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Précis of "The number sense"
"... Number sense " is a shorthand 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 ..."
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Cited by 290 (25 self)
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Number sense " is a shorthand 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 domainspecific, biologicallydetermined 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 higherlevel 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.
Nonverbal numerical cognition from reals to integers
 Trends in Cognitive Sciences
, 2000
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NonVerbal Counting in Humans: The Psychophysics of Number Representation
 Psychological Science
, 1999
"... In nonverbal 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 ..."
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Cited by 108 (5 self)
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In nonverbal 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 nonverbal 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 nonverbal 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 nonverbal 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 (trialtotrial 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...
The Representations Underlying Infants' Choice of More: Object Files versus Analog Magnitudes
, 2002
"... A new choice task was used to explore infants' spontaneous representations of more and less. Ten and 12monthold 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 wi ..."
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Cited by 106 (18 self)
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A new choice task was used to explore infants' spontaneous representations of more and less. Ten and 12monthold 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 3versus6 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 objectfile representations, comparing mental models via total volume or surface area rather than via onetoone correspondence between object files.
DC: Mathematics and learning disabilities
 J Learn Disabil
"... Between 5 % and 8 % of schoolage children have some form of memory or cognitive deficit that interferes with their ability to learn concepts or procedures in one or more mathematical domains. A review of the arithmetical competencies of these children is provided, along with discussion of underlyin ..."
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Cited by 106 (6 self)
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Between 5 % and 8 % of schoolage children have some form of memory or cognitive deficit that interferes with their ability to learn concepts or procedures in one or more mathematical domains. A review of the arithmetical competencies of these children is provided, along with discussion of underlying memory and cognitive deficits and potential neural correlates. The deficits are discussed in terms of three subtypes of mathematics learning disability and in terms of a more general framework for linking research in mathematical cognition to research in learning disabilities. The breadth and complexity of the field of mathematics make the identification and study of the cognitive phenotypes that define mathematics learning disabilities (MLD) a formidable endeavor. In theory, a learning disability can result from deficits in the ability to represent or process information in one or all of the
Predicting risksensitivity in humans and lower animals: Risk as variance or coefficient of variation
 Psychological Review
, 2004
"... This article examines the statistical determinants of risk preference. In a metaanalysis of animal risk preference (foraging birds and insects), the coefficient of variation (CV), a measure of risk per unit of return, predicts choices far better than outcome variance, the risk measure of normative ..."
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Cited by 91 (8 self)
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This article examines the statistical determinants of risk preference. In a metaanalysis of animal risk preference (foraging birds and insects), the coefficient of variation (CV), a measure of risk per unit of return, predicts choices far better than outcome variance, the risk measure of normative models. In a metaanalysis of human risk preference, the superiority of the CV over variance in predicting risk taking is not as strong. Two experiments show that people’s risk sensitivity becomes strongly proportional to the CV when they learn about choice alternatives like other animals, by experiential sampling over time. Experiencebased choices differ from choices when outcomes and probabilities are numerically described. Zipf’s law as an ecological regularity and Weber’s law as a psychological regularity may give rise to the CV as a measure of risk. Decision making under risk and uncertainty is a topic of research in disciplines as diverse as psychology, economics, zoology, and entomology. Both the animal and the human risky choice literatures have proposed models that either predict choices in a deterministic fashion or predict risk sensitivity (i.e., the probability of choosing a riskier or less risky option) in a stochastic fashion.
Infants’ discrimination of number vs. continuous extent
 Cognitive Psychology
, 2002
"... Seven studies explored the empirical basis for claims that infants represent cardinal values of small sets of objects. Many studies investigating numerical ability did not properly control for continuous stimulus properties such as surface area, volume, contour length, or dimensions that correlate w ..."
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Cited by 86 (21 self)
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Seven studies explored the empirical basis for claims that infants represent cardinal values of small sets of objects. Many studies investigating numerical ability did not properly control for continuous stimulus properties such as surface area, volume, contour length, or dimensions that correlate with these properties. Experiment 1 extended the standard habituation/dishabituation paradigm to a 1 vs 2 comparison with threedimensional objects and confirmed that when number and total front surface area are confounded, infants discriminate the arrays. Experiment 2 revealed that infants dishabituated to a change in front surface area but not to a change in number when the two variables were pitted against each other. Experiments 3 through 5 revealed no sensitivity to number when front surface area was controlled, and Experiments 6 and 7 extended this pattern of findings to the Wynn (1992) transformation task. Infants ’ lack of a response to number, combined with their demonstrated sensitivity to one or more dimensions of continuous extent, supports the hypothesis that the representations subserving objectbased attention, rather than those subserving enumeration, underlie performance in the above tasks.
Differential Contributions of the Left and Right Inferior Parietal Lobules to Number Processing
 Journal of Cognitive Neuroscience
, 1999
"... 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 brainlesioned patients: naming, comparing, multiplying, or subtracting single digits. The results revealed the ac ..."
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Cited by 79 (16 self)
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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 brainlesioned 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.
Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability
 Child Development
, 2007
"... Using strict and lenient mathematics achievement cutoff scores to define a learning disability, respective groups of children who are math disabled (MLD, n 5 15) and low achieving (LA, n 5 44) were identified. These groups and a group of typically achieving (TA, n 5 46) children were administered a ..."
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Cited by 77 (14 self)
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Using strict and lenient mathematics achievement cutoff scores to define a learning disability, respective groups of children who are math disabled (MLD, n 5 15) and low achieving (LA, n 5 44) were identified. These groups and a group of typically achieving (TA, n 5 46) children were administered a battery of mathematical cognition, working memory, and speed of processing measures (M 5 6 years). The children with MLD showed deficits across all math cognition tasks, many of which were partially or fully mediated by working memory or speed of processing. Compared with the TA group, the LA children were less fluent in processing numerical information and knew fewer addition facts. Implications for defining MLD and identifying underlying cognitive deficits are discussed. Diagnostic criteria and thus the percentage of children with a learning disability in mathematics (MLD)