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Developmental dyscalculia is related to visuo-spatial memory
and inhibition impairment
5
Denes Szucs
a,
*, Amy Devine
a
, Fruzsina Soltesz
b
, Alison Nobes
a
and Florence Gabriel
a
a
Department of Psychology, Centre for Neuroscience in Education, University of Cambridge, Cambridge, United Kingdom
b
Department of Psychiatry, University of Cambridge, United Kingdom
article info
Article history:
Received 5 December 2012
Reviewed 22 March 2013
Revised 3 April 2013
Accepted 19 June 2013
Action editor Roberto Cubelli
Published online xxx
Keywords:
Developmental disorders
Intraparietal sulcus (IPS)
Developmental learning disability
Mathematical difficulty
Number sense
abstract
Developmental dyscalculia is thought to be a specific impairment of mathematics ability.
Currently dominant cognitive neuroscience theories of developmental dyscalculia suggest
that it originates from the impairment of the magnitude representation of the human
brain, residing in the intraparietal sulcus, or from impaired connections between number
symbols and the magnitude representation. However, behavioral research offers several
alternative theories for developmental dyscalculia and neuro-imaging also suggests that
impairments in developmental dyscalculia may be linked to disruptions of other functions
of the intraparietal sulcus than the magnitude representation. Strikingly, the magnitude
representation theory has never been explicitly contrasted with a range of alternatives in a
systematic fashion. Here we have filled this gap by directly contrasting five alternative
theories (magnitude representation, working memory, inhibition, attention and spatial
processing) of developmental dyscalculia in 9e10-year-o ld primary school children. Par-
ticipants were selected from a pool of 1004 children and took part in 16 tests and nine
experiments. The dominant features of developmenta l dyscalculia are visuo-spatial
working memory, visuo-spatial short-term memory and inhibitory function (interference
suppression) impairment. We hypothesize that inhibition impairment is related to the
disruption of central executive memory function. Potential problems of visuo-spatial
processing and attentional function in developmental dyscalculia probably depend on
short-term memory/working memory and inhibition impairments. The magnitude repre-
sentation theory of developmental dyscalculia was not supported.
ª 2013 The Authors. Published by Elsevier Ltd. All rights reserved.
Developmental dyscalculia (DD) is a learning difficulty specific
to mathematics which may affect 3e6% of the population.
Pure DD (hereafter: DD) does not have apparent co-morbidity
with any other developmental disorder, such as dyslexia or
attention deficit hyperactivity disorder (ADHD), intelligence is
normal, the only apparent weakness is in the domain of
mathematics (Shalev and Gross-Tsur, 2001). The currently
dominant neuroscience theory of DD assumes that DD is
related to the impairment of a magnitude representation (MR)
often called the approximate number system (ANS; Piazza
et al., 2010) or a ‘number module’ (Landerl et al., 2004)
residing in the bilateral intraparietal sulci (IPSs). This MR is
5
This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and source are credited.
* Corresponding author.
E-mail address: ds377@cam.ac.uk (D. Szucs).
Available online at www.sciencedirect.com
Journal homepage: www.elsevier.com/locate/cortex
cortex xxx (2013) 1e15
Please cite this article in press as: Szucs D, et al., Developmental dyscalculia is related to visuo-spatial memory and inhibition
impairment, Cortex (2013), http://dx.doi.org/10.1016/j.cortex.2013.06.007
0010-9452/$ e see front matter ª 2013 The Authors. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.cortex.2013.06.007
thought to enable the intuitive understanding of numerical
magnitude enabling number discrimination (e.g., Dehaene,
1997; Piazza et al., 2010). The MR theory of DD suggests that
an impairment of the MR per se impacts on numerical skills
leading to DD (Piazza et al., 2010; Landerl et al., 2004). The
theory expects that non-symbolic numerosity comparison
(e.g., comparing the number of items in two groups) is defi-
cient in DD children. Another version of the MR theory as-
sumes that the MR itself may be intact in DD but links between
the MR and numerical symbols are impaired. This version
expects that non-symbolic numerosity comparison is intact
but symbolic numerosity comparison is deficient in DD
(Rousselle and Noe
¨
l, 2007; De Smedt and Gilmore, 2011). The
MR theory of DD also claims support from neuro-imaging
evidence because children with DD were shown to have
lower gray matter density in the parietal cortex than controls
in structural magnetic resonance imaging (MRI) studies
(Isaacs et al., 2001; Rotzer et al., 2008; Rykhlevskaia et al., 2009)
and they sometimes show different IPS activation relative to
controls in magnitude comparison tasks in functional MRI
(fMRI) studies. Strikingly, the MR theory of DD has never been
systematically contrasted with various alternative theories
proposed by extensive behavioral research. Here we report
such a study.
The most established markers of the MR are behavioral
ratio and distance effects (Moyer and Landauer, 1967)in
symbolic (e.g., ‘Which is larger; 3 or 4?’) and non-symbolic
(e.g., ‘Do you see more dots on the left or on the right?’)
magnitude comparison tasks (ratio and distance effects refer
to the fact that it is faster and less error prone to compare
further away than closer quantities) and their correlates in the
IPS (Pinel et al., 2001). To date five fMRI studies compared
distance/ratio effects in DD and controls (Kucian et al., 2006,
2011; Price et al., 2007; Mussolin et al., 2010b; Kovas et al.,
2009) and one fMRI study compared approximate calculation
(performance on this is expected to rely on the MR of the IPS)
in DD and controls (Davis et al., 2009). Behaviorally, only Price
et al. (2007) reported a different accuracy distance effect in DD
relative to controls. None of the studies reported a different
reaction time (RT) distance effect in DD relative to controls.
Price et al. (2007; non-symbolic comparison with no control
task) and Mussolin et al. (2010b; one-digit Arabic number
comparison with color comparison control task) reported
weaker IPS distance effects in DD than in controls. Kucian
et al. (2006; non-symbolic magnitude comparison with color
comparison control task) compared activity in a greyscale
comparison control task and in a magnitude comparison task
but did not find any brain activity difference between DD and
controls in either multiple testing corrected or uncorrected
whole-brain analyses. Kovas et al. (2009; non-symbolic
magnitude comparison with five ratios; with color compari-
son control task) reported DD versus control and numerical
versus control task differences in various brain regions but not
in the IPS and, in fact did not find any ratio/distance effects in
the IPS. They concluded that the IPS based MR theory of DD
may not stand. Kucian et al. (2011; non-symbolic magnitude
comparison with no control task) observed differences be-
tween DD and controls in several brain areas but not in the
parietal lobe and concluded that DD children have difficulty in
response selection relative to control children. Davis et al.
(2009) did not find IPS differences between DD and controls
in an approximate calculation task.
In summary, evidence suggesting that abnormal IPS func-
tion is related to the MR in DD is weak. Four out of six studies
returned negative fMRI findings with regard to the IPS based
MR hypothesis of DD. Of the two positive studies, only one had
supporting behavioral evidence ( Price et al., 2007). However,
this study did not use a control task, DD showed a normal RT
distance effect, there was 17.7 points difference between DD
and control on the Wechsler Intelligence Scale for Children
(WISC) Block Design test, and memory/attention was not
tested. Mussolin et al. (2010b) had a control task but did not
have supporting behavioral evidence. The lack of behavioral
evidence and control tasks leaves it unclear whether differ-
ences in IPS structure and perhaps function relate to numer-
ical skill or to some other uncontrolled and untested function
(Poldrack, 2006). In addition, each study tested a relatively
narrow range of variables.
Purely behavioral studies arguing in favor of the MR theory
used dot comparison tasks and showed that functional
markers of comparison performance differed in DD and con-
trol participants (Piazza et al., 2010; Mazzocco et al., 2011;
Mussolin et al., 2010a). However, none of these studies used
non-numerical tasks controlling for non-numerical aspects of
comparisons. Nevertheless, evidence demonstrates that both
symbolic and non-symbolic comparison performance pri-
marily reflects domain general comparison processes rather
than properties of the number representation (Holloway and
Ansari, 2008). Hence, the omission of a control task is a sig-
nificant shortcoming and, in principle, studies without control
tasks cannot draw any number-specific conclusions. In addi-
tion, the dot comparison task is inherently confounded by
non-numerical parameters which cannot be controlled in
each particular trial (Gebuis and Reynvoet, 2011, 2012; Szucs
et al., 2013). Further, when tracking both numerical and non-
numerical parameters in dot comparison tasks, event-
related brain potentials (ERPs) only showed sensitivity to
non-numerical parameters but not to numerical parameters
(Gebuis and Reynvoet, 2012). Hence, in the dot comparison
task participants’ supposedly numerical judgments can rely
on non-numerical parameters in each particular trial. This
problem also affects fMRI studies using non-symbolic magni-
tude comparison. It is noteworthy that Landerl et al. (2004) is
one of the most often cited studies in support of the MR theory.
However, that study merely demonstrated that DD have slower
magnitude comparison speed than controls which can happen
for many reasons. The distance effects did not differ in DD and
controls and DD only showed a marginally steeper counting
range RT curve than controls (pp. 117 and 119e120). In fact, the
distance effect was not significant even in controls which
suggests lack of power. In an extensive follow-up study
Landerl and Kolle (2009) could not detect any robust basic
number processing difference between DD and controls and
they concluded that they ‘did not find strong evidence that DD
children process numbers qualitatively differently from chil-
dren with typical arithmetic development’ (ibid., abstract).
While the MR theory of DD currently dominates neurosci-
ence research, behavioral research identified several cognitive
functions which play an important role in mathematical
development and proposed several alternative theories of DD
cortex xxx (2013) 1e152
Please cite this article in press as: Szucs D, et al., Developmental dyscalculia is related to visuo-spatial memory and inhibition
impairment, Cortex (2013), http://dx.doi.org/10.1016/j.cortex.2013.06.007
which have mostly been neglected by neuro-imaging
research. First, a large volume of studies found deficient ver-
bal and/or visuo-spatial WM function in DD (e.g., Hitch and
McAuley, 1991; Passolunghi and Siegel, 2001, 2004; Keeler
and Swanson, 2001; Bull et al., 2008; Swanson, 2006; Geary,
2004) and longitudinal studies confirmed that WM function
is related to mathematical performance (Geary, 2011;
Swanson, 2011; Passolunghi and Lanfranchi, 2012). WM
serves as a limited capacity mental workspace for operands,
operators, and retrieved numerical facts which have to be
mobilized even during the simplest calculations (Geary, 1993;
Ashcraft, 1995). Hence, its impairment can have detrimental
consequences for mathematical function. Second, some
studies reported spatial processing problems in DD (Rourke
and Conway, 1997; Rourke, 1993) which may be related to
visuo-spatial WM problems. Spatial processes can be poten-
tially important in mathematics where explicit or implicit
visualization is required, like when imagining operations
along the number line or visualizing functional relationships.
Third, others found deficient inhibitory function in DD
and/or a relationship between inhibitory function and math-
ematical development (Bull and Scerif, 2011; Bull et al., 1999;
Pasolunghi et al., 1999; Passolunghi and Siegel, 2004;
McKenzie et al., 2003; Espy et al., 2004; Blair and Razza, 2007;
Swanson, 2011). Fourth, similar findings were reported with
regard to attentional function (Swanson, 2011; Ashkenazi
et al., 2009; Hannula et al., 2010). Inhibitory and attentional
processes co-ordinate which items of interest receive pro-
cessing and when and in what order they enter processing.
This also assures that (temporarily) irrelevant potential
mathematical processing events are suppressed (e.g.,
Barrouillet et al., 1997; Bull et al., 1999; Pasolunghi et al., 1999;
Passolunghi and Siegel, 2004). Such processes are extremely
important in calculations which require the continuous se-
lection and coordination of several processing steps and items
in memory. In fact, inhibitory function, attentional and
working memory (WM) processes may all be intricately
intertwined and form the core of so-called ‘central executive’
memory processes (Hasher and Zacks, 1988; Miyake et al.,
2000).
Crucially, all of the above cognitive functions have been
linked to the IPS. Hence, impairment of any of the above
functions could plausibly explain IPS abnormality in DD
which is routinely cited in support of the impaired MR theory
of DD. IPS activity has been shown to be modulated by ma-
nipulations in WM (Culham and Kanwisher, 2001; Coull and
Frith, 1998; Linden et al., 2003; Todd and Marois, 2004;
Dumontheil and Klingberg, 2011), attention (Coull and Frith,
1998; Vandenberghe et al., 2012; Santangelo and Macaluso,
2013; Davranche et al., 2011), inhibitory function (Cieslik
et al., 2011; Mecklinger et al., 2003) and spatial processing
(Yang et al., 2011) tasks. Moreover, one study demonstrated
decreased IPS function in DD children in a spatial WM task
(Rotzer et al., 2009) and another study demonstrated that
brain activity during a visuo-spatial WM task in the IPS pre-
dicts mathematical ability 2 years later (Dumontheil and
Klingberg, 2011). Hence, IPS dysfunction in DD may well be
linked to WM dysfunction. In addition, an ERP investigation of
DD found that short latency (200 msec) ERPs, probably related
to automatic magnitude discrimination, were similar in DD
and controls but later (600 msec latency) processes indexed by
the P3b wave, usually related to categorization decision,
differed (Solte
´
sz et al., 2007). These findings have been
confirmed by a recent study (Heine et al., 2012). Further, Sol-
tesz et al. (2007) found that the DD and control groups differed
in neuropsychological tests measuring executive functioning.
Hence, it was concluded that basic number processing was
intact while aspects of higher level executive memory or
attention function were impaired in DD.
Overall, a serious shortcoming of the existing literature is
that the MR theory has never been directly contrasted sys-
tematically with alternative theories of DD. That is, most
behavioral studies focusing on memory and attention function
did not use measures of the MR and most MR studies did not
use a wide range of alternative measures. Here, our intention
was to understand the complexity of DD by taking a very wide
range of measurements. This allowed us to directly contrast the
MR, WM, inhibition, attention and spatial processing theories
of DD in primary school children. We matched controls for
verbal and non-verbal IQ, socio-economic status and general
processing speed. We used five experimental measures of the
MR theory with high trial numbers. We assumed that if MR
theory is correct then there should be robust differences on MR-
related measures between DD and control participants on all of
these tasks, especially on the non-symbolic and symbolic
magnitude decision tasks which are proposed to be the most
important markers of the MR. Verbal and visuo-spatial short-
term memory (STM)/WM were tested by standardized mea-
sures. Inhibition performance was measured by detecting nu-
merical and non-numerical congruency effects in four
experiments and with a Stop-signal task. Sustained attention
and simple RT speed were tested by visual target detection
experiments. Spatial processing was measured by testing both
performance scores and solution speed on a spatial symmetry
task and on a mental rotation task.
1. Materials and methods
Methods are described in more detail in Supplementary
methods. Parental consent was obtained for all phases of
the study. The study received ethical approval from the
Cambridge Psychology Research Ethics Committee.
1.1. Screening
In a first step, 1004 children were screened for DD with age-
standardized United Kingdom National Curriculum-based
maths and reading tests, administered to whole classes. The
maths test was the Mathematics Assessment for Learning and
Teaching test (MaLT; Williams, 2005), a written test containing
questions covering all areas of the maths curriculum. This test
allows for invigilators to read the questions to the children if
required to ensure test performance reflects mathematics
ability rather than reading proficiency. Reading ability was
assessed using the Hodder Group Reading Test II, levels 1 and
2 (HGRT-II; Vincent and Crumpler, 2007). These multi-choice
tests assess children’s reading of words, sentences and pas-
sages. Characteristics of the screening sample have been
described by Devine et al. (2013).
cortex xxx (2013) 1e15 3
Please cite this article in press as: Szucs D, et al., Developmental dyscalculia is related to visuo-spatial memory and inhibition
impairment, Cortex (2013), http://dx.doi.org/10.1016/j.cortex.2013.06.007
In a second step about 200 children representing the dis-
tribution of mathematics and reading scores were invited to
take part in further study. A part of this sample consented and
a subgroup of 115 children from the original sample took part
in further screening and experimental tasks. Each child was
tested for about 7e8 h duration in multiple sessions. Children
were individually administered an additional standardized
measure of mathematical ability [the Numerical Operations
subtest of Wechsler Individual Achievement Test (WIAT-II;
Wechsler, 2005)], two additional standardized measures of
reading ability (WIAT-II Word Reading and Pseudoword
Decoding subtests), and two IQ tests [the Raven’s Colored
Progressive Matrices (Raven’s CPM; Raven, 2008) and a short
form of the WISC e 3rd Edition (WISC-III, Wechsler, 1991)]. The
WISC-III short form included the Block Design (non-verbal)
and Vocabulary (verbal) subtests. This combination of sub-
tests has the highest validity and reliability of the two-subtest
forms (r
tt
¼ .91, r ¼ .86; Table L-II, Sattler, 1992). Socio-
economic status was estimated from parents’ education
levels and occupations.
1.2. Participants
Children were defined to have DD if their mean performance
on the standardized MaLT and WIAT-II UK Numerical Oper-
ations tests was worse than mean 1SD (<16th percentile)
and their performance on the HGRT-II, WISC Vocabulary,
WIAT Word Reading, WIAT Pseudoword reading, Raven and
WISC Block Design tests was in the mean 1SD range. 18
children (15.6% of the 115 children and 1.8% of the sample of
1004 children) performed worse in mathematics than the
mean 1SD criterion. Six children had both weak mathe-
matics and reading/IQ performance (score < mean 1SD) and
were not investigated further. That is, there were 12 partici-
pants in both the DD and the Control group (DD: four girls;
Control: seven girls). Criterion test profiles with standard test
scores are shown in Fig. 1. Groups were perfectly matched on
age (DD vs Control: 110 vs 109 months, p ¼ .52), non-verbal IQ,
verbal IQ and socio-economic status [parental occupation
(mean and standard error (SE) for DD vs Controls: 4.0 .6 vs
3.7 .4) and parental education (4.7 .4 vs 4.9 .3); Man-
neWhitney U test for both p > .71]. Groups differed only on the
MaLT and WIAT Numerical Operations tests. It is important to
point out that many studies do not match groups perfectly
along variables which may affect group differences in the
dependent variable and instead rely on analysis of covariance
(ANCOVA) to supposedly ‘correct for’ group differences.
However, this is a statistically invalid procedure and therefore
an improper use of ANCOVA (see e.g., Miller and Chapman,
2001; Porter and Raudenbush, 1987). Hence, it is necessary to
match experimental groups tightly as done here if it is theo-
retically important.
1.3. Further tests
WM: Children were administered five subtests of the Auto-
mated Working Memory Assessment (AWMA; Alloway, 2007);
which included two measures of verbal STM: Digit Span and
Word Recall; one measure of visuo-spatial STM: Dot Matrix;
one measure of verbal WM: Listening Span; and one measure
of visuo-spatial WM: Odd One Out (OOO). Raw and standard-
ized recall scores for all subtests, as well as processing scores for
Listening Span and OOO were measured.
Trail-making task: Trail-making tests A and B were admin-
istered. Each received a score (2 ¼ no errors or self corrected,
1 ¼ one error, 0 ¼ two or more errors) and solution speed was
measured in seconds.
Mental rotation: Three separate worksheets with different
stimuli types (objects/animals, letters and hands) were pre-
sented to the children; each worksheet had seven items. For
each item within a worksheet, a target stimulus was pre-
sented, along with three comparison stimuli, two of which
were mirror images and one was identical to the target. All
three comparison images were rotated by various angles. The
children were required to identify and circle the stimulus
identical to the target. Children’s accuracy and time to com-
plete all seven items were recorded for each worksheet.
Spatial symmetry: Children were presented with two pages
which contained six half drawn shapes against a grid back-
ground. A dashed line indicated the line of symmetry. Chil-
dren were required to draw the other half of the shape for each
item. Shapes (and lines of symmetry) were presented verti-
cally on one page and horizontally on the other. The total time
to complete the 12 shapes was recorded and the accuracy of
items was scored with one point for every correct line
segment.
1.4. Computerized experimental tasks
The following tasks were presented by the Presentation pro-
gram of Neuro-behavioral Systems using a laptop computer.
Unless described otherwise, RT and accuracy were recorded
for all trials. See Supplementary methods for further details.
Fig. 1 e Group profiles on standardized screening tests.
Group means and 95% confidence intervals are shown.
Means permutation p and independent t-test p values are
given below the X axis. For display purposes only the WISC
Vocabulary and Block Design scores were rescaled to
mean [ 100 and SD [ 15; analyses were done on original
values which are shown numerically.
cortex xxx (2013) 1e154
Please cite this article in press as: Szucs D, et al., Developmental dyscalculia is related to visuo-spatial memory and inhibition
impairment, Cortex (2013), http://dx.doi.org/10.1016/j.cortex.2013.06.007
Simple RT: Children pressed a key in response to a white
square which appeared after 1000, 2500 or 4000 msec (delay
factor). There were 60 trials.
Sustained attention: Children were required to attend to a
stimuli stream (letters) and to detect a target sequence (A B
C) and to withhold responses to other sequences containing
the target letters (‘deceiver trials’; e.g., A B D) or sequences
contai ning no target letters (‘non-target trials’; e.g., D H F).
The number of hits and misses for targets, th e RT for target
hits, the number of correct rejections and false alarms for
deceivers and non-target trials, were recorded. Children
were presented with 80 triads of the three different trial
types.
Stop-signal task: A white arrow, pointing left or right, was
shown on a black background in the middle of the screen. The
arrow was either followed by a sound, the stop signal, or there
was no sound. Children were required to indicate the direction
of the arrow using a key press during ‘go’ trials, and to with-
hold their responses during ‘stop’ trials. The ratio of ‘go’ and
‘stop’ trials was 2:1. For each trial we measured RT, Stop signal
RT (defined as the RT e average stop signal delay), and the
number of times the child responded to the arrow incorrectly.
180 trials were presented.
Animal Stroop: Stimuli were colored pictures of two ani-
mals. Children were instructed to press a button on the
keyboard on the side corresponding to the animal which was
bigger in real life (Sz
}
ucs et al., 2009; Bryce et al., 2011). In the
congruent condition the animal which was larger in real life
was presented in a larger picture than the animal which was
smaller in real life. In the incongruent condition the animal
which was larger in real life was presented in a smaller picture
than the animal which was smaller in real life. 96 trials were
presented.
Numerical magnitude comparison Stroop task: Stimuli were
pairs of white Arabic digits shown simultaneously on black
background. There were four possible number pairs, with two
different numerical distances. Children were instructed to
decide which item of the pair was numerically larger than the
other one and pressed a key where they detected the
numerically larger stimulus. Numerical and physical size in-
formation could be neutral, congruent or incongruent with
each other in equal proportions (congruency factor). In the
congruent condition the numerically larger digit was also
physically larger than the other one. In the incongruent con-
dition the numerically larger digit was physically smaller than
the other one. In the neutral condition both digits were of the
same physical size. Numerical distance between stimuli was
either 1 or 7 (numerical distance factor). 192 trials were
presented.
Physical size comparison Stroop task: This task was identical
to the numerical magnitude Stroop task, with the exception
that the task was to respond to the physically larger stimulus.
In neutral trials the digits differed in physical size but were
numerically identical. 192 trials were presented.
Subitizing: Arrays containing one to six black dots appeared
on a white background and children were instructed to say the
number of dots as quickly as possible. Dot stimuli were pre-
sented in canonical and, where possible, non-canonical ar-
rangements. RTs were measured using a voice-key. 60 trials
were presented.
Symbolic magnitude comparison: Children decided whether
visually presented digits were smaller or larger than 5. Chil-
dren pressed a button on the keyboard with their left hand if
the number was smaller than 5 and another button with their
right hand if the number was larger than 5. 80 trials were
presented.
Non-symbolic magnitude comparison: Two sets of black dots
were presented simultaneously on a white background. The
children’s task was to decide which set contained more dots
and press the button on the side of the larger set. Dot size was
varied between sets. The following factors were manipulated
in the construction of the stimuli sets: (1) The ratio of the
number of dots in the two sets (1:2, 3:5, 2:3); (2) The numerical
distance between the number of dots in the two sets; (3) The
type of the physical control variable; (4) The congruity of
physical control variables and numerosity; (5) The overall
numerical sum of items in a display. See Supplementary
methods for further details. 128 trials were presented.
1.5. Statistics
First, DD minus control difference scores were computed for
tests and for the most important experimental contrasts (see
details in Supplementary material): simple RT; animal Stroop
task congruency; numerical and physical size Stroop task
numerical distance effect, facilitation and interference; subi-
tizing slope (numbers 1e3), counting slope (numbers 4e6);
non-symbolic comparison slope and congruency effect, sym-
bolic comparison slope; Stop-signal task hit and correct
rejection performance.
Difference score data was assessed by robust non-
parametric permutation testing (Ludbrook and Dudley, 1998).
Dependent variables were test scores, accuracy and median
RT. Procedure followed Chihara and Hesterberg (2011).DD
minus control group difference scores were computed for all
measures and the whole pool of participants were randomly
divided into two groups of 12 participants one million times.
Two-tailed significance values were determined with six
decimal digits precision. In order to provide an estimate of
effect size, empirical 95% confidence intervals for difference
scores were also determined by bootstrap resampling pro-
ducing one million bootstrap samples with replacement for
each group.
Second, all experimental data was also analyzed by ana-
lyses of variance (ANOVAs) with full factorial designs. Third,
while permutation tests provide extremely stringent criteria
and groups were perfectly matched on several factors, dif-
ference scores showing significant permutation testing effects
were nevertheless further analyzed by ANCOVAs with a group
factor and with covariates of verbal intelligence (WISC Vo-
cabulary), non-verbal intelligence (Raven) and simple RT
speed (median RT from the Simple RT task). With matched
groups this procedure can further increase power (Miller and
Chapman, 2001). Fourth, simultaneous multiple regression
analysis was used to study the relative weight of variables
which significantly discriminated between the DD and control
groups and were correlated with maths performance (the
mean of the MaLT and WIAT Numerical Operations scales).
Regressions are described further in Results. Analyses were
programmed in Matlab.
cortex xxx (2013) 1e15 5
Please cite this article in press as: Szucs D, et al., Developmental dyscalculia is related to visuo-spatial memory and inhibition
impairment, Cortex (2013), http://dx.doi.org/10.1016/j.cortex.2013.06.007
2. Results
2.1. Memory
Fig. 2 summarizes significant DD versus control group differ-
ences in standardized test scores. The two groups differed on
measures of visuo-spatial STM (Dot Matrix) and WM (OOO
Recall, OOO Processing). 95% bootstrapped confidence in-
tervals were robustly below zero for each measure showing a
significant group difference (i.e., the DD group performed
worse than the control group). For comparison, means and
confidence intervals for non-significant verbal STM (Digit
Recall, Word Recall) and WM measures (Listening Recall and
Processing) are also presented. Table 1 shows F and p values
from ANCOVAs for significant tests taking verbal IQ, non-
verbal IQ and processing speed as covariates.
2.2. Accuracy measures
Fig. 3A summarizes main DD minus control group differences
in accuracy. The figure shows permutation and t-test statistics
outcomes and bootstrapped 95% confidence intervals for ef-
fect sizes. Detailed experimental results and results of facto-
rial ANOVAs are shown in Supplementary Fig. 1. Table 2
shows F and p values from ANCOVAs for significant tests
taking verbal IQ, non-verbal IQ and processing speed as
covariates. There were significant group differences in three
measures. First, in the subitizing task counting-range slope
was less steep in DD than in controls in the 4e6 number range.
This was due to a larger drop in accuracy for number 6 in
controls than in DD (see star in Supplementary Fig. 1D). Sec-
ond, there was a larger congruency effect in DD than in control
participants in non-symbolic magnitude comparison (see
star in Supplementary Fig. 1F). Third, correct rejection
performance was worse in DD than in controls in the Stop-
signal task (see star in Supplementary Fig. 1E). In ANOVAS
there was an additional marginal group congruency inter-
action in the animal size Stroop task due to a marginally larger
congruency effect in DD than in controls (Supplementary
Fig. 1B). The trail-making task was scored on a 0e2 scale.
Accuracy was practically the same in both groups in both trail-
making A/B: All DD participants and all but one control scored
maximum on trail-making A (a single control scored 0). Scores
were also matched on trail-making B (number of DD/Control
participants with particular scores: Score 2: 8/7; Score 1: 2/2;
Score 0: 2/3). Importantly, both permutation testing and con-
fidence interval estimation showed that symbolic and non-
symbolic slope was a highly non-discriminative parameter
between groups. Fig. 3 shows effect sizes. In detail, in the non-
symbolic discrimination task the mean ratio effect was
1.75 .5% (mean and SE; accuracy for each ratio: 97.2 1.1,
95.6 1.4 and 93.7 1.6%) in the DD group and 1.70 .4% in
the control group (accuracy for each ratio: 97.7 .9, 95.2 1.8
and 94.3 1.8%). In the symbolic discrimination task the mean
distance effect was 3.26 1.4% (distance 1 minus distance 4)
in the DD group and 5.24 1.4% in the control group (accu-
racy for each level of distance: DD: 91.5 1.9 and 94.8% 1.3;
controls: 89.0 2.3 and 94.2 1.6%).
2.3. Median RT
Fig. 3B summarizes main findings in RT with permutation
testing and t statistics and bootstrapped 95% confidence in-
tervals for effect sizes. Detailed experimental results and re-
sults of factorial ANOVAs are shown in Supplementary Fig. 2.
Table 3 shows F and p values from ANCOVAs for significant
tests taking verbal IQ, non-verbal IQ and processing speed as
covariates. There were significant group differences in four
measures. First, there was a larger facilitation effect in the
Fig. 2 e Permutation test results and bootstrap confidence intervals for standardized test scores. DD minus control
difference scores are shown. Circles show the mean DD minus control group differences. Filled circles and stars denote
significant group differences. Bars represent bootstrapped 95% confidence intervals. The upper number next to circles
represents the permutation test p value for group differences. The middle number represents the independent sample t-test
p value. The bottom number is the mean effect size in test score. Both standard and raw scores are shown for tests with
significant effects. Only standard scores are shown for tests with non-significant effects (verbal STM D WM). Significant
correlations between test scores and maths performance are shown below stars.
cortex xxx (2013) 1e156
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numerical Stroop task in DD than in control participants
(Supplementary Fig. 2G). The negative effect means that RT
sped up more in the congruent relative to the neutral condi-
tion in DD than in control participants. This means that task-
irrelevant physical size information had a larger effect on RT
in DD than in controls. As optimal task performance requires
focusing on the task-relevant numerical dimension, larger
facilitation from physical size information reflects the
intrusion of the task-irrelevant stimulus dimension into pro-
cessing. Hence, this effect is a marker of failure to inhibit the
task-irrelevant stimulus dimension. Second, there was a
larger distance effect in DD than in controls in the physical
size decision Stroop task (Supplementary Fig. 2H). This means
that task-irrelevant numerical information had a larger effect
on RT in DD than in controls. Third and fourth, trail-making A
(Mean/SE: DD ¼ 58.3 5.4 sec; Control ¼ 41.3 2.0 sec) and
Fig. 3 e Permutation test results and bootstrap confidence intervals for (A) accuracy and (B) median RT measures. DD minus
control difference scores are shown. Permutation and t-test p values and mean effect sizes (accuracy and RT) are shown
below figures. Significant correlations between measures and maths performance are shown in the figure if significant or
marginal (r and p values). Significant group differences are marked by red bars, text and stars. Marginal results are marked
by orange bars, text and crosses.
Table 1 e ANCOVA results for WM tests.
Dot
Matrix
OOO
recall
OOO
processing
Raw dot
matrix
Raw OOO
recall
Raw OOO
processing
Correcting for verbal IQ F(1,21)¼ 5.13 9.89 7.88 4.23 8.19 6.05
p value .0348 .0051 .0108 .0529 .0096 .0232
Correcting for non-verbal IQ
(Raven) F(1,21)¼
5.69 15.18 13.20 6.15 17.73 13.66
p value .027 .0009 .0016 .0221 .0004 .0014
Correcting for processing speed
(Simple RT task) F(1,21)¼
5.45 7.82 6.47 4.81 6.23 4.72
p value .03 .0111 .0193 .04 .0214 .0419
Correcting for all three
factors F(1,19)¼
7.21 14.41 12.18 8.1 15.14 10.58
p value .0146 .0012 .0024 .0103 .0009 .0041
Significant p values are in bold. Marginally significant p values are in bold italics.
cortex xxx (2013) 1e15 7
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mental rotation (DD ¼ 66.7 4.4 sec; Control ¼ 56.0 3.5 sec)
solution times were longer in DD than in controls. Further,
there was a marginally larger congruency effect in the animal
size decision Stroop task in DD than in controls
(Supplementary Fig. 2B). This means that task-irrelevant
physical size information had marginally larger effect on RT
in DD than in controls. Again, both permutation testing and
confidence interval estimation showed that symbolic and
non-symbolic slope was a highly non-discriminative param-
eter between groups. There were no effects in coefficient of
variation (see Supplementary Fig. 3).
2.4. Regression
Regression analysis was used to study the relative weight of
variables which significantly discriminated between DD and
control and correlated with maths performance. The three
visuo-spatial memory measures (Dot Matrix, OOO Recall and
Processing) were averaged to form a single ‘Visuo-spatial
memory’ measure. The RT facilitation effect from the nu-
merical Stroop task and the RT distance effect from the
physical size decision Stroop task were averaged to form an
‘Inhibition’ score because only these measures showed a sig-
nificant correlation with maths performance (see correlations
in Figs. 2 and 3). The counting-range slope from accuracy data
was also used because this also showed a significant correla-
tion with maths performance. Correlations between the above
variables and maths scores are shown in Table 4. The above
three variables were entered into the analysis simultaneously.
The regression had a significant fit [R
2
¼ .583, F(20,3) ¼ 9.30,
p < .0001]. Visuo-spatial WM [Standardized Beta (b) ¼ .48,
t(20) ¼ 3.2, p ¼ .0045] was a significant predictor and Inhibition
[b ¼ .36, t(20) ¼ 2.06, p ¼ .0522] was a marginally significant
predictor. Subitizing slope was a non-significant predictor
[b ¼.17, t(20) ¼1.02, p ¼ .31]. When only Visuo-spatial WM
and Inhibition were entered into the regression the overall fit
remained unchanged: [R
2
¼ .561, F(21,2) ¼ 13.39, p < .0001].
Visuo-spatial WM: b ¼ .48, t(21) ¼ 3.24, p ¼ .0039. Inhibition:
b ¼ .45, t(21) ¼ 3.00, p ¼ .0068. When verbal IQ (WISC Vocab-
ulary), Raven score and processing speed were added to the
regression, the overall fit increased [R
2
¼ .633, F(20,3) ¼ 9.30,
p < .0001] but only Visuo-spatial WM [b ¼ .61, t(20) ¼ 3.60,
p ¼ .0020] and Inhibition [b ¼ .35, t(20) ¼ 2.18, p ¼ .0421] were
individually significant predictors. Subitizing slope remained
a non-significant predictor when it was entered into the
regression with only the Inhibition ability measure [R
2
¼ .368,
F(21,2) ¼ 6.13, p ¼ .0080; Subitizing: b ¼.19, p ¼ .34; Inhibition:
b ¼ .48, p ¼ .0297].
3. Discussion
We have contrasted five theories of DD using several mea-
sures of the MR theory and alternatives. We found robust
Table 2 e ANCOVA results for accuracy measures.
Subitizing slope
4e6
Non-symbolic comparison
congruency effect
Stop-signal task
correct rejection
Correcting for verbal IQ F(1,21)¼ 7.86 9.33 7.62
p value .0109 .0062 .012
Correcting for non-verbal IQ
(Raven) F(1,21)¼
8.79 7.9 6.86
p value .0076 .0107 .0164
Correcting for processing speed
(Simple RT task) F(1,21)¼
7.01 8.45 6.53
p value .015 .0084 .0184
Correcting for all three
factors F(1,19)¼
9.49 7.88 5.69
p value .0061 .0112 .0276
Significant p values are in bold. Marginally significant p values are in bold italics.
Table 3 e ANCOVA results for RT measures.
Animal Stroop Number Stroop
facilitation
Physical size Stroop
distance effect
Trail-making
A speed
Mental
rotation speed
Correcting for verbal IQ F(1,21)¼ 5.19 16.27 4.57 10.12 3.71
p value .0338 .00 06 .0449 .0046 .0682
Correcting for non-verbal
IQ (Raven) F(1,21)¼
4 13.04 4.44 10.74 3.53
p value .0591 .0017 .0477 .0037 .0747
Correcting for processing speed
(Simple RT task) F(1,21)¼
4.39 12.96 4.94 8.02 3.18
p value .0489 .00 18 .0378 .0102 .0895
Correcting for all three
factors F(1,19)¼
5.14 11.23 3.74 8.08 3.75
p value .035 .0033 .068 .0103 .0676
Significant p values are in bold. Marginally significant p values are in bold italics.
cortex xxx (2013) 1e158
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evidence for impaired visuo-spatial WM and STM in DD and also
found evidence for impaired inhibition function in DD. Data did
not support the MR theory of DD.
3.1. There were robust visuo-spatial WM and visuo-
spatial STM impairments in DD
In contrast, verbal STM/WM were intact including both digit
and word span. Several studies reported similar dissociation
between spatial and verbal STM/WM in DD (McLean and
Hitch, 1999; Andersson and Ostergren, 2012; Schuchardt
et al., 2008; Ashkenazi et al., 2012; Passolunghi and
Mammarella, 2010). Other studies reported impaired verbal
STM/WM in DD (e.g., Geary et al., 1991, 2012). A potential
dissociating feature seems to be that studies not reporting
verbal WM differences noted that they attempted to match DD
and control groups on reading and/or verbal performance
(McLean and Hitch, 1999; van der Sluis et al., 2005; Schuchardt
et al., 2008; Andersson and Ostergren, 2012; Ashkenazi et al.,
2012; Passolunghi and Mammarella, 2010). Our DD group
also only included children with pure DD with no dyslexia and
with normal reading/verbal IQ. This probably explains the lack
of verbal memory differences. In fact, Schuchardt et al. (2008)
tested both visual and spatial STM in DD, dyslexic,
DD þ dyslexic and normal populations and found only visual
STM impairment in DD and only verbal STM impairment in
dyslexics. Hence, it seems that when reading and verbal
function is preserved, that is, in pure DD, a crucial impairment
concerns visuo-spatial WM and/or STM.
At least three neuro-imaging studies provide supporting
evidence to our findings. Rotzer et al. (2009) demonstrated
weaker IPS activation in a spatial WM task in DD than in
controls. Rykhlevskaia et al. (2009) reported reduced gray
matter density in DD not only in the IPS but also in the fusi-
form, lingual, parahippocampal gyri and in the hippocampus,
areas which may be related to encoding complex visual
stimuli. Davis et al. (2009) did not find any IPS differences
between DD and controls in an approximate calculation task
but reported differences in various brain regions associated
with WM and cognitive control functions. Visuo-spatial
memory probably provides a mental workspace for various
transformations and operations crucial for mathematics.
Visuo-spatial strategies and heuristics can be used even in
seemingly non-visual tasks, e.g., when adding or subtracting
numbers, operations and operands can be imagined/concep-
tualized along a number line. Our and other findings reviewed
above suggest that this important general visuo-spatial
workspace does not function properly in DD.
An important question concerns that most studies re-
ported only visual STM (McLean and Hitch, 1999; van der Sluis
et al., 2005; Schuchardt et al., 2008; Ashkenazi et al., 2012;
Passolunghi and Mammarella, 2010) impairment in DD while
only one of the above studies reported WM impairment
(Andersson and Ostergren, 2012). A conspicuous factor
explaining this discrepancy is that in fact only Andersson and
Ostergren (2012) used WM tasks in the visual modality. The
other studies did not measure specific visuo-spatial WM
because they relied on the classical WM model of Baddeley
(1986) which assumes that the so-called central executive
function underlying WM performance is amodal. Hence, most
studies measured WM (central executive) performance with
purely verbal tasks or some tasks may have included spatial
elements but with a strong simultaneous verbal component
(Schuchardt et al., 2008). However, there is accumulating ev-
idence that WM function may in fact dissociate by stimulus
modality and cannot be considered dependent on amodal
central executive resources (Shah and Miyake, 1996; Jarvis and
Gathercole, 2003). In fact, our study provides further evidence
for dissociation between verbal and visual WM systems.
Hence, it seems crucial to measure STM and WM capacity
separately in the verbal and visual modalities.
3.2. Five findings point to impaired inhibitory function
in DD
There were larger congruency effects in DD than in controls in
the non-symbolic magnitude decision task (from the intrusion
of non-numerical parameters) and in the animal Stroop task
(from the intrusion of physical size). In the numerical Stroop
task DD were more affected by task-irrelevant physical size. In
the physical size decision Stroop task DD were more affected
by task-irrelevant numerical magnitude and hence had a
larger automatic numerical distance effect than controls.
First, this finding demonstrates that the automatic processing
of numerical magnitude happened in DD. Second, it is unlikely
that DD had a larger involuntary distance effect than controls
because DD processed magnitude more efficiently than con-
trols. Rather, in the context of generally larger congruency
effects in DD findings suggest that DD could not resist the
intrusion of task-irrelevant stimulus dimensions as efficiently
as controls. Similar data was reported by Landerl and Kolle
(2009) who found larger unit/decade compatibility effects in
DD than in controls and concluded that this was due to worse
interference suppression in DD than in controls (again, the
unlikely alternative explanation could be that DD are better in
interpreting multi-digit numbers than controls). They also
Table 4 e Correlation matrix for variables in the regression analysis. Marginal p values are in parentheses. The correlation of
WISC Vocabulary ( p [ .31), Raven score ( p [ .77) and processing speed ( p [ .26) with maths was not significant.
Maths Counting-range slope Visuo-spatial WM
Counting-range slope r .45
p .0263
Visuo-spatial WM r .61 .18
p .0016 n.s. (.4)
Inhibition r .58 .53 .27
p .0028 .0076 n.s. (.2)
cortex xxx (2013) 1e15 9
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reported a smaller size congruity effect in DD than in controls
in the physical size decision Stroop task. Here we did not find
such an effect while using more than five times as many trials
(192 vs 36) than Landerl and Kolle (2009). The difference may
also be due to the fact that the DD group in Landerl and Kolle’s
(2009) study performed worse than controls in word and non-
word reading and the Block Design tasks. The poorer correct
rejection performance in the Stop-signal task suggests diffi-
culty in withholding an inaccurate response.
Overall, our data from five different experiments suggests
that DD were more susceptible to the effect of task-irrelevant
information than controls. Similar to our findings, interfer-
ence suppression weakness was reported in DD children/
adults and in children with weak mathematical skills in the
Wisconsin Card Sorting Task (Bull et al., 1999) and arithmetic
tasks (Pasolunghi et al., 1999; Passolunghi and Siegel, 2004; De
Visscher and Noe
¨
l, 2013). In addition, tasks with interference
suppression demands have been shown to be strongly related
to mathematical development (e.g., Bull and Scerif, 2011; Espy
et al., 2004; Blair and Razza, 2007; Swanson, 2011; Marzocchi
et al., 2002). Inhibition function impairment could lead to
mathematical problems because Numerical Operations
require the temporal and spatial (in imagination) coordination
of several processes and the retrieval of several highly similar
facts e impaired inhibition probably interferes with the or-
ganization of these processes. In addition, various theories of
WM function assume that inhibitory processes and specif-
ically interference suppression play an important role, and/or
are crucial components of the central executive function of
WM (e.g., Hasher and Zacks, 1988; May et al., 1999; Miyake
et al., 2000; Caretti et al., 2004). Hence, we suggest that the
WM and inhibition impairments detected in our study may be
related to each other and the inhibition impairment may have
led to impaired visuo-spatial WM performance. Were this
hypothesis true, DD could be attributed to the specific
impairment of visuo-spatial STM and to the specific impair-
ment of the inhibitory processes crucial to visuo-spatial cen-
tral executive WM function. In fact, the IPS has been
demonstrated to be involved in interference resolution
(Mecklinger et al., 2003; Cieslik et al., 2011). Hence, DD versus
control differences in at least some functional and structural
MRI IPS data may be related to differences in interference
resolution rather than to MR/ANS function.
Our results seem to fit into a wider framework of data re-
ported with regard to learning disabilities. Several studies
found that children with poor reading comprehension show
deficits in interference suppression in verbal WM tasks (De
Beni et al., 1998; Pimperton and Nation, 2010) but not in
visuo-spatial WM tasks (Pimperton and Nation, 2010). Inter-
ference suppression deficits in verbal WM tasks were also
reported in children with ADHD (Cornoldi et al., 2001;
Palladino, 2006; Palladino and Ferrari, 2013). Importantly,
while all the above studies found decreased verbal WM per-
formance in children with dyslexia and ADHD, our study did
not find any general verbal WM difference between DD and
control children. In contrast, here we found a robust visuo-
spatial WM difference. On the other hand, Pasolunghi et al.
(1999) and Passolunghi and Siegel (2004) did report both ver-
bal WM differences and interference suppression difficulties
in DD children. Both of these studies matched DD and control
children in verbal IQ and Passolunghi and Siegel (2004) also
matched reading performance, and the studies used DD
diagnosis cutoff scores at the 20th and 30th percentiles,
respectively. Hence, diagnosis was more permissive than in
our study and a further difference seems to be that diagnosis
relied on a standardized test in which eight out of 12 problems
were word problems (e.g., ‘On Pascoli Street there are 45
shops. 3/5 of them sell clothes. How many clothes shops are
there in Pascoli Street?’; Pasolunghi et al., 1999; p. 781). In
contrast, our study relied on two tests with overwhelmingly
Arabic digit computational problems. Hence, speculatively,
perhaps the content of the tests used to identify the DD chil-
dren affected results. In fact, Passolunghi and Siegel (2004)
report a .38SD reading score difference between their DD
and control populations. Assuming standard deviation
(SD) ¼ 15 this is equivalent to 5.7 score difference between
groups. As shown in Fig. 1 in our sample differences in reading
scores ranged between .2 and 2 scores, so DD and control
populations were slightly better matched which may affect
verbal WM results. Further, Pasolunghi et al. (1999) and
Passolunghi and Siegel (2004) did not measure visual STM and
WM function. Overall, this comparison points to the impor-
tance of matching diagnostic instruments across studies and
testing both verbal and visual WM. In addition, future studies
should explore the exact nature of potential interference
suppression deficits in DD in visuo-spatial STM/WM tasks and
investigate whether interference suppression deficits in
different learning disabilities are the consequence of similar
impaired mechanisms manifesting in different modalities.
3.3. Preserved but slow spatial processing and slow
trail-making speed in DD may be secondary to WM/
inhibition impairment
Accuracy equaled in DD and controls in the spatial symmetry
task and in the mental rotation task. We detected slower so-
lution times in DD than in controls on the trail-making A task,
which confirms some previous findings (McLean and Hitch,
1999; Solte
´
sz et al., 2007; Andersson, 2010), as well as on the
mental rotation task. The accurate performance on the sym-
metry and rotation tasks suggests that spatial skills were
available to DD albeit at a slower speed than to controls.
Hence, we conclude that slower rotation speed and the slow
trail-making performance (this task is usually thought to be
very dependent on WM central executive function) relate to
WM and inhibition function impairment in DD.
3.4. None of our findings support the MR theory
The lack of positive findings with regard to the MR theory of
DD is in sharp contrast with robust visuo-spatial STM/WM
and inhibition-related findings. We have a number of reasons
to assume that the lack of group measure interactions in MR
measures was not due to lack of power. First, our study clearly
had enough power to detect all expected experimental effects
in all nine experiments. Most importantly, we detected all ex-
pected ratio and congruency effects in the symbolic and non-
symbolic magnitude discrimination tasks and detected other
group measure interactions at good significance levels.
cortex xxx (2013) 1e1510
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Second, in order to achieve high intra-individual p ower
our study deliberately had a large number of trials in each
experim ent. There were 40 trials for e ach level of symbolic
numerical distance in the symbolic discrimination t ask (80
stimuli all together) and 40 trials for each level of ratio in the
non-symbolic discrimination task (120 stimuli all together).
That is, across the study we collected 12 40 ¼ 480 trials for
each ratio level in the DD group. In c omparison to studies
with positive MR results our study had 1.66e4 times as
many trial s per ratio level than other studies: Price et al.
(2007) p resented 12 trials per ratio level ( 24 stimuli, eight
DD children, i.e., 96 trials for each ratio across th e whole
study), Mazzocco et al. (2011) used 20 trials per ratio level (80
stimuli, 10 DD children, i.e., 200 trials per ratio level across
the whole study), Mussolin et al. (2010a, 2010b) u sed 24 trials
per r atio level (96 stimuli for each presentation format, 15
DD children, 360 trials per ratio level for each presentation
format across the whole study), Piazza et al. (2010) used 10
trials per ratio level (80 stimuli, 23 DD children i ncluding 12
dyslexic children, i.e., 230 trials per ratio level across t he
study). In addition our study had 12 DD children which is
more than the number of DD children in two out of the
above four studies. Even when factoring in the larger
number of DD children in the two remaining studies
(Mussolin et al., 2010a, 2010b; Piazza et al., 2010) our study
collected 1.33e2.08 times more tr ials per ratio level for each
presentation format than other studies. This is advanta-
geous because the larger number of trials effectively sup-
presses the amount of noise inherent to the data which
increases power.
Third, the impaired MR theory predicts that ratio effects
in non-symboli c number discrimination will differ in DD
relative to controls (Piazza et al., 2010; Mazzocco et al., 2011;
Price et al., 2007). In our study the between group difference
in the mean ratio effect was .1%. In a similar non-symbolic
number discrimination task Price et al. (2007) observed a
2.5% difference between groups in the ratio effect with the
DD group showing a larger effect than controls because DD
children were less accurate than controls at close ratios
(close vs far ratio differen ce in controls: 3.87%, DD: 6.37%;
accuracy for close vs far ratios in controls: 95.75% vs. 99.62%.
In DD: 92.75% vs. 99.12%). In that study the standard dev ia-
tion of the error data was about 1.65% and the group dif-
ference in the ratio effect was about 1.51SD. For the 12
subjects in our study this gives a Power estimate of Power
> .99 . I n our study comparable accuracy values were found
(both cont rols and DD: 93.7e97.7%) with a ratio effect of
comparable effect size (1.7%) with larger SD (2.97%). How-
ever, considering the similar size of the overall accuracy and
distance effects in relation to Pr ice et al. (2007), in our study
the .1% between gr oup ratio ef fect difference we found can
be considered practically ze ro. This is confirmed by the fact
that the bootstrap 95% confidence int erval of the non-
symbolic comparison ratio effect was clearly focused on
zero (see Fig. 3.), the very small confidence intervals wer e
approximately symmetric around zero and SEs were very
small, about .4%. All the above su ggests tha t there was not
much variability or directional bias in our data and that
there was not even an indication of a difference in the ratio
effect between the groups.
Fourth, regarding the symbo lic magnitude comparison
task the mean of the between group difference was 2% and
the SD of the data was about 5.71%. The DD group showed a
smaller absolute value distance effect than the control group
(3.26% vs 5.24%). Crucially, DD actually showed slightly better
performance on th e task t han the co ntrols while RTs were
practically identical. This makes it unlikely that DD had
impaired access to MRs in this task. Nevertheless, in the data
from the Arabic number comparison ta sk of Mussolin et al.
(2010a, 2010b) the overall mean distance effect (calculated
for all four ratios used; see ibid . Table 2) was actually exactly
the same in the control and DD g roups (2.76%) and the dif-
ference between the most extreme distance levels was also
the same in both groups (8.3%). The DD and the control group
showed a difference because the c losest levels of distance
differed more in the DD than in the contr ol group. However,
this means that the DD group was .6% less accurate at the
closest level of distance while it was actually 1.1% more ac-
curate than the controls at the second closest leve l of d is-
tance. The difference between the groups was 1.7% (controls:
2.7%; DD: 4.4%) and the SD of the data was about 1.75% (this is
not very clear as the table reports exactly the same standard
deviation values for both groups which is probably a
mistake). Hence, the group difference was .97SD. For our 12
subject s such an effect size would give Power > .99. ( It is to
note that crucial analysis results in Mussolin et al. (2010)
relied on trials collected from 5 different stimulus formats
(5 24 ¼ 120 trials for each level of distance) rather t han from
an individual stimulus format.) However, we only measured
a 2% (.33SD) between group difference in the distance effect.
In addi tion, as noted abo ve, the som ewhat hi gher accura cy in
the DD than in the contro l group also makes it unlikely th at
our DD group had problems with accessing the magnitude of
single Arabic di gits.
Fifth, it is important to emphasize the difference between
the robustness (large effect size) of WM and inhibition results
in contrast to MR-related results. Our data definitely did not
give any indication of a non-symbolic ratio effect discrepancy
between groups and while it is naturally hard to exclude that
perhaps a significant symbolic distance effect difference could
have emerged by using more trials from more participants,
WM and inhibition-related findings appeared clearly. In
contrast, any potential MR-related effects seem harder to
detect and fragile relative to the variability in data. The
robustness of WM/inhibition results is an extremely impor-
tant factor to consider when it comes to testing theories and
diagnosing children at the individual level and remediation of
DD.
Sixth, our study joins several studies with negative results
with regard to the MR theory of DD. To date eight studies could
not detect any distance/ratio effect discrepancy between DD
and controls (Landerl et al., 2004; Kucian et al., 2006, 2011;
Rousselle and Noe
¨
l, 2007; Solte
´
sz et al., 2007; Landerl and
Kolle, 2009; Mussolin et al., 2010b; Kovas et al., 2009) while
four studies reported such a difference (Price et al., 2007;
Mussolin et al., 2010a; Piazza et al., 2010; Mazzocco et al.,
2011). However, as noted before, none of these four studies
used non-numerical control tasks and their crucial non-
symbolic number comparison diagnostic task is inevitably
confounded by visual stimulus parameters (Gebuis and
cortex xxx (2013) 1e15 11
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Reynvoet, 2011, 2012) which particularly seriously affects the
computation of ‘w’, a proposed measure of the MR (Sz
}
ucs et al.
2013). It is also important to note that sometimes simply
worse accuracy on MR tasks in DD than controls is considered
evidence for impaired MR in DD. However, obviously, worse
accuracy (especially when there is no control task) can appear
for various reasons (see e.g. Sz
}
ucs et al., 2013). Hence,
decreased accuracy cannot be considered evidence for specific
MR impairment. Overall, we conclude that DD and control
groups were practically indistinguishable on measures of the
MR while other tasks strongly and clearly discriminated these
groups.
3.5. Subitizing and counting
The only piece of data from our study which could perhaps
call for number-specific explanations is that the counting-
range slope (4e6 number range) in accuracy in the subitizing
task was less steep in DD than in controls. However, first, this
finding appeared because DD children were more accurate for
number 6 than controls. Second, there were no effects in RT
which is usually considered the main measure in subitizing
tasks. Third, when counting-range slope accuracy and the
Inhibition measure were entered into a regression together,
counting-range slope was a non-significant predictor of
mathematical performance. When only WM and Inhibition
were entered into regression, the model fit remained practi-
cally unchanged. WM and Inhibition were significant pre-
dictors even when entered with verbal and non-verbal IQ
measures and with processing speed. WM and Inhibition
scores were not correlated which suggests their indepen-
dence. In contrast, counting-range slope correlated with In-
hibition and remained a non-significant predictor when
inhibition was included in the regression. Hence, as no other
MR-related measure discriminated between groups, counting-
range slope findings seem to be related to inhibition ability
and not to MR function.
3.6. Diagnosis issues
It is important to point out that there is substantial variation
across studies in defining children with DD due to the fact that
there is no agreed definition of DD. The range of cutoffs used
to define DD in demographic studies ranges from performance
below the 3rd percentile to performance below the 25th
percentile (2SDe.68SD below the mean; for review see Devine
et al., 2013). Here we used very stringent criteria to assure that
children only had mathematical difficulties. We screened 1004
children and diagnosed DD if performance on two standard-
ized mathematical measures was worse than 1SD while there
was no ADHD and dyslexia, verbal IQ/reading was normal on
four different tests and non-verbal IQ was normal on two
tests. For example, Price et al. (2007) screened 55 children and
WISC block-design performance differed by more than 1SD
between DD and controls. In Piazza et al. (2010) about half the
DD group was diagnosed with dyslexia. Mussolin et al. (2010a)
screened 187 children and diagnosed DD if performance was
worse than 1SD (15th percentile) on a multiplication test.
However, multiplication relies heavily on verbal memory
(Ashcraft, 1982). Mazzocco et al. (2011) screened 161 children
and diagnosed 10 children below 1.3SD (10th percentile) with
DD and children below .65SD (25th percentile) as low maths
achievers without using any other criteria. Various tests were
used as covariates in analyses. However, the tests were
recorded in various years during a 7-year long period and as
noted above, ANCOVAs cannot ‘correct for’ major differences
along independent variables (Miller and Chapman, 2001;
Porter and Raudenbush, 1987). Obviously, definition and
measurement discrepancies can contribute to disagreeing
findings across studies.
3.7. Conclusion
In summary, there is evidence that IPS morphology and
perhaps function differ between DD and control partici-
pants (Isaacsetal.,2001;Rotzeretal.,2008;Priceetal.,
2007; Mussolin et al., 2010b). However, there is insuffi-
cient evidence for the argument that IPS dysfunction in DD
can be linked to MR dysfunction: (1) Only one out of six
fMRI studies found supporting behavioral data (Price et al.,
2007). (2) The frequently used dot comparison task is seri-
ously compromised by non-numerical confounds (Gebuis
and Reynvoet, 2011; 2012; Sz
}
ucs et al., 2013). (3) Several
behavioral and fMRI DD studies focusing on t he MR theory
of DD do not have non-numerical control conditions. (4)
Adding to several negative findings (see above) our study
used several measures of the MR but could not detect any
clear MR impairment effects in DD. The fall ibility of evi-
dence for the MR theory of DD is in sharp contrast with the
robust nature of the visuo-spatial STM/WM difference
between DD and control groups in our data which is in
agreement with various studies. Verbal WM/STM is
probably only impaired if DD is accompanied by reading/
verbal difficulties (e.g., with dyslexia).
We conclude that the MR theory of DD which is currently
dominant in neuroscience research is insufficient to explain
pure DD. Hence, there is a need for a pa radigm shift in DD
research; neuro-imaging studies s hould now take alterna-
tive theories of DD, defined by extensive behavioral
research, seriously. Crucially, rather than aiming at recon-
firming a single theory of DD, studies should test theories
against each other. Our data suggests that the most robust
dysfunction in DD is that of visuo-spatial STM and WM with
the impairment of inhibitory function (interference sup-
pre ssion). Both of th ese functions have been linked to the
IPS. Hence, we suggest that IPS dysfunction in DD is prob-
ably related to WM and inhibition impairment. We hy-
pothesize that the WM and inhibition impairments are
related to each other and t he inhibition function impair-
ment reflects the disruption of a crucial processes of central
executive memory function. That is, pure DD could be
characterized by the specific impairment of visuo-spatial
STM and by the specific impairment of the inhibitory pro-
cesses crucial to visuo-spatial central executive memory
function r esulting in poor WM. Future imaging studies of DD
should take these cognitiv e functions into account. Inter-
vention studies could explore whether the above functions
can be improved in DD. Spati al processing seems intact in
DD albeit slowly accessible which is probably a consequenc e
of memory/inhibition impairment.
cortex xxx (2013) 1e1512
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Acknowledgments
This work was supported by Medical Research Council grant
G90951 (D.S.). D.S., F.S., A.D. and A.N. designed the study. F.G.
contributed to design. F.S. programmed experimental para-
digms. A.D., A.N. and F.G. collected the data. F.S. prepared the
data for analysis. D.S. wrote analysis programmes, analyzed
the data and wrote the manuscript.
Supplementary data
Supplementary data related to this article can be found at
http://dx.doi.org/10.1016/j.cortex.2013.06.007.
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