the individual is able to re-enact quantitative operations to generate or anticipate figurative material that is no longer available. However, like C1, in our experience, students are able to provide evidence of C2 by what amounts to mimicking previous activity in order to produce absent figurative material. The production of that material often provides them source material for reflection and comparison to recalled previous activity and its results. Although the individual might enact quantitative operations in reflection and with the assistance of the material they have produced, this is not consistent with C2 as the operations enacted in reflection were not driving the production of the figurative material. Whereas C2 involves an individual re-enacting quantitative operations in contexts previously experienced, C3 captures instances of an individual accommodating previously enacted quantitative operations to account for novel contexts. Much like a researcher determining what counts as “the same task” is dependent on their model of a student’s mathematics, what a researcher determines to be “a novel context” is also dependent on their model of a student’s mathematics. A novel context to a student could be a new phenomenon, a graphical representation, a graphical representation under a different coordinate orientation, or a graphical representation under a different coordinate system, to name a few (Fig. 4.16) (see the following studies for a collection of tasks designed with C3 in mind: Liang & Moore, 2021; Moore et al., 2013, 2019a, b; Paoletti et al., 2018). In some instances, such as changing a phenomenon or graphical coordinate system, what is novel may be the phenomenon and quantities under consideration. In other instances, such as considering different coordinate orientations, what is novel may be the figurative orientation of the quantities due to a change of referent. Regardless, a task designed with C3 in

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Fig. 4.16  Various task-design options in order to transition from a previously experienced context to a potentially novel context

mind affords the student the opportunity to re-enact previously enacted quantitative operations in the presence of different quantities or figurative orientations with the goal of perceiving an invariance across the differing contexts.9  We note that C2 can be incorporated when designing tasks with C3 in mind. As an example, a researcher might intend to gain insights into whether a student can conceive a graphical representation as quantitatively equivalent to some relationship they conceive in a phenomenon. Taking C2 into account, the researcher might ask the student to construct that graphical representation, or they might provide figurative material in the form of displayed graph or graphs and prompt the student to determine which graph(s) accurately represent the relationship. 9

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Fig. 4.17  A displayed graph emerging in a way that is invariant with that of the phenomenon

A hallmark of operative thought, per Thompson’s (1985) framing, is that the coordination of mental operations and their transformation dominate an individual’s activity. For that reason, indications of C3 provide a researcher with a strong evidentiary basis for claiming a student’s reasoning foregrounds quantitative and covariational reasoning. Due to figurative differences across previously experienced and novel contexts, a researcher implementing tasks with C3 in mind is positioned to gain insights into whether a student can conceive quantitative invariance across those contexts despite figurative differences in their actions (Fig. 4.17). A researcher is simultaneously positioned to gain insights into when a student is mimicking previous activity. In instances in which a student provides a contraindication of C3, it suggests that there is some aspect of their meaning tied to figurative aspects of the contexts in which they previously enacted that meaning. This is not to say that the student’s reasoning in those prior contexts was not quantitative, but rather that some aspect of their reasoning was reliant on particular figurative features of that context

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and they thus looked to maintain those figurative features. For instance, in transitioning from a phenomenon like a Ferris wheel ride, the student might produce a graph from memory and then try to maintain partitions along a curve because there are partitions along a curve in the phenomenon (Fig. 4.18). Here, the graph does not emerge as a product of quantitative operations, but rather is produced to provide material to mimic activity or enact operations. As we discuss in the following section, students’ attempts to maintain both quantitative and figurative aspects of their actions often lead to experienced perturbations due to incompatibilities between those maintained aspects. Because C4 depends on an individual’s capacity to anticipate some structure of quantitative operations, designing tasks incorporating this principle is complex. Furthermore, our research team has not concentrated on designing such tasks to the extent we have with that of C1–C3. In working with PSTs, one fruitful approach is designing tasks to include hypothetical student work that is nonnormative. Doing so provides a researcher insight into whether or not the PST anticipates that the hypothetical student might have produced the work through the viable enactment of the relevant operations. In the case that the PST does anticipate such an event, they may then attempt to determine that viable enactment (i.e., C3). In the case that the PST does not anticipate such an event, the PST is likely to enact meanings that identify some incorrect element of the student’s work. Annika’s response to Fig.  4.2 and determining numerous viable ways the figure is a displayed graph of y = 3x is an example of the former. An example of the latter would be a PST immediately rejecting Fig. 4.1b as a potential displayed graph of y = 3x because of it sloping downward left-to-right, or their rejecting Fig.  4.1a because the axes are incorrect (see Moore et al., 2019a for other examples). Because C4 itself does not entail the enactment of quantitative operations, we do not consider indications of it to be evidence for quantitative or covariational reasoning. Rather, indications of C4 suggest that the individual has abstracted meanings that foreground quantitative and covariational reasoning to the extent that the relevant system of operations has become a way of thinking (Harel, 2008a, b; Thompson

Fig. 4.18  Student work in which they drew a graph from memory and then attempted to re-­ produce partitions that maintained orientations and placements along segments or curves (Liang & Moore, 2021, p. 306)

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et al., 2014). On the other hand, contraindications of C4 imply the individual has abstracted meanings that could entail quantitative and covariational reasoning, but they are such that they foreground some other aspect or form of thought. As the individual then enacts that meaning to make sense of the task, the researcher gains insights into the operations and associations constituting that meaning.

Moving Forward The progress of mathematics education research is contingent on researchers pushing into new areas. In some cases, such pursuits require adopting or developing new theoretical perspectives or constructs. In other cases, such pursuits involve testing and adapting available theoretical perspectives or constructs. The work described in the previous sections is an example of the latter. Such an orientation toward theory stresses that theoretical perspectives and constructs should be pushed into new areas of study in an attempt to put that theory to the test. Doing so enables testing the viability of the theory and expanding its generalizability through adaptations that respond to constraints experienced in applying or developing the theory. Thus, our general suggestion for future work is that mathematics education researchers look to test the viability of the figurative and operative distinction within whatever area is of interest to them. Relatedly, we underscore that in pursuing such work, researchers keep in mind that they should not merely look to apply the distinctions as other researchers have, but rather view the notions of figurative and operative thought as malleable to their needs and experiences carrying out empirical work. Synthesizing the opening quotes by Dewey and Thompson, the weight of theory is in its ability to aid researchers’ pursuits of problems and explanations of phenomena. With respect to specific suggestions for research, we provide two potentially productive avenues. Our first suggestion is to continue the extensions taken by Tasova (2021) and Stevens (2019) to investigate the ways in which the figurative and operative distinction is relevant to students’ construction of mathematical ideas and other representational systems. We envision at least two pivotal insights such research might provide. Research along the suggested lines will provide insights into how and the extent to which students construct meanings that transition from foregrounding figurative aspects of thought to those meanings in which operative aspects of thought dominate figurative entailments. Relatedly, such research will provide insights into those experiences that support or inhibit students’ construction of meanings that foreground either figurative or operative notions of thought. Research along the suggested lines might also provide insights into individuals’ construction of incompatible or competing meanings. Tasova (2021) introduced the term competing meanings when expanding on Moore, Stevens, et  al.’s (2019b) observation of a PST (Polly) having enacted two graphical meanings, one that foregrounded operative notions of thought and one that foregrounded figurative notions

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of thought. In the case of Moore, Stevens, et  al. (2019b), the PST perceived her enacted meanings to be incompatible with each other, which generated a perturbation and led to the PST rejecting the displayed graph she had constructed via reasoning emergently due to her having drawn the graph from left-to-right; she claimed that drawing a graph that way was “backwards.” Similarly, in his work with middle-­ school students, Tasova (2021) noted that students enact multiple graphical meanings. In some cases, the enacted meanings were incompatible with each other, leading to a student experiencing a perturbation and ultimately having to make an accommodation to their meanings or abandon one of their meanings. In other cases, a student’s enacted meanings were not necessarily incompatible with each other, but instead, each provided different avenues to solving the perceived problem. Regardless, Tasova’s (2021) notion of competing meanings generates a few questions for research that could benefit from incorporating the figurative and operative distinction: What are the different meanings students construct for a particular topic, and to what extent are those meanings compatible or incompatible? To what extent are particular competing meanings epistemological obstacles (Harel & Sowder, 2005) or natural to cognitive development, as opposed to artifacts of canonical educational approaches? How might competing meanings be productively addressed and leveraged by educators and researchers? With respect to this last question, there is the potential to generate productive learning experiences via students reflecting on competing meanings, particularly when such reflection is engendered by a moment of perturbation. Our second suggestion for a research line moving forward is also an extension, and one in the direction of upper-level mathematics (e.g., undergraduate mathematics and above). This suggestion is motivated by personal conversations with Anderson Norton, Shiv Smith Karunakaran, David Plaxco, and Paul Dawkins. These conversations revolved around the potential role of the figurative and operative distinction in the areas of proof and proving, logic, and the study of advanced mathematical objects. In short, those conversations have revolved around the tendencies of mathematicians and mathematics educators to reduce those areas to formalisms, including syntactical rules, proof structures or templates, logic models, algebraic systems, and other objects that they take to encapsulate operations. As Norton explained, such a framing emphasizes “the conceptualization of formal mathematics, as opposed to formalism deriving from the conceptualization of operations” (Personal communication). A productive line of research, which is suggested by Norton’s (2022) recently published book, may instead be to consider how the coordination of mental operations and meanings rooted in such provide a springboard for the formalizations required in upper-level mathematics. Acknowledgments  This material is based upon work supported by the National Science Foundation under Grant Nos. DRL-1350342, DRL-1419973, and DUE-1920538. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or our respective universities. Thank you to PME-NA and SIGMAA on RUME for the opportunity to present a previous version of this manuscript. Thank you to Anderson Norton for several conversations that shaped and contributed to this chapter.

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References Byerley, C. (2019). Calculus students’ fraction and measure schemes and implications for teaching rate of change functions conceptually. The Journal of Mathematical Behavior, 55, 100694. https://doi.org/10.1016/j.jmathb.2019.03.001 Carlson, M.  P., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378. https://doi.org/10.2307/4149958 Clement, J. (1989). The concept of variation and misconceptions in Cartesian graphing. Focus on Learning Problems in Mathematics, 1(1–2), 77–87. Dewey, J. (1929). The sources of a science of education. Liveright Publishing. Ellis, A. B. (2007a). The influence of reasoning with emergent quantities on students’ generalizations. Cognition and Instruction, 25(4), 439–478. Ellis, A.  B. (2007b). A taxonomy for categorizing generalizations: Generalizing actions and reflection generalizations. Journal of the Learning Sciences, 16(2), 221–262. https://doi. org/10.1080/10508400701193705 Ellis, A. B., Özgür, Z., Kulow, T., Williams, C. C., & Amidon, J. (2015). Quantifying exponential growth: Three conceptual shifts in coordinating multiplicative and additive growth. The Journal of Mathematical Behavior, 39, 135–155. https://doi.org/10.1016/j.jmathb.2015.06.004 Ellis, A. B., Lockwood, E., Tillema, E., & Moore, K. C. (2022). Generalization across multiple mathematical domains: Relating, forming, and extending. Cognition and Instruction, 40(3), 351–384. https://doi.org/10.1080/07370008.2021.2000989 Harel, G. (2008a). DNR perspective on mathematics curriculum and instruction, part I: Focus on proving. ZDM: The International Journal on Mathematics Education, 40, 487–500. Harel, G. (2008b). DNR perspective on mathematics curriculum and instruction, part II: With reference to teacher’s knowledge base. ZDM: The International Journal on Mathematics Education, 40, 893–907. Harel, G., & Sowder, L. (2005). Advanced mathematical-thinking at any age: Its nature and its development. Mathematical Thinking and Learning, 7(1), 27–50. Johnson, H. L. (2012). Reasoning about variation in the intensity of change in covarying quantities involved in rate of change. The Journal of Mathematical Behavior, 31(3), 313–330. https://doi. org/10.1016/j.jmathb.2012.01.001 Johnson, H. L. (2015). Together yet separate: Students’ associating amounts of change in quantities involved in rate of change. Educational Studies in Mathematics, 1-22. https://doi.org/10.1007/ s10649-­014-­9590-­y Liang, B., & Moore, K. C. (2021). Figurative and operative partitioning activity: A student’s meanings for amounts of change in covarying quantities. Mathematical Thinking & Learning, 23(4), 291–317. Monk, S. (1992). Students’ understanding of a function given by a physical model. In G. Harel & E.  Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 175–193). Mathematical Association of America. Montangero, J., & Maurice-Naville, D. (1997). Piaget, or, the advance of knowledge. In In. L. Erlbaum Associates. Moore, K.  C. (2013). Making sense by measuring arcs: A teaching experiment in angle measure. Educational Studies in Mathematics, 83(2), 225–245. https://doi.org/10.1007/ s10649-­012-­9450-­6 Moore, K. C. (2014a). Quantitative reasoning and the sine function: The case of Zac. Journal for Research in Mathematics Education, 45(1), 102–138. Moore, K. C. (2014b). Signals, symbols, and representational activity. In L. P. Steffe, K. C. Moore, L. L. Hatfield, & S. Belbase (Eds.), Epistemic algebraic students: Emerging models of students’ algebraic knowing (pp. 211–235). University of Wyoming.

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Moore, K. C. (2021). Graphical shape thinking and transfer. In C. Hohensee & J. Lobato (Eds.), Transfer of learning: Progressive perspectives for mathematics education and related fields (pp. 145–171). Springer. Moore, K.  C., & Thompson, P.  W. (2015). In T.  Fukawa-Connelly, N.  Infante, K.  Keene, & M. Zandieh (Eds.), Shape thinking and students’ graphing activity (pp. 782–789). Proceedings of the Eighteenth Annual Conference on Research in Undergraduate Mathematics Education. Moore, K. C., Paoletti, T., & Musgrave, S. (2013). Covariational reasoning and invariance among coordinate systems. The Journal of Mathematical Behavior, 32(3), 461–473. https://doi. org/10.1016/j.jmathb.2013.05.002 Moore, K. C., Silverman, J., Paoletti, T., Liss, D., & Musgrave, S. (2019a). Conventions, habits, and U.S. teachers’ meanings for graphs. The Journal of Mathematical Behavior, 53, 179–195. https://doi.org/10.1016/j.jmathb.2018.08.002 Moore, K.  C., Stevens, I.  E., Paoletti, T., Hobson, N.  L. F., & Liang, B. (2019b). Pre-service teachers’ figurative and operative graphing actions. The Journal of Mathematical Behavior, 56. https://doi.org/10.1016/j.jmathb.2019.01.008 Moore, K. C., Liang, B., Stevens, I. E., Tasova, H. I., & Paoletti, T. (2022). Abstracted quantitative structures: Using quantitative reasoning to define concept construction. In G. Karagöz Akar, İ. Ö. Zembat, S.  Arslan, & P.  W. Thompson (Eds.), Quantitative Reasoning in Mathematics and Science Education (pp.  35–69). Springer International Publishing. https:// doi.org/10.1007/978-­3-­031-­14553-­7_3 Norton, A. (2014). In T. Fukawa-Connelly, G. Karakok, K. Keene, & M. Zandieh (Eds.), The construction of cohomology as objectified action (pp. 957–969). Proceedings of the Seventeenth Annual Conference on Research in Undergraduate Mathematics Education. Norton, A. (2022). The psychology of mathematics: A journey of personal mathematical empowerment for educators and curious minds. Routledge. Paoletti, T., & Moore, K. C. (2018). A covariational understanding of function: Putting a horse before the cart. For the Learning of Mathematics, 38(3), 37–43. Paoletti, T., Stevens, I.  E., Hobson, N.  L. F., Moore, K.  C., & LaForest, K.  R. (2018). Inverse function: Pre-service teachers’ techniques and meanings. Educational Studies in Mathematics, 97(1), 93–109. https://doi.org/10.1007/s10649-­017-­9787-­y Piaget, J. (1969). The mechanisms of perception. Routledge & Kegan Paul. Piaget, J. (1970). Genetic epistemology. W. W. Norton & Company, Inc. Piaget, J. (2001). Studies in reflecting abstraction. Psychology Press Ltd.. Piaget, J., & Inhelder, B. (1971). Mental imagery in the child: A study of the development of imaginal representation. Routledge & Kegan Paul. Piaget, J., & Inhelder, B. (1973). Memory and intelligence. Basic Books. Piaget, J., Inhelder, B., & Szeminska, A. (1981). The child’s conception of geometry (E. A. Lunzer, Trans.). W. W. Norton & Company. (1960). Saldanha, L.  A., & Thompson, P.  W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berensen, K. R. Dawkings, M. Blanton, W.  N. Coulombe, J.  Kolb, K.  Norwood, & L.  Stiff (Eds.), Proceedings of the 20th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 298–303). ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Smith, J. P., III, & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95–132). Lawrence Erlbaum Associates. Steffe, L.  P. (1991a). The learning paradox: A plausible counterexample. In L.  P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 26–44). Springer-Verlag. https:// doi.org/10.1007/978-­1-­4612-­3178-­3_3 Steffe, L.  P. (1991b). Operations that generate quantity. Journal of Learning and Individual Differences, 3(1), 61–82.

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Steffe, L. P., & Norton, A. (2014). Perspectives on epistemic algebraic students. In L. P. Steffe, K.  C. Moore, L.  L. Hatfield, & S.  Belbase (Eds.), Epistemic algebraic students: Emerging models of students’ algebraic knowing (pp. 317–323). University of Wyoming. Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. Springer. Stevens, I. E. (2019). Pre-service teachers’ constructions of formulas through covariational reasoning with dynamic objects [Ph.D. Dissertation]. University of Georgia. Stevens, I. E. (in press). “A=2πrh is the surface area for a cylinder”: Figurative and operative thought with formulas. Proceedings of the Twenty-Fourth Annual Conference on Research in Undergraduate Mathematics Education. Tasova, H.  I. (2021). Developing middle school students’ meanings for constructing graphs through reasoning quantitatively [Ph.D. Dissertation]. University of Georgia. Tasova, H.  I., & Moore, K.  C. (2020). Framework for representing a multiplicative object in the context of graphing. In A.  I. Sacristán, J.  C. Cortés-Zavala, & P.  M. Ruiz-Arias (Eds.), Mathematics education across cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mexico (pp. 210–219). Cinvestav/PME-NA. Thompson, P. W. (1985). Experience, problem solving, and learning mathematics: Considerations in developing mathematics curricula. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 189–243). Erlbaum. Thompson, P.  W. (1990). A cognitive model of quantity-based algebraic reasoning. Annual Meeting of the American Educational Research Association. 1990, March 27–31. Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165–208. Thompson, P. W. (1994a). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics. SUNY Press. Thompson, P.  W. (1994b). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26(2–3), 229–274. https://doi. org/10.1007/BF01273664 Thompson, P. W. (2011). In S. Chamberlin, L. L. Hatfield, & S. Belbase (Eds.), Quantitative reasoning and mathematical modeling (pp. 33–57). New perspectives and directions for collaborative research in mathematics education: Papers from a Planning Conference for WISDOM^e. Thompson, P.  W. (2013a). Constructivism in mathematics education. In S.  Lerman (Ed.), Encyclopedia of mathematics education (pp.  96–102). Springer. https://doi. org/10.1007/978-­94-­007-­4978-­8_31 Thompson, P.  W. (2013b). In the absence of meaning. In K.  Leatham (Ed.), Vital directions for research in mathematics education (pp.  57–93). Springer. https://doi. org/10.1007/978-­1-­4614-­6977-­3_4 Thompson, P.  W. (2016). Researching mathematical meanings for teaching. In L.  English & D.  Kirshner (Eds.), Third handbook of international research in mathematics education (pp. 435–461). Taylor and Francis. https://doi.org/10.4324/9780203448946-­28 Thompson, P.  W., & Carlson, M.  P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). National Council of Teachers of Mathematics. Thompson, A.  G., & Thompson, P.  W. (1996). Talking about rates conceptually, part II: Mathematical knowledge for teaching. Journal for Research in Mathematics Education, 27(1), 2–24. Thompson, A.  G., Philipp, R.  A., Thompson, P.  W., & Boyd, B.  A. (1994). Calculational and conceptual orientations in teaching mathematics. In A. Coxford (Ed.), 1994 yearbook of the NCTM (pp. 79–92). NCTM.

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Thompson, P. W., Carlson, M. P., Byerley, C., & Hatfield, N. (2014). Schemes for thinking with magnitudes: A hypothesis about foundational reasoning abilities in algebra. In L.  P. Steffe, K.  C. Moore, L.  L. Hatfield, & S.  Belbase (Eds.), Epistemic algebraic students: Emerging models of students’ algebraic knowing (Vol. 4, pp. 1–24). University of Wyoming. von Glasersfeld, E. (1981). An attentional model for the conceptual construction of units and number. Journal for Research in Mathematics Education, 12(2), 83–94. von Glasersfeld, E. (1982). Subitizing: The role of figural patterns in the development of numerical concepts. Archives de Psychologie, 50, 191–218. von Glasersfeld, E. (1987). Preliminaries to any theory of representation. In C.  Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp.  215–225). Lawrence Erlbaum. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Falmer Press. https://doi.org/10.4324/9780203454220

Chapter 5

Figurative and Operative Imagery: Essential Aspects of Reflection in the Development of Schemes and Meanings Patrick W. Thompson, Cameron Byerley, and Alan O’Bryan

Our purpose in this chapter is to clarify the role of imagery in students’ mathematical learning and reasoning. In pursuit of this goal, we address interdependencies among imagery, reflection, scheme, and meaning as theoretical constructs and illustrate their interdependence by examples. Our motive for this expanded charge is that while the notion of schemes and scheme development is sometimes discussed in studies of students’ mathematical learning, the role of imagery in that process is often neglected, yet it is central to the development of productive mathematical meanings. Often people doing mathematics education research pay insufficient attention to the very nuanced ways in which people understand a situation—what their image of the situation is. As a counterpoint, Thompson (1996) spoke of students’ images of a solution to an algebraic equation. “Their image of solving equations often is of activity that ends with something like ‘x = 2.’ So, when they end with something like ‘2 = 2’ or ‘x = x,’ they conclude without hesitation that they must have done something wrong” (Thompson, 1996, p. 274).

P. W. Thompson (*) Arizona State University, Tempe, AZ, USA e-mail: [email protected] C. Byerley University of Georgia, Athens, GA, USA A. O’Bryan Rational Reasoning LLC, Gilbert, AZ, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 P. C. Dawkins et al. (eds.), Piaget’s Genetic Epistemology for Mathematics Education Research, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-47386-9_5

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Imagery We state at the outset that by “imagery” we mean far more than “visualization.” Imagining something one has seen, or producing something as if seen, indeed, falls within the category of imagery. But recalling something one has said, thought, done, or felt also falls within the category of imagery. In an act of re-presentation,1 a person recalls having an experience. The experience might have been having a thought or feeling, having done something in some context, having interpreted something in a particular way, recalling having recalled something, or other acts of recollection. In general, by “image,” we mean the re-presentation of experience. We use “imagery” to refer to images collectively. While we say imagery exceeds visualization, we acknowledge that “visualization” has been used sometimes with broad meaning, including what we call “creative imagery.” Creative imagery is the construction of an image one has not experienced but is grounded in experience. One example is Galileo’s thought experiment of repeatedly dropping two iron balls of different sizes joined by ever-thinner filaments to infer all masses in a vacuum fall identically (Clement, 2018; Miller, 1996). Another is Einstein’s thought experiment of a person falling in an elevator to infer gravity is akin to acceleration within an inertial frame of reference (Miller, 1996). Neither Galileo nor Einstein had experiences that were recalled as such. Rather, they assembled images from experience in novel ways. Experiences are never recalled veridically. Our distinction between creative imagery and recalled experience is therefore muddied by the fact that initial experiences and recalled experiences are constructed using schemes one currently has, which can change in the interim between initial experience and recollection. For example, Piaget (1968a) presented experimental evidence of children’s memories of a visual image improving over time. The improvement, Piaget claimed, was due to students having developed more coherent schemes by which they remembered (re-­ constructed) their original perception. The experiment had first-grade children look for a short period of time at a display of 10 bars aligned vertically in ascending length (Fig. 5.1a). Piaget and colleagues asked children to draw what they remembered seeing 1  week later and 6  months later. Children’s drawings 1  week later tended to show local groups of ascension, but not uniform increase in length (Fig. 5.1b). Their drawings 6 months later, with no mention of the initial episode having been made, were considerably improved—many more children’s drawings resembled the figure presented 6 months earlier. Piaget explained that, in the intervening 6  months, children had developed schemes for order that included transitivity.2 In other words, their recollections improved because the schemes by

 Glasersfeld (1991) distinguished between “to represent” and “to re-present” this way. To represent an experience is to take one thing as standing for another. To re-present an experience is to bring to mind a record of the experience. 2  One way Piaget inferred that children’s ordering schemes included transitivity was to have them insert a stick into a series they had constructed, ordered left-to-right ascending by height. If the 1

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Fig. 5.1 (a) The figure as presented to children. (b) Type of drawing common among children 1 week later

which they re-presented their initial experience of the bars had become more advanced. Another source for our position regarding imagery is our appreciation of action as the foundation of Piaget’s genetic epistemology. To Piaget, actions were cognitive activity that might (or not) be expressed in behavior. “One can say that all action—that is to say all movement, all thought, or all emotion—responds to a need” (Piaget, 1968b, p. 6). Actions are the foundation of experience, so the phrase “recall an experience,” to be in line with Piaget’s genetic epistemology, forces us to include recalled movement, thought, or emotion as imagery. We also avoid tacitly equating “action” with observable behavior, which happens often when people use the phrase “reflection on activity.” Powers (1973a) addressed this when he explained that living organisms cannot organize themselves around how their behavior is perceived by others, but instead according to the effects of the organism’s actions as discerned by the organism (Powers, 1973b, p. 418). Powers’ message for us is that we cannot take students’ behavior at face value— it is but an expression of their actions—where we use “action” as Piaget intended. This is at the root of Steffe and Thompson’s (2000) distinction between students’ mathematics (their mathematical realities) and mathematics of students—an observer’s understanding of how students might be thinking to behave as they did or might do. Students’ mathematics is the dark matter and dark energy of mathematics education. Devising a viable mathematics of students is therefore a core mission of mathematics education research. Understanding students’ imagery while engaged in instruction and while recalling their experiences outside of instruction is central to that mission. On a related note, it is common for instructors and instructional designers to include visual presentations to supplement prose or have students engage in some form of activity. The thought is that activities or visual presentations help students child found the first stick taller than the one they were to insert and inserted the new stick to the left of the first-taller stick, the child exhibited transitivity. They knew all sticks to the left of that position were necessarily shorter than the one they inserted and all sticks to the right were necessarily taller than the one they inserted (Piaget, 1965).

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understand the goal of instruction. Our emphasis on imagery as recalled experience is important for putting these efforts in proper light. By taking seriously the stance that imagery is rooted in recalled experience, we are forced to be mindful that students recall their experience of activities or visual presentations; they cannot recall what we understand as having been presented to them. Students’ experience of an activity or presentation is conditioned by the schemes through which they understood it and recall it. Since learners are, by definition, new to the ideas being taught, their experience of activities or presentations will be substantially different from the originator’s intentions.

Imagery, Schemes, and Meanings Having spoken of schemes in relation to images repeatedly, we feel obliged to say what we mean by a scheme and speak to the role of imagery in scheme development. Piaget’s use of “scheme” was quite utilitarian. It allowed him to speak of mental organizations that supported flexible reasoning across seemingly disparate situations. Montangero and Maurice-Naville (1997, p. 155) presented a compendium of six ways Piaget used “scheme.” They ranged from “[Schemes are] organized totalities whose internal elements are mutually implied” (Piaget, 1936, p.  445) to “A scheme is the structure or the organization of actions which is transferred or generalized when this action is repeated in similar or analogous circumstances” (Piaget & Inhelder, 1966, p. 11, footnote not translated in the English version). Piaget’s statement, “organized totalities whose internal elements are mutually implied,” derives from his stance that actions are implicative. When someone engages in an action, it creates a new experiential context that could be the trigger for other actions. An action in a context implies other actions. Thus, “… elements are mutually implied” means that a scheme constitutes a locally closed system in which any of its assumed conditions could activate the scheme in its totality. An example is when someone has a mature constant speed scheme. They are aware that a time and a speed are involved when they know an object traveled some distance, that a distance and a speed are involved when they know it traveled some time, and that a time and a distance are involved when they know it traveled at some speed. “Mutual implications” of time, distance, and speed in a person’s constant speed scheme is evidenced when they understand that there are implied distances in “A car drove 60 mi/hr for 3 hours and then 40 mi/hr for 5 hours. What was the car’s average speed?” Likewise, the statement, “… organization of actions which is transferred when this action is repeated …” was Piaget’s way to account for how a scheme (as an organization of actions) can be activated in seemingly different circumstances. An example of this is when someone uses their constant speed scheme (constant rate of change of distance with respect to time) to understand constant rate of change of volume of a fluid in a container with respect to its height in the container.

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Fig. 5.2  Steffe’s (2010, p. 23) characterization of schemes

Cobb and Glasersfeld (1984) and Glasersfeld (1995, 2001) proposed that, to Piaget, a scheme was a three-part mental structure: an internal condition that would trigger a scheme, an action or system of actions, and an anticipation of what the action would produce. Steffe (2010) expanded Cobb and Glasersfeld’s definition to include a scheme’s goal (Fig.  5.2). Steffe’s motivation for including a generated goal in his definition of scheme was The Generated Goal can be regarded as the apex of a tetrahedron. The vertices of the base of the tetrahedron constitute the three components of a scheme. The double arrows linking the three components are to be interpreted as meaning that it is possible for any one of them to be in some way compared or related to either of the two others. The dashed arrow is to be interpreted as an expectation of the scheme’s result. (Steffe, 2010, p. 23)

A main feature Piaget communicated in his characterizations of scheme is they are cognitive structures that can express themselves in action or behavior. Cobb’s, Glasersfeld’s, and Steffe’s definitions address Piaget’s intention well. However, another important aspect of schemes’ functioning to Piaget, not captured by those definitions, is that people employ schemes to comprehend situations, and to give meaning to situations (Piaget & Garcia, 1991). To us, a definition of scheme must support interpretations of a person’s attempt to understand situations in terms of meanings they have for constituent elements and relationships among them. Thompson and Saldanha (2003), for example, spoke of a mature fraction scheme as a network of relationships among schemes for measure, multiplication, division, and relative size, each of which entails aspects of proportionality, as a means to understand the broad array of situations one can understand as involving fractions. To this end, Thompson et al. (2014) expanded Steffe’s, Cobb’s, and Glasersfeld’s definitions of scheme. A scheme as an organization of actions, operations, images, or schemes—which can have many entry points that trigger action—and anticipations of outcomes of the organization’s activity.3 (Thompson et al., 2014, p. 11)

We point out that, in this definition, “entry points” often are images of contexts, and anticipated outcomes are most definitely images. Unlike Steffe’s definition, and like Glasersfeld’s and Cobb’s, Thompson et al.’s (2014) definition of scheme does not  This definition is recursive, not circular. A scheme might recruit other schemes when activated.

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include goals. In a larger context, though, it is like Steffe’s definition in that a person can generate a goal in the activity of implementing a scheme or might activate a scheme because a generated goal matches a scheme’s anticipated outcome. Thompson et al.’s definition of scheme is at the root of Thompson and Harel’s (Harel, 2021; Thompson et al., 2014) attempt to give coherence among meanings for understandings, meanings, and ways of thinking (Fig. 5.3). This system for the use of “understanding,” “meaning,” and “way of thinking” aligns with Harel’s and Thompson’s quest to decouple “understand” and “understand correctly.” They do this by resting their system on the idea of assimilation. They rely on Piaget’s characterization of assimilation as, in effect, giving meaning. Assimilating an object to a scheme involves giving one or several meanings to this object, and it is this conferring of meanings that implies a more or less complete system of inferences, even when it is simply a question of verifying a fact. In short, we could say that an assimilation is an association accompanied by inference. (Johnckheere et al., 1958, p. 59, as translated by Montangero & Maurice-Naville, 1997, p. 72)

The first entry (Understanding in the moment) in Fig. 5.3 describes a person who has an understanding of something said, written, or done in the moment of understanding it. Technically, all understandings are understandings-in-the-moment. Some understandings might be a state that the person has struggled to attain at that moment through functional accommodations to existing schemes (Steffe, 1991) and is easily lost once the person’s attention moves on. This type of understanding is typical when a person is making sense of an idea for the first time. The meaning of an understanding is the space of implications that the current understanding mobilizes—actions or schemes that the current understanding implies, that the current understanding brings to mind with little effort. An understanding is stable if it is the result of an assimilation to a scheme. A scheme, being stable, then constitutes the space of implications resulting from the person’s assimilation of anything to it. The scheme is the meaning of the understanding that the person constructs in the moment. As an aside, schemes provide the “way” in Harel’s “way of understanding.” Finally, Harel and Thompson characterize “way of

Fig. 5.3  Meanings of “understanding,” “meaning,” and “way of thinking” (Thompson et  al., 2014, p. 13)

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thinking” as when a person has developed a pattern for utilizing specific meanings or ways of thinking in reasoning about particular ideas or situations.

Images and Schemes Images enter our definition of scheme, and therefore meaning, in three ways: Images can be contexts that activate a scheme, they can be waypoints in a scheme’s activity, and they can be anticipations of a scheme’s result within the context of the scheme’s activation. Thompson (1994a, 1996) explained the ways in which the notion of image is intertwined with Piaget’s concept of scheme. He pointed out three levels of imagery in Piaget’s work. The first level of imagery is when a child engages in deferred imitation. Deferred imitation is when a child enacts the imitated behavior to assimilate (understand) it. The second level of imagery (figurative) is an image of an initial state and actions that are associated with it, but the actions and image are tied tightly—such as a student accustomed to drawing altitudes in a triangle with all angles less than 90 degrees being confused when asked to draw all altitudes in a triangle with one angle greater than 90 degrees. The third level of imagery is what Piaget called “operative.” [This is an image] that is dynamic and mobile in character … entirely concerned with the transformations of the object. … [The image] is no longer a necessary aid to thought, for the actions which it represents are henceforth independent of their physical realization and consist only of transformations grouped in free, transitive and reversible combination … In short, the image is now no more than a symbol of an operation, an imitative symbol like its precursors, but one which is constantly outpaced by the dynamics of the transformations. Its sole function is now to express certain momentary states occurring in the course of such transformations by way of references or symbolic allusions. (Piaget, 1967, p. 296)

The three levels of imagery do not differ in type. They are all re-presented experiences. Instead, the levels are differentiated by the ways images are integrated into individuals’ reasoning and the types of reasoning into which they are integrated. We unpack the three levels in the following paragraphs. First-Level Imagery (Deferred Imitation) Piaget’s examples of deferred imitation are often about infants or toddlers mimicking their experience of others (e.g., opening their mouth to mimic their mother) or engaging in play to mimic social interactions. But deferred imitation is a broader phenomenon. In psychotherapy, it is called re-experiencing (Joseph & Williams, 2005)—the replaying of a traumatic event to assimilate (understand) it by either adjusting one’s understanding of a world in which such a thing could occur or adjusting one’s understanding of one’s place in the world that makes the event

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sensible. The role of deferred imitation in mathematics learning is unclear to us. This is not to claim it is unimportant. We just say we are unclear as to its role. Second-Level (Figurative) Imagery Second-level (figurative) imagery aligns with what Davis et  al. (1978) called “visually-­moderated sequences”—activity sequences triggered by a current visual or cognitive state (e.g., seeing an equation and thinking to add something to both sides) that end in a new visual or cognitive state (e.g., an expression with no constant terms) that triggers another activity sequence (e.g., dividing both sides by the same number). Each activity sequence results in a new state, but upon arriving at a new state it is simply the end of the activity sequence. It is not a goal toward which the student strove, and thus the end state is not an anticipated result of the activity, and the actual result does not, to the student, entail an image of the activity leading to it. Frank (2017) provided an excellent example of a student whose activity ends with a result that, for her, did not entail an image of the activity that led to it. The student (Ali) constructed a graph to represent two quantities’ values as they varied simultaneously in an animation of the quantities and their magnitudes. Ali ended with what Frank considered an appropriate graph. But the graph, to Ali, did not entail the covariational reasoning in which she engaged while making it. Ali spoke of the graph which she had just made as if it was a static shape as if a piece of wire. At the outset of this study, I thought that if Ali made a graph by simultaneously tracking two magnitudes, then she engaged in emergent shape thinking. I had not considered that Ali’s meaning for her sketched graph might not reflect the thinking she engaged in to make the graph. (Frank, 2017, p. 193)

Research by Lobato, Stump, and Moore provide additional examples of students operating with figurative imagery. Lobato and Thanheiser (2002) reported children thinking the slope of a ramp leading to a platform is affected by the width of the platform. They included the platform as part of the “over” image in their “up and over” slope scheme. Stump (2001) reported a student who thought a slope of −5/6 is different from a slope of 5/ − 6 because they entail different images of “up and over.” Moore and colleagues (2014, 2019) reported students became confused about the slope of a line when x- and y-axes were switched. They wanted the line to have the same slope because, to them, the line still pointed in the same direction. Their slope scheme depended on an image of a line’s direction, and a slope value, to them, was an index of directionality. Anyone operating with figurative imagery can lose track of their reasoning easily. Byerley and Thompson (2017) report several instances of teachers moving from one meaning of a situation to a contrary meaning within seconds as they employed schemes that used images figuratively. Figure  5.4 presents an item from the Mathematical Meanings for Teaching secondary mathematics (MMTsm) inventory (Thompson, 2016). Its design was motivated by the ambiguity with which teachers

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Fig. 5.4  Meaning of “over” (Byerley & Thompson, 2017). (© 2014 Arizona Board of Regents. Used with permission)

in their samples used the word “over” when speaking of rates of change. Thompson and his team could not tell whether teachers used “over” to convey an interval during which an event unfolded or to convey a spatial arrangement of numerator and denominator with respect to a vinculum. For this item, the team took “during” to be a high-level answer to Part A and f(x0 + Δx) − f(x0) = 4 grams, f(x + Δx) − f(x) = 4 grams, or even Δm = 4 to be a high-­ level answer to Part B. Byerley and Thompson reported 113 (45%) of 251 US teachers said “over” meant during or something equivalent in response to Part A, and 71 (28%) of 251 teachers said “over” meant the same as divide in response to Part A. In response to Part B, only 18 (16%) of the 113 teachers who said “over” meant “during” for Part A represented the statement as a difference or change in mass. Forty-­ one percent (41%) of the 113 teachers who said “over” meant during for Part A responded to Part B by representing the statement as a quotient involving mass and time, writing something like (change in mass)/(change in time) = 4 grams. In other words, when reading the statement as a plain-language description of a situation, the word “over” for these 113 teachers suggested an image of something happening in time. But upon reading the same statement as something to be represented symbolically, the word “over” suggested an image of numerator and denominator separated by a vinculum. An interview with James, who had a B.Sc. in mathematics education and was an experienced teacher of algebra, geometry, and precalculus, illustrates how figurative imagery leads to conflicts between schemes. James: [Over means] during or duration. You could also think of it as a ratio, so change in mass over, yeah so during or duration, so in your math class when they say, “something over something”, they always mean a divide sign so a ratio. Int: Do you think they are both saying the same thing? James: Well, yeah, I think that. Well yeah, they are saying. I think the during or duration is more saying conceptually what is going on, and the divided by or over I see the reason behind that, I think I’m more pointing out mathematically what we mean when we say over with no explanations as to why, it is just the way it is. Int: So is the mass, the change in mass divided by the change in time, is that how you write the idea of duration?

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James: Can you repeat the question? Int: Is the “delta mass divided by delta x” a mathematical way of saying duration? James: I want to say the change in x is the way of saying duration. I want to say the change in x is representing duration. But maybe we could include the division sign. So no, I would not say that “delta mass over delta x” is a way of saying duration. So this is funny. (Byerley & Thompson, 2017, pp. 188–189) James never reconciled his conflict between “over” suggesting Δx as a representation of elapsed time and “over” suggesting the quotient Δm/Δx. We explain his conflict by appealing to the imagery he apparently evoked in relation to his different purposes for reading the statement. In Part A, his purpose was to read the statement as a plain-language description of a phenomenon, for which “over” suggested an image of something happening as time elapsed. In Part B, James’ purpose was to represent mathematically a situation described in plain language, which suggested an image of two numbers or expressions separated by a vinculum. What was new for James is that the interviewer asked him to compare competing implications of his two assimilations of the same word. The images James evoked were figurative— they were tied tightly to the schemes he evoked in the contexts of his different purposes. We present a second example from Byerley and Thompson (2017) of figurative imagery leading to competing assimilations of “the same” context. Figure 5.5 shows another item from the MMTsm. Its purpose was to tease out whether teachers interpreted graphs as showing amounts of a quantity despite it being stated explicitly in two ways that the graph showed rates of change of one quantity with respect to another. Parts 2 and 3 of this item (not shown here) allowed teachers to reveal their level of commitment to their initial interpretations of the graph. Thirty-six percent (36%) of 239 high school mathematics teachers appropriately chose (c) for Part 1 of this item; 49% chose (a). The following excerpt shows a teacher who slipped from one interpretation of the item to a contrary interpretation while explaining her answer to the question.

Fig. 5.5  Part 1 of the three-part item “Increasing or decreasing from rate” (Byerley & Thompson, 2017). (© 2014 Arizona Board of Regents. Used with permission)

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Annie: [Reads problem aloud, emphasizes “grams per hour”.] We interpret increasing. umm...let’s see the function gives the rate of change in grams per hour... and so umm...what we are going to look at I would look at the rate of change being positive or negative, if we have a positive rate of change the grams per hour the mass is increasing per hour, is getting larger Annie: So I look at where I have a positive rate of change, and I try to identify where I have no rate of change [highlights maximum at (1.25, 5) where the rate of change is approximately positive 5, but the acceleration is zero], this is telling me where the mass is staying the same, and then I have a negative slope so mass is getting small down to a zero rate of change so I’m not getting any smaller or larger... [Annie chooses (a)] We do not have direct evidence of where Annie looked as she spoke, but it seems plausible she gazed at the text during her first utterances and gazed at the graph during her second utterances. It appears that in the first utterances, Annie had in mind values of the function f as values of the rate of change of the culture’s mass with respect to elapsed time, whereas in the second utterances, Annie had in mind slopes of tangents to the graph as values of the rate of change of the culture’s mass with respect to elapsed time. In other words, Annie slipped from one scheme (values of the function give the rate of change) to a different scheme (slope of tangents to the graph gave the rate of change), and the “slip” was prompted by her different imagistic contexts (text vs. graph). For Annie, with rate of change functions, values of the function give rates of change; with graphs, slopes of tangents give rates of change. We note in passing that Annie felt no conflict between her two schemes because she did not think of the function having positive values in places where its graph had negative slopes. Third-Level (Operative) Imagery At the third level of imagery, students’ schemes are not dependent upon specific images. Instead, images serve as arbitrary “momentary states” in a scheme’s implementation. Thompson and Dreyfus (1988) reported two sixth graders’ (Kim’s and Lucy’s) advancing from second-level imagery, thinking of an integer such as −5 as a location on a number line and later thinking of −5 as a displacement from any starting point. Their imagery moved from Piaget’s second level to his third level. The children’s schemes no longer needed a definite starting point. They knew they could start anywhere to enact −5. They could then think of +3 + −5 as a composition of two displacements that produced a net displacement of −2, where the second displacement started wherever the first ended. Though they enacted the displacement of +3 from a specific place on a number line, they did not feel required to use a specific place from which they must enact it. Their image of an actual starting place was not more general. Rather, it was their scheme that became more general. It did not require a specific starting place, thus, specific locations on the number line served as “momentary states” in the activity of their integer composition schemes as they conceptualized the net displacement (sum) of several displacements.

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While Kim and Lucy developed operative imagery regarding composition of d­ isplacements, only Kim thought of number in −number as itself being a net dis placement. Understanding expressions like 90   30 was unproblematic for Kim









because the expression 90   30 fit within her image of things constituting number—it was a net displacement. For Lucy, only whole numbers fit her image of things constituting number. Evaluating the negation of complex expressions was often effortful for Kim, but she nevertheless knew what she was supposed to end with—the negation of a net displacement. In sum, Kim and Lucy had operative imagery with respect to composition of displacements, while only Kim had developed operative imagery for negation. Our main point here is that you categorize the level of students’ images as second-level (figurative) or third-level (operative) according to your judgment of how necessary those specific images are in students’ schemes as they employ them. Summary The examples above show our use of “image-level” relatively. Deferred imitation can happen when anyone re-plays an event or collection of actions that they did not fully assimilate—they did not fully understand. Young children go home from school and play school as an attempt to assimilate their new experience of a teacher who attempts to control their thinking. Graduate students in mathematics replay specific aspects of a lecture in their attempt to assimilate them—to develop an understanding. Movement of an hour hand on a circular clock is often offered as a foundational image for understanding cyclical groups. For persons who must think of a clock to do arithmetic in a cyclical group, their clock image is figurative because their scheme for addition in a cyclical group requires it. A person who uses a clock whose hour hand varies from 0 to 2π hours to think of equivalent angle measures on a number line, but can also see the repetition as hops on the number line or as the number line collapsing into equivalence classes by the mapping ℝ  →  ℝ/2π,4 is employing images operatively. Any image they employ in their reasoning about arguments to a trigonometric function is a matter of convenience because it fits their purpose in the moment. Piaget’s notion of image is useful because, in developing a scheme, a student must reflect on her reasoning. To reflect on her reasoning, she must create, as best she can, images of having reasoned in the way she did. This means she must develop recollections of “momentary states” in having reasoned. To construct a scheme, students must repetitively engage in variations of the reasoning that will become solidified in that scheme and re-present it as best they can to reflect upon it. Sometimes reflection occurs during moments of confusion, sometimes after having  The mapping maps every real number x to the non-negative remainder of x divided by 2π. This has the same effect as all numbers on the number line falling simultaneously, like molecules of water vapor, onto their equivalent positions in the interval [0,2π). 4

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engaged in a chain of interpretations, inferences, and decisions. Nevertheless, in the process of constructing a scheme, images of having reasoned become students’ objects of reflection (Cooper, 1991; Harel, 2008a, b, 2013). It is worth noting that the number of schemes students develop is immense. Any word that is meaningful to them is meaningful because hearing or seeing it activates a scheme. Any symbol or symbolic expression that is meaningful to them is meaningful because seeing it or thinking of it activates a scheme. Any diagram or picture that is meaningful to them is meaningful because they assimilate it into one or more schemes. Any time you assess students’ thinking or interview students, they are interpreting both the setting and your actions through the activation of schemes. Moreover, any combination of the above that proves meaningful to a student is meaningful because of minor or major accommodations in their schemes in the moments of activating them. Any time someone puzzles about the meaning of a word or phrase and resolves their puzzlement has engaged in some form of reflection that engendered an accommodation to their schemes. When you create a scheme as a model of student thinking and impute that scheme to students, you must be cognizant that you have most certainly omitted a vast number of schemes that were at play in students’ thinking that might turn out to be important for understanding their thinking. The art of using scheme and image as explanatory constructs is to find the appropriate level of analysis that produces tractable explanations of students’ successes and difficulties. The prior paragraph points to a methodological aspect of scheme as a theoretical construct. On one hand, we say schemes are organizations of a person’s mental activity that express themselves in what an observer sees as behavior. From this perspective, schemes reside in individuals. On the other hand, we say scheme is a theoretical construct that researchers impute to individuals to explain their behavior. They are a researcher’s construct. This is much like stances taken by natural scientists. They realize anything they say is based on models built from theory-laden observations, but in doing their science they act as if their models describe reality— until observations force them to step back and question their assumptions and their models. Likewise, we infer schemes from students’ behavior in response to carefully defined probes. We impute schemes to students to form explanations of their behavior and to design supports we think will advance their thinking. We step back and question ourselves when our explanations become inconsistent or inadequate, or our designed supports do not have their intended effects.

Imagery, Schemes, and Reflective Abstraction In this section we expand our earlier discussion of imagery, schemes, and meanings to address what we mean by reflection and the role it plays in a person’s construction of schemes. John Dewey placed reflection at the center of his understanding of thinking and placed thinking at the center of the development of a critically informed democracy.

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Dewey defined reflective thought as, “Active, persistent, and careful consideration of any belief or supposed form of knowledge in the light of grounds that support it, and the further conclusions to which it tends” (Dewey, 1910, p.  6). Dewey also anticipated coherence as a characteristic of reflective thought that Piaget came to see as central to his genetic epistemology. Reflection on one’s thinking leads to the organization of facts and conditions which, just as they stand, are isolated, fragmentary, and discrepant, the organization being effected through the introduction of connecting links, or middle terms. (Dewey, 1910, p. 79)

Dewey was also in line with Piaget as to one’s motive for reflection. Demand for the solution of a perplexity is the steadying and guiding factor in the entire process of reflection—i.e., reflection serves a regulatory function. (Dewey, 1910, p. 11)

The key aspect of Dewey’s account of reflection is that “to reflect” means to think about thinking. This is in line with our prior discussions of imagery with respect to schemes when one considers that people construct schemes by creating images of having reasoned and taking those images as their objects of thought. A vast difference between Dewey’s and Piaget’s accounts of reflection is that Dewey thought of reflection as a conscious activity, whereas Piaget thought of conscious reflection as the tip of an iceberg. He posited processes of unconscious reflection that must precede anything resembling Dewey’s characterization. Piaget took the stance that one can be aware only of images one operates upon. You cannot be aware of the operations you use to operate on an image—to an extent. You can, however, project your operations of thought to a level where they become images upon which you operate. But that is different from being aware of the operations of thought you employ while using them. As explained by Ellis et al. (Chap. 6) and Tallman and O’Bryan (Chap. 8), the idea of reflection, or thinking about one’s thinking, has been on philosophers’ minds at least since the time of Aristotle. They also explain that Piaget was the first to break the notion of reflection down into constituent cognitive processes. We will not add to their extensive discussions. Instead, we will highlight essential aspects of reflective abstraction to complete our picture of how imagery and schemes (and therefore meanings) develop and interact in students’ thinking across their mathematical development. Piaget posited five types of abstractive processes: empirical, pseudo-empirical, reflecting, metareflection, and thematization. Empirical abstraction is the process of extracting common properties of sensory experience. To empirically abstract a property, the person comes to distinguish between objects having and not having the property. It is important to understand that “the property” is the person’s construction, not a “real” property. A child abstracting the property of having four legs to apply the word “dog” might at first also apply “dog” to what we call cats. Pseudo-empirical abstraction is the process of taking the results of one’s activity as objects of empirical abstraction. For example, a child constructed the sequence of arrays in Fig. 5.6. Her actions were to make a vertical line of dots, one more than

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Fig. 5.6  A child’s construction of a sequence of arrays

was already there, and then complete the figure by making the same number of horizontal dots as in the current array. When the child was asked about the number of dots in each array, she noticed that each array was a square, so she said, “1, 4, 9, 16, 25.” When asked how many would be in the next array, she squared six to say “thirty-six.” In other words, when asked about the number of dots in each array, she took the arrays as if given to her. Her reasoning did not reflect that, by her construction method, the sixth square would have 52 + 6 + 5 dots, or that the (n + 1)st array would have n2 + (n + 1) + n dots, which she could then connect to her observation that each array is a square to conclude that (n + 1)2 = n2 + (n + 1) + n. This is not to diminish the child’s accomplishment. Instead, it is to point out the difference between abstracting one’s reasoning from the activity of producing the sequence and abstracting an empirical pattern from the products of one’s reasoning. Reflecting abstraction was already exemplified in the example of pseudo-­ empirical abstraction. A person engages in reflecting abstraction when she brings to mind (re-presents), as best she can, the reasoning in which she engaged in a prior occasion. A successful process of reflecting abstraction produces a new action, but one that does not need the specific contexts accompanying the original. Successful reflecting abstractions produce reflected abstractions. Metareflection and thematization are constructs Piaget introduced to account for the ever-increasing level of persons’ abstraction of logical, mathematical, and scientific structures. He posited that reflecting abstraction produces reflected abstractions, which then can themselves become objects of reflection. He used “metareflection” to capture a person reflecting on reflected abstractions (Piaget, 2001, pp. 82–84). A person thematizes a scheme via metareflection by developing an image of its major elements, how they work together, and how they might unfold in the context of actual situations. This is not unlike the way storytellers thematize a story. They come to think of the story’s major elements, how they work together, and how each element might be unpacked into its details. As Harel (2008c, 2013, 2021) explains, thematization of a scheme evolves over repeated occasions of employing the scheme

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and reflecting on the activity of employing it. Depending on the scheme’s complexity, this could happen over many years. It is worth noticing that when a scheme is well-formed at a reflected level, one can have the experience of having implemented that scheme. Thus, the processes of metareflection and thematization can operate on images of schemes as well as on images of schemes’ constituent elements. Dawkins and Norton (2022) draw upon the construct of metareflection (reflecting on reflected abstractions) to account for students’ development of universally quantified conditional statements as logical structures. “We propose populating, inferring, expanding, and negating as four mental actions that, upon becoming reversible and composable, can give rise to the logic of universally quantified conditional statements. We adopt the view that logic is a metacognitive activity in which people abstract content-general relationships by reflecting across their content-­ specific reasoning activity.” (Dawkins & Norton, 2022, p. 1). We (the authors) had the experience of a calculus student sharing with us her thematization of an approximate accumulation function in DIRACC calculus (Thompson et al., 2019). We were in a staff meeting, the conference room door was open, and the statements in Fig. 5.7 were written on a whiteboard visible from the hallway. The student stepped into the room, pointed at the board, and proudly announced, I can tell you what every line on that board means and how it all works together! You have a function that gives the rate of change of accumulation for every value of x. You want to approximate the accumulation from a to x, so you cut up the x-axis into parts all of length Δx starting from a and define a function so its value for everything in a Δx interval is the accumulation’s rate of change at the beginning of that interval. Then you assume the accumulation happens at that constant rate over the whole Δx interval. You do that for complete intervals from a to x, and then you add the accumulation through the partial interval from left(x) to x. That’s how a value of A(x) gives an approximate accumulation from a to x.

While the student omitted some details, such as how the definition of left(x) works the way she said and how the summation in the last line gives the accumulation she claimed, she accomplished essentially what she set out to do. She explained each element in her approximate accumulation scheme and how they worked together to

Fig. 5.7  Statements defining an approximate accumulation function in DIRACC calculus

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produce an approximate accumulation from a to x for every value of x when all one knows is the accumulation’s rate of change at every moment. It is our experience that metareflection and thematization are the least emphasized of all the forms of reflection in Piaget’s genetic epistemology—in both instruction and in research. In instruction, we do not envision a teacher commanding his students to metareflect or to thematize. Rather, a teacher could promote metareflection and thematization by emphasizing schemes’ stories. While the teacher’s students must thematize their own schemes, the teacher emphasizing the story of a scheme can open students to the possibility that there is a story to understand. In research, it is inherently difficult to investigate metareflection and thematization as explanatory constructs or as phenomena to investigate for two reasons. First, these are processes that, even if a student engages in them, occur largely outside of instruction, perhaps even in their sleep, during what Hadamard (1954) called periods of “incubation” and during periods of what Steffe (1991) called “metamorphic accommodation”—accommodations to schemes that persist over situations and time. Second, metareflection and thematization are difficult to investigate methodologically. Tallman and O’Bryan (Chap. 8) suggest an approach to engendering metareflection in which researchers engage students in what appear to the students as very different situations, but which nevertheless can be understood as similar at a reflected level. Thompson (1994b) used essentially this approach with some success to investigate a fifth grader’s construction of a reflected constant rate of change scheme. Research on metareflection and thematization will be a fruitful area for future advances in theory and practice.

Case Studies We provide two case studies to illustrate the interconnections among imagery, schemes, meanings, and reflective abstraction. The first case is a seventh-grader learning a mathematical game. It will illustrate a youngster’s construction of schemes regarding the game and the crucial role his imagery played in developing them. The second case is of an adult who reasons initially at a figural level about a mathematical task and who projects the situation to a level of existing reflected abstractions upon becoming confused by an unexpected outcome.

Imagery in the Construction of a Nim Scheme According to Jorgensen (2009), Nim is one of the oldest games in the world. It is a game for two players. In one version, they start with the target number 21 and current total of zero. Players take turns adding a number from 1 to 3 to the current total. The player ending with 21 wins. Diego was a 12-year-old rising seventh grader. We asked him and his parents to let us teach him Nim and allow us to record our Zoom sessions. They agreed. Diego

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played against a computer program that embedded a general winning strategy that enabled it to win whenever the opportunity arose. Diego played against two versions of the program: (1) a fixed target of 21 and selecting numbers from 1 to 3. He could choose which player goes first; and (2) an arbitrary target and arbitrary range of numbers from which to choose. Diego could choose the numbers and which player goes first. Sessions were conducted using Zoom. We met for two sessions— June 17 and 28, 2020. The long break was when Diego attended surf camp. We report these sessions to highlight the central role of imagery in Diego’s construction of a general Nim scheme by way of the gamut of reflective processes. Diego’s early imagery was figurative, based on re-presenting states of his play. He focused on moves he might make based on the game’s current state. He refined his strategy eventually by taking his reasoning as his object of thought as opposed to states of the game as his object of thought. His re-presentations of prior reasoning became the images upon which he operated. Session 1: June 17, 2020 21 and 3 Pat explained the game’s rules to Diego and explained that Diego would be playing against a computer program. Pat shared his screen to show the program, which itself explained the game and gave an example (Fig. 5.8). Diego said he would go first. He chose numbers from 1 to 3 at random. When the computer played to reach a total of 17, Diego paused. After a few seconds, he said, “I lost. No matter what I choose, the computer will get to 21.” In the second game, Diego went first, again choosing numbers at random. Diego paused when the computer played to reach 13. “Darn, I’m going to lose again.” He remembered that the computer reaching 17 led to it winning, and he couldn’t stop the computer from reaching 17. In the third game, Diego developed the strategy of “goal numbers.”

Fig. 5.8  The computer screen in a game of Nim

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Excerpt 1 Diego: To get to 21 I have to get to 17. To get to 17 I have to get to 13. To get to 13 I have to get to … nine. To get to nine I have to get to … five. To get to five I have to get to one. To get to one I have to go first. Diego’s image of his participation in the first two games was simply to select numbers from 1 to 3 and add his selection to the current total. We say he operated with figurative imagery because each of his activities (adding a new number) resulted in a new state (a new total number), but the new total number was simply the end of an activity sequence (his turn). He was not striving for a particular goal, and when he considered a new total number, his thinking did not entail an image of the reasoning he used to get to it. He reasoned about specific numbers in specific contexts but not yet re-enacting his reasoning to reflect on it. Diego’s scheme for Nim-21 developed as he reflected on being “blocked.” In the first game, he experienced being “blocked” from reaching a desired number (21) because of the total the computer gave him (17). He concluded by trying all possibilities, he could not reach 21; the computer would win regardless of his choice of number when he is given 17. In the second game, Diego had a similar experience of being “blocked” when the computer presented him with 13—he saw that the computer would reach 17 regardless of his choice, and it would therefore win for the same reason he experienced before. In the third game, Diego employed his image of being “blocked” to devise a strategy in which he could block the computer from reaching a desired goal number. Diego employed his “blocking” image to block the computer from reaching 21 by him reaching 17, then blocking the computer from reaching 17 by him reaching 13, and so on. Diego’s images of being blocked were at first dependent on thinking about the computer reaching 17 and the possible moves he could make from 17 to 21. As Diego repeatedly reasoned about being blocked and additional blocking numbers his image became less dependent on specific game states. As Diego’s blocking number scheme became more stable and less dependent on specific states of the game, his imagery of being blocked moved from figurative to operative. It became less necessary for Diego to consider a specific game state when he reflected on being blocked. We say that Diego developed a blocking number scheme to determine goal numbers and used his goal numbers to ensure he won. He coordinated his blocking number scheme and list of goal numbers in playing the game. Diego had developed a Nim-21 scheme. 38 and 8 Pat then ran a program that played Nim with arbitrary target and range numbers, suggesting they change target and range. Diego chose 38 as the target and 1–8 as the range. Diego applied his blocking scheme to determine goal numbers of 38, 29, 20, 11, and 2. He chose to go first, selected 2, and won the game. Diego’s behavior suggested he used more than a Nim-21 blocking scheme to determine his goal numbers. He generalized his Nim-21 scheme of “subtract four”

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to “subtract one more than the largest range number.” This had the effect of generalizing his Nim-21 scheme to a Nim scheme. In using his Nim scheme, Diego made a play to reach a goal number, then awaited the total given to him by the computer to determine his next choice. He paid no attention to the number the computer chose. He attended only to the current total given to him. The computer’s choice did not play into his thinking. His underlying image was to await the computer’s total, then use it to reach the next goal number. Pat raised the matter of Diego’s number in relation to the computer’s number in the context of reaching the next goal number. Excerpt 2 (After Winning 38 and 8) Pat: Do you notice a relationship between what the computer chooses and what you choose to get to the next goal? Diego: No, not really. Pat: What happens when you are at a goal number so that you get to the next goal number? Diego: I add a number. Pat: What about the computer? Diego: It adds a number, too. Pat: What about those two numbers? What has to be true about them so you get to the next goal? Diego: (28 second pause.) They have to add up to 9! Pat: Why is that? Diego: Because if I’m at a goal number … if I’m at a goal number the computer will choose a number and then I’ll choose … I’ll choose … I’ll choose another number to get to the next goal number … and the next goal number is 9 away from the [goal] number I have. Diego’s initial responses to Pat’s question confirmed our hypothesis about his underlying image of a play. It was not an image that combined his and the computer’s moves into one move that satisfies the requirement of reaching the next goal number. Instead, his image of a play was to take the number presented to him and to figure out the number needed to reach the next goal he’d already determined. At the beginning of Excerpt 2, Diego’s image of goal numbers was operative because it was not tied to a specific state of the game—he was able to discuss goal numbers generally. Diego’s scheme for goal numbers made it possible for him to use images of goal numbers decoupled from specific game states. His image underlying his decision on the next play was figurative because it was tied to a specific state of the game. He did not have a combined play scheme that would allow him to decouple his image of his next move from the current game state. Pat’s question, “What about those two numbers? What has to be true about them so you get to the next goal?” was instrumental in providing Diego an occasion to reflect on the relationships among the current goal, computer’s play, his play, and next goal. We suspect that he organized, at least temporarily, the states of the current and next goal as being connected by the combination of the computer’s and his plays.

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We interpret Diego’s 28-second pause as him re-playing the computer’s and his move in succession. This gave him an occasion to consider the two moves together, as one. We cannot know whether Diego would have thought of this himself, but the fact that he assimilated the question and resolved it suggests that, in re-playing the game, he modified his image of a “play,” at least in that moment, to include both players’ moves that together would move the total from one goal number to the next. Diego’s modification of his image of a “play” in this way was a step towards him developing an operative image of a “play.” The development of a stable combined play scheme went along with the development of an operative image of a play that was not tied to a specific game state. 33 and 7 Pat suggested one last game. Diego chose 33 and 1–7 as target and range, respectively. Diego went through his working-back strategy to determine goal numbers 33, 25, 17, 9, and 1. He concluded that to reach 1, he had to go first. Despite the insight Diego stated after the previous game, he still focused on the number presented to him and what he had to add to reach the next goal. Excerpt 3 (in the Midst of 33 and 7) Pat: How are you deciding your number? Diego: I’m looking for the number to add to get the next goal number. After Diego won, Pat asked again if there was a relationship between the computer’s number and his number when reaching the next goal. Diego quickly noted that the sum of his and the computer’s plays needed to be 8—for the same reason he stated earlier. However, it is important to note that, with 33 and 7, Diego did not use the insight he’d stated earlier (regarding the sum of his and computer’s moves) in deciding his play. Instead of Number Computer Plays + My Number = 8, Diego’s image of play continued to be Current Total Given Me + My Number = Next Goal. Pat ended the session by thanking Diego for participating and suggested he play the game with his family before the next meeting. Pat texted Diego late the next day to ask if he’d played the game with his family. Diego replied, “No, I’m still trying to get my head around it.” We presumed by “it,” Diego meant “strategy.” Session 2: July 28, 2020 Diego’s attendance at surf camp led to an 11-day break between sessions. He had not played the game with anyone, but he had thought about the game. Pat asked Diego what he remembered. Excerpt 4 Pat: Do you remember the game we played? Diego: Yeah. Nim. Yeah.

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Pat: Do you remember how it is played? Diego: Yeah. Pat: How is it played? Diego: So … first person to get to 21 wins. If you choose one you go first, if you choose two you go second. And the first person to get to 21 wins. Pat: How do you get to 21? Diego: So, you gotta go first, and you choose … two … no. You have to choose// Pat: //What numbers are you choosing from? Diego: Oh. One, two, and three. Pat: I think you were trying to remember the strategy you used? Diego: Yeah. Pat: What was that strategy? Diego: (4 second pause.) His number and your number have to add up to four. Pat: Why is that? Diego:  So then that, so then (yawns) so you keep getting to the places where you know you’re able to get to 21. Pat: Okay. Diego: I know … I know that sounds confusing. Pat: Well, instead of just telling me about it let’s play a game. Diego: Okay. We were struck by Diego’s recollection of his strategy (in bold). What had been a transitory observation in his 7/17/20 session, an observation he never employed, had become a defining feature of his strategy on 7/28/20, despite no interaction in the interim regarding the game with us or between Diego and his family. Moreover, his strategy of C + D = 4 entailed a reason for it—this strategy blocked the computer from reaching any goal numbers. This suggests to us that on 7/17 Diego engaged in what Steffe (1991) called functional accommodation—the modification of a scheme in the context of using it—and that in the interim, he engaged in what Steffe (1991) called metamorphic accommodation, which we understand as largely equivalent to Piaget’s meaning of projecting images and actions to a reflected level. In Excerpt 4, Diego had an operative image of combined play that was not tied to a specific game state. The development of this operative image was possible due to his reflection upon combined plays and the development of a stable combined play scheme. Diego’s combined play image was part of his combined play scheme—the scheme also included the entry points that trigger action and anticipations of action. As Diego’s combined play scheme became more stable, his images of combined play became more mobile and less dependent on specific aspects of the game. We say more about this in the discussion. 21 and 3 In playing the first game with 21 and 3, Diego employed his strategy of working backward to determine the first goal number. However, he miscalculated 21 minus 4, saying “sixteen,” getting goal numbers of 21, 16, 12, 8, 4, and 0. He chose the

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computer to go first and selected his number according to his new rule that the sum of the computer’s and his number must be 4. Diego realized he would lose when the computer presented him with 17. Excerpt 5 Diego: (Total is 17; D pauses for 21 seconds) Ooohhh. (Pause) Pat: Why the long pause? Diego: Um … cause seventeen … if I put three it’s twenty and he wins, if I put two its nineteen and he wins, and one he could just put three and he wins. Pat: What do you suppose the problem is? Diego: (Pauses for 26 seconds; whistles while thinking.) Oh, I went too far back. I was supposed to get to 17 first. So I lost. So just put two [just to finish the game]. Pat: So, you were supposed to get to 17, but you said 16? Diego: Yeah. We interpret Diego’s 21-second pause as his imagining all the moves he could make in combination with the computer’s subsequent move and his 26-second pause as re-presenting (re-playing) his original reasoning to get his list of goal numbers. He recalled thinking “21 − 4 = 16” and realized he should have said “seventeen.” We also note that this episode confirms Diego’s confidence that applying the condition C + D = 4 for each pair of moves was sufficient for him to win. In the next game (21 and 3, again), Diego insisted on enacting his working-back strategy “just to make sure.” ending at one and deciding he should go first. He played appropriately, getting to each of his desired goal numbers. When the computer presented Diego with a total of 10, Diego chose 3 to reach 13. Pat asked about how he was deciding on his choice of numbers. Excerpt 6 Diego: (Computer presents a total of 10) Three. Pat: How are you figuring out what number to pick? Diego: Whatever number he chooses, I just need to pick the number that adds up to four. Diego’s “C + D = 4” scheme was now at a reflected level. He knew applying it would necessarily land him at the next goal number without having to think of what the next goal number was. 36 and 5 Diego again employed his working-back strategy, but this time just to determine the starting number. He counted 36, 30, 24, 18, 12, 6, 0, then said, “But we cannot get to 0, so the computer needs to go first.” He did not mention a goal number while playing the game. Instead, as the computer played its number, Diego played a number, so the sum of the computer’s and his numbers was 6.

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Excerpt 7 Comp: (Plays 1) Diego: Okay, five (total is six; computer chooses 5, total is 11). One Pat: How are you deciding what number to put in? Diego: Same thing. Whatever number they choose, I just have to choose another number that adds up to six. Pat: Why is six special? Diego: It’s one more than the number we can pick … so that they can’t go over the number we have to get to but we also can’t … err … and also we … it’s too … and also it’s not too much for us to get to. Diego went on to win the game. He again confirmed that his strategy no longer relied on the total given him or the next goal number. It relied only on him determining the first number to play, using this number to decide whether he or the computer should go first (computer first if first goal is 0; otherwise, him first), and knowing what the sum of his and the computer’s play must be. Diego felt confident that attending to these conditions would produce a winning strategy. He had developed a Nim scheme. General Nim Pat suggested one last game—77 as the target and 1–10 as the selection range. Diego again worked backward from the target by 11’s, getting 0 as the first goal number. Diego chose the computer to go first and won the game. Afterward, Pat asked about his general strategy. Excerpt 8 Pat: Let me ask you a question. Diego: Yeah. Pat: It seems … and correct me if I’m wrong. It seems you start with the target number, and then go back to find the next smallest target number// Diego: //Yeah. That’s what happened. Pat: And you keep going back [Diego: Yeah] until you find the first target number [Diego:Yeah] and that tells you whether or not you go first? [Diego: Yep] Is that right? Diego: Yep. If it gets to one, then you have to go first, but if it gets to zero then they have to go first. Pat: What if you get to two? Diego: You can still go first … unless … the boundary … the number … unless the number limit you have to choose from is less than two … which would be kinda boring. Diego’s last statement shows he not only reasoned generally about his strategy, but also considered the implications of alternative conditions (“unless the number limit is less than two …”). We take this as evidence that Diego’s goal number images

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and combined play images were now operative. Because of the generality of Diego’s summary, Pat decided to raise the issue of efficiency. Excerpt 9 Pat: That’s very good! Now … could you think of a way to figure out what the first number should be without having to go back step by step? Diego: I guess that (pause) if … oh, hold on, I’m just noticing. When it was 21, they had to go first. Oh no. We had to go first. So I feel like, if it’s 21 … if the ­number limit is divisible by the number limit you’re allowed to choose, then you have to go first. Pat: So 21 is divisible by 3// Diego: //Oh, no. But 36 isn’t divisible by 5. And do you remember … can you go back and see if they had to go first on 36? Scroll up to see the previous game. Pat: (Scrolls back to the game of 36 and 5.) You had the computer go first and it chose one. Diego: Yeah, they went first. If the target number is divisible by the highest number you choose from, then you have to go first. Pat: But 36 is not divisible by five. Diego: And the computer went first! (Pause) So if it’s not divisible by the highest number you’re allowed to choose from, then the other person has to go first! Pat: Why do you suppose it works that way? Diego: (Looking into the air) Because that … it’s always gonna leave … … like … it’s always gonna leave … a higher number than … it’s alw … ahh … it’s always gonna end up like … it’s never gonna end up … hmm, hold on. (Pause) Yeah. That makes no sense. (Pause.) Here. Let’s do … so … so … from what we’ve seen right now it’s um an odd number has to make the computer go first and an even number has to make … wait no, not even number. A number divisible by that means I have to go first. So let’s just see … make … do … do like 30 … do like 32 and then do 4 … 32 is divisible by 4, yeah. Diego’s response to Pat’s question “Could you think of a way to figure out what the first number should be without having to go back step by step?” has earmarks of what Piaget called pseudo-empirical abstraction. By this, we mean that Diego looked for a pattern that related game conditions (target number and range) and the decisions he had made in light of them. He reflected on the products of his reasoning, not the reasoning in which he engaged to create those products. This is not a criticism. Rather, it is an observation. Diego tested his hypothesis on 32 as the target number and 1–4 as the range— and lost. Pat suggested he try his working back strategy again. Diego did this—32, 27, 22, 17, 12, 7, and 2—noting he had to go first and start with 2. Pat determined it would be a long struggle for Diego to refine his strategy further, so he used an intervention common in exploratory teaching interviews (Castillo-Garsow, 2010; Moore, 2010; Steffe & Thompson, 2000). This was to offer a suggestion to see how Diego

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might understand it and how he might subsequently use it. An intervention move within an exploratory teaching interview can unveil the nature of and boundaries of the interviewee’s schemes. Excerpt 10 Pat: So going back … I’m going to remind you of something. Each time you went back, you subtracted five, right? [Diego: Yeah] You subtracted one more than the upper limit. [Diego: Hmm hmm] Repeated subtraction is like division. [Diego: Hmm hmm] So you went back some number of fives and you got to two. [Diego: Yeah] So, if you divide 32 by 5, what remainder do you get? [Diego: Two] (5 second pause) And what’s special about 2? Diego: You could go first and get to 2, and then you could get to 7 first, you just … you kinda win. Pat: So let’s try that. This time I’m going to try 45 and 6. Now, without using target numbers, see if you can find the first target number. Diego: Seven divided by 45 is … (40 second pause; D looks in the air). The remainder would be three. Because the closest number to that number that is divisible by seven is 42, and 42 divided by seven is six. And then the remainder would be three. So the first target number is three. Pat: So who goes// Diego: //So I would have to go first. Pat: And when you say 42 divided by seven is six, what does that six mean? Diego: (19 second pause) It means I would go back by seven six times. It appeared Diego easily understood that his going back strategy entailed repeated subtraction, and he appeared to understand Pat’s statement, “Repeated subtraction is like division.” He also inferred that the remainder after dividing 32 by 5 would be the first target number. Diego then applied a “division and remainder” strategy to a game with target 45 and range 1–6, concluding he had to start with 3 and the sum of plays had to be 7. Diego also understood the relationship between dividing 45 by 7 and his working-back strategy—dividing 45 by 7 would give the number of times he would go back by 7 to get the starting number, and the remainder would be the starting number. Afterward, Diego celebrated. Excerpt 11 Diego: So now I know how to do it without having to go back step by step! Pat: Isn’t that cool that you can figure out [where to start and] what number to add without knowing any of the target numbers except the last one? Diego: Yeah. Pat: So, just … okay, we can finish this now. But if you could, tell me what your general strategy would be no matter what numbers I pick. Diego: Umm. Get to the remainder first. Pat: Remainder of what? Diego: Get to the remainder of the number … get to the remainder of the goal number divided by the highest choice number plus one. Pat: And that remainder tells you what?

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Diego: And then … and then … that remainder tells you if you should go first or if they should go first and then you always have to get … you have to see what their number is and add another number to get the number … to get … to get to the highest number plus one. Pat: When should you have the computer go first? When should you decide to have the computer go first? (Zoom connection fails) Diego? Diego? The broken Zoom connection was because Diego’s phone battery expired. Excerpt 12 shows their exchange of text messages following the Zoom failure. Excerpt 12

Excerpts 11 and 12 provide evidence that Diego made metamorphic accommodations to his Nim scheme. His final Nim strategy was much more efficient than his first winning strategy. His final strategy was more efficient and qualitatively different from his first winning strategy because he integrated his division scheme to avoid needing to repeatedly subtract to find each goal number. Diego reflected on images of goal numbers and combined play to accommodate his Nim scheme to be more general. Discussion It is important to note that Diego did not write anything down, nor was there a visible record of his thinking. There was only what the computer presented after each player played. Diego had no visual record of his reasoning to aid his reflections. He only had what he could recall of his reasoning and its results as images to reflect upon.5  It is plausible that had Diego created a written record, his writing might have prompted him to engage more frequently in pseudo-reflective abstraction. 5

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Diego’s imagery for the game rested in his re-presentations of his reasoning, and sometimes re-presentations of his conclusions. The role of Diego’s images evolved as his schemes evolved. His initial imagery consisted of his reasoning about specific numbers in specific contexts. Diego re-enacted his reasoning in order to think about it. This is an instance of deferred imitation. Early on, Diego’s remembered decisions did not entail the reasoning leading to them, consistent with his scheme at that time employing imagery figuratively. His decisions were simply the last step in his chain of actions. Later, Diego differentiated between his decisions and the reasoning leading to them, which allowed him to begin projecting his reasoning to a reflected level. Finally, Diego generalized his images from specific game states to arbitrary game states, consistent with his imagery operating at a third level because he had created his Nim scheme at a reflected level. We provide a detailed summary of Diego’s development of his Nim and General Nim schemes below. Nim Scheme • An image of being blocked. Diego experienced “I was blocked” twice. He developed the image, “The computer gave me a number that kept me from reaching a goal number. There are numbers that ‘block’ a player from winning.” • A “Blocking” scheme—find all the blocking numbers. A blocking number keeps the computer from reaching the next goal number and ensures he can reach the next goal number. Diego’s blocking scheme relied on his image of a blocking number. • A “First number” scheme—if the first goal number is 0, the computer goes first. If first goal number is not zero, go first and select that number. This scheme relied on Diego having an image of having executed his blocking scheme. • A “Combined play” image—a combined play takes the game from one goal number to the next. In Excerpts 2 and 3, Diego reasoned that the sum of his and computer’s play had to be a certain number (9 in one instance, 8 in the other), but he seemed not to have had a “combined play” image. He did not recall these conclusions upon playing the next game. It was after the 11-day break that Diego seemed to have an image of the computer’s play and his play as one play. • A “Combined play” scheme—decide what to play based solely upon the computer’s play and the selection range. • A Nim scheme—we saw in Excerpt 7 that Diego coordinated his First Number scheme with his Combined Play scheme to form a Nim scheme. He was confident his Nim scheme would produce a winning strategy regardless of the target number and selection range. Diego had developed a scheme for Nim. He developed it over time by coordinating his blocking scheme, first number scheme, and combined play scheme to make a strategy for winning the game regardless of the target number and selection range. His coordination of schemes was enabled by having projected each to a reflected

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level. We are comfortable saying this because he could articulate each in general terms, not reliant on any specific game. In each case, the projection happened as a result of reflecting on images of his reasoning that he gained from playing the game. Diego’s images became more mobile and flexible. Figurative images became operative as Diego’s schemes developed through reflection and repeated reasoning. One of Piaget’s defining characteristics of operating at a reflected level is that the person is aware of their schemes and uses them to explain their reasoning. That Diego coordinated his blocking, first number, and combined play schemes, and explained how they worked together confirms to us he had indeed projected them to a reflected level. General Nim Scheme Diego went beyond his initial Nim scheme. He assimilated Pat’s suggestion to think of repeated subtraction as division and used that idea to develop more than a winning strategy. He employed already-developed schemes for division (as repeated subtraction) and remainder to understand Pat’s suggestion and saw how it related to his blocking scheme (repeated subtraction) in determining the first goal number. He modified his Nim scheme by incorporating the scheme “First Number =  mod (N, M).”6 Had Diego not had a well-developed division scheme, he would not have seen the relevance of Pat’s suggestion. We note in closing that Diego’s speech over time gives an indication of how he generalized his imagery. His earlier statements were about specific numbers. Later statements were about “computer’s number,” “my number,” “goal number,” and “one more than the biggest number we can choose.” We take Diego’s use of literal names for states as enabling him to differentiate his reasoning from the specific contexts in which his reasoning occurred and from the specific conclusions he drew. This also supported Diego’s projection of his reasoning to a reflected level. His use of literal names for objects upon which he acted supported his focus on the actions he took to get from one state to another. We interviewed Diego again after a lapse of 15 months. He did not recall Nim nor how it is played. We reminded him of the rules and played a game of 21. After a short pause, he recalled his Nim scheme (finding goal numbers and first number) and used it to win. We then played 45 and 1–8, and he again employed his Nim scheme to have the computer go first, and he won. In the third game (67 and 7), he recalled his generalized Nim scheme, found the remainder of 67 ÷ 8, chose to go first, gave 3 as his first number, and won. This all happened in less than 20 minutes. The rapidity with which Diego reconstructed his generalized Nim scheme suggests to us we are correct to have called it a scheme.

 This is our description, not Diego’s.

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Implications for Math Education We shared Diego’s case study specifically to highlight that students’ images of having reasoned are the primary fodder for productive reflection and hence for productive scheme formation. Students’ images of their reasoning become transformed into operations of thought they can apply outside specific contexts in which their reasoning occurred. This fact has implications for mathematics teaching and mathematics education research. Implications for Mathematics Teaching To help students form images of having reasoned so that they may reflect upon them is not the same as asking, “Why did you do that?” or “How do you know that?” Those questions often sound to students like they are being policed. A teacher highlights reasoning instructionally by engendering reflective discourse (Cobb et  al., 1997; Stein et al., 2008) with and among students, and by creating didactic objects to support reflective discourse (Thompson, 2002). Designing instruction to bring students’ imagery into the open and to support reflective discourse means to orient students to discuss ways they are understanding situations (“What do you see going on in this situation?” “Share with us what you imagined about this situation when you said that?”), meanings and reasoning they are trying to convey (“Help us understand what you meant by that?”), ways they are understanding diagrams and animations (“What do you see happening in this animation?” “What do you see this diagram depicting?”), and their reasoning in the context of solving a problem (“Please share your strategy, if you can.” “What stood out to you when you decided to divide?”). Of course, those cannot be idle questions. A teacher conveys genuine interest by incorporating students’ answers into the classroom conversation. Fostering reflective discourse also entails having students attempt to understand others’ reasoning and reflect on meanings they might intend. Reflective discourse takes students’ imagery, meanings, and reasoning as objects of class discussion. Instructors fostering reflective discourse continually demonstrate that they care about and value students’ understandings—instructors convey to students that they are interested in students’ images and meanings-in-the-moment (Hackenberg, 2010). Cobb et al. (1997) and Clark et al. (2008) reported that teachers’ consistent support of reflective discourse can positively affect students’ attitudes and classroom atmospheres. Under a teacher’s guidance, sharing the foundations of one’s thinking, and expecting the same of others, becomes a classroom mathematical practice (Yackel & Cobb, 1996). Although Pat’s interactions with Diego were in an interview setting, Pat’s questions to Diego did have an instructional effect. For example, Pat’s questions, “How are you deciding what number to select?” and “Do you see a relationship between the computer’s number and your number?” appear to have caused Diego to reflect on his reasoning when he might not have done so otherwise. Also, Pat’s questions prompted Diego to formulate responses to those questions. As noted by Inhelder and Piaget (1964), the attempt to express one’s reasoning to oneself, or to

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communicate one’s reasoning to another person, is a primary stimulant for reflection. Questions like these, offered at timely moments, can be incorporated into instruction. Implications for Mathematics Education Research Diego’s case exemplifies a methodological focus on students re-presenting their reasoning to themselves as necessary for reflection. This focus entails timely probes about what the situation under discussion is to the student. It also exemplifies a focus on asking questions that prompt students to explain their decisions. But probes into students’ decision-making processes must be crafted carefully. They must not appear to the student as demands for justification. Instead, you want probes to convey to students that you are genuinely interested in how they are thinking. “Help me understand how you thought about this?” exemplifies a genre of questions that can be useful in gaining insight into students’ imagery and reasoning. Diego’s case also highlights our stance that it is students’ images of having reasoned that provide the fodder for reflection. To live this stance in your research requires that you take students’ verbal statements and symbolic work as a clouded window into their thinking—that you must probe to gain insight into what they meant when they said or wrote what they did and how they imagined their actions being relevant to the situation as they conceived it.

Imagery in the Projection from Figurative to Reflected Thought Piaget spoke of two kinds of reflecting abstraction. The first is to construct schemes at a reflected level while the second is to reason at first with schemes at a figural level and then move one’s reasoning to counterpart schemes that have been created at a reflected level.7 This case is a study of the latter. Michael and Robert are mathematics educators. Michael shared with Robert a task for students asking them to predict the graph of y =  sin (3x + 1) as a transformation of the graph of y =  sin (x). Robert first thought about the problem in terms of slots—(_ + b), largely as a figural generalization of (x + b). He envisioned the graph being transformed according to y  =    sin  (_) and then translated according to the value of b. Robert knew y =  sin (3x) would “compress” the graph of y =  sin (x) by moving each value of sin(x) from above the value of x to above the value of x/3. He knew “+1” would translate the graph of y =  sin (3x) one to the left by putting each value of sin(3x) above the value of 3x  −  1. Robert concluded that the graph of y =  sin (3x + 1) is the graph of y =  sin (3x) shifted by −1. Michael said, “That’s not correct.”

 We have seen this distinction translated in different ways by different translators and we do not know which terms to use for them. 7

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Robert was immediately puzzled—he was sure his reasoning was correct. He first wondered how to test his claim. He knew, according to his theory of the situation, that all points on the graph of y =  sin (3x) would be shifted to the left by 1. He decided to produce both graphs on the same axes and examine one point on the graph of y =  sin (3x) shifted like he thought it should. Robert’s investigation confirmed that the graph of y =  sin (3x + 1) is not the graph of y =  sin (3x) shifted by 1 (Fig. 5.9). Robert puzzled about why the graph of y =  sin (_  + 1) is not necessarily the graph of y =  sin (_) shifted by 1. He used the mouse to highlight points on the graph and noticed the graph seemed shifted to the left by 1/3 instead of 1. He wondered, “How is it possible for three as a coefficient of x to affect the effect of adding 1?” Robert eventually asked himself, “What am I doing when I shift a graph by changing its argument?” Focusing on the idea of argument opened him to think of sin(3x + 1) as a composite function. He then considered sin(3x + 1) as sin(ax + b) and sin(ax + b) as the composite function h(k(x)). Robert employed similar imagery as initially to understand how a graph is shifted when a function’s argument is itself a function: Start with a value x = c (Fig. 5.10a), move the value x = c on the x-axis by the function k to arrive at x = k(c) (Fig. 5.10b), “pick up” the value of h(k(c)) (Fig. 5.10c), move from x = k(c) back to x = c by k−1(k(c)) (Fig. 5.10d), then plot the value of h(k(c)) above x = c (Fig. 5.10e). He concluded that the graph of y = h(k(x)) will appear to be the graph of y = h(x) but with each value h(x0) plotted above or below x = k−1(x0), provided k−1(x0) exists. Robert then applied this result to the graph of y =  sin (k(x)) where k(x) = ax + b to see that the graph of y =  sin (x) is “shifted” by the function k−1 so that each value of sin(x) appears above or below k−1(x) = (x − b)/a. In this particular case, the graph of y =  sin (3x + 1) is the graph of y =  sin (x) with each value of sin(x) plotted above or below (x − 1)/3. He explained his new approach as, You want to anticipate the graph of y = h(k(x)) given the graph of y = h(x). Imagine standing on the graph of y = h(x) at, say, x = x0. Where did this value x0 come from? It came from a value x = c so that x0 = k(c). Where will the value of h(x0) appear on the graph of y = h(k(x))? It will appear above or below c = k−1(x0). In the case of y =  sin (x) and y =  sin (3x + 1), any value sin(x0) will appear at a value x = c so that x0 = 3c + 1, or c = (x0 − 1)/3. This tells me that to get the graph of y =  sin (3x + 1) start with the graph of y =  sin (x), shift it to the left by 1, then compress that graph by 1/3. But this will be true of any function f. The graph of y = f(3x + 1) will be the graph of y = f(x) but with each value f(x) appearing above or below (x − 1)/3.

Robert shared his conclusion with Michael, who agreed that the graph of y =  sin (3x + 1) is the graph of y =  sin (x) shifted by 1, then compressed by a factor of 1/3. Michael, however, had not considered the general case of composite functions that Robert used to arrive at this specific result. How do we possess a record of Robert’s inner thoughts? Because Robert was Pat Thompson and Michael was Alan O’Bryan. Pat, realizing this was a potentially important event, wrote a log of his thoughts as he puzzled through his confusion. Figure 5.10 is our rendition of his unorganized drawings. The central point of this example is that Pat first employed imagery at a figural level—the level of action regarding an existing scheme. He initially assimilated the

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Fig. 5.9  Robert’s test of his initial claim

question to a well-developed scheme for transforming graphs, which allowed him to think primarily in terms of imagining specific actions on a graph in relation to the specific symbolic form y =  sin (_ + 1). He thought of y =  sin (_ + 1) as y =  sin (x + 1), concluding that the graph of y =  sin (_ + 1) would be the graph of y =  sin (_) shifted by 1 to the left. He moved to a level of reflection only upon being faced with the invalidity of his reasoning. Even at a reflected level, Pat employed images like what he conjured at a figural level. He envisioned how the original and new graphs are related by picking an arbitrary value on the x-axis where a value of the composite function would be plotted and moving to a location on the x-axis (by evaluating a function’s argument) where the original function would be evaluated. These images initially were figural with respect to Pat’s “translate a graph” scheme that employed them. They became operative when he moved to a reflected level to think of y =  sin (3x + 1) as y = h(k(x)). At a reflected level, he thought of a function and its graph, but not a specific function or a specific graph. Any graph would support his thinking of movements on the x-axis by an arbitrary function k and its inverse. He also thought of an arbitrary argument to the composite function. His images were arbitrary while still providing a context for the transformations he employed—evaluating the composite function’s argument to get a location for evaluating the original function, then using the argument’s inverse to move that value of the function back to where it would be plotted as a value of the composite function. He then thought of this transformation as being applied to every value in the domain of the argument. Pat resolved his initial confusion about the graph of y =  sin (3x + 1) in relation to the graph of y =  sin (x) by answering a general question about the graph of a composite function y = f(g(x)) in relation to the graph of y = f(x) for any functions f and g. He also understood his reasoning applied only where g has an inverse function. It is important to understand that this is not a story about Pat constructing a higher level scheme, like Diego, through the abstractive phases of empirical, pseudo-empirical, and reflecting abstractions. He already possessed the schemes for functions, function notation, function inverse, and function graph he eventually

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Fig. 5.10  Our rendition of Robert’s drawings and his reasoning about them

employed. Instead, this is a story of someone projecting their figurative reasoning to an already-present reflected level—a level of already-existing operative schemes. Pat understood all along he was solving the original problem, but with the additional understanding that he was solving a general version of that specific problem. Pat’s attention was focused initially on the graph of y =  sin (3x + 1) in relation to the graph of y =  sin (x). He re-imagined the problem to be about the graph of y = h(k(x)) in relation to the graph of y = h(x) for arbitrary functions h and k. It is also important to note that the role of Pat’s images changed from his initial to final thoughts. Initially, Pat’s imagery provided a template for his assimilation of the problem, assimilating y =  sin (3x + 1) as y =  sin (_ + 1), where “_ + 1” was a figurative generalization of x  +  1. Even though his imagery for y  =    sin  (_  +  1) involved movement, it was figural regarding the scheme to which he assimilated it—a scheme in which the added constant specified the direction and distance the

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function’s graph would be translated. Pat’s images at a reflected level served as arbitrary states of transformations—moving an arbitrary displacement on a number line with respect to an arbitrary graph. Pat understood movements along an axis being “caused” by evaluating functions and understood locations on the number line as related by being values of an argument or its inverse. This role of imagery is in line with Piaget’s third form, describing the role of images in relation to operative thought (see quotation on page 135). They served as “momentary states” in Pat’s reasoning about transformations from original graph to desired graph. The specific images Pat employed were not necessary to invoke the transformations that related them. Although Pat did not construct a higher level scheme through elaborate abstractive processes of differentiation, integration, and so on, he did construct a new scheme—and therefore a new meaning—for “transform a graph.” He connected (assembled) existing schemes in, for him, a novel way. Initially, his meaning for “transform a graph” was to think of the form of a function’s argument and what it implied about how a function’s graph changes. His meaning at the end was crystalized as how the original function’s values are “re-positioned” on the independent axis by the inverse of the original function’s argument when the argument is viewed as a function. Implications for Mathematics Education We shared Robert’s (Pat’s) case study specifically to highlight again that images of having reasoned are the primary fodder for productive reflection. This time, unlike Diego’s case, “productive reflection” meant to project the situation as Pat originally conceived it to a reflected level of already-existing schemes. That imagery of having reasoned is important even when moving to reflected mathematical thought has implications for mathematics teaching and mathematics education research. Implications for Mathematics Teaching and Mathematics Education Research Solving a specific problem by solving a generalized version of it is a standard move in higher mathematics. At the same time, it is a rare move in school mathematics and a move made without students’ noticing it in undergraduate mathematics. We are unaware of research into this phenomenon from a genetic epistemology foundation. The closest we know is research on problem posing (see Cai et  al., 2015), especially the early work by Brown and Walter (1983, 1993). Problem posing, as originally crafted by Brown and Walters, is to provide facts about a situation and ask students to craft problems from these facts. Cai et al. (2015) surveyed research on problem posing as an instructional technique, concluding that employing it in instruction has a positive impact on students’ problem-solving abilities. We suspect students in problem-posing studies engaged in various forms of reflection to create their problems. Reports that “more able” students pose more and

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more complex problems than “less-able” students (Cai et al., 2015; Marsh & Yeung, 1998) suggests to us that “more able” students were engaging in projection to reflected thought. However, we are unaware of problem-posing studies that examined students’ behaviors from a perspective of images they formed and their reflections on those images. One technique employed by Brown and Walters is ripe for research on students’ imagery and reflection. It is to have students revisit a problem repeatedly, to relax constraints in each iteration so that they produce ever more general versions of the original problem. Generating generalized problems, and discussing their generalization processes, could provide opportunities for reflection. Researching their solving activities for the problems they generate could then provide occasions to explore the connections they actually make between their underlying imagery and reflective processes.

Discussion Our thesis throughout this chapter has been that images of having reasoned are the foundation for reflection and scheme development. We stressed that imagery includes visualization but includes far more than visualization. We recapped Piaget’s levels of imagery and expanded their meaning to make them useful for modeling mathematical scheme formation at any level of sophistication. We included the case study involving composite functions specifically to show Piaget’s constructs can be used to model higher level mathematical thinking. We illustrated the interplay among imagery, reflection, and scheme formation through two case studies and explained the implications of each for mathematics teaching and research. In this discussion, we will highlight an aspect of Diego’s case study that was central to the work with Diego but remained tacit in our accounts. It is our preparation for the interviews—task selection and design together with a conceptual analysis of the game. We settled on Nim as the context of our interviews for two reasons. First, we wanted to avoid as much as possible Diego’s need to create written records of his work. We are not suggesting that symbolizing is unimportant—far from it. Our past research and teaching, however, convinced us that one effect of students’ mathematical schooling is they often engage in premature symbolization. We say “premature” because students often create inscriptions that then turn into the objects of their attention. Their focus on past inscriptions then diverts their attention from the reasoning that led to them. We also considered that readers might think a case study of learning the game of Nim is unrelated to learning school mathematics. This would be true if we have in mind standard school mathematics. But reflective discourse is not standard in school mathematics, and we wished to highlight that it is students’ reflection on their images of prior reasoning that is central to their advancement. We argue that the case of a student constructing a fairly complex scheme without recourse to pseudo-empirical abstractions from written work is highly relevant to ways students could learn mathematics in school. As the last interchange with Diego

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showed, he did symbolize his thinking, but he did so only after his thinking advanced to a state where the symbols retained their meaning within his reasoning. Glasersfeld defined conceptual analysis by a question, “What mental operations must be carried out to see the presented situation in the particular way one is seeing it?” (Glasersfeld, 1995, p. 78). Thompson expanded Glasersfeld’s meaning of conceptual analysis to include four uses: 1. in building models of what students actually know at some specific time and what they comprehend in specific situations, 2. in describing ways of knowing that might be propitious for students’ mathematical learning, 3. in describing ways of knowing that might be deleterious to students’ understanding of important ideas and in describing ways of knowing that might be problematic in specific situations, 4. in analyzing the coherence, or fit, of various ways of understanding a body of ideas. Each is described in terms of their meanings, and their meanings can then be inspected in regard to their mutual compatibility and mutual support. (Thompson, 2008, p. 59) We employed conceptual analysis according to #1 in our analysis of Diego’s interviews. We employed it according to #2 in our preparations for the interview—we analyzed the game of Nim according to what someone must understand to play it at the highest level and what schemes might be necessary to get there. We also drew on our own experience playing the game and from watching others play it. For any game with target N and range 1–M, we considered My  First Number =  mod (N, M) coordinated with Computer′s Play + Human′s Play = M + 1 to be the most sophisticated strategy. We also anticipated that the first scheme would be Blocking and the second would be Goal Numbers. What we had to consider was how Diego might fill in the gaps between the first schemes and the final scheme. We anticipated that the Computer′s Play + Human′s Play = M + 1 scheme was crucial for Diego to advance to using division, for without that scheme he would not see that successive goal numbers are generated by repeatedly subtracting his and the computer’s combined play. This was why Pat repeatedly asked Diego, “How are you deciding your number?” and later asked, “Do you see a relationship between the computer’s number and your number?” We also anticipated it would be crucial that Diego see a connection between repeatedly subtracting the combined play to determine the first number and dividing the target by one more than the range to find a remainder. When Pat determined this insight would be long in coming, he pointed out to Diego that he was using repeated subtraction and suggested the connection between repeated subtraction and division to see what Diego would make of it. Finally, we decided not to suggest Diego write anything down for the reasons we explained earlier. The case of Robert (Pat) illustrates a second form of reflection that turns figurative imagery into operative imagery—the projection of a context assimilated figurately to schemes that employ images operatively. The trigger for this projection was Pat thinking of the “input” to the sine function as an argument to the sine

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function, which then led him to think of the original context more generally as a composite function. The case of Robert is different from Diego’s in that Pat already possessed the schemes to which he projected the context. We argued that Pat did not construct a scheme that employed images operatively. He already possessed those schemes. Instead, Pat constructed a new meaning for “transform a graph.” In conclusion, we say again that our main purpose was to highlight three arguments. The first is that imagery, as re-presentations of experience, includes far more than visualization. The second is that a main function of imagery in students’ mathematical learning is that they form images of having reasoned. This includes the kind of reasoning in which Diego engaged, it includes reasoning students use to interpret diagrams or animations, and it includes reasoning they engage in to comprehend a problem situation along with reasoning they engage in to solve it. The third is that imagery, as a construct, does not stand alone. Imagery as a construct is useful only to the extent that it allows one to focus on the contexts of students’ schemes and meanings and to employ reflective abstraction as a construct for explaining and investigating students’ mathematical learning.

References Brown, S. I., & Walter, M. I. (1983). The art of problem posing. Franklin Institute Press. Brown, S. I., & Walter, M. I. (1993). Problem posing: Reflections and applications. Erlbaum. Byerley, C., & Thompson, P. W. (2017). Secondary teachers’ meanings for measure, slope, and rate of change. Journal of Mathematical Behavior, 48(2), 168–193. Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem posing research in mathematics: Some answered and unanswered questions. In F. M. Singer, F. M. Ellerton, & J. Cai (Eds.), Mathematical problem posing (pp. 3–34). Springer. Castillo-Garsow, C. C. (2010). Teaching the Verhulst model: A teaching experiment in covariational reasoning and exponential growth [Dissertation, Arizona State University]. http://goo. gl/9Jq6RB Clark, P.  G., Moore, K.  C., & Carlson, M.  P. (2008). Documenting the emergence of “speaking with meaning” as a sociomathematical norm in professional learning community discourse. The Journal of Mathematical Behavior, 27(4), 297–310. https://doi.org/10.1016/j. jmathb.2009.01.001 Clement, J.  J. (2018). Reasoning patterns in Galileo’s analysis of machines and in expert protocols: Roles for analogy, imagery, and mental simulation. Topoi. https://doi.org/10.1007/ s11245-­018-­9545-­5 Cobb, P., & von Glasersfeld, E. (1984). Piaget’s scheme and constructivism. Genetic Epistemology, 13(2), 9–15. Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28(3), 258–277. Cooper, R.  G. (1991). The role of mathematical transformations and practice in mathematical development. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 102–123). Springer-Verlag. Davis, R. B., Jockusch, E., & McKnight, C. (1978). Cognitive processes involved in learning algebra. Journal of Children’s Mathematical Behavior, 2(1), 10–320. Dawkins, P. C., & Norton, A. (2022). Identifying mental actions for abstracting the logic of conditional statements. The Journal of Mathematical Behavior, 66, 100954. https://doi.org/10.1016/j. jmathb.2022.100954

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Dewey, J. (1910). How we think. DC Heath. Frank, K. M. (2017). Examining the development of students’ covariational reasoning in the context of graphing [Ph.D. dissertation]. Arizona State University. Hackenberg, A.  J. (2010). Mathematical caring relations in action. Journal for Research in Mathematics Education, 41(3), 236–273. Hadamard, J. (1954). The psychology of invention in the mathematical field. Courier Corp. Harel, G. (2008a). DNR perspective on mathematics curriculum and instruction, part I: Focus on proving. ZDM  – Mathematics Education, 40, 487–500. https://doi.org/10.1007/ s11858-­008-­0104-­1 Harel, G. (2008b). DNR perspective on mathematics curriculum and instruction, part II: With reference to teacher’s knowledge base. ZDM – Mathematics Education, 40, 893–907. https:// doi.org/10.1007/s11858-­008-­0146-­4 Harel, G. (2008c). What is mathematics? A pedagogical answer to a philosophical question. In R. B. Gold & R. Simons (Eds.), Current issues in the philosophy of mathematics from the perspective of mathematicians (pp. 265–290). Mathematical Association of America. Harel, G. (2013). Intellectual need. In K. Leatham (Ed.), Vital directions for research in mathematics education (pp. 119–151). Springer. Harel, G. (2021). The learning and teaching of multivariable calculus: A DNR perspective. ZDM – Mathematics Education, 53(3), 709–721. https://doi.org/10.1007/s11858-­021-­01223-­8 Inhelder, B., & Piaget, J. (1964). The early growth of logic in the child: Classification and seriation. Routledge & Kegan Paul. Johnckheere, A., Mandelbrot, B. B., & Piaget, J. (1958). La lecture de l’expérience [Observation and decoding of reality]. P. U. F. Jorgensen, A.  H. (2009). Context and driving forces in the development of the early computer game Nimbi. EEE Annals of the History of Computing, 31(3), 44–53. Joseph, S., & Williams, R. (2005). Understanding posttraumatic stress: Theory, reflections, context and future. Behavioural and Cognitive Psychotherapy, 33(4), 423–441. https://doi.org/10.1017/ S1352465805002328 Lobato, J., & Thanheiser, E. (2002). Developing understanding of ratio-as-measure as a foundation for slope. In B. Litwiller (Ed.), Making sense of fractions, ratios, and proportions: 2002 yearbook of the NCTM. National Council of Teachers of Mathematics. Marsh, H., & Yeung, A.  S. (1998). Longitudinal structural equation models of academic self-­ concept and achievement: Gender differences in the development of math and English constructs. American Educational Research Journal, 35(4), 705–738. Miller, A. I. (1996). Metaphors in creative scientific thought. Creativity Research Journal, 9(2–3), 113–130. Montangero, J., & Maurice-Naville, D. (1997). Piaget or the advance of knowledge (A. Curnu-­ Wells, Trans.). Lawrence Erlbaum. Moore, K. C. (2010). The role of quantitative reasoning in precalculus students learning central concepts of trigonometry [Dissertation]. Arizona State University. Moore, K. C., Silverman, J., Paoletti, T., & LaForest, K. (2014). Breaking conventions to support quantitative reasoning. Mathematics Teacher Educator, 2(2), 141–157. https://doi.org/10.5951/ mathteaceduc.2.2.0141 Moore, K. C., Silverman, J., Paoletti, T., Liss, D., & Musgrave, S. (2019). Conventions, habits, and U.S. teachers’ meanings for graphs. The Journal of Mathematical Behavior, 53, 179–195. https://doi.org/10.1016/j.jmathb.2018.08.002 Piaget, J. (1936). The origins of intelligence in children (2637414). W. W. Norton. Piaget, J. (1965). The child’s conception of number. W. W. Norton. Piaget, J. (1967). The child’s concept of space. W. W. Norton. Piaget, J. (1968a). On the development of memory and identity (E.  Duckworth, Trans.). Barre Publishers. Piaget, J. (1968b). Six psychological studies. Vintage Books. Piaget, J. (2001). Studies in reflecting abstraction (R. L. Campbell, Trans.). Psychology Press.

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Piaget, J., & Garcia, R. (1991). Toward a logic of meanings. Lawrence Erlbaum. Piaget, J., & Inhelder, B. (1966). The psychology of the child. Basic Books. Powers, W. (1973a). Behavior: The control of perception. Aldine. Powers, W. (1973b). Feedback: Beyond behaviorism. Science, 179, 351–356. Steffe, L. P. (1991). The learning paradox. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 26–44). Springer-Verlag. Steffe, L. P. (2010). Perspectives on children’s fraction knowledge. In Children’s fraction knowledge. Springer. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267–307). Erlbaum. Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340. https://doi.org/10.1080/10986060802229675 Stump, S.  L. (2001). Developing preservice teachers’ pedagogical content knowledge of slope. The Journal of Mathematical Behavior, 20(2), 207–227. https://doi.org/10.1016/ S0732-­3123(01)00071-­2 Thompson, P. W. (1994a). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26(2–3), 229–274. Thompson, P. W. (1994b). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179–234). SUNY Press. Thompson, P. W. (1996). Imagery and the development of mathematical reasoning. In L. P. Steffe, P.  Nesher, P.  Cobb, G.  A. Goldin, & B.  Greer (Eds.), Theories of mathematical learning (pp. 267–283). Erlbaum. Thompson, P.  W. (2002). Didactic objects and didactic models in radical constructivism. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 197–220). Kluwer. Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 31–49). PME. Thompson, P. W. (2016). Researching mathematical meanings for teaching. In L. D. English & D.  Kirshner (Eds.), Handbook of international research in mathematics education (3rd ed., pp. 435–461). Taylor & Francis. Thompson, P.  W., & Dreyfus, T. (1988). Integers as transformations. Journal for Research in Mathematics Education, 19, 115–133. Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), Research companion to the principles and standards for school mathematics (pp. 95–114). National Council of Teachers of Mathematics. Thompson, P. W., Carlson, M. P., Byerley, C., & Hatfield, N. (2014). Schemes for thinking with magnitudes: A hypothesis about foundational reasoning abilities in algebra. In L.  P. Steffe, L. L. Hatfield, & K. C. Moore (Eds.), Epistemic algebra students: Emerging models of students’ algebraic knowing (Vol. 4, pp. 1–24). University of Wyoming. http://bit.ly/1aNquwz Thompson, P. W., Ashbrook, M., & Milner, F. A. (2019). Calculus: Newton, Leibniz, and Robinson meet technology. Arizona State University. http://patthompson.net/ThompsonCalc von Glasersfeld, E. (1991). Abstraction, re-presentation, and reflection: An interpretation of experience and Piaget’s approach. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 45–65). Springer-Verlag. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Falmer Press. von Glasersfeld, E. (2001). Scheme theory as a key to the learning paradox. In A.  Tryphon & J. Vonèche (Eds.), Working with Piaget: Essays in honour of Bärbel Inhelder (pp. 139–146). Psychology Press. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–476.

Chapter 6

Empirical and Reflective Abstraction Amy Ellis, Teo Paoletti, and Elise Lockwood

The Enduring Attention to Abstraction Philosophers in the empiricist tradition have long appealed to the notion of abstraction as a way to understand how people develop knowledge. Over 2000 years ago, Aristotle developed his concept of abstraction as a way to distinguish and recognize different aspects of objects. In his account, one can consider a particular attribute of an object and isolate it, creating a new idea based on the attribute alone (Bäck, 2006). Much later, abstraction as a concept appeared in John Locke’s An Essay Concerning Human Understanding (1690/1975), in which he stated that it was the source of all general ideas: “This is called Abstraction, whereby ideas taken from particular beings become general representations of all the same kind” (Book II, Ch. X, §9). Locke emphasized the role of abstraction in producing general ideas, as well as the importance of abstract general ideas to knowledge. A century later, Kant wrote in his Lectures on Logic (1800/1992) that by comparing objects and attending to the feature they have in common, one can abstract from all other things to form a concept: To make concepts out of representations one must be able to compare, to reflect and to abstract, for these three logical operations of the understanding are essential and universal conditions for generation of every concept whatsoever. I see, e.g., a spruce, a willow and a A. Ellis (*) Department of Mathematics, Statistics, and Social Studies Education, University of Georgia, Athens, GA, USA e-mail: [email protected] T. Paoletti College of Education and Human Development, University of Delaware, Newark, DE, USA E. Lockwood Department of Mathematics, Oregon State University, Corvallis, OR, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 P. C. Dawkins et al. (eds.), Piaget’s Genetic Epistemology for Mathematics Education Research, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-47386-9_6

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linden. By first comparing these objects with one another I note that they are different from another in regard to the trunk, the branches and the leaves, etc.; but next I reflect on that which they have in common among themselves, trunk, branches and leaves themselves, and I abstract from the quantity, the figure, etc., of these; thus I acquire a concept of a tree. (9:94-5)

In Critique of Pure Reason (1781/2003), Kant also distinguished between the capacity to construct representations of specific objects by means of the senses, which he called intuition, and the capacity to form abstract and general representations, which he called concepts. Similarly, Wilhelm von Humboldt (1795/1907), in sharing a number of observations about reflection, noted that the mind must “stand still for a moment in its progressive activity, must grasp as a unit what was just presented, and thus posit it as an object against itself” (p.  581). Glasersfeld (1991) pointed out that von Humboldt was describing a kind of abstraction: “Focused attention picks a chunk of experience, isolates it from what came before and from what follows, and treats it as a closed entity” (p. 2). Although other philosophers, such as Berkeley and Hume, critiqued the notion of abstraction as a source of concept development, it nevertheless became a lasting theme in philosophy (Laurence & Margolis, 2012). Despite its importance, however, within the empiricist tradition, philosophers did not typically articulate precisely how the process of abstraction is supposed to work. That changed with Piaget’s body of work. As Glasersfeld (1991) remarked, “Few, if any, thinkers in this century have used the notion of abstraction as often and insistently as did Piaget” (p. 8). Early in his career, Piaget distinguished empirical abstraction, which concerns properties of objects, from another type of abstraction, which he termed reflective. Over the course of his writings, Piaget continued to expand and refine his ideas about abstraction, ultimately identifying and characterizing three types of reflective abstraction (pseudo-empirical, reflecting, and reflected). Below, we explore these different types of abstraction and examine how Piaget and other scholars made sense of the distinctions between them. In doing so, we will also consider how researchers and educators can benefit from considering the role of abstraction in fostering student learning.

 onsidering Abstraction When Making Sense C of Student Reasoning Why might it be useful to distinguish different types of abstraction in Piaget’s genetic epistemology? In order to illustrate this utility, consider an example of two students’ reasoning on an interview task called the Growing Rectangle Problem (Singleton & Ellis, 2020). For this task, students were asked to compare a rectangle’s growth in length and corresponding growth in area (Fig. 6.1). Both students, Angelo and Willow, produced the same general statement relating the rectangle’s length (L) and its area (A), writing the equation A = 1.5 L. However, the type of abstraction each student engaged in to develop the general statement

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Fig. 6.1  The growing rectangle problem

differed. Furthermore, we will see that these different forms of abstraction ultimately afforded different opportunities for generalizing and justifying their conclusions.

Angelo Angelo, an eighth grade student, wrote “6” for the length and “8” for the area when identifying values to go in the blanks. In justifying this pair, he explained, “Well, from what I saw, it looks like this box [pointing to the growth part in dotted lines] is a little bigger than this one [pointing to the original rectangle with solid lines], so I guessed that it was 2 centimeters bigger.” In doing so, Angelo compared the relative sizes of the original rectangle’s length and area compared to the rectangle’s growth in the dotted portion of the figure. In an attempt to help Angelo produce a correct length-area pair, the interviewer asked him to determine the rectangle’s area if the length grew by 4 cm: Int: So, let’s say that this grew out by 4 centimeters. So, it didn’t quite go a ways as far as where these dotted lines are. The length just grows by 4 centimeters. How much do you think the area would grow? Angelo: (Long pause). I think that it would grow by 12, because if four is doubling, and if that’s 4 centimeters (pointing to the original rectangle’s length) and that’s 4 centimeters (gesturing along the growing length), then that (the length) would equal 8 centimeters. And then another 6 centimeters would equal 12 centimeters.

Angelo realized that if the length were doubled, the area would also be doubled. Under the interviewer’s direction, Angelo was then able to double again to produce a length-area pair of 8:12, halve to produce 2:3, and halve again to produce 1:1.5. The interviewer then asked Angelo to determine the rectangle’s added area for two more added length values, ¾ cm and 6 cm. Angelo reasoned that for the original length-area pair, 4:6, 4 × 1.5 = 6, and therefore he could multiply other length values by 1.5 to find their corresponding areas.

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Fig. 6.2  Angelo’s list of length-area pairs for the growing rectangle

The interviewer then asked Angelo to make a generalization. If the rectangle were to grow by any length of x cm, by how much would the area grow? Angelo looked across the results of his prior calculations (Fig. 6.2), tracing his finger across the pairs, and wrote “1.5(x).” He explained: Angelo: Well, I looked at my other answers, and then I looked at one. And 1 multiplied by 1.5 is still 1.5 (gestures at the middle row of his table, the pair 1 and 1.5). And with this one, 1.5 would equal that (gestured at the row of 6 and 9). And then I was thinking of the other ones as well and then that's what I got for those. So I’m guessing they’re all multipliable by 1.5.

When asked whether all of the pairs would necessarily follow the pattern he identified, Angelo was unsure. He began to check each pair, beginning with the 8:12 pair, then the 4:6 pair, and then the other pairs. The only way Angelo had to explain his general statement was via an empirical argument that multiplying the length by 1.5 worked for all of the length-area pairs he had tried so far. Angelo was unable to provide a justification that was deductive, and the value of 1.5 did not hold a quantitative meaning for him in terms of the rectangle situation.

Willow Willow, a sixth grade student, initially made an additive comparison. She compared the rectangle’s area of 6 cm2 to its length of 4 cm and concluded that the area “would always be 2 more” than the length. She then, however, made a sudden shift in how she was thinking about the rectangle growing, by imagining it as dynamic and moving, rather than static: Willow: Unless it will start at zero. Because if you start it at zero, if you start it from zero to find out the actual growth, then say this is like the first they grew and this, kind of, so, this grew by four first (gestures along the length) and then this grew by six (gestures to the

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whole figure, indicating area). So, this could grow by four again (gestures along the length to continue the figure past the original drawing), and this could grow by six again (gestures to the whole figure to indicate the area similarly ‘coming along’ with the length). Even though they start here they could still be like, this (the area) grew by six, this (the length) grew by four.

When she said the figure “starts at zero,” Willow appeared to imagine the rectangle to be moving from left to right. Like Angelo, she then engaged in the mental action of doubling. Willow imagined the rectangle doubling in length and then understood that area would also have to double. This understanding appeared to be connected to a mental image of the rectangle dynamically growing. Willow then said, “It would always be plus four and plus six, so if you said when the length grows by eight, then the area grows by 12.” Building on her original mental action of doubling, Willow reasoned that when the length grew by 16 cm, the area would grow by 24 cm2, because “16 is four times four, so it grew four times the original length kind of, then the area grows four times.” Using similar reasoning, Willow determined that the area would grow by 3 cm2 when the length grew by 2 cm because “two is half of four, three is half of six.” Willow also decided that if the length grew by 1 cm, the area would grow by 1.5 cm2 because “one goes into four four times” and “1.5 goes into six four times.” At this point, Willow had not identified a direct multiplicative relationship between length and area, even though she had written down the length-area pair 1:1.5. Specifically, she had not yet explicitly determined that the area was 1.5 times the length. Instead, Willow continued to reason with the original 4 cm:6 cm2 pair, finding equivalent ratios by relating each new length with the original 4 cm length to determine the multiplicative factor and then using that factor to find the new area value. In this manner, Willow identified the pair 3 cm:4.5 cm2 because three is ¾ of 4 and 4.5 is ¾ of 6. Similarly, she identified the pair 5 cm:7.5 cm2 because five is 1¼ of 4 and 1¼ of 6 would be five 1.5 s, which she added up to get 7.5 (Fig. 6.3): At this point, Willow stopped and said, “Oh, I think I see something.” She explained, “One point five is ¼ of six. So, one would be ¼ (of four), so it (the added area for an added length of 1 cm) would be 1.5. And then this (the length) grew by one.” Willow concluded, “It’s like 1.5 bigger each time.” The interviewer asked Willow to explain further, and she said, “Each time the growth in length goes up by

Fig. 6.3  Willow’s method of adding five 1.5 s to calculate 1¼ of 6

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one, the growth in area, I think the growth in area equals -” [writes “A = 1.5 x L”]. Willow was then able to think of other growth increments in relation to the unit ratio, imagining other positive increases in length as a scaled version of some factor times 1 cm, and correspondingly the area must be that same factor times 1.5 cm2.

The Basis for Angelo and Willow’s Abstractions Angelo’s first pair was based on his perceptual judgment of the relative sizes of the original and extended rectangles. However, he was then able to create new length-­ area pairs through the action of doubling the original 4:6 length-area pair. This then supported Angelo in repeating his doubling action, then halving, and ultimately noticing that because 6 = 4 × 1.5, he could multiply other length values by 1.5 to find their corresponding area values. It was after noticing a pattern across all of his correct length-area pairs that Angelo was able to produce the expression 1.5(x); this expression was a representation of the pattern of outcomes across multiple pairs in his written record. Because it represented only the pattern of outcomes, Angelo did not have confidence in the accuracy or generalizability of his expression. He felt compelled to check other length-area pairs and did not see his expression as necessarily useful for determining new pairs beyond what was already in his table. For Willow, the expression “A = 1.5 x L” represented something different than a pattern of outcomes. Even when she had produced the length-area pair 1:1.5, Willow did not yet see it as a unit relationship between length and area. It was only after determining 1¼ of 6 as five groups of 1.5 that she suddenly realized that the 1.5 could represent the rectangle’s added area corresponding to each additional unit increment of length: “It’s like 1.5 bigger each time.” By reflecting on her activity of coordinating the number of groups of area (1.5 cm2) with the corresponding number of groups of length (five groups of 1 cm), Willow was able to then re-conceive the ratio 1:1.5 as an amount of area coupled with a 1-cm increase in length. Her expression, A = 1.5 x L, was therefore a representation of Willow’s image of the rectangle’s growth in area for a unit growth in length. Once Willow had this expression, she knew she could use it to determine a corresponding area for a given length. Angelo and Willow each engaged in a form of abstraction to produce the generalization A  =  1.5  L. However, the type of abstraction they engaged in differed. Angelo abstracted based on reviewing the outcome of his activity and identifying a pattern in those outcomes, a form of result-pattern generalization (Harel, 2001). Willow, in contrast, abstracted based on her coordinating groups of area with corresponding groups of length, a form of process-pattern generalization. Understanding the different types of abstraction in Piaget’s genetic epistemology can help us make a better sense of students’ reasoning and ultimately design meaningful instruction that can support students’ thinking. In the sections below, we introduce and describe the different types of abstraction Piaget discussed and then introduce extended data episodes to characterize examples of these abstraction types.

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Empirical Abstraction Empirical abstraction concerns the abstraction of properties of objects, with sensorimotor experience providing the raw material for such abstractions. Piaget described this process as “extracting the inherent properties of the object” (1948/1967, p. 25) and “deriving the common characteristics from a class of objects” (1961/1966, p. 189). In his early work, Piaget referred to abstractions from objects as “simple abstractions” (e.g., Piaget, 1961/1966). However, he later began to use the phrase empirical abstraction, in part due to the fact that engaging in such abstractions is not, in fact, simple (Piaget, 1980). From the actor’s perspective, knowledge derived via an empirical abstraction may seem to originate from the external objects themselves. However, several researchers (e.g., Dubinsky, 2002; Montangero & Maurice-­ Naville, 2013; Steffe, 1991; Thompson, 1985), as well as Piaget himself (2001), have been careful to note that such knowledge derives from internal constructions made by the actor regarding a property of an object: Let us note right away that this type of abstraction, even in its most elementary forms, cannot be a pure “read-off” of data from the environment. To abstract any property whatsoever from an object, such as its weight or its color, the knowing subject must already be using instruments of assimilation (meanings and acts of putting into relation) that depend on sensorimotor or conceptual schemes. And such schemes are constructed in advance by the subject, not furnished by the objects (pp. 29–30).

We elaborate on the implications of this interplay at the conclusion of the chapter. Some of the observables central to empirical abstraction may come from objects themselves (e.g., color, texture), such as in the example Kant (1800/1992) described regarding the concept of a tree. Other observables can be derived from the material of the actor’s actions (e.g., pushing, lifting). As such, the empirical abstraction then “provides a conceptualization which, in a way, is descriptive of the observable features of the action’s material characteristics” (Piaget, 1976, p. 351). Describing an example of such an empirical abstraction, Piaget (1970) remarked: A child, for instance, can lift objects in her hands and realize that they have different weights. Usually, big things weigh more than little ones, but that is not always true. She finds this experientially, and her knowledge is abstracted from objects themselves. This is simple (empirical) abstraction (pp. 16–17).

In both of the examples of weight and tree, the empirical abstraction entails extracting some property from a set of objects and then classifying the objects or features of the objects, on that basis. Thompson (1985) described this process as a “separating of the object or object’s composition into similarities and differences” (p. 196), and Simon et al. (2004) described it as a “generalization of properties of objects” (pp. 312–313). We draw from the second author’s everyday life to provide an example of an empirical abstraction. Before the age of 2, his child learned that coins could be used to unlock bedroom doors in his house. The locks had approximately 1-cm-long slots that could be turned with any rigid object a few millimeters in width. The child began to test other objects (e.g., thick cardboard books, toy tongs) to explore which

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objects could (and could not) turn these locks. Through this process, the child created a concept of “objects that can open locks” via an empirical abstraction that relied on the perceptual size and rigidity of objects within the set. In doing so, the child extracted “a character x, from a set of objects and then classif[ied] them together on this basis alone, a procedure we shall refer to as simple abstraction and generalization” (Piaget, 1961/1966, p. 317). Whereas empirical abstractions, such as this one, concern properties of objects, we next turn to reflective abstractions, which concern the learner’s mental operations.

Reflective Abstraction Piaget (1976) distinguished reflective abstraction as a type of abstraction that is qualitatively different from empirical abstraction: While it is clear that any abstractions from objects is thus ‘empirical’, the action’s pole gives rise to both types: empirical with regard to observable features of the action, that is, concerning a material process (a movement, a position of the hand, and so forth), and ‘reflexive’ with regard to inferences drawn from the coordinations themselves. (pp. 345–346)

Note that in the above quote, Piaget used the term “reflexive,” which is typically translated as “reflective.” Reflective is the term we will use in this chapter. As Montangero and Maurice-Naville (2013) described, reflective abstraction is not a property of “reality” such as weight, color, or temperature, but rather is a property of the learner’s activities. In fact, in defining reflective abstraction, Piaget emphasized its origin as being the learner’s actions or coordinations: “A reflexive abstraction is one that derives its information from the subject’s actions or, more specifically, from their coordinations” (1976, p. 270). Reflective abstraction, Glasersfeld (1982b) noted, occurs when the learner abstracts mental operations from the sensorimotor context that may have given rise to them. Succinctly characterizing this difference, Piaget (1977) explained, “Empirical abstractions concern observables and reflective abstractions concern coordinations” (p. 319). Reflective abstraction is, therefore, an endogenous process (driven by internal factors), in contrast to empirical abstraction, which is exogenous in origin (driven by external factors) (Dubinsky, 2002; Montangero & Maurice-Naville, 2013). As we mentioned above, even empirical abstraction draws on the products of reflective abstraction. Montangero and Maurice-Naville (2013) noted, “Piaget constantly stressed the greater importance of reflective abstraction. The latter is responsible for the creation of forms of knowledge (categories or classes, comparisons) that make empirical abstraction possible” (p. 61). Indeed, reflective abstraction is the process Piaget proposed for the development of new knowledge; it is how more advanced concepts can be built out of existing concepts (Glasersfeld, 1982a; Simon et al., 2004). Piaget wrote that reflective abstraction “permits the derivation from those coordinations of new assimilatory frameworks and new structures by combining reflection onto new cognitive levels with reorganizing réflexion” (1980, p. 111).

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Further, he stressed that empirical abstraction is limited to sensorimotor data, whereas reflective abstraction is structuring at all levels. Reflective abstraction is, therefore, a critically important process to understand and elaborate, and yet, simultaneously, it has been difficult for the field of mathematics education to define and characterize this process. The introduction of reflective abstraction is, to our knowledge, unique to Piaget’s writings. It appeared beginning in the 1940s, but there were hints of these ideas even prior to that. In his 1928 work, Judgment and Reasoning in the Child, for instance, Piaget discussed the idea that knowledge is enriched when transferred to the level of thought; he would not formally define this process, however, until much later (Montangero & Maurice-Naville, 2013). In The Child’s Conception of Space (1948), Piaget distinguished a type of abstraction that is drawn from the coordination of the learner’s actions rather than from an object, and he again made this distinction in his 1950 Introduction to Genetic Epistemology. Later, Piaget began to provide additional specifications to the concept of reflective abstraction. In particular, beyond simply differentiating reflective from empirical abstraction, Piaget (1977) introduced the distinction between subtypes of reflective abstractions – pseudo-­empirical, reflecting, and reflected – which we discuss in more detail below. Before describing these subtypes, however, we first introduce an important separation of reflective abstraction into two phases, which are useful for making sense of the different subtypes.

Two Phases of Reflective Abstraction Piaget used two terms when discussing reflective abstraction, réfléchissement and réflexion. Réfléchissement was Piaget’s word for taking an activity or mental operation developed on one level, abstracting from that level of operating, and raising it to a higher level (Glasersfeld, 1995). This term has been translated not only as “reflection” but also as “projection” (e.g., Moessinger & Poulin-Dubois, 1981). Piaget (1970) noted that the term réfléchissement has at least two meanings: In its physical sense a reflection refers to such a phenomenon as the reflection of a beam of light off one surface onto another surface. In a first psychological sense abstraction is the transposition from one hierarchical level to another level (for instance, from the level of action to the level of operation). (pp. 17–18)

Glasersfeld (1982b) depicted the process of projection in this manner: “It lifts the construction pattern out of a sensorimotor configuration, leaving behind the actual sensory material - in other words, holding on to the connecting, relating, integrating operations, and disregarding the stuff that was connected by them” (p.  18). Montangero and Maurice-Naville (2013) referred to this projection phase as “the abstraction proper” (p.  60), the process of extracting forms of organization from one’s knowledge and reflecting what was abstracted onto a higher level. Others have also referred to lifting one’s reasoning onto a higher level or a higher plane of

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thought (e.g., Dubinsky, 2002). What Piaget meant by a higher level is a matter for some debate and elucidation, which we discuss in more detail in the sections below. The second phase Piaget (1977) discussed is réflexion, in which the abstracted structure is reorganized at the higher level: “‘Reflection’ involving a mental act of reconstruction and of reorganization on this higher level of that which has been thus transferred from the lower one” (p. 303). Both nouns, réfléchissement and réflexion, were formed from the verb réfléchir, whose present participle is réfléchissante. Piaget used abstraction réfléchissante as a generic term for both phases, which is likely why the distinction between réfléchissement and réflexion was lost when the more generic term “reflective abstraction” took hold (Glasersfeld, 1995). Campbell (2001) noted that Piaget’s emphasis of the optical meaning of reflection with réfléchissement and the cognitive meaning with the term réflexion works better in French than in English; consequently, many call the first phase projection and the second phase reflection, reconstruction, or reorganization. We will use the terms projection and reorganization throughout this chapter. Both phases, projection and reorganization, must occur for us to consider an abstraction to be a reflective abstraction. On the higher plane of thought, more powerful mental actions are present, which enables a process of constructing new combinations by a conjunction of abstractions (Dubinsky, 2002). Piaget considered this construction process to be key to mathematical development, as did others (e.g., Dubinsky, 1991; Gallagher & Reid, 1981; Steffe, 1991). Reorganization essentially enables the elaboration of new mental actions: “It leads to a generalization that is a novel composition, preoperatory or operatory because it involves a new scheme that has been elaborated by means of elements borrowed from prior schemes by differentiation” (Campbell, 2001, p. 11). For some, this new scheme is simply one that is more flexible or is composed of higher-level operations (e.g., Campbell, 2001; Thompson, 1985); for others, this scheme must entail new actions that are coordinated as operations within a reversible and composable system (e.g., Norton, 2018) (see Chap. 3 for an elaboration of Piaget’s scheme theory). Regardless of what one requires of the novel scheme, the reorganization phase is cyclical, with the resulting schemes becoming “purer and purer thanks to its internal mechanism of reflection on its reflections” (Piaget, 1977, p.  319). This further distinguishes reflective abstraction from empirical abstraction, as reflective abstraction allows engagement in thought experiments that do not rely on the presence of sensorimotor material. Through this two-phase process, reflective abstraction begins with old cognitive structures and generates new ones (Campbell, 2001). Unlike empirical abstraction, reflective abstraction elevates all of the learner’s cognitive activities – coordinations of actions, operations, schemes, and cognitive structures – and separates particular characteristics of those activities and uses them for new constructions (Piaget, 2001). For this reason, reflective abstraction is an essential focus in mathematics education (Simon et  al., 2004). Many have turned to reflective abstraction as “a guiding heuristic in a search for insight into mathematical learning” (Steffe, 1991, p. 43) and, indeed, some even consider it to be the goal of instruction (Simon, 2016). We have established that forms of abstraction that are not empirical fall under the umbrella of reflective abstraction, but within this umbrella, there are three subtypes.

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Glasersfeld (1995) characterized the subtypes in this manner: “The first of the three reflective abstractions projects and reorganizes, on another conceptual level, a coordination or pattern of the subject’s own activities or operations” (p. 105). This first subtype is called “reflecting abstraction.” Glasersfeld went on to explain that: The next is similar in that it also involves patterns of activities or operations, but it includes the subject’s awareness of what has been abstracted and is therefore called ‘reflected abstraction’. The last is called ‘pseudo-empirical’ because, like empirical abstractions, it can take place only if suitable sensorimotor material is available. (p. 105)

Thus, pseudo-empirical abstraction, reflecting abstraction, and reflected abstraction are all forms of reflective abstraction, and we discuss each in turn below.

Pseudo-empirical Abstraction Although it was not until the last period of Piaget’s work that he specified and enriched the different types of reflective abstraction, as early as 1948, he began to make implicit distinctions between the types of abstraction. For instance, Piaget (1948/1967) noted that “the ‘abstraction of shapes’ is not carried out solely on the basis of objects perceived as such, but is based to a far greater extent on the actions which enable objects to be built up in terms of their spatial structure” (p. 68). In this case, geometrical shape is not a property of an object like its weight or color, but rather a phenomenon that results from the learner’s physical or mental actions in perceiving an object coordinated from their inception of the shape grasped as a single whole. Piaget’s remark pointed to a key difference between empirical and pseudo-empirical abstractions, even though he did not develop the latter term until much later: pseudo-empirical abstraction draws coordination not from the objects alone, but rather from the learner’s physical or mental actions exerted on objects (Montangero & Maurice-Naville, 2013). The objects, such as the shapes Piaget referred to, are indispensable, but the coordination is from the learner’s actions, such as the imposition of spatial structuring. In particular, Piaget clarified this distinction in his 1977 book, Recherches sur L’Abstraction Réfléchissante: When the object has been modified by the subject’s actions and enriched by the properties drawn from their coordinations (e.g., when arranging the elements of a collection in a sequence) the abstraction bearing upon these properties is called ‘pseudo-empirical’ because, while it concerns the object and its actual observable traits as in empirical abstraction, the facts it reveals concern, in reality, the products of the coordination of the subject’s actions: this is, then, a particular case of reflective abstraction and not at all a derivative of empirical abstraction. (p. 303)

Thus, according to Piaget, abstraction that requires perceptual material but draws on one’s actions imposed on that material is pseudo-empirical (Glasersfeld, 1995). Piaget (1980) clarified that pseudo-empirical abstraction entails “a reading of the objects involved, but reading which is really concerned with properties due to the

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action of the subject himself” (p. 92, emphasis ours). He referred to this as an initial form of reflective abstraction and emphasized that it plays a fundamental psychogenetic role in all logico-mathematical learning, in that humans need to manipulate physical objects in order to build up structures that are otherwise too abstract. As an example, Montangero and Maurice-Naville (2013) shared a case of a child who is just learning to count. The child took ten stones, counted them, and placed them in a row. He then decided to check whether there would still be ten stones regardless of the order in which he counted or arranged them. In rearranging the stones in different orders and configurations, he found that the total was always 10, regardless of whether he placed them in a circle or a row or whether he counted from left to right or from right to left. The child’s abstraction of the conservation of number was drawn not from a property of the stones, per se, but instead from the organization the child introduced. But, in what is critical to making this an example of pseudo-empirical abstraction, the stones themselves were a necessary support to the child’s activity. Number and order are abstractions not based solely on perception, and thus they are pseudo-empirical rather than empirical (Campbell, 2001). In another example, Thompson (1985) distinguished the abstraction of weight from the abstraction of conservation of weight. As described above, through empirical abstraction, one can determine that different objects have different weights by holding them and can conclude that larger objects typically weigh more than smaller objects. Weight, being a property of an object, is an empirical abstraction. In contrast, it is through pseudo-empirical abstraction that one determines that the weight of an object must remain the same under transformations of elongation or deformation, as long as nothing is added or taken away. One’s abstraction of this conservation depends on their actions on objects rather than on properties of the objects themselves; it is pseudo-empirical, however, in that it still requires the objects in order to make the abstraction. Number and conservation of weight are fairly straightforward examples addressing young children’s early abstractions, but is pseudo-empirical abstraction a useful construct for more advanced mathematical learning? Montangero and Maurice-­ Naville (2013) seemed skeptical of this notion, pointing out that pseudo-empirical abstractions are numerous early in life but then, as people age, gradually decrease as reflecting and reflected abstractions increase. Some researchers do not bother to distinguish pseudo-empirical abstraction as a subtype of reflective abstraction (e.g., Simon et al., 2004). Others, however, emphasize the subtlety in determining what constitutes perceptual material or “an observable.” For instance, Campbell (2001) pointed out that even if one accepts the idea that number is a pseudo-empirical abstraction, it could then itself become an observable: “Even if number did have to be inferentially constructed by the child at level N, why should this prevent the child at level N + 1 from gathering empirical data about different numbers of objects under different circumstances, and engaging accordingly in empirical abstraction?” (pp. 9–10). Thus, Campbell argued that what is a coordination at one level can then function as an observable at the next higher level. Hence, what constitutes a pseudo-­ empirical abstraction depends on what we take to be an observable versus a construction at the current level.

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Campbell’s point introduces some flexibility in what we mean by “observables that are external, on the one hand, but that are constructed by reflecting abstraction, on the other” (Piaget, 2001, p. 31). Some researchers, such as Moore (2014) and Simon (2006), even go so far as to admit a wholly internal phenomenon as an observable, as long as it is the product of one’s mental actions, rather than the coordination of those actions. From this perspective, an observable could be, for instance, the outcome of one’s listing activity when solving a combinatorics problem or a representation of a recursive pattern that is now taken as source material for a new abstraction. We follow Moore’s (2014) stance by broadening the category of what can be admitted as a pseudo-empirical abstraction. Namely, we consider it useful to broaden what constitutes an “observable” to include the product of mental activity as source material for a new abstraction. This extension is useful for making sense of students’ learning in advanced mathematics but also for learning in the earlier grades. From this perspective, we can consider Angelo’s abstraction, which he expressed as 1.5(x), to be a pseudo-empirical abstraction for two reasons. Firstly, Angelo relied on the observable that was the two columns of numbers representing the ordered pairs he had generated (Fig. 6.2). It was in looking across these pairs of numbers and noticing a consistent pattern that he was able to extract the relationship that each area value was 1.5 times its corresponding length value. Secondly, and this speaks to our deliberate extension of what we consider to be an observable, we would admit Angelo’s abstraction as pseudo-empirical because the source material he relied on was the outcome of his action of creating equivalent ratios, as instantiated by the observed pattern in the set of ordered pairs. We would still consider this to be a pseudo-empirical abstraction even if Angelo had not relied on the available perceptual material of the two columns of numbers, because the source material for his abstraction was not a coordination of his activity, but rather the outcome of his activity. In the extended examples below, we continue to rely on this broader stance of what constitutes pseudo-empirical abstraction to examine students’ sensemaking with graphs and with combinatorial tables.

Reflecting Abstraction Reflecting abstraction is a form of reflective abstraction that, like pseudo-empirical abstraction, is drawn from the coordinations of actions or operations. The key distinction between pseudo-empirical abstraction and reflecting abstraction, according to Piaget, is that reflecting abstraction does not require perceptual material. Like pseudo-empirical abstraction, the source of reflecting abstraction is endogenous, but it does not require the presence of an object or other sensorimotor materials: “We are now dealing with the subject himself and no longer with external objects” (Piaget, 1980, p. 90). As Campbell (2001) noted:

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The abstraction of a mental characteristic that qualifies some action scheme and is destined to bring this characteristic into a more complex scheme (not just into a simple descriptive concept of internal experience) is reflecting abstraction. Calling it reflecting indicates that abstraction transforms the very conduct by differentiating it and consequently adds something to the quality that has been isolated by abstraction. (pp. 10–11)

Such abstraction from mental activity is necessarily constructive. A prior cognitive structure is what is projected to the higher developmental level, which is then reorganized into a new structure (Campbell, 2001; Tallman, 2021). Given our broader interpretation of what can be counted as a pseudo-empirical abstraction, namely, including as an observable the product of mental activity, this necessarily restricts somewhat the activity that we include as reflecting abstraction. For instance, Angelo’s abstraction, expressed as 1.5(x), relied on the product of his mental activity in creating equivalent ratios, rather than on a coordination of actions. In contrast, we characterize Willow’s abstraction as a reflecting abstraction. Recall that Willow realized that the rectangle grew 1.5 cm2 in area for each 1-cm increase in length, which she expressed as A = 1.5 x L. The source material for Willow’s abstraction was not a pattern of observable outcomes across her set of equivalent ratios, as it was for Angelo; rather, it was her activity of coordinating the number of groups of area with the corresponding number of groups of length. Willow did not require perceptual material to make her abstraction, and she drew, instead, on a coordination of actions, rather than on an outcome of actions. As such, Willow progressed through what Tallman (2021) described as three phases: (1) the differentiation of a sequence of actions from their outcome, (2) the projection of the actions from the level of activity to the level of representation (the reflected level), and (3) the reorganization of the projected actions. As Tallman (2021) described: A subject must differentiate (dissociate) actions from their effects before they can construct an internalized representation of them, what Piaget called projecting actions to the level of representation (i.e., the level of cognition). Additionally, the subject must coordinate the actions that produced the effect before they can project and represent them on a higher cognitive level. Once a subject differentiates actions from their effect and then coordinates them, they are prepared to project these coordinated actions to the reflected level, where they are organized into cognitive structures. (p. 3)

We posit that Willow, in making a group of five 1.5 s, reflected on the coordination between the number of 1.5 s and the number of 1 s: Each group was 5 because 1 was ¼ of 4 and 1.5 was ¼ of 6, and Willow needed to determine the amount of area corresponding to 5 cm in length. Willow engaged in the sequence of actions of combining five groups of 1.5 cm2 to get 7.5 cm2 for the area. In doing so, she differentiated the sequence of actions, which was coordinating the number of 1.5 s with the number of 1 s, from the outcome, which was 7.5. She then projected the actions from the level of activity, adding up five 1.5 s, to the level of internalized representation. This enabled Willow to then reorganize her projected actions. Specifically, she realized that she could now directly couple growth in area with growth in length for a 1-cm length increment: “Each time the growth in length goes up by one, the growth in area, I think the growth in area equals [writes ‘A = 1.5 x L’].”

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Reflected Abstraction Reflected abstraction is the third type of reflective abstraction, and it differs from other types of abstraction in that it entails conscious awareness: “We call ‘reflected’ abstraction the result of a reflective abstraction when it has become conscious” (Piaget, 1977, p. 303). Reflected abstractions are conscious products of other reflective abstractions, resulting in a reflection of the thought on itself (Piaget, 1976). In contrasting reflected abstraction from pseudo-empirical and reflecting abstraction, Piaget (1976) explained: By contrast, the reflexive abstraction can become conscious, particularly when the subject compares two steps that he has carried out and tries to discern common factors…In this second case, we shall refer to ‘reflected abstraction’, the past participle denoting the result of the ‘reflexive’ process. (p. 346)

Reflected abstraction thus entails the learner consciously examining their prior experiences in order to reorganize them along with their current activity (Glasersfeld, 1987). It can enable a formulation or even a formalization of the elements that have been abstracted (Montangero & Maurice-Naville, 2013). Furthermore, reflected abstraction “can completely free itself from any relationship with material objects” (Piaget, 1980, p. 92) and indeed can drive much of the activity that gives rise to higher mathematics. When engaging in pseudo-empirical or reflecting abstraction, one uses some structure in their operational compositions, and that use is implicit. As Glasersfeld (1995) explained, these types of reflective abstraction may or may not involve the learner’s awareness, and in fact, throughout history, mathematicians have used thought structures without conscious awareness: “A classic example: Aristotle used the logic of relations, yet ignored it entirely in the construction of his own logic” (Piaget & Garcia, 1983, p. 37, as cited in Glasersfeld, 1995, p. 106). A learner can be aware of what they are cognitively operating on, without also being aware of the operations being carried out. When engaging in reflected abstraction, however, the learner gains conscious awareness of their operations. This is not to say that reflected abstraction only occurs in the context of, say, formal mathematics. It can be observed even at the level of a child’s verbal expression of an action, such as “I press on the button, and the bell rings” (Montangero & Maurice-Naville, 2013, p.  59). One must, however, consciously operate on one’s actions at the level of representation. At the stage of reflected abstraction, the structure of operational compositions is “teased out” (Piaget, 1980, p. 99) and can support the establishment of a formal theory. Operations that were initially instruments of calculation become, at the stage of reflected abstraction, differentiated objects of thought in their own right, a process Piaget referred to as thematization. Tallman (2021) stated that this suggests that the learner has symbolized coordinated actions at a higher cognitive level: Reflected abstraction thus relies on what Piaget called the semiotic function, or the subject’s capacity to construct mental symbols to represent aspects of their experience. Reflected abstraction entails symbolizing coordinated actions at the level of representation so as to reify the material actions the symbol represents into a form that can be used as an object of

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thought at the level of representation. On this higher cognitive level, the subject can consciously manipulate these symbols independently of re-presenting the coordinated actions they signify, all the while being capable of doing so. The semiotic function is thus the essential mechanism by which reflecting abstraction becomes reflected abstraction. (p. 4)

Tallman explained that engaging in conceptual operations on the symbols that represent coordinated actions at the level of representation ultimately supports “increasingly organized and differentiated cognitive structures” (p. 4), i.e., learning. For an elaborated account of reflected abstraction, see Tallman and O’Bryan (this issue). Other researchers may not have the same bar for symbolizing coordinated actions (e.g., Montangero & Maurice-Naville, 2013). Piaget himself referred to explicit comparison of one’s actions or operations across cases as evidence for reflected abstraction (e.g., 1976, 2001); we will similarly consider those cases in the following data episodes.

Data Episodes In order to exemplify the different types and stages of abstraction as well as discuss the standards of evidence we hold for identifying instances of abstraction, we present extended data episodes from each of two tasks. The first is a covariation task called the Faucet Task, and the second is a series of combinatorial tasks called the Passwords Activity. For each task, we present two sets of data, and in doing so, we identify specific instances of abstraction and discuss evidence for categorizing them according to type.

The Faucet Task We draw from data from two different teaching experiments in which students addressed the Faucet Task (Paoletti, 2019; see Fig.  6.4a for a screenshot of the Faucet Task activity and https://bit.ly/36jy0Dn for the task itself). In the Faucet Task, students are initially asked to coordinate how turning different faucet knobs influences two quantities: the amount of water leaving the faucet and the water’s temperature. Eventually, students are tasked with coordinating how the relationship between these two quantities could be represented graphically. We use the first example, with a 4-year-old student, Mario, to exemplify empirical and reflective abstraction. We then use a second example with two middle-school students, Kendis and Camila, to articulate more detailed differences between pseudo-empirical, reflecting, and reflected abstractions.

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Fig. 6.4 (a) A screenshot of the Faucet Task and (b) Mario’s hand feeling the water below an actual faucet

Mario Engages in Empirical and Reflective Abstractions Prior to engaging with the Faucet Task, the teacher-researcher (TR) first provided Mario with opportunities to consider several scenarios with a real faucet (see Fig. 6.4b). Specifically, the TR prompted Mario to predict how the temperature and amount of water would change for four scenarios, each starting with each knob halfway on. When he asked Mario to predict how the water temperature and amount of water would change when he turned the cold knob on, Mario predicted that there would be “less water and colder.” Upon turning the cold knob on and putting his hand under the water, Mario noted that although the water was colder, there was more water leaving the faucet, instead of less water as he predicted. A similar interaction occurred when the TR asked Mario how the two quantities would change when turning the cold knob off. Mario predicted there would be “more water and hotter.” However, when putting his hand under the water after turning the cold knob off, Mario quickly observed there would be “less [water]…but it’s definitely hotter.” In this initial interaction, Mario was developing a conception of how quantities change together in the faucet scenario via an empirical abstraction; the water coming from the faucet provided sensorimotor data that Mario used to determine whether there was more or less water leaving the faucet, as well as whether the temperature increased or decreased. By physically engaging with the knobs, and by observing and feeling these changes, Mario was empirically abstracting the relationships between turning knobs, the changing amount of water, and the water’s changing temperature. Although Mario’s initial conception of the relationships between quantities in the faucet scenario was grounded in an empirical abstraction, in his next activity, there is evidence that he began to project his meanings to a higher conceptual level. After addressing all four scenarios with the actual faucet (Fig. 6.4b), the TR presented the faucet applet to Mario, which allowed him to turn both knobs on and off for a digital faucet (shown in Fig. 6.4a). In the applet, the changing width of the rectangle below the spigot is intended to represent the amount of water leaving the faucet, and the changing water color is intended to represent the varying

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temperature. After Mario familiarized himself with the applet, the TR asked him to predict how the water temperature and the amount of water would change in each of the four scenarios, with each knob starting halfway on. In each case, Mario was able to accurately predict, without testing, how the water temperature and the amount of water would change for each scenario. We take this as evidence that Mario projected the outcome of his activity with the actual faucet to the level of mental representation (rather than physical experience). Then, at this level, he engaged in a reorganization by coordinating his current activity with the results of his prior activity in such a way that he was able to anticipate how turning each knob would simultaneously affect the water temperature and the amount of water. Mario could therefore predict how turning any knob, in any of the scenarios, would change both the amount of water leaving the faucet and the relative amount of hot and cold water, thereby affecting temperature. Hence, we infer that he engaged in reflective abstraction as he differentiated his prior actions of turning knobs and observing water amounts and water temperature from the outcomes of those actions, projected that differentiation to the level of representation, and reorganized his conception of the faucet situation at this level. Of note, the extent to which Mario engaged in a pseudo-empirical abstraction versus a reflecting abstraction is an open question. As the Faucet applet was perceptually available, Mario may have been leveraging this representation to support his reasoning that leveraged his prior experiences with the physical faucet. If this perceptual material was necessary for Mario, we would contend that his meanings were grounded in a pseudo-empirical abstraction. However, if Mario could predict how each quantity would change without needing to rely on any available perceptual material, we would infer that his meanings were grounded in a reflecting abstraction.

 endis and Camila Engage in Pseudo-empirical, Reflecting, K and Reflected Abstractions We present a second example with the Faucet Task to illustrate nuances in different forms of reflective abstraction, drawing on two eighth grade students’ activity, Kendis and Camila (see Paoletti et al., 2021, for more detail). In this version of the task, one goal included supporting students in reasoning about graphs as representing relationships between two covarying quantities. As such, students interacted with a GeoGebra applet showing a faucet (Fig. 6.5, https://www.geogebra.org/m/ rdxkrwek), along with a coordinate system that included dynamic vertical and horizontal segments representing two quantities, water temperature and amount of water leaving the faucet. Additionally, there was a point corresponding to the endpoints of the two segments (Fig. 6.5). The task is designed to support students conceiving of a point as a multiplicative object (Saldanha & Thompson, 1998), simultaneously representing the magnitudes of the two quantities, and eventually conceiving of graphs as being produced via an emergent trace of this point (i.e., reasoning emergently; see Moore, 2021).

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Fig. 6.5  A screenshot from the GeoGebra faucet applets that included a graph representing water temperature and amount of water leaving the faucet on the vertical and horizontal axes, respectively

Prior to the interactions below, Kendis and Camila had worked collaboratively to describe how each segment on the axes would vary based on how the quantities varied (e.g., when the cold knob is turned off, the vertical red segment would get longer because the temperature increases, and the horizontal pink segment would get shorter because there is less water leaving the faucet). Further, Kendis had described that the point “stays in line with both of them,” with “both of them” referring to the endpoints of the segments. When addressing how the point would move when turning the hot knob on, Kendis was aware that the point’s final position would be up and to the right of its initial starting location. However, she initially imagined the point’s movement as dictated by the two segments changing sequentially rather than simultaneously. Specifically, when describing how the point would move in this case, Kendis argued: Kendis: It’s going this way (tracing to the right along the horizontal axis from the endpoint of the pink segment, (1) in Fig. 6.6) and, look, it’s going to stay in a line with it (pointing to the top of the red segment), so it’s just going to move over and up (traces sequentially to the right (2), then up (3) in Fig. 6.6).

In (1), moving from left to right, Kendis described how the segment representing the amount of water increased. She then attended to sequential changes in both quantities as she indicated that the horizontal, left-to-right, motion near the point in the plane was followed by a vertical upward motion created by an increase in temperature. Paoletti et  al. (2021) contended that such sequential reasoning is consistent with the developmental nature of covariational reasoning in which a student thinks of one quantity, then the next, then back to the first, and so on (Saldanha & Thompson, 1998). After making this prediction, Kendis observed the point’s diagonal up-and-to-theright motion in the applet when turning the hot water on. Such sensorimotor

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Fig. 6.6  Kendis’s hand motions describing how the point would move when the hot knob is turned on

experience provided Kendis an opportunity to experience a pseudo-empirical abstraction. Specifically, she differentiated the results of the point moving (i.e., its final position) from the sequence of actions that cause the point to move; she explicitly coordinated how the simultaneously changing quantities would be represented via simultaneously changing segment lengths. Kendis then argued that the point would move diagonally because of the covariation of the two quantities. For example, discussing the point’s motion as the hot knob is turned off, Kendis described the point’s movement as representing two simultaneously changing situational quantities: Kendis: So this (pointing to the top of the red segment along the vertical axis) is going to go down (drags finger down). It’s going to go down (repeats downward motion along axis), and then (points to the horizontal axis) it’s less water also. So it’s going to go diagonal (making a diagonal cutting motion with her hand).

Kendis then immediately engaged in a series of movements without talking. First, she motioned horizontally left from the point to indicate a decreasing amount of water (represented by (1) in Fig. 6.7), and then she motioned down from the endpoint of her first motion to indicate a decreasing water temperature (represented by (2) in Fig. 6.7). Critically—and differing from her earlier activity—Kendis lastly motioned diagonally down and to the left (represented by (3) in Fig. 6.7). Based on this activity, we infer that Kendis’s prior experience observing the motion of the point supported her in engaging in a pseudo-empirical abstraction. Specifically, Kendis projected the sensorimotor experience of the point’s diagonal motion to a higher level, which supported her in reorganizing her conception of the point’s horizontal and vertical movement from sequential to simultaneous. She now conceived the point as a multiplicative object that simultaneously represented the two covarying quantities’ magnitudes. At this point, due to Kendis’s motions focusing first on the horizontal and vertical motions on the graph on the computer screen prior to the diagonal motion, we infer that the perceptual material was important for

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Fig. 6.7  Kendis’s hand motions when describing how the point would move when the hot knob is turned off

Kendis’s conception of the point’s movement, which is why we conclude that she engaged in a pseudo-empirical abstraction rather than a reflecting abstraction. By introducing multiple scenarios (e.g., turning the hot water on, turning the hot water off, etc.), the TR intended to provide the students with opportunities to conceive of numerous graphs as being produced via the trace of a point constrained by covarying quantities’ magnitudes. He considered that such opportunities may support them in reflecting across their reasoning in these different cases, thereby potentially fostering the development of reflecting abstractions. These abstractions, in turn, could then support the goal of the students becoming explicitly aware of their conceiving graphs as the product of a trace of a point that represented the magnitudes of two covarying quantities. We would take such an explicit awareness as an indication of a student having engaged in a reflected abstraction. To explore the extent to which the students may have engaged in reflecting and reflected abstractions in the Faucet Task, the TR provided them with five completed graphs, with no applet available. He invited the students to describe situations that would produce each graph, with several graphs involving more than one turn, which was novel relative to the previous situations. We contend that a student describing a series of turns that would produce a given graph provides evidence they have stripped away the particular faucet situations that were central to their prior reasoning because they would now be able to both anticipate traces that would produce a given graph and describe a situation that would produce such a trace. Providing evidence of such reasoning, when describing the graph in Fig. 6.8a, Camila imagined the graph tracing from the top-left point (traversing the arc labeled “a” and then the arc labeled “b” in Fig. 6.8a). Camila then explained that the situation started with cold half on and hot all the way on. She explained that turning the cold knob the rest of the way on would produce the first arc, as “it’s going down in temperature and to the right, so it means you’re increasing water and it’s going down, so it means you have to be adding cold water.” She then described that turning the hot knob off would produce the arc labeled “b” in Fig. 6.8a. Hence, Camila provided evidence

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Fig. 6.8  Two graphs Camila interpreted

that her repeated prior experiences with constructing graphs that represented different faucet situations supported her in reorganizing her meanings for graphs at a higher level. In particular, as she stripped away the specific faucet situations, she projected her reasoning about graphs as traces to a higher level, and she then engaged in a reorganization at this level to anticipate that any graph could be produced via the trace of a dynamic point representing two covarying quantities. Further, Camila later provided evidence that she was consciously aware of her meaning that graphs are produced via emergent traces. In this instance, Camilla discussed the graph in Fig.  6.8b. This graph can be interpreted as representing a scenario in which both knobs start in the off position, then the cold knob is turned on (temperature is constant while the amount of water increases), and then the hot knob is turned on (temperature and amount of water increase). However, the graph could also be interpreted another way. Namely, it could represent a scenario in which both knobs start in the on position, then the hot knob is turned all the way off (temperature and amount of water decrease), and then the cold knob is turned all the way off (amount of water decreases, while temperature remains constant). Camila was able to conceive of the same final graph as being producible via two different action scenarios. When asked to describe a situation that could produce the graph, Camila provided one explanation, but then the TR spontaneously asked her if she could interpret the graph a second way: Camilla: (With both knobs turned off) First step is to turn the cold on (motions hand from left to right as if tracing the straight part of the graph in Fig. 6.6d) then turn the hot one on (motions an arc with hand from left to right). TR: So they’re both starting completely off, turning cold on then turning hot on… So in terms of the two quantities, how did you know that was (trails off). Camilla: Well, [the graph] continued to go to the right (motions with hand from left to right) so it means [amount of water]’s increasing in quantity (repeats motion) and then after the second transition [the graph]’s going up in temperature (motions as arc with hand from left to right) which means you’re going to be adding hot water, so the first one we started off as cold adding it (traces a horizontal line motion from left to right) and then

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we had to add more of hotter temperature (see Fig.  6.9 for a recreation of Camila’s imagined actions). TR: Cool, yeah, so this one going straight tells you what was happening with the temperature? Camilla: It means it’s not changing. TR: Not changing, right? So it’s got to just be cold water coming on, but then we get here and the temperature increases, the amount of water increases, which [inaudible]. Could there be another way this, this plays out? Camilla: Hot water off. TR: Hot water, so you start with both of them on, turn hot water off, get to here [crosstalk]. Camilla: [crosstalk] And then the cold is at halfway and then you could also turn it off (traces fingers in air straight across from right to left; see Fig. 6.10 for a recreation of Camila’s imagined actions).

In this interaction, Camila described two different action scenarios that resulted in the same outcome (i.e., the final graph). Such activity is consistent with Piaget’s (1976) description of reflected abstraction as comparing two processes one has carried out while discerning common factors across those processes. We take this to be evidence that Camila engaged in a reflected abstraction with regard to her meanings for graphs as being producible via traces. That is, Camila’s activity entails a(n at least implicit) comparison of two different sets of actions she carried out as resulting in the same common final graph (i.e., the actions depicted in Figs. 6.9 and 6.10). We take this implied comparison as evidence of her having engaged in a reflected

Fig. 6.9  The first scenario Camila imagined producing the graph in Fig. 6.8

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Fig. 6.10  The second scenario Camila imagined producing the graph in Fig. 6.8

abstraction such that she was explicitly aware of her understanding that a graph could be produced via the trace of a point representing covarying quantities.

The Passwords Activity We now present two examples of data from the domain of combinatorics, each of which relate to the same set of tasks we call the Passwords Activity. We first use an example from a single interview with an undergraduate vector calculus student, Tyler, to demonstrate pseudo-empirical and reflecting abstraction. Then, we use a second example with three undergraduate vector calculus students, Carson, Aaron, and Anne-Marie, to demonstrate reflecting and reflected abstraction. Broadly, the set of tasks involves counting passwords of varying length and with certain constraints; the overall goal of the tasks is to motivate deep understanding of the binomial theorem (for more information on this activity, see Lockwood & Reed, 2016; Ellis et al., 2022a). In the Passwords Activity, students consider the question, “How many three-­ character passwords can be made using the letters A and B?”, and we explicitly

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Table 6.1 (a, b) The three-­character and four-character AB tables

direct students to organize their work by completing tables according to the number of As in the password (Table 6.1a). We aim to have students fill out each row by listing passwords. Once students complete the three-character passwords problem, they then repeat this process for passwords of lengths 4 and 5 (the four-character generating table is seen in Table 6.1b). The goal is to have the students build (via partial or complete listing) the tables to see how they would use them when progressing to the next part of the tasks. Then, students move on to passwords involving the number 1 and the letters A and B (which we call AB1 passwords). Students are asked to make tables for three-­ character and four-character AB1 passwords, organized according to the number of 1 s in the passwords (the four-character AB1 table is in Table 6.21). The table for the number of four-character passwords that use 1, A, and B can be completed by first thinking of counting the number of ways of placing the 1 s and then considering the number of options for the remaining non-1 positions (the positions that are not 1 s must be As or Bs, which reduces the problem to a previous one involving AB passwords).

Tyler Engages in Pseudo-empirical and Reflecting Abstraction Tyler was a first-year college student enrolled in vector calculus, and he participated in a single, 60-minute individual interview (we have shared excerpts from Tyler’s work elsewhere, such as in Ellis et al., 2022b). Tyler’s method of solving the tasks typically involved organized and systematic listing. That is, Tyler’s listing activity became a mechanism by which he could progress through and solve the problems; ultimately, both the outcomes of that listing activity and the listing activity itself would become sources of abstraction for Tyler.

 Note that the right column of Table 6.1b represents a way of writing the expressions that highlights the relationship with the AB tables; in designing the tasks, we hoped students would eventually recognize this structure, although they could also simply write the totals in the right column. 1

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Fig. 6.11 (a) Tyler’s list of the eight three-character AB passwords and (b) his three-character AB table

Tyler’s work on the AB password tasks. We began by asking Tyler for the total number of three-character AB passwords. In response, Tyler created a list of outcomes (Fig. 6.11a). Tyler had a strategic, organized way of listing outcomes, and he used his list to create the correct table for three-character AB passwords (Fig. 6.11b).2

 We do not return to Tyler’s specific systematic listing strategy in this paper, but for the curious reader, he created the list in Fig. 6.11a by first creating the column of AAA, AAB, and ABB and then creating the column of BBB, BBA, and BAA. He noted that these had consecutive groups of letters, and then he added the ABA and BAB as passwords that would “mix them up.” When asked about his strategy, Tyler noted that he listed all the passwords that started with A and then all the passwords that started with B. The takeaway is that he was systematic in his listing strategy. 2

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We then asked Tyler to create the four-character AB table, and he again generated the table (Fig. 6.12) by systematically listing and then counting the outcomes for each row. We note the 1, 4, 6, 4, 1 in the right column; as we will describe, he would later refer to the 6 in this table when creating a four-character AB1 table. An important episode occurred when Tyler tried to fill in the entry of the three-­ character AB1 table with zero 1 s: Int.: How about for zero? Tyler: Zero, um, it’s not gonna be one this time. Int.: Okay. Tyler: (Writes AAA, then pauses) A, A, A, um, eight maybe? Int.: Okay, and why, why’d you guess that? Tyler: Um, because for this one with just; what I’m doing now I guess is just two letters, um, so it’s the exact same one I did here, isn’t it (points to the length 3, AB table in Fig. 6.11b). Yeah, so I’m going to go with eight just because that’s the exact same thing.

We contend that this is an instance of pseudo-empirical abstraction for two reasons. Firstly, we posit that the initial act of writing AAA served as a reminder of Tyler’s prior listing activity, and he directed his attention toward the written table on the page, which provided available sensorimotor material. Tyler first listed out AAA, and then he paused before he guessed that there were eight AB1 passwords with zero 1s. This suggests that the physical act of writing down the AAA password triggered for Tyler a reminder of his previous listing. In this way, Tyler engaged in a reflective abstraction that was tied to sensorimotor material, making it an instance of pseudo-empirical abstraction. Furthermore, we also draw on our broadened category of what might be considered an “observable” (based on Moore, 2014), where here the 8 served as the outcome of Tyler’s prior mental activity of listing. We see this as pseudo-empirical abstraction because Tyler drew on the results of his Fig. 6.12 Tyler’s four-­character AB table

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activity, rather than on the coordination of his actions in producing the original table. Thus, the outcome of his listing activity served as the observable on which Tyler relied to abstract the table structure. Specifically, Tyler projected the result of his initial listing activity (the eight total outcomes he produced in the three-­character AB table), which he then fit into the AB1 situation, reorganizing his current activity with the AB1 table to incorporate the outcome of his prior activity. Tyler could then conceive of that 8 not just as the result of his work on the AB table but as the total number of possibilities in his AB1 table in the case where there was no 1. In this episode, then, Tyler recognized that when he was counting the AB1 passwords with zero 1s, he was in a situation that was identical to the previous situation involving only As and Bs. Tyler connected the current situation with the previous situation, and he used his activity and his previous table to make sense of (and find a solution for) the new situation. In particular, he seemed to understand that in his prior listing activity, he had generated a total of eight three-character AB passwords, and so finding himself in a similar situation, he could use the result of that prior activity. Next, we asked Tyler to fill out a four-character AB1 table, organized according to the number of 1s. In working on the row for one 1, Tyler did something unexpected – he introduced a way of describing a general outcome involving 1s and xs. Specifically, he wrote out four general outcomes, 1xxx, x1xx, xx1x, and xxx1 (Fig.  6.13a), and he used those general outcomes to fill out the new table. Tyler discussed his reasoning in the following exchange, and as he did so, he referred back to his four-character AB table (Fig. 6.13b). Tyler: And then the 1, so what I was thinking – what I was saying earlier, how there is only a certain amount of spots for it. Like it has to be … like I’m just going to use x cause, um, has to be in one of these spots ... (writes the combinations of 1 and xs in Fig. 6.13). Int.: Great. Tyler: So there’s, now there’s just three xs, um, and I know that for ... three spots with two different letters there’s going to be eight different ways to do it (points back to the ­previous 3-character AB table, see Fig. 6.13b)…Um, so I guess eight … there’s eight different of each of those just using this same table (points to the three-character AB table). Um, there’s just 32 so I want to say there’s going to be um, 32 for just the one. Int.: Okay and you got, you’re thinking of that as kind of the 4 times 8? Tyler: Yeah I, just adding them all up.

Fig. 6.13 (a) Tyler’s list of one 1 and three xs, and (b) Tyler explicitly refers back to the AB table

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This was a key moment in Tyler’s work. He articulated a generalized outcome, expressing four-character AB1 passwords as arrangements of xs and 1s. Tyler continued to use 1s and xs in filling out the rest of the four-character AB1 table. Figure 6.14a shows his list of xs and 1s in the four-character AB1 case, with exactly two 1s. There are exactly six of them, and the following exchange demonstrates Tyler’s meaning of those six general outcomes as they relate back to his previous work. Specifically, Tyler seemed to understand why six such outcomes would make sense, because he could recognize that he was in a situation of arranging two distinct objects, which is what his previous work involving AB passwords also entailed. He ultimately arrived at the correct table for four-character AB1 passwords (Fig. 6.14b). Tyler: Yeah there you go. Is that all of them? Yeah so six, because that would make sense… Int.: Does that six make sense? Tyler: Does it? Uh, well that would – that’s um, two variables like instead of doing three things there’s two, um, with the four combo, so two, was six over here (points back to the 6 in the correct entry of the four character AB password table, Fig. 6.12), so that’s why I thought it made sense.

This was a crucial revelation in terms of Tyler’s work and his ultimate success on the activity. We note that Tyler could see the arrangements of 1s and xs and the arrangements of As and Bs as being essentially “the same.” One example of this phenomenon was when Tyler was able to justify why there were six arrangements of two 1s and two xs. As seen previously, Tyler said that the 6 made sense by noting, “Like instead of doing three things there’s two, um, with the four combo, so two was six over here [points back to the 6  in the correct entry of the four-character AB table], so that’s why I thought it made sense.” Tyler’s response suggests that he

Fig. 6.14 (a) Tyler’s six combinations of two 1s and two xs and (b) his four-character AB1 table

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recognized that he was in a situation of arranging two 1s and two xs, and he explicitly related the result of that activity to the number of ways of arranging two As and two Bs, which he had solved previously. Tyler extrapolated a similarity between arranging AABB and 11xx, as he recognized that, ultimately, he was just rearranging two kinds of things. In doing so, Tyler stripped away the specifics (e.g., it did not matter whether they were As and Bs or 1s and xs) as he focused on the coordination of his actions to make a generalization across the two scenarios. In this case, we infer that Tyler engaged in a reflecting abstraction, as he began to consider a general way to represent ways to count the number of passwords with one 1. Here, the six ways to list two As and two Bs were not just the result that he viewed as separate from (the outcome of) a particular process; rather, he reflected on the process of listing those outcomes themselves. He reflected on the repeated action of arranging two types of two characters, and the particulars of what those characters are were stripped away. We infer that Tyler projected his specific listing activity of arranging two As and two Bs in the AB passwords case to the higher level of the AB1 case (i.e., a more general level of thinking of listing structurally). He then engaged in a reorganization, which involved making sense of his current activity in the AB1 case, listing two 1s and two xs; he thus coordinated his current specific activity with a generalized structure of arranging two of each of two types of characters. Thus, to summarize the episodes we have described with Tyler, we contend that he engaged in both pseudo-empirical and reflecting abstractions, and the nature of his initial activity facilitated each abstraction. In the first case, he abstracted 8, the result of a listing process for three-character AB passwords, and that 8 was tied to the sensory material of the table he had written and the physical listing process he had engaged in. In the second case, he looked at the result of that process (the 6), but he did not abstract that result but rather the process itself. We follow Moore (2014) in considering the difference between abstracting the result of a process and abstracting a coordinated process itself as being a key distinguishing feature between pseudo-empirical and reflecting abstractions.

 Group of Students Engage in Pseudo-empirical, Reflecting, A and Reflected Abstraction We now present an example in which a small group of four first-year college mathematics students (Carson, Anne-Marie, Aaron, and Josh) were working through the Passwords Activity. We highlight a slightly different aspect of their work, focusing particularly on their reasoning about symmetry within the tables, but in doing so, we demonstrate instances of pseudo-empirical, reflecting, and reflected abstraction. We have reported aspects of these data with a different focus elsewhere, specifically examining a construct we call empirical re-conceptualization (Ellis et al., 2022a), as

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well as exploring students’ notions of equivalence (Lockwood & Reed, 2020) and transfer (Lockwood & Reed, 2021). In prior work creating tables for five-character AB passwords, the students had determined that there were 25 = 32 total five-character AB passwords, which they could justify using the multiplication principle. The tables for these passwords yielded a column of 1, 5, 10, 10, 5, 1, as seen in Aaron and Carson’s respective work (Fig.  6.15). In the following excerpt, we see that both Aaron and Carson used numerical patterns to complete the table. They reasoned combinatorially for the 1, 5, 5, and 1 entries (note, e.g., Aaron wrote there are five passwords with one A because “A goes into one of five slots”). They knew the total had to be 32, and they hypothesized that, because of prior symmetry they had observed, the two middle numbers would be the same, and the total would need to add to 32. As Aaron explained, “I just knew that it had to add up to 32, so, you know, 20 added up to what was already there.” Carson agreed with him and said the following: Carson: I did the exact same thing. I started with the symmetry again where you know you have one option for each of the monogamous sets, and then five options for sets where one of the things is different than all of the others, and then two empty slots. And because it’s symmetric, then we know that those two slots need to be the same number, and knew that the total had to go up to 32 just based off of, you know, if you [have] five slots and two options for each slot, you’re going to get 25. [...] So, there was a remainder of 20, so half of 20 is 10, so 10 for each of those slots.

We claim that Aaron and Carson’s statement that rows for 2 and 3 would be 10 was a case of pseudo-empirical abstraction. They abstracted the pattern of symmetry they had observed by looking across previous tables. We argue that their abstraction was pseudo-empirical because, like Angelo, they reflected on the outcomes of their initial activity, as represented as a pattern in the prior table, rather than on a coordination of actions that they engaged in while listing or creating the table. The outcomes were available to them as sensorimotor material in the prior table, but even if the prior table had been absent, we would still consider this a pseudo-empirical abstraction due to the students’ reliance on the results of their activity. The interviewer asked the students to explain how they became aware of the symmetry and why it would make sense. Anne-Marie and Carson’s response to this question demonstrates having made a reflecting abstraction:

Fig. 6.15 (a) Aaron’s and (b) Carson’s five-character tables

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Anne-Marie: But, I just see it as, like, it’s where the As and Bs flip. So, like, when I was writing out the list, I was moving the A through the matrix of Bs. And, like, so that’s when…the symmetry…I think it was kind of when you were moving the B instead of the A. Int.: Okay, nice. Carson: Yeah, so just, it’s…asking to move one A around in a group of Bs is the same thing asking to move one B around a group of As. It’s going to return the same number of results. Int.: Okay. Carson: So, it’s like that flip she was talking about, you know, where you were going down this list of how many As you have. Eventually, it’d be easier to ask how many Bs you have, because it’ll be the same list, just backwards.

In this excerpt, Anne-Marie described her listing process as writing outcomes with fewer As and “moving the A” through the Bs and then changing to “moving the B instead of the A.” Carson expanded on Anne-Marie’s statement, and he described moving one (general) character through a string of another (general) character. Carson’s language suggests that he was engaging in reflecting abstraction as he considered a general listing activity based on the coordination of his actions. In particular, the source material of the abstraction was the students’ reflection on their general process of listing rather than the outcome of the listing itself. Here, we interpret that Carson extracted the regularity he observed in his listing activity  – specifically, the idea of moving one character through a string of other characters – and he projected that listing activity into a new situation in which he was working with different characters (now a B instead of an A) and different strings (moving through a string of As instead of a string of Bs). In a final example, we describe an episode in which Carson engaged in a reflected abstraction. Here, the interviewer again asked the students about the symmetry: “Someone else just tell me a little bit more about why that’s the case. [...] Two As and three Bs, three As and two Bs, like, yeah, why is it the same number of things?” Carson responded by bringing up a connection to a problem that the students had solved in an earlier session, an “arrangement with restricted repetition” problem, which involves arranging characters, some of which are identical (or repeated). For example, such problems would be to arrange letters in the word MAMA or RACCOON. Such problems can be solved by arranging all letters and then dividing by the number of arrangements of repeated letters. In the following excerpt, Carson describes a new approach to thinking about the AB passwords problems, which was different from the work he had done so far on the Passwords Activity. Carson looked at the third row of the AB table and considered the original solution as a problem involving arranging three As and two Bs: Carson: Another way you could look at this [the fourth row in the 5-character AB table, Fig. 6.15b] is if you have the A, A, A, B, B, how many different ways can you arrange the letters in that word? So that’s going to 5!, which is the total number of letters over the repeat letters, so 3!, 2! (writes “AAABB” and “5!/(3!2!)”). Right? Int.: Mm hmm. Carson: And that’s the same equation there. That will hold true whether the repea letters are three A’s and two B’s, or three B’s and two A, or three C’s and two D’s. Right?

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Int.: Okay. Carson: And I mean, that’s a mathematical relationship we’ve talked about before, because you’re just pulling out the redundant, um, redundant arrangements given repeated letters. Int.: Okay, good. Carson: So, I mean, yeah, it doesn’t really matter what the letters are as long as they’re as those ratios.

This exchange shows evidence that Carson engaged in a reflected abstraction. Carson recognized that the eq. 5!/(3!2!) was the same as the current situation he was in, and he also made a more general statement that indicated that he realized a broader phenomenon. Specifically, Carson understood that although in the Passwords Activity, they were counting AB passwords with a certain number of As, this activity was actually representative of a type of activity that could be done with any two kinds of characters. Carson abstracted a process/operation (arranging two kinds of characters) across multiple situations – one involving counting the number of ways to arrange letters in a given word (which he had solved previously) and one involving determining the number of AB passwords with three As and two Bs. Furthermore, Carson was consciously aware of this similar structure and could describe it to the interviewer. Carson’s articulation of an isomorphism across the two situations (the fact that both cases entail the same process of arranging characters with repetition) suggests that he had engaged in a reflected abstraction.

Discussion Our aim in this chapter has been to describe and provide illustrative examples of the levels of abstraction as described by Piaget. Below, we discuss the standards of evidence we find useful for guiding our thinking when examining students’ abstractions. We then describe the cyclical and interrelated nature of different types of abstraction and conclude by offering some final thoughts on the ways in which distinguishing between types of abstraction can be useful to researchers.

Standards of Evidence Having provided a number of data examples demonstrating various types of abstraction, we now reflect back across those examples to summarize what we take as standards of evidence for deciding how to classify an abstraction by type. To classify an abstraction as empirical, it must be of properties of objects that are, to the actor, inherent in the objects themselves. Generalizing about properties such as color, weight, size, or, in the case of Mario, the amount of water leaving a faucet would all be potential examples of empirical abstractions. Once there is evidence

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that the actor is relying on coordinations of their physical or mental actions exerted on the objects, we no longer consider the abstraction to be empirical. It is now a pseudo-empirical abstraction, even if one is, for instance, making generalizations about (or classifying objects based on) properties that appear to be inherent to the object, from the actor’s perspective. In some cases, such as with Tyler’s listing activity, we adopt Moore’s (2014) stance and consider abstractions to be pseudo-­ empirical when they stem from the product of activity, rather than from coordinations of actions, regardless of whether or not one relies on sensorimotor material. By expanding what counts as pseudo-empirical abstractions in this manner, we open up possibilities for conducting more nuanced analyses of students’ thinking that is common in school mathematics, such as the type of pattern finding we saw with Angelo, as well as in higher-level mathematics. When determining whether a student has made a reflecting abstraction, we look for evidence of them coordinating actions or operations, rather than relying solely on the outcomes of their actions or operations. In research contexts, this typically occurs when students engage in multiple tasks over time, such as in teaching experiment settings, because it offers opportunities for us to observe students’ scheme or operation construction. Specifically, we can look for evidence that students have engaged in similar coordinations across tasks in such a manner that they have ultimately stripped away specific details from any individual experience and instead have created a more general structure, such as in Tyler’s activity when he used 1 s and xs to represent any two-digit code. As we noted above, some researchers only require that the scheme be one that is composed of operations more advanced than the prior operations being used (e.g., Campbell, 2001), which is our standard of evidence as well. For instance, we saw evidence of more advanced operations when Willow shifted from grouping sets of 1.5 cm2 to coordinating the number of 1.5cm2s with the number of 1 cms. It is worth noting that others, such as Norton (2018), hold a higher standard by requiring the presence of operations arising through a coordination of actions within a reversible and composable system. When considering whether an abstraction is reflected, we adopt the criterion Piaget (2001) often used, namely, that one has compared commonalities across different activities and discerned common factors. Such comparisons, as Piaget (1976) noted, require the actor to explicitly reflect on their coordinations, a reflection of thought on itself; in this manner, we can infer conscious awareness. We saw evidence of this when Carson saw the Passwords Activity as involving the same operations as a prior class of arrangement problems, and he was able to articulate that both were simply the process of arranging two kinds of characters, represented as 5!/(3!2!). We note that this criterion may also be less strict than what other researchers require (e.g., Tallman, 2021). For a more elaborated account of reflected abstraction specifically, see Tallman and O’Bryan (Chap. 8). In general, evidence for reflecting and reflected abstractions can be challenging to observe and may require gathering data over time in order to document changes in students’ activity.

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The Cyclical Nature of Abstraction In this chapter, we have introduced a somewhat artificial separation of each type of abstraction in order to discuss each in turn; in actuality, however, the processes of abstraction are not so neatly separated. For instance, we have discussed that empirical abstractions presuppose prior reflective abstractions (Montangero & Maurice-­ Naville, 2013). As an example, for Mario to recognize changes in water temperature via an empirical abstraction, he must already have constructed schemes via reflective abstraction by which to describe temperature. The necessity of reflective abstraction for empirical abstractions both underscores Piaget’s unease with the term “simple” while also highlighting the cyclical nature of the different forms of abstraction in general. There are also cyclical relationships between the different types of reflective abstraction; sometimes, these relationships are opaque, and it can be difficult to determine precisely what kind of abstraction has occurred. Both Kendis and Camila’s repeated experiences with the Faucet Task and Tyler’s list-making activity highlight that pseudo-empirical abstractions can lay an important foundation for future reflecting (and reflected) abstractions. By repeatedly engaging in pseudo-­ empirical abstractions, the students were able to develop operations that ultimately supported further abstractions based on coordinations of actions. With the Passwords Activity, for instance, the designed set of tasks intentionally grew in complexity and generality. This supported students to initially engage in systematic listing of outcomes and filling out tables, offering opportunities to reflect on the results of their listing activity via pseudo-empirical abstraction. But then, as the complexity of each task grew, students were able to reflect more generally on their operations of arranging a certain number of types of characters, no longer focusing on the specifics of what the characters were. This afforded reflecting and reflected abstractions that supported students’ understanding of combinatorial listing strategies, as well as combinatorial justifications for the number of passwords with certain constraints. It is important to include two caveats here. Firstly, it is not our intention to imply that every reflecting abstraction presupposes a pseudo-empirical abstraction. One can certainly engage in reflecting abstraction without relying on a prior pseudo-­ empirical abstraction. The relationship between the different types of abstraction can be complex, and it is not necessarily the case that individuals shift in some ladder-like fashion from empirical to pseudo-empirical to reflecting to reflected abstractions. Our second caveat is that a pseudo-empirical abstraction does not necessarily need to lead to subsequent reflecting or reflected abstractions. As we saw with Angelo’s determination of a rule based on a pattern of outcomes, students may complete individual tasks – or entire series of tasks – without ever making reflecting (or reflected) abstractions. Repeatedly engaging in different situations that elicit schemes grounded in reflective abstractions can, in turn, support students in comparing their activity across situations to ultimately form reflected abstractions. Camila’s description of two situations that could produce the same graph via different emergent traces is

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one example of how a student can build on prior pseudo-empirical and reflecting abstractions in order to produce a reflected abstraction. Operations that were initially instruments of calculation, such as creating a trace of a point as a representation of covarying quantities, then became differentiated objects of thought in their own right, enabling Camila to identify different scenarios that would result in the same outcome. As such, reflected abstractions are powerful in that they can support students in formalizing their own mathematics. Reflected abstractions (in contrast to other types of abstraction) do necessarily entail prior reflective abstractions, even though it may be challenging or even impossible for a researcher to tease apart those myriad prior abstractions. This is the case because reflected abstractions are defined as conscious products of reflective abstractions (Piaget, 1976). Repeated activity that elicits meanings grounded in reflecting abstractions, in particular, can be particularly powerful for supporting students to ultimately develop reflected abstractions (Oehrtman, 2008).

The Value of Abstraction as a Construct How can identifying and characterizing instances of abstraction be useful to our goals as researchers? Adopting the lens of abstraction, and in particular, characterizing the processes of projection and reorganization, offers a powerful mechanism to explain students’ learning over time. Certainly, abstraction and other aspects of Piaget’s genetic epistemology are not the only possible ways to do this, but we find that it provides a meaningful structure to organize our thinking, as researchers, as we build second-order models (Steffe et al., 1983) of students’ mathematics. These models, in turn, enable us to make theory-driven predictions about students’ understanding and behavior. For instance, consider Willow and Angelo’s activity with the Growing Rectangle Problem. Both students produced the same general statement, A = 1.5 L. Examining the nature of each student’s abstraction, however, provided important insights into their understanding, as well as the likely generativity of their generalization. Generative generalizations are ones that can be extended to accommodate new cases and can be justified through deductive arguments (Ellis et  al., 2022b). By identifying Willow’s abstraction as reflecting, we could predict that she would likely be able to justify her general statement, as well as use it to produce new length-area pairs, even if we provided non-integer length values. Correspondingly, we could predict that Angelo might need additional support in justifying his general statement beyond an empirical argument, using it to predict new pairs, or adjusting it to respond to a different-sized rectangle. Across our data sets, we have found that generalizations based on reflecting abstractions are more generative than the ones that emerge from pseudo-empirical abstractions. This, in turn, can inform our design process by constructing sequences of tasks that can engender reflecting abstractions. This was the case in our work with Kendis and Camila. Conceiving a graph as a trace is a non-trivial concept that is seldom intentionally developed in US school mathematics (Thompson & Carlson, 2017). However, such an understanding is possible, even for middle-school students, if they are intentionally supported in

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gradually engaging in higher levels of abstraction (Paoletti et al., 2021). For instance, we anticipated that a series of pseudo-empirical abstractions could support students in subsequently developing the necessary reflecting abstractions to ultimately conceive and anticipate graphs as representing dynamic traces. In the case of Kendis and Camila, Kendis’s conceiving of a point as a multiplicative object (Lee, 2016; Saldanha & Thompson, 1998) via a pseudo-empirical abstraction laid the foundation for her eventually engaging in a reflecting abstraction to anticipate the point as a representation of covarying quantities. Then, in reflecting on their operations, both students ultimately became consciously aware of their understanding that graphs are traces representing two covarying quantities, in particular contexts. This process of reflected abstraction then enabled the students to more generally conceive of graphs as representations of covariation, even in novel contexts. For example, Paoletti et al. (2021) described Kendis and Camila graphically representing a relationship between the side length and the area of a triangle in the sessions immediately following the Faucet Task. Hence, their reflected abstraction in one context supported their graphing activity in a novel context. Similarly, building on our findings from the Growing Rectangle Problem, we developed a sequence of tasks that would foster shifts from pseudo-empirical to reflecting abstraction. We did so by encouraging students to reason with multiple growing rectangles with different height values and ultimately with growing rectangles in which they had to determine their own height values. Students who initially reasoned as Angelo did, generalizing based on the outcome of observed patterns, began to reflect on their activity of coordinating growth in length with growth in area for different rectangles. Doing so with new rectangles with unspecified height values further encouraged this form of reflection, ultimately supporting the development of a constant rate of change (Ellis et al., 2020). As discussed earlier, we also designed the Passwords Activity to engender opportunities to engage in repeated abstractions within and across tasks, supporting students in making pseudo-empirical abstractions that could ultimately be leveraged to foster reflecting and reflected abstractions. Using the construct of abstraction provides a level of precision in characterizing students’ learning and cognition, and it allows us to go beyond merely describing behavior. It offers a way to make sense of what is at the core of observable distinctions (such as the distinction between result-pattern generalization and process-­ pattern generalization (Harel, 2001)) and explain the cognitive mechanisms responsible for those distinctions. Piaget’s notion of abstraction is just one construct of many to explain learning. Models of learning can offer insight into how to design better task sequences, as well as inform our thinking about powerful instructional practices that can foster meaningful mathematical engagement. Characterizing the nature and content of students’ abstractions when reasoning about particular mathematical ideas has proved helpful, not only in making sense of students’ thinking in the moment but also in then being able to predict future reasoning and responsively craft subsequent activities to more effectively support conceptual development. By understanding students’ processes of abstraction, we are able to better understand the nature of teaching and learning mathematics.

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Chapter 7

Groups and Group-Like Structures Anderson Norton

In the introduction to this book, we noted that Piaget’s genetic epistemology adopts a Kantian perspective in that it explains how cognitive (rational) structures shape empirical experiences to form knowledge of the world. Although Kant’s (1998) rational empiricism greatly influenced Piaget, the two thinkers diverged when it came to assumptions about innate structures. Specifically, Kant had taken time, space, and number for granted, and in this, he was not alone: Our knowledge of the first principles, such as space, time, motion, and number, is as certain as any knowledge we obtain by reasoning. As a matter of fact, this knowledge is provided by our hearts and instinct is the necessary basis on which our reasoning has to build its conclusions. (Blaise Pascal, 1966/1670 (in Pensées, line 110)).

In contrast, Piaget took the construction of time, space, number, and even logic, as research foci. He learned from children how those structures emerge through human activity, beginning with reflexes and sensorimotor activity. With a few close colleagues—especially Barbel Inhelder—he reported these results in separate books on each topic (Inhelder & Piaget, 1969; Piaget, 1969/1946; Piaget & Inhelder, 1967/1948; Piaget & Szeminska, 1952). It was important to Piaget’s genetic epistemology that he study children because, as evident in Pascal’s quote, we, as adults, take for granted many of these primitive structures. These early constructions are so fundamental to our organizations of experience that we assume their certainty and necessity. They are the logico-­ mathematical structures from which we build models of the worlds we experience: Mathematics may be defined as the study of shape and number, or as the science of patterns (Resnik, 1981; Steen, 1988). Piaget dug deeper into questions of shape, number, and patterns in general, by considering their psychological underpinnings. He demonstrated the A. Norton (*) Department of Mathematics, Virginia Tech, Blacksburg, VA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 P. C. Dawkins et al. (eds.), Piaget’s Genetic Epistemology for Mathematics Education Research, Research in Mathematics Education, https://doi.org/10.1007/978-3-031-47386-9_7

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unity of mathematics by identifying commonalities in their structures. Specifically, he defined mathematics as the coordination of reversible and composable mental actions. That is, he defined it as an organization of operations: ‘An operation may be defined as an action which can return to its starting point, and which can be integrated with other actions also possessing this feature of reversibility’. (Piaget & Inhelder, 1967/1948, p. 36)

In building models of children’s mathematics, Piaget described the organization of operations within structures. In this chapter, we make a distinction between two kinds of structures used repeatedly in Piaget’s epistemology of mathematics. Chapter 3 addressed the first kind of structure, that of a scheme, which Piaget used to describe how children sequence operations in service of a goal. These structures have proved fruitful within research in mathematics education, across mathematical domains from counting to calculus. The second kind of structure—groups and group-like structures—has received less attention in mathematics education research. As defined in detail later in the chapter, a group is a mathematical structure that describes how elements of a set can be combined. Piaget used groups to describe how a set of mental actions might be combined and organized as reversible and composable (qua) operations. When Piaget referred to a group, he referred to the formal mathematical structure studied in abstract algebra. However, as he never tired of reminding his readers, he did not assert that students are aware of such a structure: “Let us note that at these [unformalized] levels, even if the operations used are more and more conscious, the structured wholes remain completely alien to the subject’s conscious reflection” (Beth & Piaget, 1966, p. 246). After all, both schemes and groups are researcher constructs used to build models of students’ mathematics (Glasersfeld & Steffe, 1991). We infer them from students’ behaviors to explain how they might reason. We rely on them because they prove useful in explaining and predicting how students might operate mathematically. For example, Piaget and Inhelder (1967/1948) relied on a “group of displacements” to describe how children construct space on the basis of their own activity. Through self-locomotion (e.g., crawling), children learn to alter the worlds they perceive by moving within them. They can continue their motion in a given direction or move in a new direction, composing those two displacements to reach some new vantage point. They can also reverse their motion to return to a starting point. These two properties of motion—composability and reversibility—satisfy the primary conditions of a mathematical group. Piaget argued that this group structure explains how children build the totality of space, in which to move and to posit objects even when removed from perceptual experience. The purpose of this chapter is to examine how Piaget used groups and group-like structures to model mathematical development as a coordination of mental actions. By extension, we consider ways that mathematics education researchers can use such structures today, not only to build models of children’s mathematics in various domains but also to describe the nature of mathematics itself. The chapter begins by distinguishing the general structure of a group from that of a scheme and by elucidating their relationship. It proceeds to introduce formal properties of groups and group-like structures Piaget (1972a) called “groupings.” We elaborate on examples

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of “the splitting loope” (Wilkins & Norton, 2011) and “the splitting group” (Norton & Wilkins, 2012) to demonstrate how group-like structures have, in fact, proved useful in mathematics education. After noting formal connections to mathematical programs carried out by Felix Klein and the Bourbaki, the chapter closes with specific suggestions for applying groups and group-like structures to future research in mathematics education.

Two Kinds of Structure Glasersfeld (1995) described Piagetian schemes as three-part structures composed of a recognition template, a sequence of actions, and a goal. For example, consider the iterative fraction scheme (IFS), which renders fractions “numbers in their own right” (Hackenberg, 2007). When a student operating with an IFS sees a fraction— even an improper fraction, like “7/5”—that fraction symbolizes a sequence of actions they could perform (Tzur, 1999). Specifically, they can assimilate 7/5 as seven iterations of the unit fraction, 1/5, which results from partitioning a whole into five equal parts. Moreover, they can reverse that sequence of actions to reproduce the whole from 7/5 of it. Figure 7.1 illustrates a simplified schematic model of the IFS. Glasersfeld (1995) was well aware that the three-part model oversimplifies matters (as all models do); it fits better with sensorimotor schemes than operative schemes (see Chap. 3). In a sensorimotor scheme, actions are carried out in sequence, physically or in imagination. In an operative scheme, the three-part structure, along with all of its actions, collapses into a single logico-mathematical concept: A schema of action is, in fact, only the form of a series of actions that take place successively without a simultaneous perception of the whole. Reflective abstraction, on the other hand, upgrades it to the form of an operational schema, that is, of a structure such that, when one of the operations is used, its combination with others becomes deductively possible through a reflection going beyond the momentary action… and these operations can sooner or later be carried out symbolically without any further attention being paid to the objects which were in any case ‘any whatever’ from the start. (Beth & Piaget, 1966, p. 234)

Reflective abstraction is the subject of Chap. 6, so we won’t elaborate on its role here except to emphasize that it is the process by which we “upgrade” actions to reversible and composable operations, so it is inherently tied to the construction of group structures. Consider the remainder of Beth and Piaget’s comments with regard to the IFS, wherein improper fractions, like 7/5, become mathematical

Fig. 7.1  Three-part model of a scheme

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objects (numbers in their own right). The sequence of actions used to produce 7/5 (partitioning a whole into five equal parts and iterating one of them seven times) no longer needs to be carried out, as 7/5 symbolizes them all at once. This scheme is called operational because its actions have become interiorized as operations. Actions become interiorized as operations when we organize them within groups or group-like structures. Therein, operations achieve reversibility and composability. Composition of actions within a group does not depend on any particular sequence of actions or any particular scheme. All possible compositions and all inverse operations are determined by the structure of the group. Note here that an action could serve as its own inverse (e.g., reflecting the plane over a line), but generally, actions have other actions as their inverses (e.g., partitioning and iterating or rotating the plane about a point in either of two directions). Moreover, in mathematics, we are generally interested in composing actions with many other actions to produce large sets of mathematical objects (e.g., producing all isometries of the plane by composing reflections over various lines). So, to operationalize an action, we generally need a group that contains other actions (including the trivial action of doing nothing). Throughout this chapter, “group” refers to a formal mathematical structure, often as a structure for modeling students’ mathematical ways of operating. The structure of groups pertains to the organization of logico-mathematical operations themselves, an organization that takes time and requires experience and reflection. Within groups, operations achieve simultaneity and logical necessity. Simultaneity refers to the sense that the operations do not need to be carried out in sequence. Their organization depends on the structuring of the group as a whole. As we will see in the next section, logical necessity depends upon the reversibility and composability of the operations. Schemes achieve their own reversibility thanks to the reversibility and composability of their operations. For example, the IFS is a reversible scheme because its principal operations are reversible and composable. Specifically, partitioning and iterating become reversible operations within a group-like structure called the splitting loope (Wilkins & Norton, 2011). This structure prescribes ways partitions and iterations might be composed with themselves and one another. We return to this example later in the chapter after carefully defining groups and their properties.

Groups Groups serve as fundamental structures both in mathematics and in Piaget’s epistemology. They frame Piaget’s characterizations of logico-mathematical operations in particular. Indeed, the appearance of groups in psychology as well as mathematics is foundational to his epistemology of mathematics. Piaget explains the coincidence as a consequence of the nature of mathematics itself. We have, throughout the history of mathematics, been operating within laws of reversibility and composability,

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and we “discovered” groups when we became explicitly aware of these ways of operating: …for it is only at the end of a sufficiently long series of reflective abstractions that the subject discovers the most profound characteristic of operations: that of being connected together in structures which have their own laws of totality. That is why we had to wait for E. Galois to discover the group concept which Viète or Descartes, nevertheless, constantly used unconsciously in their algebra. (Beth & Piaget, 1966, p. 294)

A formal definition of group did not appear until the 1850s,1 but the group structure was implicit in the work of mathematicians for many centuries and not just in algebra. As so happens, we can characterize the field of abstract algebra as the study of our own mathematical ways of operating across various domains, from number theory to geometry and complex analysis. For example, when mathematicians of the Italian Renaissance encountered imaginary roots, they implicitly understood that accepting them as solutions of equations meant including them within a closed system for adding and multiplying them with other numbers. Bombelli’s use of a + bi, in particular, described an extension from real numbers to complex numbers that, today (thanks to abstract algebra), we recognize as a field extension. Among the structures of abstract algebra (e.g., rings, fields, and ideals), the structure of groups is one of the most commonly recognized. The near ubiquity of groups in mathematics owes to the simplicity of their general structure. A set of mathematical actions or objects only need to satisfy a short list of conditions to constitute a group: 1. Closure under composition: Every element of the set can be composed with any other element of the set to produce some element of the set; this composition could take the form of addition, multiplication, function composition, or any other binary operation, so long as it results in another element of the set. 2. An identity element: There is some element of the set that, when composed with any other element of the set, yields that other element of the set; in the real numbers under addition, this element is 0, and under multiplication, 1. 3. Reversibility: Every element of the set has an inverse element in the set, such that their composition yields the identity element (e.g., 3 × 1/3 = 1). 4. Associativity: Given the composition of any three elements, the order in which the (binary) composition is performed does not matter (i.e., (a  ×  b)  ×  c  =  a × (b × c)); this property should not be confused with commutativity, which concerns the order of the elements themselves (rather than the order of composition) and is not a necessary property of groups. We elaborate on each of these conditions/properties, both mathematically and psychologically, in separate subsections below. Familiar examples of organized actions that we can model with a group structure include the rotations of a wheel and

 Historians credit Arthur Cayley for providing the first definition of an abstract group, in 1954. See Kleiner (1986) for a thorough and engaging review of its history, including contributions from Viète, Descartes, and (especially) Galois. 1

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manipulations of the Rubik’s cube. Formal mathematical examples of groups include isometries of the plane, permutations of an ordered set, positive rational numbers under multiplication, and n-dimensional vectors under vector addition. As noted by Beth and Piaget (1966) in the quote above, the group structure formalized in the nineteenth century had implicitly permeated mathematical thinking for centuries before, from number theory, to geometry, to algebra. The apparent ubiquity of groups in mathematics underscores the simplicity of the structure and the unity of mathematics as a product of human thought.

Closure Closure refers to composability. Not only does it imply that any two elements in the group can be composed but also that the result of their composition is an element of the group. For example, consider the group of integers under addition. Whenever we add two integers, their sum is another integer. In fact, closure under addition is what defines the integers as a set of numbers. “Need we remind the reader that a whole number exists, psychologically as well as logically (in spite of Russell2), only by virtue of being an element of a sequence of numbers (engendered by the operation +1)” (Piaget, 2001/1947, p. 39). This perspective has roots in the work of French mathematician Henri Poincaré (1952/1905): “Mathematicians do not study objects but relations between objects” (p. 40). We can observe its lasting influence in modern philosophies of mathematics3 and even policy documents, such as those produced by the National Research Council: “numbers do not exist in isolation” (NRC, 2009, p. 30). Beginning from units of 1 and − 1, we can define every integer through iterative addition of those units: 5 is five iterations of 1, and − 7 is seven iterations of −1. We rely on such recursion to produce an infinite set of integers. Formally, we say that 1 and − 1 generate the group of integers under addition. Psychologically, we can think about the closure property as describing a kind of “wholeness” (Piaget, 1970, p.  6). Students construct whole numbers as nested sequences. Whole numbers are whole, not only in the sense that each number in the sequence consists of undivided units (1 s) but also because combining any two numbers in the sequence results in another number in the sequence. As with all mathematical objects, we construct numbers as coordinations of action and then transform them through further action. Binary operations, like addition, represent this action on objects as a composition of two like objects. For example, what we symbolize in writing 2 + 2 is the composition of the actions that define  Here, Piaget is referring to Bertrand Russell’s circular definition of number, which Piaget (1971) critiques in Genetic Epistemology (see pp. 36–37). 3  “We are not given mathematical objects in isolation but rather in structures. That 13 is a prime number is not determined by some internal property of 13 but rather by its place in the structure of the natural numbers” (Resnick, 1981, p. 529). 2

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2; namely, 1 and 1 (two iterations of the unit 1). The composition yields 1 and 1, and 1 and 1, or four iterations of 1 (also known as 4). Note that whole numbers under addition do not form a group because they lack reversibility until students extend them, as directed quantities, in two directions: positive and negative (Ulrich, 2012; Wessman-Enzinger, 2019). Students might count backward from 4, but for closure, when they reach 0, they would need to produce negative numbers. Moreover, they would need to reconcile these new, negative numbers with the familiar whole numbers, to form a group. If we understand mathematical objects as coordinations of action, and if we understand composition of mathematical objects as a composition of the actions that define those objects, then closure refers to the scope of possible coordinations of action. It defines a space for operating. In the case of number, this space consists of an entire number system defined by our ways of operating (e.g., the set of integers, defined by iterations of a unit in either of two directions: 1 or − 1). In the case of geometry, this space could comprise isometries of the plane, all linear transformations of n-dimensional space, or all possible projections of the plane. As we will see toward the end of this chapter, Felix Klein (1893) formally classified geometries in this way, based on groups of transformations. In the example of integers, we took addition as the binary operation for composing elements in the group. As formal operations, binary operations differ from many of the (logico-mathematical) operations Piaget described. Whereas Piaget defined operations as reversible and composable mental actions, binary operations refer to ways we might continue or combine the coordinations of action that define a pair of mathematical objects. In this, logical-mathematical operations and binary operations have the following in common: “Operations are a continuation of actions; they express certain forms of co-ordination which are general to all actions” (Inhelder & Piaget, 1969, p. 291). In integer addition, we continue the iteration of units (1 or − 1), as described above. In integer multiplication, we continue a transformation of units. We can transform the unit 1 into any composite unit, say 3. We can then distribute the units of 1 within a second composite unit, say 5, over the units of 1 in that first composite unit: 3 × 5. This is what Steffe (1992) referred to as a unit coordination. It is equivalent to transforming each of the three units of 1 in the first composite unit into the second composite unit, 5. Several mathematics education researchers have used the iteration, distribution, and transformation of units to describe such recursive behavior in students’ mathematics (e.g., Confrey & Smith, 1995; Davydov, 1992; Steffe, 1992). In the case of finite groups, we can represent closure with a Cayley table, which shows all possible combinations of elements from the set. For example, consider the group of symmetries of a rectangle. This group is formed by the four linear transformations of the plane that leave the rectangle fixed (in its original position in the plane). They include reflections over the two lines of symmetry shown on the left side of Fig. 7.2 (rm and rn), a 180-degree rotation about their intersection (ρπ) and a trivial transformation (i.e., doing nothing or, equivalently, rotating 360 degrees). The possible combinations of these symmetries are shown in the Cayley table on the

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Fig. 7.2  Symmetries of the rectangle

right side of Fig. 7.2. Closure is represented in the Cayley table in that no combination of the four symmetries produces anything but one of those four symmetries.

Identity and Reversibility In addition to the composability that closure describes, reversibility is the key criterion of logico-mathematical operations, but formally speaking, we could not define reversibility without an identity element (e.g., the trivial transformation in the example above). We might think about the identity element as the starting point to which we return when we compose an element and its inverse. If we think about these elements as actions (or operations), the identity itself is an action, just one that has no effect when composed with other actions. Once we have identified an identity element, we can define inverse elements as pairs of elements (possibly the same element, as its own inverse) whose composition has the same null effect as the identity element. Psychologically, reversibility provides a kind of balance, or stability, to our ways of operating: “Reversibility is the very criterion of equilibration” (Piaget, 2001/1947, p. 12). It guarantees that, when acting on an object, we can always return to the starting point. As such, it renders mathematics completely reliable. In science, reliability is repeatability: the possibility of returning to a starting point, repeating the same sequence of actions, and achieving the same result. However, science can never achieve perfect reliability because the conditions of an experiment can never be replicated with perfect precision. In contrast, owing to the closure and reversibility of its actions, mathematics provides for “complete compensation” (Piaget, 1985/1975, p. 133). It “anticipates all transformations and pre-­ corrects errors” (p.  133) by never introducing anything but compositions of reversible actions. Piaget (1985/1975) distinguished three forms of reversibility: “any operation always involves relationships of inversion, of reciprocity, or of correlativity with certain other operations” (p. 133). The first form (inversion) corresponds with the inverse criterion of groups. Reciprocity refers to ordering relations, as in A

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 B, B  A (in Beth & Piaget, 1966). Both inversion and reciprocity appear in Piaget’s research on children’s construction of number, and their coordination led Piaget to introduce “groupings,” discussed in a later section. The final form (the correlative) appears only in Piaget’s INRC group—a meta-group that we will examine in a still-later section. Math education researchers have recognized the importance of developing reversible concepts across K-12 mathematics (Greer, 2011; Hackenberg, 2010; Simon et al., 2016). Although we can identify examples of non-reversibility in formal mathematics, such as multiplying by 0 or non-invertible matrices, reversibility re-emerges when we consider the mental actions that undergird them (Norton, 2016).

Associativity Formally, associativity refers to the property that if f, g, and h are elements of a group, then (h∘g)∘f = h∘(g∘f). We tend to take this property for granted, and we can when considering functions, mappings, and other transformations. In those cases, the property only stipulates that one transformation picks up where the prior one leaves off. Consider the illustration in Fig. 7.3. From the identity element, i, in the lower left corner of the figure, we can follow any of three paths in composing h∘g∘f. First, we could move up by way of f, to f∘i, which is just f; then move diagonally by g, to get g∘f; and finally, move up by h to get h∘g∘f. But, by closure, h∘g and g∘f are themselves transformations, and it should not matter which one we substitute into h∘g∘f. Indeed the diagram shows that it does not matter whether we substitute for h∘g, taking the shortcut from f to h∘g∘f or whether we substitute for g∘f, taking the shortcut from i to g∘f. Either way, we end up at h∘g∘f. Looking at Fig. 7.2, we can see why Piaget (2001/1947) described the property of associativity as an independence of path. He probably had in mind his group of displacements, which he often used as an example. In that example, paths refer to paths traveled in space, and shortcuts, or “detours,” refer to alternative paths that lead to the same destination (Beth & Piaget, 1966). In a composition of displacements, one displacement always picks up where the prior displacement left off. Fig. 7.3 Independence of path

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Fig. 7.4 Twenty-five marbles

Associativity, as independence of path, explains why we often find multiple solutions to a single mathematical problem. In a study of fourth-grade students in the USA and Japan, Silver et al. (1995) found that children readily produce multiple solutions to enumeration tasks. The researchers asked students to determine the number of marbles shown in Fig.  7.4 in as many different ways as they could. Students ordered, subdivided, and grouped the marbles in various ways and then counted, added, and multiplied to reach the same result: 25. Interestingly, the researchers found that children in Japan relied more on multiplication in their solutions, whereas children in the USA relied more on addition. This finding buttresses another Piagetian idea: that children use the mathematical structures they have available to assimilate tasks, including the patterns they see. Multiple solution paths are similarly evident in solutions to probability problems wherein solutions involve determining numbers of possible combinations. Likewise, in algebra, we can manipulate equations variously to solve them, and in geometry, we can construct squares in numerous ways. In each of these domains, we ultimately rely on the associativity of mental actions that define mathematical objects.

Group-Like Structures In some of the examples given heretofore, group elements have been described as actions or operations. In other examples, they have been described as numbers. This is not problematic so long as we recognize that numbers themselves are the products of mental actions, such as unitizing and iterating, as specified by Steffe (1992). As such, the combination of two numbers constitutes a combination of the mental actions that define them; just consider the prior example of 2 + 2 = 4. However, in describing the group-like structure of “groupings,” Piaget sometimes referred to elements as classes (sets of objects that share a property). These classes are not necessarily numbers, and their status as mathematical objects (arising from the coordination of actions) is not clearly specified. They appear more closely related to the development of set theory, or logic (see Chap. 10), than mathematics. On the other hand, logic and mathematics are intertwined within Piaget’s logico-­ mathematical operations.

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Whereas groups satisfy four criteria (closure, identity, reversibility, and associativity), some algebraic structures satisfy only a subset of these criteria or other weaker criteria. For example, consider the set of positive rational numbers, closed under the binary operation of division. There is an identity element, 1, and every element of the set has itself for an inverse (e.g., 5/7÷5/7 = 1). However, this structure does not satisfy the associativity criterion because, for example, (5÷7)÷(5/7) ≠ 5÷(7÷5/7). Instead, it satisfies a weaker condition known as the Latin squares property. In the case of finite sets, the Latin squares property specifies that every element appears exactly once in each row and column of the Cayley table (think Sudoku), as it does in the table on the right side of Fig. 7.2.4 This structure is known as a loop. Piaget consistently included composability and reversibility (with the implicit inclusion of an identity) in his descriptions of logico-mathematical operations, but he often neglected associativity. It could be that he took this property for granted, as we are wont to do, but he also wanted to broaden his consideration of algebraic structures to include non-associative group-like structures. Specifically, he loosened the associativity criterion for the grouping structure.

Properties of Groupings Readers might find it difficult to understand Piaget’s concept of groupings, for a few reasons. First, unlike the group structure, which is central to abstract algebra, the grouping structure “is essentially only of psychological interest, and this is due to its elementary character as well as to its own restricted nature” (Beth & Piaget, 1966, p. 172); it seems to have arisen to accommodate Piaget’s models of children’s construction of number (Piaget & Szeminska, 1952). Second, although groupings appear in many of Piaget’s books, the most comprehensive definitions appear in books not yet translated to English, especially Essai de Logique Opératoire (1972a).5 Finally, as a few critics have noted, even when he does elaborate on his meaning, Piaget sometimes mischaracterizes or ambiguously describes their formal logic. Piaget’s distinction between groupings and groups aligns with his distinction between concrete operations and formal operations: groupings describe the pairwise combination of concrete operations, and groups describe the complete combinatorial system of formal operations. Whereas the integers form a group under the operation of addition, they form only a grouping under ordering relations (2 is less than 3) and class inclusions (even numbers are contained in the class of integers).  Note that all finite groups satisfy the Latin squares property because they are associative, which is an even stronger condition. We can use associativity and the other properties of a group to prove that every element in a finite group appears exactly once in each row and column of the Cayley table. 5  Among books that have been translated to English, Psychology of Intelligence (2001/1947) and Mathematical Psychology and Epistemology (1966) stand out, so those are used as chief references on groupings. 4

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Orderings and class inclusions form a lattice, which describes the nested relationships of numbers or classes. Piaget (1972a, p. 92) designed the grouping structure as a hybrid of a lattice and a group: So the problem is the characterization of a structure that reconciles the proper reversibility of a group and a system of nestedness, properly found in the lattice. It’s this double existence that fills the notion of a “groupement” [grouping]. One can, in effect, conceive of the grouping as a lattice rendered reversible thanks to a game of dichotomies or complementary hierarchies (e.g., a set A and its complement A’ within a larger set B). The distinction between groups and groupings is evident in the properties Piaget specified for groupings, in comparison to the four criteria for groups, listed above. The following five properties of groupings appear in The Psychology of Intelligence (2001/1947): 1. Combinativity (composability): Any two concrete operations in the grouping can be combined (or composed) to form a new operation in the grouping. This property resembles the closure criterion for groups, but note the distinction in language here: combining a pair of operations generates a new operation in the grouping. Because groups anticipate all possible combinations, the composition of elements within a group produces another element of the group. In other words, the group is a complete structure, but the grouping structure is actively built up “step by step” by coordinating pairs of concrete operations (Beth & Piaget, 1966, p. 173). 2. Reversibility: Here, reversibility refers to reversing the composition under the binary operation. Combined concrete operations can be separated again by reversing the binary operation used to combine them. For example, if we combine the concrete operations that determine the ordering relations A