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.

nthesis of these findings, the tension between the discrete and the continuous, which has been central to the historical development of mathematical thought, is rooted in the non-verbal foundations of numerical thinking, which, it is argued, are common to humans and non-verbal animals. In this view, the non-verbal representatives of number are mental magnitudes (real numbers) with scalar variability. Scalar variability means that the signals encoding these magnitudes are "noisy;" they vary from trial to trial, with the width of the signal distribution increasing in proportion to (scaled to) its mean. In short, the greater the magnitude, the noisier its representation. These noisy mental magnitudes are arithmetically processed--added, subtracted, multiplied, divided and ordered. Recognition of the importance of arithmetically processed mental magnitudes in the non-verbal representation of number has emerged from a convergence of results from human and animal studies. This is comparative

Although animals of many species have been shown to discriminate between visual-spatial arrays or auditory-temporal sequences on the basis of numerosity, most of the evidence for numerosity discrimination comes from experiments involving extensive laboratory training. Under these conditions, animals' discrimination of two numerosities depends on their ratio and is independent of their absolute value. It is an open question whether any untrained nonhuman animal spontaneously represents number in this way as do human children and adults. Here we present the results of habituation-discrimination experiments on cotton-top tamarin monkeys (Saguinus oedipus) that provide evidence for numerosity discrimination in the absence of training. Presented with auditory stimuli (speech syllables) controlled for the continuous variables of sequence duration, item duration, inter-stimulus interval, and overall energy, tamarins readily discriminated sequences of 4 vs 8, 4 vs 6, and 8 vs 12 syllables. In contrast, tamarins failed to discriminate sequences of 4 vs 5 and 8 vs 10 syllables, providing evidence that their numerosity discrimination is approximate and shows the set-size ratio signature of numerosity discrimination in humans and trained non-human animals. These results provide strong support for the hypothesis that representations of large, approximate numerosity are evolutionarily ancient and spontaneously available to non-human animals.

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.

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.

ous (uncountable) quantities is the system of real numbers. It includes the irrational numbers, like 2, and the transcendental numbers, like p. It is used by modern humans to represent many distinct systems of continuous quantity--duration, length, area, volume, density, rate, intensity, and so on. Because the system of real numbers is isomorphic to a system of magnitudes, the terms real number and magnitude are used interchangeably. Thus, when we refer to "mental magnitudes" we are referring to a real number system in the brain. Like the culturally specified real number system, the real number system in the brain is used to represent both continuous quantity and numerosity. Magnitudes and real numbers have the property that there is no way to pick out a successor, the next number in the sequence. Given a line of some length, there is no procedure whereby one could pick out the next longer line. Similarly, given a real number, like, say, 2, there is no procedure that picks out the next

This study examined the perceptions that Virginia public high school counselors have towards the Junior Reserve Officers' Training Corps (JROTC) program in their schools. Specifically, four areas of research questions were addressed: (1) knowledge; (2) beliefs and attitudes about benefits to students; (3) the extent to which JROTC is recommended to all students; and (4) the appropriateness of JROTC for particular students. Data for this study were obtained from high school counselors in Virginia who were working in a school which offered JROTC as an elective. A questionnaire containing statements about the claims of JROTC was generated. The questionnaire contained a combination of true/false questions, Likert-type scale questions, a checklist of student characteristics, open-ended questions, and a final section on demographics. The investigator conducted follow-up interviews with school counselors at selected schools. Data analysis were relational and descriptive. Results indicated that school counselors were knowledgeable about the relationship between JROTC and the military. School counselors indicated that they were in general agreement with the claims of benefits to students by JROTC, and indicated a positive attitude about these benefits to students. School counselors identified specific characteristics of students for whom they would recommend JROTC. School counselors also indicated for whom JROTC is an appropriate elective, and for whom it may not be appropriate or feasible. It was concluded that school counselors have a generally positive perception towards JROTC in Virginia public schools. Recommendations for future research were presented. iv This dissertation is dedicated in memory of my grandmother: Antoinette Vezina LeBlanc December 22, 1909 - October ...

What is the nature of our mental representation of quantity? We find that human adults show no performance cost of comparing numerosities across vs. within visual and auditory stimulus sets, or across vs. within simultaneous and sequential sets. In addition, reaction time and performance in such tasks are determined by the ratio of the numerosities to be compared; absolute set size has no effect. These findings suggest that modality-specific stimulus properties undergo a non-iterative transformation into representations of quantity that are independent of the modality or format of the stimulus. q 2002 MIT Published by Elsevier Science B.V. All rights reserved.

A Grey parrot (Psittacus erithacus) that was able to quantify 6 item sets (including subsets of heterogeneous groups, e.g., blue blocks within groupings of blue and green blocks and balls) using English labels (I. M. Pepperberg, 1994a) was tested on comprehension of these labels, which is crucial for numerical competence (K. C. Fuson, 1988). He was, without training, asked “What color/object [number]?” for collections of various simultaneously presented quantities (e.g., subsets of 4, 5, and 6 blocks of 3 different colors; subsets of 2, 4, and 6 keys, corks, and sticks). Accuracy was greater than 80% and was unaffected by array quantity, mass, or contour. His results demonstrated numerical comprehension competence comparable to that of chimpanzees and very young children. He also demonstrated knowledge of absence of quantity, using “none” to designate zero.

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