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.

In non-verbal counting tasks derived from the animal literature, adult human subjects repeatedly attempted to produce target numbers of key presses at rates that made vocal or subvocal counting difficult or impossible. In a second task, they estimated the number of flashes in a rapid, randomly timed sequence. Congruent with the animal data, mean estimates in both tasks were proportional to target values, as was the variability in the estimates. Converging evidence makes it unlikely that subjects used verbal counting or time durations to perform these tasks. The results support the hypothesis that adult humans share with non-verbal animals a system for representing number by magnitudes that have scalar variability (a constant coefficient of variation). The mapping of numerical symbols to mental magnitudes provides a formal model of the underlying non-verbal meaning of the symbols (a model of numerical semantics). Animal subjects represent both number and duration by mental magnitudes (Church, 1984; Gibbon, Church, & Meck, 1984; Meck & Church, 1983). These representations are formally analogous to points on the real number line. Meck and Church (1983) proposed such a representation to account for animal non-verbal counts of objects or events (Figure 1). According to their theory each item is enumerated by an impulse of activation which is added to an accumulator. The magnitude in the accumulator at the end of the count is read into memory, where it represents the number of the counted set. The noise (trial-to-trial variability) in these remembered magnitudes is proportional to the magnitude, a property that Gibbon (1977) called scalar variability . Mathematical modeling of psychophysical data from a variety of tasks indicates that memory is the dominant source of trial-...

unseen stimuli

We measured cerebral activation with functional magnetic resonance imaging at 3 Tesla while eight healthy volunteers performed various number processing tasks known to be dissociable in brain-lesioned patients: naming, comparing, multiplying, or subtracting single digits. The results revealed the activation of a circuit comprising bilateral intraparietal, prefrontal, and anterior cingulate components. The extension and lateralization of this circuit was modulated by task demands. The intraparietal and prefrontal activation was more important in the right hemisphere during the comparison task and in the left hemisphere during the multiplication task and was intensely bilateral during the subtraction task. Thus, partially distinct cerebral circuits with the dorsal parietal pathway underlie distinct arithmetic operations.

Abstract not found

Subjects classified visible 2-digit numbers as larger or smaller than 55. Target numbers were preceded by masked 2-digit primes that were either congruent (same relation to 55) or incongruent. Experiments 1 and 2 showed prime congruency effects for stimuli never included in the set of classified visible targets, indicating subliminal priming based on long-term semantic memory. Experiments 2 and 3 went further to demonstrate paradoxical unconscious priming effects resulting from task context. For example, after repeated practice classifying 73 as larger than 55, the novel masked prime 37 paradoxically facilitated the “larger ” response. In these experiments task context could induce subjects to unconsciously process only the leftmost masked prime digit, only the rightmost digit, or both independently. Across 3 experiments, subliminal priming was governed by both task context and long-term semantic memory. This research started by asking how much semantic analysis occurs unconsciously in response to visually masked numbers. Experiment 1 set out specifically to resolve a discrepancy between two recently reported findings. When it became apparent that Experiment 1’s methods could address additional interesting questions about subliminal priming, those additional questions became

This article explores the effect of external representations on numeric tasks. Through several minor modifications on the previously reported two-digit number comparison task, we obtained different results. Rather than a holistic comparison, we found parallel comparison. We argue that this difference was a reflection of different representational forms: the comparison was based on internal representations in previous studies but on external representations in our present study. This representational effect was discussed under a framework of distributed number representations. We propose that in numerical tasks involving external representations, numbers should be considered as distributed representations and the behavior in these tasks should be considered as the interactive processing of internal and external information through the interplay of perceptual and cognitive processes. We suggest that theories of number representations and process models of numerical tasks should consider external representations as an essential component.

The SNARC effect specifically relates small magnitudes to the left hand side and larger magnitudes to the right hand side (e.g. Dehaene et al., 1990; Dehaene et al., 1993). It is certain that cultural characteristics define the SNARC effect: In western cultures small and large numbers are coded in a left-right direction while in Arabic countries magnitude information is coded from right to left (Dehaene et al., 1993; Zebian, in press). In this sense, reading and writing direction have been considered to be the main determinants of the SNARC effect. Indeed, a number of recent studies support the idea that the mastering of a language (and thus reading and writing direction) biases scanning habit in a favourable direction (e.g. Chatterjee et al., 1999; Padakanaya et al., 2002). If this is indeed the case, it is not surprising that the SNARC effect has been found with stimuli other than numbers (Gevers et al., 2004). Related to this, future research could address the question if magnitude and ordinal information are processed by the same mechanism or by different mechanisms with similar properties. However, reading and writing direction alone are not sufficient to explain all SNARC related findings. For instance, it does not allow for an explanation of a

Could our use of numbers reflect a shared mental model of numbers? If so, heuristics based on a cognitively-inspired treatment of numbers could be used to improve automatic interpretation of numerical data drawn from domains about which little is known.We conjecture that number labels are applied to quantities and to numerousness or counts in different ways, for example, in references to 3 kilograms of apples as opposed to 3 apples. We use the device of a magnitude space to model the treatment of numbers as counts and as quantities, and show how this can be used practically in interpreting numbers. The application area of interest to us is categorization on small sample sets.

Subjects classified visible 2-digit numbers as larger or smaller than 55. Target numbers were preceded by masked 2-digit primes that were either congruent (same relation to 55) or incongruent. Experiments 1 and 2 showed prime congruency effects for stimuli never included in the set of classified visible targets, indicating subliminal priming based on long-term semantic memory. Experiments 2 and 3 went further to demonstrate paradoxical unconscious priming effects resulting from task context. For example, after repeated practice classifying 73 as larger than 55, the novel masked prime 37 paradoxically facilitated the "larger" response. In these experiments task context could induce subjects to unconsciously process only the leftmost masked prime digit, only the rightmost digit, or both independently. Across 3 experiments, subliminal priming was governed by both task context and long-term semantic memory. Long-Term Semantic Memory Versus Contextual Memory in Unconscious Number Processing This research started by asking how much semantic analysis occurs unconsciously in response to visually masked numbers. Experiment 1 set out specifically to resolve a discrepancy between two recently reported findings. When it became apparent that Experiment 1 's methods could address additional interesting questions about subliminal priming, those additional questions became the focus of Experiments 2 and 3.