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:
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:
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.
2. 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.
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:
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:
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
]
3. 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
]
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
]
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.
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.
5. 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
]
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.
6. 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.
