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.

Abstract not found

This article presents a step in the establishment of the following hypothesis:

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.

Four experiments used a preferential looking method to investigate six-month-old infants' capacity to represent numerosity in visual-spatial displays. Building on previous findings that such infants discriminate between arrays of 8 vs. 16 discs, but not 8 vs. 12 discs (Xu & Spelke, 2000), Experiments 1 and 2 investigated whether infants' numerosity discrimination depends on the ratio of the two set sizes with even larger numerosities. Infants successfully discriminated between arrays of 16 vs. 32 discs, but not 16 vs. 24 discs, providing evidence that their discrimination shows the set-size ratio signature of numerosity discrimination in human adults, children, and many non-human animals. Experiments 3 and 4 addressed a controversy concerning infants' ability to discriminate large numerosities (observed under conditions that control for total filled area, array size and density, item size, and correlated properties such as brightness: Brannon, 2002; Xu, 2003; Xu & Spelke, 2000) vs. small numerosities (not observed under conditions that control for total contour length: Clearfield & Mix, 1999). To investigate the sources of these differing findings, Experiment 3 tested infants' large-number discrimination with controls for contour length, and Experiment 4 tested small-number discrimination with controls for total filled area. Infants successfully discriminated the large-number displays but showed no evidence of discriminating the small-number displays. These findings provide evidence that infants have robust abilities to represent large numerosities. In contrast, infants may fail to represent small numerosities in visual-spatial arrays with continuous quantity controls, consistent with the thesis that separate systems serve to represent large vs. small numerosities. A we...

Developmental research suggests that some of the mechanisms that underlie numerical cognition are present and functional in human infancy. To investigate these mechanisms and their developmental course, psychologists have turned to behavioral and electrophysiological methods using briefly presented displays. These methods, however, depend on the assumption that young infants can extract numerical information rapidly. Here we test this assumption and begin to investigate the speed of numerical processing in 5-month-old infants. Infants successfully discriminated between arrays of 4 vs. 8 dots on the basis of number when a new array appeared every 2 seconds, but not when a new array appeared every 1.0 or 1.5 seconds. These results suggest alternative interpretations of past findings, provide constraints on the design of future experiments, and introduce a new method for probing infants' enumeration process. Further experiments using this method provide initial evidence that infants' enumeration mechanism operates in parallel and yields increasingly accurate numerical representations over time, as does the enumeration mechanism used by adults in symbolic and nonsymbolic tasks. Over the past two decades, a wealth of research has focused on the nature and origins of numerical knowledge. Although reports that infants represent small numbers of objects have been interpreted in multiple ways (e.g., Carey, 2001; Clearfield & Mix, 1999; Feigenson, Carey & Spelke, 2002; Simon, 1997; Starkey & Cooper, 1980; Treiber & Wilcox, 1984; Wynn, 1992; Wynn, Bloom & Chiang, 2002), recent research provides clear evidence that infants as young as 6 months represent the approximate cardinal values of large sets of entities. In studies using a looking time method, for example, 6-month-old infant...

Abstract not found

Mathematicians frequently evoke their “intuition” when they are able to quickly and automatically solve a problem, with little introspection into their insight. Cognitive neuroscience research shows that mathematical intuition is a valid concept that can be studied in the laboratory in reduced paradigms, and that relates to the availability of “core knowledge” associated with evolutionarily ancient and specialized cerebral subsystems. As an illustration, I discuss the case of elementary arithmetic. Intuitions of numbers and their elementary transformations by addition and subtraction are present in all human cultures. They relate to a brain system, located in the intraparietal sulcus of both hemispheres, which extracts numerosity of sets and, in educated adults, maps back and forth between numerical symbols and the corresponding quantities. This system is available to animal species and to preverbal human infants. Its neuronal organization is increasingly being uncovered, leading to a precise mathematical theory of how we perform tasks of number comparison or number naming. The next challenge will be to understand how education changes our core intuitions of number.

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,