Mathematical Cognition and Learning
Acquisition of Complex
Arithmetic Skills
and Higher-Order
Mathematics Concepts
Edited by
David C. Geary
Psychological Sciences
University of Missouri
Columbia, MO, United States
Daniel B. Berch
Curry School of Education
University of Virginia
Charlottesville, VA, United States
Robert J. Ochsendorf
Directorate for Education and Human Resources
National Science Foundation
Arlington, VA, United States
Kathleen Mann Koepke
Eunice Kennedy Shriver National Institute of Child
Health and Human Development (NICHD)
National Institutes of Health (NIH)
Bethesda, MD, United States
,Contents
Contributors xiii
Foreword: Build It and They Will Come xv
Robert S. Siegler
Preface xxi
1. Insights from Cognitive Science on Mathematical
Learning
David C. Geary, Daniel B. Berch, Robert J. Ochsendorf,
Kathleen Mann Koepke
On the Nature of Theories and Models in Cognitive Psychology 2
The Role of Theories in Cognitive Psychology 2
Theory Testing and Validation 4
Methodological Considerations 4
Why? 5
What, When, How, and Who? 6
Challenges for Instruction 11
Conclusions and Future Directions 13
References 14
Part I
Complex Arithmetic Processing
2. The Understanding of Additive and Multiplicative
Arithmetic Concepts
Katherine M. Robinson
Introduction 21
What is Conceptual Knowledge of Arithmetic? 22
The Importance of Conceptual Knowledge 23
A Brief History of Research on Conceptual Knowledge 23
The Importance of Multiplicative Concepts and the State
of Current Research 26
Additive Versus Multiplicative Concepts 26
The Inversion Concept 27
The Associativity Concept 27
v
,vi Contents
Are Additive and Multiplicative Concepts the Same? 28
Inversion 29
Associativity 30
Inversion Versus Associativity 31
Individual Differences and Factors in the Use of
Conceptually-Based Shortcuts 33
Individual Differences 34
Factors Relating to Conceptually-Based Shortcut Use 36
Computational Skills and Age 36
Working Memory 37
Inhibition and Attention 37
Attitudes 38
Educational Experiences 40
Conclusions and Future Directions 41
References 42
3. Arithmetic Word Problem Solving: The Role of Prior
Knowledge
Catherine Thevenot
Introduction 47
The Role of Daily Life Experience in Solving Arithmetic Word
Problems 48
The Role of Stereotypic Representations About Problem
Solving in School 50
The Role of Problem Schemata Stored in Long-Term Memory 52
The Use of Schemata Versus Situation-Based Models 58
How Can We Help Students 60
Conclusions and Future Directions 62
References 63
4. Neurodevelopmental Disorders as Model Systems
for Understanding Typical and Atypical Mathematical
Development
Marcia A. Barnes, Kimberly P. Raghubar
Introduction 67
Spina Bifida as a Model System for Understanding Mathematical
Learning Disabilities 68
Longitudinal Approaches to the Study of Mathematical
Development and Disability 72
Sources of Mathematical Disability 73
Longitudinal Studies of Mathematical Cognition in Children
with SBM and Their Typically Developing Peers 79
Are Domain-General Cognitive Abilities Related to
Number Knowledge? 79
Longitudinal Mediation of School-Age Mathematics
Achievement 81
, Contents vii
What is the Relation of Different Preschool Domain-General
Cognitive Abilities to Different Mathematics Outcomes at
School Age? 83
Do the Longitudinal Mediators Differ for Math and
Reading? 85
What is the Relation of Early Domain-General Abilities and
Domain-Specific Number Knowledge to Later Mathematical
Achievement? 86
What are the Implications of the Findings for Assessment
and Intervention? 87
Conclusions and Future Directions 89
Acknowledgments 90
References 90
Part II
Rational Number Processing
5. The Transition from Natural to Rational Number
Knowledge
Jo Van Hoof, Xenia Vamvakoussi, Wim Van Dooren,
Lieven Verschaffel
The Importance of Rational Numbers 102
Rational Numbers: A Challenge for Learners and for
Mathematics Education 102
The Interference of Natural Number Knowledge in Rational
Number Tasks 103
The Size of Rational Numbers 104
The Effect of Arithmetic Operations 104
The Dense Structure of Rational Numbers 104
Representation of Numbers as an Intersecting Difficulty 105
The Natural Number Bias 105
Theoretical Frameworks for Studying the Natural
Number Bias 106
The Conceptual Change Perspective 106
The Dual Process Perspective on Reasoning 108
Combining the Conceptual Change Theory and Dual
Process Perspective to Study Mathematical Thinking
and Learning 109
Overview of Our Studies Using Both Conceptual Change
Theory and Dual Process Perspective 109
Size 110
Operations 111
Density 112
How are the Three Aspects Related to Each Other? 113
Conclusions and Future Directions 115
Future Directions 117
References 120
,viii Contents
6. Fraction Development in Children: Importance of
Building Numerical Magnitude Understanding
Nancy C. Jordan, Jessica Rodrigues, Nicole Hansen, Ilyse Resnick
Integrated Theory of Numerical Development 126
Understanding of Fractions Involves Both Conceptual and
Procedural Knowledge 128
Fraction Development in Early Childhood 129
Early Fraction Calculation Ability 129
Equal Sharing 130
Early Knowledge of Proportionality 130
Early Misconceptions 130
Fraction Development Between Third and Sixth Grade:
Findings from the Delaware Longitudinal Study 131
Predictors of Fraction Knowledge 132
Growth in Fraction Magnitude Understanding 134
Helping Students Who Struggle with Fractions 136
Acknowledgment 137
References 137
7. Numbers as Mathematical Models: Modeling Relations
and Magnitudes with Fractions and Decimals
Melissa DeWolf, Miriam Bassok, Keith J. Holyoak
Understanding Rational Numbers 141
Introduction 141
Prior Research on Magnitude Assessment and Misconceptions
About Rational Numbers 142
Student Misconceptions 142
Magnitude Representations for Rational and Natural
Numbers 143
Relational Affordances of the Fraction Notation 144
Using Mathematics to Model Relations 144
Semantic Alignment 145
Modeling with Rational Numbers 147
Alignments Between Rational Numbers and Quantity Types 148
Discrete/Continuous Ontological Distinction 148
Modeling Discrete and Continuous Quantities with
Fractions and Decimals 149
Modeling Magnitude with Decimals 152
Connections Between Rational Numbers and Other Math
Concepts 154
Multiplicative Reasoning and Fraction Understanding 154
Differential Contributions of Magnitude and Relational
Knowledge to Learning Algebra 156
Conclusions and Future Directions 158
References 160
, Contents ix
Part III
Algebraic, Geometric, and Trigonometric Concepts
8. Understanding Children’s Difficulties with Mathematical
Equivalence
Nicole M. McNeil, Caroline Byrd Hornburg, Mary Wagner Fuhs,
Connor D. O’Rear
Introduction 167
Children’s Difficulties with Mathematical Equivalence
Problems 168
Popular Accounts of Children’s Mathematics Learning
Difficulties 171
The Symbol Misunderstanding Account 171
The Deficient Working Memory System Account 174
The Poor Number Knowledge Account 178
The Change-Resistance Account 181
Conclusions and Future Directions 187
Acknowledgments 188
References 188
9. Learning and Solving More Complex Problems:
The Roles of Working Memory, Updating, and Prior
Skills for General Mathematical Achievement
and Algebra
Kerry Lee, Swee Fong Ng, Rebecca Bull
Introduction 197
Algebra and Earlier Mathematics Skills 199
Relational Tasks 200
Algebra and Arithmetic 202
Arithmetic and Algebraic Word Problems in the Singapore
Curriculum 203
General Mathematics Achievement, Algebra, and
Relations with Domain-General and Domain-Specific
Influences 205
The Present Study 209
Study Design 210
General Mathematical Achievement, Domain-Specific
and Domain-General Precursors 212
Mathematical Relational Skills and Arithmetic Word
Problems 212
Algebraic Problems, Earlier Mathematical Skills, and
Domain-General Capacities 213
Conclusions and Future Directions 214
Future Directions 216
References 217
,x Contents
10. Learning Geometry: The Development of Geometrical
Concepts and the Role of Cognitive Processes
Irene C. Mammarella, David Giofrè, Sara Caviola
Classical Studies on Geometry 222
Core Intuitive Principles of Geometry 223
Academic Achievement in Geometry 225
The Development of Geometrical Knowledge 230
Cognitive Processes Involved in Geometry 232
Educational Implications 236
Conclusions and Future Directions 240
References 241
11. The Unit Circle as a Grounded Conceptual Structure
in Precalculus Trigonometry
Kevin W. Mickey, James L. McClelland
Grounded Conceptual Structures in Mathematical
Cognition 248
The Unit Circle as a Grounded Conceptual Structure
for Trigonometry 252
Preliminary Investigations 254
Preliminary Study: Observing Use and Success of the
Unit Circle 257
Study 2: Comparing a Unit Circle Lesson to a Rules Lesson
and Baseline Knowledge 259
Challenges in Learning the Unit Circle 261
Unit Circle Instruction for Students Without Prior
Precalculus Trigonometry 262
Internalizing the Unit Circle 264
The Role of Epistemic Belief in Acquiring an Integrated
Conceptual Representation 265
Conclusions and Future Directions 266
References 266
Part IV
Instructional Approaches
12. The Power of Comparison in Mathematics Instruction:
Experimental Evidence from Classrooms
Bethany Rittle-Johnson, Jon R. Star, Kelley Durkin
Introduction 273
Short-Term, Researcher-Led Classroom Research 274
Instructional Materials 274
Studies on Comparing Methods 276
, Contents xi
Studies on Comparing Problems 280
Summary of Researcher-Led Classroom Studies and
Proposed Guidelines 283
Year-Long Study Helping Teachers Use Comparison
in Algebra I Classrooms 284
Supplemental Curriculum Materials 285
Implementation and Evaluation 288
Discussion of Year-Long Study 290
Conclusions and Future Directions 291
Acknowledgments 291
References 292
13. Evidence for Cognitive Science Principles that Impact
Learning in Mathematics
Julie L. Booth, Kelly M. McGinn, Christina Barbieri,
Kreshnik N. Begolli, Briana Chang, Dana Miller-Cotto,
Laura K. Young, Jodi L. Davenport
Introduction 297
Scaffolding Principle 299
Evidence from Laboratory Studies 300
Evidence from Classroom Studies 300
Recommendations for Further Research 300
Distributed Practice Effect 301
Evidence from Laboratory Studies 302
Evidence from Classroom Studies 302
Recommendations for Further Research 302
Feedback Principle 303
Evidence from Laboratory Studies 303
Evidence from Classroom Studies 304
Recommendations for Further Research 304
Worked Example Principle 304
Evidence from Laboratory Studies 305
Evidence from Classroom Studies 305
Recommendations for Further Research 306
Interleaving Principle 306
Evidence from Laboratory Studies 306
Evidence from Classroom Studies 307
Recommendations for Further Research 308
Abstract and Concrete Representations Principles 308
Evidence from Laboratory Studies 308
Evidence from Classroom Studies 309
Recommendations for Further Research 310
Error Reflection Principle 310
Evidence from Laboratory Studies 311
Evidence from Classroom Studies 311
Recommendations for Further Research 312
,xii Contents
Analogical Comparison Principle 312
Evidence from Laboratory Studies 313
Evidence from Classroom Studies 313
Recommendations for Further Research 314
Conclusions and Future Directions 314
General Recommendations 316
Acknowledgments 317
References 317
Index 327
, Contributors
Christina Barbieri, University of Delaware, Newark, DE, United States
Marcia A. Barnes, Department of Special Education & Meadows Center for
Preventing Educational Risk, University of Texas, Austin, TX, United States
Miriam Bassok, Department of Psychology, University of Washington, Seattle,
WA, United States
Kreshnik N. Begolli, Temple University, Philadelphia, PA, United States
Daniel B. Berch, Curry School of Education, University of Virginia, Charlottesville,
VA, United States
Julie L. Booth, Temple University, Philadelphia, PA, United States
Rebecca Bull, National Institute of Education, Singapore, Singapore
Sara Caviola, Department of Psychology, University of Cambridge, Cambridge,
United Kingdom
Briana Chang, Temple University, Philadelphia, PA, United States
Jodi L. Davenport, WestEd, San Francisco, CA, United States
Melissa DeWolf, Department of Psychology, University of California, Los Angeles,
CA, United States
Kelley Durkin, Peabody Research Institute, Vanderbilt University, Nashville,
TN, United States
Mary Wagner Fuhs, University of Dayton, Dayton, OH, United States
David C. Geary, Psychological Sciences, University of Missouri, Columbia,
MO, United States
David Giofrè, Liverpool John Moores University, Natural Sciences and Psychology,
Liverpool, United Kingdom
Nicole Hansen, Fairleigh Dickinson University, Teaneck, NJ, United States
Keith J. Holyoak, Department of Psychology, University of California, Los Angeles,
CA, United States
Caroline Byrd Hornburg, University of Notre Dame, Notre Dame, IN, United States
Nancy C. Jordan, University of Delaware, Newark, DE, United States
Kathleen Mann Koepke, Eunice Kennedy Shriver, National Institute of Child Health
and Human Development (NICHD), National Institutes of Health (NIH), Bethesda,
MD, United States
xiii
Acquisition of Complex
Arithmetic Skills
and Higher-Order
Mathematics Concepts
Edited by
David C. Geary
Psychological Sciences
University of Missouri
Columbia, MO, United States
Daniel B. Berch
Curry School of Education
University of Virginia
Charlottesville, VA, United States
Robert J. Ochsendorf
Directorate for Education and Human Resources
National Science Foundation
Arlington, VA, United States
Kathleen Mann Koepke
Eunice Kennedy Shriver National Institute of Child
Health and Human Development (NICHD)
National Institutes of Health (NIH)
Bethesda, MD, United States
,Contents
Contributors xiii
Foreword: Build It and They Will Come xv
Robert S. Siegler
Preface xxi
1. Insights from Cognitive Science on Mathematical
Learning
David C. Geary, Daniel B. Berch, Robert J. Ochsendorf,
Kathleen Mann Koepke
On the Nature of Theories and Models in Cognitive Psychology 2
The Role of Theories in Cognitive Psychology 2
Theory Testing and Validation 4
Methodological Considerations 4
Why? 5
What, When, How, and Who? 6
Challenges for Instruction 11
Conclusions and Future Directions 13
References 14
Part I
Complex Arithmetic Processing
2. The Understanding of Additive and Multiplicative
Arithmetic Concepts
Katherine M. Robinson
Introduction 21
What is Conceptual Knowledge of Arithmetic? 22
The Importance of Conceptual Knowledge 23
A Brief History of Research on Conceptual Knowledge 23
The Importance of Multiplicative Concepts and the State
of Current Research 26
Additive Versus Multiplicative Concepts 26
The Inversion Concept 27
The Associativity Concept 27
v
,vi Contents
Are Additive and Multiplicative Concepts the Same? 28
Inversion 29
Associativity 30
Inversion Versus Associativity 31
Individual Differences and Factors in the Use of
Conceptually-Based Shortcuts 33
Individual Differences 34
Factors Relating to Conceptually-Based Shortcut Use 36
Computational Skills and Age 36
Working Memory 37
Inhibition and Attention 37
Attitudes 38
Educational Experiences 40
Conclusions and Future Directions 41
References 42
3. Arithmetic Word Problem Solving: The Role of Prior
Knowledge
Catherine Thevenot
Introduction 47
The Role of Daily Life Experience in Solving Arithmetic Word
Problems 48
The Role of Stereotypic Representations About Problem
Solving in School 50
The Role of Problem Schemata Stored in Long-Term Memory 52
The Use of Schemata Versus Situation-Based Models 58
How Can We Help Students 60
Conclusions and Future Directions 62
References 63
4. Neurodevelopmental Disorders as Model Systems
for Understanding Typical and Atypical Mathematical
Development
Marcia A. Barnes, Kimberly P. Raghubar
Introduction 67
Spina Bifida as a Model System for Understanding Mathematical
Learning Disabilities 68
Longitudinal Approaches to the Study of Mathematical
Development and Disability 72
Sources of Mathematical Disability 73
Longitudinal Studies of Mathematical Cognition in Children
with SBM and Their Typically Developing Peers 79
Are Domain-General Cognitive Abilities Related to
Number Knowledge? 79
Longitudinal Mediation of School-Age Mathematics
Achievement 81
, Contents vii
What is the Relation of Different Preschool Domain-General
Cognitive Abilities to Different Mathematics Outcomes at
School Age? 83
Do the Longitudinal Mediators Differ for Math and
Reading? 85
What is the Relation of Early Domain-General Abilities and
Domain-Specific Number Knowledge to Later Mathematical
Achievement? 86
What are the Implications of the Findings for Assessment
and Intervention? 87
Conclusions and Future Directions 89
Acknowledgments 90
References 90
Part II
Rational Number Processing
5. The Transition from Natural to Rational Number
Knowledge
Jo Van Hoof, Xenia Vamvakoussi, Wim Van Dooren,
Lieven Verschaffel
The Importance of Rational Numbers 102
Rational Numbers: A Challenge for Learners and for
Mathematics Education 102
The Interference of Natural Number Knowledge in Rational
Number Tasks 103
The Size of Rational Numbers 104
The Effect of Arithmetic Operations 104
The Dense Structure of Rational Numbers 104
Representation of Numbers as an Intersecting Difficulty 105
The Natural Number Bias 105
Theoretical Frameworks for Studying the Natural
Number Bias 106
The Conceptual Change Perspective 106
The Dual Process Perspective on Reasoning 108
Combining the Conceptual Change Theory and Dual
Process Perspective to Study Mathematical Thinking
and Learning 109
Overview of Our Studies Using Both Conceptual Change
Theory and Dual Process Perspective 109
Size 110
Operations 111
Density 112
How are the Three Aspects Related to Each Other? 113
Conclusions and Future Directions 115
Future Directions 117
References 120
,viii Contents
6. Fraction Development in Children: Importance of
Building Numerical Magnitude Understanding
Nancy C. Jordan, Jessica Rodrigues, Nicole Hansen, Ilyse Resnick
Integrated Theory of Numerical Development 126
Understanding of Fractions Involves Both Conceptual and
Procedural Knowledge 128
Fraction Development in Early Childhood 129
Early Fraction Calculation Ability 129
Equal Sharing 130
Early Knowledge of Proportionality 130
Early Misconceptions 130
Fraction Development Between Third and Sixth Grade:
Findings from the Delaware Longitudinal Study 131
Predictors of Fraction Knowledge 132
Growth in Fraction Magnitude Understanding 134
Helping Students Who Struggle with Fractions 136
Acknowledgment 137
References 137
7. Numbers as Mathematical Models: Modeling Relations
and Magnitudes with Fractions and Decimals
Melissa DeWolf, Miriam Bassok, Keith J. Holyoak
Understanding Rational Numbers 141
Introduction 141
Prior Research on Magnitude Assessment and Misconceptions
About Rational Numbers 142
Student Misconceptions 142
Magnitude Representations for Rational and Natural
Numbers 143
Relational Affordances of the Fraction Notation 144
Using Mathematics to Model Relations 144
Semantic Alignment 145
Modeling with Rational Numbers 147
Alignments Between Rational Numbers and Quantity Types 148
Discrete/Continuous Ontological Distinction 148
Modeling Discrete and Continuous Quantities with
Fractions and Decimals 149
Modeling Magnitude with Decimals 152
Connections Between Rational Numbers and Other Math
Concepts 154
Multiplicative Reasoning and Fraction Understanding 154
Differential Contributions of Magnitude and Relational
Knowledge to Learning Algebra 156
Conclusions and Future Directions 158
References 160
, Contents ix
Part III
Algebraic, Geometric, and Trigonometric Concepts
8. Understanding Children’s Difficulties with Mathematical
Equivalence
Nicole M. McNeil, Caroline Byrd Hornburg, Mary Wagner Fuhs,
Connor D. O’Rear
Introduction 167
Children’s Difficulties with Mathematical Equivalence
Problems 168
Popular Accounts of Children’s Mathematics Learning
Difficulties 171
The Symbol Misunderstanding Account 171
The Deficient Working Memory System Account 174
The Poor Number Knowledge Account 178
The Change-Resistance Account 181
Conclusions and Future Directions 187
Acknowledgments 188
References 188
9. Learning and Solving More Complex Problems:
The Roles of Working Memory, Updating, and Prior
Skills for General Mathematical Achievement
and Algebra
Kerry Lee, Swee Fong Ng, Rebecca Bull
Introduction 197
Algebra and Earlier Mathematics Skills 199
Relational Tasks 200
Algebra and Arithmetic 202
Arithmetic and Algebraic Word Problems in the Singapore
Curriculum 203
General Mathematics Achievement, Algebra, and
Relations with Domain-General and Domain-Specific
Influences 205
The Present Study 209
Study Design 210
General Mathematical Achievement, Domain-Specific
and Domain-General Precursors 212
Mathematical Relational Skills and Arithmetic Word
Problems 212
Algebraic Problems, Earlier Mathematical Skills, and
Domain-General Capacities 213
Conclusions and Future Directions 214
Future Directions 216
References 217
,x Contents
10. Learning Geometry: The Development of Geometrical
Concepts and the Role of Cognitive Processes
Irene C. Mammarella, David Giofrè, Sara Caviola
Classical Studies on Geometry 222
Core Intuitive Principles of Geometry 223
Academic Achievement in Geometry 225
The Development of Geometrical Knowledge 230
Cognitive Processes Involved in Geometry 232
Educational Implications 236
Conclusions and Future Directions 240
References 241
11. The Unit Circle as a Grounded Conceptual Structure
in Precalculus Trigonometry
Kevin W. Mickey, James L. McClelland
Grounded Conceptual Structures in Mathematical
Cognition 248
The Unit Circle as a Grounded Conceptual Structure
for Trigonometry 252
Preliminary Investigations 254
Preliminary Study: Observing Use and Success of the
Unit Circle 257
Study 2: Comparing a Unit Circle Lesson to a Rules Lesson
and Baseline Knowledge 259
Challenges in Learning the Unit Circle 261
Unit Circle Instruction for Students Without Prior
Precalculus Trigonometry 262
Internalizing the Unit Circle 264
The Role of Epistemic Belief in Acquiring an Integrated
Conceptual Representation 265
Conclusions and Future Directions 266
References 266
Part IV
Instructional Approaches
12. The Power of Comparison in Mathematics Instruction:
Experimental Evidence from Classrooms
Bethany Rittle-Johnson, Jon R. Star, Kelley Durkin
Introduction 273
Short-Term, Researcher-Led Classroom Research 274
Instructional Materials 274
Studies on Comparing Methods 276
, Contents xi
Studies on Comparing Problems 280
Summary of Researcher-Led Classroom Studies and
Proposed Guidelines 283
Year-Long Study Helping Teachers Use Comparison
in Algebra I Classrooms 284
Supplemental Curriculum Materials 285
Implementation and Evaluation 288
Discussion of Year-Long Study 290
Conclusions and Future Directions 291
Acknowledgments 291
References 292
13. Evidence for Cognitive Science Principles that Impact
Learning in Mathematics
Julie L. Booth, Kelly M. McGinn, Christina Barbieri,
Kreshnik N. Begolli, Briana Chang, Dana Miller-Cotto,
Laura K. Young, Jodi L. Davenport
Introduction 297
Scaffolding Principle 299
Evidence from Laboratory Studies 300
Evidence from Classroom Studies 300
Recommendations for Further Research 300
Distributed Practice Effect 301
Evidence from Laboratory Studies 302
Evidence from Classroom Studies 302
Recommendations for Further Research 302
Feedback Principle 303
Evidence from Laboratory Studies 303
Evidence from Classroom Studies 304
Recommendations for Further Research 304
Worked Example Principle 304
Evidence from Laboratory Studies 305
Evidence from Classroom Studies 305
Recommendations for Further Research 306
Interleaving Principle 306
Evidence from Laboratory Studies 306
Evidence from Classroom Studies 307
Recommendations for Further Research 308
Abstract and Concrete Representations Principles 308
Evidence from Laboratory Studies 308
Evidence from Classroom Studies 309
Recommendations for Further Research 310
Error Reflection Principle 310
Evidence from Laboratory Studies 311
Evidence from Classroom Studies 311
Recommendations for Further Research 312
,xii Contents
Analogical Comparison Principle 312
Evidence from Laboratory Studies 313
Evidence from Classroom Studies 313
Recommendations for Further Research 314
Conclusions and Future Directions 314
General Recommendations 316
Acknowledgments 317
References 317
Index 327
, Contributors
Christina Barbieri, University of Delaware, Newark, DE, United States
Marcia A. Barnes, Department of Special Education & Meadows Center for
Preventing Educational Risk, University of Texas, Austin, TX, United States
Miriam Bassok, Department of Psychology, University of Washington, Seattle,
WA, United States
Kreshnik N. Begolli, Temple University, Philadelphia, PA, United States
Daniel B. Berch, Curry School of Education, University of Virginia, Charlottesville,
VA, United States
Julie L. Booth, Temple University, Philadelphia, PA, United States
Rebecca Bull, National Institute of Education, Singapore, Singapore
Sara Caviola, Department of Psychology, University of Cambridge, Cambridge,
United Kingdom
Briana Chang, Temple University, Philadelphia, PA, United States
Jodi L. Davenport, WestEd, San Francisco, CA, United States
Melissa DeWolf, Department of Psychology, University of California, Los Angeles,
CA, United States
Kelley Durkin, Peabody Research Institute, Vanderbilt University, Nashville,
TN, United States
Mary Wagner Fuhs, University of Dayton, Dayton, OH, United States
David C. Geary, Psychological Sciences, University of Missouri, Columbia,
MO, United States
David Giofrè, Liverpool John Moores University, Natural Sciences and Psychology,
Liverpool, United Kingdom
Nicole Hansen, Fairleigh Dickinson University, Teaneck, NJ, United States
Keith J. Holyoak, Department of Psychology, University of California, Los Angeles,
CA, United States
Caroline Byrd Hornburg, University of Notre Dame, Notre Dame, IN, United States
Nancy C. Jordan, University of Delaware, Newark, DE, United States
Kathleen Mann Koepke, Eunice Kennedy Shriver, National Institute of Child Health
and Human Development (NICHD), National Institutes of Health (NIH), Bethesda,
MD, United States
xiii
