Piaget’s Genetic Epistemology for Mathematics Education Research

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