REVIEW
Developmental dyscalculia
Karin Kucian & Michael von Aster
Received: 18 June 2014 /Revised: 5 November 2014 /Accepted: 9 November 2014 /Published online: 23 December 2014
# Springer-Verlag Berlin Heidelberg 2014
Abstract Numerical skills are essential in our everyday life,
and impairments in the development of number processing
and calculation have a negative impact on schooling and
professional careers. Approximately 3 to 6 % of children are
affected from specific disorders of numerical understanding
(developmental dyscalculia (DD)). Impaired development of
number processing skills in these children is characterized by
problems in various aspects of numeracy as well as alterations
of brain activation and brain structure. Moreover, DD is
assumed to be a very heterogeneous disorder putting special
challenges to define homogeneous diagnostic criteria. Finally,
interdisciplinary perspectives from psychology, neuroscience
and education can contribute to the design for interventions,
and although results are still sparse, they are promising and
have shown positive effects on behaviour as well as brain
function.
Conclusion: In the current review, we are going to give an
overview about typical and atypical development of numerical
abilities at the behavioural and neuronal level. Furthermore,
current status and obstacles in the definition and diagnostics of
DD are discussed, and finally, relevant points that should be
considered to make an intervention as successful as possible
are summarized.
Keywords Developmentaldyscalculia .Numberprocessing .
Calculation . Learning disability . Therapy . Learning
Abbreviations
DD Developmental dyscalculia
fMRI Functional magnetic resonance imaging
IPS Intraparietal sulcus
MRI Magnetic resonance imaging
Introduction
There is a widespread misunderstanding of the importance of
numerical understanding in everyday life and a lack of appreciation
of how important math learning is for young children.
Numerical abilities are essential in daily routine, and they are
becoming even more crucial with the increasing role of technology
in contemporary society. Low numeracy skills have a
negative impact on the employment prospects and mental and
physical health of individuals and on the economic status of
countries [37, 72]. Importantly, profound difficulties with
numeracy are very common with a prevalence rate between
3 and 6 % [52, 89]. Despite the relatively high occurrence of
specific numerical learning disorders, only few research projects
focus on this clearly high-priority area. It is important to
gain a clear understanding about typical as well as atypical
development of numerical competencies on behavioural and
neuronal levels to foster the acceptance as a disorder and raise
public awareness for the need to provide targeted educational
and therapeutic support tailored to affected children.
Communicated by: Peter de Winter.
Revisions received: 29 October 2014/5 November 2014
K. Kucian (*) :M. von Aster
Center for MR-Research, University Children’s Hospital Zurich,
Steinwiesstrasse 75, 8032 Zurich, Switzerland
e-mail: karin.kucian@kispi.uzh.ch
M. von Aster
e-mail: m.aster@drk-kliniken-westend.de
K. Kucian :M. von Aster
Children’s Research Center, University Children’s Hospital Zurich,
Zurich, Switzerland
M. von Aster
Department of Child and Adolescent Psychiatry, DRK-Hospital
Westend Berlin and Ernst von Bergmann Hospital Potsdam,
Spandauer Damm 130, 14050 Berlin, Germany
Eur J Pediatr (2015) 174:1–13
DOI 10.1007/s00431-014-2455-7
In the following sections, we are going to review the typical
developmental trajectories of cognitive number representations
and contrast these behavioural characteristics to atypical
development associated with developmental dyscalculia
(DD). In this regard, we are outlining general and critical
points that have to be considered in the definition and diagnosis
of DD. Since recent findings suggest that DD is a brainbased
disorder [e.g. see meta-analysis by Kaufmann [53]],
neuronal representations of numbers in the adult and developing
brain as well as corresponding neuronal underpinnings
of DD are reviewed. Finally, based on current behavioural and
neuronal knowledge about DD, a possible intervention method
and future outlook are presented.
Ontogenesis and phylogeny of number representations
In our daily life, we are confronted with a huge amount of
numbers in various formats—Arabic digits (3), number words
(three), Roman numbers (III), time (3 pm), magnitudes (•••),
finger signs, words with numeric meaning (triplet, trio), or
with ordinal (third) or temporal (e.g. waltz time) information.
The expression ‘number representation’ describes the mental
image or construct of such numbers or magnitudes [36]. A
challenge during development is to learn to differentiate as
well as link these different number representations [13].
Different findings point to the existence of an innate basic
number sense that enables even newborns to compare and
discriminate between concrete magnitudes approximately
[30]. Moreover, a recent study could even demonstrate that
foetuses in the last trimester are attentive to numerosity [84].
This basic number system seems to be evolutionary and not
only present in humans, but also in many animal species such
as bees, fish, birds, monkeys, salamander to mention some
among many others [1, 35, 38, 40, 56, 81]. Therefore, these
basic numerical abilities seem not dependent on enculturation
or language as is demonstrated in a study with an indigene
tribe of the Amazon has shown as well. This Brazilian tribe
lives isolated in the jungle and has developed number words
only for collections of one, two, three, four, and five items—
everything beyond is simply labelled as many. Behavioural
tests have shown that these people are able to perform approximate
numerical tasks with concrete magnitudes comparable
to us including magnitudes larger than five [73]. Finally,
another very basic numeric capacity called subitizing is the
simultaneous perception of small quantities up to five without
counting [94]. However, recent findings have shown that the
subitizing range in preschool children was only up to three and
that the range can be larger by pattern recognition when dots
are canonically presented [43, 55]. These numeric precursor
abilities serve as foundation for the development of more
sophisticated and enculturated numerical concepts (please
see also Table 1).
Furthermore, numerical skills and representations are assumed
to be acquired in a hierarchical and dynamic fashion
[102]. Based on the representation of magnitudes, a verbal
representation (number words) during preschool is followed
by the establishment of a symbolic representation (Arabic
digits) during schooling. Finally, spatial number representation
is assumed to develop successively which is thought to
resemble a number line in our mind and demonstrates that
quantity is represented in a spatial nature [24]. In addition,
behavioural evidence suggests that alike many other nonnumerical
(cognitive and non-cognitive) factors influence
the maturation of numerical thinking such as working memory,
language, spatial skills or attention which all are tightly
linked to numerical development [for review, please see
Kaufmann [49]]. A recent study has shown for instance that
the level of executive functions and spatial skills in 3-year-old
children predicts 70 % of the variance in later mathematics
performance [100]. Moreover, Pina et al. [74] highlighted that
different components of working memory (spatial and verbal)
relate to different mathematical areas and that non-verbal
intelligence and language may have different relationships
depending on the mathematical area assessed.
The representation of the ‘mental number line’ is assumed
to be influenced by cultural factors, like counting routine [34],
the numerical system [85, 86], or reading direction [24, 106],
as well as spatial routine, sensory and motor experiences [33,
66] and neuronal processes [34, 82]. Recent findings have
even demonstrated that infants aged 7 months, who are far
from gaining symbolic knowledge and mathematics education,
show a preference for increasing magnitude displayed in
a left-to-right spatial orientation [20]. Accordingly, most people
in western cultures show a left to right oriented mental
number line with a logarithmic scaling [23, 24]. However,
with increased development and expertise in a certain number
range, this scaling follows rather a linear function reflecting a
more mature representation of numbers [58, 91].
During the early stages of the development of arithmetic
competences, children solve arithmetic tasks by inefficient
counting strategies also often reflected by the use of finger
counting. Later on, after practicing and automatization, results
can rather be retrieved from memory [92].
Taken together, present findings clearly show that numerical
representations in children are modified and formed according
to experience, increasing expertise and maturation of
domain-specific and domain-general neuro-cognitive processes
(please see also Table 1).
Definition of developmental dyscalculia
DD is a specific learning disability affecting the development
of arithmetical skills with estimated prevalence rates between
3 and 6 % [52, 89] (see also Table 1). Recent studies claim that
2 Eur J Pediatr (2015) 174:1–13
more girls than boys are affected [32], but literature is still
controversial, and also, opposite findings have been reported
[27, 76]. Independent of gender, DD seems to be an enduring
learning disorder persisting into adulthood [54, 88] with
Table 1 Summary of typical and atypical development of numerical skills, neuronal correlates of typical and atypical number processing, as well as
relevant points in the definition and etiology, the diagnosis, and intervention of developmental dyscalculia
Development of number processing
Typical Atypical (Dyscalculia)
Behaviour Precursor
skills
▪ Innate number sense (discrimination between
small quantities with larger distances)
▪ Impaired innate number sense
▪ Increased quantity discrimination skills
(larger quantities, smaller distances)
▪ Problems in quantity estimation
▪ Subitizing ▪ Reduced subitizing range
Number
skills
▪ Mapping of different number representations
(concrete quantity, number words, Arabic
digits etc.)
▪ Inaccurate mapping and transfer of different number
representations (more problems with symbolic representations)
▪ Counting ▪ Counting difficulties, like backward counting
▪ Place-value system ▪ No understanding of the place-value system
▪ Mental number line representation
(logarithmic -> linear function)
▪ Impaired development of or access to the mental number line
Calculation
skills
▪ Arithmetic by counting ▪ Stick to counting strategies by the usage of their fingers
▪ Arithmetic by retrieval ▪ Often very limited retrieval
▪ Arithmetic by decomposition
(e.g. 6*8 = 5*8 = 40+8 = 48)
▪ Lack of understanding how to decompose difficult problems into
easer ones
▪ Acquisition of different calculation procedures
and concepts
▪ No understanding of calculation procedures and concepts
Brain ▪ Complex fronto-parietal neuronal network:
Parietal lobes (IPS), prefrontal cortices, regions associated with
the dorsal and ventral visual pathways, sub-cortical areas
and cerebellum.
▪ Maturation:
Increased and more focal recruitment of brain areas
associated with automated number processing, like the
IPS or with numerical fact retrieval on left hemispheric
perisylvian brain areas and the contribution of
hippocampal-prefrontal circuits.
Decreased reliance on supporting brain functions mainly
located in the frontal lobes.
▪ Aberrant neuronal network:
Differences in brain activation, grey and white matter volume, and
fibre connections.
Activation pattern less precise and main functional and structural
deficits in core regions for number processing (parietal regions).
Other cortical and subcortical regions that contribute to numerical
cognition can be affected.
Stronger recruitment of supporting areas associated with working
memory, attention, monitoring, updating or finger representation
might reflect compensatory mechanisms.
Developmental dyscalculia
Definition &
Etiology
▪ Heterogeneous learning impairment affecting numerical and/or arithmetic functioning at behavioural, psychological and neuronal
levels
▪ Persistent into adulthood
▪ Hereditary and environmental components are hypothesised as possible causes
▪ Prevalence: 3–6 %
▪ Gender distribution: females > males, but findings are still inconsistent
▪ Comorbidities: very common, like attentional problems (ADHD), dyslexia, anxiety, visuospatial impairments, spatial working
memory problems etc.
Diagnosis ▪ Early identification, diagnosis, and treatment is very important
▪ Detailed description of strengths and weaknesses by multi-dimensional assessments tracking:
◦ Numerical and arithmetical processes
◦ Domain-general abilities
◦ Neurological, sensory, and motor processes
◦ Socio-emotional and social functioning
◦ Medical, educational, and family history
▪ Difficulties must persist over longer time periods
Intervention ▪ Adapted to individual profile of performance levels and difficulties
▪ One-to-one training
▪ Structured and hierarchically built
▪ Basic non-curricular and curricular numerical topics.
▪ Many repetitions
▪ Motivation is stimulated by reward and reduction of math anxieties
Eur J Pediatr (2015) 174:1–13 3
hereditary as well as environmental components being
discussed (see also Fig. 1) [2, 8, 87]. A recent first largescale
genetically sensitive investigation of number sense has
even revealed that basic numerical understanding is only
modestly heritable (32 %), with individual differences being
rather explained by environmental influences (68 %) [97].
DD is a heterogeneous disorder resulting from individual
deficits in numerical or arithmetic functioning at behavioural,
cognitive/neuropsychological and neuronal levels [50, 80].
However, it has to be considered that arithmetic difficulties
can reflect individual differences in both numerical and nonnumerical
functions (see Fig. 1). Children with DD have
problems in the mastery of a wide range of numerical understanding
such as counting skills, magnitude processing, arithmetic,
transcoding between number words, digits and quantities,
the spatial number representation, or more domain
general skills like working memory or attentional processes
(see also Fig. 1). In particular, visuo-spatial
working memory seems to be impaired in children with
DD [5, 78, 96] (Vignette 1).
Generally, multiple problems are most common and pure
DD apply to a minority of cases only. Rubinsten and Henik
[80] propose different frameworks for the origin of DD or
mild learning disabilities and their cognitive deficits. First, a
unique core deficit in processing numerosities is assumed
because of a unique pathophyhsiology. Butterworth et al.
[14] similarly emphasize a core deficit in understanding sets
and their numerosities, which the authors claim to be fundamental
to all aspects of elementary school mathematics. However,
present research challenges a unique cognitive deficit as
possible cause of DD. For instance, Rouselle and Noel [79]
found that children with mathematical disabilities were only
impaired when comparing Arabic digits, but not when comparing
non-symbolic magnitudes. Also, in typical development,
children’s efficiency to process symbols seems to be
important for the development of their arithmetic fluency
above and beyond the influence of non-symbolic number
processing skills [9]. Finally, a single restricted biological
deficit as origin of DD is not in line with current neurobiological
findings which indicate functional and structural abnormalities
in a widespread network across cortical and subcortical
brain regions [31, 60]. Therefore, other frameworks
taking the heterogeneous behavioural deficits, the common
occurrence of co-morbidities and neuronal underpinnings of
Fig. 1 Potential factors and different levels on which developmental dyscalculia can manifest itself (Fig. 1 is adapted from [52])
4 Eur J Pediatr (2015) 174:1–13
DD into account are rather plausible. As alternative origin,
Rubinsten and Henik (2009) suggest that multiple cognitive
deficits in DD might be explained by multiple brain dysfunctions
or that dysfunction in a central brain area could result in
multiple cognitive or behavioural manifestations (Vignette 2).
As described, arithmetical difficulties are often associated
with other learning problems (dyslexia, ADHD). Probably as a
consequence of these multiple learning difficulties found in
DD, many children develop additional psychiatric disorders,
like anxieties, depression or aggressive behaviour. It has to be
taken into account that especially such secondary symptoms of
DD can lead to general refusal and impairments in other school
topics (see Kohn, Richtmann et al. 2013 for a new math anxiety
questionnaire for German speaking children). Finally, also on a
longer timescale, it has been shown that low math abilities have
serious negative impact on professional careers [72].
In sum, number processing comprises a multitude of different
numerical and non-numerical competencies, and children
with DD differ in their individual profiles of strength and
weaknesses in these skills. Therefore, also, the diagnosis of
DD should be based on multi-dimensional assessments tracking
different numerical and arithmetical processes and relevant
domain-general abilities as well as neurological and
socio-emotional functioning.
Diagnosis of developmental dyscalculia
According to Kaufmann and von Aster [52], a detailed diagnostic
evaluation is needed when dyscalculia is suspected in
order to take proper account of the complexity of this learning
disorder and to produce an accurate picture of the affected
child’s particular strengths and weaknesses in the area of
numbers and calculations (see also Table 1). In view of the
multiplicity of the functional components participating in this
learning disorder and the wide range of potential mental and
neuropaediatric comorbidities, it is clear that the diagnostic
evaluation must go beyond the strictly mathematical components
to include a thorough personal, familial and scholastic
developmental history. The evaluation of the child’s general
cognitive development must consider non-numerical cognitive
domains, social factors, emotional well-being and, where
appropriate, findings from neurophysiology (neurology and
neuroimaging). Furthermore, not only at school age, where
most of the children who struggle with numbers and calculation
are diagnosed, but at a much earlier time point, potential
children at risk could be identified. In children of kindergarten
age, specific precursor skills can be identified that have been
found to be reliable predictors of later calculation abilities [7,
95]. For instance, the understanding of quantity, subitizing,
mastery of counting skills or the identification of Arabic
numerals can already be impaired in preschool children. However,
not every child that is conspicuous in any of these
number competencies at preschool age must develop DD. At
Vignette 1 Impaired magnitude representation—dyscalculic boy,
11 years, 5th grade. This boy showed severe problems in the estimation
of concrete quantities. The task was to estimate the number of dots, balls
or cups, respectively, which have been shown only for 2 or 5 s. He was
able to estimate small quantities like nine dots, but failed with larger
quantities. For instance, he estimated 57 balls as 900, or 89 cups as
10,000. This clearly shows that this boy has no understanding of large
quantities and big numbers. They are just a lot, and 900 or 10,000 are a
big number. Illustrated is a copy of the original protocol of the subtest
number 9 ‘quantity estimation’ of the ZAREKI-R [103]
Vignette 2 Impaired transcoding and place value system—dyscalculic
girl, 8 years, 3rd grade. Numbers were dictated verbally and the girl had to
write down the corresponding Arabic digit. For illustration, the verbally
given numbers are listed on the left side to the notes of the girl. She shows
difficulties in the verbal to Arabic transcoding and the understanding of
the numerical place value system. Illustrated is a copy of the original work
sheet of the subtest number 3 ‘transcoding’ of the test battery ZAREKI-R
[103]
Eur J Pediatr (2015) 174:1–13 5
this early age, it has to be taken into account that the scope of
development is wide, and therefore, children differ also in the
mastery of numerical skills. Hence, some children might
simply be slow learners but not necessarily dyscalculic
(Vignette 3).
Current revisions of international classification manuals take
the heterogeneous character of DD into account which is
reflected in changes in conceptualizing learning disorders in
general. In the International Classification of Diseases (ICD
10th revision) of the World Health Organization [71] as well
as the former version of the Diagnostic and Statistical Manual
of Mental Disorders (DSM 4th revision) of the American
Psychiatric Association [3], DD has been defined as a
domain-specific learning disorder that emerges at an early stage
of development and cannot be explained by inappropriate
schooling or deficient learning opportunities. Domain specificity
commonly has been described by a discrepancy between
general intelligence and specific ability. This concept implies a
rather static and endogenous origin of the disorder, which is
challenged by recent research and theory that addresses the
nature of general intelligence (IQ) and the power of environmental
factors, ranging from cultural circumstances (e.g. nature
and extent of schooling, nature of the counting system) to the
effects of pre-/post-natal illness or socio-emotional adversity
[50, 60]. In addition, there is growing empirical evidence that
children with numerical difficulties and average as well as
below average intellectual levels show identical patterns of
difficulties in math learning [29, 67]. It has also to be kept in
mind that numerical understanding, mathematical capabilities
and general cognitive skills measured by intelligence tests are
partly overlapping constructs [for more detailed discussion
please see Ehlert [29]]. Many intelligence test batteries include
sub-tests requiring numerical abilities. Therefore, it is very
difficult, if not impossible, to disentangle cognitive functions
related to number processing and/or testmetric intelligence. In
addition, the constructs of testmetric intelligence used in IQ
tests differ widely according to the extend they require number
knowledge, arithmetic or mathematical reasoning. Finally, children
with low general intelligence scores could hardly meet a
discrepancy criterion because they would need to have extremely
low math performance scores. Accordingly, in the
recently published 5th revision of the DSM [3], the specific
developmental disorders of scholastic skills have been grouped
together as dimensions within a single category, in each of
which there can be a greater or lesser degree of impairment;
the psychometric demonstration of a discrepancy to otherwise
normal intelligence is no longer required. This change has been
introduced to take better account of the heterogeneity of these
disorders with respect to performance profiles and comorbidities,
and thus to improve the clinical utility of the diagnoses.
A recent approach focused on diagnosing DD on the basis
of reliable single case brain imaging methods [28]. Despite the
promising results obtained in this study, the authors emphasized
that brain-based diagnostics are not as good as standard
diagnostic tests that are based on behavioural measures only,
but as the following sections demonstrate, might be a valuable
extension in near future.
Neuronal correlates
Subsequent sections should provide insights into the neuronal
correlates of typical adult number representations and of arithmetical
processing as well as the important differences to
developmental neuronal systems in children. In the last section,
neuronal underpinnings of DD will be pinpointed which
demonstrates that DD is related to atypical development of
numerical representations in the brain.
Numbers in the brain
During the last few years, a clearer picture has emerged of
functional processes in the typical adult brain during number
Vignette 3 Impaired number meaning—girl, 8 years, 2nd grade. ‘At
second grade, teachers told Irma’s parents that their 8-year-old child had
severe dyscalculia, verified by a test of mathematical achievement. Reading
and spelling skills had been excellent. Irma developed a deep aversion
to mathematics and was absolutely convinced that she would never learn
arithmetic. She also showed increasing symptoms of depression and
reluctance to attend school. She was, however, able to compare the
magnitude of different amounts of objects, indicating that core-system
abilities were intact. Irma had above average verbal and nonverbal
intelligence and she engaged in rich and imaginative play. At about 3 to
4 years of age Irma started to invent a fantasy play, a kind of fairy tale in
which the protagonists, unfortunately, had numbers for names. During
therapy sessions she would paint and give detailed biographies of all the
persons in her fantasy world: ‘Three’ was a lovely boy, but sometimes
cheeky to his mother. He had two friends in his neighbourhood, ‘nine’
and ‘twenty-three’, with whom he did a lot of silly tricks, and so on…’
[102]. A closer look at her drawings also revealed that they were full of
numbers. No wonder she got into trouble when she was asked by the
teacher to subtract 3 from 9. Irma’s case demonstrates her extensive use of
numbers for non-numerical assignments and associations were only
detected by careful exploration of her learning history and that a conventional
dyscalculia screener might would have classified her as dyscalculic
by mistake
6 Eur J Pediatr (2015) 174:1–13
processing and calculation by means of contemporary brainimaging
techniques. These findings revealed an entire neuronal
network necessary for number processing. Dehaene et al.
[22, 26] proposed in his ‘triple-code model’ the distinction
between an analogue magnitude representation mediating semantic
number processing (i.e. numerosity) in the parietal
lobe, a verbal-phonological number representation supporting
verbal counting and number fact retrieval in left perisylvian
areas and a visual-Arabic number representation represented
by strings of digits including areas associated with the ventral
visual stream.
Within this distributed network, the intraparietal sulcus
(IPS) has been confirmed as the core locus of numerical
processing. The IPS is activated whenever numerical magnitude
is implicated, even in the absence of numerical task
demands [for review please see references 15, 48, 70]. Furthermore,
recent findings of single-cell recordings in monkeys
revealed neurons in the IPS, which fire predominantly for a
certain numerosity. Moreover, groups of neurons encode either
the number in one or another modality (e.g. auditory
pulses vs visual items), or both. Neurons responding
to numerosity irrespective of the sensory modality support
the notion of a supramodal neuronal code of numerical
quantity [69].
As a nature of numerical cognition, the neuronal network
engages also brain regions associated rather with domaingeneral
capacities like memory, attentional, perceptual, motor
and spatial functions. Particularly, the prefrontal cortex has
been associated with general cognitive processes with
emphasis on its role in monitoring, strategy choice,
planning or manipulating information, as required in
calculation tasks [18].
Taken together, number processing and calculation are a
demanding cognitive ability which is processed by a complex
neuronal network. In addition to the key areas for numerical
cognition located in the parietal lobes, prefrontal cortices,
regions associated with the dorsal and ventral visual pathways,
as well as sub-cortical areas and the cerebellum play a
significant role in numerical tasks [for review see reference 4].
In Fig. 2, top view of the typical fronto-parietal activation
pattern found in adults is illustrated on the right side.
Typical development of number representations in the brain
During development, neurocognitive systems are still immature,
and functional differentiation of specific brain regions
has not yet taken place or is not yet completed in children [10,
44, 49, 59]. With respect to the development of numerical
cognition, it has been argued that with increasing age and
experience, number representations and their interconnections
undergo qualitative changes [59]. It seems plausible that a
core region for number processing might consist of highly
interconnected neurons for different numerical inputs. If so,
activation of one neuronal population could quickly spread to
other populations, leading to cross-notational activation. With
development and higher numerical proficiency, these
cross-notational activations probably increase, reflecting
automatization of numerical processes. Therefore, a direct
comparison of mature and developing brain systems
may not be feasible due to considerable differences
regarding brain structure and function.
Converging evidence suggests an anterior-posterior shift of
brain activity associated with number processing and arithmetic
during development (please see also Table 1). These
changes are thought to reflect decreasing reliance on domaingeneral
supporting (frontal) processing mechanisms and increasing
functional specialization of number-relevant (frontoparietal)
brain regions [49]. Moreover, the effectiveness of
calculation potentially depends on the development of more
sophisticated strategies requiring automated retrieval of arithmetical
facts and processing of quantities as well as Arabic
digits. These maturational changes are reflected in differences
in performance and accompanied brain functions. On the one
hand, an increased and more focal recruitment of brain areas
associated with automated number processing, like the IPS, is
suggested due to more automatic processing of numbers. And,
on the other hand, the stronger reliance on numerical fact
retrieval is assumed to be reflected by an increased activation
of left hemispheric perisylvian brain areas and the contribution
of hippocampal-prefrontal circuits [16, 17].
The brain of dyscalculics
Convergent evidence is growing that DD has a particular
neuronal correlate [for review see book chapter, e.g. by
Kucian 60]. Despite the relatively high prevalence of DD,
only very little imaging studies have addressed the question
of neuronal attributes of this learning deficit. Nevertheless,
existing findings could demonstrate that in children suffering
from DD, the typical developmental trajectories seem to be
Fig. 2 Top view of typical fronto-parietal activation network for number
processing in children (left) and adults (right). An ontogenetic shift from
mainly frontal to task specific regions in the intraparietal sulcus (IPS) can
often be observed. a anterior, p posterior, l left, r right. Illustration is based
on data of Kucian, von Aster, Loenneker, Dietrich, and Martin [63]
Eur J Pediatr (2015) 174:1–13 7
deficient and might be explained by neuronal deficits (please
see also Table 1).
On the one hand, neuronal underpinnings of DD are designated
by aberrant brain activation [75, for review see book
chapter, e.g. references 6, 60, 61] and macro- [77, 83] as well
as micro-structural [57, 83] deficits in core areas for number
processing mainly located in parietal areas [for review see
reference 53]. On the other hand, also compensatory mechanisms
can be observed in children with DD which are
reflected by increased reliance on supporting brain functions
related to domain general cognitive abilities [51, 62].
Aberrant brain functions
Regarding brain activation, there is consistent evidence that
DD is associated with abnormal parietal activity patterns
indicating a deficient neuronal representation of numerosity.
A couple of studies showed a general reduced brain activation
in these core areas for number processing [11, 58, 61, 78].
Additionally, there is growing evidence that impaired numeracy
might also be due to deficits in other brain regions. In
particular, several imaging studies reported reduced brain
activation in distributed areas of the frontal lobe, occipital
areas and deep brain structures [53, 58, 61, 62, 68, 75, 78].
Although reported findings base on different numerical tasks
which range from rather basic number sense [non-symbolic
distance effect 75], to the understanding of ordinality [58], to
symbolic number comparison [68], to arithmetic [multiplication
or approximate addition, 11, 61], or to non-numerical
skills that are processed by overlapping networks [spatial
working memory, 78], coincident reduction of brain activation
in the fronto-parietal network has been reported.
In contrast to reduced brain activation, children with DD
also tend to show increased brain activation in mainly frontal
areas, but also parietal regions, probably as a consequence of
compensatory processes due to their deficits [51, 62, 68].
Again, the development of possibly compensatory mechanisms
seems to be independent of tasks, since both
non-symbolic [51, 62] as well as symbolic number
processing [68] evoked such increases in brain activation
in children with DD.
Numerical tasks with varying difficulty levels induce usually
an increased brain activation for more complex problems
[e.g. 75]. However, brain activation in people with DD seems
not do be modulated in the same way as found in typically
developing subjects. Some studies demonstrated a lack of
modulation of parietal brain activation related to task complexity
with greater activation for more difficult problems in
DD [6, 21, 68, 75, 93]. Furthermore, similar to the lack of
brain modulation in parietal areas, Berteletti et al. [11] described
an absence of brain modulation by problem size in
verbal regions, suggesting that children with math difficulties
do not consistently retrieve solutions verbally even for smaller
problems. In terms of reported difficulties in spatial working
memory in DD, Ashkenazi et al. [5] showed that children with
math difficulties show no modulation brain activity during
arithmetic problem solving related to visuo-spatial working
memory performance. The authors conclude that, unlike typically
developing peers, children with math difficulties do not
use visuo-spatial working memory resources appropriately
during arithmetic problem solving.
Differences in grey and white matter volume
Besides of abnormal brain functions, also morphometric differences
between typically developing children and children
with DD have been reported [57, 77, 83]. Regarding macrostructural
differences, DD is characterized by reduced grey
and white matter volume in areas (parietal areas, in particular
the IPS) that are supposed to play a key role for domainspecific
development of numerosity. In addition, reduced grey
and white matter volume was found in regions (frontal and
subcortical areas) important for the development of domain
general abilities [77, 83].
Atypical neuronal fibre connections
Successful cognitive performance relies on the development
and formation of well-organized networks in the
brain. Fast and adequate connections between different
brain regions are crucial for efficient transfer and adjustment
of information. Micro-structural deficits are related to
aberrant fibre connections between brain regions. To date,
only very little knowledge is available regarding specific
impairments of brain connections as possible neuronal
correlates of DD [57, 83]. However, existing findings
highlighted a connection deficit between parietal and frontal
areas in children with DD. In particular, the superior
longitudinal fasciculus (SLF) seems to be affected in parts
that are adjacent to the IPS [57]. Reasons for microstructural
alterations in the SLF are speculative but might
reflect deficient myelination of fibres rather than axonal
development [for review please see references 64, 104].
Nevertheless, results clearly demonstrate that not only the
IPS but also the intact connection of parietal areas with the
frontal and temporal cortex through the SLF seems to be
essential for number representation [57, 98, 99].
In sum, regarding this wide range of reported abnormalities
in neural networks for number processing and calculation in
DD, it is difficult to draw a clear picture, but upon existing
literature, it can be concluded that the activation pattern of
children with DD is less precise and main functional and
structural deficits are apparent in core regions for number
processing, which mainly comprise parietal regions. However,
also other cortical and subcortical regions that contribute to
numerical cognition can be affected. The stronger recruitment
8 Eur J Pediatr (2015) 174:1–13
of supporting areas associated with working memory, attention,
monitoring, updating or finger representation are supposed
to reflect compensatory mechanisms in dyscalculic
children, but could also reflect deficits in these domaingeneral
skills which might contribute to the development of
DD. As a consequence, observed heterogeneity and comorbidity
in DD are a natural outcome of such a
multicomponent neuronal system responsible for efficient
number processing [31].
Intervention
Ambitious efforts are needed to transform basic knowledge
about neurocognitive development of number processing
into evidence-based practical applications for special
needs education and therapy. So far, only few attempts
have been conducted to incorporate current neuropsychological
knowledge into the development of interventions
for children with math difficulties [for review see
reference 19]. However, as concluded by these authors,
interdisciplinary approaches that include psychology, neuroscience,
and education could contribute to optimal designs
for interventions targeting neurocognitive mechanisms
and can furthermore evaluate the efficacy of such
interventions at the behavioural and brain level. In general,
specific intervention of children or adolescents with
math difficulties can help if the intervention is designed in
an effective way. Ise and Schulte-Körne [42] conducted a
meta-analysis to distinguish which factors make an intervention
effective. The authors distinguish between curricular
trainings, focussing on school material and noncurricular
interventions, which mainly train basic numerical
understanding. Results provide evidence that an intervention
is most successful (see also Table 1):
& In a single training (not group wise or in class)
& When it is adapted to individual performance levels
& When it is structured and hierarchically built
& When it includes basic non-curricular as well as curricular
numerical topics
& When it consists of many repetitions
& When motivation is stimulated by reward and reduction of
math anxieties.
Regarding the two latter factors, computer-based interventions
may provide valuable contributions as they have the
advantage that children generally like to work with the computer
and that this setting is free from any social pressures
fostering math anxieties.
Based on our previous research, we have developed and
evaluated a computer-based training program for children
with DD with the aim of improving number representations
and strengthening the link between numbers and spatial processes
on the internal mental number line [58]. Results have
indicated that children with and without DD improved their
arithmetical abilities and spatial number representations. Additionally,
the training resulted in a modulation of brain functions
(see Fig. 3): After completion of the training, a reduction
in the recruitment of brain regions relevant for numeracy,
including mainly frontal areas, left IPS and the left fusiform
gyrus was observed in both control children and children with
DD. A decrease of brain activation in these regions and
particularly of the frontal lobe is assumed to reflect automatization
of cognitive processes necessary for mathematical reasoning
[105]. Furthermore, a significant increase of activity in
bilateral parietal areas, including the IPS, was found in children
with DD 5 weeks after completion of the training. In
conclusion, domain-specific game-like interventions are associated
with neuroplasticity in functional circuitry that is impaired
in children with DD, and moreover, they can positively
influence brain activation that is atypical into more typical
brain activation.
Based on these positive results, we have further developed
and extended the training. The new version is called
Calcularis and combines the training of basic numerical cognition
with the training of arithmetical abilities in a wider
number range [45–47, 101]. Calcularis includes multiple
games in a hierarchical structure which aim to train different
numerical and arithmetical aspects based on current
neurocognitive models of numerical cognition [22, 25, 59,
65, 102]. Competencies such as subitizing, non-symbolic
numerical magnitude judgements, understanding the number
system, transcoding between number words, magnitudes and
Arabic digits, number line comprehension, expressions like
bigger/smaller, more/less or add/subtract, ordinality
Fig. 3 Modulation of brain activation after computer-based intervention
in children. Blue decrease of brain activation in predominantly supporting
functions in the frontal lobules in children with and without DD. Red
increase of brain activation in task-related regions in the parietal lobules
in children with DD after a 5-week rest period. Illustration is based on
data of Kucian, Grond et al. [58]
Eur J Pediatr (2015) 174:1–13 9
principles, addition, subtraction, multiplication, or division are
trained. Based on so-called dynamic Bayes-nets, Calcularis is
highly able to adapt to the particular needs of each individual
child and in that way offers supporting learning conditions on
the grounds of individual profiles of numerical problem solving
skills. This is also reflected in the outcome after 6 or
12 weeks of training: Children improved their arithmetical,
especially subtraction skills supporting the notion of better
mathematical understanding and a shift to increased and more
sophisticated fact retrieval strategies. In addition, Käser et al.
[45] reported that children performed better in a number line
task after the training. In the number range between 0 and 10,
the deviation from the correct position was reduced by 33 %
after 6 weeks. Children especially also reduced the variance of
the distance between the correct and the indicated location of
the number on the number line (medium-large effect size).
The authors argue that better performance in the number line
task indicates refinement of the internal mental number line
and more accurate access to it and confirms the results of
previous studies [12, 39, 90] which demonstrated significant
correlations between arithmetical learning and the quality of
numerical magnitude representation.
Taken together, intervention programs which are carefully
developed adaptive to individual learning profiles and based
on current knowledge of neuropsychology and brain imaging
findings of numerical cognition provide a helpful tool for
children with math difficulties or DD. They have the power
to improve the understanding of numerical concepts and modulation
of corresponding brain functions. However, these
computer interventions should be seen as supplementation to
conventional learning therapies since individual therapy by
trained dyscalculia therapists seems still more effective [41],
and the focus should mostly lie on optimal teaching and
learning environment for children in general.
Open questions and future directions
As outlined in the present review, our knowledge and understanding
of behavioural and neuronal characteristics, as well
as, the general awareness of DD in our society, have constantly
increased over the last decade. Although we are gaining a
clearer picture of this learning disability, still many questions
are unsolved, reported findings are sometimes contradictory
and generate new questions.
Generally, we think that we have arrived at a stage, where
we should expand our view angles on dyscalculia:
First, the artificial restriction of most research on pure DD
is missing the point that DD children with additional comorbidities
are rather the rule and not the exception. Although it is
essential to keep your examination cohorts as homogenous as
possible to draw clear conclusions, future research should
more focus on children who reflect rather reality, namely
DD children with comorbid disorders. In the same context, it
might push the level of understanding DD to a next step by not
pooling all children with DD together. Instead of, according to
the heterogeneous character of DD, a division into different
sub-groups with more homogenous difficulty profiles would
probably pinpoint underlying causes and behavioural consequences
more precisely.
Second, current findings from neuroscience clearly illustrate
that number processing and calculation is processed by
the integration of different co-activated networks in our brain.
On the one hand, empirically grounded knowledge highlighted
the important role of the IPS. On the other hand, there is a wide
range of reported, but up to now mostly neglected, differences
in brain function or structure in many other cortical and subcortical
brain areas between typical and atypical developmental
trajectories. Therefore, future studies should bear also other
brain regions apart from the parietal lobe in their focus.
Third, when we are thinking of atypical developmental
trajectories present in DD, longitudinal studies would provide
the potential of delineating the course of DD in a more realistic
way than currently conducted cross-sectional studies. Ideally,
such studies would tap the state of behavioural and neuronal
progress at several time points to follow the developmental
function, which might not be linearly, as accurate as possible.
Fourth, it is still unclear to what extent the link between
included brain regions in number processing and calculation
might explain the neuronal deficits in DD. In other words,
future studies should go beyond the mere localization of
abnormalities and offer information on connectivity between
brain networks and how such networks differ between groups
of individuals.
Fifth, present methods allow us to gain insights into DD
from different perspectives, such as brain function, brain
structure, brain metabolism or time course of neuronal processes
by means of electrophysiology. The challenge now will
be to integrate these findings by multi-dimensional approaches
in DD. Such examinations will be very demanding
but provide the chance to define clearer models of underlying
problems.
Sixth, effects of secondary symptoms like specific math
anxiety have been disregarded to a vast degree in behavioural
studies, the investigation of neuronal underpinnings and also
in intervention studies. However, on the grounds that many
children with DD suffer from accompanied anxieties, one
should not neglect the significant negative effects of anxiety
symptoms on learning. Moreover, it is unclear to what extent
observed brain activation increases or decreases, or even brain
structural alterations are due to elevated anxiety levels or
corresponding emotional control mechanisms.
Finally, our future efforts should pursue the goal to improve
the appropriate support of children with DD. This would start
with a uniform definition and diagnosis of DD, followed by a
regulated compensation of any disadvantages and individual
10 Eur J Pediatr (2015) 174:1–13
therapy which takes the heterogeneous profile of each child
into account.
Conclusion
DD is a complex and heterogeneous phenomenon that affects
different components of mental development and requires
interdisciplinary research, remediation and therapy. The
neurocognitive development of number representations and
calculation abilities is interconnected with the development of
other cognitive domains and domain general abilities like
attentional and behavioural control and working memory,
but is also associated and intertwined with individual experiences
in individual environments. Therefore, assessments of
different numerical and non-numerical skills should be completed
including a careful exploration of the individual learning
history.
Conflict of interest We, Karin Kucian and Michael von Aster, certify
that there is no conflict of interest with any financial organization regarding
the material discussed in the manuscript.
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