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Dyscalculia: From Brain to Education
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Dyscalculia: From Brain to Education
Recent research in cognitive and developmental neuroscience is providing a new approach to
the understanding of dyscalculia that emphasizes a core deficit in understanding sets and their
numerosities, which is fundamental to all aspects of elementary school mathematics. The neural
bases of numerosity processing have been investigated in structural and functional neuroimaging
studies of adults and children, and neural markers of its impairment in dyscalculia have been
identified. New interventions to strengthen numerosity processing, including adaptive software,
promise effective evidence-based education for dyscalculic learners.
Developmental dyscalculia is a mathemat-
ical disorder, with an estimated prevalence
ofabout5to7%(1), which is roughly the
same prevalence as developmental dyslexia (2).
A major report by the UK government con-
cludes, “Developmental dyscalculia is currently
the poor relation of dyslexia, with a much lower
public profile. But the consequences of dyscalculia
are at least as severe as those for dyslexia”[(3),
p. 1060]. The relative poverty of dyscalculia fund-
ing is clear from the figures: Since 2000, NIH has
spent $107.2 million funding dyslexia research but
only $2.3 million on dyscalculia (4).
The classical understanding of dyscalculia as
a clinical syndrome uses low achievement on
mathematical achievement tests as the criterion
without identifying the underlying cognitive
phenotype (5–7). It has therefore been unable to
inform pathways to remediation, whether in
focused interventions or in the larger, more
complex context of the math classroom.
Why Is Mathematical Disability Important?
Low numeracy is a substantial cost to nations, and
improving standards could dramatically improve
economic performance. In a recent analysis, the
Organisation for Economic Co-operation and De-
velopment (OECD) demonstrated that an improve-
ment of “one-half standard deviation in mathematics
and science performance at the individual level im-
plies, by historical experience, an increase in an-
nual growth rates of GDP per capita of 0.87%”[(8),
p. 17]. Time-lagged correlations show that improve-
ments in educational performance contribute to
increased GDP growth. A substantial long-term
improvement in GDP growth (an added 0.68% per
annum for all OECD countries) could be achieved
just by raising the standard of the lowest-attaining
students to the Programme for International Stu-
dent Assessment (PISA) minimum level (Box 1).
In the United States, for example, this would
mean bringing the lowest 19.4% up to the min-
imum level, with a corresponding 0.74% increase
in GDP growth.
Besides reduced GDP growth, low numeracy
is a substantial financial cost to governments and
personal cost to individuals. A large UK cohort
study found that low numeracy was more of a
handicap for an individual’s life chances than
low literacy: They earn less, spend less, are more
likely to be sick, are more likely to be in trouble
with the law, and need more help in school (9). It
has been estimated that the annual cost to the UK
of low numeracy is £2.4 billion (10).
What Is Dyscalculia?
Recent neurobehavioral and genetic research
suggests that dyscalculia is a coherent syndrome
that reflects a single core deficit. Although the
literature is riddled with different terminologies,
all seem to refer to the existence of a severe dis-
ability in learning arithmetic. The disability can
be highly selective, affecting learners with normal
intelligence and normal working memory (11),
although it co-occurs with other developmental
disorders, including reading disorders (5)andat-
tention deficit hyperactivity disorder (ADHD) (12)
more often than would be expected by chance.
There are high-functioning adults who are severe-
ly dyscalculic but very good at geometry, using
statistics packages, and doing degree-level com-
puter programming (13).
There is evidence that mathematical abilities
have high specific heritability. A multivariate
genetic analysis of a sample of 1500 pairs of
monozygotic and 1375 pairs of dizygotic 7-year-
old twins found that about 30% of the genetic
variance was specific to mathematics (14). Al-
though there is a significant co-occurrence of
dyscalculia with dyslexia, a study of first-degree
relatives of dyslexic probands revealed that nu-
merical abilities constituted a separate factor,
with reading-related and naming-related abilities
being the two other principal components (15).
These findings imply that arithmetical learning is
at least partly based on a cognitive system that is
distinct from those underpinning scholastic at-
tainment more generally.
This genetic research is supported by neuro-
behavioral research that identifies the representa-
tion of numerosities—the number of objects in a
Centre for Educational Neuroscience and Institute of Cognitive
Neuroscience, University College London, Psy chologica l Sc i-
ences, Melbourne University, Melbourne VIC 3010, Aus-
Department of Educational Psychology, University of
Minnesota, Minneapolis, MN 55455, USA.
Centre for Educa-
tional Neuroscience and London Knowledge Lab, Institute of
Education, University of London, London WC1N 3QS, UK.
*To whom correspondence should be addressed. E-mail:
Box 1: PISA question example
AtLevel1,studentscananswerquestions involving familiar contexts in which all relevant
information is present and the questions are clearly defined. They are able to identify information and
carry out routine procedures according to direct instructions in explicit situations. They can perform
actions that are obvious and follow immediately from the given stimuli. For example:
Mei-Ling found out that the exchange rate between Singapore dollars and South African rand was 1
SGD = 4.2 ZAR
Mei-Ling changed 3000 Singapore dollars into South African rand at this exchange rate.
How much money in South African rand did Mei-Ling get?
79% of 15-year-olds were able to answer this correctly.
Box 2: Dyscalculia observed
Examples of common indicators of dyscalculia are (i) carrying out simple number comparison and
addition tasks by counting, often using fingers, well beyond the age when it is normal, and (ii)finding
approximate estimation tasks difficult. Individuals identified as dyscalculic behave differently from
their mainstream peers. For example:
To say which is the larger of two playing cards showing 5 and 8, they count all the symbols on
To place a playing card of 8 in sequence between a 3 and a 9, they count up spaces between the two
to identify where the 8 should be placed.
To count down from 10, they count up from 1 to 10, then 1 to 9, etc.
To count up from 70 in tens, they say “70, 80, 90, 100, 200, 300…”
They estimate the height of a normal room as “200 feet?”
www.sciencemag.org SCIENCE VOL 332 27 MAY 2011 1049
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set—as a foundational capacity in the develop-
ment of arithmetic (16). This capacity is impaired
in dyscalculic learners even in tasks as simple as
enumerating small sets of objects (11)orcompar-
ing the numerosities of two arrays of dots (17).
The ability to compare dot arrays has been cor-
related with more general arithmetical abilities
bothinchildren(18) and across the age range
(17,19). This core deficit in processing numer-
osities is analogous to the core deficit in phono-
logical awareness in dyslexia (Box 2) (2).
Although there is little longitudinal evidence,
it seems that dyscalculia persists into adulthood
(20), even among individuals who are able in
other cognitive domains (13). The effects of ear-
ly and appropriate intervention with dyscalculia
have yet to be investigated. This also leaves open
the question as to whether there is a form of dys-
calculia that is a delay, rather than a deficit, that
will resolve, perhaps with appropriate educational
Converging evidence of dyscalculia as a dis-
tinct deficit comes from studies of impairments in
the mental and neural representation of fingers. It
has been known for many years that fingers are
used in acquiring arithmetical competence (21).
This involves understanding the mapping be-
tween the set of fingers and the set of objects to
be enumerated. If the mental representation of
fingers is weak, or if there is a deficit in under-
standing the numerosity of sets, then the child’s
cognitive development may fail to establish the
link between fingers and numerosities. In fact,
developmental weakness in finger representation
ability (22). Gerstmann’s Syndrome, whose symp-
toms include dyscalculia and finger agnosia, is
due to an abnormality in the parietal lobe and, in
its developmental form, is also associated with
poor arithmetical attainment (23).
Numbers do not seem to be meaningful for
dyscalculics—at least, not meaningful in the
way that they are for typically developing learn-
ers. They do not intuitively grasp the size of a
number and its value relative to other num-
bers. This basic understanding underpins all
work with numbers and their relationships to
What Do We Know About the Brain
The neural basis of arithmetical abilities in the
parietal lobes, which is separate from language
and domain-general cognitive capacities, has been
broadly understood for nearly 100 years from re-
search on neurological patients (24). One par-
ticularly interesting finding is that arithmetical
concepts and laws can be preserved even when
facts have been lost (25), and conversely, facts
can be preserved even when an understanding of
concepts and laws has been lost (26).
Neuroimaging experiments confirm this pic-
ture and show links from the parietal lobes to the
left frontal lobe for more complex tasks (27,28).
One important new finding is that the neural or-
ganization of arithmetic is dynamic, shifting from
one subnetwork to another during the process of
learning. Thus, learning new arithmetical facts
primarily involves the frontal lobes and the intra-
parietal sulci (IPS), but using previously learned
facts involves the left angular gyrus, which is also
implicated in retrieving facts from memory (29).
Some of the principal links are summarized in
Fig. 1. Even prodigious calculators use this net-
work, although supplementing it with additional
brain areas (30) that appear to extend the capacity
of working memory (31).
There is now extensive evidence that the IPS
supports the representation of the magnitude of
symbolic numbers (32,33), either as analog mag-
nitudes or as a discrete representation that codes
cardinality, as evidenced by IPS activation when
processing the numerosity of arrays of objects
(34). Moreover, when IPS functioning is dis-
turbed by magnetic stimulation, the ability to esti-
mate discrete magnitudes is affected (35,36). The
critical point is that almost all arithmetical and
numerical processes implicate the parietal lobes,
especially the IPS, suggesting that these are at the
core of mathematical capacities.
Patterns of brain activity in 4-year-olds and
adults show overlapping areas in the parietal
lobes bilaterally when responding to changes in
numerosity (37). Nevertheless, there is a devel-
opmental trajectory in the organization of more
complex arithmetical abilities. First, the organi-
zation of routine numerical activity changes with
age, shifting from frontal areas (which are as-
sociated with executive function and working
memory) and medial temporal areas (which are
associated with declarative memory) to parietal
areas (which are associated with magnitude
processing and arithmetic fact retrieval) and
occipito-temporal areas (which are associated
with processing symbolic form) (38). These
changes allow the brain to process numbers more
efficiently and automatically, which enables it to
carry out the more complex processing of arith-
metical calculations. As A. N. Whitehead ob-
served, an understanding of symbolic notation
relieves “the brain of all unnecessary work …
and sets it free to concentrate on more advanced
This suggests the possibility that the neural
specialization for arithmetical processing may
arise, at least in part, from a developmental inter-
action between the brain and experience (40,41).
Thus, one way of thinking about dyscalculia is
that the typical school environment does not
provide the right kind of experiences to enable
the dyscalculic brain to develop normally to learn
Of course, mathematics is more than just sim-
ple number processing and retrieval of previously
learned facts. In a numerate society, we have to
digits and facts
Fig. 1. Causal model of possible inter-relations between biological, cognitive, and simple behavioral
levels. Here, the only environmental factors we address are educational. If parietal areas, especially the
IPS, fail to develop normally, there will be an impairment at the cognitive level in numerosity rep-
resentation and consequential impairments for other relevant cognitive systems revealed in behavioral
abnormalities. The link between the occipitotemporal and parietal cortex is required for mapping number
symbols (digits and number words) to numerosity representations. Prefrontal cortex supports learning new
facts and procedures. The multiple levels of the theory suggest the instructional interventions on which
educational scientists should focus.
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learn more complex mathematical concepts, such
as place value, and more complex procedures,
such as “long”addition, subtraction, multiplica-
tion, and division. Recent research has revealed
the neural correlates of learning to solve com-
plex, multidigit arithmetic problems (24). Again,
this research shows that solving new problems
requires more activation in the inferior frontal
gyrus for reasoning and working memory and the
IPS for representing the magnitudes of the num-
bers involved, as compared with retrieval of pre-
viously learned facts (42).
The striking result in all of these studies is the
crucial role of the parietal lobes. That the IPS is
implicated in both simple and complex calcu-
lations suggests that the basic representations of
magnitude are always activated, even in the re-
trieval of well-learned single-digit addition and
multiplication facts (43). This is consistent with
the well-established “problem-size effect,”in
which single-digit problems take longer to solve
the larger the operands, even when they are well
known (44). It seems that the typically devel-
oping individual, even when retrieving math
facts from memory, cannot help but activate the
meaning of the component numbers at the same
time. If that link has not been established, cal-
culation is necessarily impaired.
What Do We Know About the Brain
The clinical approach has identified behavioral
deficits in dyscalculic learners typically by per-
formance on standardized tests of arithmetic.
However, even in primary school (K1 to 5), arith-
metical competence involves a wide range of
cognitive skills, impairments in any
of which may affect performance,
including reasoning, working mem-
ory, language understanding, and
spatial cognition (45). Neural differ-
ences in structure or functioning may
reflect differences in any of these cog-
nitive skills. However, if dyscalculia
is a core deficit in processing nu-
merosities, then abnormalities should
be found in the parietal network that
supports the enumeration of small
sets of objects (11) and the compar-
ison of numerosities of arrays of
The existence of a core deficit in
processing numerosities is consist-
ent with recent discoveries about dys-
calculic brains: (i) Reduced activation
has been observed in children with dyscalculia
during comparison of numerosities (46,47),
comparison of number symbols (46), and arith-
metic (48)—these children are not using the
IPS so much during these tasks. (ii) Reduced
gray matter in dyscalculic learners has been ob-
served in areas known to be involved in basic
numerical processing, including the left IPS (49),
the right IPS (50), and the IPS bilaterally (51)—
these learners have not developed these brain
And (iii) differences in connectivity among the
relevant parietal regions, and between these
parietal regions and occipitotemporal regions
associated with processing symbolic number form
(Fig. 2), are revealed through diffusion tensor im-
aging tractography (51)—dyscalculic learners
have not sufficiently developed the structures
needed to coordinate the components needed
Figure 1 suggests that the IPS is just one
component in a large-scale cortical network that
subserves mathematical cognition. This network
different dyscalculia patterns of presenting symp-
toms when the core deficit is combined with
other cognitive deficits, including working mem-
ory, reasoning, or language (52).
Moreover, the same presenting symptom could
reflect different impairments in the network. For
example, abnormal comparison of symbolic num-
bers (53,54) could arise from an impairment in
the fusiform gyrus associated with visual process-
ing of number symbols, an impairment in IPS
associated with the magnitude referents, analog or
numerosity, of the number symbols, or reduced
connectivity between fusiform gyrus and IPS.
How Is This Relevant to Maths Education?
There have been many attempts to raise the
performance of children with low-numeracy, al-
though not specifically dyscalculia. In the United
States, evidence-based approaches have focused
on children from deprived backgrounds, usually
low socioeconomic status (46,47). The current
National Strategy in the United Kingdom gives
special attention to children with low numeracy
by (i) diagnosing each child’s conceptual gaps in
understanding and (ii) giving the child more
individual support in working through visual,
verbal, and physical activities designed to bridge
each gap. Unfortunately, there is still little quan-
titative evaluation of the effectiveness of the
strategy since it was first piloted in 2003: “the
evaluations used very diverse measures; and
most did not include ratio gains or effect sizes
or data from which these could be obtained”
[(55), p. 13].
It has not been possible to tell, therefore,
whether identifying and filling an individual’s
conceptual gaps with a more individualized ver-
sion of the same teaching is effective. A further
problem is that these interventions are effective
when there has been specialist training for teaching
assistants, but not all schools can provide this (55).
These standardized approaches depend on
curriculum-based definitions of typical arithmet-
ical development, and how children with low
numeracy differ from the typical trajectory. In con-
trast, neuroscience research suggests that rather
than address isolated conceptual gaps, remediation
should build the foundational number concepts
first. It offers a clear cognitive target for assess-
ment and intervention that is largely independent
of the learners’social and educational circum-
stances. In the assessment of individual cognitive
capacities, set enumeration and comparison can
supplement performance on curriculum-based
standardized tests of arithmetic to differentiate
dyscalculia from other causes of low numeracy
In intervention, strengthening the meaningful-
ness of numbers, especially the link between the
math facts and their component meanings, is cru-
cial. As noted above, typical retrieval of simple
arithmetical facts from memory elicits activation
of the numerical value of the component numbers.
Without specialized intervention, most dys-
calculic learners are still struggling with basic
arithmetic in secondary school (20). Effective
early intervention may help to reduce the later
impact on poor numeracy skills, as it does in
dyslexia (58). Although very expensive, it prom-
ises to repay 12 to 19 times the investment (10).
What Can Be Done?
Although the neuroscience may suggest what
should be taught, it does not specify how it should
be taught. Concrete manipulation activities have
been used for many decades in math remediation
because they provide tasks that make number
concepts meaningful (59), providing an intrinsic
Fig. 2. Structural abnormalities in young dyscalculic brains suggesting the critical role for the IPS. Here, we show
areas where the dyscalculic brain is different from that of typically developing controls. Both left and right IPS are
implicated, possibly with a greater impairment for left IPS in older learners. (A) There is a small region of reduced
gray-matter density in left IPS in adolescent dyscalculics (41). (B) There is right IPS reduced gray-matter density
(yellow area) in 9-year-olds (42). (C) There is reduced probability of connections from right fusiform gyrus to other
parts of the brain, including the parietal lobes (43).
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relationship between a goal, the learner’s action,
and the informational feedback on the action (60).
Educators recognize that informational feedback
provides intrinsic motivation in a task, and that is
of greater value to the learner than the extrinsic
motives and rewards provided by a supervising
Experienced special educational needs (SEN)
teachers use these activities in the form of games
with physical manipulables (such as Cuisenaire
rods, number tracks, and playing cards) to give
learners experience of the meaning of number.
Through playing these games, learners can dis-
cover from their manipulations, for example,
which rod fits with an 8-rod to match a 10-rod.
However, these methods require specially trained
teachers working with a single learner or a small
group of learners (64–66) and are allotted only
limited time periods in the school schedule.
A promising approach, therefore, is to con-
struct adaptive software informed by the neuro-
science findings on the core deficit in dyscalculia.
Such software has the potential to reduce the
demand on specially trained teachers and to tran-
scend the limits of the school sched-
ule. There have been two examples
of adaptive games based on neuro-
The Number Race (67)targets
the inherited approximate numerosity
system in the IPS (68)thatmay
support early arithmetic (18). In dys-
calculics, this system is less precise
(17), and the training is designed to
improve its precision. The task is to
select the larger of two arrays of
dots, and the software adapts to the
learner, making the difference between
the arrays smaller as their performance
improves, and provides informational
feedback as to which is correct.
Another adaptive game, Grapho-
game-Maths, targets the inherited
system for representing and manip-
ulating sets in the IPS, which is im-
paired in dyscalculia (45). Again,
the basis of the game is the compar-
ison of visual arrays of objects, but
here the sets are small and can be
counted, and the progressive tasks
are to identify the link between the
number of objects in the sets and their
verbal numerical label, with infor-
mational feedback showing which is
The effectiveness of the two
games was compared in a carefully
controlled study of kindergarten chil-
dren (aged 6 to 7 years) who were
identified by their teachers as need-
ing special support in early maths.
After 10 to 15 min of play per school
day for 3 weeks, there was a signif-
icant improvement in the task prac-
ticed in both games—namely, number
comparison—but the effect did not generalize to
counting or arithmetic. Graphogame-Maths ap-
peared to lead to slightly better and longer-lasting
improvement in number comparison (69).
Although the Number Race and Graphogame-
Maths are adaptive games based on neuroscience
findings, neither requires learners to manipulate
numerical quantities. Manipulation is critical for
providing an intrinsic relationship between task
goals, a learner’s actions, and informational feed-
back on those actions (60). When a learning en-
vironment provides informational feedback, it
enables the learner to work out how to adjust their
actions in relation to the goal, and they can be their
own “critic,”not relying on the teacher to guide
(61–63). This is analogous to the “actor-critic”mod-
el of unsupervised reinforcement learning in neuro-
science, which proposes a critic element internal
to the learning mechanism—not a guide that is ex-
ternal to it—which evaluates the informational
feedback in order to construct the next action (70).
A different approach, one that emulates the
manipulative tasks used by SEN teachers, has
been taken in adaptive software that, driven by
the neuroscience research on dyscalculia, focuses
on numerosity processing (Fig. 3) (56). The in-
formational feedback here is not an external critic
showing the correct answer. Rather, the visual
representation of two rods that match a given
distance, or not, enables the learner to interpret
for themselves what the improved action should
be—and can serve as their own internal critic.
An additional advantage of adaptive software
is that learners can do more practice per unit time
than with a teacher. It was found that for “SEN
learners (12-year-olds) using the Number Bonds
game [illustrated in Fig. 3], 4–11 trials per minute
were completed, while in an SEN class of three
supervised learners only 1.4 trials per minute
were completed during a 10-min observation”
[(56), p. 535]. In another SEN group of 11 year
olds, the game elicited on average 173 learner
manipulations in 13 min (where a perfect perfor-
mance, in which every answer is correct, is 88 in
5 min because the software adapts the timing ac-
cording to the response). In this way, neuroscience
research is informing what should be targeted in
the next generation of adaptive software.
Stage 4: Digits and colors; the learner has to identify which rod fits the gap before the stimulus rod reaches the stack
Stage 6: Digits only; the learner has to identify which digit makes 10 before the stimulus number reaches the stack
Which number bonds make 10?
Fig. 3. Remediation using learning technology. The images are taken from an example of an interactive, adaptive
game designed to help the learner make the link between digits and their meaning. The timed version of the number
bonds game elicits many learner actions with informational feedback, and scaffolds abstraction, through stages 1, 5
colors + lengths, evens; 2, 5 colors + lengths, odds; 3, all 10 colors + lengths; 4, digits + colors + lengths; 5, digits +
lengths; and 6, digits only. Each rod falls at a pace adapted to the learner’s performance, and the learner has to click
the corresponding rod or number to make 10 before it reaches the stack (initially 3 s); if there is a gap, or overlap, or
they are too slow, the rods dissolve. When a stack is complete, the game moves to the next stage. The game is available
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At present, it is not yet clear whether early
and appropriately targeted interventions can turn
a dyscalculic into a typical calculator. Dyscalculia
may be like dyslexia in that early intervention can
improve practical effectiveness without making
the cognitive processing like that of the typically
What Is the Outlook?
Recent research by cognitive and developmental
scientists is providing a scientific characterization
of dyscalculia as reduced ability for understand-
ing numerosities and mapping number symbols
to number magnitudes. Personalized learning
applications developed by educational scientists
can be targeted to remediate these deficits and
can be implemented on handheld devices for
independent learning. Because there are also in-
dividual differences in numerosity processing in
the normal range (17,19,71),thesameprograms
can assist beginning mainstream learners, so one
can envisage a future in which all learners will
benefit from these developments.
Although much progress has been made, a
number of open questions remain.
(i) The possible existence of a variety of dys-
calculia behavioral patterns of impairment raises
the interesting question of whether one deficit—
numerosity estimation—is a necessary or suffi-
cient for a diagnosis of dyscalculia. This is critical
because it has implications for whether a single
diagnostic assessment based on numerosity is
sufficient, or whether multiple assessments are re-
quired. It also has implications for whether nu-
merosity should be the focus of remediation, or
whether other (perhaps more symbolic) activities
must also be targeted.
(ii) Does the sensitivity of neuroscience mea-
sures make it possible to identify learners at risk for
dyscalculia earlier than is possible with behav-
ioral assessments, as is the case with dyslexia (72)?
(iii) Further research is needed on the neural
consequences of intervention. Even where inter-
vention improves performance, it may not be
clear whether the learner’s cognitive and neural
functioning has become more typical or whether
compensatory mechanisms have developed. This
would require more extensive research, as in the
case of dyslexia, in which functional neuroimag-
ing has revealed the effects of successful behavioral
interventions on patterns of neural function (2).
(iv) Personalized learning applications enable
fine-grain evaluation of theory-based instruc-
tional interventions. For example, the value of
active manipulation versus associative learning
on one hand, and of intrinsic versus extrinsic
motivation and feedback on the other, can be as-
sessed by orthogonally manipulating these features
of learning environments. Their effects can be as-
sessed behaviorally in terms of performance on the
tasks by target learners. Their effects can also be
assessed by investigating neural changes over time
in target learners, through structural and functional
neuroimaging. Classroom-based but theory-based
testing holds great potential for the development
of educational theory and could contribute in turn
to testing hypotheses in neuroscience.
(v) At the moment, dyscalculia is not widely
recognized by teachers or educational authorities
nor, it would seem, by research-funding agencies.
Recognition is likely to be the basis for improved
prospects for dyscalculic sufferers.
There is an urgent societal need to help failing
learners achieve a level of numeracy at which
they can function adequately in the modern
workplace. Contemporary research on dyscalcu-
lia promises a productive way forward, but it is
still a “poor relation”in terms of funding (4),
which means there is a serious lack of evidence-
based approaches to dyscalculia intervention. An
understanding of how the brain processes under-
lying number and arithmetic concepts will help
focus teaching interventions on critical concep-
tual activities and will help focus neuroscience
research on tracking the structural and functional
changes that follow intervention. Learning more
about how to help these learners is driving, and will
continue to drive, where the science should go next.
1. R. S. Shalev, in Why Is Math So Hard for Some
Children? The Nature and Origins of Mathematical
Learning Difficulties and Disabilities, D. B. Berch,
M. M. M. Mazzocco, Eds. (Paul H. Brookes Publishing,
Baltimore, MD, 2007), pp. 49–60.
2. J. D. E. Gabrieli, Science 325, 280 (2009).
3. J. Beddington et al., Nature 455, 1057 (2008).
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