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BioMed Central

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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 ranges

Figure 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 TieSmallLarge

Addition: Accuracy (%)

Pre

Post

Reaction Time (msec)

Pre

Post

100 (0)

97 (3)

76 (5)

88 (5)

92 (2)

89 (2)

61 (10)

64 (9)

1995 (245)

1694 (229)

2974 (293)

2908 (618)

3172 (365)

3244 (368)

6892 (347)

6427 (454)

Subtraction: Reaction Time

(msec)

Pre

Post

2564 (172)

2428 (179)

6032 (512)

6724 (725)

8010 (834)

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 training

Figure 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%

RuleSmall 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 training

Figure 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 comparison reaction time (n=9)

600

800

1000

1200

1400

1600

0 0.2 0.40.6 0.8

Numerical distance ( = |log(A/B)| )

RT (msec)

Pre

Post

Non symbolic comparison accuracy (n=9)

75%

80%

85%

90%

95%

100%

0 0.20.40.6 0.8

Numerical distance ( = |log(A/B)| )

Accuracy

Pre

Post

Symbolic comparison accuracy (n=9)

75%

80%

85%

90%

95%

100%

0 0.2 0.4 0.60.8

Numerical distance ( = |log(A/B)| )

Accuracy

Pre

Post

Symbolic comparison reaction time (n=9)

1000

1200

1400

1600

1800

2000

00.20.40.60.8

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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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.

Page 15

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.

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.

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.

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.

Kopera-Frye K, Dehaene S, Streissguth AP: Impairments of

number processing induced by prenatal alcohol exposure.

Neuropsychologia 1996, 34:1187-1196.

Rourke BP: Arithmetic disabilities, specific and otherwise: A

neuropsychological perspective. J Learn Disabil 1993,

26:214-226.

Jordan NC, Hanich LB: Mathematical Thinking in Second-

Grade Children with Different Forms of LD. J Learn Disabil

2000, 33:567-578.

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.

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.

Mazzocco MlMM, Myers GF: Complexities in Identifying and

Defining Mathematics Learning Disability in the Primary

School-Age Years. Annals of Dyslexia 2003, 53:218.

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.

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.

Jordan NC, Hanich LB, Kaplan D: Arithmetic fact mastery in

young children: A longitudinal investigation. J Exp Child Psychol

2003, 85:103-119.

Temple CM: Procedural dyscalculia and number fact dyscalcu-

lia: Double dissociation in developmental dyscalculia. Cogni-

tive Neuropsychology 1991, 8:155-176.

Dehaene S: Précis of the number sense. Mind and Language 2001,

16:16-36.

Dehaene S: The Number Sense: How the Mind Creates Mathematics

Oxford: Oxford University Press; 1997.

Berch DB: Making Sense of Number Sense: Implications for

Children With Mathematical Disabilities. J Learn Disabil 2005,

38:333-339.

Feigenson L, Dehaene S, Spelke E: Core systems of number.

Trends Cogn Sci 2004, 8:307-314.

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.

Pica P, Lemer C, Izard V, Dehaene S: Exact and approximate

arithmetic in an Amazonian indigene group. Science 2004,

306:499-503.

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.

Dehaene S, Piazza M, Pinel P, Cohen L: Three parietal circuits for

number processing. Cognitive Neuropsychology 2003, 20:487-506.

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.

Koontz KL, Berch DB: Identifying simple numerical stimuli:

Processing inefficiencies exhibited by arithmetic learning

disabled children. Mathematical Cognition 1996, 2:1-23.

Rubinsten O, Henik A: Automatic Activation of Internal Magni-

tudes: A Study of Developmental Dyscalculia. Neuropsychology

2005, 19:641.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

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.

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.

Gersten R, Jordan NC, Flojo JR: Early Identification and Inter-

ventions for Students With Mathematics Difficulties. J Learn

Disabil 2005, 38:293-304.

Mazzocco MMM, Thompson RE: Kindergarten Predictors of

Math Learning Disability. Learning Disabilities Research and Practice

2005, 20:142-155.

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.

Butterworth B: The Mathematical Brain London: Macmillan; 1999.

Gersten R, Chard D: Number Sense: Rethinking Arithmetic

Instruction for Students with Mathematical Disabilities. J

Spec Educ 1999, 33:18.

Mazzocco MMM: Challenges in Identifying Target Skills for

Math Disability Screening and Intervention. J Learn Disabil

2005, 38:318-323.

Dehaene S: Varieties of numerical abilities. Cognition 1992,

44:1-42.

Dehaene S, Cohen L: Towards an anatomical and functional

model of number processing. Mathematical Cognition 1995,

1:83-120.

Cohen L, Dehaene S: Calculating without reading: Unsuspected

residual abilities in pure alexia. Cognitive Neuropsychology 2000,

17:563-583.

Lemer C, Dehaene S, Spelke E, Cohen L: Approximate quantities

and exact number words: Dissociable systems. Neuropsycholo-

gia 2003, 41:1942-1958.

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.

Wechsler D: Echelle d'intelligence de Wechsler pour enfants,

troisième édition (WISC-III). Paris: Les Editions du Centrede

Psychologie Appliquée; 1991.

Sattler JM: Assessment of Children 3rd edition. San Diego, CA: Jerome

M. Sattler, Publisher, Inc; 1992.

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.

Schneider W, Eschman A, Zuccolotto A: E-Prime Users Guide.

Pittsburgh: Psychology Software Tools Inc.; 2002.

Mandler G, Shebo BJ: Subitizing: An analysis of its component

processes. J Exp Psychol Gen 1982, 111:1-21.

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.

Moyer RS, Landauer TK: Time Required for Judgements of

Numerical Inequality. Nature 1967, 215:1519-1520.

Ashcraft MH: Cognitive arithmetic: A review of data and the-

ory. Cognition 1992, 44:75-106.

Sekuler R, Mierkiewicz D: Children's judgments of numerical

inequality. Child Dev 1977, 48:630-633.

Green CS, Bavelier D: Action video game modifies visual selec-

tive attention. Nature 2003, 423:534-537.

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.

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.

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.

Eden GF: The role of neuroscience in the remediation of stu-

dents with dyslexia. Nat Neurosci 2002, 5:1080-1084.

Griffin SA, Case R, Capodilupo A: Teaching for understanding:

The importance of the central conceptual structures in the

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

77.