We present data from a preferential looking method to investigate when infants have mapped singular and plural markers in English onto the semantic distinction between singleton sets and sets with more than 1 individual. Twenty- to 36-month-old children heard sentences that marked number in 1 of 2 ways: (a) redundantly with verb morphology, lexical quantifiers, and noun morphology (“Look, there ARE SOME blicketS”/“Look, there IS A blicket”) or (b) only with noun morphology (“Look at the blicketS”/“Look at the blicket”). Twenty-four-month-old infants, but not 20-month-old infants, looked at the screen that matched the carrier sentence with respect to singular–plural distinction when number was expressed on the verb, on the noun, and with quantifiers. Detailed looking-time analyses suggest that the arrays begin to be differentiated on the child’s hearing are or is. Twenty-four-month-olds failed when number was marked on the noun alone, whereas 36-month-olds suc-Correspondence should be addressed to Sid Kouider, Laboratoire de Sciences Cognitives et

Mathematics is a system for representing and reasoning about quantities, with arithmetic as its foundation. Its deep interest for our understanding of the psychological foundations of scientific thought comes from what Eugene Wigner called the unreasonable efficacy of mathematics in the natural sciences. From a formalist perspective, arithmetic is a symbolic game, like tic-tac-toe. Its rules are more complicated, but not a great deal more complicated. Mathematics is the study of the properties of this game and of the systems that may be constructed on the foundation that it provides. Why should this symbolic game be so powerful and resourceful when it comes to building models of the physical world? And on what psychological foundations does the human mastery of this game rest? The first question is metaphysical—why is the world the way it is? We do not treat it, because it lies beyond the realm of experimental behavioral science. We review the answers to the second question that experimental research on human and non-human animal cognition suggests.

Can human adults perform arithmetic operations with large approximate numbers, and what effect, if any, does an internal spatial–numerical representation of numerical magnitude have on their responses? We conducted a psychophysical study in which subjects viewed several hundred short videos of sets of objects being added or subtracted from one another and judged whether the final numerosity was correct or incorrect. Over a wide range of possible outcomes, the subjects ’ responses peaked at the approximate location of the true numerical outcome and gradually tapered off as a function of the ratio of the true and proposed outcomes (Weber’s law). Furthermore, an operational momentum effect was observed, whereby addition problems were overestimated and subtraction problems were underestimated. The results show that approximate arithmetic operates according to precise quantitative rules, perhaps analogous to those characterizing movement on an internal continuum. Human adults possess an ability to estimate and manipulate approximate numerical magnitudes, which has been termed number sense (Dehaene, 1997). This ability appears to be largely independent of language and other symbol systems, since it is present in both infants (Xu & Spelke, 2000) and other animal species (Brannon & Roitman, 2003;

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Abstract: Research on human infants, adult nonhuman primates, and children and adults in diverse cultures provides converging evidence for four systems at the foundations of human knowledge. These systems are domain specific and serve to represent both entities in the perceptible world (inanimate manipulable objects and animate agents) and entities that are more abstract (numbers and geometrical forms). Human cognition may be based, as well, on a fifth system for representing social partners and for categorizing the social world into groups. Research on infants and children may contribute both to understanding of these systems and to attempts to overcome misconceptions that they may foster.

entities that can be represented by a numeral, a word, a number of lines on a scorecard, or a sequence of chimes from a clock. This abstract, notation-independent appreciation of numbers develops gradually over the first several years of life. Here, using functional magnetic resonance imaging, we examine the brain mechanisms that 6- and 7-year-old children and adults recruit to solve numerical comparisons across different notation systems. The data reveal that when young children compare numerical values in symbolic and nonsymbolic notations, they invoke the same network of brain regions as adults including occipito-temporal and parietal cortex. However, children also recruit inferior frontal cortex during these numerical tasks to a much greater degree than adults. Our data lend additional support to an emerging consensus from adult neuroimaging, nonhuman primate neurophysiology, and computational modeling studies that a core neural system integrates notationindependent numerical representations throughout development but, early in development, higher-order brain mechanisms mediate this process. &

Children’s sense of numbers before formal education is thought to rely on an approximate number system based on logarithmically compressed analog magnitudes that increases in resolution throughout childhood. School-age children performing a numerical estimation task have been shown to increasingly rely on a formally appropriate, linear representation and decrease their use of an intuitive, logarithmic one. We investigated the development of numerical estimation in a younger population (3.5- to 6.5-year-olds) using 0–100 and 2 novel sets of 1–10 and 1–20 number lines. Children’s estimates shifted from logarithmic to linear in the small number range, whereas they became more accurate but increasingly logarithmic on the larger interval. Estimation accuracy was correlated with knowledge of Arabic numerals and numerical order. These results suggest that the development of numerical estimation is built on a logarithmic coding of numbers—the hallmark of the approximate number system—and is subsequently shaped by the acquisition of cultural practices with numbers.

ABSTRACT — ‘‘The Number Race’ ’ is an adaptive game designed to improve number sense. We tested its effectiveness using a cross-over design in 53 low socioeconomic status kindergarteners in France. Children showed improvements in tasks traditionally used to assess number sense (numerical comparison of digits and words). However, there was no improvement on non-symbolic measures of number sense, suggesting that rather than being in number sense per se, the improvement was in number sense access; or links between symbolic and non-symbolic representations of number. Focused adaptive interventions such as this may contribute to reducing the socioeconomic gap in math achievement. Computer-aided instruction can be a useful tool in early mathematics education, even in preschool and kindergarten (Clements, 2002). Adaptive computer games designed to behaviorally train a particular aspect of cognition hold particular promise, especially for children disadvantaged by learning difficulties or socioeconomic status (SES). Not only

Previous studies have suggested that typically developing 6-month-old infants are able to discriminate between small and large numerosities. However, discrimination between small numerosities in young infants is only possible when variables continuous with number (e.g. area or circumference) are confounded. In contrast, large number discrimination is successful even when variables continuous with number are systematically controlled for. These findings suggest the existence of different systems underlying small and large number processing in infancy. How do these develop in atypical syndromes? Williams syndrome (WS) is a rare neurocognitive developmental disorder in which numerical cognition has been found to be impaired in older children and adults. Do impairments of number processing have their origins in infancy? Here this question is investigated by testing the small and large number discrimination abilities of infants and toddlers with WS. While infants with WS were able to discriminate between 2 and 3 elements when total area was confounded with numerosity, the same infants did not discriminate between 8 and 16 elements, when number was not confounded with continuous variables. These findings suggest that a system for tracking the features of small numbers of object (object-file representation) may be functional in WS, while large number discrimination is impaired from an early age onwards. Finally, we argue that individual differences in large number processing in infancy are more likely than small number processing to be predictive of later development of numerical cognition.

doi: 10.3389/fnint.2012.00007 Developmental neuroscience of time and number: implications for autism and other neurodevelopmental disabilities