Are abstract representations of number -- representations that are independent of the particular type of entities that are enumerated -- a product of human language or culture, or do they trace back to human infancy? To investigate these questions, four experiments investigated whether human infants discriminate between sequences of actions (jumps of a puppet) on the basis of numerosity. At 6 months, infants successfully discriminated 4- vs. 8-jump sequences, when the continuous variables of sequence duration, jump duration, jump rate, jump interval and duration and extent of motion were controlled and rhythm was eliminated. In contrast, infants failed to discriminate 2- vs. 4-jump sequences, suggesting that infants fail to form cardinal number representations of small numbers of actions. Infants also failed to discriminate between sequences of 4 vs. 6 jumps at 6 months, and succeeded at 9 months, suggesting that infants' number representations are imprecise and increase in precision with age. All of these findings agree with those of studies using visual-spatial arrays and auditory sequences, providing evidence that a single, abstract system of number representation is present and functional in infancy. Infants' Enumeration of Actions: Numerical Discrimination and its Signature Limits Recent research provides evidence that human infants discriminate between large sets of elements on the basis of numerosity, when a variety of continuous quantitative variables are controlled. For example, 6-month-old infants discriminate visual arrays of 8 vs. 16 dots when array size and density, dot size, summed area and brightness, and summed contour length are equated either during habituation or during test (e.g., Brannon, 2002;Brannon, Abbott, & Lutz, in press; Xu & Spelke, 2000; Xu...

What are the brain and cognitive systems that allow humans to play baseball, compute square roots, cook soufflés, or navigate the Tokyo subways? It may seem that studies of human infants and of non-human animals will tell us little about these abilities, because only educated, enculturated human adults engage in organized games, formal mathematics, gourmet cooking, or map-reading. In this chapter, we argue against this seemingly sensible conclusion. When human adults exhibit complex, uniquely human, culture-specific skills, they draw on a set of psychological and neural mechanisms with two distinctive properties: they evolved before humanity and thus are shared with other animals, and they emerge early in human development and thus are common to infants, children, and adults. These core knowledge systems form the building blocks for uniquely human skills. Without them we wouldn’t be able to learn about different kinds of games, mathematics, cooking, or maps. To understand what is special about human intelligence, therefore, we must study both the core knowledge systems on which it rests and the mechanisms by which these systems are orchestrated to permit new kinds of concepts and cognitive processes. What is core knowledge? A wealth of research on non-human primates and on human

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...

What are the origins of abstract concepts such as “seven, ” and what role does language play in their development? These experiments probed the natural number words and concepts of 3-year-old children who can recite number words to ten but who can comprehend only one or two. Children correctly judged that a set labeled eight retains this label if it is unchanged, that it is not also four, and that eight is more than two. In contrast, children failed to judge that a set of 8 objects is better labeled by eight than by four, that eight is more than four, that eight continues to apply to a set whose members are rearranged, or that eight ceases to apply if the set is increased by 1, doubled, or halved. The latter errors contrast with children’s correct application of words for the smallest numbers. These findings suggest that children interpret number words by relating them to 2 distinct preverbal systems that capture only limited numerical information. Children construct the system of abstract, natural number concepts from these foundations.

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.

for assistance, and Nora Newcombe and Elliott Blass for advice and comments on the manuscript. Above all, I am grateful to Ariel Grace and Kristin Shutts for their unending support and after-hours labor on this project. Draft, 4/20/05. This paper has not yet been peer reviewed. Please do not copy or cite without author's permission. This report considers three prominent claims that boys and men have greater natural aptitude for high-level careers in mathematics and science. According to the first claim, males are more focused on objects and mechanical systems from the beginning of life. According to the second claim, males have a profile of spatial and numerical abilities that predisposes them to greater aptitude in mathematics. According to the third claim, males show greater variability in mathematical aptitude, yielding a preponderance of males at the upper end of the distribution of mathematical talent. Research on cognitive development in human infants and preschool children, and research on cognitive performance by students at all levels, provides evidence against these claims. Mathematical and scientific reasoning develop from a set of biologically based capacities that males and females share. From these capacities, men and women appear to develop equal talent for mathematics and science.

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...

& Behavioral and brain imaging research indicates that human infants, humans adults, and many nonhuman animals represent large nonsymbolic numbers approximately, discriminating between sets with a ratio limit on accuracy. Some behavioral evidence, especially with human infants, suggests that these representations differ from representations of small numbers of objects. To investigate neural signatures of this distinction, event-related potentials were recorded as adult humans passively viewed the sequential presentation of dot arrays in an adaptation paradigm. In two studies, subjects viewed successive arrays of a single number of dots interspersed with test arrays presenting the same or a different number; numerical range (small numerical quantities 1–3 vs. large numerical quantities 8–24) and ratio difference varied across blocks as continuous variables were controlled. An early-evoked component (N1), observed over widespread posterior scalp locations, was modulated by absolute number with small, but not large, number arrays. In contrast, a later component (P2p), observed over the same scalp locations, was modulated by the ratio difference between arrays for large, but not small, numbers. Despite many years of experience with symbolic systems that apply equally to all numbers, adults spontaneously process small and large numbers differently. They appear to treat smallnumber arrays as individual objects to be tracked through space and time, and large-number arrays as cardinal values to be compared and manipulated. &

Numerical abilities are thought to rest on the integration of two distinct systems, a verbal system of number words and a non-symbolic representation of approximate quantities. This view has lead to the classification of acalculias into two broad categories depending on whether the deficit affects the verbal or the quantity system. Here, we test the association of deficits predicted by this theory, and particularly the presence or absence of impairments in non-symbolic quantity processing. We describe two acalculic patients, one with a focal lesion of the left parietal lobe and Gerstmann's syndrome and another with semantic dementia with predominantly left temporal hypometabolism. As predicted by a quantity deficit, the first patient was more impaired in subtraction than in multiplication, showed a severe slowness in approximation, and exhibited associated impairments in subitizing and numerical comparison tasks, both with Arabic digits and with arrays of dots. As predicted by a verbal deficit, the second patient was more impaired in multiplication than in subtraction, had intact approximation abilities, and showed preserved processing of non-symbolic numerosities.

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. &