An Integrative Theory of Numerical Development

Abstract

Understanding of numerical development is growing rapidly, but the volume and diversity of findings can make it difficult to perceive any coherence in the process. The integrative theory of numerical development posits that a coherent theme does exist-progressive broadening of the set of numbers whose magnitudes can be accurately represented-and that this theme unifies numerical development from infancy to adulthood. From this perspective, development of numerical representations involves four major acquisitions: (a) representing magnitudes of nonsymbolic numbers increasingly precisely, (b) linking nonsymbolic to symbolic numerical representations, (c) extending understanding to increasingly large whole numbers, and (d) extending understanding to all rational numbers. Thus, the mental number line expands rightward to encompass larger whole numbers, leftward to encompass negatives, and interstitially to include fractions and decimals. KEYWORDS-numerical development; numerical magnitudes; mathematical development; fractions; negative numbers; number line Research on numerical development is expanding rapidly, with large literatures emerging on numerical development in infancy, childhood, and adolescence; development of subitizing, counting, estimation, and arithmetic; knowledge of whole numbers, fractions, decimals, and negatives; and nonsymbolic and symbolic representations. Researchers have also examined conceptual and procedural knowledge; underpinnings of numerical development in evolutionary processes, neural processes, cognitive processes, and emotional processes; longitudinal stability of individual differences; and numerical competence in normal and special populations. The list does not end there: Researchers have also looked at relations to numerical knowledge of variations in economic status, culture, language, and instruction; relations among numerical, spatial, and temporal knowledge; relations of numerical knowledge to more advanced mathematics; and relations of interventions that improve numerical knowledge to subsequent learning, to name a subset of areas within the field (see Table S1 in the online Supporting Information for references for each area). Discoveries in these areas attest to the health of the field of numerical development. However, the sheer number of discoveries and areas can make it difficult to perceive any coherence in the developmental process. Is there such coherence or is numerical development just one thing after another? THE INTEGRATED THEORY OF NUMERICAL DEVELOPMENTThe integrated theory of numerical development proposes that the continuing growth of understanding of numerical magnitudes provides a unifying theme for numerical development. Within this perspective, numerical development is a process of broadening the set of numbers whose magnitudes, individually or in arithmetic combination, can be accurately represented. The theory identifies four main trends in numerical development: (a) representing increasingly precisely the magnitudes of numbers expressed nonsymbolically,...

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