Number sense " is a short-hand for our ability to quickly understand, approximate, and manipulate numerical quantities. My hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domain-specific, biologically-determined ability are reviewed: the presence of evolutionary precursors of arithmetic in animals; the early emergence of arithmetic competence in infants independently of other abilities, including language; the existence of a homology between the animal, infant, and human adult abilities for number processing ; and the existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. I postulate that higher-level cultural developments in arithmetic emerge through the establishment of linkages between this core analogical representation (the " number line ") and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution.

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 object files.

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Evolutionary approaches to judgment under uncertainty have led to new data showing that untutored subject reliably produce judgments that conform to may principles of probability theory when (a) they are asked to compute a frequency instead of the probability of a single event, and (b) the relevant information is expressed as frequencies. But are the frequencycomputation systems implicated in these experiments better at operating over some kinds of input than others? Principles of object perception and principles of adaptive design led us to propose the individuation hypothesis: that these systems are designed to produce wellcalibrated statistical inferences when they operate over representations of “whole ” objects, events, and locations. In a series of experiments on Bayesian reasoning, we show that human performance can be systematically improved or degraded by varying whether a correct solution requires one to compute hit and false-alarm rates over “natural ” units, such as whole objects, as opposed to inseparable aspects, views, and other parsings that violate evolved principles of object construal. The ability to make well-calibrated probability judgments depends, at a very basic level, on the ability to count. The

In two experiments, a manual search task explored 12- to 14-month-old infants' representations of small sets of objects. In this paradigm, patterns of searching revealed the number of objects infants represented as hidden in an opaque box. In Experiment 1, we obtained the set-size signature of object-file representations: infants succeeded at representing precisely 1, precisely 2, and precisely 3 objects in the box, but failed at representing 4 (or even that 4 is greater than 2). In Experiment 2, we showed that infants' expectations about the contents of the box were based on number of individual objects, and not on a continuous property such as total object volume. These findings support the hypothesis that infants maintained representations of individuals, that object-files were the underlying means of representing these individuals, and that object-file models can be compared via one-to-one correspondence to establish numerical equivalence.

Previous research has shown that children go through a stage in which they know that the number words each refer to a distinct numerosity, yet do not know which numerosity each number word picks out (Wynn, 1992). How do children attain this level of knowledge? We explore the possibility that particular properties of how number words are used within sentences inform children of the semantic class to which they belong. An analysis of transcripts of the spontaneous speech of three one- and two-year-old children and their parents (from the CHILDES database; MacWhinney & Snow, 1990) suggests that the relevant cues are available as input in parents ’ speech to children, and that children generally honour these properties of number words in their own speech. Implications of this proposal for word learning more generally are discussed.

Studies using operant training have demonstrated that laboratory animals can discriminate the number of objects or events based on either auditory or visual stimuli, as well as the integration of both auditory and visual modalities. To date, studies of spontaneous number discrimination in untrained animals have been restricted to the visual modality, leaving open the question of whether such capacities generalize to other modalities such as audition. To explore the capacity to spontaneously discriminate number based on auditory stimuli, and to assess the abstractness of the representation underlying this capacity, a habituation-discrimination procedure involving speech and pure tones was used with a colony of cotton-top tamarins. In the habituation phase, we presented subjects with either two- or three-speech syllable sequences that varied with respect to overall duration, intersyllable duration, and pitch. In the test phase, we presented subjects with a counterbalanced order of either two- or three-tone sequences that also varied with respect to overall duration, inter-syllable duration, and pitch. The proportion of looking responses to test stimuli differing in number was significantly greater than to test stimuli consisting of the same number. Combined with earlier work, these results show that at least one non-human primate species can spontaneously discriminate number in both the visual and auditory domain, indicating that this capacity is not tied to a particular modality, and within a modality, can accommodate differences in format.

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The learning of mathematics is of universal interest across all OECD countries. In modern cultures, all children are expected to master up to at least 15 years of formal instruction in mathematics. Innumeracy has serious negative consequences for society. Mathematics is important not only for its own sake but also because it allows citizens to operate more effectively in their day-to-day lives. People need to balance checkbooks, make payments on items, and plan for future retirement. The topics studied by professional mathematicians extend far beyond everyday arithmetic or algebra. For example, some mathematicians devote themselves to the study of topology, others are interested in analysis or non-Euclidian geometries. The limitations of brain-imaging technologies, however, require researchers to simplify the cognitive demands facing subjects and to focus on simpler (but more common) mathematics topics. As we have seen in the chapter on reading, brain-imaging technologies can model relatively simple tasks. Thus, even though researchers who study the relationship between brain learning and mathematics hope someday to find brain activity correlates for all of mathematics, the current focus is on the learning of small whole numbers,