Piaget’s Genetic Epistemology for Mathematics Education Research 9783031473869, 9783031473852
The book provides an entry point for graduate students and other scholars interested in using the constructs of Piaget’s
125 10 22MB
English Year 2024
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
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