How do our mental representations of number change over development? The dominant view holds that children (and adults) possess multiple representations of number, and that age and experience lead to a shift from greater reliance upon logarithmically organized number representations to greater reliance upon more accurate, linear representations. Here we present a new theoretically motivated and empirically supported account of the development of numerical estimation, based on the idea that number-line estimation tasks entail judgments of proportion. We extend existing models of perceptual proportion judgment to the case of abstract numerical magnitude. Two experiments provide support for these models; three likely sources of developmental change in children's estimation performance are identified and discussed. This work demonstrates that proportion-judgment models provide a unified account of estimation patterns that have previously been explained in terms of a developmental shift from logarithmic to linear representations of number.

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Mental representations of numerical magnitude are commonly thought to undergo discontinuous change over development in the form of a "representational shift." This idea stems from an apparent categorical shift from logarithmic to linear patterns of numerical estimation on tasks that involve translating between numerical magnitudes and spatial positions (such as number-line estimation). However, the observed patterns of performance are broadly consistent with a fundamentally different view, based on psychophysical modeling of proportion estimation, that explains the data without appealing to discontinuous change in mental representations of numerical magnitude. The present study assessed these 2 theories' abilities to account for the development of numerical estimation in 5- through 10-year-olds. The proportional account explained estimation patterns better than the logarithmic-to-linear-shift account for all age groups, at both group and individual levels. These findings contribute to our understanding of the nature and development of the mental representation of number and have more general implications for theories of cognitive developmental change.

Educated humans use language to express abstract number, applying the same number words to seven apples, whistles, or sins. Is language or education the source of numerical abstraction? Claims to the contrary must present evidence for numerical knowledge that applies to disparate entities, in people who have received no formal mathematics instruction and cannot express such knowledge in words. Here we show that preschool children can compare and add large sets of elements without counting, both within a single visual-spatial modality (arrays of dots) and across two modalities and formats (dot arrays and tone sequences). In two experiments, children viewed animations and either compared one visible array of dots to a second array or added two successive dot arrays and compared the sum to a third array. In further experiments, a dot array was replaced by a sequence of sounds, so that participants had to integrate quantity information presented aurally and visually. Children performed all tasks successfully, without resorting to guessing strategies or responding to continuous variables. Their accuracy varied with the ratio of the two quantities: a signature of large, approximate number representations in adult humans and animals. Addition was as accurate as comparison, even though children showed no relevant knowledge when presented with symbolic versions of the addition tasks. Abstract knowledge of number and addition therefore precedes, and may guide, language-based instruction in mathematics.cognition ͉ development ͉ numeracy ͉ quantitative skills