Chapter Three - Foundations of Children's Numerical and Mathematical Skills: The Roles of Symbolic and Nonsymbolic Representations of Numerical Magnitude

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Abstract

Numerical and mathematical skills are critical predictors of academic success. The last three decades have seen a substantial growth in our understanding of how the human mind and brain represent and process numbers. In particular, research has shown that we share with animals the ability to represent numerical magnitude (the total number of items in a set) and that preverbal infants can process numerical magnitude. Further research has shown that similar processing signatures characterize numerical magnitude processing across species and developmental time. These findings suggest that an approximate system for nonsymbolic (e.g., dot arrays) numerical magnitude representation serves as the basis for the acquisition of cultural, symbolic (e.g., Arabic numerals) representations of numerical magnitude. This chapter explores this hypothesis by reviewing studies that have examined the relation between individual differences in nonsymbolic numerical magnitude processing and symbolic math abilities (e.g., arithmetic). Furthermore, we examine the extent to which the available literature provides strong evidence for a link between symbolic and nonsymbolic representations of numerical magnitude at the behavioral and neural levels of analysis. We conclude that claims that symbolic number abilities are grounded in the approximate system for the nonsymbolic representation of numerical magnitude are not strongly supported by the available evidence. Alternative models and future research directions are discussed.

Introduction

Numerical information informs our everyday behavior. Consider a glance at the alarm clock in the morning, counting change in the line-up at the coffee shop, or reading about the latest election polls—each of these common situations places demands on the ability to process numerical information. Research has shown that the ability to process numbers and use them in mathematical operations (such as calculation) is a critical predictor of an individual's economic and social success (e.g., Bynner & Parsons, 1997). Longitudinal studies investigating the predictors of academic achievement reveal that school-entry numerical and mathematical skills are a strong predictor of later academic achievement. School-entry math skills are a stronger predictor of later achievement than both school-entry reading and attentional skills (e.g., Duncan et al., 2007). Findings such as these demonstrate the critical role that numerical and mathematical knowledge and skills play in children's academic development and outcomes.

What do we know about how numbers are represented in the brain and mind and how such representations change over the course of learning and development? The past three decades have seen a surge in the empirical study of number representation and processing in multiple species and at different levels of analyses (for reviews, see Ansari, 2008, Dehaene, 1997, Nieder and Dehaene, 2009). The aim of this chapter is to provide an overview of this research and to synthesize what is currently known, as well as to discuss open questions and future research directions.

Section snippets

An Approximate System for the Representation of Numerical Magnitude

Much of the research on how we represent and process numerical information has been focused on uncovering the foundational systems that underpin the development of complex numerical and mathematical abilities. In particular, there has been a focus on understanding the representations of numerical magnitude, or the total number of items in a set. The representation of processing numerical magnitude has been investigated from infancy onward at both the behavioral and brain levels of analysis. In

The Symbolic Representation of Numerical Magnitude

The literature just reviewed suggests the existence of an approximate representation of numerical magnitude that humans share with other species and that can be measured very early in human development. All of the findings reviewed thus far relied upon nonsymbolic representations of numerical magnitude (e.g., dot arrays) to glean insights into the representation and processing of numerical magnitude. In contrast to nonhuman primates, however, humans who grow up in literate cultures acquire

The Relationship Between Symbolic and Nonsymbolic Representations of Numerical Magnitude

The mixed evidence concerning the association between nonsymbolic numerical magnitude processing and children's arithmetic achievement casts doubt on the assumption that symbolic number skills are scaffolded on their nonsymbolic counterparts. Specifically, one influential proposal suggests that the very meaning of number symbols is determined by direct reference to the corresponding nonsymbolic magnitude: “When we learn number symbols, we simply attach their arbitrary shapes to the relevant

Summary and Conclusions

Numbers play a critical role in our everyday lives, and acquiring numerical and mathematical skills is one of the central goals of formal education across the globe. Over the past three decades, researchers from the fields of Cognitive Science, Psychology, and Neuroscience have investigated how numbers are represented and processed in the brain and mind. A particular focus of this line of research has been on better understanding the foundations upon which the development of numerical and

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