Developmental Dyscalculia and Cognitive...

DEVELOPMENTAL NEUROPSYCHOLOGY, 1994, 70(4), 413-441Copyright © 1994, Lawrence Erlbaum Associates, Inc.

Developmental Dyscalculia andCognitive Neuropsychology

Scott M. SokolNeurolinguistics Laboratory

Institute of Health ProfessionsMassachusetts General Hospital and

Department of NeurologyHarvard Medical School

Paul Macaruso and Tamar H. GollanNeurolinguistics Laboratory

Institute of Health ProfessionsMassachusetts General Hospital

In this article we present a study investigating the cognitive impairments exhib-ited in developmental dyscalculia, or arithmetic learning disability. We begin byproviding a brief review of the literature on the subject, including both single-case and group designs. In our view both approaches, at least as thus faremployed, are associated with theoretical and/or methodological shortcomings.We suggest that the cognitive neuropsychological approach may offer a betterparadigm for research in this domain. Accordingly we present a recent cognitiveneuropsychological model of numeric processing that has been informed by thepatterns of impairment evidenced in individuals with acquired brain damage.We then present preliminary results from several developmental dyslexic stu-dents who have numeric processing impairments and attempt to demonstratehow their performance patterns conform to the expectations of the model.Finally, in the discussion, we consider certain methodological issues related tothis study and suggest possible directions for future research.

In contrast to the extensive literature on developmental dyslexia, there hasbeen relatively little work investigating developmental dyscalculia, or arith-

Requests for reprints should be sent to Scott M. Sokol, Graduate Program in CommunicationSciences and Disorders, MGH Institute of Health Professions, 101 Merrimac Street, Boston,MA 02120. e-mail: [email protected]

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414 SOKOL, MACARUSO, GOLLAN

metic learning disability. This imbalance is unfortunate, given that problemsin using numbers and learning arithmetic appear to be as prevalent in schoolpopulations as problems in learning to read (Badian, 1983); and, as withreading difficulties, problems in numeric processing often persist into adult-hood (Hitch, 1978).

In this article we present some preliminary research investigating devel-opmental dyscalculia, which is informed by the recent theoretical and meth-odological advances in cognitive neuropsychology. We begin by providing abrief overview of the extant literature on developmental dyscalculia andpoint to some inherent problems with the approaches taken thus far. Next, wepresent a cognitive model of numeric and arithmetic processing that hasemerged from cognitive neuropsychological research on acquired dys-calculia. We then describe our initial findings with several developmentallyimpaired students and attempt to demonstrate how our results support thistheoretical model. Finally, in the discussion, we consider two methodologi-cal issues relevant to our study and conclude by outlining some directions forfuture research.

OVERVIEW OF RESEARCH ONDEVELOPMENTAL DYSCALCULIA

Previous studies of developmental dyscalculia have made use of both single-case and group designs.

Single-Case Studies

In one of the earliest investigations of developmental dyscalculia, Guttman(1937) presented a series of case histories of children with numeric process-ing difficulties. For example, one of his subjects was described as having a"constructional disturbance of figure writing" (Guttman, p. 21). The subjectproduced errors in writing numbers to dictation (e.g., "three thousand twohundred twenty-eight" was produced as 302028) and in solving writtenarithmetic problems (e.g., 68 - 29 = 30). Guttman remarked that many of theimpairments seen in congenital cases were similar to those found in acquireddyscalculia.

More recently Badian (1983) investigated developmental dyscalculia,using a classification scheme devised by Hecaen, Angelergues, and Houillier(1961) for the study of acquired dyscalculia. Three broad classes of dys-calculia were described in the Hecaen et al. scheme: alexia and agraphia fornumbers (i.e., impaired calculation resulting from deficits in reading andwriting numbers), spatial acalculia (i.e., impaired calculation resulting from

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DEVELOPMENTAL DYSCALCULIA 415

spatial processing deficits), and anarithmetia (i.e., disruption of calculationability per se). Badian found many cases of spatial dyscalculia and an-arithmetia in the developmental population but little evidence of alexia andagraphia for numbers. The greatest number of cases were placed in a fourthcategory called attentional-sequential dyscalculia. Children with an atten-tional-sequential disorder make careless errors in carrying out proceduresand have great difficulty learning and recalling multiplication facts.

Other single-case studies of developmental dyscalculia have been de-scribed by Cohn (1968, 1971) and Slade and Russell (1971). Cohn's work isinformative in that it reveals the diversity of impairments that may arise indevelopmental dyscalculia, including malformed number symbols, poor sin-gle-digit addition, failure to discriminate specific order characteristics ofmultidigit numbers, inability to remember multiplication facts, and errors incarrying numbers. Slade and Russell studied four cases of developmentaldyscalculia in which poor multiplication skill was the core symptom. Threeof the subjects had a "faulty grasp of basic multiplication tables" (Slade &Russell, p. 295). In a qualitative analysis of performance, Slade and Russelluncovered a number of methods used by the subjects to overcome their lackof factual knowledge. For example, one subject solved 8 x 7 by multiplying2 x 7 = 14 and adding four 14s.

The single-case approach has also been used to document cases of devel-opmental Gerstmann syndrome. According to Gerstmann (1940), the neu-ropsychological disorders of dyscalculia, dysgraphia, finger agnosia, andright-left disorientation reflect a unitary cognitive impairment associatedwith left parietal damage. Although the Gerstmann syndrome has been identi-fied most often in acquired cases (e.g., Kinsbourne & Warrington, 1962;Roeltgen, Sevush, & Heilman, 1983; Sobota, Restum, & Rivera, 1985), cases ofdevelopmental Gerstmann syndrome, in which constructional dyspraxia isadded to the symptom tetrad, have been discovered (e.g., Grigsby, Kemper, &Hagerman, 1987; Kinsbourne & Warrington, 1963; PeBenito, 1987; PeBenito,Fisch, & Fisch, 1988).1

Group Studies

Most studies of developmental dyscalculia have employed group designs.For example, in an early study Kosc (1974) employed screening tests toidentify three groups of children with mathematical disabilities. One groupperformed poorly on numeric tests (e.g., basic addition, subtraction, and

1Our citation of work on the Gerstmann syndrome does not imply a belief that such asyndrome has any theoretical or clinical validity. On the contrary, we find little in the way ofscientific motivation to support its use in the clinical or experimental literatures (see Benton,1961, for a similar position). We mention this work here solely on account of its ratherwidespread use in describing certain arithmetic difficulties in acquired and developmental cases.

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4 1 6 SOKOL, MACARUSO, GOLLAN

multiplication), a second group had difficulty with performance tests (e.g.,dividing geometric figures into specific shapes), and a third group haddifficulty on both sets of tests. Kosc then showed that the three groupsperformed differently on a series of psychological tests. For example, thelowest scores on a test of word reading and writing were obtained by the twogroups with poor numeric skills.

Group studies have been used extensively by Rourke and his associates intheir neuropsychological investigations of subtypes of arithmetic learningdisability (e.g., Ozols & Rourke, 1988; Rourke & Finlayson, 1978; Strang &Rourke, 1983; see Strang & Rourke, 1985, for a review). They first identifysubgroups of arithmetic learning disability (e.g., low arithmetic scores—aver-age reading/spelling scores; low arithmetic scores—lower reading/spellingscores) and then compare the subgroups on various measures of auditory-perceptual, visual-perceptual, and linguistic functioning. Their findings in-dicate that, in general, students with specific arithmetic disability displaypoor right-hemisphere functions (e.g., visual—spatial and tactile—perceptualskills), whereas students with arithmetic plus reading/spelling disabilitydisplay poor left-hemisphere functions (e.g., linguistic and auditory—percep-tual processing). It should be noted, however, that a number of studies haveobtained results inconsistent with this dichotomy (see, e.g., Nolan, Ham-meke, & Barkley, 1983; Rosenberger, 1989; Share, Moffitt, & Silva, 1988).

Some researchers have employed group studies to examine the memoryskills of students with arithmetic learning disability. For example, Siegel andLinder (1984) found that children with arithmetic plus reading disabilityshowed short-term memory problems with both auditorily and visually pre-sented strings of letters, whereas children with specific arithmetic disabilityhad difficulty mainly with visual presentation (see Siegel & Feldman, 1983,and Siegel & Ryan, 1988, for similar results; but see Webster, 1979, for acontradictory result). In a subsequent study, Siegel and Ryan (1989) testedreading-disabled and specific arithmetic-disabled students on two working-memory tasks, one that required supplying the final word to a series ofsentences and then recalling the set of final words, and another that requiredcounting the number of yellow dots in random patterns containing blue andyellow dots and then recalling how many yellow dots there were in eachpattern. They found that reading-disabled students had difficulty on bothverbal (sentences) and nonverbal (dots) memory tasks, whereas specificarithmetic-disabled students displayed below normal performance only onthe nonverbal task (see also Fein, Davenport, Yingling, & Galin, 1988;Fletcher, 1985).

Many group studies have investigated the performance of learning-disabledchildren in solving basic arithmetic problems. For example, Russell and Gins-burg (1984) found that fourth graders experiencing "mathematics difficulty"fell well behind their normally achieving peers in rapid recall of basicaddition facts. In contrast, they often displayed adequate performance inother areas of mathematics, such as estimation, relative magnitude judg-ments, knowledge of base-ten concepts, enumeration skills, and simple prob-lem solving. Similarly, Fleischner, Garnett, and Shepherd (1982) and Garnett

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DEVELOPMENTAL DYSCALCULIA 4 1 7

and Fleischner (1983) found that learning-disabled children produced fewercorrect responses than nondisabled children in a speeded test of single-digitaddition, subtraction, and multiplication (see also Pritchard, Miles, Chinn, &Taggart, 1989); and Ackerman, Anhalt, and Dykman (1986) showed thatmany students with attention deficit disorders displayed both slow and inac-curate performance in solving simple addition and subtraction problems.Ackerman et al. concluded that learning-disabled students have difficulty inautomatization of basic skills, such as number facts (see also Garnett &Fleischner, 1983); and they argued that poor sequential memory abilitiescontribute greatly to these difficulties.

Lastly, a number of group studies have examined the information-process-ing strategies employed by learning-disabled children in solving single-digitaddition problems. For example, Svenson and Broquist (1975) compared"subnormal" children in Grades 4 through 6 with normal third graders andfound that a counting model in which a counter is set to the value of thelarger operand and incremented a number of times equal to the value of thesmaller operand can describe the solution time data for both groups. Incomparison with normal children, subnormal children count at a slower rateand on occasion have difficulty identifying the larger operand as a startingpoint for the counter. A recent series of studies by Geary and his associateshave also examined single-digit addition in arithmetic-disabled children. Forexample, Geary, Widaman, Little, and Cormier (1987) found that, althoughnormal children typically shift from counting in Grade 2 to direct memoryretrieval in Grades 4 and 6, most arithmetic-disabled children rely on count-ing strategies in all three grades (see also Geary, Brown, & Samaranayake,1991) and that arithmetic-disabled second graders count slower, producemore errors, and are more variable in their application of counting strategiesthan normal second graders. In a subsequent study, Geary (1990) showedthat when memory retrieval is used instead of counting, arithmetic-disabledfirst and second graders produced a higher proportion of retrieval errors andless systematic solution times than normal children (see also Geary &Brown, 1991). Goldman, Pellegrino, and Mertz (1988) found, however, thatextended practice can lead to more efficient counting solutions and/or in-creased reliance on memory retrieval in many arithmetic-disabled children.

Evaluation of Single-Case and Group Studies

Although the single-case and group studies reviewed here have heightenedour awareness of the disorders associated with developmental dyscalculia,there are several theoretical and methodological shortcomings associatedwith these approaches. Considering the single-case studies first, few at-tempts have been made to characterize the deficits seen in developmentaldyscalculics in terms of a well-specified cognitive model of numeric andarithmetic processing. As an example, consider the study by Slade and

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Russell (1971). They identified, through a comparison of performance acrossthe four arithmetic operations (+, - , x, -*-), that multiplication—particularlyknowledge of basic multiplication facts—was a source of difficulty in eachof their four cases of developmental dyscalculia. However, their determina-tion of the locus of impairment in these cases remains incomplete, becausethey failed to systematically evaluate the various processes associated withsolving arithmetic problems. For example, they provided little informationabout the numeral comprehension and numeral production abilities of theirsubjects.

Identification of the locus or loci of impairment in arithmetic-disabledchildren is given even less attention in group studies. In a typical study,subjects are first assigned to groups on the basis of performance on a cursorystandardized math test and then administered one or more tasks designed totap global processing abilities (e.g., visual-spatial skills, nonverbal mem-ory). Low group performance on certain tasks (e.g., a memory test usingrandom dot patterns) is taken as evidence that the group suffers from a globaldisorder (e.g., poor nonverbal memory). In actuality, however, without at-tempting to evaluate numeric processing skills in greater detail, there is noway to determine what sorts of numeric processing deficiencies the subjectssuffer from or how these deficiencies relate to the global processing disorder.

Another fundamental problem with group studies is that conclusions arebased on averaged data. Unless within-group variability is negligible, whichis unlikely given the rough measures used to determine group membership,mean performance will not adequately reflect the ability levels of mostmembers of the group.

The failure to provide a detailed evaluation of numeric processing skillsdoes not apply to the group studies that have examined single-digit addition(e.g., Geary, 1990; Geary et al., 1987; Goldman et al., 1988; Svenson &Broquist, 1975). One of the highlights of these studies is that they draw uponinformation-processing models of the normal development of arithmeticknowledge (e.g., Siegler & Shrager, 1984) to identify the source of process-ing failure in these children. The study by Goldman et al. (1988) is exem-plary in this regard in that it also stresses individual differences in use ofprocessing strategies. However, most of these studies are concerned mainlywith single-digit addition at the exclusion of an assessment of other poten-tially relevant numeric processing skills. One exception is the study byRussell and Ginsburg (1984), who examined a diverse set of numeric pro-cessing skills in a group of children experiencing mathematics difficulty.Unfortunately, Russell and Ginzburg did not spend much time discussingindividual variation within their group; neither did they base their investiga-tion on an explicit cognitive model of the processes underlying the skills inquestion.

2The averaging problem in group studies has been debated extensively in the cognitiveneuropsychological literature. For detailed discussion, the reader is referred to the special issueof Cognitive Neuropsychology (Caramazza, 1988) and references therein.

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COGNITIVE NEUROPSYCHOLOGY ANDNUMERIC PROCESSING

We propose that the study of developmental dyscalculia would benefit sub-stantially from the methods employed in cognitive neuropsychology. Overthe past two decades cognitive neuropsychology has received growing ac-ceptance as a useful tool in developing models of normal cognitive systemsand in understanding acquired cognitive disorders (e.g., Caramazza, 1986;Coltheart, 1984; Shallice, 1979). In this approach individuals with cognitiveimpairments are investigated on a single-case basis to determine the locus ofbreakdown within an information-processing model of the normal cognitivesystem. The approach assumes that a cognitive skill (e.g., arithmetic) can beconceptually decomposed into a number of modular processing componentsand that brain damage may selectively disrupt one or more of these compo-nents without modifying other components. A cognitive deficit is describedin terms of functional damage to either the representations or processesimplicated in normal performance.

A Model of Numeric Processing

In 1985, McCloskey, Caramazza, and Basili proposed a cognitive model ofnumeric processing and reported data from several brain-damaged patientsin support of the model's componential architecture (for a modified versionof this model, see Figure 1). Since that time, members of our research teamhave made extensive use of this model and more generally the cognitiveneuropsychological approach in the study of acquired dyscalculia (e.g.,Caramazza & McCloskey, 1987; Macaruso, McCloskey, & Aliminosa, 1993;McCloskey, 1992; McCloskey, Aliminosa, & Sokol, 1991; McCloskey &Caramazza, 1987; McCloskey, Sokol, & Goodman, 1986; McCloskey,Sokol, Goodman-Schulman, & Caramazza, 1990; Sokol & McCloskey,1988, 1991; Sokol, McCloskey, & Cohen, 1989; Sokol, McCloskey, Cohen,& Aliminosa, 1991).

At the most general level, the McCloskey et al. (1985) model draws adistinction between numeral processing mechanisms and calculation mecha-nisms. Within numeral processing the model separates numeral comprehen-sion and numeral production mechanisms. Numeral comprehensionmechanisms are used to convert numeric inputs into central semantic repre-sentations for use in subsequent processing, such as calculation; and numeralproduction mechanisms translate central semantic representations of num-bers into specific forms for output. The numeral comprehension and numeralproduction systems are further subdivided into components for processingArabic numerals (e.g., 328) and components for processing verbal numerals(e.g., three hundred twenty-eight). Arabic numeral comprehension is re-quired to read a price tag, whereas writing a check makes use of both Arabicand verbal numeral production.

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Calculation Mechanisms

Verbal Numeral Comprehension

•Ighttimesthree

lexical processing

phonological

graphemic

syntacticprocessing

Arabic Numeral Comprehension

8 x 3

lexicalprocessing

syntacticprocessing

OperationProcessing

symbol

word

Procedures

additionsubtractionmultiplication

division

Facts

additionsubtractionmultiplication

division

Verbal Numeral Production

CENTRALSEMANTIC

REPRESENTATION

lexical processing

phonological

graphemic

syntacticprocessing

twenty-• four

Arabic Numeral Production

lexicalprocessing

syntacticprocessing

2 4

FIGURE 1 A model of numeric processing.

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Within each of the numeral comprehension and numeral production com-ponents, a distinction is drawn between lexical and syntactic processing.Lexical processing refers to comprehension or production of the individualelements in a numeral (e.g., 7 or seven), whereas syntactic processing in-volves processing of relations among elements (e.g., digit or word order) tocomprehend or produce a numeral as a whole. A final distinction is made inlexical processing for both verbal numeral comprehension and production.Phonological processing mechanisms are postulated for processing spokennumber words and graphemic processing mechanisms for written numberwords. For example, production of the spoken number word "ninety" re-quires retrieval of a phonological representation from a phonological outputlexicon, whereas written production of ninety requires retrieval of a graphe-mic representation from a graphemic output lexicon.

To perform calculations, special mechanisms are required in addition tonumeral comprehension and numeral production. The model posits threedistinct types of mechanisms: comprehension of operation symbols (e.g., +,- , x) and words (e.g., plus, minus, times), retrieval of arithmetic facts (e.g.,4 x 9 = 36), and execution of calculation procedures (e.g., in multidigitaddition, start at the far right column, compute the sum of the digits in thecolumn, write the ones digit of the sum at the bottom of the column, carry thetens digit, if any, etc.).

More than simply specifying the components of processing and flow ofinformation among these components, this model also posits many of thespecific computational operations carried out by subsumed components.However, discussion of these computational features falls outside of thescope of this article; interested readers are referred to McCloskey et al.(1986), McCloskey, Harley, and Sokol (1991), Sokol and McCloskey (1988),and Sokoletal. (1991).

Evidence

A substantial amount of data has been accumulated from studies of acquireddyscalculia, showing that cognitive numeric processing may be fractionatedby brain damage and that for the most part the observed dissociationsconform to those expected by the McCloskey et al. (1985) model. Forexample, several researchers have reported dissociations between numeralprocessing and calculation (e.g., Grewel, 1969; Hecaen et al., 1961; McClos-key et al., 1985). Dissociations have also been found between numeralcomprehension and numeral production (e.g., Benson & Denckla, 1969;McCloskey et al., 1986; Singer & Low, 1933), between Arabic and verbalnumeral processing (e.g., Berger, 1926; Grafman, Kampen, Rosenberg,Salazar, & Boiler, 1989; Macaruso et al., 1993; Noel & Seron, 1993), andbetween lexical and syntactic processing (e.g., Benson & Denckla, 1969;Singer & Low, 1933; Sokol & McCloskey, 1988). Within the domain ofcalculation, dissociations have been reported between operation symbolcomprehension and other calculation abilities (Ferro & Botelho, 1980),

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between retrieval of arithmetic facts and execution of calculation procedures(e.g., Sokol, McCloskey, et al., 1991; Warrington, 1982), and between re-trieval of arithmetic facts associated with different operations (Dagenbach &McCloskey, 1992).

Cognitive Neuropsychology and DevelopmentalDyscalculia

Although as stated above there have been a number of case studies ofdevelopmental dyscalculia reported in the behavioral neurology and neu-ropsychology literatures, few have attempted a cognitive neuropsychologi-cal approach to the problem. One exception is the work of Temple (1989,1991). In the past Temple and her colleagues have employed a cognitiveneuropsychological approach to study developmental language impairments(e.g., Temple, 1984, 1985, 1986; Temple & Marshall, 1983). Recently Tem-ple has extended this approach to the study of numeric processing im-pairments. In Temple (1989) she reported a case of an 11-year-old malesubject, with no known neurological abnormalities, who evidenced selectiveimpairments in numeral processing. Her second study (Temple, 1991) re-ported a double dissociation in two subjects, the first of whom had primarydifficulty with arithmetic procedures but no difficulty retrieving arithmeticfacts and the second of whom had primary difficulty with fact retrieval butintact procedural knowledge.

In describing the cognitive performance of her developmentally impairedsubjects, Temple noted that her results could be taken as general support forthe McCloskey et al. (1985) model. Shalev, Weirtman, and Amir (1988) alsoadopted this model in their developmental neuropsychological research,though its application for them was more on the order of a general concep-tual framework rather than an explicit explanatory model. Although resultssuch as these are promising with respect to the use of a model such as theMcCloskey et al. (1985) model in studies of developmental dyscalculia,clearly more work is needed. In particular it seems important to determine ifthe range of functional dissociations predicted by the model and obtained incases of acquired dyscalculia (e.g., syntactic vs. lexical numeral processing),is present in developmental cases. The present study was designed with thisgoal in mind.

METHOD

Subjects

Twenty students (16 boys, 4 girls) attending the Landmark School in PridesCrossings, Massachusetts, served as subjects. They ranged in age from 13 to20 years old, with a median age of 15 years old. (The ages of individual

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students are given in the Appendix.) All students enrolled at the LandmarkSchool have been diagnosed with developmental dyslexia. These studentsscore within the normal range on standard intelligence tests and do not sufferfrom a primary perceptual, neurological, or psychiatric disorder. Studentswere selected for our study on the basis of teacher referrals and test scoresindicating weaknesses in basic math skills.

Materials and Procedures

Each student was administered a modified version of the Johns HopkinsUniversity Dyscalculia Battery. The battery consists of tasks designed toprobe systematically the numeral processing and calculation mechanismsspecified in the McCloskey et al. (1985) model. Magnitude comparison tasks(e.g., which is larger, twenty or thirteen, 83,497 or 84,398?) are used to probenumeral comprehension. The battery includes separate magnitude compari-son tasks for Arabic numerals, spoken-verbal numerals, and written-verbalnumerals. Transcoding tasks (e.g., converting "eighteen" into 18, converting4,601 into four thousand six hundred one) are used to probe both numeralcomprehension and numeral production. The battery includes six transcod-ing tasks, one for each of the six possible conversions among Arabic, spo-ken-verbal, and written-verbal numerals. Each magnitude comparison andtranscoding task contains 20 items. Ten of the items involve numeralssmaller than 100 and 10 involve numerals in the range 101 through 99,999.The battery also includes single-digit arithmetic problems (e.g., 4 + 7, 8 x 5),which are used to probe arithmetic fact retrieval, and multidigit problems(e.g., 617 - 328, 23 x 86), which probe both fact retrieval and execution ofcalculation procedures. There are 20 single-digit and 10 multidigit problemsfor each arithmetic operation: addition, subtraction, multiplication, and divi-sion. (For further details on the Johns Hopkins University Dyscalculia Bat-tery, see Macaruso, Harley, & McCloskey, 1992).

Subjects were tested individually in one or more sessions lasting less than1 hr per session. In addition to completing the battery, some of the studentswere administered follow-up tasks. These tasks were selected on an individ-ual basis to obtain more information about the status of particular processingmechanisms. In most cases the follow-up tasks were identical in form tothose used on the battery. Any exceptions are described in the results section.

RESULTS

Overall the students displayed a wide range of performance levels on thebattery. Listed in the Appendix are individual students' error rates (expressedin percentage of errors) on the transcoding and arithmetic tasks from the

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battery. Students are listed according to how well they performed on thebattery, with more successful students listed first. Six of the students per-formed quite well, producing more than 10% errors on only one or two tasks.At the other extreme, three of the students performed rather poorly, produc-ing more than 10% errors on all or nearly all of the tasks. These studentsappeared to suffer from widely distributed impairments in numeric process-ing. The remaining 11 students produced more than 10% errors on three tofive tasks. These students appeared to have one or more selective distur-bances in numeric processing. For the present purposes we discuss theperformance of students who displayed relatively circumscribed forms ofimpairment in numeric processing.

Numeral Processing Versus Calculation

At the broadest level, the McCloskey et al. (1985) model proposes that thecognitive mechanisms involved in numeral comprehension and numeralproduction are functionally distinct from the mechanisms used to performcalculations. Support for this distinction can be seen in the performancepatterns of Rob and Tom. Rob produced errors at a rate of 33% (13/40) intranscoding numerals in the range 101 through 99,999 across four transcod-ing tasks (spoken-verbal to Arabic, spoken-verbal to written-verbal, Arabicto spoken-verbal, and Arabic to written-verbal).3 Examples of his errorsinclude 372 being produced as three seven hundred two, and "eight thousandtwo hundred seventeen" became 8,2017. Rob's poor performance on trans-coding tasks suggests a general impairment in numeral processing. In con-trast, Rob performed extremely well on arithmetic tasks, respondingcorrectly to 92% (110/120) of the single- and multidigit problems on thebattery. His only errors occurred in multidigit division, a source of difficultyfor virtually every student in the study. Thus, Rob's calculation mechanismsappeared relatively intact.

According to the McCloskey et al. (1985) model, solving arithmeticproblems implicates both numeral comprehension and numeral productionprocesses, as well as calculation mechanisms. Thus, one might have ex-pected Rob's poor numeral processing skills to affect his arithmetic perfor-mance. However, Rob's difficulties in numeral processing occurred mainlyfor numerals greater than 100. He made just one error in transcoding numer-als less than 100 across the four transcoding tasks. Because the component

3Typically we do not consider tasks that present numerals in written-verbal form whenassessing numeral processing skills in dyslexic students. Errors on tasks with written-verbalstimuli may arise as a result of their reading difficulties and thus not reflect a specific deficit inprocessing numerals. Also, given that dyslexic students often have spelling difficulties, weoverlook spelling errors when scoring written-verbal responses.

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processes involved in calculation (e.g., arithmetic fact retrieval) typicallymanipulate numerals smaller than 100, Rob's impaired ability to processlarge numerals did not affect his arithmetic performance.

Thus, Rob showed relatively intact calculation in the face of impairednumeral processing. The opposite pattern was displayed by Tom. He pro-duced just one error in transcoding numerals smaller than 100 across the fourtranscoding tasks, and his error rate for numerals in the range 101 through99,999 was only 10% (4/40). These results suggest at most only minordifficulties in numeral comprehension and numeral production. In contrastTom erred on 28% (33/120) of the single- and multidigit arithmetic problemson the battery. Of his 32 errors, 28 occurred in multiplication and division.For example, Tom wrote 50 in response to 4 x 5 and responded, "six", to"fifty-six divided by seven." On follow-up tests of single-digit multiplica-tion and division, Tom erred on 28% (140/495) of multiplication problemsand 44% (115/259) of division problems. In summary Tom displayed poorcalculation skills in the face of basically intact numeral processing abilities.

Numeral Comprehension Versus Numeral Production

The McCloskey et al. model also draws a distinction between numeralcomprehension and numeral production processes. The performance ofChuck on tasks involving Arabic numerals provides support for this distinc-tion. Chuck's comprehension of Arabic numerals was excellent, as evi-denced by his perfect performance in magnitude comparisons involvingArabic numerals in the range 101 through 999,999 (28/28 correct).

In contrast Chuck performed poorly on tasks requiring the production ofArabic numerals in the same range. In particular his error rate was 44%(29/66) in spoken-verbal-to-Arabic transcoding and 52% (34/66) in written-ver-bal-to-Arabic transcoding. Table 1 gives examples of his errors on these tasks.

As with all transcoding tasks, it is necessary to rule out comprehensiondifficulties in establishing a locus of impairment in the production system.Chuck's comprehension of spoken verbal numerals appeared intact, based onhis excellent performance in magnitude comparison for spoken verbal nu-merals in the range 101 through 999,999 (28/30 correct). Thus, his difficul-ties on the spoken-verbal-to-Arabic transcoding task may not be attributed toa spoken-verbal comprehension impairment but rather to an Arabic produc-tion impairment. However, in the case of written-verbal-to-Arabic transcod-ing it is not possible to rule out the contribution of a written-verbalcomprehension impairment because Chuck performed less well on written-ver-bal magnitude comparison for numerals in the same range (13/20 correct).

4Although Chuck frequently misplaced commas in his Arabic responses, we did not score aserrors those responses which were correct except for misplaced commas.

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TABLE 1Examples of Chuck's Errors on Numeral Transcoding Tasks

Stimulus Response

Spoken-verbal-to-Arabic"nine thousand nine hundred thirty" 9,9030"one hundred ninety-four thousand five" 100,945"five hundred six thousand one" 5,061"four hundred fifty-five thousand forty" 4,554,0

Written-verbal-to-Arabicthree thousand five hundred two 3,0502seventy thousand six hundred sixty 76,86ninety-one thousand four hundred seventeen 91,401,7one hundred three thousand five hundred 100,305

Arabic Numeral Processing Versus Verbal NumeralProcessing

According to the McCloskey et al. (1985) model, the processing of Arabicnumerals is subserved by separate mechanisms from the processing of verbalnumerals. Evidence for this distinction can be seen again in the transcodingperformance of Chuck. In the previous subsection we showed that Chuckwas impaired in production of Arabic numerals. His error rate was 44%(29/66) in spoken-verbal-to-Arabic transcoding for numerals in the range101 through 99,999. In contrast, Chuck had much less difficulty in produc-tion of verbal numerals. His error rate was only 18% (12/66) in Arabic-to-spoken-verbal transcoding for numerals in the same range, %2(1, N = 132) =12.72, p < .01.5 These results support a distinction between Arabic and verbalnumeral production.

Additional support for this distinction can be seen in Joe's performancepattern. Whereas Chuck was impaired in Arabic numeral production, Joe haddifficulty in verbal numeral production. Joe made no errors on the battery inspoken-verbal-to-Arabic transcoding ofnumerals in the range 101 through99,999, which suggests intact comprehension of spoken-verbal numerals andintact production of Arabic numerals. In contrast he responded incorrectly to40% (4/10) of the items in spoken-verbal-to-written-verbal transcoding fornumerals in the same range. For example, in response to "eighteen thousandone hundred forty-five," Joe wrote, "eight thosd seven hundd forty five."Because Joe's comprehension of spoken verbal numerals appeared intact, hispoor performance in spoken-verbal-to-written-verbal transcoding suggestsan impairment in written-verbal numeral production. Converging evidencefor this claim is seen in the Arabic-to-written-verbal task. On this task hemade 50% (5/10) errors on numerals in the same range as that previously

5In the previous subsection we presented evidence to suggest that Chuck's comprehension ofthe Arabic and spoken-verbal stimuli employed in these transcoding tasks was intact.

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noted, despite apparently intact Arabic comprehension. Given a rather lim-ited amount of data, however, our conclusions concerning Joe's performancepattern must remain tentative.

Lexical Processing Versus Syntactic Processing

The distinction between lexical and syntactic processing is readily apparentin the errors produced by Robyn on transcoding tasks requiring the produc-tion of Arabic numerals. For numerals in the range 101 through 999,999,Robyn's error rate was 40% (21/52) in spoken-verbal-to-Arabic transcodingand 48% (25/52) in written-verbal-to-Arabic transcoding. As illustrated inTable 2, her errors nearly always preserved the lexical identity of the non-zero digits but were syntactically ill-formed (e.g., "one hundred ninety-fourthousand five" was produced as 19405). Ninety-six percent (44/46) of hererrors were purely syntactic errors in which the nonzero digits were correctbut the response was of the wrong order of magnitude. Only two responsescontained a lexical error (e.g., "sixty-seven thousand three hundred four"became 6Q_,304). The fact that nearly all of Robyn's errors were purelysyntactic strongly supports the distinction between lexical and syntacticmechanisms in numeral processing. (Power and Dal Martello (1990) alsofound a large discrepancy between syntactic and lexical errors in a study ofspoken-verbal-to-Arabic transcoding in second-grade children. Like Robyn,the children produced many syntactic but few lexical errors, such as "twohundred eighty-one" became 20081).

Arithmetic Fact Retrieval Versus Execution ofCalculation Procedures

In the domain of calculation, the McCloskey et al. (1985) model separatesthe processes involved in retrieval of arithmetic facts from those involved inthe execution of calculation procedures. Support for this distinction is evi-dent when we compare the arithmetic performance of Matt and Robyn. Mattmade many errors in fact retrieval but had few problems in executing calcu-lation procedures. Robyn, on the other hand, showed excellent knowledge ofarithmetic facts but made numerous procedural errors in solving multidigitproblems.

The double dissociation is most evident when we compare Matt andRobyn's multiplication performance. Matt's success rate in solving single-

6In addition to not considering spelling errors (see footnote 2), we also try to give subjects thebenefit of the doubt when interpreting handwriting. Joe, in particular, presented with very poorhandwriting; therefore, all three authors scored his written-verbal output. When in doubt about aparticular letter, he was considered to have correctly produced it.

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TABLE 2Examples of Robyn's Errors on Numeral Transcoding Tasks

Stimulus

Spoken-verbal-to-Arabic"five hundred eleven""eight thousand sixty-seven""one hundred ninety-four thousand five""eight hundred seven thousand three hundred

twenty-three*Written-verbal-to-Arabic

nine thousand sixJive hundred four thousandthree hundred thousand foureight hundred five thousand nine hundred forty

Response

5011867

19405

87323

90654000

30485940

digit multiplication problems was only 66% (342/518). For problems in therange of 6 x 6 through 9 x 9 , his success rate dropped to 17% (14/82).Examples of his errors include "fifty-one" in response to 6 x 9, "forty-six"to "seven times seven," and 65 to 8 x 9.

In contrast to his poor fact retrieval performance, Matt was quite success-ful in carrying out the procedures required to solve multidigit problems. Forinstance, he responded correctly to 60% (6/10) of the multidigit multiplica-tion problems on the battery. Only one of his error responses was clearly dueto a problem in executing the multiplication procedure. Two were due to anerror in fact retrieval and one was ambiguous. More to the point, when Mattwas provided with a "crib sheet" containing the 100 multiplication facts withanswers, his success rate in solving multidigit problems rose to 83% (15/18).Of his three errors, none could be clearly attributed to a procedural difficulty.Two of his errors were due to faulty fact retrieval, and one was ambiguous.The fact retrieval errors occurred because Matt sometimes forgot to consulthis facts sheet.

The left column of Figure 2 shows examples of Matt's fact retrieval errorsin the context of accurately carrying out the multiplication procedure. In thefirst problem Matt retrieved the incorrect answer—35—to 6 x 7 (correctlycarrying the 1); in the second problem he again incorrectly retrieved 35, thistime as the product of 5 x 6; finally in the third example, he retrieved 27 asthe product of 7 x 3. These results show that Matt's difficulties in solvingmultiplication problems arose from an impairment in fact retrieval and notfrom difficulties in execution of the multiplication procedure.

The opposite pattern is seen in the multiplication performance of Robyn.In response to 120 single-digit multiplication problems, Robyn's successrate was 93%. All of her errors except one were instances in which Robynresponded N to a 0 x N problem (e.g., 0 x 6 = 6). Answers to problems like0 x N are presumably solved by rule (i.e., 0 x N = 0), and thus errors like 0x N = N appear to reflect the application of the wrong rule rather than aproblem in fact retrieval per se (Sokol et al., 1991). Although Robyn dis-played excellent fact retrieval abilities, she had a great deal of difficulty

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MATT ROBYN

I72

x 60

Q

I3

806X ,35

193x 17

FIGURE 2 Examples of fact versus procedural errors in Matt and Robyn.

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solving multidigit multiplication problems. Her success rate in solving thesame problems presented to Matt was only 39% (11/28). Of her 17 errorresponses, 12 contained at least one mistake in applying the multiplicationprocedure. Only one contained a clear multiplication fact error and four wereambiguous.

Examples of her errors in executing the multiplication procedure areshown in the right column of Figure 2. In the first problem, although shecorrectly retrieved the products of 6 x 2 and 6 x 7 , she made severalprocedural errors that included neglecting to carry the 1 from the partialproduct 12, failing to shift the second row of partial products, and omittingthe required final addition procedure. In the second problem, regardless ofthe product obtained, she apparently carried a 1; then she neglected the 3 x0 product, carrying the 1 to the 3 x 8 product; also, she misaligned her partialproducts, thus leaving out the left-most zero in the first row's partial productduring the addition procedure. In the final example, she inappropriatelyadded the carry digit 6 from the first row of partial products to the second(i.e., to the product of 1 x 1).

In summary, the contrast between Matt and Robyn's multiplication per-formance provides strong support for the processing distinction betweenarithmetic fact retrieval and execution of calculation procedures proposed inthe McCloskey et al. (1985) model. (As noted above, Temple, 1991, alsofound a double dissociation between fact retrieval and calculation proce-dures in developmental dyscalculia.)

Calculation Versus Approximation

Although not made explicit in the McCloskey et al. (1985) model, a distinc-tion between calculation and approximation has been included in othermodels of numeric processing (e.g., Dehaene, 1992). Evidence in support ofthis distinction can be found in both acquired and developmental dys-calculia. For example, Dehaene and Cohen (1991) discussed a patient withacquired dyscalculia who produced numerous errors in solving simple calcu-lations (e.g., 2 + 2 = 3) but quickly rejected implausible answers on averification task (e.g., 2 + 2 = 9). Dehaene and Cohen concluded that thepatient lost the ability to carry out precise arithmetic operations but retainedthe ability to activate approximate quantity information (see alsoWarrington, 1982, for discussion of a similar case). Russell and Ginsburg(1984) reported that children with mathematics difficulty who showed sub-par performance on single- and multidigit calculations were able to judgewhether an answer is close in magnitude to the correct answer on an estima-tion task (e.g., is 926 close or far away from the real answer to 53 + 28?).

The opposite dissociation—excellent calculation in the face of weakapproximation skills—can be seen in the performance of Doug. In solvingsingle-digit addition, subtraction, multiplication, and division problems,Doug responded with a 97% (551/570) success rate. Doug also correctly

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solved 90% (27/30) of the multidigit addition, subtraction, and multiplica-tion problems on the battery and solved an additional set of multidigitmultiplication problems with 89% (16/18) accuracy. Overall Doug's calcula-tion performance was superior to that of nearly all of the dyslexic students inthe study.

In contrast to his excellent calculation skills, Doug had a great deal ofdifficulty on tasks that required approximating or estimating numeric an-swers. On one task Doug was asked to select the correct answer (out of four)to an arithmetic problem without performing any calculations (e.g., 40 x 20= 80, 800, 8000, or 80,000). He was presented with 54 multiplication prob-lems, 10 division problems, 5 addition problems, and 5 subtraction prob-lems. His success rate was 41% (22/54) for multiplication and 50% (10/20)across the other operations.

To gauge how poorly Doug performed on this task, we compared hisperformance with Tom's on the same task. Recall that Tom produced numer-ous errors in single- and multidigit calculations, especially for multiplicationand division. Despite his poor calculation skills, Tom outperformed Doug onthe approximation task. Tom's success rate of 63% (34/54) for multiplicationwas significantly higher than Doug's 41% success rate, % (1, N = 108) =5.34, p < .05. Tom also scored higher than Doug across the other operations(14/20 correct).

When Doug was given the four-choice task a second time with scrap paperto help him derive the correct answer, his performance improved dramati-cally. He responded correctly to 91% (49/54) of the multiplication problemsand to 90% (18/20) of the other problems. Examples of Doug's incorrectselections on the four-choice task later on, together with his correct solutionsto the same problems using scrap paper, are shown in Figure 3.

On a second task Doug was asked to provide numeric answers to tenestimation questions (e.g., "About how many miles is it from Boston to LosAngeles?"). Many of Doug's responses were extremely inaccurate. For com-parative purposes we asked the same questions of 38 age-matched controlsubjects. Doug's response fell completely outside of the control range onthree of the ten questions (e.g., Doug responded "ninety-six" to the Bos-ton/Los Angeles question) and at the fringes of the control range on fourothers (e.g., When asked, "About how many pounds does an average brickweigh?", Doug responded, "Twenty-four"). In summary, these results showthat Doug's poor numeric approximation and estimation skills dissociatedgreatly from his excellent calculation skills. (See Sokol, Macaruso, &Gollan, 1991, for description of several other dyslexic students with estima-tion problems similar to Doug's.)

DISCUSSION

In this article we first briefly reviewed the literature on developmentaldyscalculia, citing examples of both group and single-case studies. Despitethe interest that such studies have sparked in the area, each approach appearsto suffer from certain theoretical and/or methodological shortcomings. As an

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COro

10 I 90 -

109 190 900

1.000 « 10 •

100,000 10,000 1,000.000

\ pot* -

I O6OY

40X20.

W ) 800 8.000 80.000

5x62.

110 //4| MO 5«O

7^~0 •n

4*

FIGURE 3 Examples of Doug's poor approximation ability for correctly solved problems.

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alternative, we chose to investigate developmental dyscalculia from a cogni-tive neuropsychological perspective. In so doing, we utilized the modularinformation-processing model initially proposed by McCloskey et al.(1985). Two interrelated points can be made from our preliminary empiricalstudy: (1) the patterns of impaired performance exhibited by subjects withdevelopmental impairments appear to conform closely to those of acquireddyscalculic patients; and (2) the McCloskey et al. model, which itself wasinformed by data from acquired dyscalculia, provides a useful theoreticalframework for understanding developmental dyscalculia.

In the sections that follow, we take up two issues that relate to themethodology of this study and conclude with a brief discussion of directionsfor future research.

Dyslexia and Dyscalculia

Despite the apparent match between our data and the McCloskey et al.(1985) model, at least one methodological aspect of our study could becalled into question, namely, all of our subjects had impairments not only tonumeric processing, but to reading as well. That is, these students were allenrolled in a private school specializing in the remediation of written lan-guage impairments, and each subject had at least a concurrent if not primarydiagnosis of developmental dyslexia. Is it valid, therefore, to claim that ourresults are in any way generalizable to students with specific arithmeticdisability (i.e., without reading disability)?

We claim that at the level of analysis with which we have been concerned,the answer is yes. The issue under discussion is, after all, not whethernumeric processing impairments sometimes or even often co-occur withother cognitive impairments, but rather how one can best understand thenumeric impairments themselves, regardless of comorbidity. Marshall(1989) discussed this topic at some length with respect to the correlation ofdyslexia with other cognitive disorders (see pp. 78-80). His conclusionregarding reading summarizes our own regarding numeric processing:

The basic issue of describing the patterns of impaired and preserved perfor-mance within the domains of reading and writing remains crucial to any even-tual theoretical understanding of the mechanisms involved. It is at bestunfortunate that the question of patterns of association and dissociation withinreading and writing skills was for so long confused by the issue of associationsbetween reading or writing deficits and deficits in other areas of cognitiveperformance, (p. 80)

We add that, although the relationship between numeric processing andother cognitive systems such as reading is a potentially interesting one, it isfar from straightforward on either theoretical or empirical grounds (seeMcCloskey et al., 1986, pp. 324-325, for further discussion on this point).

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Our aim in this article therefore has simply been to demonstrate how a modelof cognitive numeric processing is informative of patterns of impairment indevelopmentally impaired students. With this model as a basis for formulat-ing specific hypotheses, it may be possible in future research to explore inmore detail the comorbidity of arithmetic and other cognitive impairments(e.g., reading).

Developmental Cognitive Neuropsychology

There has been considerable controversy in the literature involving compar-ison of acquired and developmental impairments and more generally regard-ing the use of data from developmentally impaired subjects to inform modelsof normal cognition (e.g., Ellis, 1985; Marshall, 1989; Morton, 1989). Atissue is whether the surface similarities noted between the performance ofdevelopmentally impaired subjects and those who have suffered acquiredbrain damage are more than just "spurious, arising, as they do, from thestructure of the material (in this case reading material) rather than from thestructure of the disorders" (Morton, 1989, p. 43). In the case of the presentstudy, the argument could be made that, although our data do bear closeresemblance to those reported in studies of acquired dyscalculia, it is still nota foregone conclusion that the identical computational systems are at workin developmental and acquired cases.

To be sure, there are compelling arguments to be found on both sides ofthis issue- and we do not presume to add much to the theoretical debate atthis time. What we do minimally claim, however, is that the models devel-oped to account for patterns of impaired performance in acquired cases mayprove good first approximations of models of arrested cognitive develop-ment within the same domain. Furthermore, we argue that the basic approachof cognitive neuropsychology, which views behavioral deficits as resultingfrom specific impairments to an otherwise normal cognitive system, can beas useful in understanding developmental disorders as it has been in under-standing acquired ones. Such an approach not only holds much theoreticalpromise in informing models of cognitive development but also provides afoundation for more precise assessment of cognitive impairments (seeMargolin, 1992, on this point and particularly the chapter by Macaruso et al.on assessment of dyscalculia). Accurate assessment in turn allows for thestructuring of motivated remedial programs (Byng & Coltheart, 1986; How-ard & Patterson, 1989; Deloche, Seron, & Ferrand, 1989).

7For the record we tend to side with Marshall (1989) on this point. That is, in a modularcognitive system such as the one we are claiming is at work in numeric processing, it is likelythat basic architectural components are available to the developing cognitive system but thattheir internal operations must be triggered by specific experience. In this way, one might expectsimilar patterns of impairment to emerge either as a result of arrested cognitive development oracquired brain damage.

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Future Directions

In this article we have focused almost exclusively on the basic architectureof cognitive numeric processing, ignoring for the most part specific compu-tational issues (e.g., in what way precisely is syntactic numeral processingimpaired in Robyn). This focus seemed to us the logical first step in determiningthe adequacy of the model under use in the study of developmental cases, a stepsimilar to that taken by McCloskey et al. (1985) in their initial presentation ofthe model and its relationship to patterns of acquired dyscalculia.

Although continuing to gather data that address cognitive architectureissues, we have now begun to focus our attention on three related areas.First, we are conducting several in-depth, single-case studies examining theinternal mechanisms of individual cognitive components of the model. Oneexample is the Arabic production component. As noted previously, bothRobyn and Chuck make specific errors in Arabic production. Through sys-tematic analysis of these errors, we hope to learn more about the mechanismsof Arabic numeral production, especially syntactic aspects and how thesemay go awry in cognitive development.

A second area of focus is consideration of elements of numeric processingnot specifically included in the McCloskey et al. (1985) model. We havealready touched upon one such area in this article, namely estimation abili-ties. We are currently engaged in a more comprehensive study of estimationimpairments in several developmentally impaired students, which we hopeto relate to an emerging body of work on number estimation abilities in thenormal cognitive literature (see Brown & Siegler, 1993, for review).

Finally, given the potential of functional plasticity in cases of develop-mental impairment, it seems incumbent upon researchers working within thisfield to link their theoretical and empirical work to the problem of cognitiveremediation. Along these lines, we have begun to include a remediationcomponent in our investigations of developmental dyscalculia. Given theprevalence of arithmetic fact retrieval impairments in developmental cases,we have targeted this area in one of our remediation studies. In addition toattempting to strengthen fact representations in our subjects, we are concur-rently addressing at least two theoretical issues in this work: (a) Is the effectof remediation sensitive to the modality of stimulus presentation (i.e., verbalor Arabic input)? and (b) Are there qualitative differences in the learningparameters displayed by developmentally impaired students compared withyounger (normally developing) children at the same baseline level of factknowledge? Answers to these questions should not only impact on futureremediation attempts but should also inform basic scientific theory in nu-meric and arithmetic processing.

ACKNOWLEDGMENTS

A subset of these data was previously reported at the annual meeting of theSociety for Neuroscience in New Orleans, LA, in 1991.

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This research was supported in part by a grant from the McDonnellFoundation awarded to the Massachusetts General Hospital NeurolinguisticsLaboratory.

The first two authors shared equal effort on this project.Tamar Gollan is now in the Department of Psychology, University of

Arizona, Tucson.The authors wish to thank the students at the Landmark School for their

participation, as well as Christopher Woodin and Mary Jane McCready fortheir cooperation and referrals. We also thank Valentina Ruzecki and KaraSullivan for assistance in collecting and analyzing data at Landmark.

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APPENDIXPercent Errors on the Transcoding and Arithmetic Tasks From the Dyscalcuila Battery

Student

DougSteveBrendaHeather R.JasonBradGlenJoeIvanJonChrisTomT.RobHeatherTomChuckMattDanaRobynSean

Age

1313131418141714131618152018141320161818

A-SV

00500

1000000000

1000

101515

A-WV

0000550

25555

1030300

250

151560

Transcoding

SV-A

05

1505

10105

101510• 51555

155

101545

SV-WV

101550050

255

10100

2515101030450

55

WV-A

50005

25155

201520302025253020701580

WV-SV

0500

10100

105

105

1000

255

20350

75

Addition

30333707

103

17003333

131760

Arithmetic

Subtraction Multiplication Division

3303

17037

2033

1307

1377

231763

1030

1330

3313231713130

17471720233097

3313172323203337303333233320473337705390

Note. A-SV = Arabic-to-spoken-verbal; A-WV = Arabic-to-written-verbal; SV-A = spoken-verbal-to-Arabic; SV-WV = spoken-verbal-to-written-verbal; WV-A = written-verbal-to-Arabic; WV-SV = written-verbal-to-spoken-verbal.

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Developmental Dyscalculia and Cognitive...

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