Acquisition of Complex Arithmetic Skills and Higher-Order Mathematics Concepts

David C. Geary, Daniel B. Berch, Robert Ochsendorf, Kathleen Mann Koepke

• Publisher: Academic Press
• Year: 2017
• Print ISBN:
  9780128050866
• eBook ISBN:
  9780128133682
• Topic: Psychology
• Subtopic: Cognitive Psychology & Cognition
• Index: Psychology

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

Chapter 1

Insights from Cognitive Science on Mathematical Learning^*

David C. Geary^*

Daniel B. Berch^**

Robert J. Ochsendorf^†

Kathleen Mann Koepke^‡
* Psychological Sciences, University of Missouri, Columbia, MO, United States
^** Curry School of Education, University of Virginia, Charlottesville, VA, United States
^† Directorate for Education and Human Resources, National Science Foundation, Arlington, VA, United States
^‡ Eunice Kennedy Shriver, National Institute of Child Health and Human Development (NICHD), National Institutes of Health (NIH), Bethesda, MD, United States

Abstract

Research in the cognitive sciences has yielded unique and critical insights into how the human mind represents, processes, and learns mathematical concepts and procedures. The chapters throughout this volume illustrate these insights for complex arithmetic, rational numbers, algebra, geometry, and trigonometry. In this chapter, we review and discuss several basic themes, such as the importance of prior knowledge for learning that cut across most or all of the other chapters, and highlight several challenges to translating these findings into instructional approaches and policies.

Keywords

cognitive science

mathematical cognition

mathematical learning

development

instruction

education

For the most part, Volumes 1 and 2 of this series focused on basic nonsymbolic quantitative processing and symbolic numerical processing, including their evolutionary origins, early development, neural substrates, and genetic influences. As such, generally the most complex quantitative processing treated in the series thus far has been concerned with single-digit arithmetic. In contrast, the present volume covers higher-level mathematical processing and operations, including the development of multidigit arithmetic skills, learning to understand fractions and other rational number concepts and procedures, engaging in algebraic reasoning, acquiring geometric principles and competencies, and visualizing trigonometric expressions. As a consequence, the theoretical underpinnings of this work are more complex, multilayered, and wide-ranging in nature, drawing not only on some of the more rudimentary foundations of natural number processing, but also other domain-specific and domain-general skills needed for processing higher-order quantitative relations among multidigit Arabic numerals, rational numbers, number words, spatial relations, and abstract mathematical symbols.

In this first section, we provide the reader with the foundational components of cognitive theories that undergird research in these areas, and also preview the kinds of theoretical accounts our authors advance when interpreting the evidence they review. In the second and third respective sections, we overview how basic research in cognitive science informs studies of mathematical learning and how a few core principles and general empirical findings help to tie together the basic questions being addressed with the research described across the chapters of these volumes, and how the answers to these questions set the stage for future basic and applied (e.g., design of interventions) research.

On the nature of theories and models in cognitive psychology

Theories in the cognitive and behavioral sciences are composed of any more or less formalized sets of propositions (Marx, 1963). Theoretical propositions are statements—in the form of sentences or equations—of relations between constructs. Constructs, in turn, comprise symbolic representations of relations among variables, which consist of classes of objects, events, or their properties. For example, the construct of “memory” is frequently defined as the encoding (mental representation), storage, and retrieval of information supplied by the environment (e.g., phone numbers, names, facial expressions, or multiplication facts), and as theoretical constructs are by definition ...

Table of contents

• Cover
• Title page
• Table of Contents
• Copyright
• Contributors
• Foreword: Build It and They Will Come
• Preface
• Chapter 1: Insights from Cognitive Science on Mathematical Learning
• Part I: Complex Arithmetic Processing
• Part II: Rational Number Processing
• Part III: Algebraic, Geometric, and Trigonometric Concepts
• Part IV: Instructional Approaches
• Index