Chapter 17 – Neurocognitive Start-Up Tools for Symbolic Number Representations*

Publisher Summary This chapter reviews the features of two preverbal systems for numerical quantification, the approximate number system (ANS) and the object tracking system (OTS). This chapter also critically assesses their role in cultural learning of symbolic numbers. The most important defining feature of the ANS is that it represents number in an approximate and compressed fashion, in such a way that two sets can be discriminated only if they differ by a given numerical ratio. The OTS is a mechanism by which objects are represented as distinct individuals that can be tracked through time and space. This core system for representing objects centers on the spatio-temporal principles of continuity and contact. One of the defining properties of this system is that it is limited in capacity to three to four individuals at a time. Event-related potential (ERP) source reconstruction was not performed in this study, and this does not allow any conclusions on anatomical dissociation between the ANS and the OTS. In sum, although somewhat inconsistently across studies, the OTS seems to be associated with regions of the posterior parietal and occipital cortices that do not seem to overlap with regions involved in the ANS. The electrophysiological signatures of the two systems also appear to be distinct.

[1]  Fei Xu,et al.  Linking Visual Attention and Number Processing in the Brain: The Role of the Temporo-parietal Junction in Small and Large Symbolic and Nonsymbolic Number Comparison , 2007, Journal of Cognitive Neuroscience.

[2]  Daniel Ansari,et al.  Mapping numerical magnitudes onto symbols: the numerical distance effect and individual differences in children's mathematics achievement. , 2009, Journal of experimental child psychology.

[3]  J. Ziegler,et al.  Reading acquisition, developmental dyslexia, and skilled reading across languages: a psycholinguistic grain size theory. , 2005, Psychological bulletin.

[4]  Marco Zorzi,et al.  A Computational Model of Number Comparison , 2020, Proceedings of the Twenty First Annual Conference of the Cognitive Science Society.

[5]  C. Gallistel,et al.  Preverbal and verbal counting and computation , 1992, Cognition.

[6]  Wim Fias,et al.  Representation of Number in Animals and Humans: A Neural Model , 2004, Journal of Cognitive Neuroscience.

[7]  H. Coslett,et al.  Simultanagnosia. To see but not two see. , 1991, Brain : a journal of neurology.

[8]  David C Burr,et al.  Subitizing but not estimation of numerosity requires attentional resources. , 2010, Journal of vision.

[9]  S. Dehaene Origins of Mathematical Intuitions , 2009, Annals of the New York Academy of Sciences.

[10]  Daniel Ansari,et al.  Age-related Changes in the Activation of the Intraparietal Sulcus during Nonsymbolic Magnitude Processing: An Event-related Functional Magnetic Resonance Imaging Study , 2006, Journal of Cognitive Neuroscience.

[11]  Melissa E. Libertus,et al.  Stable individual differences in number discrimination in infancy. , 2010, Developmental science.

[12]  David J. Freedman,et al.  Dynamic population coding of category information in inferior temporal and prefrontal cortex. , 2008, Journal of neurophysiology.

[13]  Michael Andres,et al.  Mode-dependent and mode-independent representations of numerosity in the right intraparietal sulcus , 2010, NeuroImage.

[14]  D G Gadian,et al.  Calculation difficulties in children of very low birthweight: a neural correlate. , 2001, Brain : a journal of neurology.

[15]  N. R. Franks,et al.  Chimpanzees and the mathematics of battle , 2002, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[16]  Susan Carey,et al.  Cognitive Foundations of Arithmetic: Evolution and Ontogenisis , 2001 .

[17]  R. Johnston,et al.  Exploring the roles of the visual‐spatial sketch pad and central executive in children's arithmetical skills: Views from cognition and developmental neuropsychology , 1999 .

[18]  S. Luck,et al.  The development of visual short-term memory capacity in infants. , 2003, Child development.

[19]  V. Michel,et al.  Recruitment of an Area Involved in Eye Movements During Mental Arithmetic , 2009, Science.

[20]  K. Wynn Children's acquisition of the number words and the counting system , 1992, Cognitive Psychology.

[21]  M. Brysbaert,et al.  Semantic priming in number naming , 2002, The Quarterly journal of experimental psychology. A, Human experimental psychology.

[22]  E. Spelke,et al.  Language and Conceptual Development series Core systems of number , 2004 .

[23]  S. Dehaene,et al.  Functional and Structural Alterations of the Intraparietal Sulcus in a Developmental Dyscalculia of Genetic Origin , 2003, Neuron.

[24]  Stanislas Dehaene,et al.  Distinct Cerebral Pathways for Object Identity and Number in Human Infants , 2008, PLoS biology.

[25]  Bahador Bahrami,et al.  A Candidate for the Attentional Bottleneck: Set-size Specific Modulation of the Right TPJ during Attentive Enumeration , 2011, Journal of Cognitive Neuroscience.

[26]  P. Bryant,et al.  Categorizing sounds and learning to read—a causal connection , 1983, Nature.

[27]  M. Chun,et al.  Dissociable neural mechanisms supporting visual short-term memory for objects , 2006, Nature.

[28]  Bertrand Thirion,et al.  Deciphering Cortical Number Coding from Human Brain Activity Patterns , 2009, Current Biology.

[29]  Justin Halberda,et al.  Developmental change in the acuity of the "Number Sense": The Approximate Number System in 3-, 4-, 5-, and 6-year-olds and adults. , 2008, Developmental psychology.

[30]  Elizabeth S. Spelke,et al.  Symbolic arithmetic knowledge without instruction , 2007, Nature.

[31]  Andreas Nieder,et al.  Semantic Associations between Signs and Numerical Categories in the Prefrontal Cortex , 2007, PLoS biology.

[32]  E. Spelke,et al.  Large number discrimination in 6-month-old infants , 2000, Cognition.

[33]  Elizabeth S Spelke,et al.  Origins of Number Sense , 2003, Psychological science.

[34]  P. Cavanagh,et al.  The Capacity of Visual Short-Term Memory is Set Both by Visual Information Load and by Number of Objects , 2004, Psychological science.

[35]  Elizabeth M. Brannon,et al.  The Neural Development of an Abstract Concept of Number , 2009, Journal of Cognitive Neuroscience.

[36]  Stanislas Dehaene,et al.  Calibrating the mental number line , 2008, Cognition.

[37]  Andrea Facoetti,et al.  Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia , 2010, Cognition.

[38]  Brian Butterworth,et al.  Developmental dyscalculia and basic numerical capacities: a study of 8–9-year-old students , 2004, Cognition.

[39]  S. Dehaene,et al.  A Magnitude Code Common to Numerosities and Number Symbols in Human Intraparietal Cortex , 2007, Neuron.

[40]  Elizabeth S. Spelke,et al.  Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling , 2010, Cognition.

[41]  Jonathan I. Flombaum,et al.  Rhesus monkeys (Macaca mulatta) spontaneously compute addition operations over large numbers , 2005, Cognition.

[42]  D. Ansari Effects of development and enculturation on number representation in the brain , 2008, Nature Reviews Neuroscience.

[43]  S. Dehaene,et al.  Cultural Recycling of Cortical Maps , 2007, Neuron.

[44]  Z. Pylyshyn Visual indexes, preconceptual objects, and situated vision , 2001, Cognition.

[45]  Steven J Luck,et al.  Rapid Development of Feature Binding in Visual Short-Term Memory , 2006, Psychological science.

[46]  G J Hitch,et al.  Working memory impairments in children with specific arithmetic learning difficulties. , 1999, Journal of experimental child psychology.

[47]  J. Jay Todd,et al.  Capacity limit of visual short-term memory in human posterior parietal cortex , 2004, Nature.

[48]  Philippe Pinel,et al.  Tuning Curves for Approximate Numerosity in the Human Intraparietal Sulcus , 2004, Neuron.

[49]  Karen Wynn,et al.  Children's understanding of counting , 1990, Cognition.

[50]  Melissa E. Libertus,et al.  Electrophysiological evidence for notation independence in numerical processing , 2007, Behavioral and Brain Functions.

[51]  Stanislas Dehaene,et al.  PSYCHOLOGICAL SCIENCE Research Article Does Subitizing Reflect Numerical Estimation? , 2022 .

[52]  Susan Carey,et al.  One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles , 2007, Cognition.

[53]  Elizabeth M. Brannon,et al.  Monkeys match and tally quantities across senses , 2008, Cognition.

[54]  Daniel Ansari,et al.  Neural correlates of symbolic number processing in children and adults , 2005, Neuroreport.

[55]  Justin Halberda,et al.  Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia). , 2011, Child development.

[56]  Elizabeth M Brannon,et al.  Basic Math in Monkeys and College Students , 2007, PLoS biology.

[57]  L. Rips,et al.  From numerical concepts to concepts of number , 2008, Behavioral and Brain Sciences.

[58]  Justin Halberda,et al.  Individual differences in non-verbal number acuity correlate with maths achievement , 2008, Nature.

[59]  S. Dehaene,et al.  Single-trial classification of parallel pre-attentive and serial attentive processes using functional magnetic resonance imaging , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[60]  Jesse Snedeker,et al.  When is Four Far More Than Three? Children’s Generalization of Newly-Acquired Number Words , 2009 .

[61]  B. Scholl Objects and attention: the state of the art , 2001, Cognition.

[62]  ROBERT S. MOYER,et al.  Time required for Judgements of Numerical Inequality , 1967, Nature.

[63]  Bert Reynvoet,et al.  Children's representation of symbolic magnitude: the development of the priming distance effect. , 2009, Journal of experimental child psychology.

[64]  Brian Butterworth,et al.  Core information processing deficits in developmental dyscalculia and low numeracy. , 2008, Developmental science.

[65]  M. Coltheart,et al.  Is there a causal link from phonological awareness to success in learning to read? , 2004, Cognition.

[66]  M. Posner,et al.  Brain mechanisms of quantity are similar in 5-year-old children and adults. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[67]  Karen Wynn,et al.  Infants’ auditory enumeration: Evidence for analog magnitudes in the small number range , 2009, Cognition.

[68]  Bruce E. Lyon,et al.  Egg recognition and counting reduce costs of avian conspecific brood parasitism , 2003, Nature.

[69]  Elizabeth M. Brannon,et al.  Monkeys Match the Number of Voices They Hear to the Number of Faces They See , 2005, Current Biology.

[70]  Elizabeth M. Brannon,et al.  Nonverbal representations of time and number in animals and human infants. , 2003 .

[71]  David J. Freedman,et al.  Categorical representation of visual stimuli in the primate prefrontal cortex. , 2001, Science.

[72]  Elizabeth M Brannon,et al.  Spontaneous analog number representations in 3-year-old children. , 2010, Developmental science.

[73]  T. Verguts,et al.  Dissecting the symbolic distance effect: Comparison and priming effects in numerical and nonnumerical orders , 2008, Psychonomic bulletin & review.

[74]  C. Packer,et al.  Roaring and numerical assessment in contests between groups of female lions, Panthera leo , 1994, Animal Behaviour.

[75]  Gavin R. Price,et al.  Impaired parietal magnitude processing in developmental dyscalculia , 2007, Current Biology.

[76]  S. Dehaene,et al.  Exact and Approximate Arithmetic in an Amazonian Indigene Group , 2004, Science.

[77]  W. Meck Functional and neural mechanisms of interval timing , 2003 .

[78]  M. Noël,et al.  Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing , 2007, Cognition.

[79]  Stanislas Dehaene,et al.  Dissociable mechanisms of subitizing and counting: Neuropsychological evidence from simultanagnosic patients. , 1994 .

[80]  E. Spelke,et al.  Newborn infants perceive abstract numbers , 2009, Proceedings of the National Academy of Sciences.

[81]  Maro G. Machizawa,et al.  Neural activity predicts individual differences in visual working memory capacity , 2004, Nature.

[82]  Scott T. Grafton,et al.  Neural Evidence Linking Visual Object Enumeration and Attention , 1999, Journal of Cognitive Neuroscience.

[83]  S. A. Rose,et al.  Visual short-term memory in the first year of life: capacity and recency effects. , 2001, Developmental psychology.

[84]  J. Cantlon,et al.  Shared System for Ordering Small and Large Numbers in Monkeys and Humans , 2006, Psychological science.

[85]  Stanislas Dehaene,et al.  The neural basis of the Weber–Fechner law: a logarithmic mental number line , 2003, Trends in Cognitive Sciences.

[86]  Daniel C. Hyde,et al.  All Numbers Are Not Equal: An Electrophysiological Investigation of Small and Large Number Representations , 2009, Journal of Cognitive Neuroscience.

[87]  S. Vecera,et al.  Psychoanatomical substrates of Bálint's syndrome , 2002, Journal of neurology, neurosurgery, and psychiatry.

[88]  Daniel Ansari,et al.  Mathematics anxiety affects counting but not subitizing during visual enumeration , 2010, Cognition.

[89]  Stanislas Dehaene,et al.  Development of Elementary Numerical Abilities: A Neuronal Model , 1993, Journal of Cognitive Neuroscience.

[90]  Marie-Pascale Noël,et al.  Symbolic and nonsymbolic number comparison in children with and without dyscalculia , 2010, Cognition.

[91]  Manuela Piazza,et al.  How Humans Count: Numerosity and the Parietal Cortex , 2009, The Neuroscientist : a review journal bringing neurobiology, neurology and psychiatry.