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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 sugg ..."
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Cited by 155 (21 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.
Modulation of Parietal Activation by Semantic Distance in a Number Comparison Task
 NeuroImage
, 2001
"... INTRODUCTION How do we go from seeing a word to accessing its meaning? Classical models of word processing postulate that words are initially recognized in modalityspecific input lexicons before contacting a common semantic representation (Caramazza, 1996; Morton, 1979). This predicts that areas wh ..."
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Cited by 49 (19 self)
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INTRODUCTION How do we go from seeing a word to accessing its meaning? Classical models of word processing postulate that words are initially recognized in modalityspecific input lexicons before contacting a common semantic representation (Caramazza, 1996; Morton, 1979). This predicts that areas which are engaged in semanticlevel processing should activate in direct correlation with the amount of semantic manipulation required by the task and do so independent of the modality of presentation of the concept (Chao et al., 2000; Perani et al., 1999; Vandenberghe et al., 1996). Here, we attempt to identify the cerebral areas engaged in the coding and internal manipulation of an abstract semantic content, the meaning of number words. Although numbers can be written in multiple notations, such as words or digits, the parietal lobes are thought to comprise a notationindependent representation of their semantic content as quantities. According to the "triplecode model" of number process
Synaesthesia  A Window Into Perception, Thought and Language
, 2001
"... We investigated graphemecolour synaesthesia and found that: (1) The induced colours led to perceptual grouping and popout, (2) a grapheme rendered invisible through `crowding' or lateral masking induced synaesthetic colours  a form of blindsight  and (3) peripherally presented graphemes did ..."
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Cited by 43 (1 self)
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We investigated graphemecolour synaesthesia and found that: (1) The induced colours led to perceptual grouping and popout, (2) a grapheme rendered invisible through `crowding' or lateral masking induced synaesthetic colours  a form of blindsight  and (3) peripherally presented graphemes did not induce colours even when they were clearly visible. Taken collectively, these and other experiments prove conclusively that synaesthesia is a genuine perceptual phenomenon, not an effect based on memory associations from childhood or on vague metaphorical speech. We identify different subtypes of numbercolour synaesthesia and propose that they are caused by hyperconnectivity between colour and number areas at different stages in processing; lower synaesthetes may have crosswiring (or crossactivation) within the fusiform gyrus, whereas higher synaesthetes may have crossactivation in the angular gyrus. This hyperconnectivity might be caused by a genetic mutation that causes defective pruning of connections between brain maps. The mutation may further be expressed selectively (due to transcription factors) in the fusiform or angular gyri, and this may explain the existence of different forms of synaesthesia. If expressed very diffusely, there may be extensive crosswiring between brain regions that represent abstract concepts, which would explain the link between creativity, metaphor and synaesthesia (and the higher incidence of synaesthesia among artists and poets). Also, hyperconnectivity between the sensory cortex and amygdala would explain the heightened aversion synaesthetes experience when seeing numbers printed in the `wrong' colour. Lastly, kindling (induced hyperconnectivity in the temporal lobes of temporal lobe epilepsy [TLE] patients) may explain the purp...
ORIGINS OF NUMBER SENSE: LargeNumber Discrimination in Human Infants
, 2003
"... Four experiments investigated infants' sensitivity to large, approximate numerosities in auditory sequences. Prior studies provided evidence that 6monthold infants discriminate large numerosities that differ by a ratio of 2.0, but not 1.5, when presented with arrays of visual forms in which many c ..."
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Cited by 29 (3 self)
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Four experiments investigated infants' sensitivity to large, approximate numerosities in auditory sequences. Prior studies provided evidence that 6monthold infants discriminate large numerosities that differ by a ratio of 2.0, but not 1.5, when presented with arrays of visual forms in which many continuous variables are controlled. The present studies used a headturn preference procedure to test for infants' numerosity discrimination with auditory sequences designed to control for element duration, sequence duration, interelement interval, and amount of acoustic energy. Sixmonthold infants discriminated 16 from 8 sounds but failed to discriminate 12 from 8 sounds, providing evidence that the same 2.0 ratio limits numerosity discrimination in auditorytemporal sequences and visualspatial arrays. Ninemonthold infants, in contrast, successfully discriminated 12 from 8 sounds, but not 10 from 8 sounds, providing evidence that numerosity discrimination increases in precision over development, prior to the emergence of language or symbolic counting.
The mental number line and the human angular gyrus
 Neuroimage
, 2001
"... To investigate the hemispheric organization of a languageindependent spatial representation of number magnitude in the human brain we applied focal repetitive transcranial magnetic stimulation (rTMS) to the right or left angular gyrus while subjects performed a number comparison task with numbers b ..."
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Cited by 15 (0 self)
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To investigate the hemispheric organization of a languageindependent spatial representation of number magnitude in the human brain we applied focal repetitive transcranial magnetic stimulation (rTMS) to the right or left angular gyrus while subjects performed a number comparison task with numbers between 31 and 99. Repetitive TMS over the angular gyrus disrupted performance of a visuospatial search task, and rTMS at the same site disrupted organization of the putative “number line. ” In some cases the pattern of disruption caused by angular gyrus rTMS suggested that this area normally mediates a spatial representation of number. The effect of angular gyrus rTMS on the number line task was specific. rTMS had no disruptive effect when delivered over another parietal region, the supramarginal gyrus, in either the left or the right hemisphere. © 2001 Academic Press
Mathematical cognition
 In
, 2005
"... 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 scie ..."
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Cited by 15 (2 self)
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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 tictactoe. 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 nonhuman animal cognition suggests.
Longterm semantic memory versus contextual memory in unconscious number processing
 Journal of Experimental Psychology: Learning, Memory, and Cognition
, 2003
"... Subjects classified visible 2digit numbers as larger or smaller than 55. Target numbers were preceded by masked 2digit 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 vis ..."
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Cited by 13 (2 self)
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Subjects classified visible 2digit numbers as larger or smaller than 55. Target numbers were preceded by masked 2digit 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 longterm 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 longterm 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
A Neural Model of How the Brain Represents and Compares MultiDigit Numbers: Spatial and Categorical Processes
, 2003
"... Both animals and humans represent and compare numerical quantities, but only humans have evolved multidigit placevalue number systems. This article develops a Spatial Number Network, or SpaN, model to explain how these shared numerical capabilities are computed using a spatial representation of nu ..."
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Cited by 11 (5 self)
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Both animals and humans represent and compare numerical quantities, but only humans have evolved multidigit placevalue number systems. This article develops a Spatial Number Network, or SpaN, model to explain how these shared numerical capabilities are computed using a spatial representation of number quantities in the Where cortical processing stream, notably the inferior parietal cortex. Multidigit numerical representations that obey a placevalue principle are proposed to arise through learned interactions between categorical language representations in the What cortical processing stream and the Where spatial representation. Learned semantic categories that symbolize separate digits, as well as place markers like `ty,' `hundred,' and `thousand,' are associated through learning with the corresponding spatial locations of the Where representation. Such WhattoWhere auditorytovisual learning generates placevalue numbers as an emergent property, and may be compared with other examples of multimodal crossmodality learning, including synesthesia. The model quantitatively simulates error rates in quantification and numerical comparison tasks, and reaction times for number priming and numerical assessment and comparison tasks. In the Where cortical process, transient responses to inputs are integrated before they activate an ordered spatial map that selectively responds to the number of events in a sequence and exhibits Weber law properties. Numerical comparison arises from activity pattern changes across the spatial map that define a `directional comparison wave.' Variants of these model mechanisms have elsewhere been used to explain data about other Where stream phenomena, such as motion perception, spatial attention, and target tracking. The model is compared wi...
Making sense of number sense: Implications for children with mathematical disabilities
 Journal of Learning Disabilities
, 2005
"... Drawing on various approaches to the study of mathematics learning, Gersten, Jordan, and Flojo (in this issue) explore the implications of this research for identifying children at risk for developing mathematical disabilities. One of the key topics Gersten et al. consider in their review is that of ..."
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Cited by 9 (1 self)
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Drawing on various approaches to the study of mathematics learning, Gersten, Jordan, and Flojo (in this issue) explore the implications of this research for identifying children at risk for developing mathematical disabilities. One of the key topics Gersten et al. consider in their review is that of “number sense. ” I expand on their preliminary effort by examining in detail the diverse set of components purported to be encompassed by this construct. My analysis reveals some major differences between the ways in which number sense is defined in the mathematical cognition literature and its definition in the literature in mathematics education. I also present recent empirical evidence and theoretical perspectives bearing on the importance of measuring the speed of making magnitude comparisons. Finally, I discuss how differing conceptions of number sense inform the issue of whether and to what extent it may be teachable. Gersten, Jordan, and Flojo (in this issue) judiciously review and appraise relevant empirical evidence and theoretical perspectives pertaining to issues of both the early identification of mathematical disabilities (MD) and their remediation.
Nonverbal arithmetic in humans: Light from noise
"... manipulation of these nonverbal magnitude representations is sparse and lacking in depth. This study uses the analysis of variability as a tool for understanding properties of these combinatorial processes. Human subjects participated in tasks requiring responses dependent upon the addition, subtrac ..."
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Cited by 7 (1 self)
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manipulation of these nonverbal magnitude representations is sparse and lacking in depth. This study uses the analysis of variability as a tool for understanding properties of these combinatorial processes. Human subjects participated in tasks requiring responses dependent upon the addition, subtraction, or reproduction of nonverbal counts. Variance analyses revealed that the magnitude of both inputs and answer contributed to the variability in the arithmetic responses, with operand variability dominating. Other contributing factors to the observed variability and implications for logarithmic versus scalar models of magnitude representation are discussed in light of these results. Humans and other animals appear to compute descriptive statistics in a variety of domains—from language (e.g., Aslin, Saffran, & Newport, 1999), to foraging (Gallistel, 1990), to vision (e.g., Ariely, 2001) and motor skills (Trommershäuser, Maloney, & Landy, 2003). These statistics may derive from mental magnitudes representing elementary abstractions like number, duration, and distance (Gallistel, Gelman, & Cordes, 2006). These magnitude