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Origins of Mathematical Intuitions  The Case of Arithmetic
 THE YEAR IN COGNITIVE NEUROSCIENCE
, 2009
"... 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 parad ..."
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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.
Nonsymbolic approximate arithmetic in children: Abstract addition prior to instruction
, 2008
"... Do children draw upon abstract representations of number when they perform approximate arithmetic operations? In this study, kindergarten children viewed animations suggesting addition of a sequence of sounds to an array of dots, and they compared the sum to a second dot array that differed from the ..."
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Do children draw upon abstract representations of number when they perform approximate arithmetic operations? In this study, kindergarten children viewed animations suggesting addition of a sequence of sounds to an array of dots, and they compared the sum to a second dot array that differed from the sum by 1 of 3 ratios. Children performed this task successfully with all the signatures of adults ’ nonsymbolic number representations: accuracy modulated by the ratio of the sum and the comparison quantity, equal performance for within and crossmodality tasks and for addition and comparison tasks, and performance superior to that of a matched subtraction task. The findings provide clear evidence for nonsymbolic numerical operations on abstract numerical quantities in children who have not yet been taught formal arithmetic.
Representations of the magnitudes of fractions
 Journal of Experimental Psychology: Human Perception and Performance
, 2010
"... We tested whether adults can use integrated, analog, magnitude representations to compare the values of fractions. The only previous study on this question concluded that even college students cannot form such representations and instead compare fraction magnitudes by representing numerators and den ..."
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We tested whether adults can use integrated, analog, magnitude representations to compare the values of fractions. The only previous study on this question concluded that even college students cannot form such representations and instead compare fraction magnitudes by representing numerators and denominators as separate whole numbers. However, atypical characteristics of the presented fractions might have provoked the use of atypical comparison strategies in that study. In our 3 experiments, university and community college students compared more balanced sets of singledigit and multidigit fractions and consistently exhibited a logarithmic distance effect. Thus, adults used integrated, analog representations, akin to a mental number line, to compare fraction magnitudes. We interpret differences between the past and present findings in terms of different stimuli eliciting different solution strategies.
Core multiplication in childhood
 Cognition
, 2010
"... A dedicated, nonsymbolic, system yielding imprecise representations of large quantities (Approximate Number System, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5–7yearold children without formal schooling in multiplication and divis ..."
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A dedicated, nonsymbolic, system yielding imprecise representations of large quantities (Approximate Number System, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5–7yearold children without formal schooling in multiplication and division were given a task requiring a scalar transformation of large approximate numerosities, presented as arrays of objects. In different conditions, the required calculation was doubling, quadrupling, or increasing by a fractional factor (2.5). In all conditions, participants were able to represent the outcome of the transformation at abovechance levels, even on the earliest training trials. Their performance could not be explained by processes of repeated addition, and it showed the critical ratio signature of the ANS. These findings provide evidence for an untrained, intuitive process of calculating multiplicative numerical relationships, providing a further foundation for formal arithmetic instruction.
Developmental Change in Numerical Estimation
"... Mental representations of numerical magnitude are commonly thought to undergo discontinuous change over development in the form of a “representational shift. ” This idea stems from an apparent categorical shift from logarithmic to linear patterns of numerical estimation on tasks that involve transla ..."
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Mental representations of numerical magnitude are commonly thought to undergo discontinuous change over development in the form of a “representational shift. ” This idea stems from an apparent categorical shift from logarithmic to linear patterns of numerical estimation on tasks that involve translating between numerical magnitudes and spatial positions (such as numberline estimation). However, the observed patterns of performance are broadly consistent with a fundamentally different view, based on psychophysical modeling of proportion estimation, that explains the data without appealing to discontinuous change in mental representations of numerical magnitude. The present study assessed these 2 theories’ abilities to account for the development of numerical estimation in 5 through 10yearolds. The proportional account explained estimation patterns better than the logarithmictolinearshift account for all age groups, at both group and individual levels. These findings contribute to our understanding of the nature and development of the mental representation of number and have more general implications for theories of cognitive developmental change.
SpaceTime Interdependence and Sensory Modalities: Time Affects Space in the Hand But Not in the Eye
"... Time and space are intimately related, but what is the real nature of this relationship? Is time mapped metaphorically onto space, or do the two domains share a common representational format? In the present paper, participants touched (but could not see) physical sticks while listening to an audito ..."
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Time and space are intimately related, but what is the real nature of this relationship? Is time mapped metaphorically onto space, or do the two domains share a common representational format? In the present paper, participants touched (but could not see) physical sticks while listening to an auditory note. Judgements of stick length were affected by concurrent note duration, but not vice versa. When participants were allowed to see as well as touch the sticks, however, the effects reversed. These findings run counter to the spatial metaphor account of time, which claims that effects of space on time should always be stronger than those of time on space. Rather, our findings support the spatial representation account, in which time and space share a common neural substrate that may be affected by concurrent temporal or spatial information, depending on the perceptual acuity of the modality used to perceive space.
Assessing “economic value”: Symbolicnumber mappings predict risky and riskless valuations
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Address for correspondence:
"... Theoretical models of unsupervised category learning postulate that humans “invent ” categories to accommodate new patterns, but tend to group stimuli into a small number of categories. This “Ockham’s razor ” principle is motivated by normative rules of statistical inference. If categories influence ..."
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Theoretical models of unsupervised category learning postulate that humans “invent ” categories to accommodate new patterns, but tend to group stimuli into a small number of categories. This “Ockham’s razor ” principle is motivated by normative rules of statistical inference. If categories influence perception, then one should find effects of category invention on simple perceptual judgments. In two experiments, we tested this prediction by asking participants to estimate the number of colored circles on a computer screen, with the the number of circles drawn from a colorspecific distribution. When the distributions associated with each color overlapped, paricipants ’ estimates were biased towards values intermediate between the two means, indicating that they grouped the stimuli into a single category. These data suggest that humans favor simpler explanations of sensory inputs. In contrast, when the distributions associated with each color overlapped minimally, the bias was reduced, indicating that sensory evidence for more complex explanations can override the simplicity bias.
Summary
"... have implicated human parietal cortex in numerical processing, and macaque electrophysiology has shown that intraparietal areas house neurons tuned to numerosity. Yet although the areas responding overall during numerical tasks have been well defined by neuroimaging, a direct demonstration of indivi ..."
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have implicated human parietal cortex in numerical processing, and macaque electrophysiology has shown that intraparietal areas house neurons tuned to numerosity. Yet although the areas responding overall during numerical tasks have been well defined by neuroimaging, a direct demonstration of individual number coding by spatial patterns has thus far been elusive. Results: We used multivariate pattern recognition on highresolution functional imaging data to decode the information content of finescale signals evoked by different individual numbers. Parietal activation patterns for individual numerosities could be accurately discriminated and generalized across changes in lowlevel stimulus parameters. Distinct patterns were evoked by symbolic and nonsymbolic number formats, and individual digits were less accurately decoded (albeit still