Humans share with non-human primates1–6, birds7,8 and
amphibians9 the ability to discriminate numerical magnitudes.
The parameters that underlie number-discrimination
abilities have been clearly delineated and are similar
across species and across development10,11 (FIG. 1). The so-
called ‘number sense’ (ReFs 11,12) therefore probably has
a long evolutionary history. The ability to discriminate
numerical quantity might have served functions that are
important to survival, such as making foraging decisions
or discerning the number of approaching predators13.
However, the basic ability to discriminate numeri-
cal quantities cannot fully explain the entire extent of
human numerical and mathematical skills. In human
history, a large set of uniquely human competencies
emerged that provided us with the ability to process
abstract numerical symbols (such as number words and
Arabic numerals) and perform mental arithmetic using
these new ‘mental tools’.
Most of the current research into the neuroscience
of numbers is focused on the neural correlates of the
basic representation of numerical quantity in the brain;
there is also growing exploration of the relationship
between numerical processing of quantity and visuo-
spatial processing14–16. Thus far, investigations into the
basic neural representation and processing of numerical
magnitude and into the acquisition of symbolic repre-
sentations and calculating ability have been conducted in
relative isolation from one another, and it has often been
assumed that symbolic representations are mapped onto
pre-existing non-symbolic representations of number.
This Review provides a synthesis of these different
strands of research and reassesses the hypothesis that
symbolic number representations acquire their numeri-
cal meaning through being mapped onto non-symbolic
representations. In addition, the Review integrates data
from computational modelling, functional neuroimaging
and single-cell recording studies to formulate hypotheses
for future studies that aim to increase our understanding
of the complex interactions that occur between culture
and biology in the construction of the brain processes
that underlie mathematical skills.
The IPS and numerical-magnitude processing
Data from brain-damaged patients, single-cell recordings
and functional neuroimaging have implicated the intra-
parietal sulcus (IPS) as the crucial area for the processing
of numerical magnitude. In order to understand how
such parietally mediated processes might shape the
acquisition of numerical symbols and mental arithme-
tic skills, I first discuss the current understanding of the
functioning of this brain region during basic number
Neural correlates of quantity representation. When
adults determine which of two Arabic numerals is
numerically larger, their reaction times and error rates
are inversely related to the numerical distance between
the numbers17,18. Moreover, when the distance is held
constant but the absolute size of the two numerical
quantities is increased, reaction time increases and
Laboratory, Department of
Psychology and Graduate
Program in Neuroscience,
University of Western
Ontario, Ontario N6G 2K3,
12 March 2008
The total number of items in a
set. It can be either exact or
approximate, depending on
whether the sets are counted
or the total number of items is
The difference between two
numbers. For example, the
numerical distance between
eight and five is three. Many
studies show that numerical
distance has a profound effect
on the time it takes to make a
Effects of development and
enculturation on number
representation in the brain
Abstract | A striking way in which humans differ from non-human primates is in their ability to
represent numerical quantity using abstract symbols and to use these ‘mental tools’ to perform
skills such as exact calculations. How do functional brain circuits for the symbolic
representation of numerical magnitude emerge? Do neural representations of numerical
magnitude change as a function of development and the learning of mental arithmetic?
Current theories suggest that cultural number symbols acquire their meaning by being mapped
onto non-symbolic representations of numerical magnitude. This Review provides an
evaluation of this contention and proposes hypotheses to guide investigations into the neural
mechanisms that constrain the acquisition of cultural representations of numerical magnitude.
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Reaction time (ms)
Reaction time (ms)
Mean looking time (s)
16 vs 32
16 vs 24
Nature Reviews | Neuroscience
0.20.40.6 0.8 1.0
accuracy decreases19–21. These distance and size effects
have become litmus tests for determining the nature of
basic representations of numerical quantity22.
Functional MRI (fMRI) studies19–21 have revealed
that activation in bilateral regions of the inferior parietal
lobules — specifically, the IPS (FIG. 2) — negatively cor-
relates with numerical distance. This finding has been
frequently replicated23–26, and recent data suggest that
the relationship between numerical distance and IPS
activity is disrupted in children with developmental
dyscalculia23 (BOX 1).
Numerical distance also negatively correlates with
activation in bilateral prefrontal and precentral regions,
suggesting the involvement of fronto-parietal networks
in the processing of numerical magnitude27,28. Although
some studies29 have attempted to delineate the simi-
larities and differences between prefrontal and parietal
activation during numerical-magnitude processing, the
focus has primarily been on the IPS30. Methodological
advances such as Dynamic Causal Modelling31 and
Grainger Causality Modelling32 of fMRI data should
facilitate research on network-based representations of
Single-cell recordings from neurons in the prefrontal
and parietal cortices33,34 in monkeys (FIG. 2c) have also
provided significant insight into the neural correlates
of numerical-magnitude representation. In a delayed
match-to-sample task (FIG. 2d), monkeys were trained to
judge whether a sample numerosity (an array of dots) dif-
fered from a target numerosity. Populations of neurons
in the prefrontal cortex (PFC) and IPS preferentially
fired during the presentation of a particular numeros-
ity. Hence, these cells seem to represent cardinal values or
specific places on the ‘mental number line’ (also known as
place coding or ‘labelled line’ coding). Interestingly, the
firing rate of these neurons decreased monotonically as
the numerical distance between the preferred and the
presented numerical magnitude increased (FIG. 2e). The
firing properties of these ‘number neurons’ in the mon-
key PFC and IPS explain the distance effect, as the tuning
curves of neurons with preferred numerosities that are
separated by a small numerical distance overlap more
than those of neurons with preferred numerosities that
lie far apart. Consistent with the size effect, neurons with
a high preferred numerical magnitude have wider tun-
ing curves than neurons with a low preferred numerical
magnitude33,34 (FIG. 2f).
Number neurons in the prefrontal and parietal corti-
ces have similar basic response properties, but their tem-
poral response properties differ34: IPS neurons respond
earlier than PFC neurons, suggesting that numerical
quantity might be extracted in the IPS and sent forward
to the PFC for the implementation of number-related
Modules in the brain or distributed patchworks? There
has been much debate over the extent to which the IPS
contains a number module35. In humans, processing
of symbolic (for example, Arabic numerals) and non-
symbolic (for example, arrays of dots) representations of
numerical magnitude triggers similar neural activation
in the IPS, indicating that this brain region might house
a number module23,36. However, it is unclear whether
the IPS only processes numerical magnitude — in other
words, whether there is domain-specific representation
of numerical magnitude in the IPS. Indeed, in a series
Figure 1 | Ontogenetic and phylogenetic continuity in basic signatures of
numerical-magnitude representation. a | When children and adults compare which of
two numerical stimuli is numerically larger, there is an inverse relationship between
numerical distance and reaction time; this is known as the distance effect17. Furthermore,
reaction times are positively correlated with the absolute size of the numerical stimuli
(the size effect). The distance effect exists in both children and adults, but it decreases in
strength with age128,129. b | When six-month-old infants are presented with repeated
presentations of large numbers of dots, their looking time decreases (habituation). When
looking times to old (habituated) and new numerosites are subsequently compared, the
infants’ ability to distinguish between old and new numerosities is dependent on the
distance between the numerical magnitudes130,131. For example, Xu132 found that 6-month-
old infants are sensitive to the difference between 16 and 32, but fail to discriminate
between 16 and 24. c | Rhesus macaque monkeys can learn to order pairs of numerosities
by numerical magnitude by touching them sequentially on a touchscreen. The accuracy
and speed with which the animals can do this depends on the ratio of the numerosities.
The similarity in the performances of monkeys and humans suggests that there is a high
degree of phylogenetic continuity in basic representations of numerical magnitude.
Part c reproduced, with permission, from ReF. 3 (2006) Blackwell Publishers.
NATuRe RevIeWS | neurOscience
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Reaction time (ms)
Normalized response (%)
Numerical distance from
Nature Reviews | Neuroscience
A term used to describe non-
symbolic representations of
numerical magnitude (such as
arrays of dots or squares).
The last number in a sequence;
cardinal numbers represent the
total number of items in a set.
Mental number line
A metaphor for the mental
representation of numerical
quantity, based on findings
that support an association
between space and number.
How single neurons or, in the
case of fMRI, large populations
of cells are tuned to respond
to a particular stimulus rather
than to other, similar stimuli.
A neuron might respond
preferentially to three items
but also fire during the
presentation of one or two
When brain regions respond
more to a stimulus from one
domain of cognitive processing
(for example, faces) than they
do to another (for example,
houses). Regions that exhibit
such domain-specific response
properties are thought to be
biologically determined to
represent and process stimulus
categories from a particular
of three fMRI experiments reported in one paper37, IPS
activation was similar during non-symbolic numerical-
magnitude processing and non-numerical control tasks,
suggesting that the IPS might not house a domain-
specific representation of numerical magnitude.
The idea of domain-specific brain representa-
tion rests on the assumption that a single region (in
this case the IPS) is involved in a particular type of
stimulus processing and responds to the stimulus cat-
egory regardless of the presentation format (symbolic,
Figure 2 | neural correlates of basic numerical-magnitude representation in the human and monkey brain.
a | Functional neuroimaging studies have implicated the bilateral intraparietal sulcus (IPS; shown in yellow) in numerical-
magnitude processing. b | Activation in the IPS is negatively correlated with numerical distance during a number-comparison
task27. This inverse relationship is similar to the behavioural distance effect (see red line). c | Using a delayed match-to-sample task
in which the animal has to indicate whether a sample numerosity matches a test display, the ‘number sensitivity’ of single
neurons in the macaque prefrontal33 and parietal cortices34 (d; shown in orange and yellow, respectively) was tested. The data
revealed the existence of ‘number neurons’, which fire preferentially to the presentation of a particular number of dots.
e | Although number neurons prefer a particular numerical magnitude, their response to other numerosities is related to the
numerical distance between the presented and the preferred numerical magnitude, revealing a distance effect at the level of
single neurons. Thus, a neuron that was found to prefer a numerical magnitude of three fired slightly less during the presentation
of four dots and less still during the presentation of five dots or one dot. f | Consistent with the size effect, the degree to which
number neurons respond to other numerosities (their filter bandwidth) increases as a function of the size of their preferred
numerical magnitude. A similar bandwidth function could be observed for the behavioural data, where monkeys’ incorrect
classification of non-matching target numerosities increased as a function of the sample numerosity. Part b reproduced, with
permission, from ReF. 27 (2001) Academic Press. Part c reproduced, with permission, from ReF. 55 (2005) Macmillan Publishers
Ltd. Parts d and e reproduced, with permission, from ReF. 33 (2002) American Association for the Advancement of Science.
280 | APRIl 2008 | voluMe 9
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Nature Reviews | Neuroscience
A term from cognitive science
that refers to the notion that
different cognitive domains (for
example, language, visuo-
spatial cognition and social
cognition domains) have
principles and are represented
in encapsulated modules.
non-symbolic, auditory or visual). As pointed out by
Nieder37, this hypothesis is derived from psychological
theories of modularity38, and it is not clear how well such
assumptions transfer to brain-based modularity of cog-
nitive processing. Furthermore, to establish that a brain
region is specifically involved in a particular cognitive
process, one needs to compare the engagement of that
region during the cognitive process of interest with its
activation in response to an infinite number of other
stimulus categories and processes — an impossible
Two recent fMRI39,40 studies offer a different view on
the relative degree of specialization of the IPS for numer-
ical magnitude. Both revealed substantial overlap in IPS
activity for numerical and non-numerical comparisons
of quantity, but both also identified cortical IPS regions
that showed relatively stronger activation for numerical
than for non-numerical magnitudes. This suggests that
the IPS contains a patchwork of areas that have over-
lapping biases for a particular stimulus category, rather
than neatly segregated modules. In the future, multi-voxel
pattern analysis41–43 might help to elucidate the overlap
between representations of non-numerical and numeri-
cal quantity in the IPS, the distribution of local biases
for particular types of comparison and the reliability of
these biases across subjects.
The notion that numerical and non-numerical
quantities are represented in overlapping areas of the
IPS has been confirmed in single-cell recording stud-
ies44. Monkeys were trained to discriminate either dif-
ferent numerosities or different line lengths (FIG. 3a); IPS
neurons responded to length, numerical magnitude or
both, indicating that there is both discrete (numerical)
and continuous (non-numerical) coding of magnitude in
the IPS, even at the single-cell level. Furthermore, these
results suggest that there is no clear topographical seg-
regation between populations of neurons that respond
to either discrete or continuous magnitude (FIG. 3b).
Another study45 found similar common and segregated
coding of sequential and simultaneously presented
numerical magnitude in the monkey IPS.
Although most current evidence suggests that there
are both distributed and overlapping representations
of numerical and non-numerical magnitude, one fMRI
study46 had different results: participants were presented
with either different numbers of blue and green squares
(discrete, numerical quantities) or with the same display
transformed into smoothly changing distributions of
green and blue hues (continuous, non-numerical quanti-
ties), and were asked to judge whether they saw more blue
or more green. There was greater bilateral activation of
the IPS during discrete than during continuous quantity
judgements, suggesting that at least some parts of the IPS
are specific for the representation of discrete quantity.
Importantly, the resolution of fMRI is still severely lim-
ited in comparison with that of single-unit recordings and,
as discussed above44, such recordings have already revealed
highly distributed representations of numerical and non-
numerical magnitude, as well as format-specific and
format-general representation45 in the IPS. Thus, it is
possible that the fMRI findings that suggest that parts of
the IPS are specialized for numerical magnitude might
simply reflect the existence of areas in which relatively
more neurons code for numerical than non-numerical
magnitude. Therefore, even data that show greater
engagement for numerical versus non-numerical mag-
nitude processing might be consistent with the notion
of distributed networks of activation with local biases.
Similarly, a region that responds to both symbolic and
non-symbolic representations of numerical magnitude
might nevertheless contain neurons that code specifically
for one format. In other words, it is currently difficult
Box 1 | Neural basis of developmental dyscalculia
It is a little-known fact119,120 that approximately 5% (although estimates vary between 3
and 11%) of children who exhibit normal intelligence present a specific and persistent
difficulty with calculation and mental arithmetic, called developmental dyscalculia.
Compared with developmental dyslexia, there has been little research into the
behavioural and neural basis of developmental dyscalculia121, but behavioural studies
suggest that it is associated with impairments of basic numerical-magnitude
processing122. The few studies that have investigated the neural basis of developmental
dyscalculia consistently suggest that there is a pattern of structural and functional
alterations in the intraparietal sulcus (IPS) and the prefrontal cortex (PFC) (see figure:
yellow indicates the IPS; red, blue and green indicate where structural and anatomical
abnormalities have been found in the left (a) and right (b) hemispheres). Isaacs et al.123
used structural MRI to compare the brains of children who had low birth weight and who
exhibited calculation difficulties with those of children of similar low birth weight but
with normal scores on a calculation test. Children with specific calculation difficulties
were found to have less grey matter in the left IPS (red area in the figure). A recent
comparison of the structural neuroanatomy of children with and without
developmental dyscalculia124 revealed that children with developmental dyscalculia
have less right-parietal grey matter and have grey matter abnormalities in regions of
the frontal cortex (blue areas in the figure). A functional neuroimaging study of
developmental dyscalculia125 did not find any abnormalities in brain activation during
number-comparison or calculation in children with dyscalculia, but the children did
show lower overall activation during approximate calculation in regions of the PFC
and IPS. Another functional MRI study126 compared children with dyscalculia with a
group of age-matched typically developing peers, using a non-symbolic numerical-
magnitude comparison task. Although the non-symbolic numerical distance task
significantly modulated activity in the right IPS in the typically developing participants,
this effect was absent in the children with developmental dyscalculia (green area in the
figure). Consistent with the notion of a right-parietal dysfunction, Cohen-Kadosh et al.127
found that virtually lesioning the right parietal cortex in healthy adults by means of
transcranial magnetic stimulation impaired automatic number processing.
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Length protocol Numerosity protocol
Length and numerosity selectivity
No quantity selectivity
Quantity selectivity for each
Nature Reviews | Neuroscience
Multi-voxel pattern analysis
fMRI data are typically
analysed using voxel-wise
statistics; multi-voxel pattern
analysis uses pattern-
classification algorithms to
decode fMRI activity that is
distributed across multiple
The rank–order relationships
between numbers (for example,
the third in line).
to unequivocally localize modules in the brain for par-
ticular domains of cognitive processing or for particular
stimulus types using fMRI. In addition, given that the
parietal cortex is a higher-level association area of
the brain with heterogeneous functions47–49 that include,
among others, grasping50, visual attention51 and work-
ing memory52,53, it might be futile to search for spatially
localized and dissociable domain-specific representation.
It might instead be more productive to investigate how
the representation and processing of numerical magni-
tude in the parietal cortex interacts with other parietally
mediated cognitive functions14,54.
Magnitude or order? one should also consider how
brain activation during numerical-magnitude process-
ing differs from the representation of other information
that can be embodied by numerical stimuli, such as
sequential order55. Indeed, tasks that require magnitude
processing also involve processing of order-related infor-
mation56. Two recent papers57,58 directly compared the
neural bases of order and magnitude processing by con-
trasting brain activation during numerical-magnitude
comparisons with brain activation during comparisons
of non-numerical stimuli that carry order information
(letters and months). They found that the anterior IPS
responds equally to both numerical and non-numerical
order, suggesting a role for this anterior region of the IPS
in the abstract representation of ordinal information that
is not number-specific.
Although these data suggest that there is a link
between cardinal and ordinal representations of
numerical magnitude, data from brain-damaged
Figure 3 | Distributed and overlapping representations of numerical and non-numerical quantity in the
intraparietal sulcus. a | Macaque monkeys were trained on a delayed match-to-sample task in which they judged either
whether a sample line presented after a delay matched the length of a previously presented line (left panel) or whether the
number of dots in a sample array presented after a delay matched the number of dots in a previously presented array (right
panel). Responses of single cells were recorded from the intraparietal sulcus (IPS). The data suggest that there are cells that
are selective for either numerical magnitude or length, but they also reveal that there are neurons that respond to both.
b | Interestingly, no topographic organization of numerical-magnitude- or length-sensitive cells could be detected,
demonstrating that even at the level of individual neurons in the parietal cortex there is no neat spatial segregation
between the representation of numerical and non-numerical quantity. Figure reproduced, with permission, from ReF. 44
(2007) National Academy of Sciences.
282 | APRIl 2008 | voluMe 9
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248 16 3224
Number of items
Number of items
Firing rate (spikes/s)Firing rate (spikes/s)
Left IPSRight IPS
Nature Reviews | Neuroscience
patients have revealed double dissociations between
the two59,60. Furthermore, a recent event-related poten-
tial (eRP) study suggested that order and magnitude
processing might have different time-courses in the
left and right IPS and in prefrontal regions61. As was
recently suggested62, single-cell neurophysiology stud-
ies might help us to better understand the extent to
which both overlapping and segregated representations
of cardinality and ordinality exist at the single-cell
level. Nevertheless, the data suggest that the IPS might
be involved in domain-general representations of
order that are recruited during numerical-magnitude
Symbolic representations of numerical magnitude
verguts and Fias63 have put forward a model (FIG. 4a)
which proposes that symbolic and non-symbolic inputs
are transformed into internal place-coded (cardinal)
representations by different pathways. Consistent with
an earlier computational model64, their model predicts
that non-symbolic numerosities are transformed by an
intermediate step (referred to as ‘summation coding’)
into a summed representation that can be represented
internally as a place code. Consistent with the accumulator
model of magnitude representation65, the noise in the sum-
mation field is proportional to the number of inputs
that are being summed. The verguts and Fias model
Figure 4 | Different processing pathways for symbolic and non-symbolic numerical magnitude.
a | A computational model63 predicting independent input fields for symbolic and non-symbolic numerical stimuli. Unlike
symbolic inputs, non-symbolic stimuli need to be summed before they can be mapped onto the number field. This
summation yields size and distance effects because the similarity between input vectors decreases with distance and
increases with their relative size. In this model, representations of symbolic numerical magnitude are learned through the
simultaneous presentation of symbolic and non-symbolic inputs. Results of simulations suggest that there are sharper
representations for symbolic inputs, as the intermediate (hidden layer) summation coding is not required. b | Consistent
with this model63, neurons in the monkey lateral intraparietal cortex (LIP) exhibit responses that might reflect summation
coding: their responses either increase or decrease monotonically as a function of the size of presented non-symbolic
numerosities. c | Recent fMRI adaptation data67 support the notion that symbolic stimuli have sharper representations in
the left intraparietal sulcus (IPS) than non-symbolic stimuli. Although both the left and the right IPS responded to symbolic
and non-symbolic numerical deviants, the left IPS responded equally to close and far non-symbolic deviants following the
adaptation to symbolic deviants. These findings suggest that the coarse representation of both close and far non-symbolic
deviants differs from the sharp tuning of the adapted representations of symbolic numerical quantity. Part a reproduced,
with permission, from ReF. 63 (2004) MIT Press. Part b reproduced from ReF. 66. Part c reproduced, with permission, from
ReF. 67 (2007) Elsevier Science.
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Accumulator model of
A model of numerical-
magnitude processing in
which enumeration involves
the passing of impulses
through a gate into a
This summed representation
can be likened to a measuring
cup: in this analogy the level of
the accumulated impulses
represents the total number of
enumerated impulses. This is
also referred to as ‘summation
implicitly predicts how children learn symbolic repre-
sentation — namely, the representations are acquired
after the representational system has learned to associ-
ate non-symbolic inputs with internal representations
of numerical magnitude. This assumption is based on
the finding that even young infants can discriminate
between non-symbolic numerical magnitudes, and the
model thus proposes that symbolic representations are
learned through the simultaneous presentation of sym-
bolic and non-symbolic inputs. This has the consequence
that although symbolic inputs do not go through the
intermediate summation coding, their representations
in the number field are influenced by the simultaneous
non-symbolic inputs in such a way that, for example, the
presentation of a symbol for the place code ‘3’ leads to
slight co-activation of the place codes ‘2’ and ‘4’. In other
words, because symbolic place-code representations are
acquired by mapping them onto the coarse (or approxi-
mate) place-code representation of non-symbolic mag-
nitudes, the symbolic representations ‘inherit’ some of
the overlapping nature of the non-symbolic place codes.
This, according to verguts and Fias, explains why the
distance effect still occurs when symbolic representa-
tions of numerical magnitude are compared. However,
because the influence of the noisy summation coding on
symbolic representations is indirect, the tuning curves
for symbolic place codes on the number field are thought
to be sharper and the distance effect thus smaller. Recent
studies have provided support for some of the differ-
ent predictions (summation and place coding) of this
Evidence for summation coding in the lateral IPS. Recent
evidence suggests that there are cells in the monkey lat-
eral intraparietal cortex (lIP; located on the lateral bank
of the IPS) that have firing rates that either increase or
decrease monotonically with the number of visual ele-
ments presented to the monkey66 (FIG. 4b). The response
properties of these neurons nicely meet the prediction of
the ‘accumulator model’ (ReFs 65,68) and are consistent
with the idea of summation coding63,64. Consistent with
these models, it is possible that these summation-coding
neurons in the lIP serve as inputs to the place-
coding neurons in the IPS33,34,69, but how summation
and place coding are interrelated in the brain remains
an open question.
A significant strength of the above study66 is that,
unlike most single-cell recording studies, it did not
require the monkeys to be trained to discriminate
numerosity, and therefore the measured neuronal
responses are unlikely to be the result of training.
Association of symbolic and non-symbolic magnitude in
the monkey PFC. Although non-human primates do not
use numerical symbols, they can be trained to associ-
ate symbols with numerical magnitudes70,71. A recent
study with macaque monkeys72 used a delayed match-
to-sample method (FIG. 2c) to train the monkeys to both
match identical numerosities (arrays of dots) to each
other and match numerical symbols (Arabic numerals)
to numerosities. In other words, monkeys learned to
associate Arabic numerals with numerosities. Consistent
with previous studies33, neurons in the PFC preferred
particular quantities and exhibited size and distance
effects. Importantly, many of these cells preferred a
particular quantity in both the numerosity–numerosity
and the symbol–numerosity matching paradigms, sug-
gesting that prefrontal regions might have a role in map-
ping abstract numerical symbols onto their quantitative
referents. Consistent with the prediction of the model
by verguts and Fias63 that symbolic representations are
more distinct and less noisy, the tuning curves of PFC
neurons were narrower when symbols (Arabic numer-
als) were matched with non-symbolic magnitudes than
when non-symbolic magnitudes were matched with
By contrast, although neurons in the IPS showed a
preference for either symbolic or non-symbolic rep-
resentations of a particular magnitude, almost none
responded to both symbolic and non-symbolic numeri-
cal magnitudes72. Indeed, only PFC neurons seemed
to be involved in the association of non-symbolic and
symbolic representations of numerical magnitude. These
results might seem to be at odds with the notion that the
IPS is crucial for the representation of numerical magni-
tude in adult humans73. However, it is possible that the
association between non-symbolic and symbolic repre-
sentations of numerical magnitude in the adult human
IPS is the result of learning and developmental processes.
Consistent with this hypothesis are recent neuroimaging
studies28,74 which show that children recruit more pre-
frontal regions during symbolic number processing than
adults. Thus, during human brain development and the
acquisition of linguistic competencies, parietal regions
might specialize for the representation and processing
of the association between numerical magnitudes and
abstract symbols. It remains to be seen whether such
a shift can also be revealed in non-human primates
following more extensive training.
It should be noted that the type of training that was
used in this study with monkeys assumes that symbolic
representations of numerical magnitude emerge as a
function of their pairing with non-symbolic representa-
tions, such as arrays of dots (as does the verguts and
Fias model). The extent to which these findings can be
generalized to the brain mechanisms that subserve the
acquisition of numerical symbols in humans depends
entirely on the validity of this assumption.
Representation of symbolic magnitude in the human
brain. The above data suggest that, in monkeys, neurons
that code for non-symbolic magnitudes have slightly dif-
ferent response properties to those that code for symbolic
magnitude. Specifically, the prefrontal and parietal neu-
rons that respond preferentially to a symbolic numerical
magnitude have narrower tuning curves than neurons
that repond to a non-symbolic magnitude. Similarly, a
recent neuroimaging study67 provides evidence for more
precise coding of symbolic than non-symbolic numeri-
cal magnitude in humans; however, in contrast to mon-
keys, humans had narrower tuning curves for symbolic
representations in the left IPS rather than in the PFC.
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A phenomenon whereby
repeated presentation of a
particular stimulus leads to
reductions in the fMRI signal in
brain regions that are involved
in representing and processing
that stimulus. It is also referred
to as ‘repetition suppression’.
Participants were presented with trains of either arrays
of dots (non-symbolic magnitudes) or Arabic numerals
(symbolic magnitudes) of a constant numerical quantity.
Infrequently, numerical deviants of either format were
presented. The repetition of a given numerical quantity
led to a reduced response in the bilateral IPS regard-
less of the presentation format (FIG. 4c). Consistent with
earlier findings75, the deviants led to a recovery in IPS
activation that was proportional to the numerical dis-
tance between the habituation and the deviant numerical
magnitude. Interestingly, this distance-dependent recov-
ery occurred both when deviant Arabic numerals were
presented among dots and when deviant numbers of dots
were presented among Arabic numerals, suggesting that
adaptation is notation-invariant.
In addition, there was a hemispheric difference:
following adaptation to an Arabic numeral, deviant
dot arrays that were far removed (in terms of numeri-
cal quantity) from the repeated Arabic numerals and
deviant dot arrays that were of similar (close) quantity
induced an equal amount of dishabituation in the left IPS
but not the right IPS (FIG. 4c). This suggests that there is
greater precision of coding (or narrower tuning curves)
for symbolic representations of numerical magnitude in
the left IPS than in the right IPS.
More evidence suggesting that there might be nota-
tion-specific symbolic representation of numerical
quantity in the parietal cortex comes from another
fMRI adaptation study76 in which activation of the left
IPS decreased as a function of the number of repeated
presentations of the same numerical magnitude regard-
less of format, whereas repetition-induced suppression
of activity in the right parietal cortex was found only for
Arabic numerals (not for number words).
In summary, the symbolic representation of numeri-
cal magnitude in the human left IPS might be more
precise than the representation of non-symbolic numeri-
cal magnitude. Moreover, symbolic representations in
the parietal cortex might be format-specific. These
data therefore suggest that the acquisition of symbolic
representations of numerical magnitude either changes
pre-existing representations or leads to the construction
of novel, format-specific representations of numerical
magnitude in the IPS.
How do symbolic representations emerge?
Taken together, the data suggest that the symbolic
representation of numerical magnitude involves proc-
esses that are different in subtle ways from those that
are engaged by non-symbolic numerical stimuli, and
that there might be multiple representations for dif-
ferent symbolic formats (such as Arabic numerals
and number words). These models and data therefore
demand careful re-evaluation of the commonly held
assumption10,77,78 that representations that are tapped
by non-symbolic stimuli are the sole building blocks
for higher-level, symbolic representations. Thus far,
both computational models63,78 and training studies72
have assumed that the development of the ability to
represent and process numerical symbols involves a
process by which symbolic representations of numerical
magnitude are mapped onto pre-existing non-symbolic
representations. A plausible alternative possibility, how-
ever, is that non-symbolic and symbolic representations
of numerical magnitude draw on different neurocog-
nitive processes. It has been argued that approximate
representations of numerical magnitude might not
serve as adequate representational precursors to exact
representations of the integer list79.
Some data already suggest that symbolic representa-
tions of numerical magnitude might be different from
non-symbolic representations. For example, Polk et al.80
reported that a patient with damage to the left supramar-
ginal gyrus (SMG) was severely impaired on symbolic
but not non-symbolic magnitude processing. Moreover,
a recent cortical-stimulation study81 revealed that the
SMG is associated with Arabic-number reading, sug-
gesting that this area has a role in processing symbolic
representations of numerical magnitude.
Furthermore, a recent investigation82 in two-to-
four-year-old children suggested that the process of
approximate numerical-magnitude representation is
independent from a child’s developing understanding
of the meaning of counting (the ‘cardinality principle’).
Specifically, leCorre and Carey82 found that some chil-
dren can count out exactly eight and nine objects when
asked to do so but did not use the number words ‘eight’
or ‘nine’ when asked to estimate rapidly presented arrays
of eight-to-twelve dots, indicating that the children had
not yet approximately mapped number words onto large
non-symbolic numerosities. In other words, such map-
ping does not seem to be a necessary and sufficient pre-
requisite for children’s developing understanding of the
functional significance of the count sequence. Instead,
it has been argued that the ability to rapidly enumerate
small numbers of objects (‘subitizing’) might be a cru-
cial scaffold for the acquisition of an exact, integer-list
representation of number79,82.
This study also raises an often neglected point: chil-
dren acquire the meaning of number words before they
learn the visual symbols that represent numerical mag-
nitudes (that is, Arabic numerals); it might therefore be
the case that children map visual symbols onto earlier-
developed representations of auditory number words,
rather than directly onto non-symbolic representations
of numerical magnitude.
More evidence for potential qualitative differences
between symbolic and non-symbolic representations
of numerical magnitude is provided by a recent study
which showed that children with developmental dyscal-
culia are impaired on symbolic but not non-symbolic
Interestingly, the ‘numerosity code’ (ReFs 84,85)
computational model of numerical-magnitude repre-
sentation does not assume that there is a tight coupling
between symbolic and non-symbolic representations
of numerical magnitude and might therefore model
symbolic representations that are not a direct conse-
quence of mappings onto non-symbolic representations
of numerical magnitude. This model posits that each
numerical magnitude is represented as a set of activated
units (summation coding). However, in contrast to other
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Impairment of the ability to
distinguish between fingers. It
is associated with damage to
the left angular gyrus and
frequently co-occurs with
An arithmetic problem
involving two numbers (for
example, 12 x 45).
models63,64 that have incorporated summation coding,
this model posits that summation-coded representations
of numerical magnitude are not noisy or approximate,
but discrete, linear and exact. Such properties fit well
with the symbolic representations that are used to com-
pute the exact results of, for example, mental-arithmetic
problems. According to this model, distance and size
effects emerge as a consequence of a nonlinear decision
process rather than from an approximate representation
of numerical magnitude and noisy summation codes.
Note that this model does not assume that place coding
of numerical magnitude is the representation of exact
number, but instead proposes that summation coding is
discrete. The numerosity code currently does not offer
a developmental perspective and hence it is unclear how
this representation develops as a function of experience
with numerical symbols.
Neural correlates of mental arithmetic
once symbols (spoken number words and numerical
symbols (for example, Arabic numerals)) have been
mapped onto internal representations of numerical
quantities, these symbolic systems can be applied to
invented algorithms to solve, for example, arithmetic
Work on the brain mechanisms that underlie mental
arithmetic has a long history. In 1919 Henschen reported
a strong association between damage to the left parietal
cortex and calculation deficits86. Furthermore, in 1940
Gerstman87 described a group of symptoms (which
together comprise Gerstmann syndrome), including acal-
culia, finger agnosia and left–right disorientation, which
resulted from damage to the left angular gyrus, which lies
adjacent to the IPS. Since then, a large number of papers
have confirmed the role of the angular and supramar-
ginal gyri in calculation88–91. Given the involvement of the
left temporoparietal cortex in language and reading92, it
has been proposed that the activation of these gyri dur-
ing calculation might reflect the involvement of verbal
processes in calculation93. Specific learning difficulties
in mathematics (developmental dyscalculia) frequently
co-occur with impairments of reading (developmental
dyslexia); therefore, it is possible that a common impair-
ment of functions that are subserved by the angular gyrus
might underlie this co-morbidity. It is plausible that it is
symbol-referent mapping that is impaired, such as map-
ping between arithmetic problems and their solutions
(which are stored in memory), grapheme–phoneme
mappings or mappings between abstract numerical
symbols and the magnitudes that they represent.
An intriguing aspect of Gerstmann syndrome is
the co-occurrence of finger agnosia and calculation
deficits following damage to the left angular gyrus. A
potential link between finger discrimination and mental
arithmetic has been investigated in both children and
adults94,95. A recent transcranial magnetic stimulation
(TMS) study96 demonstrated that transient disruption
of the angular gyrus results in impairment to the access of
finger schemas, as well as impairment in numerical-
magnitude processing. This suggests that childrens’
use of finger counting strategies in the early stage of
arithmetic development might leave a ‘cortical trace’
that is reactivated during mental arithmetic even when
retrieval strategies are being used.
Understanding the significance of deactivation during
calculation. In most fMRI studies of calculation, the
activation that is observed in regions in the left temporo-
parietal cortex, such as the angular and supramarginal
gyri, is in fact due to a difference in deactivation relative
to baseline97,98. Thus, greater activation of the angular
gyrus — for example, during single- versus multi-
digit calculation — is mostly reflective of significantly
less deactivation during single-digit multiplication99.
Interestingly, bilateral regions of the temporoparietal
cortex are thought to be part of the so-called ‘default’
or ‘resting-state’ network100,101, which exhibits decreases
in blood flow relative to baseline during various goal-
directed, active tasks. Although deactivations are
frequently left undiscussed, it has been shown that
the degree of deactivation of brain regions can be
tightly coupled to individual differences in perform-
ance102,103. Future investigations should provide a better
understanding of the role of deactivation in the left
temporoparietal cortex during calculation and number
processing, its relationship to the ‘resting-state’ network
and its functional significance for symbolic number
processing and calculation.
Training-related changes. A recent series of studies inves-
tigated the changes in functional neuroanatomy that
occur as individuals learn arithmetic problems58,104–106.
In one study104, adults were extensively trained on a set
of complex multiplication problems. When they subse-
quently attempted to solve both trained and untrained
problems, activation in the left intraparietal and left
inferior frontal regions was greater during untrained-
problem solving. The reverse contrast revealed greater
activation in the left angular gyrus, indicating a training-
related shift in activation from left intraparietal regions
to the left angular gyrus (FIG. 5b).
Another study105 investigated whether relative shifts
in activation differ as a function of the particular training
method. Specifically, ‘training by drill’ (rote learn-
ing the result of a two-operant problem) was compared
with ‘training by strategy’ (applying an instructed
algorithm). Greater activation of the angular gyrus
was found during the solving of problems learned by
drill than during the solving of those trained using the
strategy algorithm. Furthermore, another study found
that although the angular gyrus was activated more by
trained than by untrained multiplication problems, it
did not exhibit training effects for subtraction106. Thus,
the type of instruction and the particular arithmetic
operation dynamically modulate the relative activation
of intraparietal and left temporoparietal regions during
Individual differences in arithmetic and the left temporo-
parietal cortex. The strong association between activ-
ity in the left angular gyrus and the learning of exact
arithmetic raises the question of whether individual
286 | APRIl 2008 | voluMe 9
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Nature Reviews | Neuroscience
Novel > repeated
Repeated > novel
The process (encompassing
education, learning, et cetera)
by which an individual
becomes a fully functioning
member of his or her culture.
differences in the recruitment of this region are related
to individual differences in mathematical achievement.
A recent study97 indicated that this is indeed the case.
Specifically, adults who scored low on a standardized test
of mathematics but who had otherwise normal intelli-
gence showed less activation of the angular gyrus during
both single- and multi-digit multiplication than their
mathematically more able peers97. Consistent with the
group difference, individuals’ standardized mathematics
scores correlated with calculation-related activation of
the angular gyrus. This association remained significant
even after the effect of individual differences in accuracy
and response times were controlled for, strengthening
the claim that relative levels of modulation of the angular
gyrus during calculation have functional significance
and do not merely reflect differences in performance.
Effects of culture and education. To what extent do edu-
cation and culture lead to changes in the connectivity
and relative degree of activation of the brain circuits
that underlie numerical and mathematical processing?
A recent comparison of native english and Chinese
speakers107 showed different activation and functional
connectivity between brain regions during simple
addition and number comparison. Specifically, native
english speakers recruited more regions in left peri-
sylvian regions, including Broca’s area, whereas native
Chinese speakers activated more premotor regions dur-
ing the same task. exactly which factors (for example,
approach to reading and differences in learning strate-
gies and education) bring about these differences in
activation remains to be determined. With reference to
the above discussion of the neural circuits that underlie
symbolic number processing, it is of particular interest
to note that cultural differences in brain function were
even found for numerical-magnitude comparison of
single-digit Arabic numerals107. Together with studies
from other cognitive domains108–110, these findings reveal
the striking transformative effect of cultural variability
on brain function. Future studies should also take into
account the way in which number words differ across
languages and whether such differences influence the
representation and processing of numerical magnitude
in the brain.
Another pathway by which to investigate the influ-
ence of enculturation and education is to compare the
neural activation that is associated with different
methods and strategies for approaching mathematical
problems. In one fMRI study111, two groups of adult par-
ticipants were asked to memorize prototypes of algebra
problems in either a verbal format (for example, “Brian
earns $7 an hour and gets $9 in tips.”) or in a symbolic,
equation format (for example, 7H + 9 = e). During the
acquisition of the fMRI data, participants were asked to
calculate results using the information contained in the
prototypes, by presenting them with problems such as
‘hours = 3, earning = ?’. The two groups’ behavioural
performance was equivalent, but there were striking
differences in the brain regions that were recruited dur-
ing the problem solving. Specifically, participants in the
‘verbal’ group recruited left prefrontal regions, whereas
those in the ‘symbolic’ group activated regions of the
posterior parietal cortex.
In another study112, fMRI was used to compare the
neural correlates of solving algebraic problems by two
methods that are taught in schools in Singapore: the
‘symbol’ method and the ‘model’ method. In the sym-
bol method, children are taught to transform algebraic
problems into symbolic equations, whereas in the model
method they are taught to create diagrams that represent
the information contained in the word problems. When
adults were asked to use these methods to verify solu-
tions to algebraic word problems, the two approaches
Figure 5 | The calculating brain changes dynamically as a function of learning.
a | Training for arithmetic problems leads to decreasing engagment of the inferior
parietal cortex (shown in yellow) and increasing recruitment of the angular gyrus103–105
(shown in blue). Specifically, the left angular gyrus is more activated during the solving of
trained than untrained problems. Furthermore, its relative activation is modulated by the
type of instruction104 and the type of arithmetic operation that is being learned105.
b | During the learning of arithmetic problems, increases in the recruitment of the
angular gyrus for repeated problems occur rapidly. The figure shows a moving time
window of 200 scans (the scan ranges are indicated below the brain images) and reveals
that there are significant changes in activity of the angular gyrus (shown in green) after
only approximately 8 repetitions of a problem (corresponding to the time window
between 100 and 299 scans), with increasing activation of this region for repeated
compared with novel problems. In addition, an area in the left middle frontal gyrus
(shown in red) was found to increase in its response to novel compared with repeated
problems as a function of training. These findings suggest that the neural correlates of
mental arithmetic rapidly undergo dynamic changes as a function of learning. Part b
reproduced, with permission, from ReF. 98 (2007) Academic Press.
NATuRe RevIeWS | neurOscience
voluMe 9 | APRIl 2008 | 287
© 2008 Nature Publishing Group
Nature Reviews | Neuroscience
experiments that compare
different groups of participants
(for example, children of
different ages) rather than
individuals in a single group.
yielded equal behavioural performance but use of the
symbol method was associated with greater activation
of the superior parietal lobules and the precuneus. It is
possible that the symbol method imposes greater atten-
tional demands and thus is subjectively more difficult.
In summary, recent data indicate that different instruc-
tional methods might result in different patterns of brain
Neural correlates of number development
In a pioneering study, Rivera et al.113 revealed age-related
increases in the recruitment of the left inferior parietal
cortex (specifically the left SMG) during calculation in a
cross-sectional sample of people aged between 8 and 19. As
noted above, the SMG (together with other regions of the
left temporoparietal cortex, such as the angular gyrus)
is involved in mental arithmetic in adults. These devel-
opmental data therefore suggest that the adult outcome
is the result of a process of developmental specialization
of this region for mental arithmetic. Such developmental
increases also occur in the left lateral occipito–temporal
cortex, and they were coupled with reductions of activity
in bilateral regions of the frontal cortex, the hippocam-
pus and the basal ganglia113. Thus, a frontoparietal shift
in activity (FIG. 6a) might underlie the development of
arithmetic skills. The increase in the recruitment of the
left SMG and the lateral occipitotemporal cortex sug-
gests that there is an ontogenetic specialization of these
regions for mental arithmetic and for the processing of
symbolic visual stimuli, respectively. The decrease in
the reliance on frontal regions might relate to reduced
reliance on processes of cognitive control, attention and
working memory with age, whereas the decreases in the
hippocampus suggest that there is increasing consolida-
tion of arithmetic facts into long-term memory. These
findings indicate that dynamic increases and decreases
in activation occur in a large network of regions, again
highlighting the importance of considering networks of
activation rather than focusing on a select set of brain
regions. evidence for a frontoparietal shift has also been
shown for more basic tasks such as symbolic28,74 and non-
symbolic114 magnitude comparison: the modulation of
IPS activation by numerical distance for both symbolic
and non-symbolic magnitude comparisons increases
with age28,115. Taken together, the data suggest that
although mental arithmetic leads to specialization of the
left temporoparietal cortex, basic magnitude processing
involves the ontogenetic specialization of the IPS.
Similar to monkeys72, children use more prefrontal
regions during numerical processing than adults. This
might suggest that the prefrontal cortex subserves the
early association between external and internal repre-
sentations of numerical magnitude in both non-human
primates and human children. Importantly, however,
there are also similarities between children and adults
in the brain regions that underlie numerical-magnitude
processing. one fMRI adaptation study investigated the
neural correlates of numerical-magnitude representation
in 4-year-old children and adults115. Participants were
repeatedly presented with stimuli containing a particular
number of dots (for example, 16). Interspersed into this
train of repeated arrays of dots were stimuli containing
Figure 6 | Ontogenetic differences and similarities in the neural correlates of mental arithmetic and magnitude
processing. a | Participants ranging from 8 to 19 years old were asked to verify the correctness of arithmetic equations
(such as 5 + 3 = 9) while brain activation was recorded using fMRI. Regions in red indicate areas in which activity
during calculation increases with age. These included the left supramarginal gyrus, which has been shown to subserve
adult calculation29. Regions in blue indicate areas in which activation during calculation was negatively correlated with age.
The engagement of frontal regions decreased as a function of age, perhaps reflecting increasing automaticity in the
processing of arithmetic equations and less reliance on frontally mediated cognitive mechanisms. b | In an fMRI study114,
4-year-old children and adults were presented with trains of stimuli containing the same non-symbolic numerical
magnitude (for example, 16 dots). Repetition of the numerical magnitude led to a decrease in the fMRI response of brain
regions that encode numerical magnitude. Interspersed in the trains of repeated stimuli, stimuli that deviated in either
numerical magnitude (for example, 32 dots instead of 16) or shape (for example, 16 squares instead of 16 dots) were
presented. The presentation of deviants caused a recovery of the fMRI response. The intraparietal sulcus responded more
to numerical-magnitude deviants in both 4-year-old children and adults. Regions of activation in adults are shown in green,
regions of activation in children are shown in purple and the overlap in activation between children and adults is shown in
red. Part a reproduced, with permission, from ReF. 113 (2005) Oxford University Press. Part b reproduced from ReF. 115.
288 | APRIl 2008 | voluMe 9
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culturally-invented mental tools, such as abstract
Non-symbolic stimuli, like symbolic stimuli, are
external representations of numerical magnitude that
need to be mapped onto internal representations; the
processes that enable such a mapping are not well under-
stood. Current theories and computational models sug-
gest67,78 that the acquisition of symbolic representations of
numerical magnitude is grounded in the representations
of non-symbolic numerical magnitude. Alternatively,
symbolic and non-symbolic external representations of
numerical magnitude might be mapped onto different
internal representations that have higher-order similari-
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should evaluate the extent to which qualitatively different
representations of symbolic and non-symbolic numeri-
cal magnitude might exist and should determine how
such representations are constructed during develop-
ment. Rather than continuing to examine the common
areas that are activated by symbolic and non-symbolic
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children are systematically (through formal education)
introduced to symbolic representations of numeri-
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in symbolic representations of numerical magnitude
(such as the difference in the way in which the teens
are represented in the Chinese and english languages117)
affect brain representations.
on a broader level, such investigations will provide
a greater understanding of the mechanisms that allow
cultural inventions to be represented and processed by
the brain through both the recruitment of existing rep-
resentation118 and the ontogenetic construction of new
systems of representation.
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I would like to thank three anonymous reviewers for their
valuable comments on an earlier version of this manuscript.
I would like to thank I. Lyons, G. Price, I. Holloway and
M. Zorzi for helpful discussions of many of the issues dis-
cussed in the paper. I would like to thank L. van Eimeren for
help with the figures. This research was supported by grants
from the Natural Science and Engineering Council of Canada,
the Canada Research Chairs Program, The Canada
Foundation for Innovation and the Ontario Ministry of
Research and Innovation.
Daniel Ansari’s homepage:
Numerical Cognition Laboratory:
Numeracy and Literacy Research Group: http://www.
Brannon Lab: http://www.duke.edu/web/mind/level2/
Cognitive Neuroimaging Unit: http://www.unicog.org/
Computational Cognitive Neuroscience Laboratory:
Primate Neurocognition Laboratory:
Laboratory for Developmental Studies:
Cognitive & Systems Neuroscience Laboratory:
Number Processing and Calculation Research Group:
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