There is currently considerable discussion about the relative influences of evolutionary and cultural factors in the development of early numerical skills. In particular, there has been substantial debate and study of the relationship between approximate, nonverbal (approximate magnitude system, AMS) and exact, symbolic (symbolic number system, SNS) representations of number. Here we examined several hypotheses concerning whether, in the earliest stages of formal education, AMS abilities predict growth in SNS abilities, or the other way around. In addition to tasks involving symbolic (Arabic numerals) and non-symbolic (dot arrays) number comparisons, we also tested children's ability to translate between the two systems (i.e., mixed-format comparison). Our data included a sample of 539 Kindergarten children (mean=5.17yrs, SD=0.29yrs), with AMS, SNS and mixed comparison skills assessed at the beginning and end of the academic year. In this way, we provide, to the best of our knowledge, the most comprehensive test to date of the direction of influence between the AMS and SNS in early formal schooling. Results were more consistent with the view that SNS abilities at the beginning of Kindergarten lay the foundation for improvement in both AMS abilities and the ability to translate between the two systems. Importantly, we found no evidence to support the reverse. We conclude that, once one acquires a very basic grasp of exact number symbols, it is this understanding of exact number (and perhaps repeated practice therewith) that facilitates growth in the AMS. Though the precise mechanism remains to be understood, these data challenge the widely held view that the AMS scaffolds the acquisition of the SNS.

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Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children’s ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children’s performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children’s ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1–3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics.

In the present study we examined whether children with Developmental Dyscalculia (DD) exhibit a deficit in the so-called 'Approximate Number System' (ANS). To do so, we examined a group of elementary school children who demonstrated persistent low math achievement over 4 years and compared them to typically developing (TD), aged-matched controls. The integrity of the ANS was measured using the Panamath (www.panamath.org) non-symbolic numerical discrimination test. Children with DD demonstrated imprecise ANS acuity indexed by larger Weber fraction (w) compared to TD controls. Given recent findings showing that non-symbolic numerical discrimination is affected by visual parameters, we went further and investigated whether children performed differently on trials on which number of dots and their overall area were either congruent or incongruent with each other. This analysis revealed that differences in w were only found between DD and TD children on the incongruent trials. In addition, visuo-spatial working memory strongly predicts individual differences in ANS acuity (w) during the incongruent trials. Thus the purported ANS deficit in DD can be explained by a difficulty in extracting number from an array of dots when area is anti-correlated with number. These data highlight the role of visuo-spatial working memory during the extraction process, and demonstrate that close attention needs to be paid to perceptual processes invoked by tasks thought to represent measures of the ANS.

The neural foundations of arithmetic learning are not well understood. While behavioral studies have revealed relationships between symbolic number processing and individual differences in children's arithmetic performance, the neurocognitive mechanisms that bind symbolic number processing and arithmetic are unknown. The current fMRI study investigated the relationship between children's brain activation during symbolic number comparison (Arabic digits) and individual differences in arithmetic fluency. A significant correlation was found between the numerical ratio effect on reaction times and accuracy and children's arithmetic scores. Furthermore, children with a stronger neural ratio effect in the left intraparietal sulcus (IPS) during symbolic number processing exhibited higher arithmetic scores. Previous research has demonstrated that activation of the IPS during numerical magnitude processing increases over the course of development, and that the left IPS plays an important role in symbolic number processing. The present findings extend this knowledge to show that children with more mature response modulation of the IPS during symbolic number processing exhibit higher arithmetic competence. These results suggest that the left IPS is a key neural substrate for the relationship between the relative of precision of the representation of numerical magnitude and school-level arithmetic competence.