Basic number processing and difficulties in single-digit arithmetic: Evidence from...

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Research report

Basic number processing and difficulties in single-digit

arithmetic: Evidence from Velo-Cardio-Facial Syndrome

Bert De Smedta,*, Bert Reynvoetb, Ann Swillenc, Lieven Verschaffeld,Bart Boetsa and Pol Ghesquierea

aCentre for Disability, Special Needs Education, and Child Care, University of Leuven, BelgiumbLaboratory for Experimental Psychology, University of Leuven, BelgiumcCentre for Human Genetics, University of Leuven, BelgiumdCentre for Instructional Psychology and Technology, University of Leuven, Belgium

a r t i c l e i n f o

Article history:

Received 5 March 2007

Reviewed 4 June 2007

Revised 15 June 2007

Accepted 18 June 2007

Action editor Roberto Cubelli

Published online 1 February 2008

Keywords:

Basic number processing

Velo-Cardio-Facial Syndrome

Mathematical disabilities

Cognitive phenotype

Developmental dyscalculia

a b s t r a c t

It has been suggested that mathematical disabilities (MD) emerge as a consequence of

impairments in basic number processing skills. The aim of the present study was to inves-

tigate basic number processing skills in children with Velo-Cardio-Facial Syndrome (VCFS),

a common genetic disorder with a high prevalence of MD, and to examine whether these

basic low-level skills account for their performance in single-digit arithmetic. Twenty-five

children with VCFS and 25 individually matched controls (age range: 6–12 years) partici-

pated. They all completed two basic number processing tasks (number reading, number

comparison) and three single-digit arithmetic tasks comprising addition, subtraction and

multiplication. In the latter tasks, strategy use was recorded next to accuracy and speed.

Our data revealed that children with VCFS were significantly slower than controls on

number comparison but not on number reading. Analysis of the single-digit arithmetic

data revealed that children with VCFS performed more poorly than controls on large

addition and subtraction problems. Both groups did not differ on multiplication and small

additions and subtractions. At the strategy level, children with VCFS were significantly

slower in executing backup strategies in addition and subtraction, but showed preserved

retrieval of arithmetic facts. Taken together, children with VCFS show a consistent pattern

of deficits at the level of number representations, arithmetic operations and strategy use,

which suggests an impaired quantity subsystem in terms of Dehaene et al.’s model (2003).

Most importantly, the correlational analyses showed that basic number processing skills

directly accounted for single-digit arithmetic performance and strategy use in the children

of the present study.

ª 2007 Elsevier Srl. All rights reserved.

* Corresponding author. Centre for Disability, Special Needs Education, and Child Care, University of Leuven, Vesaliusstraat 2, bus 3765,B-3000 Leuven, Belgium.

E-mail address: [email protected] (B. De Smedt).

ava i lab le at www.sc ienced i rec t . com

journa l homepage : www.e lsev ier . com/ loca te / cor tex

0010-9452/$ – see front matter ª 2007 Elsevier Srl. All rights reserved.doi:10.1016/j.cortex.2007.06.003

c o r t e x 4 5 ( 2 0 0 9 ) 1 7 7 – 1 8 8


1. Introduction

Numbers and arithmetic are so much part of our modern life

that it is essential for children to acquire basic mathematical

competencies. Unfortunately, 3–6% of primary school children

show large difficulties in acquiring these competencies (e.g.,

Shalev et al., 2000). These difficulties have often been referred

to as mathematical disabilities (MD) (Geary, 2004) or mathe-

matics difficulties (Jordan et al., 2003). In line with Mazzocco

(2005), we will use the term MD to refer to both conditions.

One of themajor concerns of researchers in the field is to char-

acterize the various cognitive processes that are implicated in

the development of MD (e.g., Geary, 2004). This unraveling of

the cognitive characteristics of MD might provide an impor-

tant basis for designing and realizing effective educational

interventions for children with MD.

It is widely agreed that most children with MD have diffi-

culties in retrieving facts from memory and/or problems in

understanding and executing calculation procedures (Geary,

2004; Jordan et al., 2003). Several general cognitive explana-

tions for these difficulties have been put forward, like prob-

lems with working memory (Bull et al., 1999; McLean and

Hitch, 1999; Passolunghi and Siegel, 2004; Swanson and

Sachse-Lee, 2001) or with retrieval of information from long-

term memory (Bull and Johnston, 1997; Geary, 2004; Swanson

and Beebe-Frankenberger, 2004). Although these general cog-

nitive skills are known to contribute to math achievement,

their role in the development of MD remains a debated issue

(Landerl et al., 2004; Temple and Sherwood, 2002).

Another theoretical perspective proposes that MD emerge

as a consequence of impairments in numerical representa-

tions (Ansari and Karmiloff-Smith, 2002; Butterworth, 1999,

2005; Landerl et al., 2004; Noel et al., 2005), which should be

reflected in difficulties in basic number processing tasks, like

number reading and comparing numerical magnitudes. Land-

erl et al. (2004) provided some of the first systematic empirical

evidence for this hypothesis by showing that children with

MDperformed significantlymore poorly than controls on a va-

riety of basic number processing tasks, such as number nam-

ing, number comparison and reciting counting sequences.

Interestingly, children with MD did not differ from controls

on working memory and naming speed tasks. Consistent

with this alternative perspective are also the results of several

other studies in children with MD that have reported difficul-

ties with the identification of numbers (Bull and Johnston,

1997; Hitch and McAuley, 1991; van der Sluis et al., 2004) and

with number comparison (Passolunghi and Siegel, 2004;

Rousselle and Noel, 2007). Durand et al. (2005) showed that

the speed of comparing numbers was a unique predictor of in-

dividual differences in arithmetic in primary school children.

In line with this, a recent review of longitudinal research by

Gersten et al. (2005) indicated that tests that assess children’s

abilities to represent and manipulate number seem to be the

most robust predictors of subsequent mathematical skills.

At the neural level, functional neuroimaging studies in

adults (Dehaene et al., 2003, 2004, for a review) and children

(Ansari et al., 2005; Temple and Posner, 1998) have identified

brain regions within the inferior parietal cortex clearly

dedicated to basic symbolic number processing. These areas

appear to be consistently active during arithmetical tasks

(e.g., Dehaene et al., 2003, 2004; Rivera et al., 2005; Stanescu-

Cosson et al., 2000). Most importantly, inferior parietal abnor-

malities have been reported in children with MD (Eliez et al.,

2001; Isaacs et al., 2001; Molko et al., 2003). Taken together,

the abovementioned studies support the view that impair-

ments in basic number processing might represent an impor-

tant cognitive correlate of MD.

This hypothesis is in accordance with current neurocogni-

tive theories that attribute a pivotal role of basic number

processing to the development of arithmetical competencies

(Dehaene et al., 2003). The Dehaene et al. (2003) model pro-

poses that mathematical performance is made possible by at

least two functional and anatomical distinct components of

basic number processing (see also Dehaene and Cohen,

1995). First, there is a verbal system, located in the left perisyl-

vian language areas, which hosts the verbal representation of

number and is related to (rote) verbal memory. Second, there

is a quantity system, located in the intraparietal sulcus, which

covers the quantity representation of number. The model fur-

ther predicts that the distinct components of basic number

processing are related to different single-digit arithmetic op-

erations: the verbal system is thought to underlie retrieval-

based operations, like multiplication, whereas the quantity

system is assumed to subserve operations that typically in-

volve semantic manipulations of quantity, such as subtrac-

tion. The case of addition is, however, more complex as

number size determines the system of basic number process-

ing that is predominantly involved: small additions (i.e.,

sum< 10) mainly involve the verbal system, whereas large

problems (i.e., sum> 10) dependmore heavily upon the quan-

tity system (Stanescu-Cosson et al., 2000). It should be noted

that the suggested distinction in terms of arithmetic opera-

tions or problem types is not an absolute one. For example,

some subtractions might be automatically retrieved from

long-term memory; conversely, multiplications might be

solved by quantity-based semantic elaboration strategies,

like decomposing problems into simpler facts (e.g., 3� 5¼

5þ 5þ 5). This indicates that single-digit arithmetic perfor-

mance might be characterized more into detail in terms of

the strategies that people apply during single-digit arithmetic

problem solving. Such problems are solved either by retrieving

the answer from long-term memory or by using a backup

strategy, like counting or decomposing a problem into smaller

facts (e.g., Siegler, 1996). These procedural backup strategies

draw heavily upon the quantity system of number processing,

whereas direct retrieval is related to the verbal system

(Dehaene et al., 2003). The Dehaene et al. (2003) model further

proposes that subtypes of MD should be traceable to differen-

tial impairment in theabovementioned subsystemsofnumber

processing (i.e., verbal vs quantity subtype), which should be

reflected in their single-digit arithmetic performance patterns

and strategy use. Although such patterns of impairment have

been observed in adults with acalculia (e.g., Lemer et al., 2003),

this proposal remains largely untested in children.

An important window onto MD is provided by research on

the cognitive phenotype of children with a clearly defined ge-

netic disorder who are at high risk for developing MD (Ansari

and Karmiloff-Smith, 2002;Mazzocco, 2001). MDhave been as-

sociated with several genetic disorders, like Turner Syndrome

c o r t e x 4 5 ( 2 0 0 9 ) 1 7 7 – 1 8 8178


and Fragile X Syndrome (e.g., Mazzocco and McCloskey, 2005).

The present study examines MD in another common genetic

disorder with a high prevalence of MD, namely Velo-Cardio-

Facial Syndrome (VCFS).

VCFS or chromosome 22q11.2 deletion syndrome is the

most frequent known microdeletion syndrome, with an inci-

dence of 1 in 4000 live births (Scambler, 2000). The syndrome

is caused by a submicroscopic deletion in the q11 region of

chromosome 22, which is detectable with 100% accuracy by

means of fluorescence in situ hybridization (FISH). The dele-

tion mainly occurs as a de novo event, but familial inheritance

is noted in about 15% of the patients (Swillen et al., 1998).

Although the clinical presentation of VCFS is quite variable,

the major characteristics of the syndrome are velopharyngeal

abnormalities, congenital cardiac anomalies, mild facial

dysmorphism and learning disabilities (for a review see

Shprintzen, 2000).

Intellectual abilities of children with VCFS are generally

lower than average (mean IQ¼ 75), with about 55% having

a borderline to normal intelligence (IQ> 70) and about 45%

having (mild) mental retardation. The IQ profile is often char-

acterized by a verbal IQ (VIQ)> performance IQ (PIQ) discrep-

ancy (Swillen et al., 1997). At the neuropsychological level, it

has been shown that children with VCFS have relatively pre-

served verbal short-term memory skills but show difficulties

in visuospatial short-term memory, working memory and in-

hibitory control (Bearden et al., 2001; Sobin et al., 2004; for a re-

view see De Smedt et al., 2006a).

Most important for the scope of the present study is the

consistent report of MD in children with VCFS, compared to

relatively normal reading and spelling achievement (Moss

et al., 1999; Wang et al., 2000; Woodin et al., 2001). There is

converging neuroimaging evidence for minor abnormalities

within the fronto-parietal network that subserves mathemat-

ical cognition (Dehaene et al., 2003) in children with VCFS.

Structural imaging studies have reported volumetric reduc-

tions of the inferior parietal cortex (Eliez et al., 2000; Simon

et al., 2005b) and abnormalities in the fronto-parietal neural

connectivity (Barnea-Goraly et al., 2005) in these children.

One fMRI study described an abnormal activation pattern

within the left supramarginal gyrus during a mental arithme-

tic task (Eliez et al., 2001).

Recently, efforts have been made to delineate more care-

fully themath impairment in children with VCFS at the cogni-

tive level. For example, Simon et al. (2005a) showed that

children with VCFS performed more poorly than matched

controls on counting dots and comparing magnitudes. In

a previous study, we have examined a broad range of mathe-

matical abilities with carefully designed cognitive tasks in 11

children with VCFS and 11 matched controls (De Smedt

et al., 2007). This study showed that children with VCFS did

not differ from controls in number reading and retrieving

arithmetic facts from long-term memory. However, children

with VCFS were significantly slower in number comparison

and executing procedural strategies, and significantly less ac-

curate in multidigit calculation and word problem solving. In

a follow-up study, this pattern of strengths and weaknesses

was partially replicated in 14 younger children with VCFS

(De Smedt et al., 2006b). In terms of the Dehaene et al. (2003)

model, these data provide some of the first evidence for an

impaired quantity system in children with VCFS. Interest-

ingly, this fits with the reported inferior parietal lobe anoma-

lies in these children (Barnea-Goraly et al., 2005; Eliez et al.,

2000, 2001; Simon et al., 2005b).

Although deficits in the representation of quantities have

been reported in children with VCFS (De Smedt et al., 2007;

Simon et al., 2005a), there are no studies available that have di-

rectly associated this impairment in basic number processing

with measures of higher order arithmetical cognition, like

single-digit arithmetic, as proposed in Dehaene et al.’s (2003)

model. Such an association study is needed to fully elucidate

the math impairment seen in children with VCFS. Therefore,

the present study aimed to replicate previous work on basic

number processing in children with VCFS and tried to extend

these findings by examining the direct association between

basic number processing skills and single-digit arithmetic

performance. To this end, 25 children with VCFS were com-

pared with a carefully selected control group.

First, we examined basic number processing in children

with VCFS and matched controls by means of a number com-

parison task, a classic task to evaluate the quantity represen-

tation of number, and a number reading task, which provides

ameasurement of the verbal representation of number. Based

upon the previous work, we expected that children with VCFS

would performmore poorly than controls on number compar-

ison, but not on number reading.

Second, we administered single-digit addition, subtraction

andmultiplication. Taking into account Dehaene et al.’s (2003)

model and previous studies in VCFS, we hypothesized that

children with VCFS would be impaired on tasks that draw

heavily on the quantity system, such as subtraction and large

additions. By contrast, we expected that children with VCFS

would not differ from controls on tasks that preferentially

rely on the verbal system, like multiplication and small addi-

tion problems. As performance patterns may have different

underlying strategies, strategy data were additionally col-

lected on a trial-by-trial basis. We hypothesized that children

with VCFS would differ from controls in the use of backup

strategies but not in the use of retrieval strategies.

Third, we examined the correlations between basic num-

ber processing skills and single-digit arithmetic performance.

Starting from Dehaene et al.’s (2003) model we expected that

performance on number comparison would be associated

with performance on subtraction and large addition problems,

and with application of procedural backup strategies. We

further hypothesized performance on number reading to be

associated with performance on multiplication and small ad-

dition problems, andwith application of the retrieval strategy.

Because learning disorders were the primary focus of the

present study, we only selected children with VCFS without

mental retardation (i.e., IQ> 70). Hence, our findings only

relate to a subgroup of children with VCFS.

2. Method

2.1. Participants

Fifty primary school children participated in this study. For all

of them, written informed parental consent was obtained.

c o r t e x 4 5 ( 2 0 0 9 ) 1 7 7 – 1 8 8 179


Children with VCFS were recruited from a group of patients

followed at the VCFS clinic of the University Hospital Gas-

thuisberg of Leuven (Belgium). Children with mental retarda-

tion (i.e., IQ< 70), children with severe physical disabilities or

brain damage, and bilingual children were excluded. This

yielded a sample of 25 children with VCFS (14 females, 11

males), with a mean chronological age of 9.8 years (SD¼ 1.9).

In all of them, diagnosis of VCFS was confirmed by FISH

with probe LSI Tuple1 (Vysis, Downers Grove, IL); all children

had a de novo deletion.

An extensive preparatory study was conducted to find

a control group of 25 children who individually matched the

25 children with VCFS. For each child with VCFS, a control

child of the same sex was selected from the same class within

the same school, to minimize possible effects of instruction.

Parents of eligible control children completed a short ques-

tionnaire tomake sure that these children had nomedical his-

tory, brain damage, auditory or visual problems, or familial

occurrence of developmental disorders.

We further tried to match both groups of children as

closely as possible on age, IQ and parental educational level.

To obtain a measure of IQ, all eligible control children and

children with VCFS completed the group administered Verbal

Analogies subtest of the RAKIT – a standardized Dutch intelli-

gence test for children (Bleichrodt et al., 1987). This subtest

contains 30 items of the form ‘A is to B as C is to.‘ and chil-

dren are asked to select the appropriate answer from four re-

sponse options. An estimated verbal IQ-score could be derived

based upon the standardization sample. This IQ-score has

a high internal consistency (.95) and is highly correlated

(r¼ .87) with IQ-scores derived from the conventional Wechs-

ler Intelligence Scale for Children (Bleichrodt et al., 1987). To

obtain a marker of socioeconomic status, all parents of chil-

dren with VCFS and eligible control children were asked to re-

port their educational level and all were accordingly classified

into the standardized categories of the International Standard

of Classification of Education system, developed by the

UNESCO (OECD, 1999).

Based upon the IQ data and the parent reports, we selected

those control children who represented the best individual

match to the children with VCFS on sex, IQ, age and parental

educational level. Twenty-five children were selected (13

females, 12 males), with a mean chronological age of 9.3 years

(SD¼ 1.7).

Although we attempted to match both groups in terms of

age and IQ, they both differed in age (t(24)¼ 3.41, p< .01) and

IQ (VCFS mean¼ 93.5 (13.2), control mean¼ 107.9 (14.0);

t(24)¼�4.33, p< .01). Both these variables were therefore

statistically taken into account in all subsequent analyses as

covariates. Because parental educational level was measured

at the ordinal level, Wilcoxon Signed Rank tests were used

for these comparisons. These tests revealed no group differ-

ences in parental educational level ( p> .30).

2.2. Procedures and materials

Children were individually tested on their own school in

a quiet environment. The order of task administration was

fixed for all participants in order not to confound individual

differences with task order effects.

Tasks were developed using the AFFECT 3.0 software

(Hermans et al., 2002) and were all presented on the same

Toshiba Notebook. All stimuli were presented in white on

a black background in Arial font (size 72). The experimenter

initiated each trial by pressing a control key. Each trial started

with a 200 msec fixation cross in the centre of the computer

screen accompanied by a beep of 440 Hz. After 1000 msec,

the stimulus appeared in the centre of the screen and

remained visible until the child responded. Stimuli were pre-

sented in a pseudo-random order, with the constraint that

the same integer did not appear in two consecutive trials.

Children were instructed to perform both accurately and fast.

2.2.1. Number comparison

A classic number comparison task (Sekuler and Mierkiewicz,

1977) was administered tomeasure children’s representations

of quantity. In this task, children had to indicate the numeri-

cally larger of two presented numbers, which occurred simul-

taneously – one displayed on the left, one displayed on the

right – on the computer screen. Stimuli comprised all combi-

nations of the numbers 1–9, yielding 72 trials. Children were

asked to respond by pressing a key on the side of the larger

number. Key pressesweremade on an external computer key-

board thatwas put in front of the notebook andwas connected

to it. The left response key, labeled with a blue sticker, was ‘d’;

the right response key, labeled with a yellow sticker, was ‘k’.

Response times and answers were recorded by the notebook

with millisecond precision. There were eight practice trials

to familiarize childrenwith the key assignments (four practice

trials for left response, four practice trials for right response).

2.2.2. Number reading

Sixteen numbers were successively presented on the com-

puter screen and the child was asked to name them as fast

as possible. Stimuli consisted of four units, two teens, four

decades and six multidigits (decade–unit). Responses were

verbal. When the child responded, the experimenter,1 which

was seated next to the child, immediately pressed the space-

bar of an external keyboard connected to the notebook in or-

der to register the response time. Response times were

recorded with millisecond precision. After the registration of

the response time, the child’s answer was entered on the

keyboard by the experimenter. There were two practice trials

to make children familiar with task administration.

2.2.3. Single-digit arithmetic

Single-digit arithmetic assessment consisted of an addition,

a subtraction and a multiplication task. For each task, stimuli

were drawn from all possible pairwise combinations of inte-

gers 2–9, with the exclusion of tie problems. From the remain-

ing 56 combinations, only one of each commutative pair was

randomly selected, thereby counterbalancing the position of

the largest number. That is, in addition and multiplication,

the larger value integer appeared 14 times as the first and 14

times as the second operand. In subtraction, the subtrahend

was larger than the difference in 14 problems; the difference

1 All tasks were administered by the same experimenter (BertDe Smedt) in order to avoid bias in the collection of responsetime data.

c o r t e x 4 5 ( 2 0 0 9 ) 1 7 7 – 1 8 8180


was larger than the subtrahend in the other 14 problems. This

resulted in a final set of 28 problems for each operation. All

problems were presented horizontally (like aþ b¼/). Addi-

tion and subtraction problems were divided into two problem

types depending onwhether a carry operation (large problems;

n¼ 16)was involved or not (small problems; n¼ 12);multiplica-

tion problems were divided into easy problems (i.e., the prod-

uct was lower than 20 or one of the operands was 5; n¼ 15) or

difficult problems (i.e., the remaining problems; n¼ 13)

(Lemaire and Siegler, 1995; Mabbott and Bisanz, 2003). For

each task, children were told that they could use whatever

strategy they wanted. Each task was preceded by two practice

trials, to make children familiar with task administration.

Responses were verbal. The registration of response time

and accuracy was carried out in the same way as in the num-

ber reading task. Strategy use was recorded per trial by asking

the children immediately after each response to describe how

they got the answer. It has been widely demonstrated that

such verbal reports are a reliable and valid way of assessing

children’s strategy use in single-digit arithmetic (Robinson,

2001; Siegler, 1996), but it should be noted that the use of

such reports has been criticized as well (for a discussion see

Kirk and Ashcraft, 2001). Strategies were classified into re-

trieval (¼the child responded immediately and there was no

evidence of overt calculations) or backup (¼the child calculated

the answer by means of a known fact or by using a counting

strategy). The other category was added to include ambiguous

responses, guesses, and ‘‘don’t know’’ answers. All verbal

self-reports of the children were literally transcribed and

coded by a second rater. The kappa statistic was calculated

to measure interrater reliability, showing high agreement

(kappa> .96).

The multiplication task was not administered in five chil-

dren with VCFS and their five matched controls, as all these

children were first graders and, consequently, they were not

yet instructed in multiplication. Whenever we refer to the

multiplication task, this only applies to 20 children with

VCFS and their 20 matched controls.

2.3. Statistical analysis

A series of mixedmodel analyses examined group differences

on the various measures. All data were analyzed in a pairwise

manner, to account for their dependent nature, by means of

mixed models (Littell et al., 2006). Dependency of the data

was modeled by entering matched pair as a random factor

in all analyses. Because both groups were not matched in

terms of IQ and age, both variables were taken into account

as fixed covariates in all analyses. Reaction time data were

log transformed due to skewed distributions. They were fur-

ther analyzed by linear mixed models ( proc mixed ) of the

SAS computer program (Littell et al., 2006). Because the accu-

racy data followed a binomial distribution, theywere analyzed

by generalized linearmixedmodels ( proc glimmix) by the same

computer program. Likewise, these models were used to ana-

lyze the frequencies with which strategies were executed. As-

sociations between tasks that measured basic number

processing skills and single-digit arithmetic performance

were analyzed by means of Spearman partial correlation

coefficients.

3. Results

3.1. Basic number processing

3.1.1. Number comparison

The results of the number comparison task are presented in

Fig. 1. For each numerical difference or distance between the

compared numbers, mean response times and error rates

were calculated. Due to the very low proportion of errors

(<3%) made on this task, there was no attempt at statistical

analysis of these errors. Response times were submitted to

a mixed model with group, distance, and size of the smaller

digit as fixed effects. Both groups showed the typical distance

effect (Moyer and Landauer, 1967; Sekuler and Mierkiewicz,

1977) as response times decreased when the numerical dis-

tance between two numbers increased (F(7, 3480)¼ 17.20,

p< .01). The size effect (Moyer and Landauer, 1967) was present

in both groups as for an equal distance response times in-

creased with the size of the numbers (F(21, 3480)¼ 3.19,

p< .01). There was a main effect of group (F(1, 31.8)¼ 5.84,

p¼ .02): childrenwith VCFS (1057 msec) were slower than con-

trols (874 msec). There were no significant interactions with

group ( ps> .23).

We additionally explored whether this deficit in number

comparison was also present at the individual level by looking

how many children performed abnormally low on this task.

To determine abnormal performance, we used a two-step pro-

cedure as described by Boets et al. (2006) and Ramus et al.

(2003). The threshold for abnormal performance was set at

1.65 SD below the control group mean, which corresponds to

the 5th percentile in a normal distribution. However, a control

child might occasionally show abnormal performance, which

highly influences the control mean and standard deviation,

and makes the criterion for abnormal performance much

more stringent than intended. Therefore, a two-step criterion

was applied: (1) the mean and standard deviation of the

control group were calculated and control children that

show abnormal performance were identified, (2) the control

mean and standard deviation were recomputed with abnor-

mally performing control children excluded, and these values

were used to identify children with VCFS who scored below

Fig. 1 – Number comparison (mean ± SE).

c o r t e x 4 5 ( 2 0 0 9 ) 1 7 7 – 1 8 8 181


1.65 SD. This analysis revealed that 10 out of 25 children with

VCFS (40%) showed abnormal performance, compared to 3 out

of 25 control children (12%). This difference is significant

(Fisher’s exact test: p¼ .05).

3.1.2. Number reading

Accuracy on number reading was very high (VCFS: 100%; con-

trols: 99.7%) and was not further statistically analyzed. As

expected, children with VCFS (M¼ 923 msec; SD¼ 142) did

not differ from controls (M¼ 913 msec; SD¼ 129) in the speed

of reading numbers (F(1,29.8)¼ .98, p¼ .33).

3.2. Single-digit arithmetic

3.2.1. Accuracy

Descriptive statistics for accuracy for each operation per

group per problem type are shown in Table 1. There was a sig-

nificant ( p< .01) main effect of problem type on accuracy for

each operation: small problems were solved significantly

more accurately than large problems in addition and subtrac-

tion; in multiplication, easy problems were solved more accu-

rately than difficult problems. Children with VCFS did not

differ from controls on accuracy as neither the effects of group

nor the group x problem type interactions were significant

( ps> .29).

3.2.2. Speed

Table 1 shows the descriptive statistics for response times for

each operation per group per problem type. Therewas a signif-

icant ( p< .01) main effect of problem type for each operation.

In addition and subtraction, small problems were solved sig-

nificantly faster than large problems; in multiplication, easy

problems were solved significantly faster than difficult prob-

lems. A significant group x problem type interaction was

found for both addition (F(1, 1343)¼ 9.88, p< .01) and subtrac-

tion (F(1, 1321)¼ 5.23, p¼ .02). Analysis of the simple effects

showed that in addition, childrenwith VCFS performed signif-

icantly slower on large (t(28.2)¼ 3.29, p< .01) but not on small

(t(31.1)¼ 1.43, p¼ .16) problems. A similar trend was found in

subtraction: there were no group differences on small

problems (t(33.5)¼ .61, p¼ .54), but the effect of large problems

tended to approach statistical significance (t(31.7)¼ 1.65,

p¼ .10). Interestingly, children with VCFS did not differ from

controls on speed of multiplication, as neither the effect of

group (F(1, 20.1)¼ .00, p¼ .99) nor the group x problem type

interaction (F(1, 1055)¼ .83, p¼ .36) was significant.

3.2.3. Strategy data

Strategy usewas examined to provide amore detailed account

of the performance patterns on the single-digit arithmetic

tasks. We therefore looked at both strategy distribution and

strategy efficiency (Table 2).

Strategy distribution. Strategy distribution was determined

per operation by calculating for each child the relative fre-

quency with which a particular strategy was used. Table 2

shows the strategy distribution for both groups on the three

operations. The frequency of strategies belonging to the other

category was low (<5%) and these trials were excluded from

all further analyses. There were no differences in frequencies

of retrieval and backup strategies in addition ( p¼ .45). In sub-

traction and multiplication, retrieval strategies were used

more often than backup strategies ( ps< .01) and this differ-

ence was most prominent in multiplication. No group differ-

ences in strategy distribution were found ( ps> .24).

Strategy efficiency. Accuracy and response time with which

a strategy was executed were used as indices of strategy effi-

ciency. These data are shown in Table 2. For each operation,

we analyzed accuracy and response times separately by

means of mixed models with group and strategy as fixed

effects.

Retrieval strategies were executed more accurately than

backup strategies, as evidenced by a significant main effect

of strategy for all three operations ( p< .01). There were no

group differences in strategy accuracy for addition and multi-

plication ( ps> .56). On subtraction, a significant group x strat-

egy interaction (F(1, 1348)¼ 11.47, p< .01) emerged: children

with VCFS executed backup strategies significantly less accu-

rately than controls (t(71.21)¼�2.05, p¼ .04), but did not differ

in their accuracy of the retrieval strategy (t(56.68)¼ .55,

p¼ .58).

Table 1 – Mean (SD) accuracy and speed in single-digit arithmetic

VCFS Controls

Accuracy Speed Accuracy Speed

Additiona

Small 97.00 (4.08) 2.79 (1.42) 98.33 (3.40) 2.33 (1.04)

Large 88.75 (12.24) 5.48 (2.52) 92.50 (10.36) 4.09 (1.91)

Subtractiona

Small 98.00 (4.36) 3.39 (1.73) 97.33 (4.64) 2.69 (1.43)

Large 75.75 (25.66) 6.79 (2.63) 82.75 (27.20) 5.46 (3.17)

Multiplicationb

Easy 95.00 (10.79) 2.77 (1.36) 97.00 (6.30) 2.44 (.96)

Difficult 80.04 (23.24) 6.22 (3.83) 85.20 (17.62) 5.51 (2.92)

Note: Accuracy is expressed in percentage correct. Speed is expressed in seconds.

a Group n¼ 25.

b Group n¼ 20.

c o r t e x 4 5 ( 2 0 0 9 ) 1 7 7 – 1 8 8182


With regard to strategy speed, retrieval strategies were

executed faster than backup strategies, as evidenced by a sig-

nificant main effect of strategy for all three operations

( p< .01). There was a highly significant group x strategy in-

teraction in both addition (F(1, 1362)¼ 15.27, p< .01) and sub-

traction (F(1, 1325)¼ 10.93, p< .01): children with VCFS were

significantly slower in executing backup strategies (addition:

t(33.3)¼ 3.62, p< .01; subtraction: t(54)¼ 2.27, p¼ .03) but did

not differ from controls in speed of executing retrieval strat-

egies (addition: t(31.3)¼ 1.34, p¼ .19; subtraction: t(49.4)¼ .72,

p¼ .48). There were no group differences in speed of execut-

ing strategies in multiplication as neither the effect of group

(F(1, 25.2)¼ .03, p¼ .87) nor the group x problem type interac-

tion (F(1, 1045)¼ 2.59, p¼ .11) was significant.

3.3. Correlational analyses

To examine the relationships between basic number process-

ing and single-digit arithmetic, Spearman correlations, with

the influence of age and IQ partialed out, were calculated.

For each operation, we first examined the correlations bet-

ween number reading, number comparison and single-digit

arithmetic at the problem type level (small vs large; easy vs

difficult). Next, we repeated this analysis at the strategy level

(retrieval vs backup). Correlations were calculated for the total

group and for the children with VCFS and control children

separately.

Associations between basic number processing and single-

digit arithmetic accuracy were weak (rs<�.30, ps> .05),

probably due to range restrictions. We further examined the

relationshipbetweenbasicnumberprocessingandsingle-digit

arithmetic speed. It should be noted that such associations

simply might occur as a result of a common speed factor in

each of these tasks. To rule out this possibility, correlations

with number reading were additionally partialed out for the

influence of number comparison and correlations with

number comparisonwere additionally partialed out for the in-

fluence of number reading. Thus, the reported correlations

represent theunique contributionof theparticularnumberpro-

cessing skill to single-digit arithmetic speed. The results of

these analyses are reported in Tables 3 and 4.

For the total group, number reading was significantly re-

lated to single-digit arithmetic for nearly all operations and

problem types. Number comparison was only significantly as-

sociated with subtraction. Analyses at the strategy level

(Table 4) revealed that the speed of executing the retrieval strat-

egy was significantly associated with number reading in all

three operations. Number comparison was only significantly

Table 2 – Strategy data in single-digit arithmetic per group per operation

Strategy distributiona Strategy efficiency

VCFS Controls Accuracyb Speedc

VCFS Controls VCFS Controls

Additiond

Retrieval 50.86 51.14 95.22 96.93 2.82 (1.32) 2.24 (1.08)

Backup 48.29 48.43 90.53 93.51 6.02 (2.81) 4.46 (2.03)

Other .86 .43 – – – –

Subtractiond

Retrieval 56.14 53.86 93.13 92.30 3.40 (1.35) 2.84 (1.44)

Backup 40.29 43.29 79.79 90.90 8.27 (3.70) 5.86 (3.30)

Other 3.57 2.86 – – – –

Multiplicatione

Retrieval 70.36 74.46 92.64 96.16 2.79 (1.17) 2.77 (1.16)

Backup 25.18 22.68 81.56 84.25 9.55 (5.83) 6.58 (4.44)

Other 4.46 2.86 – – – –

a Mean relative frequency with which a strategy is executed.

b Mean accuracy, expressed in percentage correct, with which a strategy is executed.

c Mean response time (SD), expressed in seconds, with which a strategy is executed.

d Group n¼ 25.

e Group n¼ 20.

Table 3 – Spearman partial correlations between basicnumber processing and single-digit arithmetic speed peroperation per problem type

Speed of numberreading

Speed of numbercomparison

Total VCFS Controls Total VCFS Controls

Additiona

Small .50** .38 .55** .19 .22 .04

Large .28 .10 .51* .32* .45** �.08

Subtractiona

Small .32* .29 .38 .39* .53* .06

Large .39** .40 .48* .54** .48* .38

Multiplicationb

Easy .54** .18 .71** .16 .21 �.04

Difficult .42* .17 .61** .23 .54* �.09

Note: *p< .05, **p< .01.

a Total: n¼ 50, VCFS: n¼ 25, controls: n¼ 25.

b Total: n¼ 40, VCFS: n¼ 20, controls: n¼ 20.

c o r t e x 4 5 ( 2 0 0 9 ) 1 7 7 – 1 8 8 183


associated with the speed of executing a backup strategy in

addition and subtraction.

Analyses at the subgroup level showed a slightly different

pattern. At the problem type level (Table 3), associations be-

tween number reading and single-digit arithmetic were most

prominent in the control group, whereas in children with

VCFS evidence for this association was limited. Conversely,

associations between number comparison and single-digit

arithmetic were clearly observed in children with VCFS on

the large problems and in subtraction, whereas in the control

group there was only amarginally significant associationwith

large subtraction problems (rs¼ .38, p¼ .07). At the strategy

level (Table 4), associations between number reading and

speed of executing the retrieval strategy were observed in

both groups. In children with VCFS, number comparison was

related to the speedof executing a backup strategy. In controls,

such relationship occurred in subtraction, but not in addition

and multiplication.

4. Discussion

The aim of the present study was to investigate basic number

processing in children with VCFS and to examine whether

these basic low-level skills account for their performance in

single-digit arithmetic. Children with VCFS were significantly

slower than controls on number comparison but not on num-

ber reading,whichfitswithprevious researchon basic number

processing in childrenwith VCFS (De Smedt et al., 2007; Simon

et al., 2005a). Starting from Dehaene et al.’s (2003) model of

number processing, these data also corroborate our hypothe-

sis of impaired quantity but preserved verbal representations

of number in children with VCFS. This is also confirmed by

the data at the individual level. Consistent with our predic-

tions, analysis of the single-digit arithmetic data revealed

that children with VCFS performedmore poorly than controls

on large addition and subtraction problems. Both groups

did not differ on multiplication and small additions and

subtractions. Similarly, children with VCFS were significantly

slower in executing backup strategies in addition and subtrac-

tion, but showed preserved retrieval of arithmetic facts. Taken

together, children with VCFS show a consistent pattern of

deficits at the level of number representations, arithmetic

operations and strategy use, which suggests an impaired

quantity subsystem in terms of Dehaene et al.’s (2003) model.

As expected on the basis of Dehaene et al.’s (2003) model,

the correlational analyses at the total group level revealed

specific associations between basic number processing and

single-digit arithmetic. Number comparison was only associ-

ated with subtraction and large addition problems, whereas

number reading was primarily related to multiplication and

small additions. This pattern of associations was additionally

confirmed when the data were re-analyzed in terms of strat-

egy use: retrieval strategies were mainly associated with ver-

bal number processing, whereas backup strategies, which

require semantic elaboration, were only associated with

quantity processing. The latter relationshipwas only observed

in addition and subtraction, but not in multiplication, proba-

bly as a result of the infrequent use of backup strategies in

multiplication.

It should, however, be noted that the pattern of correla-

tions at the subgroup level revealed a slightly distinct pattern

between children with VCFS and controls. Number compari-

son appeared to be a more important correlate of single-digit

arithmetic in childrenwith VCFS than in controls. Conversely,

number reading was a better predictor in controls than in

children with VCFS. This might indicate that single-digit

arithmetic wasmore automatized in controls than in children

with VCFS, which is in line with the delayed math develop-

ment in these children (Moss et al., 1999; Wang et al., 2000).

Stated differently, these findings suggest that control children

already usedmore their verbal system, whereas children with

VCFS were still relying more on their (impaired) quantity

system. However, this pattern was not reflected in group

differences in strategy distribution.

The differences and associations found in the present

study were only prominent in response times and not in accu-

racy data. This is probably due to the simplicity of the math

skills assessed in our study: accuracy performance in both

groups was near ceiling and yielded only limited variation,

which attenuated possible associations.

It is important to note that the reported associations be-

tween response times cannot merely be attributed to a com-

mon speed factor as all correlations with number reading

were partialed out for number comparison and all correlations

with number comparison were partialed out for number

reading. Therefore, the correlations reported in Tables 3 and

4, represent unique associations between that particular num-

ber processing skill and single-digit arithmetic. This indicates

that low-level basic number processing tasks, even when

controlled for other factors, are able to explain a significant

amount of variance in higher-level cognitive tasks like

single-digit arithmetic.

In accordance with previous studies of basic number pro-

cessing in children with MD (Gersten et al., 2005; Landerl

et al., 2004; Passolunghi and Siegel, 2004; Rousselle and Noel,

2007) and in children with VCFS (De Smedt et al., 2007; Simon

et al., 2005a), our data provide evidence for an impaired

Table 4 – Spearman partial correlations between basicnumber processing and single-digit arithmetic speed peroperation per strategy

Speed of numberreading

Speed of numbercomparison

Total VCFS Controls Total VCFS Controls

Addition

Retrievala .41** .30 .48* .16 .19 �.06

Backupa .17 �.16 .46* .32* .55** �.09

Subtraction

Retrievala .40** .60** .39 .21 .18 .04

Backupa .32* .27 .55** .54** .35 .45*

Multiplication

Retrievalb .62** .41 .61** .22 .35 .33

Backupc .22 .02 .53* .19 .57* �.35

Note: *p< .05, **p< .01.

a Total: n¼ 50, VCFS: n¼ 25, controls: n¼ 25.

b Total: n¼ 40, VCFS: n¼ 20, controls: n¼ 20.

c Total: n¼ 34, VCFS: n¼ 16, controls: n¼ 18.

c o r t e x 4 5 ( 2 0 0 9 ) 1 7 7 – 1 8 8184


quantity representation of number in childrenwith VCFS. The

nature of this impairment needs, however, to be further clar-

ified. As noted by Rousselle and Noel (2007), there are at least

three possible explanations for poor performance on a sym-

bolic Arabic number comparison task. First, this might result

from a deficit in the representation of quantity per se, as is

proposed by the defective number module hypothesis by Butter-

worth (1999, 2005) and Landerl et al. (2004). A second possibil-

ity is that children with VCFS have no impairments in

quantity processing per se but only have difficulties in access-

ing numericalmeaning (or quantity) from symbols. This access

deficit hypothesis (Rousselle and Noel, 2007) implies that chil-

dren with VCFS would only be impaired in symbolic Arabic

number comparison tasks but not in quantity comparison

tasks with no symbolic processing requirement. Although

the present study did not assess non-symbolic number pro-

cessing, a careful examination of the interactions between

group and size and/or distance on the administered number

comparison task might further shed light on this issue. If

such interactions were present, this would favour the defec-

tive number module hypothesis, as this indicates that repre-

sentations of quantity are more imprecise in the impaired

group. By contrast, the access deficit hypothesis requires an

overall slower performance of the impaired group without in-

teractions between group and distance and/or size. In the

present study, these interactions were not significant, favor-

ing the access deficit hypothesis. In an earlier study, we

showed that older children with VCFS were particularly im-

paired in small distances, but not in large ones (De Smedt

et al., 2007). A careful inspection of Fig. 1 suggests indeed

such a trend. However, the children of the present study

were both younger and of a much wider age range than in

our previous study (De Smedt et al., 2007). It might be that

the group by distance interaction might only be detectable

in older children with VCFS and might only become apparent

throughout development. Future (longitudinal) research is re-

quired to investigate this issue. A third and final explanation,

which is a more stringent version of the access deficit hypoth-

esis, might be that poor performance on a symbolic number

comparison task is due to difficulties in automatically activat-

ing corresponding quantities when presented with single-

digit numbers, even when irrelevant for the task (Rubinsten

and Henik, 2005, 2006). Such automatic activation appears to

be already present from the early grades of primary school

in typically developing children (Noel et al., 2005, for a review).

Rubinsten and Henik (2005, 2006) examined this hypothesis by

means of a Stroop paradigm, providing evidence for smaller

congruity effects in adults with dyscalculia, which suggests

difficulties in automatically activating quantity representa-

tions. A recent study by Rousselle and Noel (2007) contrasted

the three abovementioned hypotheses on quantity processing

in children with MD, and provided only evidence for the ac-

cess deficit hypothesis. Future research in children with

VCFS should therefore try to disentangle which of these possi-

bilities accounts for their impairment in representing

quantities.

Our cognitive data provide evidence for an impaired quan-

tity system in children with VCFS and this should be reflected

at the brain level by abnormalities in the intraparietal sulcus

(e.g., Isaacs et al., 2001; Molko et al., 2003). This hypothesis

fits with structural imaging studies that have documented in-

ferior parietal lobe abnormalities in children with VCFS (e.g.,

Eliez et al., 2000; Simon et al., 2005b). However, it has been

shown that both the verbal and quantity systems of number

processing are anatomically contiguous in the inferior parietal

cortex and interact closely during any calculation (Stanescu-

Cosson et al., 2000). Therefore, future imaging studies in

VCFS should subdivide the inferior parietal lobe into specific

subregions and examine how each of these contributes to dis-

tinct aspects of mathematical cognition. It is interesting to

note that the data of the present study are also in line with

the only functional MRI study on mental arithmetic in VCFS

(Eliez et al., 2001). In this study, participants were presented

with two single-digit arithmetic tasks, both containing small

problems. The simple task consisted of small two-operand ad-

ditions and subtractions, similar to the ones assessed in the

present study. In this condition no differences in activation

patterns between children with VCFS and controls were

found, which is in line with the present findings. The difficult

task consisted of three operands combining addition and

subtraction (e.g., 6þ 2� 3). Such problems are difficult to solve

with direct retrieval and require the use of procedural strate-

gies, which draw heavily on the quantity system of number

processing. Interestingly, children with VCFS showed more

activation than controls during this task in the right intrapar-

ietal sulcus, the latter of which hosts the quantity representa-

tion of number.

The results of the present study additionally suggest that

children with VCFS might benefit from interventions for chil-

dren with MD in general that particularly foster the develop-

ment of the quantity representation of number or of the

association between quantities and numbers. For example,

Wilson et al. (2006) have recently developed an adaptive

computer game targeted at the remediation of such difficul-

ties in number processing. In an open trial assessment,

they showed that, after the intervention, children with MD

showed improvements in symbolic number comparison and

subtraction.

The current findings cannot exclude that the calculation

deficit in childrenwith VCFSmight be accounted for by limita-

tions in working memory, which have been reported in chil-

dren with VCFS (Sobin et al., 2005; Woodin et al., 2001) and

children with MD (Bull et al., 1999; McLean and Hitch, 1999;

Swanson and Sachse-Lee, 2001). However, a previous study

with the same research sample makes such an explanation

rather unlikely, as children with VCFS did not perform more

poorly than controls on measures of working memory, lexical

retrieval speed, processing speed and reading ability, and

none of these general cognitive skills were related to their

math performance (De Smedt et al., unpublished data).

It should also be acknowledged that the present study was

cross-sectional in nature. Thus, it remains to be seen if impair-

ments in basic number processing are the source, or instead,

the consequence of impairments in single-digit arithmetic.

One possibility is to compare number comparison in children

with VCFSwith that of another group of younger childrenwith

similar single-digit arithmetic ability (i.e., ability-levelmatched

design; see Chiappe, 2005 for a discussion of this approach in

MD research). Another approach is to assess basic number

processing skills before or at the start of formal schooling

c o r t e x 4 5 ( 2 0 0 9 ) 1 7 7 – 1 8 8 185


and to longitudinally examine their relationship with single-

digit arithmetic throughout primary school. Such a design

would allow to determine whether the difficulties in number

comparison precede the problems in single-digit arithmetic

or not.

It remains to be seen whether the results of the present

study hold over development, especially with regard to

single-digit multiplication. For example, the second grade

children with VCFS (n¼ 6) and their controls (n¼ 6) were at

an early stage of learning single-digit multiplication. As a re-

sult, this task was rather difficult for these young children,

which might partially explain the absence of group differ-

ences in multiplication. Interestingly, a previous study on

older (age 10–12) children with VCFS and their matched

controls – in which single-digit multiplication was relatively

automatized – showed that the pattern of preserved single-

digit multiplication in children with VCFS, even for the difficult

problem types, still holds at that age (De Smedt et al., 2007).

The latter suggests that the results of the present study on

the multiplication task cannot merely be explained by effects

of task difficulty.

A possible limitation of the present study is related to the

use of a number reading task as ameasure of verbal represen-

tations of number. Arguably, number reading not only encom-

passes the verbal representation of number, but also involves

transcoding between visual and verbal representations. How-

ever, the converging evidence presented in Table 4 shows high

associations between the number reading and the verbal re-

call of number facts (Table 4). This indicates that the verbal

representation of number is probably the key aspect here.

Another limitation of our study deals with the exclusive

use of a verbal IQ measure. It should be kept in mind that

the intellectual profile of children with VCFS is often charac-

terized by VIQ> PIQ discrepancy (Swillen et al., 1997). Simon

et al. (2005a) proposed that MD in children with VCFS are re-

lated to their poor visuospatial abilities. However, visuospatial

abilities are highly related to PIQ, and therefore, we decided

a priori to control for VIQ instead of PIQ. Indeed, if MD in

VCFS are due to underlying visuospatial impairments, con-

trolling for PIQ might have removed important group differ-

ences in math performance that were of primary interest in

the context of the present study. Nevertheless, we addition-

ally collected PIQ data, by means of the Standard Progressive

Matrices (Raven et al., 1992). As expected, both groups differed

significantly on this test (VCFS mean¼ 76.9 (14.5), control

mean¼ 94.4 (15.0); t(24)¼�5.07, p< .01). All analyses of the

present study that were presented above were repeated with

this measure of PIQ as a covariate. The results of these addi-

tional analyses were, however, exactly the same as the ones

with the VIQ covariate that are reported in this manuscript.

These findings challenge, at least in the present sample, the

hypothesis that MD in VCFS derive from underlying visuospa-

tial deficits (Simon et al., 2005a).

To conclude, the data of the present study show that chil-

dren with VCFS showed a preserved verbal system but an im-

paired quantity system. This pattern was found at the level of

basic number processing, arithmetic operations, and strategy

use. Most importantly, the correlational analyses showed that

basic number processing skills directly accounted for single-

digit arithmetic performance in the children of the present

study. Future studies should investigate whether the opposite

pattern, i.e., impaired verbal number processing but a pre-

served quantity processing, exists and whether this pattern

is also reflected at the level of arithmetic operations and

strategies.

Acknowledgements

Bert De Smedt is a postdoctoral fellow of the Department of

Educational Sciences of the University of Leuven, Belgium.

We are grateful to all children and families who participated

in our research.

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