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European Child & Adolescent Psychiatry9: II/41II/57 Steinkopff Verlag 2000

M. von Aster Developmental cognitive neuropsychology of number processing and calculation:

varieties of developmental dyscalculia

Michael von Aster ( )University of ZurichDepartment of Child andAdolescent PsychiatryBergstrasse 1208708 Maennedorf/Zurich, SwitzerlandE-mail: [email protected]

Abstract This article provides a brief overview about the current state of cog-nitive developmental neuropsychology

of developmental dyscalculia (DD) aswell as results from a Zurich study thatinvestigates different subtypes of DDaccording to various aspects of numeri-cal abilities that are impaired or pre-served. The differential effects of impairments of one particular numeri-cal area on the development of othernumerical abilities are highlighted inthe case of a 17 year old boy with

severe DD and Developmental Gerst-mann Syndrome. A comprehensivemodel of developmental dynamics of

number processing and calculationabilities will be proposed in the lastsection with respect to the developmentof intelligence theory.

Key words Number processing cal-culation development informationprocessing developmental dyscalculia

subtypes theory of intelligence

Introduction

Current neuropsychological models postulate representationaland format-specic modules, located in different areas of theleft and the right cerebral hemisphere, that are relevant foradult cognitive number processing and calculation (15). Themost widely discussed models are those of McCloskey andcoworkers (43), the triple-code model of Dehaene (12) andthe encoding complex theory of Campbell and Clark (9). Thesemodels are based mainly on observations of dissociations anddouble dissociations in various aspects of number processingand calculation in adult subjects with brain trauma. Some par-ticular specications of these models remain controversial.

Taking these models to represent an end stage of childdevelopment, the following paper will focus on the processesof suspected modularization and lateralization of differentaspects of number processing, in other words, on the mean (ormeans) to that end stage and on possible bars and obstacleswhich may be encountered along the path(s). In addition toadults, very different kinds of difficulties in learning to manip-ulate numbers and acquiring arithmetic concepts have been

observed in children. Several authors described subtypes of developmental dyscalculia (DD) and attempted to relate thesespecic cognitive impairments to maturational dysfunctions of distinct modules or particular regions of the developing brain(24, 41, 51, 66).

According to the International Classication of Diseases,10 th revision (ICD-10, 74), and the Diagnostic and StatisticalManual of Mental Disorders, 4 th edition (DSM-IV, 2), the maindening criterion of DD is a signicant discrepancy betweenspecic mathematical abilities and general intelligence. There-fore, by denition, there is no major difference concerninggeneral reasoning abilities between children of the samechronological age both with and without DD. However, com-paring children with DD to children without, mathematicalabilities in children with DD are at a level expected for normalchildren of lower chronological age. This implies that childrenwith DD show a difference between their mental age and theirchronological age concerning number processing and calcula-tion abilities. The central role of psychometric intelligence indening criteria for developmental disorders makes it neces-sary to take a closer look at modern theories of intelligence in


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order to provide a conceptual frame that could account forempirical ndings from different relevant scientic domains.

The current developmental theory of intelligence that refersparticularly to empirical ndings on developmental disorders(i.e., dyslexia) is that by Mike Anderson (1). According to histheory, differences in the general cognitive performance[expressed as the intelligence quotient (IQ)] between childrenof the same chronological age (within-age differences) aredue to differences in basic information processing capacitieswhich rely mainly on the speed of processing and remain sta-ble over the course of development. However, differences incognitive abilities between children of different chronologicalage (between-age differences) are attributed to differences inthe maturation of specific functional brain modules thatemerge at certain stages of development and are either phylo-genetically preformed or ontogenetically acquired throughextensive practice according to contextual needs and environ-mental circumstances.

Andersons theory of the minimal cognitive architecture

provides a useful framework for the synthesis of different,partly contradictory, findings concerning DD and otherlearning difficulties, and to account for the various genetic,gender-related and environmental (cultural, socio-emotional,instructional) inuences that have been found to contribute towithin-age and between-age differences in cognitive abilitiesin children.

The text of this paper is organized into four sections. In therst section, some essential empirical ndings and currentpoints of theoretical debate in cognitive neuropsychology con-cerning modularization and lateralization of number-relatedabilities will be summarized. The major ideas of Andersons

theory (1) and their potential to explain neuropsychologicalndings on DD are also highlighted. In the second and thirdsections, results of a group study and a single case study on DDwill be presented and discussed. A comprehensive develop-mental model for number processing and calculation will beproposed in the last section and, conclusively, discussed withrespect to the previous sections.

Section 1: Developmental dyscalculia:decient modularization and/or lateralization of number processing and calculation abilities?

Dissociations, subtypes, patterns of cognitive functioning,localization

Evidence from functional brain-imaging studies have recentlydemonstrated that different aspects of complex cognitive abil-ities (i.e. language or number processing) are undoubtedlyrepresented in different areas of the human brain. Furthermore,

it could be demonstrated that the cortical representation of acertain ability varies with the time at which this ability isacquired. Kim et al. (36), for example, using functional mag-netic resonance imaging, compared early and late bilingualsubjects during language processing. In subjects who hadacquired the second language after the age of 11 years (latebilinguals), there was a significant distance between theregions of the Broca area activated during language process-ing in the rst and second language whereas, in bilingual sub-

jects who had acquired the second language from the rstyears of life (early bilinguals), identical patterns of activationwithin the same area of the Broca area were demonstrated.

Dehaene et al. (13) studied functional brain activity patternsin adult subjects during different number-related tasks. Theydemonstrate that tasks requiring approximate calculations (i.e.4 + 5 = 8 or 3) showed a pattern of cortical activation with amaximum in the inferior parietal regions of both cerebralhemispheres. In contrast, during tasks that required exact cal-culation (4 + 5 = 9 or 7), maximum activity was located in the

left inferior prefrontal cortex.Dehaene and colleagues (13) argue that these results sup-

port the theory for the existence of at least two of the threemodules that constitute the triple code model (Fig. 1) for num-ber processing (12), namely a module in which numbers arerepresented as number-words (verbal word frame) and amodule in which numbers are represented as an analogue locuson an internal number line (analogue magnitude represen-tation).

Within the triple-code model, abilities such as approxima-tion and number comparison are attributed to the analoguemodule whereas abilities such as counting, the use of count-

ing procedures in addition and subtraction and arithmeticalfact retrieval are attributed to the verbal module. Multi-digitoperations and parity judgements rely, according to this model,on a third module, the visual Arabic number form in whichnumbers are represented by their Arabic code. These threemodules constitute a system for number processing and cal-culation in which the modules are autonomous, interconnectedand activated according to the particular needs of a given task.

Dissociations and double-dissociations of particular num-ber processing and calculation abilities which have been exten-sively described in brain-damaged adults have only beenreported in a few cases of children with DD. For instance, Tem-ple (65) reported a developmental disorder of number pro-cessing termed digit dyslexiain an 11-year-old boy who hadage-appropriate reading and spelling skills but was unable toread Arabic numbers or write dictated Arabic numberscorrectly. Furthermore, he was unable to read and write num-ber-words correctly despite otherwise good reading skills,indicating the existence of independent modules for lexicaland syntactical processing of Arabic as well as written and spo-ken numbers. Other case reports on children who performedwell on reading and writing numbers in different formats but

II/42 European Child & Adolescent Psychiatry, Vol. 9, Suppl. 2 (2000) Steinkopff Verlag 2000


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M. von Aster II/43Varieties of developmental dyscalculia

showed severe difficulties in retrieving arithmetic facts frommemory (number-fact dyscalculia) or in using arithmeticalprocedures despite a good understanding of the concepts of thearithmetical operations (developmental procedural dyscalcu-lia) have been stated to conrm certain elements of the model

of McCloskey and coworkers (Fig. 2), in particular the exis-tence of modules for number-fact and procedural knowledge(66).

Fig. 1 The triple-code model fornumerical cognition (12).

Fig. 2 Model of the number processing system (43).


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Different subtypes of DD have also been related to eitherleft-hemisphere or right-hemisphere dysfunction according tocharacteristic decits in arithmetical abilities and underlyingdecits in phonological or visuo-spatial processing.

Geary (24), for example, related two DD subtypes to dys-functions of the left hemisphere: a memory subtype with alow frequency of arithmetical fact retrieval, often associatedwith reading difficulties, and a procedural subtype with fre-quent use of developmentally immature procedures and adelay in understanding underlying concepts. Right-hemi-sphere dysfunction is, according to Geary, related to thevisuo-spatial subtype, associated with a high frequency of errors that indicate misinterpretations of spatially representednumerical information such as place value.

Rourke and colleagues (8, 47, 53, 55, 64) dened subtypesof learning disorders in children on the basis of patterns of scholastic strengths and weaknesses. They designated twosubtypes of mathematically related decits.

Subtype RS comprises children who manifest poor

performance in mathematics and even poorer performance inreading and spelling. Subtype A, also called the nonverballearning disability (NLD), comprises children who performpoorly in mathematics but perform adequately for theirchronological age in reading and spelling. Both groups mani-fested characteristic discrepancies between verbal IQ and per-formance IQ. RS children showed higher performance IQsthan verbal IQs, whereas the exact opposite pattern was foundfor the NLD children.

Evaluation of neuropsychological functioning has revealedchildren in the NLD group to have decits in visual-spatial andtactile-kinesthetic perception as well as psychomotor decits.

These children make calculation errors more frequently andtheir errors are more diverse than those of children in the RSgroup. In the opinion of the authors, this indicates a lack of understanding of mathematical algorithms due to a distur-bance in the process of non-verbal concept formation. Theacoustic perception and memory, as well as the verbal skills of these children are developmentally age-appropriate.

The authors found the exact opposite neuropsychologicalpattern of strengths and weaknesses in RS children: goodperformance in visual-spatial and tactile-kinesthetic percep-tion and decits in acoustic perception and memory as well asverbal processing.

Rourke and coworkers attributed the decits of the childrenin the RS group to functional disorders of the left hemisphere,and those of the children in the NLD group to disorders of theright cerebral hemisphere. A study by Mattson et al. (42)supports the hypothesis of complementary lateralized dys-functions underlying the different subtypes of arithmeticaldisorders in childhood. The authors investigated the elec-troencephalograph activity (3644 Hz) in children withdyslexia, children with arithmetical disorders and in controlsubjects while they were engaged in solving verbal and non-

verbal cognitive tasks. The children with weaknesses in read-ing skills manifested less left-hemisphere activity whileperforming verbal tasks than either the children with a weak-ness in arithmetic or the control group. In contrast, while per-forming non-verbal tasks, the children with a weakness inarithmetic manifested less right-hemisphere activity thaneither the children with weaknesses in reading skills or thecontrol children.

In an overview of his research ndings, Rourke (52) ascer-tained a correspondence between the pattern of neuropsycho-logical decits in NLD and that described for the so-calleddevelopmental Gerstmann syndrome.

The Gerstmann syndrome, described for the rst time in1930 by Gerstmann, comprises four symptoms that have beenobserved in adult patients with circumscribed left-hemisphericbrain lesions in the area of the gyrus angularis: acalculia, ngeragnosia, right/left disorientation and dysgraphia. The obser-vation of these four symptoms in children without any veri-able previous brain damage prompted Kinsbourne (37) to

propose a developmental Gerstmann syndrome (DGS). Theclinical validity of this syndrome has been critically discussed,especially considering the fact that the functional disordersreferred to can, to a large extent, be explained by the concur-rent aphasic symptoms that often occur (49). A series of inves-tigations and clinical case studies (6, 27, 46, 48, 58, 61, 62, 69)contributes to the recognition that DGS is observed in childrenwith DD, both with and without concurrent disorders in theareas of language and written language, as well as in combi-nation with other neuropsychiatric symptoms. It is, therefore,impossible to singularly relegate DGS to a specic, localizedfunctional brain disorder (7). Should one follow the concept

of complementary lateralized functional brain disorders for thevarious subtypes of arithmetical disorders, it seems logical topostulate a disorder of right hemisphere function for the non-aphasic or non-dyslectic form of DGS and a disorder of lefthemisphere function for Gerstmann symptoms with associateddisorders in language processing.

The problem with those concepts which focus on basic dys-functional processes of either the left or the right hemisphereare mainly that the presence of non-verbal or verbal neuropsy-chological deficits are not likely to predict DD preciselyenough. Share et al. (60) were generally able to replicateRourkes ndings. However, in contrast to Rourke and cowork-ers, they considered gender differences in their data analysisand found that girls with a specic arithmetical disorder donot manifest the expected NLD pattern of visual-spatial weak-nesses, nor did their neuropsychological pattern of functioningdiffer from that of a control group. In our own study (70), only50 % of the children who fullled the criteria for DD showeda neuropsychological pattern that was characteristic for NLD.In contrast, there were children who indeed showed a typicalNLD pattern of neuropsychological functioning but, neverthe-less, perform well in mathematics (28, 39).



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The still somewhat confusing evidence concerning differ-ent kinds of difficulties in numerical processing and cerebralfunctioning might find a clearer conceptualization withinAndersons theory (1) of the minimal cognitive architecture.

Theory of the minimal cognitive architecture

Andersons theory, which aims to account for the various indi-vidual differences in intellectual functioning, postulates twomain developmental routes for the acquisition knowledge withthree processing mechanisms being involved in theseprocesses (Fig. 3).

Through route 1, knowledge arises by thinking, generatedby a basic processing mechanism and two specic proces-sors(SP1 and SP2) that are characterized as knowledge acqui-sition mechanisms utilizing two different modes of processing;SP1 suited to verbal (sequential/propositional) and SP2 suitedto spatial (simultaneous/holistic) material. With reference tocerebral lateralization of cognitive functions, SP1 can beviewed as a left-hemisphere processor and SP2 as a right-hemisphere processor. Concerning route 1, individual differ-ences are explained by differences in the capacity of the basicprocessing mechanism (expressed by general IQ); the fasterknowledge is acquired, the more established during a periodof time.

Possible differences in the latent power of the two specicprocessors are psychometrically expressed by differencesbetween verbal IQ and performance IQ, but only in individu-als with normal or high processing speed, since the speed of

the basic processing mechanism constrains the complexity of specic routines that can be implemented.

Through route 2, knowledge is not provided by thinking butquickly and automatically via modulesthat mature at differ-ent times during development. Their computations are uncon-strained by the speed of the basic processing mechanism.Modular processing is specied by Anderson for cognitivefunctions such as perception of the three-dimensional space,

theory of mind, syntactic parsing and phonological encoding,some of these being evolutionary preformed and geneticallytransmitted (i.e. perception of the three-dimensional space),others with an ontogenetic rather than a phylogenetic basis,established through extensive practice and automatization dur-ing development. A recent example of the latter function maybe seen in the development of a separate cortical representa-tion for a second language within the Brocas area that hasbeen documented in late bilingual subjects (36).

Anderson argued, for example, that the exceptional skillspossessed by subjects with otherwise very low psychometricintelligence (savants) can be explained by the existence of modules that are functionally independent from the basic pro-cessing mechanism. Furthermore, his theory accounts for dif-ferent types of reading failure: in children with a low generalability, a poor reading ability (as well as other scholastic skills)will be predicted by a low IQ. Therefore, reading failure (andpoor scholastic achievement) is caused by a slow basic pro-cessing mechanism. Reading failure may also be caused by apoor specic processor (SP1) being manifested psychometri-cally by a correlation between reading problems and a dis-crepancy between low verbal IQ and high performance IQ, as

Fig. 3 Theory of minimal cogni-tive architecture (1).


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also described by Rourke (51) for subtype RS. However, inchildren with an average IQ and no such discrepancy betweenverbal and performance IQ, dyslexia also occurs and, there-fore, may be caused by a defective module for phonologicalencoding.

As for reading failure, it seems attractive to also concep-tualize different kinds of learning problems in the areas of arithmetic and mathematics within Andersons framework:NLD may, therefore, correspond to a poor specic processorfor visual-spatial material (SP2), whereas children who haveaverage intelligence without a pattern characteristic of NLDand who, nevertheless, have particular problems in manipu-lating numbers and performing calculations, may suffer fromdefective modularization of number processing and calcula-tion abilities.

Section 2: Subtypes of developmental dyscalculia:

results from a Zurich study

The complexity of the modular organization of the cognitivenumber processing and calculation systems, as pictured in theneuropsychological theory, makes it necessary to assess dif-ferent aspects of the numerical domain separately in order toobtain a differential diagnosis that may be useful for planningappropriate intervention strategies. Within a Europeanresearch network, the Neuropsychological Test Battery forNumber Processing and Calculation in Children (NUCALC,72; 17) has been developed. A similar version of this battery(EC 301; (18)) has previously been developed within the same

European research network for application to adult patientswith acalculia.The NUCALC battery contains 12 subtests investigating

the following abilities.

Counting

Counting abilities are considered to be important prerequisitesin children for the acquisition of addition and subtractionskills. In a rst subtest, children have to enumerate differentsets of dots. With reference to the theoretical framework elab-orated by Gelman and Gallistel (26), four different basic abil-ities were evaluated and scored: (i) production of the conven-tional sequence of spoken verbal numbers, (ii) synchronizationbetween manual pointing and oral counting, (iii) the visualmemory dimension of the task with discrimination betweenthe dots counted and those remaining to be counted, and (iv)transcoding the last spoken verbal number to its correspondingArabic digit form (cardinality principle).

The second subtest is counting backwards. This ability isconsidered important for the acquisition of counting-down

strategies in subtraction (23). Whereas forward counting is theprototype of automatic cognitive processes, the production of the backward sequence of number-words is controlled by theworking memory system.

Number transcoding

The subtests reading Arabic numbers aloudand writing dic-tated Arabic numbers represent the most common transcod-ing processes in early school age. Both tasks operate on thesame numerical systems (Arabic digits and spoken verbalnumbers) but exchange their source and object codes. The sixitems of each subtest have been constructed with reference tocorresponding difficulty factors and verbal syntactical struc-tures. Several dissociations in transcoding responses observedmainly in adult brain-damaged patients provided the empiri-cal basis for different representational modules outlined in thepreviously mentioned models of number processing and

calculation.

Magnitude comparison

Two subtests investigate the capacities of subjects to comparethe arithmetical values of two numbers. The numbers have tobe encoded correctly according to their lexical-syntacticalstructure (linguistic and Arabic) and they have to be assessedwith respect to their internal magnitude representation. Dif-ferent aspects of difficulties are systematically implementedboth concerning the semantic representation (small and large

distances) and notational aspects (length of number-words ordigit strings). Impaired performances have been reported ingroup- and single-case studies of adult brain-damaged patientswith aphasia as well as with right-sided lesions (10).

Mental calculation

Six problems for addition and subtraction are presented orallyinvestigating simple number fact and procedural knowledge(i.e. ve plus eightor fourteen minus six). In a second sub-test (story problems), children have to solve addition andsubtraction problems embedded in a situational context. Thesefour tasks vary according to the level of difficulty and the typeof problem (change, combine and compare; see 11).

Placing Arabic numbers on an analogue number line

This subtest, which includes ve items, aims at evaluating thecomprehension of numbers and capacities of subjects fornumerical estimation, with reference to the analogue magni-



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tude representation system of the triple-code model postulatedby Dehaene (12).

Perceptual quantity estimation

Children have to estimate the numerosity of two visuallypresented sets of objects (balls and cups).

Contextual estimation

The semantic values of numbers depends not only on theirarithmetical value but also on the particular context. Childrenhave to judge if, for example, ten leaves on a tree or eightlamps in a room are few, average or a lot. Recently, Kopera-Frye et al. (40) reported that such cognitive estimation testswere the best indicators for calculation and number-process-ing impairments in patients with fetal alcohol syndrome.

A functional analysis of the NUCALC subtests with refer-ence to the different representational modules and the specicinput and output formats as well as the internal translationpathways mentioned in the triple-code model is presented inTable 1.

Normative data have been collected in a representative sam-ple of N = 279 second- to fourth-grade children from regularschools in the canton of Zurich (72). Data from children of the

same school grades have also been collected in France,Argentina, Brazil and Greece applying a French, Spanish, Por-tuguese and Greek version of the battery. Results of cross-cul-tural comparison studies have recently been published (14,73).

Signicant effects on the general test performance in theSwiss-German as well as in other populations have been foundfor age and grade level and also, as could be expected on thebasis of the literature, for gender, favoring boys (especially insubtraction, verbal comparison and story-problem tasks).

In addition, in the Zurich population, foreign native lan-guage affected general test performance: surprisingly, thislinguistic effect was only observed for the Zurich population.In contrast, in a French population, children with a native lan-guage other than French were not at a disadvantage in com-parison to their native French-speaking classmates. An expla-nation may be that the confusing irregularities of the Germannumber-word lexicon with the inversion of tens and ones [23- drei und zwanzig (literally translated: three and twenty)]

will disturb the acquisition of early transcoding routines. Bilin-gual or foreign-language children in German-speaking coun-tries have to transcode number words from both the nativelanguage and the irregular second language (German) into theArabic notational system and vice versa.

The validity of the instrument has been investigated con-cerning the external criterion of school failure in mathematics.The NUCALC clearly distinguished children who were indi-

Table 1 Functional analysis of the NUCALC subtests with reference to the triple-code model (12).


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cated by their teachers to have signicant problems in copingwith school mathematics from those children who had nolearning problems according to teacher ratings. The internalconsistency for all 63 test items of the battery expressed byCronbachs alpha coefficient was .92.

Subjects

The test results of 93 children with poor achievement in schoolmathematics, chosen partly from a normal school population(identied by school performance measures and teacher rat-ings) and partly from a child psychiatric referral population(identied by ICD-10 criteria), have been used to investigatesubgroups of children with different patterns of disablement.The mean age of the children included in this study was 9 years11 month, ranging from 8 to 10 years 11 month. The childrenfrom the clinical sample were all attending regular school cur-riculum and were of normal intelligence. The ratio of the two

sexes was 52 boys/41 girls. However, in the sub-populationextracted from the normal school population, there was a slightover-representation of girls (52.5 %) whereas, in the clinicalsub-population, girls have been underrepresented (29.4 %),which reects the general distribution of the sexes in clinical

child psychiatric referral as well as special-educationalpopulations.

Methods

For statistical analysis the test results were corrected for ageand according to the different lengths of the subscales, the raw-scores were transformed to z-values. Cluster analysis (SPSS6.0) was used to classify subjects according to similarities inperformance patterns.

Results

Applying the NUCALC test prole, a four-cluster solution wasidentied, comprising three groups of children who showed amean sum score in more that one subtest greater than 1.5 stan-dard deviations below that of the normal school population and

one larger group of children with a mean total score signi-cantly lower than that of the normal population, but not morethan one standard deviation in each subtest (Fig. 4).

The latter group has been dened as being a sub-clinicalgroup of poor performers whereas the former three groupshave been dened as representing clinical subtypes of DD.

Some (N = 13) children from the clinical sub-populationwere classied into the poor-performance cluster, indicating afalse-positive diagnosis according to ICD-10 criteria on thebasis of NUCALC test results. In contrast, 13 children from thenon-clinical sub-population were classied into one of thethree clusters with DD.

With reference to the representative sample of normalschool children, the prevalence of children with one of thethree subtypes of DD has been estimated to be 4.7 %, whichcorresponds with ndings from other recent studies (29, 38).The number of children to be classied as poor performers hasbeen estimated as being 16.5 %. More boys than girls were inthe three clinical subtype groups and a ratio of 1:1 in the poor-performance cluster.

The three subtypes of developmental dyscalculia

The rst subtype was labeled verbal subtype. Children whoclustered into this group had greatest difficulty with countingroutines. Consequently, they failed to use counting proceduresto perform mental calculations correctly, especially for sub-traction and also to store arithmetical facts. Most of these chil-dren (9 out of 11) had additional difficulties in reading andspelling and about 50% of these children (6 out of 11) also hadattention decit hyperactivity disorder (ADHD).

The second subtype was labeled Arabic subtype. Childrenwho clustered into this group showed severe difficulties in


Fig. 4 Subtypes of developmental dyscalculia.


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reading Arabic numbers aloud and in writing Arabic numbersto dictation. It seems likely that the difficulties observed in thenumber-comparison subtests (especially when items were pre-sented digitally) are conditioned by these transcoding prob-lems, considering that relating numbers to analogue positionson a scale and estimating quantities is not a problem for thesechildren. Most of the children who clustered in this subtypeonly had learning problems in the area of mathematics. How-ever, for more than 50 % of these children, German was asecond language.

The third cluster has been labeled pervasive subtype.Children who were classied into this group had severe prob-lems in almost all subtests. Nearly all of these children (9 outof 10) also had reading and spelling difficulties, and the major-ity (7 out of 10) manifested behavioral and emotional prob-lems of clinical signicance and, in some cases, also ADHD.

Discussion

The results must be taken very preliminary due to the limitedsample size. However, as insinuated before, it could be sug-gested that, concerning the three modules dened within theframework of the triple-code model, different maturationaldysfunctions are involved in the three subtypes, arising atdifferent stages of development.

Children showing the pervasive subtype pattern may sufferfrom defective maturation of an analogue magnitude module,probably caused by genetic inuences or early brain damage.According to Dehaene et al. (13), the analogue module that issupposed to be inborn represents the basic senseof numeros-

ity and encodes the semantics of numbers. This module devel-ops and elaborates from the preverbal numerical abilities (thathave shown to be present in infants) to an expanding visual-ized inner number line throughout childhood. It could bespeculated that a malfunction of this genetically preformedmodule will impede the development of later mathematicalabilities, in particular the maturation of the verbal and Arabicrepresentational modules and the associated acquisition of number fact and procedural knowledge. The difficulties chil-dren experience in lling number symbols (words and digits)with a proper corresponding visualizable number meaningmay represent the causal mechanism that leads to inabilities,not only to approximate and estimate numerical relationshipsbut also to store number facts and tables and to perform arith-metic procedures.

Children showing the verbal and the Arabic subtype patterndid not have difficulties that can be understood as a defectivebasic sense for numbers and numerosity. They are able toperceive and to compare quantities, to make judgements con-cerning approximate values and to build up an inner analoguenumber line. Children with these kinds of difficulties have par-ticular problems in learning, processing and transcoding two

culturally transmitted systems of number symbolization, (ver-bal/alphabetical and Arabic), each with characteristic lexicons,syntax rules and designations for different utilizations.

Children classied into the verbal subtype failed to countaccurately and correctly. Even if they mastered the countingword sequence, they made frequently errors in enumeratingsets of objects properly by missing one, taking one twice orviolating the one-to-one correspondence. During the develop-ment of counting strategies for solving simple addition andsubtraction tasks, these children may have miscounted fre-quently (i.e. solving the problem 3+5, they do not alwaysreach 8 but frequently also 7 or 9). As a consequence, retrievalstrategies and number-fact knowledge could not be establishedappropriately and the children rely on time-consuming, vul-nerable and immature counting strategies. In our sample, 6 outof 11 children showed ADHD and had an impulsive and hastyworking style which might have contributed to faulty count-ing. According to our clinical experience, these children willhave a relatively good prognosis if the ADHD is treated early

and successfully, and no other comorbidity is present.However, reading and spelling difficulties as well as develop-mental disorders of speech and language are also relativelycommon in this group of children. Depending on the type andthe severity of such associated disorders, the development of elaborated analogue representations may also be constrainedas described in the next section.

Children with the verbal subtype generally do not have dif-ficulties in acquiring the Arabic notational system and intranscoding Arabic digits to verbal numbers and vice versa, asdo children classied in the Arabic subtype cluster. Most of theerrors that characterize the performance of these children

occur in reading Arabic numerals, writing Arabic numerals todictation as well as in cognitive number comparison tasks,especially when the pairs of numbers are presented digitally.

Compared to the normal population, children with a mothertongue other than German were overrepresented in this group.Upon school entrance and systematic instruction of the Arabicnotational system, these children have to apply multiple rulesand routines for number transcoding between the native andthe second-language number-word sequence and betweenboth and the Arabic notation system. This may constitute asubstantial risk for the maturation of the Arabic module and forthe acquisition of arithmetical procedures that depend on thisparticular Arabic representation (written multi-digit calcula-tion). However, this kind of learning difficulty may vary withthe presence of more or less sophisticated irregularities in thenumber-word lexicons of the spoken languages, with the timeat which a child is rst exposed to a second language and theintensity of exposure. It would be quite interesting to investi-gate whether this phenomenon is specic for certain charac-teristics of number-word lexicons and, therefore, is possiblyrestricted to specic language areas (i.e. German). The cross-cultural differences in mathematical performance between


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East-Asian children and children from European countries andthe U.S. have been considered to be partly due to linguisticfactors that favor East-Asian children, since East-Asian num-ber-word lexicons correspond quite closely to the Arabicnotation system (44, 45). Another characteristic feature of theArabic number symbolization is its purely visual-spatial rep-resentation, though children with visual-spatial processingdecits (such as left-right disorientation) may also experiencedifficulties, for instance with the place value system andvertical alignments.

Section 3: Course and outcome of DD in a 17-year-old boy

Referral

HN was referred by his mother due to increasing symptoms of depression and suicidal thoughts, social anxiety, isolation andrefusal to attend school. He had a long history of learningdisabilities which were most pronounced in the areas of math-ematics but also affected reading and spelling. HN is presentlyin the 10th school year (Werkjahr) that prepares for a voca-tional apprenticeship.

History

HNs parents divorced when he was 1.5 years old. Both parents

have an academic background; the father had mild dyslexiaduring his early school years. The mothers older brother hadproblems with mathematics during primary school education.There is no other family history of mental diseases or academicfailure.

According to the mothers report, pregnancy, birth and pri-mary adaptation were without problem, HN was a quiet infantwith slightly delayed motor milestones. At the age of 2.5 years,he attended a toddler group where his mother noticed that hehad a retarded language development compared to other chil-dren of the same chronological age. At the age of 4 years, heattended preschool; for the rst few weeks, HN experiencedsevere separation anxiety and his mother had to stay with him.One year later, the mother was reported that her son was ratherhyperactive, inattentive, frequently experienced ts of rageand destroyed play material.

When HN was 6 years old, the mother moved with her sonfrom Germany to Switzerland but HN never adapted to theSwiss-German dialect. A rst child psychiatric consultationdiagnosed a developmental delay and, as HN was of normalintelligence, a special educational setting within a regularschool curriculum was proposed.

At the age of 7 years, HN attended a private school with asmall class size and a regular curriculum. According to a sec-ond neuropediatric and child psychiatric examination, mini-mal brain dysfunction with decits in attention, motor controland perception as well as a hyperkinetic disorder was diag-nosed and HN was attended psychotherapy, psychomotor andsensory-integration training. A year later, he was moved to aregular primary school, staying there for two years, manifest-ing increasing learning problems. At the end of the secondgrade, an extensive psychological assessment diagnosed adevelopmental language disorder, a developmental disorder of motor function, a combined disorder of scholastic skills(dyslexia and dyscalculia) and an attention-decit hyperactiv-ity disorder. The following results were obtained: normal intel-ligence with a discrepancy between low verbal and averageperformance (IQ); poor measures of phonological segmenta-tion and verbal memory span; average measures concerningvisual-motor coordination and visual-spatial processing.Results of an electroencephalography showed no dysfunc-

tionality.Acting on professional advice, HN was transferred to a

school specialized for children with developmental languagedisorders. Here he received special training in reading andspelling as well as in arithmetic.

In spite of these various therapies, HNs academic per-formance did not improve satisfactory. At the age of 11 years,after repeating the third grade, he was reported to be non-moti-vated, unhappy and to have a very low self-esteem. He told hismother that he was afraid of other children, that he did not likeschool and he began to refuse to attend school. A phoniatricexamination came to the following 5 diagnoses: dysnomia,

dysuencia, dyslexia, dyscalculia, dysgraphia.Due to repeatedly refusing to attend school, HN was trans-ferred to a boarding school where reading, spelling and math-ematical training as well as speech therapy and psychotherapywere continued.

At the age of 15 years, he refused to return to the boardingschool after a summer holiday spent with his mother. He livedwith his mother for the next two years and attended the 7th and8th grades at a special school for handicapped children. Dur-ing that period, he again refused to go to school over severalweeks, reported that he was afraid of other children, becamemore and more isolated and failed to make friends. His mothernoted increasing depressed moods and withdrawal.

Psychiatric examination

HN is a good-looking, fashionably dressed adolescent who isable to establish contact in a friendly and polite manner butseems slightly embarrassed and shy. He is left handed. His dif-ficulties in comprehending complex and longer sentenceswithin verbal communication were not immediately clear due



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to his well developed abilities to express non-verbally throughgesture and mimic full awareness of what had been said andto mask incomplete comprehension. His verbal comprehen-sion and reciprocal communication improved markedly if conversation was slow and used short sentence constructs. Hisown verbal expression seemed sufficient for social pragmaticbelongings, not dysgrammatical, but simple concerning syn-tactical structure and vocabulary. HN is painfully aware of hislearning difficulties and he considers himself to be stupid. Heis afraid of other adolescents, has long experiences of beingteased and he is convinced that he is completely unattractiveto other peers. During the last year, he often thought aboutkilling himself but had not followed through his thoughts inlight of the effect that his suicide would have on his mother.No symptoms of attention decit or hyperactivity could beregistered, even during psychological testing.

In summary, HN is a 17-year-old adolescent of low averagegeneral intelligence with a long history of learning problemsand school failure who has received various forms of treatment

and repeated psychological, medical and special educationalwork-ups over a number of years. He has a low self-esteem,negative self-attitudes, a social phobia and is actuallydepressed. He has developed much better reading than spellingskills, the most inferior area of school performance beingmathematics.

Investigation of number processing and calculation abilities

HN was administered the EC 301-R (18). This test batterycontains 13 subtests that investigate different abilities: count-

ing, number transcoding (spoken, written verbal and Arabicformat), mental and written calculation, approximate calcula-tion and story problems, magnitude comparisons on spoken,written verbal and Arabic numbers, contextual magnitude esti-mation, perceptual quantity estimation and placing Arabic andspoken numbers on an analogue number line. In addition, EC301-R containes subtests that investigate knowledge concern-ing arithmetic signs and numerical facts of everyday life.

Results

Except for minor difficulties in counting backwards, HN hadno problems with counting and enumerating different sets of dots and he also responded correctly to questions concerningnumerical facts of everyday life (i.e. how many days are in aweek? how many minutes are in 1 hour?).

HN had severe difficulties in transcoding written verbal ororally presented numbers into their Arabic notation, especiallywhen the number-words comprised more than two elements.(i.e. spoken verbal: eleven-thousand-six-hundred-and-thirty

110630, one-hundred-and-eighteen 818; written

verbal: ve-hundred-thousand 5000, three-hundred-and-eighty-seven 30078). The rate of errors was 55 %(near-chance performance). In contrast, he was much betterin transcoding in the reverse direction: Arabic numbers intoverbal or written verbal numbers (15 % incorrect responses).However, his handwriting was rather disorganized and hemade lots of spelling errors (Fig. 5).

This pattern of verbal/Arabic dissociation also appearedwhen HN was asked to compare two numbers according totheir magnitude. He was faultless when the numbers were pre-sented in Arabic notation but had great difficulty with the cor-

responding orally presented comparison tasks, where he failedin 50 % of the items. The type of errors indicate that his falsedecisions were related to the length of the number words andto the values of the ending elements (i.e. one-hundred-thou-sand-and-sixty-vewas indicated to be more than three-hun-dred-thousand and seven-hundred-and sixty-nine was morethan two-thousand-and-thirty-ve).

The pattern of transcoding problems indicate that HN hadmarkedly impaired verbal number comprehension mecha-nisms, much better verbal number production abilities and anearly preserved ability to process Arabic numerals. He couldcomprehend and produce arithmetical signs without problemand he was also able to arrange written calculations correctly(addition and subtraction) according to their vertical align-ments. He could solve written multidigit addition and sub-traction tasks without problem but he failed to solve writtenmultiplications correctly, due to left-right confusion in the ver-tical arrangement of the indented intermediate results fromwhich each one was calculated correctly (Fig. 6).

On simple exact mental calculation problems (addition,subtraction, multiplication and division, presented orally aswell as digitally, all with values smaller than 50), HNresponded to 87.5 % of the items correctly. Errors onlyoccurred in mental subtraction (i.e. 14-6 = 4). However, hewas unable to correctly solve simple arithmetical problems

that were embedded in an orally presented situational context(story problems).

Fig. 5 Transcoding from Arabic to the written verbal code by HN.

Fig. 6 Multiplication as written by HN.


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HN responded correctly to only 25 % of the items thatinvestigated approximate calculation (problems were pre-sented in Arabic format, i.e. 1520 - 780 = 2300 or 1450 or 400or 700). Correct answers were only registered in one additionand one subtraction problem, both items being composed of relatively small numbers with results smaller than 1000. Hefailed to approximate addition and subtraction problems thatcontained operands larger than 1000; he failed completely toapproximate multiplication and division problems.

In contrast, HN was faultless in perceptual quantity andcontextual magnitude estimations (quantities always smallerthan 100) and he could also relate Arabic as well as spokennumbers very well to analogue positions on a scale rangingfrom 0 to 100.

Discussion

HN showed psychiatric symptoms that were reported to be

most frequently associated to developmental dyscalculia:ADHD and internalizing disorders such as anxiety and depres-sion (54, 59, 71, 72). However, while ADHD had decreased, thesymptoms of anxiety and depression had increased during thecourse of HNs development. It can be speculated that attentiondecit and poor phonological awareness might have hamperedthe modularization of verbal number processing mechanismsduring the early school years. Consistent experiences of failureand of feeling different from other peers might have been fol-lowed by increasing symptoms of anxiety, depression, poormotivation and refusal to attend school which, in turn, mighthave contributed to poor learning progress.

An actual psychometric intelligence test was not done. Theresults of the previously mentioned tests correspond quite wellto the pattern that has been described by Rourke (52) as char-acteristic of subtype RS (see above). The difficulties HNexperienced in left-right orientation along with his graphomo-tor problems indicate the presence of developmental Gerst-mann syndrome (DGS) as well (however, nger agnosia wasnot evaluated). Subtype RS and DGS and also developmentallanguage disorder and dyslexia have been considered to rep-resent left-hemisphere dysfunctions. With reference to themechanisms of Andersons theory of minimal cognitive archi-tecture (1), one could postulate that HN has a poor specicprocessor for verbal material (SP1). Whether or not a poor SP1will impede the maturation of certain modules might be aninteresting theoretical question concerning possible interac-tions between the different mechanisms that constitute the tworoutes of knowledge acquisition in Andersons theory. With theassumption of format-specic representational modules fornumber processing, as proposed in the triple-code model byDehaene (12), it can be supposed that the development of twomodules has been disturbed in HN: the verbal module and theanalogue magnitude module.

As pointed out in section 2, the analogue magnitude moduleprovides an essential meaning, the semantics of numbers. Thismodule has been supposed to rely on primary inborn and pre-verbal abilities of numerical quantication and is conceptual-ized as an inner number line that is variable and heteromorphic(16). According to HNs pattern of dysfunction, it can beargued that the mental construction of this (growing) numberline depends on an appropriate mastery of verbal numbersymbolizations in order to label the tick marks that constitutethe number line.

HN was not able to build up abstract analogue representa-tions for very large quantities, due to limited abilities to rep-resent larger numbers verbally and to assign appropriate namesto larger quantities (verbal representation). He was still able toconstruct and enlarge an inner number line up to about 100,including all values he could symbolize verbally. However, hisdifficulties in comparing pairs of orally presented numbers andto approximate results of arithmetic operations composed of large numbers might indicate his inability to visualize and

locate large numbers internally. A spoken number-word suchas three-thousand-eight-hundred-and-thirty-seven might beperceived by HN as a fuzzy convolute of names, each elementsounding familiar, but too long to be xed in his workingmemory, therefore being identied as a particular symbol forone single number such as threeor twenty-eight. This, how-ever, would appear to be a necessary condition to evaluate thatthree-thousand-eight-hundred-and-thirty-seven is less thanthree-thousand-eight-hundred-and-thirty-nine, more thanthree-thousand-eight-hundred-and-thirty-four and approxi-mately half of eight-thousand.

HN showed a classical pattern of verbal/Arabic dissociation

(43). He was able to read Arabic numerals (Arabic encoding,internal translation and verbal production) and to comparepairs of Arabic numbers, even of larger size, correctly. It seemsthat he had learned the syntactic rules of the Arabic notationas well as the transcoding rules quite well. He was also able tocarry out written multidigit calculations successfully, althoughhe could not predict by estimation whether a result could log-ically be correct or not, presumably due to his limited abilitiesto locally visualize large numbers on an internal number line.

Section 4: A comprehensive model of developmental dyscalculia

Different subtypes

The study presented in section 2 describes three different sub-types of DD in children with normal intelligence that havebeen interpreted to reect impairments in the maturation of particular representational modules, as proposed in current



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cognitive-neuropsychological models such as the triple-codemodel of Dehaene (12).

The verbal subtype represents a disorder of lexical/syntac-tical verbal representations of numbers, the use of language-based arithmetical procedures (i.e. counting strategies) andarithmetic fact retrieval.

The Arabic subtype represents a disorder in the acquisitionof the second number language, the Arabic notation system,

including problems in reading, writing and comparing Arabicnumerals, for instance by violating syntactic rules such asplace value positions.

The pervasive subtype represents a lack of primary seman-tic concepts of number and numerosity, which implicates aninability to develop appropriate analogue number representa-tions such as an internal number line. Secondly, it representsalso an impaired development of verbal and Arabic represen-tations and related computational skills.

We will now try to explain these ndings as well as othermentioned concepts on DD within the theoretical frameworkof the minimal cognitive architecture proposed by Anderson(1). Figure 7 illustrates the three subtypes at the level of mod-ular dysfunction within route 2 of Andersons theory (Modus3).

With reference to Fodor (22) and Karmiloff-Smith (35),two different types of modules have been proposed by Ander-son (1); the rst, innate and given as a result of evolutionarypressure (labeled Mark I), the other being acquired duringindividual development according to learning and automati-zation processes (labeled Mark II). The analogue magnituderepresentation from Dehaenes triple-code model can be con-

ceptualized as a Mark-I-type module, whereas the verbal andthe Arabic number representations are more likely to representmodules of the Mark-II type. It can therefore be concluded thatthe pervasive subtype is likely to be caused by genetic 1 inu-ences or early brain damage, whereas the verbal and Arabicsubtype are more likely to be caused by linguistic develop-mental disorders, probably due to genetic as well as environ-mental inuences. Gearys (25) distinction between biological

primary and secondary abilities describes the same conditionfrom a cognitive-developmental perspective. Biologicalprimary (universal) quantitative abilities in Gearys conceptcorrespond to those mechanisms attributed to the analoguemagnitude module in Dehaenes model, whereas biologicalsecondary culturally transmitted abilities might correspond toverbal and Arabic representational modules.

NLD and subtype RS (see above) are the best known exam-ples for concepts that postulate lateralized hemispheric dys-function for DD which could be related to the specic proces-sors (SP1/SP2) in Andersons theory (Modus 2 in g. 7). How-ever, as reviewed in section 1 poor visual-spatial processingas well as poor language processing are not strongly predic-tive for developmental dyscalculia, but indeed these condi-tions constitute an increased risk. It seems plausible that a poorSP1 or SP2 might seriously interfere with the processes of

1 In 42 % of children with DD from an Israeli population-based study, apositive family history for learning disorders has been found, suggest-ing genetic inuences (28).

Fig. 7 Theory of minimal cogni-tive architecture (1); adapted fordisorders of number processingand calculation.


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modularization: poor capacity to process particular qualities of information (verbal/propositional, visual-spatial/simultane-ous) may restrict knowledge acquisition and result in poorautomatization and modularization of certain abilities. Thisinteraction might take place, for example, in children withdevelopmental language disorders (poor SP1). It is well knownthat these children frequently exhibit phonological dyslexia,problems in mastering a second language or in numberprocessing and school arithmetic, probably due to the limitedmaturation of linguistic modules. In contrast, children withNLD (poor SP2) might have a handicap for the modularizationof particular cognitive abilities that rely on visual-spatial pro-cessing mechanisms and frequently result in school-relatedlearning difficulties such as visual-spatial subtypes of dyslexiaand developmental dyscalculia. Poor specic processors mightalso impede the modularization of other related cognitive func-tions. For example, NLD has been reported to be frequentlyassociated not only with DD but also with disorders of empa-thy and social interaction (autistic spectrum) (39, 54). It can

be speculated that non-verbal processing decits that includepoor face recognition may hamper the maturation of theoryof mind that is supposed to play a central role in the develop-ment of social competence and that have been supposed to bemodularly organized in Andersons theory (1).

Modus 1 in Fig. 7 nally represents poor mathematicalabilities predicted by poor general intellectual capacities asexpressed by a low IQ which corresponds to a poor basic pro-cessing mechanism in Andersons theory.

Developmental dynamics

The three representational modules pictured in the triple-codemodel are supposed to be functionally autonomous, with spec-ied input, output and transcoding mechanisms, and activatedaccording to the demands of a certain problem. Taking thismodel as representing the end stage of adult number process-ing and calculation, it will now be argued that these modulesemerge and elaborate at different stages in development.Furthermore, it will be argued that the development of themodules and neural networks depend (i) on certain genetic pre-dispositions, (ii) on specic environmental inuences and (iii)with regard to each individual module, on the developmentalprogress of the neighboring number modules.

The set of inborn, probably universal numerical abilitiesdescribed by Starkey et al. (63) that constitutes the core of Dehaenes analogue semantic number representation seemsnecessary to understand how and why number-words are usedand to grasp basic numerical principles (increase/decrease,part/whole), counting principles and arithmetical concepts.Without these semantic roots, number-words and arithmeticalgorithms are learned by route and produced without reallyknowing what has been done and why. The pervasive subtype

described in section 2 has therefore been postulated to becaused by genetic factors or early brain damage.

HN was able to understand and use basic number principlesbut had severe difficulties in learning number-words that werecomposed of more than two sequential elements. It seems thatthis was due to a severe, probably also genetically transmitted,language developmental disorder. It has been argued that thedifficulties experienced by HN are due to a defective matura-tion of the module designated for verbal/alphabetical numberrepresentation. This might have been detrimental for furtherelaboration of analogue representations of larger numbers,which meant that HN failed to internally construct and repre-sent numbers that he could not name appropriately. He couldnot build up an inner number line for larger numbers and, con-sequently, he could not process them mentally while calculat-ing. HN could, however, produce larger number-words whilereading Arabic numbers because he had mastered the Arabic


Fig. 8 Developmental dynamics of modularization of number process-ing and calculation abilities.


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notation principles well and was able to use the visually rep-resented sequence of digits as an aid to articulate the corre-sponding number-word correctly but without being able tovisualize the number analogically. Therefore, the elaborationof analogue representations seems to rely on appropriate lin-guistic representations. However, this elaboration may also beinuenced by other environmental factors that could explaincross-cultural differences, i.e. differences between Europeanand South-American children when instructed to place num-bers on an analogue number line (14).

Finally, different linguistic factors that characterize num-ber-word sequences in different languages might contribute toparticular difficulties that are experienced by foreign-languageor bilingual children in acquiring the Arabic notation systemand written calculation procedures in the school environment(73). These children have to transcode from two number-wordsequences which have different linguistic irregularities, i.e.German and French (or Spanish) into the Arabic notation.These linguistic confusions but also problems in left-right

orientation and spatial processing might disturb the learning of particular symbols and syntactic rules of the Arabic notationand therefore impede the maturation of the Arabic module assupposed for children showing the Arabic subtype pattern (seesection 2).

In conclusion, it seems that the arising and elaborating cog-nitive number representations are semi-autonomous duringdevelopment, each depending on adequate developmentalprogress of the others.

Mediating and moderating environmental factors (rightside in Fig. 8) has been found to contribute to individualdifferences in mathematical performance (for current

overviews, see 24 and 19). Factors that have been of particu-lar interest in the discussion of mathematical development andDD are those of gender (24, 70), interactions between learn-

ing and psychological functioning (3, 32, 71), inuences of social class and cultural environment (29, 68), as well asaspects of formal education and instructional methods (4, 14,31, 50).

Prevalence rates for DD have often been reported to behigher for girls than for boys, which seems to reect a univer-sal gender effect on the development of mathematical skills.The fact that the prominence of this effect is highly variablebetween different countries and that this effect has decreasedover the last decades (33, 34) indicates that it is not likely tobe explained only by the biology of the female (5). Severalalternative hypotheses have been posed to explain the gendereffect (30). Differences between boys and girls in attitudes,self-esteem (67), and specic anxiety (3) might play an impor-tant role in establishing motivated and self-directed learning.Educational settings and instructional methods might also sub-tly favor boys when they do not appropriately reect differ-ences in cognitive styles, algorithmic thinking and preferredlearning environments between boys and girls (56, 57, 75). In

the Zurich study summarized previously, a higher rate for girlsamong poor-performing children from a normal populationwas also found. However, according to these results, boys arenevertheless over-represented in all three groups of childrenwith subtypes of DD, a preliminary result which has to be con-rmed by studies on larger samples.

In children with DD, behavioral and emotional symptomsas well as other learning difficulties frequently coexist, some-times preceding, sometimes proceeding experiences of schoolfailure, both of which were the case for HN. The quantity andseverity of comorbid disorders has a strong prognostic valuefor overall personality adjustment and academic outcome (20,

21). Early diagnosis and early intervention, matched to differ-ential assessment are urgently needed to prevent such negativevicious circles.

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