Development of Mathematical Cognition

Neural Substrates and Genetic Influences

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

• Publisher: Academic Press
• Year: 2015
• Print ISBN:
  9780128018712
• eBook ISBN:
  9780128019092
• Topic: Psychology
• Subtopic: Cognitive Psychology & Cognition
• Index: Psychology

Development of Mathematical Cognition: Neural Substrates and Genetic Influences reviews advances in extant imaging modalities and the application of brain stimulation techniques for improving mathematical learning. It goes on to explore the role genetics and environmental influences have in the development of math abilities and disabilities. Focusing on the neural substrates and genetic factors associated with both the typical and atypical development of mathematical thinking and learning, this second volume in the Mathematical Cognition and Learning series integrates the latest in innovative measures and methodological advances from the top researchers in the field. - Provides details about new progress made in the study of neural correlates of numerical and arithmetic cognition - Addresses recent work in quantitative and molecular genetics - Works to improve instruction in numerical, arithmetical, and algebraic thinking and learning - Informs policy to help increase the level of mathematical proficiency among the general public

Chapter 1

Introduction

How the Study of Neurobiological and Genetic Factors Can Enhance Our Understanding of Mathematical Cognitive Development

Daniel B. Berch¹; David C. Geary²; Kathleen Mann Koepke^{3 1} Curry School of Education, UVA Cognitive Science Program, University of Virginia, Charlottesville, Virginia, USA
² Department of Psychological Sciences, Interdisciplinary Neuroscience Program, University of Missouri, Columbia, Missouri, USA
³ NICHD/CDBB, Rockville, Maryland, USA

Abstract

In this introductory chapter, we examine some of the basic issues concerning the use of brain imaging methods and interpretations of the data they generate, as well as the ways in which these considerations may influence the study of mathematical cognitive development. We begin by providing a brief history of brain imaging research in this field, followed by a discussion of the importance of adopting a developmental perspective. Then we review various criticisms of functional magnetic resonance imaging (fMRI), the dominant technique employed by the authors in this volume, and summarize a recent analysis that reveals the major limitations of these claims. Next, we describe some specific ways in which neuroimaging can inform cognitive theories. Finally, we chronicle major developments in the field of genetics, describe some of the interpretive challenges associated with behavioral genetic designs, and make recommendations concerning how cognitive science and behavioral genetics can best move forward in the age of molecular genetics and neuroscience.

Keywords

Triple-code model

Brain imaging methods

Reverse inference

Modularity

Localization

Neurovascular changes

Adaptation paradigm

Multi-voxel pattern analysis

Developmental perspective

Behavioral and molecular genetics

Epigenetics

Introduction

Our understanding of developmental changes in mathematical cognition has advanced significantly over the past 40 plus years. A variety of innovative tasks have been used to study the intuitive number sense of preverbal human infants, toddlers, and preschoolers (see Geary, Berch, & Mann Koepke, 2015 for reviews), as well as the formal mathematical learning of school-age children and adolescents. The findings that have emerged from the use of these paradigms and the consistent effects they have generated are so numerous that it would be challenging to list them all here, much less to explain their significance with respect to achieving a comprehensive understanding of both the typical and atypical development of mathematical thinking and learning. That being said, it is worth providing the reader with a representative set of topics that have been studied to date. These include subitizing (i.e., the rapid and accurate apprehension of the quantity of small collections of items); magnitude comparison of nonsymbolic quantities (e.g., random dot arrays) and symbolic quantities (e.g., Arabic numerals); transcoding (i.e., translating from one numerical format to another); other relations between nonsymbolic and symbolic numerical skills; the development of counting skills, single-digit arithmetic, and multidigit arithmetic processing; the acquisition of place value; conceptual and procedural knowledge of fractions; proportional reasoning; mathematical equivalence; and the role of domain general processes (e.g., working memory, processing speed) in mathematical cognitive development. Furthermore, some findings have been so robust that they have attained the status of “effects,” such as the numerical distance effect (Moyer & Landauer, 1967); the problem-size effect (Ashcraft & Guillaume, 2009; Zbrodoff & Logan, 2005), the Spatial-Numerical Association of Response Codes (SNARC) effect (see Fischer & Shaki, 2014 for a review), and the operational momentum effect (also reviewed by Fischer & Shaki, 2014).

Neurobiological Perspectives on Mathematical Cognitive Development

Few contemporary researchers would question that developmental changes in mathematical thinking and learning are a reflection of changes in neurobiological functions, whether induced from maturation per se, environmental triggers (e.g., informal home activities, formal education), or some combination of both. But precisely what can the study of neural substrates and genetic influences bring to this field and how can we best characterize these potential contributions, outline their value-added to cognitive accounts alone, as well as delineate their limitations? In this chapter, we explore these issues to provide a foundation for the more specific treatments of circumscribed research topics that the reader will encounter in the ensuing chapters. Among other themes, we address some longstanding as well as more recent controversies concerning the uses of neuroimaging methods in general and functional magnetic resonance imaging (fMRI) in particular—both because the latter has been used for studying the neural correlates of mathematical development far more extensively than other techniques and because some of the issues are relevant to other imaging procedures. Our analysis of these concerns is designed to acquaint the reader with some of the critical assumptions underlying the use of such techniques and how they can inform cognitive theory, thereby prospectively advancing our understanding of mathematical cognitive development.

Using Neuroimaging Methods to Study Children's Mathematical Development

Ten to fifteen years ago, we knew comparatively little about the neural underpinnings of developmental changes in mathematical cognition. As such, it would have been difficult to even conceive of putting together an entire edited volume focusing on this topic. However, the technical, methodological, and statistical advances achieved in neuroimaging in recent years, along with the increasing availability of scanners for use by cognitive neuroscientists have contributed to an increasing growth in the number of empirical studies of mathematical cognition and development, meriting the broad as well as in-depth treatment of this topic that the present volume affords. That being said, as it is generally acknowledged that work in this area is still in its early stages, stepping back to take a larger perspective on this entire enterprise is warranted.

We begin by reviewing the comparatively brief history of this kind of research, after which we discuss the importance of taking into consideration differences between the developing and mature brain. Then we explore some of the assumptions made when interpreting the meaning of fMRI data with respect to both behavioral outcomes and cognitive theories, examine criticisms concerning the meaning and value of brain imaging data, review the kinds of inferences one can draw from brain imaging studies, and consider the extent to which more recent methodological developments in analyzing neuroimaging data could accelerate progress in this rapidly evolving field.

Mathematical Cognition and Development: Brain Structure and Function

Although it is sometimes suggested that brain research should provide the scientific foundation for children's education in mathematics and in other academic areas, it is too early to directly apply findings from studies of brain processes during mathematical reasoning to classroom teaching and learning. Yet promising research emerging from the field of cognitive neuroscience is permitting investigators to begin forging links between neurobiological functions and mathematical cognition. We present a few highlights in the history of this research and then touch on an issue that has been overlooked for much of this history, that is, the difference between the developing and the mature brain (Ansari, 2010).

A Brief History

Early research on the relation between brain functioning and mathematical competencies was restricted to the sequelae of brain injury due to disease (e.g., stroke) or trauma (e.g., gunshot wounds) and largely focused on number and arithmetic. As an example and based on a series of case studies of the results of brain injury, Henschen and Schaller concluded, “In a general sense, akalkulia (sic) is the inability to form combinations of numbers” (Henschen & Schaller, 1925, p. 232) and is often associated with “cipher blindness” (poor recognition of numerals) and damage to the angular gyrus. Henschen and Schaller also observed that following these injuries “an educated person will revert to simple means of expression similar to those of … stone folk” (Henschen & Schaller, 1925, p. 232); that is, they would thereafter rely on basic number words and the limited calculation abilities found in traditional populations. Their insight presaged Geary's (1995) distinction between evolved quantitative abilities and those dependent on formal education and Dehaene's number sense (2011; Dehaene, 1992; Dehaene & Cohen, 1995).

As is well known in this literature, Gerstmann (1940) defined a cluster of deficits associated with damage to the angular gyrus and adjacent regions of the occipital lobe that included calculation difficulties, along with finger agnosia (i.e., inability to recognize and differentiate different fingers), poor discrimination of the left and right side of the body, and often difficulties in understanding time (e.g., comprehending analog watches). The calculation difficulties included simple problems, as studied by Henschen and Schaller (1925), as well as a poor understanding of the meaning of numerals in different columns for more complex problems (e.g., 154 + 263, and not understanding that the “5” and “6” represent sets of 10). In his magnum opus, Higher Cortical Functions in Man, Luria (1980) described the same pattern of deficits associated with injury to the angular gyrus, and noted the importance of the prefrontal cortex for planning and enacting a problem-solving sequence. The latter would involve actually solving the problem 154 + 263, not simply understanding the magnitudes represented by the position of the columnar numerals; these processes were subsequently shown to be heavily dependent on working memory (Hitch, 1978). Luria also noted that temporal lobe lesions that disrupted language also disrupted arithmetical or other mathematical processes that were dependent on language, such as counting.

Dehaene and Cohen (1995; Dehaene, 1992) organized these early neuropsychological findings with their triple-code model of number and arithmetic processing. The model includes three forms of numerical and arith...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Contributors
• Foreword
• Preface
• Chapter 1: Introduction: How the Study of Neurobiological and Genetic Factors Can Enhance Our Understanding of Mathematical Cognitive Development
• Part I: Neural substrates
• Part II: Genetic Influences
• Index