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Arithmetics

Resources for dot stimuli generation (suitable for number adaptation experiment)

Control over non-numerical parameters in numerosity experiments

(accompanying m-files in zip format)

A survey of our research and main publications in the cognitive neuroscience of arithmetic

In The Number Sense, Stanislas Dehaene proposed that the human capacity for arithmetic finds its ultimate roots in a basic cerebral system for perception and mental manipulation of approximate numbers, very ancient in evolution. According to this theory, we share this system with many animal species, and it appears very early in human development, independently of language. Of course, it is a primitive system, capable only of basic computations such as estimation, comparison, addition and subtraction of approximate numbers. On this shared basis, various human cultures invent increasingly elaborate cultural tools such as Arabic symbols, counting routines, algorithms for exact addition, multiplication etc.

Thus, the origins of human arithmetic lie in both a universal core system of approximate quantity, and on various cultural tools for exact arithmetic

Our laboratory uses several methods to test this theory: neuroimaging, neuronal modelling, mental chronometry, studies of patients with acalculia or dyscalculia, and studies of populations in the Amazon.

For accessible reviews of our work, you might consult the following publications:

Neuroimaging

We use brain-imaging techniques such as fMRI to identify the human brain responses during arithmetic tasks. Our research has identified a brain area, the horizontal segment of the intraparietal sulcus, as playing an important role in the quantity representation.

The following paper provides a review of this line of research:

  • Stanislas Dehaene, Manuela Piazza, Philippe Pinel, and Laurent Cohen. Three parietal circuits for number processing. Cognitive Neuropsychology, 20:487--506, 2003. [PDF ]

Recent specialized publications on neuroimaging of arithmetic include:

  • Manuela Piazza, Veronique Izard, Philippe Pinel, Denis Le Bihan, and Stanislas Dehaene. Tuning curves for approximate numerosity in the human intraparietal sulcus.. Neuron, 44(3):547-55, October 2004. [WWW ] [PDF ] [Abstract]
  • Philippe Pinel, Manuela Piazza, Denis LeBihan, and Stanislas Dehaene. Distributed and overlapping cerebral representations of number size and luminance during comparative judgements. Neuron, 41(6):983-993, 2004. [PDF ]
  • Manuela Piazza, E. Giacomini, Denis LeBihan, and Stanislas Dehaene. Single-trial classification of parallel pre-attentive and serial attentive processes using functional magnetic resonance imaging. Proceeding of the Royal Society Biological Sciences, 270:1237--1245, 2003. [PDF ]
  • Olivier Simon, Jean-François Mangin, Laurent Cohen, Denis LeBihan, and Stanislas Dehaene. Topographical layout of hand, eye, calculation and language related areas in the human parietal lobe. Neuron, 33:475--487, 2002. [PDF ]
  • Philippe Pinel, Stanislas Dehaene, D. Rivière, and Denis LeBihan. Modulation of parietal activation by semantic distance in a number comparison task. Neuroimage, 14:1013--1026, 2001. [PDF ]
  • Lionel Naccache and Stanislas Dehaene. The priming method : imaging unconscious repetition priming reveals an abstract representation of number in the parietal lobes. Cerebral Cortex, 11:966--974, 2001. [PDF ]
  • R. Stanescu, Philippe Pinel, Pierre-François van de Moortele, Denis LeBihan, Laurent Cohen, and Stanislas Dehaene. Cerebral bases of calculation processes: Impact of number size on the cerebral circuits for exact and approximative calculation. Brain, 123:2240--2255, 2000.
  • Florence Chochon, Laurent Cohen, Pierre-François van de Moortele, and Stanislas Dehaene. Differential contributions of the left and right inferior parietal lobules to number processing. Journal Cognitive Neuroscience, 11:617--630, 1999. [PDF ]
  • Stanislas Dehaene, Elizabeth Spelke, Philippe Pinel, R. Stanescu, and S. Tsivkin. Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284:970--974, 1999. [PDF ]
  • Stanislas Dehaene. The organization of brain activations in number comparison: Event-related potentials and the additive-factors method. Journal of Cognitive Neuroscience, 8:47--68, 1996.
  • Stanislas Dehaene, Nathalie Tzourio, V. Frak, L. Raynaud, Laurent Cohen, Jacques Mehler, and Bernard Mazoyer. Cerebral activations during number multiplication and comparison: a PET study. Neuropsychologia, 34:1097--1106, 1996.

Neuronal modeling and cross-species comparisons

In 1993, Stanislas Dehaene and Jean-Pierre Changeux developed a neuronal network model of number processing, which made the prediction that the parietal cortex should contain “numerosity detectors”. These are neurons tuned to a specific number, and thus firing preferentially for instance to sets of 3 objects. The model explained why such a code led to classical behavioral findings such as the distance effect and Weber’s law. The model also predicted that these neurons should have a Gaussian tuning curve when plotted on a logarithmic axis of number size.

  • Stanislas Dehaene and Jean-Pierre Changeux. Development of elementary numerical abilities: A neuronal model. Journal Cognitive Neuroscience, 5:390--407, 1993.

In 2002, these predictions were beautifully verified by Andreas Nieder and Earl Miller in the macaque monkey. Their work was published in a landmark series of papers in Science, Neuron, PNAS, and Journal of Cognitive Neuroscience. For accessible descriptions of their work, you may consult:

  • Stanislas Dehaene. Single-Neuron Arithmetic. Science, 297:1652--1653, 2002. [PDF ]
  • Stanislas Dehaene. The neural basis of Weber-Fechner's law: Neuronal recordings reveal a logarithmic scale for number. Trends in Cognitive Science, 7:145--147, 2003. [PDF ]

Following Nieder and Miller’s work, we used fMRI to demonstrate a similar sensitivity to numerosity in the human intraparietal sulcus, using an fMRI adaptation method:

  • Manuela Piazza, Veronique Izard, Philippe Pinel, Denis Le Bihan, and Stanislas Dehaene. Tuning curves for approximate numerosity in the human intraparietal sulcus.. Neuron, 44(3):547-55, October 2004. [WWW ] [PDF ] [Abstract]

The possibility of looking at number at the single-neuron level opens up many possibilities for further research. Particularly important is the prospect of being able to characterize the similarities and differences between the human and animal codes for number, and to investigate the changes induces by the acquisition of language and symbols. While this research is only in its earliest stages, we have proposed homologies between human and macaque cortical maps in the following papers:

  • Stanislas Dehaene, Nicolas Molko, Laurent Cohen, and Anna Wilson. Arithmetic and the brain. Current Opinion in Neurobiology, 14:218-224, 2004. [PDF ]
  • Olivier Simon, Jean-François Mangin, Laurent Cohen, Denis LeBihan, and Stanislas Dehaene. Topographical layout of hand, eye, calculation and language related areas in the human parietal lobe. Neuron, 33:475--487, 2002. [PDF ]

In collaboration with Marc Hauser in Harvard, we have also studied behavioral paradigms for numerosity discrimination in the Tamarin monkey:

  • Marc Hauser, Stanislas Dehaene, Ghislaine Dehaene-Lambertz, and A. L. Patalano. Spontaneous number discrimination of multi-format auditory stimuli in cotton-top tamarins (Saguinus oedipus). Cognition, 86:B23--B32, 2002. [PDF ]

Mental chronometry

In spite of important advances in neuroimaging, chronometric methods remain essential to study the organization of mental representations in humans. In the past, we have published several papers that characterized the behavioral signature of the quantity representation. Following the seminal work of Moyer and Landauer (1967), we showed in particular that, even when numbers are presented as Arabic digits, a representation of their quantity is automatically activated and leads to distance effects in tasks such as number comparison or same-different judgements. In 1993, we also discovered and named the SNARC effect – a Spatial-Numerical Association of Response Codes which demonstrates that numbers are automatically associated with positions in space, thus supporting the metaphor of a spatially organized internal “number line”.

  • Stanislas Dehaene and J. F. Marques. Cognitive Euroscience : Scalar variability in price estimation and the cognitive consequences of switching to the Euro. Quaterly Journal of Experimental Psychology, 55(3):705--731, 2002. [PDF ]
  • Stanislas Dehaene and R. Akhavein. Attention, automaticity, and levels of representation in number processing. Journal of Experimental Psychology: Learning, Memory, and Cognition, 21:314--326, 1995. [PDF ]
  • Stanislas Dehaene, S. Bossini, and P. Giraux. The mental representation of parity and numerical magnitude. Journal of Experimental Psychology: General, 122:371--396, 1993. [PDF ]
  • Stanislas Dehaene, E Dupoux, and Jacques Mehler. Is numerical comparison digital ? Analogical and symbolic effects in two-digit number comparison. Journal of Experimental Psychology: Human Perception and Performance, 16:626--641, 1990. [PDF ]
  • Stanislas Dehaene. The psychophysics of numerical comparison : a reexamination of apparently incompatible data. Perception & Psychophysics, 45:557--566, 1989.

In 1998, Lionel Naccache, Etienne Koechlin and Stanislas Dehaene introduced subliminal priming combined with neuroimaging in the number domain. This work showed that Arabic numerals or spelled-out number words could be flashed for a very short duration and, in spite of their invisibility, could facilitate the processing of other numbers presented subsequently. This method has become a very useful tool to probe the organization of number representations, but also to study the differences between conscious and non-conscious processing – now a main topic of research in the lab.

  • Lionel Naccache, E Blandin, and Stanislas Dehaene. Unconscious masked priming depends on temporal attention. Psychological Science, pp 416--424, 2002. [PDF ]
  • Lionel Naccache and Stanislas Dehaene. The priming method : imaging unconscious repetition priming reveals an abstract representation of number in the parietal lobes. Cerebral Cortex, 11:966--974, 2001. [PDF ]
  • Lionel Naccache and Stanislas Dehaene. Unconscious semantic priming extends to novel unseen stimuli. Cognition, 80:215--229, 2001. [PDF ]
  • Etienne Koechlin, Lionel Naccache, Elizabeth Block, and Stanislas Dehaene. Primed numbers: exploring the modularity of numerical representations with masked and unmasked priming. Journal of Experimental Psychology: Human Perception and Performance, 25:1882--1905, 1999. [PDF ]
  • Stanislas Dehaene, Lionel Naccache, G. Le Clec'H, E. Koechlin, M. Mueller, Ghislaine Dehaene-Lambertz, P. F. van de Moortele, and Denis LeBihan. Imaging unconscious semantic priming. Nature, 395:597--600, 1998. [PDF ]

Acalculia

Deficits of number processing can be extremely informative about the cerebral organization of number processes. Stanislas Dehaene and Laurent Cohen have collaborated on many years on single-case studies of adult patients with selective dysfunctions of number processing and calculation (acalculia). This work has revealed important dissociations between operations, particularly between exact arithmetic based on rote verbal learning (e.g. multiplication tables), and other operations such as subtraction or estimation which seem to involve internal manipulations of quantities.

  • Cathy Lemer, Stanislas Dehaene, Elizabeth Spelke, and Laurent Cohen. Approximate quantities and exact number words: Dissociable systems. Neuropsychologia, 2003. [PDF ]
  • Laurent Cohen and Stanislas Dehaene. Calculating without reading : Unsuspected residual abilities in pure alexia. Cognitive Neuropsychology, 17:563--583, 2000. [PDF ]
  • Laurent Cohen, Stanislas Dehaene, Florence Chochon, Stéphane Lehéricy, and Lionel Naccache. Language and calculation within the parietal lobe: A combined cognitive, anatomical and fMRI study. Neurospychologia, 138:1426--1440, 2000. [PDF ]
  • Stanislas Dehaene and Laurent Cohen. Language and elementary arithmetic -- dissociations between operations. Brain and Language, 69:492-495, 1999.
  • Stanislas Dehaene and Laurent Cohen. Cerebral pathways for calculation: double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33:219--250, 1997. [PDF ]
  • Laurent Cohen and Stanislas Dehaene. Cerebral networks for number processing: Evidence from a case of posterior callosal lesion. NeuroCase, 2:155--174, 1996.
  • Laurent Cohen and Stanislas Dehaene. Number reading in pure alexia: the effect of hemispheric asymmetries and task demands. NeuroCase, 1:121--137, 1995.
  • Stanislas Dehaene and Laurent Cohen. Dissociable mechanisms of subitizing and counting: neuropsychological evidence from simultanagnosic patients. Journal od experimental Psychology: Human Perception and Performance, 20:958--975, 1994. [PDF ]
  • Laurent Cohen and Stanislas Dehaene. Amnesia for arithmetic facts: a single case study. Brain and Language, 47:214--232, 1994. [PDF ]
  • Laurent Cohen, Stanislas Dehaene, and Patrick Verstichel. Number words and number non-words: A case of deep dyslexia extending to arabic numerals. Brain, 117:267--279, 1994.
  • Stanislas Dehaene and Laurent Cohen. Two mental calculation systems: A case study of severe acalculia with preserved approximation. Neuropsychologia, 29:1045--1074, 1991.
  • Laurent Cohen and Stanislas Dehaene. Neglect dyslexia for numbers? A case report. Cognitive Neuropsychology, 8:39--58, 1990.

Dyscalculia

A prediction of the number sense hypothesis is that there might exist children with very early deficits of number sense. Early injuries to the parietal lobe, due to perinatal injuries or to genetic causes, should lead to a disruption of number sense and therefore to a lifelong impairment in arithmetic. Thus, our lab has a growing interest in dyscalculia – impairments of number processing and calculation in children with other normal education and intelligence. Our work has largely focused on characterizing the dyscalculia that occurs in Turner’s syndrome, a genetic disease associated with loss of one X chromosome. In collaboration with Ann Streissguth’s team in Seattle, we have also shown that foetal alcohol syndrome (exposition of the foetus to alcohol during pregnancy) is often accompanied by striking deficits of number processing.

  • Nicolas Molko, Arnaud Cachia, Denis Rivière, Jean-François Mangin, Marie Bruandet, Denis LeBihan, Laurent Cohen, and Stanislas Dehaene. Functional and structural alterations of the intraparietal sulcus in a developmental dyscalculia of genetic origin. Neuron, 40(4):847-858, 2003. [PDF ]
  • Marie Bruandet, Nicolas Molko, Laurent Cohen, and Stanislas Dehaene. A cognitive characterization of diyscalculia in Turner syndrome. Neuropsychologia, 2003. [PDF ]
  • K. Kopera-Frye, Stanislas Dehaene, and A. P. Streissguth. Impairments of number processing induced by prenatal alcohol exposure. Neuropsychologia, 34:1187--1196, 1996.

We have developed and are now testing a new tool for rehabilitation of dyscalculia: a computer game ("The Number Race") designed especially to help children practice their quantity manipulation skills and the links from quantity to Arabic numerals and number words. Anna Wilson and Stanislas Dehaene have designed an adaptive computer algorithm that can detect in which domain a child experiences difficulty, and present problems at the relevant level, difficult enough to be challenging, but easy enough to avoid discouragement. This software is open-source (under a GNU public licence) and can be downloaded here from our website.

For more information on dyscalculia, visit this page.

Amazon

Recently, a new exciting opportunity to test the number sense hypothesis has emerged with a collaboration with Pierre Pica, a linguist at CNRS (UMR 7023 “Formal Structures of Language”) and who collaborates with the linguistic section of the Brazilian National Museum (Rio de Janeiro). Pierre Pica regularly visits Brazil where he studies the indigenous populations of the Amazon. He has particularly focused on the Mundurukú, who live in an autonomous territory of the state of Para. The Mundurukú have a very limited vocabulary of number words, essentially limited to words for one, two, three, four and five. Studying this people provides a unique opportunity to investigate the capacity for arithmetic processing in the absence of a well-developed language of number words.

Of course, it is not possible to study the Mundurukú with neuroimaging or neuropsychological methods. However, Pierre Pica, Cathy Lemer, Véronique Izard and Stanislas Dehaene developed a series of behavioral tests which were run on a solar-powered computer with 55 native Mundurukú subjects. The results, which appeared on October 15th 2004 in the journal Science, indicate that, in spite of their reduced lexicon, the Mundurukú have a well-developed capacity for numerical approximation. They can estimate, add and subtract approximate numbers presented as sets of dots, even when the numbers range all the way up to 80 or more. However, their lack of a counting routine prevents them from being able to perform exact calculations even with small numbers. For instance, they were unable to report the precise outcome of 6 – 4. Thus, the results help tease apart which aspects of our arithmetic competence are universal, and which are dependent on cultural and linguistic tools.

Our original Science publication is

  • Pierre Pica, Cathy Lemer, Véronique Izard, and Stanislas Dehaene. Exact and approximate arithmetic in an Amazonian indigene group.. Science, 306(5695):499-503, October 2004. [WWW ] [PDF ] [Abstract]

A press release describing this research can be found at http://www2.cnrs.fr/en/319.htm

Several movies illustrating the difficulties that the Mundurukú have when attempting to count are available on http://video.rap.prd.fr/videotheques/cnrs/grci.html.

Explanations about these movies can be found on http://www.unicog.org/docs/MovieDocumentationCountingInMunduruku.htm

(all photos of the Amazon are © Pierre Pica and CNRS)