No abstract

Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals-these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal 'number sense' than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children's past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.

Behavioral, neuropsychological, and brain imaging research points to a dedicated system for processing number that is shared across development and across species. This foundational Approximate Number System (ANS) operates over multiple modalities, forming representations of the number of objects, sounds, or events in a scene. This system is imprecise and hence differs from exact counting. Evidence suggests that the resolution of the ANS, as specified by a Weber fraction, increases with age such that adults can discriminate numerosities that infants cannot. However, the Weber fraction has yet to be determined for participants of any age between 9 months and adulthood, leaving its developmental trajectory unclear. Here we identify the Weber fraction of the ANS in 3-, 4-, 5-, and 6-year-old children and in adults. We show that the resolution of this system continues to increase throughout childhood, with adultlike levels of acuity attained surprisingly late in development.

A new choice task was used to explore infants' spontaneous representations of more and less. Ten- and 12-month-old infants saw crackers placed sequentially into two containers, then were allowed to crawl and obtain the crackers from the container they chose. Infants chose the larger quantity with comparisons of 1 versus 2 and 2 versus 3, but failed with comparisons of 3 versus 4, 2 versus 4, and 3 versus 6. Success with visible arrays ruled out a motivational explanation for failure in the occluded 3-versus-6 condition. Control tasks ruled out the possibility that presentation duration guided choice, and showed that presentation complexity was not responsible for the failure with larger numbers. When crackers were different sizes, total surface area or volume determined choice. The infants 'pattern of success and failure supports the hypothesis that they relied on object-file representations, comparing mental models via total volume or surface area rather than via one-to-one correspondence between objectfiles.

Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. We hypothesize that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. Here we show that ninth grade students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low-, typically-, and high-achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relationship persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low- from typically-achieving students, suggesting an ANS deficit that is specific to MLD.