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E-raamat: Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts

Edited by (National Institute of Child Health and Human Development, NIH, Rockville, MD, USA), Edited by (University of Missouri, Columbia, MO, USA), Edited by (University of Virginia, Charlottesville, VA, USA), Edited by (National Science Foundation, Arlington, VA USA)
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Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts focuses on typical and atypical learning of complex arithmetic skills and higher-order math concepts. As part of the series Mathematical Cognition and Learning, this volume covers recent advances in the understanding of children’s developing competencies with whole-number arithmetic, fractions, and rational numbers. Each chapter covers these topics from multiple perspectives, including genetic disorders, cognition, instruction, and neural networks.

  • Covers innovative measures and recent methodological advances in mathematical thinking and learning
  • Contains contributions that improve instruction and education in these domains
  • Informs policy aimed at increasing the level of mathematical proficiency in the general public

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Comprehensive reference that focuses on the typical and atypical learning of complex arithmetic skills and higher-order math concepts
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(1)
The Role of Theories in Cognitive Psychology
2(2)
Theory Testing and Validation
4(1)
Methodological Considerations
4(1)
Why?
5(1)
What, When, How, and Who?
6(5)
Challenges for Instruction
11(2)
Conclusions and Future Directions
13(1)
References
14(7)
Part I Complex Arithmetic Processing
2 The Understanding of Additive and Multiplicative Arithmetic Concepts
Katherine M. Robinson
Introduction
21(1)
What is Conceptual Knowledge of Arithmetic?
22(1)
The Importance of Conceptual Knowledge
23(1)
A Brief History of Research on Conceptual Knowledge
23(3)
The Importance of Multiplicative Concepts and the State of Current Research
26(1)
Additive Versus Multiplicative Concepts
26(1)
The Inversion Concept
27(1)
The Associativity Concept
27(1)
Are Additive and Multiplicative Concepts the Same?
28(1)
Inversion
29(1)
Associativity
30(1)
Inversion Versus Associativity
31(2)
Individual Differences and Factors in the Use of Conceptually-Based Shortcuts
33(1)
Individual Differences
34(2)
Factors Relating to Conceptually-Based Shortcut Use
36(1)
Computational Skills and Age
36(1)
Working Memory
37(1)
Inhibition and Attention
37(1)
Attitudes
38(2)
Educational Experiences
40(1)
Conclusions and Future Directions
41(1)
References
42(5)
3 Arithmetic Word Problem Solving: The Role of Prior Knowledge
Catherine Thevenot
Introduction
47(1)
The Role of Daily Life Experience in Solving Arithmetic Word Problems
48(2)
The Role of Stereotypic Representations About Problem Solving in School
50(2)
The Role of Problem Schemata Stored in Long-Term Memory
52(6)
The Use of Schemata Versus Situation-Based Models
58(2)
How Can We Help Students
60(2)
Conclusions and Future Directions
62(1)
References
63(4)
4 Neurodevelopmental Disorders as Model Systems for Understanding Typical and Atypical Mathematical Development
Marcia A. Barnes
Kimberly P. Raghubar
Introduction
67(1)
Spina Bifida as a Model System for Understanding Mathematical Learning Disabilities
68(4)
Longitudinal Approaches to the Study of Mathematical Development and Disability
72(1)
Sources of Mathematical Disability
73(6)
Longitudinal Studies of Mathematical Cognition in Children with SBM and Their Typically Developing Peers
79(1)
Are Domain-General Cognitive Abilities Related to Number Knowledge?
79(2)
Longitudinal Mediation of School-Age Mathematics Achievement
81(2)
What is the Relation of Different Preschool Domain-General Cognitive Abilities to Different Mathematics Outcomes at School Age?
83(2)
Do the Longitudinal Mediators Differ for Math and Reading?
85(1)
What is the Relation of Early Domain-General Abilities and Domain-Specific Number Knowledge to Later Mathematical Achievement?
86(1)
What are the Implications of the Findings for Assessment and Intervention?
87(2)
Conclusions and Future Directions
89(1)
Acknowledgments
90(1)
References
90(12)
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(1)
Rational Numbers: A Challenge for Learners and for Mathematics Education
102(1)
The Interference of Natural Number Knowledge in Rational Number Tasks
103(1)
The Size of Rational Numbers
104(1)
The Effect of Arithmetic Operations
104(1)
The Dense Structure of Rational Numbers
104(1)
Representation of Numbers as an Intersecting Difficulty
105(1)
The Natural Number Bias
105(1)
Theoretical Frameworks for Studying the Natural Number Bias
106(1)
The Conceptual Change Perspective
106(2)
The Dual Process Perspective on Reasoning
108(1)
Combining the Conceptual Change Theory and Dual Process Perspective to Study Mathematical Thinking and Learning
109(1)
Overview of Our Studies Using Both Conceptual Change Theory and Dual Process Perspective
109(1)
Size
110(1)
Operations
111(1)
Density
112(1)
How are the Three Aspects Related to Each Other?
113(2)
Conclusions and Future Directions
115(2)
Future Directions
117(3)
References
120(6)
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(2)
Understanding of Fractions Involves Both Conceptual and Procedural Knowledge
128(1)
Fraction Development in Early Childhood
129(1)
Early Fraction Calculation Ability
129(1)
Equal Sharing
130(1)
Early Knowledge of Proportionality
130(1)
Early Misconceptions
130(1)
Fraction Development Between Third and Sixth Grade: Findings from the Delaware Longitudinal Study
131(1)
Predictors of Fraction Knowledge
132(2)
Growth in Fraction Magnitude Understanding
134(2)
Helping Students Who Struggle with Fractions
136(1)
Acknowledgment
137(1)
References
137(4)
7 Numbers as Mathematical Models: Modeling Relations and Magnitudes with Fractions and Decimals
Melissa DeWolf
Miriam Bassok
Keith J. Holyoak
Understanding Rational Numbers
141(1)
Introduction
141(1)
Prior Research on Magnitude Assessment and Misconceptions About Rational Numbers
142(1)
Student Misconceptions
142(1)
Magnitude Representations for Rational and Natural Numbers
143(1)
Relational Affordances of the Fraction Notation
144(1)
Using Mathematics to Model Relations
144(1)
Semantic Alignment
145(2)
Modeling with Rational Numbers
147(1)
Alignments Between Rational Numbers and Quantity Types
148(1)
Discrete/Continuous Ontological Distinction
148(1)
Modeling Discrete and Continuous Quantities with Fractions and Decimals
149(3)
Modeling Magnitude with Decimals
152(2)
Connections Between Rational Numbers and Other Math Concepts
154(1)
Multiplicative Reasoning and Fraction Understanding
154(2)
Differential Contributions of Magnitude and Relational Knowledge to Learning Algebra
156(2)
Conclusions and Future Directions
158(2)
References
160(7)
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(1)
Children's Difficulties with Mathematical Equivalence Problems
168(3)
Popular Accounts of Children's Mathematics Learning Difficulties
171(1)
The Symbol Misunderstanding Account
171(3)
The Deficient Working Memory System Account
174(4)
The Poor Number Knowledge Account
178(3)
The Change-Resistance Account
181(6)
Conclusions and Future Directions
187(1)
Acknowledgments
188(1)
References
188(9)
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(2)
Algebra and Earlier Mathematics Skills
199(1)
Relational Tasks
200(2)
Algebra and Arithmetic
202(1)
Arithmetic and Algebraic Word Problems in the Singapore Curriculum
203(2)
General Mathematics Achievement, Algebra, and Relations with Domain-General and Domain-Specific Influences
205(4)
The Present Study
209(1)
Study Design
210(2)
General Mathematical Achievement, Domain-Specific and Domain-General Precursors
212(1)
Mathematical Relational Skills and Arithmetic Word Problems
212(1)
Algebraic: Problems, Earlier Mathematical Skills, and Domain-General Capacities
213(1)
Conclusions and Future Directions
214(2)
Future Directions
216(1)
References
217(5)
10 Learning Geometry: The Development of Geometrical Concepts and the Role of Cognitive Processes
Irene C. Mammarella
David Giofre
Sara Caviola
Classical Studies on Geometry
222(1)
Core Intuitive Principles of Geometry
223(2)
Academic Achievement in Geometry
225(5)
The Development of Geometrical Knowledge
230(2)
Cognitive Processes Involved in Geometry
232(4)
Educational Implications
236(4)
Conclusions and Future Directions
240(1)
References
241(7)
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(4)
The Unit Circle as a Grounded Conceptual Structure for Trigonometry
252(2)
Preliminary Investigations
254(3)
Preliminary Study: Observing Use and Success of the Unit Circle
257(2)
Study 2: Comparing a Unit Circle Lesson to a Rules Lesson and Baseline Knowledge
259(2)
Challenges in Learning the Unit Circle
261(1)
Unit Circle Instruction for Students Without Prior Precalculus Trigonometry
262(2)
Internalizing the Unit Circle
264(1)
The Role of Epistemic Belief in Acquiring an Integrated Conceptual Representation
265(1)
Conclusions and Future Directions
266(1)
References
266(7)
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(1)
Short-Term, Researcher-Led Classroom Research
274(1)
Instructional Materials
274(2)
Studies on Comparing Methods
276(4)
Studies on Comparing Problems
280(3)
Summary of Researcher-Led Classroom Studies and Proposed Guidelines
283(1)
Year-Long Study Helping Teachers Use Comparison in Algebra I Classrooms
284(1)
Supplemental Curriculum Materials
285(3)
Implementation and Evaluation
288(2)
Discussion of Year-Long Study
290(1)
Conclusions and Future Directions
291(1)
Acknowledgments
291(1)
References
292(5)
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(2)
Scaffolding Principle
299(1)
Evidence from Laboratory Studies
300(1)
Evidence from Classroom Studies
300(1)
Recommendations for Further Research
300(1)
Distributed Practice Effect
301(1)
Evidence from Laboratory Studies
302(1)
Evidence from Classroom Studies
302(1)
Recommendations for Further Research
302(1)
Feedback Principle
303(1)
Evidence from Laboratory Studies
303(1)
Evidence from Classroom Studies
304(1)
Recommendations for Further Research
304(1)
Worked Example Principle
304(1)
Evidence from Laboratory Studies
305(1)
Evidence from Classroom Studies
305(1)
Recommendations for Further Research
306(1)
Interleaving Principle
306(1)
Evidence from Laboratory Studies
306(1)
Evidence from Classroom Studies
307(1)
Recommendations for Further Research
308(1)
Abstract and Concrete Representations Principles
308(1)
Evidence from Laboratory Studies
308(1)
Evidence from Classroom Studies
309(1)
Recommendations for Further Research
310(1)
Error Reflection Principle
310(1)
Evidence from Laboratory Studies
311(1)
Evidence from Classroom Studies
311(1)
Recommendations for Further Research
312(1)
Analogical Comparison Principle
312(1)
Evidence from Laboratory Studies
313(1)
Evidence from Classroom Studies
313(1)
Recommendations for Further Research
314(1)
Conclusions and Future Directions
314(2)
General Recommendations
316(1)
Acknowledgments
317(1)
References
317(10)
Index 327
David C. Geary is a cognitive developmental and evolutionary psychologist at the University of Missouri. He has wide ranging interests but his primary areas of research and scholarly work are childrens mathematical cognition and learning and Darwins sexual selection as largely but not solely related to human sex differences.

Professor Geary directed a 10-year longitudinal study of childrens mathematical development from kindergarten to ninth grade, with a focus on identifying the core deficits underlying learning disabilities and persistent low achievement in mathematics. The study was funded by the National Institutes of Health (US), including through a MERIT award to professor Geary. One result has been the identification of the school-entry number knowledge that predicts economically-relevant mathematical competencies in adolescence. As a follow-up, professor Geary is directing a second longitudinal study, funded by the National Science Foundation (US), to identify the preschool quantitative competencies that predict this school-entry number knowledge. Professor Geary has also published conceptual and theoretical articles on individual differences in childrens mathematical learning, as well as a book published by the American Psychological Association, Childrens mathematical development (1994); recently translated into Korean. Professor Geary has also contributed to applied and policy related work on this topic, serving, for instance, on the Presidents National Mathematics Advisory Panel, and chairing its learning processes task group.

Professor Gearys interests in evolution are reflected in two of his other books published by the American Psychological Association, The origin of mind: Evolution of brain, cognition, and general intelligence (2005), and Male, female: The evolution of human sex differences (1998, 2010 second edition). The corresponding empirical work ranges from the study of changes in brain volume during hominid evolution to human mate choices to hormonal responses to simulated (video game) competition. Professor Gearys current interests in this area follow from several of his collaborative studies on the effects of prenatal toxin exposure on sex differences in cognition and behavior in mice. Specifically, traits related to Darwins sexual selection are often exaggerated relative to other traits. These would include, for example, the bright plumage of the males of many species of bird that in turn is a good indicator of their behavioral and genetic health. These traits are particularly sensitive to environmental disruption, even in healthy individuals. Professor Gearys in progress book, The evolution of vulnerability, is focused on these traits in humans and how they can be used to identify at-risk populations and individuals.

Daniel B. Berch is Professor of Educational Psychology and Applied Developmental Science at the University of Virginias Curry School of Education. Prior to this position, he was Associate Dean for Research and Faculty Development at the Curry School. Before coming to the University of Virginia, Professor Berch served as Associate Chief of the Child Development and Behavior Branch at the National Institute of Child Health and Human Development, NIH. His previous federal service included a year spent as a Senior Research Associate at the U. S. Department of Education, advising the Assistant Secretary for Educational Research and Improvement. Professor Berch is a cognitive developmental psychologist with interests ranging from the development of numerical cognition and mathematical learning disabilities (MLD) to evolutionary perspectives on education. He has published articles on childrens magnitude representations, the development of number sense, and the role of working memory in MLD. He is senior editor of the book, Why is math so hard for some children? The nature and origins of mathematical learning difficulties and disabilities (co-edited by Michele Mazzocco).

Among other honors, he received the NIH Award of Merit, was elected Fellow of the American Psychological Associations Division of Experimental Psychology, served as an ex officio member of the U.S. Department of Educations National Mathematics Advisory Panel commissioned by President George W. Bush, was elected to the Evolution Institutes Scientific Advisory Board (and chairs its Education Subcommittee), was appointed to the AIM Academys Research Advisory Board, and served as a member of the Professional Advisory Board of the National Center for Learning Disabilities for six years. During the past several years, Professor Berch has been working on the implications of evolutionary theory for educational research and practice, publishing a book chapter on instructing evolved minds, serving as one of the Evolution Institutes primary organizers of a 2013 conference on evolutionary perspectives in educational research funded by the American Educational Research Association, and is co-author (with David Geary) of an article entitled Evolutionary Approaches to Understanding Childrens Academic Achievement” to be published in Wileys forthcoming online reference work, Emerging Trends in the Social and Behavioral Sciences.

Kathleen Mann Koepke, Ph.D., is Director of the Math and Science Cognition and Learning, Development & Disorders Program in the Child Devlopment and Behavior Branch (CDBB) of the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD), National Institutes of Health (NIH), Rockville, MD (kmk@nih.gov).

Dr. Mann Koepke oversees a research program focused on developing and supporting research and training initiatives to increase knowledge relevant to the development of math and science cogniton, reasoning, knowledge, and abilities, both in animals and in humans from birth through all years of formal education in diverse learners with and without disabilities. This research focus recently lead to her serving as co-Guest Editor of a special journal issue regarding the co-occurrence of math and reading disabilities (Mann Koepke, K and Miller, B. (Eds.) At the Intersection of Math & Reading Disabilities. Journal of Learning Disabilities. 2013: 46(6)).

She is a lifecourse developmental cognitive neuroscientist/psychologist with a passion to serve the cognitively challenged and/or disabled via promoting new and innovative basic research and theoretically-grounded evidence-based intervention strategies to maximuze function. Dr. Mann Koepke has served in the Division of Extramural Programs across the National Institute on Aging (NIA), the National Institute of Nursing Research (NINR), and now NICHD, overseeing research on cognitive and neurological development and disorders, including neurobiological, behavioral and caregiving research foci. She has served on numerous federal and national committees aimed at advancing research and services for young learners and persons with cognitive and/or physical disability or differences, co-authoring numerous calls for new research to close significant gaps, as well as peer-reviewed publications. Prior to coming to NIH, she was faculty in Neurology (Psychology) at Washington University-St. Louis School of Medicine where she managed the universitys Alzheimers Disease Research Center (supported in part by grant P50AG05681), and served as Director of its Education Core and its Rural Outreach Satellite. While there, expanding on her enthusiasm for the use of newly developed technologies as tools for cognition, she developed and regularly contributed to the first-ever freely available web-based online educational support system for anyone interested in Alzheimers disease (AD) and for formal and informal dementia patient caregivers; the cite has garnered numerous national and international awards. This early online educational service has been used & replicated around the globe as a model for online disease/disabilty-focused educational support service.