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Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts - 1st Edition - ISBN: 9780128050866, 9780128133682

Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts, Volume 3

1st Edition

Editors: David Geary Daniel Berch Robert Ochsendorf Kathleen Mann Koepke
eBook ISBN: 9780128133682
Hardcover ISBN: 9780128050866
Imprint: Academic Press
Published Date: 22nd December 2016
Page Count: 360
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Table of Contents

  • Contributors
  • Foreword: Build It and They Will Come
  • Preface
  • Chapter 1: Insights from Cognitive Science on Mathematical Learning
    • Abstract
    • On the nature of theories and models in cognitive psychology
    • Methodological considerations
    • Challenges for instruction
    • Conclusions and future directions
  • Part I: Complex Arithmetic Processing
    • Chapter 2: The Understanding of Additive and Multiplicative Arithmetic Concepts
      • Abstract
      • Introduction
      • What is conceptual knowledge of arithmetic?
      • The importance of conceptual knowledge
      • A brief history of research on conceptual knowledge
      • The importance of multiplicative concepts and the state of current research
      • Additive versus multiplicative concepts
      • The inversion concept
      • The associativity concept
      • Are additive and multiplicative concepts the same?
      • Individual differences and factors in the use of conceptually-based shortcuts
      • Individual differences
      • Factors relating to conceptually-based shortcut use
      • Conclusions and future directions
    • Chapter 3: Arithmetic Word Problem Solving: The Role of Prior Knowledge
      • Abstract
      • Introduction
      • The role of daily life experience in solving arithmetic word problems
      • The role of stereotypic representations about problem solving in school
      • The role of problem schemata stored in long-term memory
      • The use of schemata versus situation-based models
      • How can we help students
      • Conclusions and future directions
    • Chapter 4: Neurodevelopmental Disorders as Model Systems for Understanding Typical and Atypical Mathematical Development
      • Abstract
      • Introduction
      • Spina bifida as a model system for understanding mathematical learning disabilities
      • Longitudinal approaches to the study of mathematical development and disability
      • Longitudinal studies of mathematical cognition in children with SBM and their typically developing peers
      • Conclusions and future directions
      • Acknowledgments
  • Part II: Rational Number Processing
    • Chapter 5: The Transition from Natural to Rational Number Knowledge
      • Abstract
      • The importance of rational numbers
      • Rational numbers: a challenge for learners and for mathematics education
      • The interference of natural number knowledge in rational number tasks
      • The natural number bias
      • Theoretical frameworks for studying the natural number bias
      • Overview of our studies using both conceptual change theory and dual process perspective
      • Conclusions and future directions
    • Chapter 6: Fraction Development in Children: Importance of Building Numerical Magnitude Understanding
      • Abstract
      • Integrated theory of numerical development
      • Understanding of fractions involves both conceptual and procedural knowledge
      • Fraction development in early childhood
      • Fraction development between third and sixth grade: findings from the delaware longitudinal study
      • Helping students who struggle with fractions
      • Acknowledgment
    • Chapter 7: Numbers as Mathematical Models: Modeling Relations and Magnitudes with Fractions and Decimals
      • Abstract
      • Understanding rational numbers
      • Prior research on magnitude assessment and misconceptions about rational numbers
      • Using mathematics to model relations
      • Alignments between rational numbers and quantity types
      • Connections between rational numbers and other math concepts
      • Conclusions and future directions
  • Part III: Algebraic, Geometric, and Trigonometric Concepts
    • Chapter 8: Understanding Children’s Difficulties with Mathematical Equivalence
      • Abstract
      • Introduction
      • Children’s difficulties with mathematical equivalence problems
      • Popular accounts of children’s mathematics learning difficulties
      • The change-resistance account
      • Conclusions and future directions
      • Acknowledgments
    • Chapter 9: Learning and Solving More Complex Problems: The Roles of Working Memory, Updating, and Prior Skills for General Mathematical Achievement and Algebra
      • Abstract
      • Introduction
      • Algebra and earlier mathematics skills
      • General mathematics achievement, algebra, and relations with domain-general and domain-specific influences
      • The present study
      • Conclusions and future directions
    • Chapter 10: Learning Geometry: the Development of Geometrical Concepts and the Role of Cognitive Processes
      • Abstract
      • Classical studies on geometry
      • Core intuitive principles of geometry
      • Academic achievement in geometry
      • The development of geometrical knowledge
      • Cognitive processes involved in geometry
      • Educational implications
      • Conclusions and future directions
    • Chapter 11: The Unit Circle as a Grounded Conceptual Structure in Precalculus Trigonometry
      • Abstract
      • Grounded conceptual structures in mathematical cognition
      • Preliminary investigations
      • Preliminary study: observing use and success of the unit circle
      • Study 2: comparing a unit circle lesson to a rules lesson and baseline knowledge
      • Challenges in learning the unit circle
      • Conclusions and future directions
  • Part IV: Instructional Approaches
    • Chapter 12: The Power of Comparison in Mathematics Instruction: Experimental Evidence From Classrooms
      • Abstract
      • Introduction
      • Short-term, researcher-led classroom research
      • Year-long study helping teachers use comparison in algebra I classrooms
      • Conclusions and future directions
      • Acknowledgments
    • Chapter 13: Evidence for Cognitive Science Principles that Impact Learning in Mathematics
      • Abstract
      • Introduction
      • Scaffolding principle
      • Distributed practice effect
      • Feedback principle
      • Worked example principle
      • Interleaving principle
      • Abstract and concrete representations principles
      • Error reflection principle
      • Analogical comparison principle
      • Conclusions and future directions
      • Acknowledgments
  • Index


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.

Key Features

  • 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


Academics/researchers, graduate and undergraduate students specializing in the following disciplines: cognitive psychology; infant cognition; cognitive neuroscience; behavioral genetics; educational psychology; early childhood education; and special education


No. of pages:
© Academic Press 2017
22nd December 2016
Academic Press
eBook ISBN:
Hardcover ISBN:

Ratings and Reviews

About the Editors

David Geary

David Geary

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 children’s mathematical cognition and learning and Darwin’s sexual selection as largely but not solely related to human sex differences.

Professor Geary directed a 10-year longitudinal study of children’s 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 children’s mathematical learning, as well as a book published by the American Psychological Association, Children’s 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 President’s National Mathematics Advisory Panel, and chairing it’s learning processes task group.

Professor Geary’s 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 Geary’s 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 Darwin’s 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 Geary’s 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.

Affiliations and Expertise

University of Missouri, Columbia, MO, USA

Daniel Berch

Daniel Berch

Daniel B. Berch is Professor of Educational Psychology and Applied Developmental Science at the University of Virginia’s 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 children’s 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 Association’s Division of Experimental Psychology, served as an ex officio member of the U.S. Department of Education’s National Mathematics Advisory Panel commissioned by President George W. Bush, was elected to the Evolution Institute’s Scientific Advisory Board (and chair’s its Education Subcommittee), was appointed to the AIM Academy’s 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 Institute’s 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 Children’s Academic Achievement” to be published in Wiley’s forthcoming online reference work, Emerging Trends in the Social and Behavioral Sciences.

Affiliations and Expertise

University of Virginia, Charlottesville, VA, USA

Robert Ochsendorf

Affiliations and Expertise

National Science Foundation, Arlington, VA USA

Kathleen Mann Koepke

Kathleen Mann Koepke

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 (

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 university’s Alzheimer’s 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 Alzheimer’s 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.

Affiliations and Expertise

National Institute of Child Health and Human Development, NIH, Rockville, MD, USA