An open trial assessment of "The Number Race", an adaptive computer game for remediation of dyscalculia

Abstract

In a companion article, we described the development and evaluation of software designed to remediate dyscalculia. This software is based on the hypothesis that dyscalculia is due to a "core deficit" in number sense or in its access via symbolic information. Here we review the evidence for this hypothesis, and present results from an initial open-trial test of the software in a sample of nine 7-9 year old children with mathematical difficulties.
Children completed adaptive training on numerical comparison for half an hour a day, four days a week over a period of five-weeks. They were tested before and after intervention on their performance in core numerical tasks: counting, transcoding, base-10 comprehension, enumeration, addition, subtraction, and symbolic and non-symbolic numerical comparison.
Children showed specific increases in performance on core number sense tasks. Speed of subitizing and numerical comparison increased by several hundred msec. Subtraction accuracy increased by an average of 23%. Performance on addition and base-10 comprehension tasks did not improve over the period of the study.
Initial open-trial testing showed promising results, and suggested that the software was successful in increasing number sense over the short period of the study. However these results need to be followed up with larger, controlled studies. The issues of transfer to higher-level tasks, and of the best developmental time window for intervention also need to be addressed.

Full text

An open trial assessment of ”The Number Race”, an
adaptive computer game for remediation of dyscalculia.
Anna Wilson, Susannah Revkin, David Cohen, Laurent D. Cohen, Stanislas
Dehaene
To cite this version:
Anna Wilson, Susannah Revkin, David Cohen, Laurent D. Cohen, Stanislas Dehaene. An
open trial assessment of ”The Number Race”, an adaptive computer game for remediation
of dyscalculia.. Behav Brain Funct, 2006, 2, pp.20. <10.1186/1744-9081-2-20>.<inserm-
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BioMed Central
Page 1 of 16
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Behavioral and Brain Functions
Open Access
Research
An open trial assessment of "The Number Race", an adaptive
computer game for remediation of dyscalculia
Anna J Wilson*1, Susannah K Revkin1, David Cohen4, Laurent Cohen1,3 and
Stanislas Dehaene1,2
Address: 1INSERM-CEA Unit 562 « Cognitive Neuroimaging » Service Hospitalier Frédéric Joliot, CEA-DRM-DSV, 91401 Orsay, France, 2Collège
de France, 11 place Marcelin Berthelot, 75231 Paris Cedex05, France, 3Service de Neurologie, Hôpital de la Pitié-Salpêtrière, AP-HP, 47 bd de
l'Hôpital, 75013, Paris, France and 4Department of Child and Adolescent Psychiatry, Université Pierre et Marie Curie, Laboratoire CNRS "Du
comportement et de la cognition", Hôpital Pitié-Salpêtrière, AP-HP, 47 bd de l'Hôpital, 75013, Paris, France
Email: Anna J Wilson* - ajwilsonkiwi@yahoo.fr; Susannah K Revkin - susannahrevkin@yahoo.fr; David Cohen - david.cohen@psl.ap-hop-
paris.fr; Laurent Cohen - laurent.cohen@psl.ap-hop-paris.fr; Stanislas Dehaene - dehaene@shfj.cea.fr
* Corresponding author
Abstract
Background: In a companion article [1], we described the development and evaluation of
software designed to remediate dyscalculia. This software is based on the hypothesis that
dyscalculia is due to a "core deficit" in number sense or in its access via symbolic information. Here
we review the evidence for this hypothesis, and present results from an initial open-trial test of the
software in a sample of nine 7–9 year old children with mathematical difficulties.
Methods: Children completed adaptive training on numerical comparison for half an hour a day,
four days a week over a period of five-weeks. They were tested before and after intervention on
their performance in core numerical tasks: counting, transcoding, base-10 comprehension,
enumeration, addition, subtraction, and symbolic and non-symbolic numerical comparison.
Results: Children showed specific increases in performance on core number sense tasks. Speed
of subitizing and numerical comparison increased by several hundred msec. Subtraction accuracy
increased by an average of 23%. Performance on addition and base-10 comprehension tasks did not
improve over the period of the study.
Conclusion: Initial open-trial testing showed promising results, and suggested that the software
was successful in increasing number sense over the short period of the study. However these
results need to be followed up with larger, controlled studies. The issues of transfer to higher-level
tasks, and of the best developmental time window for intervention also need to be addressed.
Background
In the preceding accompanying article [1], we described
the development and validation of software designed to
remediate dyscalculia. In this article, we first put the use of
the software in its context with a discussion of dyscalculia
including its symptoms, causes, and possible subtypes.
We then present results from initial testing of this software
in an actual remediation setting, using a group of children
with mathematical learning difficulties.
Published: 30 May 2006
Behavioral and Brain Functions 2006, 2:20 doi:10.1186/1744-9081-2-20
Received: 08 May 2006
Accepted: 30 May 2006
This article is available from: http://www.behavioralandbrainfunctions.com/content/2/1/20
© 2006 Wilson et al; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Developmental dyscalculia
Developmental dyscalculia ("dyscalculia") is a disorder in
mathematical abilities presumed to be due to a specific
impairment in brain function [2-4]. This neuropsycholog-
ical definition is highly similar to legal definitions of
mathematical disabilities (e.g. Public Law 94–142 in the
United States). However although the theoretical defini-
tion of these constructs is generally agreed upon, their
operalization is another issue, and varying selection crite-
ria have tended to result in considerable differences
between the populations in different studies [5]. This is
essentially because much remains to be discovered about
the symptoms of dyscalculia; in fact research on dyscalcu-
lia is in its infancy compared to research on its reading
analog, dyslexia. Here we briefly review what is currently
known. For a more in-depth discussion of this complex
topic, we refer the reader to a recent chapter [6], as well as
to several excellent reviews [5,7-10].
Dyscalculic children show a variety of fundamental math-
ematical deficits. These include an early delay in under-
standing some aspects of counting [11-13], and a later
delay in using counting procedures in simple addition
[12,14-16]. They also include a persistent deficit in mem-
orizing and recalling arithmetic facts (eg. 3 + 7 = 10 or 4 ×
5 = 20) [16-23]. In addition, recent studies (discussed fur-
ther below) have suggested that dyscalculic children show
numerical deficits at an even lower level, that of the repre-
sentation of quantity and/or the ability to link quantity to
symbolic representations of number.
The causes of dyscalculia remain unknown. Several
researchers have argued for a genetic component [24].
Indeed, dyscalculia is frequently observed in several
genetic disorders such as Turner's syndrome [25], Fragile
X syndrome [26], and Velocardiofacial syndrome [27].
However, factors such as premature birth and prenatal
alcohol exposure are also associated with higher rates of
dyscalculia [28,29]. Co morbid disorders are common,
particularly attentional deficit hyperactivity disorder
(ADHD), and dyslexia [17,19,21].
Proposed subtypes of dyscalculia
It has been proposed by many authors that there are sub-
types of dyscalculia, resulting from different causes and
showing different symptom profiles. However the evi-
dence on this question remains inconsistent. A full discus-
sion of this is beyond the scope of the current paper;
however it should be noted that much work has been car-
ried out addressing this issue in the special education field
[for reviews consult [7,8,24,30]].
Two recent subtype proposals are those of Geary [8,24],
and of Jordan and colleagues [16,31-33]. Based on a
review of the cognitive, neuropsychological and genetic
literature, Geary [24] proposed that there are three sub-
types of dyscalculia. The first is a procedural subtype, due
to executive dysfunction and characterized by a develop-
mental delay in the acquisition of counting and counting
procedures used to solve simple arithmetic problems. The
second is a semantic memory subtype, due to verbal
memory dysfunction and characterized by errors in the
retrieval of arithmetic facts. This type is linked to phonetic
dyslexia. These first two subtypes fit fairly well with
observed symptoms of dyscalculia. The third proposed
subtype is due to visuospatial dysfunction; however while
this subtype is found in adult acquired dyscalculia, evi-
dence for it in developmental dyscalculia is scarce,
because few studies have examined dyscalculic children's
spatial abilities. (One exception is a recent study by Maz-
zocco and colleagues [34]).
Jordan and colleagues have also argued for a subtype
linked to dyslexia, and have conducted several studies
revealing that children who have co-morbid dyslexia
(MDRD, for math and reading disabilities) show a differ-
ent pattern of deficits in mathematics than those who
have pure dyscalculia (MD only) [16,31-33]. However,
not all studies have found different profiles for these
groups on basic numerical cognition tasks [e.g. [35,36]].
In addition the differences found are only a single dissoci-
ation (MDRD children perform worse than MD only chil-
dren on word problems and untimed fact retrieval),
leaving open the possibility that MDRD children are sim-
ply have more difficulties in general, thus showing a
quantitative rather than a qualitative difference. Jordan
and others have also conducted studies suggesting that
children showing "fact retrieval deficits" might form a par-
ticular subtype [37,38], consistent with Geary's semantic
memory subtype. Thus these two subtype proposals are
not mutually exclusive.
Number sense
Up until very recently, most research on the symptoms of
dyscalculia focused on higher level tasks such as addition,
subtraction and problem solving. The problem with this is
that multiple cognitive processes are likely to contribute
to each of these tasks, and thus they may not be ideal for
clearly illuminating key symptoms of dyscalculia. Con-
trary to this approach, research in normal adult numerical
cognition has focused on basic component processes and
used extremely simple tasks. This research has led to the
identification of a core aspect of numerical cognition:
"number sense", or the ability to represent and manipu-
late numerical quantities non-verbally [39,40]. (Note that
the phrase "number sense" is also used in the special edu-
cation community with varied and broader meanings, for
a discussion see [41]. Here we used the phrase as used in
the cognitive neuroscience literature [39]).

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Number sense is known to be present within the first year
of life, and to undergo normal development during early
childhood, in particular a progressive refining of the accu-
racy with which numerical quantities can be estimated
[42,43]. Number sense does not depend on language or
education, inasmuch as it is present in Amazonian chil-
dren and adults without formal education or a developed
number lexicon [44]. This ability has now been linked to
a particular area of the brain, the horizontal intra-parietal
sulcus (HIPS) in both adults and children [45-47].
Dyscalculia as a core deficit in number sense
Could dyscalculia, or at least one of its subtypes, be the
result of an impairment in number sense? Recently, sev-
eral studies have tested dyscalculic children's performance
on the same tasks used in adult numerical cognition
research. For instance Landerl et al. [35] found that dyscal-
culic children were slower at numerical comparison (but
not non-numerical comparison) compared to controls,
and that they showed deficits in subitizing (rapid appre-
hension of small quantities). These results confirmed a
sole earlier finding that dyscalculic children were slower
to process numbers (compared to letters) relative to non-
dyscalculic children, and that they appeared to be slower
in subitizing [48]. Rousselle and Noël [36] also found that
dyscalculic children were slow to perform number com-
parison, and that some subjects lacked the typically
observed "distance effect" (higher reaction times for
smaller numerical distances). Some evidence also suggests
that dyscalculic children may exhibit less automatic acti-
vation of quantity from Arabic digits [49] (although other
authors have failed to replicate this result [36]).
Other studies have started to look at the predictive value
of number sense measures, which have recently been pro-
posed for use in screening for dyscalculia, based on cross-
sectional correlations with mathematics performance in
kindergarten and first grade [50-53]. For instance Maz-
zocco and Thompson [53] showed that kindergarteners
who perform badly on number comparison, number con-
stancy, and reading numerals are likely to show persistent
dyscalculia in grades 2 and 3.
As of yet, the neural bases of dyscalculia have only been
investigated in special populations, but results from these
studies reveal abnormalities in the area associated with
number sense. For instance, one study showed that dys-
calculic adolescents who were born pre-term had less grey
matter in the left HIPS than non-dyscalculic adolescents
who were born pre-term [28]. Molko and colleagues
[25,54] found that young adult women with Turner's syn-
drome, which is associated with dyscalculia, showed
structural and functional abnormalities in the same area,
particularly in the right hemisphere.
Based on this data, we and others have proposed that dys-
calculia (or at least one of its subtypes) may be caused by
a core deficit, this deficit being either in number sense
itself, or in its access from symbolic number information
[6,41,52,55-57]. (Note: Although we refer to both of these
hypotheses as falling under a "core deficit", other authors
have termed the latter hypothesis the "access deficit"
hypothesis [36].) According to the triple-code model of
numerical cognition [58,59], numbers are represented in
three primary codes: visual (Arabic digits), verbal
(number words), and analog (magnitude representation).
The non-symbolic (analog) representation appears to
develop early in infancy [42], but children establish the
symbolic representations as a result of culture and educa-
tion [36,44]. An essential aspect of the triple-code model
is the presence of bidirectional transcoding links between
all three representations. Dyscalculia might therefore be
caused by either a) a malfunction in the representation of
numerical magnitude itself, or by b) malfunction in the
connections between quantity and symbolic representa-
tions of number.
These two hypotheses could be separated based on per-
formance on non-symbolic tasks. Direct impairment of
the quantity system would result in failure on both non-
symbolic and symbolic numerical tasks, whereas a discon-
nection should leave purely non-symbolic tasks intact. A
recent study [36] brought support to the disconnection
hypothesis, with dyscalculic children showing a deficit
only on symbolic number comparison and not on non-
symbolic comparison. However, this issue needs to be
investigated further.
How to evaluate dyscalculia?
In spite of this remaining uncertainty, the core deficit
hypothesis readily leads to a choice of tests that might be
appropriate to reveal symptoms of developmental dyscal-
culia. These symptoms should be similar to those seen in
cases of acquired acalculia where number sense is
impaired. They might therefore include a reduced under-
standing of the meaning of numbers, and a low perform-
ance on tasks which depend highly on number sense,
including non symbolic tasks (e.g. comparison, estima-
tion or approximate addition of dot arrays), as well as
symbolic numerical comparison and approximation.
With respect to simple arithmetic, we would expect that
subtraction would provide a more sensitive measure of a
core number sense deficit than either addition or multipli-
cation. This is because addition and particularly multipli-
cation problems are thought to be more frequently solved
using a memorized table of arithmetic facts than are sub-
traction problems. In adult acalculic patients, subtraction
can double-dissociate from addition and multiplication:
subtraction deficits are frequently associated with impair-

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ment in number sense tasks such as approximation, while
addition and especially multiplication deficits are fre-
quently associated with aphasia or alexia [40,46,60,61]. If
these findings can be generalized in a developmental con-
text, they would suggest that dyscalculic children of the
"core number sense" subtype should show particular dif-
ficulties with elementary subtraction problems.
Based on the adult neuropsychological data, we would
expect that more verbal tasks such as counting and fact
retrieval should be less affected. However, this latter pre-
diction is not clear-cut because it is unknown to what
extent number sense contributes to the acquisition of
counting principles (especially cardinality) and helps
bootstrap an early understanding of the meaning of addi-
tion and subtraction. Even though fact retrieval in adults
relies largely on a rote memory process, it is possible that
number sense could aid in retrieval success by increasing
the semantic content of the information being retrieved
[62].
The previously discussed low-level symptoms observed in
dyscalculia (impairments in speed of processing of
numerical information, in numerical comparison and
subitizing, and possibly in automatic access to magnitude
information) are all consistent with the core deficit the-
ory. The higher level symptoms (impairments in acquisi-
tion of counting and addition procedures and in fact
retrieval) may be a derivative of an initial dysfunction of
the core number sense system. However there are many
other possible causes of these high-level symptoms. One
way to indirectly test the core deficit theory is to test the
response of children to a primarily quantity-based inter-
vention, an undertaking which we present in the current
study.
"The Number Race" software
In the accompanying article [1], we described in detail the
development and validation of "The Number Race", an
adaptive game designed for the remediation of dyscalcu-
lia. The software was designed with both of the possible
causes of a core deficit in mind. In order to enhance
number sense, it provides intensive training on numerical
comparison and emphasizes the links between numbers
and space. In order to cement the links between symbolic
and non-symbolic representations of number, it uses scaf-
folding and repeated association techniques whereby Ara-
bic, verbal and quantity codes are presented together, and
the role of symbolic information as a basis for decision
making is progressively increased. In addition, higher lev-
els of the software are designed to provide training on
small addition and subtraction facts, although this train-
ing is restricted to a small range of numbers and provides
conceptually oriented, concrete representations of opera-
tions rather than drilling of arithmetic facts.
The current study
The current study was carried out as the first step in an
ongoing series of tests of the efficacy of the "Number
Race" software. In this first study, we used an open-trial
design, analogous to the first stage of testing of a new
medical therapy: We identified a group of children who
had learning difficulties in mathematics, and tested their
performance on a battery of numerical tasks before and
after remediation. This design had two primary goals: 1)
to determine whether performance improves significantly
between the pre- and post-training periods, a minimal
requirement before proceeding with larger and more
expensive studies; and 2) to identify which measures of
arithmetic performance are most sensitive to training.
An analysis of children's improvement profiles across the
tasks tested at pre and post remediation also allowed for
an assessment of the coherence of this pattern with the
core deficit hypothesis. The tasks tested were based on
previous work in adults with acquired acalculia as well as
on previous work in developmental dyscalculia. They cov-
ered the main cognitive processes involved in numerical
processing as well as the basic academic skills relevant to
schoolwork in early elementary school. They included ver-
bal counting, transcoding, base-10 comprehension, enu-
meration (permitting measurement of subitizing and
counting), addition, subtraction, and symbolic and non
symbolic number comparison. We hypothesized that chil-
dren would show the largest improvement on tasks which
draw more heavily on number sense, such as number
comparison, subtraction, and to a lesser extent, addition.
Conversely, we did not expect to see much improvement
in counting or transcoding, because these tasks do not
depend much, if at all, on number sense. If a core deficit
is caused by difficulties in linking symbolic and non-sym-
bolic information, we should see greater improvement in
symbolic rather than non-symbolic number comparison.
Many more specific patterns could be expected within par-
ticular tasks based on research in normal subjects and
adult acalculic patients; we discuss these below in the con-
text of each task.
Methods
Sample
Twenty two French children aged 7–10 years were
recruited from three participant schools in Paris by
teacher recommendation; which was based on the obser-
vation of persistent and/or severe difficulties in mathe-
matics. We carried out exclusion screening for these
children, of whom 13 were selected for the study. All chil-
dren and their families gave informed consent prior to
screening.
Children were tested by a native French speaker and
trained neuropsychologist using a WISC-III [63] short

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form [64] consisting of vocabulary, picture completion,
and arithmetic. Children were excluded if they had an esti-
mated IQ of less than 80 using the non-arithmetic WISC-
III subtests (3 children), lack of a below average score on
the WISC arithmetic (2 children), or behavioral difficul-
ties (1 child). Three children were excluded due to diverse
other reasons (teacher withdrawal, failure to meet age
requirements, and visual problems). Four of the children
who participated in the study were eventually excluded
from the final sample due to extended absences (two chil-
dren), disruptive behavior (one child), and the discovery
that French was not the child's first language (one child).
The final sample for the study was thus nine children
between the ages of 7 and 9 (average age = 8.1 years). Of
the final sample, three of the children were repeating a
school year (this is common practice in France for chil-
dren who are not making adequate progress). The arith-
metic subtest scores of the WISC for the final sample
ranged from 1st to 37th percentile, with the average score at
the 12th percentile, confirming children's low mathemat-
ics performance. It should be noted that in the absence of
recruitment from a large population and the use of a strict
cut-off procedure, the current sample is best described as
children with mathematical learning difficulties rather
than dyscalculia stricto sensu.
Procedure
The study took place at school during school hours over a
period of 10 weeks. Screening and pre-testing occurred in
the first two weeks, children were on vacation during the
next two weeks, and then completed their remediation in
the fifth to ninth weeks. This consisted of one half-hour
session using the software for each child four days a week,
supervised by the authors (AJW & SKR), thus for a maxi-
mum of ten hours (due to absences, the average was eight
hours). During the tenth week, the children were post-
tested. (One child fell ill during this period, and had to be
tested three weeks later.)
Testing battery
Children were tested in three half-hour sessions, primarily
using a computerized testing battery. Tasks included were
enumeration, symbolic and non-symbolic numerical
comparison, addition, and subtraction. These tasks were
designed to measure basic components of numerical cog-
nition, and were based on work in adult acalculic patients
[61], as well as in recent work in developmental dyscalcu-
lia [35]. We supplemented the computerized battery with
three non-computerized tasks (counting, transcoding and
understanding of the base-10 system), which were sub-
tests drawn from the TEDI-MATH battery [65]. This bat-
tery was developed for the assessment and profiling of
dyscalculia.
Non-computerized tasks
The three non-computerized tests took a half-hour to
complete and were administered by a native French
speaker and trained neuropsychologist. Because the TEDI-
MATH battery is not available in English we describe them
briefly below.
Counting
The test included 6 items, each worth 2 points. Children
were asked to count as high as they were able (the experi-
menter stopped them at 31), to count up to particular
numbers (9 and 6), to count starting at particular number
(3 and 7), to count from one number to another number
(5 to 9 and 4 to 8), to count backwards from a number (7
and 15), and to count by 2 s and by 10 s.
Transcoding
This test was designed to measure children's ability to read
and write Arabic digits. In the first part of the test, children
were dictated 20 numbers, ranging from 1 to 3 digits,
which they wrote on a sheet of paper. In the second part
of the test, children were asked to read 20 written Arabic
numbers, also ranging from 1 to 3 digits.
Base-10 comprehension
In the first part of this test (11 points), children were
shown small plastic rods arranged in bundles of ten, as
well as a stack of individual rods. The tester showed three
combinations of rods and bundles and the child had to
say how many rods were in each combination (20, 24,
and 13). Then children were given a verbal quantity and
asked how many bundles and rods would be needed to
make this quantity (14, 20, 8, and 36). Finally they were
told that the tester had a certain quantity of rods, and
wanted to give another quantity to a friend; would she
need to break open a bundle to do this? (15-7, 29-6, 16-5,
and 32-4).
In the second part of the test (6 points), children were
given two types of round tokens, which they were told
represented money (1 euro and 10 euros). They were
asked to show how many tokens would be required to buy
a toy which cost a particular amount (17, 13, 19, 23, 15
and 31 euros).
Finally, in the third part of the test (10 points), children
were shown a sheet of paper with written numerals, and
asked to circle the ones column (28, 13, 10, 520, and 709)
or the tens column (20, 15, 37, 650, 405).
Computerized tasks
The computerized tasks were administered in two half-
hour sessions. In the first session, children completed dot
enumeration, addition and subtraction. In the second ses-
sion they completed the two comparison tasks (symbolic

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and non-symbolic). Children were given instructions at
the beginning of each task, and then completed several
training trials (using different stimuli from experimental
trials where possible). The experimenter sat next to the
child throughout the task to ensure that they were paying
attention, and gave the child breaks of several min as
needed.
All tasks were presented on a Celeron laptop running E-
Prime software [66], with a 14 inch screen set to 600 × 800
pixel resolution. We measured accuracy and reaction time
for all tasks. For those requiring a voice response, reaction
time was measured using a microphone connected to a
serial response box, and the child's responses were
recorded by the experimenter, who also coded for micro-
phone errors. If children had difficulties with the micro-
phone, a back-up system was used, in which the
experimenter pressed a key as soon as they responded. In
tasks requiring a manual response, children pressed one
of the two touch-pad mouse buttons to indicate the side
of the screen (left or right) of the correct stimulus. Chil-
dren were given no feedback on their accuracy. In all tasks
children had to respond within 10 sec (except in addition
and subtraction, for which they had 15 sec).
Dot enumeration
In this task, based on Mandler and Shebo [67], we exam-
ined children's subitizing and counting performance by
measuring verbal reaction times for enumeration of sets of
one to eight dots. Children were told to count whenever
they needed to. They completed 64 trials in two blocks,
which took around 10 min. Stimuli consisted of 64 square
350 by 350 pixel white images, each containing a set of 1
to 8 randomly arranged black 36 pixel diameter dots (8
images for each numerosity 1 through 8). They were gen-
erated using a Matlab program which generated dot dis-
plays controlled for overall occupied area (thus distance
between the dots was larger for smaller numerosities). The
time course of each trial was as follows: The trial began
with an auditory alerting signal (a beep) concurrent with
the appearance of a visual alerting signal (a green fixation
symbol composed of the two signs <> presented in the
centre of the screen). The dot display appeared in the cen-
tre of the screen 1500 msec after this signal, and remained
on screen for 10 sec, or until the child responded. The
screens had a black background throughout the experi-
ment. As soon as the child responded they were presented
with a "reward" image for one second, which was an
attractive square tile filled with abstract color patterns.
The same image appeared on all trials, regardless of the
accuracy of the response. After the offset of this image,
there was a two second inter-trial interval.
A normal pattern of results is a reaction time curve which
is almost constant over the numerosities 1–3 (reflecting
subitizing), and then increases steadily for numerosities
4–8 (reflecting counting). Previous work suggests that
dyscalculic children show slowing in both the subitizing
and counting range, although more markedly in the subi-
tizing range [35,48]. In addition, in adult acquired acalcu-
lia patients, subitizing deficits have been associated with
number sense deficits [61]. We thus might expect that
children's performance in both of these ranges would
improve after remediation, but more so in the subitizing
range, which provides the purest measure of quantity
processing.
Addition
In this task we measured verbal responses to single digit
addition problems. Children were told they could use
their fingers if they needed to. They completed 32 trials in
two blocks, which took around 10 min. Each problem was
coded by type, which had four categories: tie, rule, nor-
mal-small, and normal-large. Tie items consisted of prob-
lems with identical operands (e.g. 5 + 5). Rule items
consisted of sums which can be solved by the application
of a simple rule, in this case x + 0 = x. Normal items were
items which did not fall in either of these categories. These
were divided into two groups according to problem size,
which had two categories: "small problems" (sum 10 and
under) and "large problems" (sum 11 or over). Stimuli
consisted of 32 single digit addition problems, with the
larger digit always listed first. The stimulus selection proc-
ess was thus as follows: all possible single digit addition
pairs were listed, and coded for type and magnitude. A full
cross of the factors of type and magnitude was possible,
except that for the type "rule", there were no large prob-
lems. The cell size for each type was reduced so that there
were 16 normal problems (8 small, 8 large), 8 rule prob-
lems, and 8 tie problems (4 small, 4 large). The time
course of each trial was the same as in the dot enumera-
tion task, except that children had 15 sec to respond.
Problems were presented in 18 point courier font and
colored white.
A normal pattern of results is for rule (ie. x + 0 = x) and tie
(ie. x + x = y) items to show faster response times, and for
small items to show faster response times than large items
("magnitude effect", see [68] for a historical review). As
previously mentioned, dyscalculic children tend to show
a developmental delay in the use of addition procedures
involving counting and also in recall of addition facts.
Addition procedures such as finger counting, or counting
up from the larger addend, were not included in the soft-
ware, nor was practice on facts with a sum larger than ten.
The software did of course emphasize understanding and
manipulation of quantities, and thus to the extent that
addition involves this, we might expect children to show
some improvement. However, the role of number sense in
addition is not clear-cut. In adults at least, exact addition

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appears to involve primarily rote memory processes. Thus
it was not clear whether children would show improve-
ment in addition or not.
Subtraction
In this task we measured verbal responses to single digit
subtraction problems. Children were told they could use
their fingers if they needed to. They completed 36 trials in
two blocks, which took around 10 min. There were three
types of problem: "rule", "small subtrahend" and "large
subtrahend". Rule items consisted of items which can be
solved by the application of a simple rule, such as x – 0 =
x, or x – x = 0. All other items were items which did not
fall in this category. Small subtrahend items had a subtra-
hend of 2–4 (inclusive) and large subtrahend items had a
subtrahend between 5–8 (inclusive). Stimuli consisted of
36 single digit subtraction problems, with the larger digit
always listed first. The stimulus selection process was as
follows: all possible single digit subtraction pairs were
listed, and coded for type (rule vs. non-rule and small vs.
large subtrahend). The cell size for each type was reduced
so that there were 18 normal problems (10 small subtra-
hend, 8 large subtrahend), and 18 rule problems. The
time course of each trial was the same as in the addition
task. Problems were presented in 18 point courier font
and colored white.
A normal pattern of results is for rule items (x – 0 = x, or x
– x = 0) to show faster response times (because quantity
manipulation is not required to resolve them), and for
items with a large subtrahend to show slower reaction
times than those with a small subtrahend (reflecting
counting backwards, or counting up from the subtra-
hend). As previously mentioned, dyscalculic children also
show a developmental delay in the use of subtraction pro-
cedures involving counting. Unlike addition, however, in
adults subtraction (of non-rule items) is thought to
depend primarily on the ability to understand, represent
and manipulate quantity [46]. Therefore we would expect
children to show considerable progress in subtraction (at
least on non-rule items).
Numerical comparison: symbolic
In this task, based on Moyer and Landauer [69], children
were presented with two digits of different numerical
sizes, and had to indicate which was the largest. Children
completed 36 trials in two blocks, which took around 5
min. Stimuli consisted of all of the possible pairs of one
digit numbers (excluding zero) irrespective of order, giv-
ing a total of 36 pairs. The side of the largest number was
varied randomly from trial to trial. The time course of
each trial was the same as in the dot enumeration task.
The pairs of digits were presented in white on a black
screen, offset 40 pixels from fixation.
A normal pattern of results is the classical "distance
effect", in which reaction times are longer and accuracy
lower to compare numbers which are closer than numbers
which are further away [69]. As previously mentioned,
dyscalculic children are slower at symbolic number com-
parison than non-dyscalculic controls [35]. Given that
number comparison is the primary task in the software, at
the least we expected children to show a general increase
in performance. Furthermore, we expected children to
show a change in the shape of the distance effect, reflect-
ing an increase in precision of quantity representation.
Numerical comparison: non-symbolic
In this task, children were presented with two arrays of
dots of two different numerosities, and had to indicate
which was the largest. The numerosities used were exactly
the same as those used in the symbolic task. Children
completed 36 trials in two blocks, which took around 5
min. Stimuli were 250 pixel diameter white circles, con-
taining arrays of black dots. All possible pairs of one digit
numerosities (excluding zero) were included, giving a
total of 36 pairs. Half of the stimuli pairs were equalized
for total occupied area (of the array) and dot size, but var-
ied in total luminance and density, and the other half
were equalized for total luminance and density, but varied
in total occupied area and dot size. Dot arrays presented
in a given run of the experiment were randomly drawn
(without replacement) from a pool of arrays double the
size needed. The side of the largest number was varied ran-
domly from trial to trial. The time course of each trial was
the same as in the symbolic comparison task, except that
in order to avoid counting, the pair of dot arrays were
flashed on screen for only 840 msec, in conjunction with
a central yellow fixation star (*), which then remained on-
screen for 9160 msec, or until the child responded. Stim-
uli were offset 130 pixels from fixation.
The interest of including this task was to compare chil-
dren's progress on it to that on the symbolic task. If a core
deficit is caused primarily by difficulties in linking sym-
bolic and non-symbolic information, we should see an
improvement in symbolic number comparison, but con-
siderably less improvement in non-symbolic number
comparison.
Results
For all computerized tasks, mean accuracy and median
reaction time (RT; for correct responses only) were calcu-
lated for each subject within each condition. These values
were then entered in repeated measures ANOVAs, to com-
pare pre and post differences. We tested for all main
effects and interactions.

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Enumeration performance in the subitizing and counting rangesFigure 1
Enumeration performance in the subitizing and counting ranges. a) Average enumeration reaction times and accu-
racy in the subitizing range, showing a significant improvement (p = 0.003) at post-test. b) Average enumeration reaction times
and accuracy in the counting range. No difference is shown between pre and post-test.
Enumeration: Counting range reaction time
and error rate (n = 8)
1000
2000
3000
4000
5000
6000
45678
Numerosity
RT (msec)
0%
20%
40%
60%
80%
100%
Error rate
Pre
Post
Enumeration: Subitizing range reaction time
and error rate (n = 8)
600
800
1000
1200
1400
1600
123
Numerosity
RT (msec)
0%
20%
40%
60%
80%
100%
Error rate
Pre
Post
a.
b.

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Non-computerized tasks
Children showed marginally significant improvements in
the counting subtest of the TEDI-MATH, with an average
pre and post scores of 9/12 and 11/12 respectively (t(8) =
2.24, p = 0.056). They also showed a small significant
improvement in transcoding, with average pre and post
scores of 36/40 and 39/40 respectively (t(8) = 2.42, p =
0.04). In both of these tasks about half the children were
below their age and/or class level at pre-testing, and all
had reached at least their class level at post testing. How-
ever understanding of base-10 system showed no
improvement; even though four children were considera-
bly below age-level at pre-testing.
Computerized tasks
Dot enumeration
Dot enumeration results are shown in Figures 1a (subitiz-
ing range) and 1b (counting range). Results showed a
large improvement in speed for the subitizing range but
not the counting range. Following previous authors [35],
we analyzed the data in two separate ANOVAs, a 2 × 3
ANOVA for the subitizing range, and a 2 × 5 ANOVA for
the counting range. Data from one subject was excluded
from both of the analyses, because she showed a highly
abnormal pattern at post-testing, and because 5 out of her
6 medians in the subitizing range were outliers in the
group distribution (a distance of over 2 times the inter-
quartile range from the median). In the subitizing range,
children's RT decreased an average of nearly 300 msec in
the second session relative to the first, and the RT analysis
showed significant main effects of session (F(1,7) = 19.1,
p = 0.003), and of numerosity (F(2,14) = 6.18, p = 0.01).
There was no significant session x numerosity interaction.
Children were slightly less accurate in the second session
(96% in the second session vs. 98% in the first), however
this difference was non- significant (F(1,7) = 2.33, p =
0.17), thus it is unlikely that it indicates a speed/accuracy
trade-off. In the counting range, children's overall per-
formance showed almost no change. Both the RT and
accuracy analyses revealed only highly significant main
effects of numerosity (F(4,28) = 66.2, p < 0.001, F(4,28) =
4.65, p = 0.005, respectively), but no change in the mean
or slope of performance as a function of numerosity.
Addition
Addition results are shown in Table 1. The data were ana-
lyzed using 2 × 4 ANOVAs (session × problem type). The
main effect of session was not significant for either accu-
racy or RT, which stayed at a similar level from pre to post
testing. The main effect of problem type was significant in
both the accuracy and RT analyses (F(3,24) = 14.9, p <
0.001; F(3,24) = 92.7, p < 0.001, respectively), however
this simply reflected normal differences between rule and
tie vs. "normal" problems, as well as a magnitude effect
(worse performance with larger operands). This effect did
not show an interaction with session, and no improve-
ment was seen in any of the four categories of problems.
Subtraction
Subtraction accuracy is shown in Figure 2, and reaction
time in Table 1. Children's performance, which was ini-
tially low, showed a large pre-post change. The data were
analyzed using 2 × 3 ANOVAs (session × problem type).
In the accuracy analysis, the main effect of session was sig-
nificant (F(1,8) = 6.51, p = 0.03), and the main effect of
problem type was highly significant (F(2,16) = 6.85, p =
0.007). The overall interaction between session and prob-
lem type was not significant, however the largest improve-
ments were seen in the non-rule-based problems:
accuracy increased to from 58% to 87% for small subtra-
hend problems (p = 0.07 using a post-hoc t-test) and from
50% to 67% for large subtrahend problems (p = 0.08),
while performance on rule problems, in contrast, showed
no significant change across session. In the RT analysis
(Table 1), only a highly significant effect of problem type
was observed (F(2,12) = 28.9, p < 0.001). (Note: 4 out of
54 observations in the analysis had missing data, due to
Table 1: Mean accuracy and reaction time for addition task, and reaction time for subtraction task.
Rule Tie Small Large
Addition: Accuracy (%)
Pre 100 (0) 76 (5) 92 (2) 61 (10)
Post 97 (3) 88 (5) 89 (2) 64 (9)
Reaction Time (msec)
Pre 1995 (245) 2974 (293) 3172 (365) 6892 (347)
Post 1694 (229) 2908 (618) 3244 (368) 6427 (454)
Subtraction: Reaction Time
(msec)
Pre 2564 (172) 6032 (512) 8010 (834)
Post 2428 (179) 6724 (725) 8575 (1049)
Note. Parentheses contain standard error.

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the lack of any correct responses available to contribute to
the median reaction time.)
Number comparison: symbolic
The symbolic number comparison data are shown in Fig-
ures 3a and 3b, and were analyzed by 2 × 4 (session × dis-
tance) ANOVAs. For each trial, the numerical distance
between the two stimuli (A and B) was evaluated as their
log ratio |log(A/B)|. Trials were grouped into four catego-
ries of increasing distance, maintaining roughly equal cell
sizes. Graphs show the (weighted) mean Weber fraction
for each category. The accuracy analysis showed no signif-
icant main effect for session, although as was expected
there was a highly significant main effect for distance
(F(3,24) = 7.51, p = 0.001). The session × distance inter-
action fell short of significance (F(3,24) = 2.41, p = 0.09).
As can be seen from Figure 3a, this reflects a slight change
in the shape of the accuracy curve from pre to post testing,
consistent with the post-testing curve becoming steeper.
This suggests that the precision of children's numerical
representation may have increased.
In addition, children were overall much faster in respond-
ing at post test (by 468 msec), reflected in a significant
main effect of session (F(1,8) = 19.7, p = 0.002) in the RT
analysis. The main effect of distance was also significant
(F(3,24) = 7.08, p = 0.001). Children tended to show
slightly less of a change in RT across distance in the post
test, although this interaction did not reach significance
(p = 0.17). This trend is similar to changes which occur as
a result of normal development [71], which are an overall
increase in speed and a reduction in the slope of the dis-
tance effect.
Number comparison: non-symbolic
The non-symbolic number comparison data were ana-
lyzed in exactly the same way as the symbolic comparison
data and are shown in Figures 3c and 3d. Children's initial
performance on this task was less accurate overall than in
the symbolic task, but at post testing there was an overall
increase in accuracy (+5%), reflected in a significant main
effect of session (F(1,8) = 9.81, p = 0.01), again suggesting
a higher precision after training. The main effect of dis-
tance was highly significant (F(3,24) = 6.70, p = 0.002).
The session × distance interaction was not significant.
Children's initial speed of response for this task was much
faster than for the symbolic comparison task. Neverthe-
less, they still showed a significant speed increase of 226
msec across the remediation period (main effect of ses-
sion significant, F(1,8) = 13.8, p = 0.006). The main effect
of distance was also significant (F(3,24) = 6.56, p =
0.002). There was no significant interaction, reflecting the
fact that the distance effect was around the same size at pre
and post testing.
In order to test whether the effects of remediation were
different for the two types of comparison tasks (symbolic
vs. non-symbolic), we ran post-hoc 2 × 2 × 4 (task × ses-
sion × distance) ANOVAs for both reaction time and accu-
racy. For reaction time, a main effect of task was observed
(F(1,8) = 19.24, p = 0.002), reflecting the fact that chil-
dren were faster on average at the non-symbolic task.
There was a significant task × session interaction (F(1,8) =
6.77, p = 0.03), reflecting the larger improvement in reac-
tion time for the symbolic comparison task. For accuracy,
however, there was a trend towards a task × session inter-
action (F(1,8) = 3.99, p = 0.08) but in the converse direc-
tion, reflecting a larger increase in accuracy for the non-
symbolic task. This limits interpretation of the task × ses-
sion interaction in reaction time, because it might be due
to a speed-accuracy trade-off. In the accuracy analysis, the
task × session × distance interaction was significant
(F(3,24) = 3.17, p = 0.04), reflecting an increase in accu-
racy for the non-symbolic task at far distances, whereas
the symbolic task showed an increase only at one close
distance.
Discussion
Children's results from pre and post testing showed
improvement on several tasks, suggesting that the remedi-
ation was successful in producing an improvement in
basic numerical cognition. Of course the use of an open-
trial design limits the conclusions which can be made, due
to the lack of a control group. It is important for future
studies to allow a separation of which effects are due to
Performance in subtraction before and after trainingFigure 2
Performance in subtraction before and after training.
Subtraction average accuracy (significant main effect of ses-
sion, p = 0.04). "Rule" items were items such as x-x = 0 or x-
0 = x. Small subtrahend items had a subtrahend from 2–4
inclusive, and large subtrahend items had a subtrahend from
5–8 inclusive.
Subtraction accuracy (n=9)
30%
40%
50%
60%
70%
80%
90%
100%
Rule Small Subtrahend Large Subtrahend
Accuracy
Pre
Post

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the software, and which can be attributed to general atten-
tional or motivational improvements, practice with com-
puters, test-retest effects, or regression to the mean.
Notwithstanding, we argue that many of the specific
improvements seen are unlikely to be due to such general
factors. For example, in order to propose that the increase
in enumeration speed was the result of general attentional
improvements or regression to the mean, one would have
to explain why it only occurred within a specific range of
small numerosities 1–3.
Consistency of results with the core deficit hypothesis
To what extent are the results observed consistent with the
core deficit theory of dyscalculia? As predicted, children
showed improvements in performance on classical
number sense tasks. This was most marked in numerical
comparison (both symbolic and non-symbolic), in which
Performance in symbolic and non-symbolic comparison before and after trainingFigure 3
Performance in symbolic and non-symbolic comparison before and after training. a) Symbolic comparison (Arabic
digit) accuracy, plotted as a function of the distance between the numbers (measured by their log ratio). A slight change in
shape in the post curve suggests an increase in precision of the quantity representation. b) Symbolic comparison (Arabic digit)
reaction time. A significant decrease is seen between pre and post curves in overall RT (p = 0.002). c) Non-symbolic compari-
son (dot clouds) accuracy. A significant increase in accuracy at post test (p = 0.01) suggests a more precise representation of
numerosity. d) Non-symbolic comparison (dot clouds) reaction time. A significant decrease is seen in overall RT (p = 0.006).
Note: Error bars indicate one standard error.
Non symbolic compar ison reaction time (n=9)
600
800
1000
1200
1400
1600
0 0.2 0.4 0.6 0.8
Numerical distance ( = |log(A/B)| )
RT (msec)
Pre
Post
Non symbolic compar ison accuracy (n=9)
75%
80%
85%
90%
95%
100%
0 0.2 0.4 0.6 0.8
Numerical distance ( = |log(A/B)| )
Accurac
y
Pre
Post
Symbolic comparison accuracy (n=9)
75%
80%
85%
90%
95%
100%
0 0.2 0.4 0.6 0.8
Numerical distance ( = |log(A/B)| )
Accurac
y
Pre
Post
Symbolic comparison reaction time (n=9)
1000
1200
1400
1600
1800
2000
0 0.2 0.4 0.6 0.8
Numerical distance ( = |log(A/B)| )
RT (msec)
Pre
Post
c. d.
a. b.

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performance was 200–450 msec faster at post test. There
were also some indications that in the symbolic task the
quantity information extracted by children might have
become more precise: in accuracy the distance effect
tended towards a steeper curve, and in reaction time the
distance effect showed a trend towards being smaller at
post test. A similar effect was not seen for the non-sym-
bolic task. The latter results seem consistent with the idea
that a core deficit might be caused by impairment in the
links between non-symbolic and symbolic representa-
tions; however they should be interpreted with caution
since they were only marginally significant.
Results from the enumeration task supported the core def-
icit hypothesis, with children's speed of enumeration
increasing by 300 msec for numerosities in the subitizing
range of 1–3, whilst showing no change in the counting
range of 4–8. This is consistent with findings that dyscal-
culic children show impaired subitizing [35,48], and with
the association between subitizing and number sense def-
icits seen in adult patients [61]. Previous research has
demonstrated plasticity in subitizing performance in
young adult video game players [72]. The present results
suggest that this plasticity in numerical cognition can be
harnessed by simpler computer games and be put to use
for the remediation of dyscalculia.
Also consistent with the core deficit hypothesis was the
large increase in children's subtraction accuracy (an aver-
age of 23% on non-rule problems), which suggests an
improvement in the ability to manipulate or conceptual-
ize quantities. This increase was not seen for addition,
consistent with the idea that subtraction draws more
strongly on quantity representation and manipulation
than addition does [46]. More generally, as predicted,
children showed little or no improvement on tasks which
do not rely heavily on number sense; rule-based items in
addition and subtraction as well as counting and trans-
coding. In counting there was no improvement seen in
the 4–8 range of enumeration task, and although there
was a slight improvement seen in the non-computerized
task, this was in the more difficult aspects of the task,
counting backwards and counting by twos or tens (which
are arguably more a measure of calculation). There was
only a small improvement seen in the non-computerized
task of transcoding (8%). These last two tasks had a close
to ceiling performance at initial testing, however this was
not the case for the computerized enumeration task.
One task which showed a lack of improvement despite
low initial performance was base-10 comprehension. We
had no particular a priori hypothesis for this task, because
there is no research on the relationship between base-10
comprehension and number sense, even in adults. In
purely empirical terms these results cannot be seen as sur-
prising because the software did not include two-digit
numbers, or explicit training on the base-10 system. How-
ever, this does not exclude the possibility of an eventual
benefit at a later stage (see transfer discussion below).
The dissociation between improved subtraction and
unchanged addition requires further discussion. We have
stressed the possibility that subtraction is more quantity-
based than addition [46]. In adults, this is likely because
few subtractions are memorized (thus they have to be
solved by manipulating the quantities involved). In con-
trast, addition problems are often solved by verbal mem-
ory recall. Thus the failure of children to improve in
addition could be seen as caused by a lower involvement
of number sense in this process. However it should be
noted that it is unknown to what extent dyscalculic chil-
dren of this age solve addition problems by verbal mem-
ory.
An alternative possibility is that there was an initial qual-
itative difference in children's prior knowledge of subtrac-
tion vs. addition. In our sample, children seemed much
less familiar with subtraction at the beginning of the
study. Based on our observations during testing and a
post-hoc error analysis, we found that in subtraction chil-
dren made more errors which were conceptually rather
than procedurally based; such as application of an incor-
rect rule (x-x = x, or x-0 = 0), adding instead of subtracting,
or simply not knowing how to proceed to calculate a
response (particularly not knowing how to use their fin-
gers to do so). In contrast, all children were familiar with
addition and the procedures of adding on their fingers or
by verbal "counting up" (even if they did not do this fast
or accurately). The types of errors made in addition indi-
cated that children had grasped the general idea of adding.
The majority of incorrect responses were actually non-
responses due to children running out of time to execute
slow finger counting procedures. The small amount of
other errors were due to mis-counts, bridging errors (e.g.
8 + 6 = 4) or memory (e.g. 2 + 2 = 8) errors. Thus,
improvement in subtraction may have resulted from
improved conceptualization as a result of increased
number sense. This may not have occurred for addition
because children already had a reasonably solid concept
of addition, and their difficulties lay more in the fast and
accurate execution of counting procedures. The idea that
the improvement seen in calculation tasks was due to con-
ceptual rather than memory retrieval improvement fits
with other findings that fact fluency in dyscalculics seems
particularly resistant to remediation [73].
A third and final possibility is that training on particular
problems failed to generalize to other problems. One
important difference between the addition and subtrac-
tion tests was that the addition test included problems

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with a sum greater than 10 which were not trained in the
software, whereas the subtraction task did not. The addi-
tion problems included in the software would have fallen
into the "small" category only (sum < 10); however chil-
dren's pre-remediation performance on "small normal"
problems was already at 90% accuracy with a fairly fast
response time of around 3 sec, thus there was not a large
amount of room for improvement.
The preceding discussion brings up an important question
for future research: To what extent does an improvement
in number sense (or in its access via symbols) generalize
to higher level mathematical tasks? This is an issue which
is obviously of great importance for educators, because if
a remediation of a number sense core deficit is to make a
difference to dyscalculic children's performance in the
classroom, this type of transfer must occur. Research in
both the reading and mathematics domains suggests that
transfer should take place, at least over developmental
time. In the reading domain, training on the core proc-
esses of phonological discrimination and grapheme-pho-
neme correspondences generalizes to higher-level reading
tasks [74-76]. In the mathematical domain, recent studies
have shown that measures of number sense in the kinder-
garten years predict later performance in mathematics
[e.g. [53]]. Earlier work by Sharon Griffin and colleagues
[77,78] also suggests that training number sense in at-risk
children in the kindergarten years can produce long last-
ing effects on mathematical performance as a whole.
The possible mechanisms for transfer necessarily remain
speculative, because there is much we do not know about
the development of numerical cognition, and in particu-
lar the role of number sense in aiding acquisition of
higher level mathematical concepts and procedures. How-
ever, based on the hypothesis that number sense provides
the semantics of number (or our internal "mental number
line") [40,58], improving children's number sense and/or
its access from symbolic information ought to, over time,
produce general increases in comprehension of all aspects
of mathematics. For instance, more accurate and faster
access to number sense might be critical for conceptual
understanding of addition and subtraction [56,77,78].
The attentional resources freed up by improved access to
number semantics might also enhance children's ability
to develop more efficient strategies such as breaking down
problems into simpler steps, monitoring problem steps
and solutions for errors, and relying less on concrete sup-
ports [36]. Facts might become easier and more efficient
to memorize because of their higher semantic content
[62]. Increases in accuracy in calculation might also
reduce the likelihood of forming false associations in long
term memory [36].
Assuming that transfer can occur, it is also crucial to know
its timeframe. The current study suggests that transfer
appears to be somewhat limited at the age of 7–9 and over
a short period of two months. However, it is possible that
better transfer would be seen at a longer post-training
delay (and in fact this has been the case with phonemic
awareness reading interventions [75]). There might also
be a more propitious "developmental window" at a
younger age, during which intervention is maximally
effective.
Achieving the instructional goals of the software
How well were the instructional goals of the software
achieved? As presented in the accompanying article [1],
these were 1) enhancing number sense, 2) cementing
links between representations of number, and 3) concep-
tualizing and automatizing arithmetic.
We have seen that goals 1) and 2) appear to have been
achieved to some degree. However, success on goal 3) was
mixed. While the sub-goal of conceptualizing arithmetic
may have been well achieved (see above discussion), the
lack of improvement in addition suggests that that of
automatizing arithmetic was not. This may have been par-
tially due to the functioning of the software. Fact training
was only present at the higher levels of difficulty, and the
software took too long to reach these levels. The result was
that addition and subtraction facts were only tested in a
small percentage of trials (17% and 11% of game trials, or
72 and 43 problems per child on average, respectively). In
future versions of the software, we plan to alter the pro-
gram so that these higher levels can be reached much
faster if the child is progressing well enough.
Limitations of the study
The current results satisfy the goals of this initial study,
which were to test for the presence of improvement and to
identify sensitive pre-post measures. However several
important limitations should be noted. Firstly, the study
used a very small sample (n = 9). Secondly, in the absence
of a control group, the observed effects could be due to
specific classroom or home activity during the period of
the study, or to an interaction between this activity and
participation in the remediation program. Therefore our
results should be taken with caution, as a first positive
finding on the path towards establishing efficacy, and in
need to be backed up by larger, controlled studies.
It should also be noted that the criteria for identifying dys-
calculia are a subject of on-going debate and research, and
vary widely from study to study. Thus with any study in
this field it is difficult to say to what extent the sample is
characteristic of the disorder. Our sample is clearly most
parsimoniously described as having "mathematical learn-
ing difficulties". To what extent these children and their

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Page 14 of 16
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responses to intervention are similar to those selected
using a stricter criteria remains to be established, as does
whether different subtypes of dyscalculia might respond
differently.
Conclusion
In the first section of this paper, we described what is
known about dyscalculia, and discussed the core deficit
hypothesis, which proposes that dyscalculia is due to an
underlying deficit in number sense or in its access via sym-
bolic information. The pre and post testing results of our
open-trial study showed that after using software designed
to enhance number sense and its access via symbolic
information for a total of 10 hours over a five week
period, children made progress in several core areas of
numerical cognition: number comparison, subitizing,
and subtraction. These results are unlikely to be caused by
general motivation or attentional effects because they
were specific to particular tasks and to particular condi-
tions within tasks.
We emphasize that our results are only a first step in the
series of studies required to prove efficacy of the software.
The inclusion of a control group is a critical next step, and
the generality and duration of the effects found also need
to be tested. Furthermore, much work remains to be done
to establish to what extent basic training on number sense
produces transfer to other higher level tasks, over what
time period this occurs, the underlying brain mecha-
nisms, and whether there is an optimal developmental
time window during which intervention has the most
impact.
Competing interests
The author(s) declare that they have no competing inter-
ests.
Authors' contributions
Collection and analysis of data for the open-trial study
was carried out by AJW and SKR, with assistance from DC,
SD and LC. All authors contributed intellectually to
designing the open-trial study and interpreting its data,
and all authors read and approved the final manuscript,
which was prepared by AJW and SD.
Acknowledgements
Funding for this research was provided by a Fyssen Foundation postdoc-
toral award to Anna Wilson, a European Union Marie Curie (NUMBRA)
doctoral grant to Susannah Revkin, and a McDonnell Foundation centennial
grant to Stanislas Dehaene. In addition, we gratefully acknowledge a consul-
tancy contract to Anna Wilson from the Brain and Learning Project of the
OECD.
Thanks to the many other people who contributed in some way to this
project: Céline Amy, Dominique Chauvin and staff, Ghislaine Dehaene-
Lambertz, Chantal Germain, Mme. Malotkoff, Nicole Martin, Philippe
Mazet, Bruce McCandliss, Monique Plaza, Michael Posner, Pekka Rasanen,
and Catherine Soares.
We are also grateful to the Academy of Paris, in particular Inspector Cham-
peyrache, as well as participating pupils and families of the Alexandre
Dumas, Cité Voltaire, and Daumesnil elementary schools.
References
1. Wilson AJ, Dehaene S, Pinel P, Revkin SK, Cohen L, Cohen D: Prin-
ciples underlying the design of "The Number Race", an adap-
tive computer game for remediation of dyscalculia. Behavioral
and Brain Functions 2006, 2:19.
2. Kosc L: Developmental Dyscalculia. J Learn Disabil 1974,
7:164-177.
3. Shalev RS, Gross-Tsur V: Developmental Dyscalculia and Medi-
cal Assessment. J Learn Disabil 1993, 26:134-137.
4. Shalev RS, Gross-Tsur V: Developmental dyscalculia. Pediatr Neu-
rol 2001, 24:337-342.
5. Butterworth B: Developmental dyscalculia. In Handbook of Math-
ematical Cognition Edited by: Campbell J. New York: Psychology Press;
2005.
6. Wilson AJ, Dehaene S: Number sense and developmental dys-
calculia. In Human Behavior and the Developing Brain 2nd edition.
Edited by: Coch D, Dawson G, Fischer K. Guilford Press in press.
7. Dowker A: Early Identification and Intervention forStudents
With Mathematics Difficulties. J Learn Disabil 2005, 38:324-332.
8. Geary DC: Mathematics and learning disabilities. J Learn Disabil
2004, 37:4-15.
9. Shalev RS: Developmental dyscalculia. J Child Neurol 2004,
19:765-771.
10. Butterworth B: The development of arithmetical abilities. J
Child Psychol Psychiatry 2005, 46:3-18.
11. Geary DC, Bow-Thomas CC, Yao Y: Counting knowledge and
skill incognitive addition: A comparison of normal and math-
ematically disabled children. J Exp Child Psychol 1992, 54:372-391.
12. Geary DC, Hamson CO, Hoard MK: Numerical and arithmetical
cognition: A longitudinal study of process and concept defi-
cits in children with learning disability. J Exp Child Psychol 2000,
77:236-263.
13. Geary DC, Hoard MK, Hamson CO: Numerical and arithmetical
cognition: Patterns of functions and deficits in children at
risk for a mathematical disability. J Exp Child Psychol 1999,
74:213-239.
14. Geary DC: A componential analysis of an early learning deficit
in mathematics. J Exp Child Psychol 1990, 49:363-383.
15. Geary DC, Brown SC, Samaranayake VA: Cognitive addition: A
short longitudinal study of strategy choice and speed-of-
processing differences in normal and mathematically disa-
bled children. Dev Psychol 1991, 27:787-797.
16. Jordan NC, Montani TO: Cognitive arithmetic and problem
solving: A comparison and children with specific and general
mathematics difficulties. J Learn Disabil 1997, 30:624-634.
17. Badian NA: Persistent Arithmetic, Reading, or Arithmetic and
Reading Disability. Annals of Dyslexia 1999, 49:45-70.
18. Ginsburg HP: Mathematics Learning Disabilities: A View from
Developmental Psychology. J Learn Disabil 1997, 30:20-33.
19. Gross-Tsur V, Manor O, Shalev RS: Developmental dyscalculia:
prevalence and demographic features. Dev Med Child Neurol
1996, 38:25-33.
20. Kirby JR, Becker LD: Cognitive Components of Learning Prob-
lems in Arithmetic. Remedial and Special Education 1988, 9:7-16.
21. Lewis C, Hitch GJ, Walker P: The prevalence of specific arithme-
tic difficulties and specific reading difficulties in 9- to 10-year
old boys and girls. J Child Psychol Psychiatry 1994, 35:283-292.
22. Ostad SA: Developmental differences in addition strategies: A
comparison of mathematically disabled and mathematically
normal children. Br J Educ Psychol 1997, 67:345-357.
23. Ostad SA: Developmental progression of subtraction strate-
gies: A comparison of mathematically normal and mathe-
matically disabled children. European Journal of Special Needs
Education 1999, 14:21-36.
24. Geary DC: Mathematical disabilities: Cognitive, neuropsycho-
logical and genetic components. Psychological Bulletin 1993,
114:345-362.

Behavioral and Brain Functions 2006, 2:20 http://www.behavioralandbrainfunctions.com/content/2/1/20
Page 15 of 16
(page number not for citation purposes)
25. Bruandet M, Molko N, Cohen L, Dehaene S: A cognitive charac-
terization of dyscalculia in Turner syndrome. Neuropsychologia
2004, 42:288-298.
26. Rivera SM, Menon V, White CD, Glaser B, Reiss AL: Functional
brain activation during arithmetic processing in females with
Fragile X syndrome is related to FMR1 protein expression.
Hum Brain Mapp 2002, 16:206-218.
27. Eliez S, Blasey CM, Menon V, White CD, Schmitt JE, Reiss AL: Func-
tional brain imaging study of mathematical reasoning abili-
ties in velocardiofacial syndrome (del22q11.2). Genet Med
2001, 3:49-55.
28. Isaacs EB, Edmonds CJ, Lucas A, Gadian DG: Calculation difficul-
ties in children of very low birthweight: a neural correlate.
Brain 2001, 124:1701-1707.
29. Kopera-Frye K, Dehaene S, Streissguth AP: Impairments of
number processing induced by prenatal alcohol exposure.
Neuropsychologia 1996, 34:1187-1196.
30. Rourke BP: Arithmetic disabilities, specific and otherwise: A
neuropsychological perspective. J Learn Disabil 1993,
26:214-226.
31. Jordan NC, Hanich LB: Mathematical Thinking in Second-
Grade Children with Different Forms of LD. J Learn Disabil
2000, 33:567-578.
32. Hanich LB, Jordan NC, Kaplan D, Dick J: Performance across dif-
ferent areas of mathematical cognition in children with
learning difficulties. J Educ Psychol 2001, 93:615-626.
33. Jordan NC, Hanich LB, Kaplan D: A longitudinal study of mathe-
matical competencies in children with specific mathematics
difficulties versus children with comorbid mathematics and
reading difficulties. Child Dev 2003, 74:834-850.
34. Mazzocco MlMM, Myers GF: Complexities in Identifying and
Defining Mathematics Learning Disability in the Primary
School-Age Years. Annals of Dyslexia 2003, 53:218.
35. Landerl K, Bevan A, Butterworth B: Developmental dyscalculia
and basic numerical capacities: a study of 8–9-year-old stu-
dents. Cognition 2004, 93:99-125.
36. Rousselle L, Noel M-P: Basic numerical skills in children with
mathematics learning disabilities: A comparison of symbolic
vs non-symbolic number magnitude processing. Cognition in
press.
37. Jordan NC, Hanich LB, Kaplan D: Arithmetic fact mastery in
young children: A longitudinal investigation. J Exp Child Psychol
2003, 85:103-119.
38. Temple CM: Procedural dyscalculia and number fact dyscalcu-
lia: Double dissociation in developmental dyscalculia. Cogni-
tive Neuropsychology 1991, 8:155-176.
39. Dehaene S: Précis of the number sense. Mind and Language 2001,
16:16-36.
40. Dehaene S: The Number Sense: How the Mind Creates Mathematics
Oxford: Oxford University Press; 1997.
41. Berch DB: Making Sense of Number Sense: Implications for
Children With Mathematical Disabilities. J Learn Disabil 2005,
38:333-339.
42. Feigenson L, Dehaene S, Spelke E: Core systems of number.
Trends Cogn Sci 2004, 8:307-314.
43. Noël M-P, Rousselle L, Mussolin C: Magnitude representation in
children: Its development and dysfunction. In Handbook of
Mathematical Cognition Edited by: Campbell J. New York: Psychology
Press; 2005.
44. Pica P, Lemer C, Izard V, Dehaene S: Exact and approximate
arithmetic in an Amazonian indigene group. Science 2004,
306:499-503.
45. Cantlon JF, Brannon EM, Carter EJ, Pelphrey KA: Functional Imag-
ing of Numerical Processing in Adults and 4-y-Old Children.
PLoS Biol 2006, 4:e125.
46. Dehaene S, Piazza M, Pinel P, Cohen L: Three parietal circuits for
number processing. Cognitive Neuropsychology 2003, 20:487-506.
47. Piazza M, Izard V, Pinel P, Le Bihan D, Dehaene S: Tuning curves for
approximate numerosity in the human intraparietal sulcus.
Neuron 2004, 44:547-555.
48. Koontz KL, Berch DB: Identifying simple numerical stimuli:
Processing inefficiencies exhibited by arithmetic learning
disabled children. Mathematical Cognition 1996, 2:1-23.
49. Rubinsten O, Henik A: Automatic Activation of Internal Magni-
tudes: A Study of Developmental Dyscalculia. Neuropsychology
2005, 19:641.
50. Chard DJ, Clarke B, Baker S, Otterstedt J, Braun D, Katz R: Using
Measures of Number Sense to Screen for Difficulties in
Mathematics: Preliminary Findings. Assessment for Effective
Intervention 2005, 30:3-14.
51. Clarke B, Shinn MR: A Preliminary Investigation Into the Iden-
tification and Development of Early Mathematics Curricu-
lum-Based Measurement. School Psych Rev 2004, 33:234-248.
52. Gersten R, Jordan NC, Flojo JR: Early Identification and Inter-
ventions for Students With Mathematics Difficulties. J Learn
Disabil 2005, 38:293-304.
53. Mazzocco MMM, Thompson RE: Kindergarten Predictors of
Math Learning Disability. Learning Disabilities Research and Practice
2005, 20:142-155.
54. Molko N, Cachia A, Riviere D, Mangin JF, Bruandet M, Le Bihan D,
Cohen L, Dehaene S: Functional and structural alterations of
the intraparietal sulcus in a developmental dyscalculia of
genetic origin. Neuron 2003, 40:847-858.
55. Butterworth B: The Mathematical Brain London: Macmillan; 1999.
56. Gersten R, Chard D: Number Sense: Rethinking Arithmetic
Instruction for Students with Mathematical Disabilities. J
Spec Educ 1999, 33:18.
57. Mazzocco MMM: Challenges in Identifying Target Skills for
Math Disability Screening and Intervention. J Learn Disabil
2005, 38:318-323.
58. Dehaene S: Varieties of numerical abilities. Cognition 1992,
44:1-42.
59. Dehaene S, Cohen L: Towards an anatomical and functional
model of number processing. Mathematical Cognition 1995,
1:83-120.
60. Cohen L, Dehaene S: Calculating without reading: Unsuspected
residual abilities in pure alexia. Cognitive Neuropsychology 2000,
17:563-583.
61. Lemer C, Dehaene S, Spelke E, Cohen L: Approximate quantities
and exact number words: Dissociable systems. Neuropsycholo-
gia 2003, 41:1942-1958.
62. Robinson CS, Menchetti BM, Torgesen JK: Toward a Two-Factor
Theory of One Type of Mathematics Disabilities. Learning Dis-
abilities: Research and Practice 2002, 17:81.
63. Wechsler D: Echelle d'intelligence de Wechsler pour enfants,
troisième édition (WISC-III). Paris: Les Editions du Centrede
Psychologie Appliquée; 1991.
64. Sattler JM: Assessment of Children 3rd edition. San Diego, CA: Jerome
M. Sattler, Publisher, Inc; 1992.
65. Van Nieuwenhoven C, Grégoire J, Noël M-P: TEDI-MATH: Test
Diagnostique des Compétences de Base en Mathématiques.
Paris, France: Les Editions du Centre de Psychologie Appliquée;; 2001.
66. Schneider W, Eschman A, Zuccolotto A: E-Prime Users Guide.
Pittsburgh: Psychology Software Tools Inc.; 2002.
67. Mandler G, Shebo BJ: Subitizing: An analysis of its component
processes. J Exp Psychol Gen 1982, 111:1-21.
68. Zbrodoff NJ, Logan GD: What everyone finds: The problem-
size effect. In Handbook of Mathematical Cognition Edited by: Camp-
bell JD. New York, NY: Psychology Press; 2005:331-346.
69. Moyer RS, Landauer TK: Time Required for Judgements of
Numerical Inequality. Nature 1967, 215:1519-1520.
70. Ashcraft MH: Cognitive arithmetic: A review of data and the-
ory. Cognition 1992, 44:75-106.
71. Sekuler R, Mierkiewicz D: Children's judgments of numerical
inequality. Child Dev 1977, 48:630-633.
72. Green CS, Bavelier D: Action video game modifies visual selec-
tive attention. Nature 2003, 423:534-537.
73. Fuchs LS, Compton DL, Fuchs D, Paulsen K, Bryant JD, Hamlett CL:
The Prevention, Identification, and Cognitive Determinants
of Math Difficulty. J Educ Psychol 2005, 97:493.
74. Torgesen JK: Intensive Remedial Instruction for Children with
Severe Reading Disabilities: Immediate and Long-term Out-
comes From Two Instructional Approaches. J Learn Disabil
2001, 34:33.
75. Ehri LC, Nunes SR, Willows DM, Schuster BV, Yaghoub-Zadeh Z,
Shanahan T: Phonemic awareness instruction helps children
learn to read: Evidence from the National Reading Panel's
meta-analysis. Reading Research Quarterly 2001, 36:250.
76. Eden GF: The role of neuroscience in the remediation of stu-
dents with dyslexia. Nat Neurosci 2002, 5:1080-1084.
77. Griffin SA, Case R, Capodilupo A: Teaching for understanding:
The importance of the central conceptual structures in the

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Behavioral and Brain Functions 2006, 2:20 http://www.behavioralandbrainfunctions.com/content/2/1/20
Page 16 of 16
(page number not for citation purposes)
elementary mathematics curriculum. Teaching for transfer: Fos-
tering generalization in learning 1995:123-151.
78. Griffin SA, Case R, Siegler RS: Rightstart: Providing the central
conceptual prerequisites for first formal learning of arithme-
tic tostudents at risk for school failure. Classroom lessons: Inte-
grating cognitive theory and classroom practice 1994:25-49.