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Journal of Experimental Child Psychology 74, 240–260 (1999)Article ID jecp.1999.2516, available online at http://www.idealibrary.com onWorking Memory Impairments in Children withSpecific Arithmetic Learning DifficultiesJanet F. McLean and Graham J. HitchLancaster University, Lancaster, United KingdomWorking memory impairments in children with difficulties in arithmetic have previouslybeen investigated using questionable selection techniques and control groups,leading to problems concluding where deficits may occur. The present study attempted toovercome these criticisms by assessing 9-year-old children with difficulties specific toarithmetic, as indicated by normal reading, and comparing them with both age-matchedand ability-matched controls. A battery of 10 tasks was used to assess different aspects ofworking memory, including subtypes of executive function. Relative to age-matchedcontrols, children with poor arithmetic had normal phonological working memory butwere impaired on spatial working memory and some aspects of executive processing.Compared to ability-matched controls, they were impaired only on one task designed toassess executive processes for holding and manipulating information in long-term memory.These deficits in executive and spatial aspects of working memory seem likely to beimportant factors in poor arithmetical attainment. © 1999 Academic PressKey Words: working memory; executive processes; arithmetic; children; learningdifficulties.There are many reasons children may fail to learn arithmetic. Examplesinclude anxiety about mathematics, lack of experience and poor motivation(Ashcraft & Faust, 1994; Levine, 1987), reading difficulties (Muth, 1984; Richman,1983), and neuropsychological damage (McCloskey, Harley, & Sokol,1991). A growing body of evidence suggests that arithmetical learning difficultiescan be associated with cognitive deficits (e.g., Bull & Johnston, 1997; Geary& Brown, 1991; Geary, Brown, & Samanayake, 1991; Hitch & McAuley, 1991;Rourke & Findlayson, 1978; Rourke & Strang, 1978; Siegel & Ryan, 1989;Temple, 1991). The present study builds on this work by focusing on cognitivedeficits in a subgroup of children within the general population who have specificdifficulties in arithmetic but normal reading.This research was conducted in partial fulfillment of a Ph.D. by the first author. The cooperationof schools in the Lancaster area and the financial support of the Economic and Social ResearchCouncil are gratefully acknowledged.Address correspondence and reprint requests to Janet McLean, Department of Psychology, FlorentineHouse, University of Glasgow, Glasgow G12 8BQ, United Kingdom.0022-0965/99 $30.00Copyright © 1999 by Academic PressAll rights of reproduction in any form reserved.240

WORKING MEMORY AND ARITHMETIC DIFFICULTIES241Describing the cognitive deficits of this subgroup can be problematic. Studiesbased on clinical samples have suggested three different types of cognitive deficitin arithmetic (see Badian, 1983, and Geary, 1993, for reviews). These arecharacterized as visuospatial deficits (Hartje, 1987; Rourke, 1993; Rourke &Findlayson, 1978; Rourke & Strang, 1978), difficulties with reading and writingof numbers (Kosc, 1974; Temple, 1989), and problems of retrieving arithmeticfacts from long-term memory (Badian, 1983; Benson & Weir, 1972; Jackson &Warrington, 1986). White, Moffitt, and Silva (1992) found that arithmetic disabledchildren performed poorly not only on the visuospatial tasks used byRourke and colleagues, but also on the Making Trails tasks (Reitan, 1958), theGrooved Pegboard (Klove, 1963), and the WISC-R Coding (Wechsler, 1974),which they interpreted as reflecting difficulties with visual–motor integration.Taken together, clinical studies suggest that difficulties with arithmetic areassociated with a variety of deficits. However, it is difficult to extrapolate fromthese observations to the general population.In a comprehensive review of cognitive deficits in arithmetic disabled childrenwhich included nonclinical studies, Geary (1993) identified two basic categorieswith different developmental trajectories. The first is manifested by the use ofdevelopmentally immature arithmetic procedures which usually regain normallevels after 2 or 3 years of schooling. This category has been associated withdeficits in counting, computational skill, and working memory (Geary, 1990;Geary, Bow-Thomas, & Yao, 1992). The second category involves a morepersistent difficulty with the representation and retrieval of arithmetic facts fromlong-term semantic memory. Fact-retrieval difficulties have been linked to deficitsin counting speed and working memory (Garnett & Fleischner, 1983; Geary,1990; Geary et al., 1991). It is interesting to note that both of these categoriesinclude working memory, the limited capacity system responsible for transformingand maintaining temporary information (see, e.g., Baddeley & Hitch, 1974).A number of previous studies have compared measures of working memory innormal children, children with arithmetic difficulties, and children with enhancedarithmetical ability (e.g., Bull & Johnston, 1997; Dark & Benbow, 1990, 1991;Geary et al., 1991; Hitch & McAuley, 1991; Siegel & Ryan, 1989; Swanson,1993, 1994). Although these studies differ in their assessment and aims, theyagree in suggesting that working memory is related to differences in the abilityto perform arithmetic. For example, Geary et al. (1991) investigated arithmeticallydisabled and normal children over 1 year. The disabled group receivedremedial education in mathematics and were below the 46th percentile onnational achievement tests. Both forward and backward digit span were significantlyreduced for the disabled group, consistent with a working memory deficit.Geary et al. (1991; see also Geary, 1990, 1993) proposed that although skills suchas counting knowledge and strategy choice are the primary areas of deficit in themathematically disabled child, working memory deficits may lead to a failure todevelop long-term memory representations of basic facts. Geary et al. (1991)

242 MCLEAN AND HITCHsuggest that if the representation of a problem’s integers is lost more rapidly fromworking memory it is less likely to be associated with the answer. However, theirarithmetic disabled children also had poor reading skills. Their children’s averagepercentile ranking for reading was actually below that for arithmetic, and wasmuch lower than the reading ranking for the control group. This is importantbecause a significant amount of language is used in mathematics, and problemsin reading and mathematics often co-occur (Muth, 1984; Richman, 1983; seeGeary, 1993, for a review). Therefore, children with more general academicdifficulties have been included in Geary and colleagues’ studies, and this leads toproblems interpreting their findings with respect to arithmetic per se.Evidence that working memory deficits differ as a function of the specificityof learning difficulties is presented by Siegel and Ryan (1989). They gavechildren two complex span tasks designed to assess working memory capacity.These were a counting span task (Case, Kurland, & Goldberg, 1982) and asentence span task (adapted from Daneman & Carpenter, 1980). Children withspecific arithmetic difficulties (i.e., WRAT Arithmetic 25th percentile andWRAT Reading 30th percentile) were impaired on counting span but not onsentence span. In contrast, children who were impaired on reading as wellas on arithmetic had significantly lower counting and sentence spans. Siegel andRyan suggested that the reading-and-arithmetic disabled group had a generalworking memory impairment, whereas the arithmetic disabled group had adomain-specific working memory deficit. In a related study, Hitch and McAuley(1991) confirmed that children with specific arithmetic difficulties were impairedon counting span but not on other complex span tasks. They also showedthat these children had significantly lower digit spans than did age-matchedcontrols.A general criticism of the studies described above is that they have employeda design in which children with arithmetic difficulties are compared to normalachievers of the same chronological age. This type of design cannot determinethe cause of the disability, as any cognitive deficits may be a consequence ratherthan a cause of low arithmetic attainment. Vellutino, Pruzek, Steger, and Meshoulam(1973) attempted to resolve this problem in the area of reading difficultiesby suggesting that a younger ability-matched control group should beincluded. The logic is that if the poor achievers perform worse on some cognitivetask than do younger, ability-matched controls, the deficit is unlikely to be asimple consequence of their low level of academic achievement. Given poorachievers’ greater age and experience, such a deficit would suggest they arearriving at similar levels of ability via different routes. However, Backman,Mamen, and Ferguson (1984) noted that age-matched controls remain useful foridentifying differences in cognitive function at the same level of exposure to theskill in question.A second concern about previous work is the sparsity of attempts to assess thevarious facets of working memory in children with poor arithmetic. If this system

WORKING MEMORY AND ARITHMETIC DIFFICULTIES243is impaired, it is clearly important to know more precisely how it is affected.There are several different ways of viewing working memory (see, e.g., Baddeley& Hitch, 1974; Daneman & Carpenter, 1980; Just & Carpenter, 1992; Turner &Engle, 1989). The Baddeley and Hitch (1974) multicomponent model was usedhere, as it explains a wide range of experimental and neuropsychological evidence(see also Baddeley, 1986) and tasks have been developed for assessingsome of its components. According to this model, working memory consists ofa central executive processor which interacts with two slave subsystems, thephonological loop and the visuospatial sketch pad. The phonological loop isspecialized for the storage and rehearsal of speech-based verbal information(Baddeley, 1992a, 1992b), whereas the sketch pad is specialized for holdingvisual and spatial material (Logie, 1986; Quinn & McConnell, 1996). In a recentdevelopment, Baddeley (1996) proposed a fractionation of the central executiveinto separate but overlapping functions of coordinating concurrent activities,switching retrieval plans, attending to inputs, and holding and manipulatinginformation in long-term memory.Evidence from studies of the normal population suggests that different componentsof working memory may have specialized roles in arithmetic (Ashcraft,1995). For example, the phonological loop appears to be involved in counting(Logie & Baddeley, 1987) and in holding information in complex calculations(Logie, Gilhooly, & Wynn, 1994; see also Fürst & Hitch, in press). Thevisuospatial sketch pad appears to be involved in multidigit problems wherevisual and spatial knowledge of column positioning is required (Heathcote,1994). Other researchers have noted the use of a visual number line (Dehaene,1992) and spatial representations of individual numbers (Hartje, 1987), butwithout linking them to working memory. The role of the central executive hasbeen noted several times (see, e.g., Ashcraft, Donley, Halas, & Vakali, 1992;Lemaire, Abdi, & Fayol, 1996; Logie et al., 1994). Ashcraft suggested that theexecutive is responsible for initiating and directing processing, comprehension,and retrieval from long-term memory. However, remarkably little is known.Accordingly, the following discussion generates tentative hypotheses about therole of different executive functions in arithmetic.The first executive function identified by Baddeley (1996) is the capacity tocoordinate performance on two or more separate tasks. Arithmetic can beconsidered a multiple task when it involves subtasks such as calculating partialtotals and keeping track of other information. However, given that the subtasksare parts of an integrated skill, the requirement for coordination is presumablylow relative to performing multiple independent tasks. The second executivefunction is switching retrieval strategies. This is clearly necessary for problemssuch as multidigit multiplication, which typically involves both multiplying andadding. Experimental evidence suggests that switching is also required forcarrying operations (Fürst & Hitch, in press). The third executive function isattending selectively to different inputs. This is clearly a feature of multidigit

244 MCLEAN AND HITCHproblems carried out as a series of subtasks, where attention is paid to selectedparts of a problem at different times. The fourth executive function is activatingand manipulating information in long-term memory. This seems likely to beinvolved in operations such as using the equivalence of two relationships (e.g.,5 2 4 3) in order to simplify a calculation. In summary, therefore, nearlyall the components of working memory seem likely to be involved in arithmeticalcalculation, each playing a somewhat different role.In light of the foregoing considerations, the present study had two aims. Thefirst was to gain a more complete picture of working memory deficits in childrenwith specific arithmetic difficulties. This was achieved by using a battery of tasksassessing different components of working memory. The second aim was toconsider whether working memory deficits are responsible for children’s difficultiesin arithmetic. This was investigated by applying the age-matched andability-matched design described above. To ensure the selectivity of arithmeticaldeficits, all three groups were matched on reading. That is, in all three groups,children’s reading was at normal, age-appropriate levels. An incidental advantageof using reading in this way is that it avoids controversial issues associatedwith matching groups on general intelligence (Siegel, 1988).The battery for assessing working memory comprised two phonological looptasks, Digit Span and Nonword Repetition (Gathercole, Willis, Baddeley, &Emslie, 1994); two visuospatial sketch pad tasks, Visual Matrix Span (Wilson,Scott, & Power, 1987) and Corsi Block Span (Milner, 1971); and five centralexecutive tasks. The central executive tasks consisted of three Making Trailstasks (Reitan, 1958), to assess switching between retrieval plans; a Crossing Outtask (Moran & Mefford, 1959), to measure selective attention; and a novel“Missing Item task,” designed as a measure of the ability to hold and manipulateinformation in long-term memory. All the tasks used numerical materials whereverpossible, in order to maximize chances of observing deficits in the poorarithmetic group. A general assessment of working memory in the arithmeticdomain was also made. This was provided by Addition Span (Adams & Hitch,1997), a measure of how complex a mental addition children could performaccurately. More detailed rationales for these tasks are given in the Methodsection.Expectations for comparisons between children with poor arithmetic (PoorArithmetic) and their age-matched peers (Age-Matched) were as follows. Followingprevious work it was expected that children with Poor Arithmetic wouldbe impaired on Digit Span (see, e.g., Geary et al., 1991; Hitch & McAuley, 1991;Siegel & Linder, 1984). Nonword Repetition should be similarly impaired if thereduction in digit span is due to a deficit associated with the phonological loop.If Rourke’s ideas generalize to the normal population, the Poor Arithmetic groupshould be impaired on the two visuospatial tasks, Corsi Blocks Span and VisualMatrix Span. Least clear of all, but perhaps most interesting, were expectationsfor the central executive tasks. Previous work demonstrating a deficit in counting

WORKING MEMORY AND ARITHMETIC DIFFICULTIES245span (Siegel & Ryan, 1989) could be interpreted as suggesting an executivedeficit. However, this conclusion may not be justified (Hitch & McAuley, 1991;see also Towse & Hitch, 1995). Furthermore, if the executive can be fractionated,then the possibility arises of deficits in specific executive functions. Of thoseconsidered, activating long-term memory seemed particularly interesting givenprevious work suggesting impaired access to long-term memory representationsin arithmetic disabled children (Koontz & Berch, 1996).No predictions were made for comparisons between the Poor Arithmetic groupand the Ability-Matched controls, as this comparison has not previously beenmade. However, any deficit in the Poor Arithmetic group would be particularlyinteresting, as it would suggest a cognitive impairment that is not simply aconsequence of low attainment.Children were selected from a large sample attending normal schools byapplying cutoffs to arithmetic and reading scores. This was considered moreappropriate than using discrepancy scores (e.g., Swanson, 1994) or regressioncriteria, given the aim of investigating children with poor arithmetic in comparisonto the chronological age norm. It also overcomes problems with poorapproximation to the normal distribution. (For a review of selection techniquessee Fletcher et al., 1989). The cutoff procedure selects children that fall below aset criterion for arithmetic ability and above one for reading ability. Children inthe poor arithmetic group were chosen from the 4th year of schooling, as it wasfelt that by this age differences between reading and arithmetic would haveemerged and it would be possible to find younger ability-matched controls.METHODParticipantsOne hundred twenty-two children (64 females and 58 males) participated inthe initial part of the study. Twenty-two were in their 3rd year of schooling (meanage 7 years 11 months), and 100 were in their 4th year of schooling (mean age9 years 0 months). They attended five local primary schools in the Lancaster area.All children were administered the Graded Arithmetic–Mathematics Test (GAM;Vernon & Miller, 1976), the Primary Reading Test (France, 1979), and six 1-minwritten tests of speeded calculation (single- plus single-digit addition, singleplustwo-digit addition, two- plus two-digit addition, subtraction, multiplication,and division; adapted from Hitch, 1978).The Graded Arithmetic–Mathematics Test is a standardized group test forchildren aged 6–12 years. It has a time limit of 30 min and contains 70 writtencomputation problems. These begin with arithmetical items and end with mathematicalitems. In the present study only arithmetic was assessed, as childrennever got as far as the mathematical items in the time limit. The Primary ReadingTest (PRT) is another standardized group test suitable for 6- to 12-year-oldchildren. It comprises 10 picture–word matching items and 38 sentence completionitems and has a time limit of 30 min. Level 2 of the test was used in the

246 MCLEAN AND HITCHpresent study, as this was suitable for children aged above 7 years. Given that thestandardized scores for the two assessment tests were calculated more than 20years ago, raw scores were used to allocate children to groups.Twelve children (7 girls and 5 boys) were selected for the Poor Arithmeticgroup on the basis that they fell in the bottom 25% of raw scores on the GradedArithmetic–Mathematics Test and the middle 50% of raw scores on the PrimaryReading Test. This number constituted 8.5% of the 4th-year sample. TwelveAge-Matched controls (6 girls and 6 boys) were selected whose arithmetic andreading raw scores were in the middle 50% of the sample. A further 12Ability-Matched controls (8 girls and 4 boys) were selected from the 3rd-yearsample on the basis that their arithmetic raw scores were similar to those of thePoor Arithmetic group and their reading was normal for their age. Descriptivestatistics for each group are shown in Table 1.T tests were performed to ensure that the groups were adequately matched.These showed that the Poor Arithmetic group and Age-Matched controls did notdiffer on reading, t(22) 1, but did on arithmetic, t(22) 10.1, p .001. ThePoor Arithmetic group and Ability-Matched controls did not differ on arithmetic,t(22) 1, but did on reading, t(22) 2.59, p .05.Working Memory BatteryPhonological LoopAuditory digit span. The forward auditory Digit Span task from the WISC-Rsubtest (Wechsler, 1974) was used as a standard measure of phonologicalshort-term memory. Children were given the test twice, and digit span wascalculated as the mean of the two scores. The reported reliability for 8- to16-year-old children on this task is 0.85 (Weschler, 1974).Nonword repetition. The Children’s Test of Nonword Repetition (Gathercoleet al., 1994) was used as a second index of phonological short-term memory. Thistest involves the immediate repetition of auditorily presented nonwords and thusrequires temporary storage of unfamiliar phonological sequences. Its reliabilityfor 7-year-old children (the eldest reported) is 0.80 (Gathercole et al., 1994).Nonword repetition scores correlate reasonably well but not highly with digitspan (Gathercole, Willis, Emslie, & Baddeley, 1992), suggesting that the twotasks tap a mixture of common and separate abilities.The test consists of 40 nonwords, with 10 items at each length of from two tofive syllables. The child was asked to listen to a tape recording of a femalespeaker reading aloud the nonwords in a neutral English accent and to repeat eachone immediately after its presentation. Items were presented in a random orderand there was a 3-s interval between the end of one item and the onset of the next.Credit was given for each correct repetition, with immediate self-correctionscredited as a correct response.

WORKING MEMORY AND ARITHMETIC DIFFICULTIES247TABLE 1Descriptive Statistics for the Selected GroupsMeasuresPoor Arithmetic(N 12)Age-Matched(N 12)Ability-Matched(N 12)Chronological ageMean 9.08 8.90 7.94SD 0.35 0.37 0.32ArithmeticRaw scoresMean 15.92 23.25 16.08SD 1.98 1.60 1.56Arithmetic ageMean 8.02 9.59 8.05SD 0.49 0.29 0.39ReadingRaw scoresMean 29.00 29.58 25.75SD 3.51 1.93 2.56Reading ageMean 9.00 9.10 8.33SD 0.68 0.33 0.54Speeded calculationTotal scoreMean 44.42 72.67 40.10SD 3.92 22.01 11.12Visuospatial Sketch PadVisual matrix span. A matrix span task adapted from Wilson, Scott, and Power(1987) was used to measure visual working memory. This task assesses immediaterecognition memory for random visuospatial patterns of increasing complexity.Stimuli are presented relatively briefly in order to minimize the possibilityof using verbal descriptions.The present version was administered using a laptop computer. Stimuli consistedof a set of 15 matrices made up of different numbers of black boxes andwhite boxes. Each box was 2 cm square, and was drawn on a dark background.

248 MCLEAN AND HITCHThe first matrix had two boxes, one black and one white, and appeared in the topleft-hand side of the screen for 2000 ms. This was followed by a blank screen for2000 ms before a similar matrix appeared. The second matrix differed from thefirst in that the black box had changed to white. Children were asked to point tothe square which had changed. Once the child had given a response, the matrixincreased by two more squares either down or across the screen. The boxes werefilled black and white randomly, and the procedure was repeated as before. Trialscontinued up to a 30 30 matrix. Span was measured as the largest matrixremembered correctly before two successive failures. Each child completed thetask twice and the two values of span were averaged.Corsi blocks span. The Corsi blocks task (Corsi, 1972), described by Milner(1971), was administered to measure spatial span. The apparatus for this task isa set of identical blocks glued to random positions on a board. On each trial theparticipant observes the experimenter tap a sequence of blocks and then attemptsto reproduce the sequence. Span is measured by gradually increasing sequencelength to find the point where performance breaks down. Evidence that performanceon this spatiotemporal task dissociates from verbal serial recall is presentedby Isaacs and Vargha-Khadem (1989) and by Pickering, Gathercole, andPeaker (1998).The apparatus consisted of nine 4.4-cm square wooden blocks which werepainted black and fastened in random positions on a black board. To aid theexperimenter, the numbers 1–9 were painted on the side of each block facingaway from the child. Blocks were tapped at a rate of one per second and no blockwas tapped more than once in a trial. The first trial was a sequence of length 2.If this was repeated correctly, then a sequence of length 3 was tapped. If an errorwas made the child was given a second chance at sequence length 2. If the childsucceeded on the second attempt, the task continued at length 3. If not, the taskwas stopped. This incremental procedure was continued to determine span, thelongest sequence correctly repeated before two successive failures. Each childcompleted the task twice and the two values of span were averaged.Central ExecutiveMaking Trails tasks. Following Baddeley (1996), the ability to switch retrievalstrategies was assessed using variants of the Making Trails task (Reitan, 1958).This task involves generating a stream of responses by alternating between twodifferent sequences. The dependent variable was the time taken to complete thetrail (or as much as could be done). There were three versions of the basic task,as follows.For Trails Written the material consisted of 22 circles on a page of A4 paper.Half the circles had a number in the center (1–11), and half had a letter (A–K).Children were asked to start at number 1 and make a trail with a pencil so thateach number alternated with its corresponding letter (e.g., 1-A-2-B-. . .-11-K; seeFig. 1). They were not prompted if a mistake was made in any of the sequences.

WORKING MEMORY AND ARITHMETIC DIFFICULTIES249FIG 1.Example of Making Trails Written.Reported reliabilities for this version of the Making Trails task lie between 0.60and 0.90 (Lezak, 1995).Trails Verbal was similar to Trails Written except that the children were askedto do the task orally. Again they began at 1 and finished with K, or (in case oferror) the 22nd response.Trails Color also was similar and was designed to minimize any effects ofreading ability. It was adapted for use with children from an adult version (D’Elia& Satz, 1989). The materials consisted of 22 circles, half of which were pink andhalf yellow, on a page of A4 paper. Each circle contained a number from 1 to 11such that each number appeared once in a pink circle and once in a yellow one.Children were asked to make a written trail starting with yellow 1, then pink 1,and so on, alternating colors with each number.Crossing Out task. A Crossing Out task adapted from Moran and Mefford(1959) was used to measure selective attention. It is similar to a standardcancellation task except that the target is repeatedly shifted. Demands on selectiveattention are considered high because ignoring nontargets involves inhibitingresponses to items previously designated as targets.Each child was given a sheet of paper with 10 rows of digits. The first digit ineach line was colored red and identified the target; the rest were black. The childwas asked to cross out all the digits in the line that were the same as the red one.

250 MCLEAN AND HITCHEach line contained four randomly positioned targets. After the first line, whichwas used as practice, the child was instructed that the number to be crossed outwas different for each line. The dependent variable was the time taken tocomplete the 9 remaining rows.Missing Item task. This was a novel task developed to measure the capacity tohold and manipulate information accessed in long-term memory. It was apaper-and-pencil task with items consisting of one addition and a second,incomplete addition (e.g., 2 3 4 ? ?). The child was asked to completethe equivalence and then the total. Three practice trials were given in which theexperimenter talked through how to do the task. After this the child completed 15more problems under instructions to be as accurate as possible. No promptingwas given for incorrect responses. The dependent variable was the time taken tocomplete all 15 problems. The rationale for this task was that the child had toaccess long-term memory to complete the first relationship (2 3 ?) and thenmaintain this information and access long-term memory again to complete thesecond relationship (5 4 ?).General ResourcesAddition Span task. This task was developed by Adams and Hitch (1997), andinvolved mentally adding a pair of numbers. The size of the numbers wasgradually increased to find a span beyond which performance breaks down.Addition Span shares the requirement to combine temporary information storageand ongoing processing with complex span tasks such as reading span andlistening span (Daneman & Carpenter, 1980). However, given evidence thatmental arithmetic loads on more than one component of working memory (Fürst& Hitch, in press; Logie et al., 1994), it was considered here as a nonspecificmeasure of working memory resources.The test was conducted orally and began with a single-digit plus single-digitaddition. If the answer was correct, a one-digit plus two-digit addition was givennext. If this was also answered correctly, the size of the next addition wasincreased by one digit according to the protocol described by Adams and Hitch(1997). Span was recorded as the largest problem answered correctly before twosuccessive failures. Each child was given the task twice and a mean span wascalculated from the two estimates. None of the additions involved carry operations,as pairs of integers making up the additions always had totals between 5and 9.ProcedureThe task battery was administered 3 months after the initial screening, andtoward the end of the school year. Each child was tested individually in a quietarea and completed the 10 tasks in two sessions about a week apart. Tasks weregiven in a different random order for each child, with the exception that the threeTrail Making tasks were given together.

WORKING MEMORY AND ARITHMETIC DIFFICULTIES251TABLE 2Performance Characteristics of the Selected Groups on the Experimental MeasuresPoor-ArithmeticAge-MatchedAbility-MatchedANOVAMeasureM SD M SD M SD df F pPhonological loopDigit Span 4.79 0.50 5.42 a 0.79 4.75 0.92 2, 33 2.92 .07Nonword Repetition 26.17 5.98 26.17 4.93 26.92 6.39 2, 33 .07 ns(max 40)Visuo-spatial sketch padMatrix Span 7.83 2.88 7.29 1.98 6.41 1.77 2, 33 1.20 nsCorsi Span 3.92 0.56 4.92*** 0.67 4.00 0.77 2, 33 8.22 .01Central executiveTrails Written (s) 75.1 16.1 53.9* 12.5 78.5 31.2 2, 33 4.54 .05Trails Verbal (s) 66.8 26.3 38.9** 13.9 61.3 30.2 2, 33 4.38 .05Trails Color (s) 52.2 13.3 33.4** 11.5 49.2 13.6 2, 33 7.51 .01Crossing Out (s) 88.3 23.6 77.7 13.8 100.9 30.8 2, 33 2.85 .07Missing Item (s) 282.1 74.5 157.1*** 38.5 215.4** 41.5 2, 33 16.10 .001General resourcesAddition Span 3.33 0.54 4.25*** 0.54 3.45 0.72 2, 33 8.06 .01Note. Asterisks apply to planned comparisons between Poor Arithmetic and Age-Matched andbetween Poor Arithmetic and Ability-Matched.a p .051.* p .05.** p .01.*** p .001.RESULTSDifferences between groups were analyzed using one-way analysis of variance(ANOVA) for each task supplemented by planned comparisons between the PoorArithmetic and Age-Matched groups, and between the Poor Arithmetic andAbility-Matched groups. Table 2 shows means and standard deviations togetherwith the results of the ANOVAs. There were a number of significant maineffects, but there were no significant effects for Nonword Repetition and VisualMatrix Span, and only marginal effects for the Digit Span and Crossing Outtasks.Planned comparisons revealed that the Poor Arithmetic group had significantlylower scores than did the Age-Matched controls on Corsi Span, F(1, 33) 13.35,partial ( 2 ) .29; Trails Written, F(1, 33) 5.73, partial ( 2 ) .15; TrailsVerbal, F(1, 33) 7.82, partial ( 2 ) .19; Trails Color, F(1, 33) 13.04,

252 MCLEAN AND HITCHpartial ( 2 ) .28; Missing Item, F(1, 33) 32.15, partial ( 2 ) .49; andAddition Span, F(1, 33) 13.70, partial ( 2 ) .29. The Poor Arithmetic groupalso had lower digit spans than did the Age-Matched controls and this differenceapproached significance F(1, 33) 4.09, p .051. Comparisons against theAbility-Matched controls revealed that the Poor Arithmetic group were significantlyimpaired on the Missing Item task, F(1, 33) 9.17, partial ( 2 ) .22, butnot on any of the other tasks.Error rates were recorded for the Crossing Out and Missing Item tasks.Generally, these were low. The overall mean percentage was 0.6% for theCrossing Out task and 6.2% for the Missing Item task. Error rates on the TrailMaking tasks were not recorded because of difficulties scoring and interpretingthem.Table 3 shows partial correlations (to control for effects of age) between the10 working memory tasks and scores on the Graded Arithmetic–MathematicsTest, the Primary Reading Test, and the Calculation Test. Five of the workingmemory measures correlated significantly with arithmetic ability. These were theMissing Item task, Addition Span, Trails Verbal, Trails Color, and Corsi Span.The Calculation Test also showed a high correlation with arithmetic ability. It isinteresting to note that none of the measures correlated with reading ability.DISCUSSIONThe results show that children with specific difficulties in arithmetic areimpaired in several, but not all, aspects of working memory. However, asexplained above, tasks on which the Poor Arithmetic group were impairedrelative to Ability-Matched controls were considered the most likely to beinformative about the basis of their learning difficulty. Accordingly, this aspectof the results is discussed first.Relative to the Ability-Matched group, the Poor Arithmetic group were significantlyimpaired only in time to complete the Missing Item task. As nodifference was noted in errors, this is unlikely to reflect a simple speed–accuracytrade-off. The Missing Item task was designed to assess the executive function ofholding and manipulating information in long-term memory. However, identificationof the critical feature of this task must be tentative bearing in mind itsnovelty and the current status of the proposed fractionation of executive function.Some information is provided by the observation that children with Poor Arithmeticdid not differ from Ability-Matched controls on the Calculation Test andAddition Span. These tasks require access to similar arithmetical knowledge butdo not appear to involve such demanding interactions with long-term memory.However, it is still open whether the children’s problem lies with resources forinteractions with long-term memory or elsewhere. An alternative explanation isthat the problem is one of understanding part–whole relations. Yet another is thatthe Poor Arithmetic group had weak conceptual and procedural knowledge ofequivalence (see, e.g., Rittle-Johnson & Alibali, 1999). Thus, even though error

WORKING MEMORY AND ARITHMETIC DIFFICULTIES253TABLE 3Partial Correlation Coefficients between GAM Score, PRT Score, Calculation Test 1 Score,and the Working Memory Measures, Controlling for Chronological Age .001 2 3 4 5 6 7 8 9 10 11 12 131. GAM Score 1.00 .28 .62*** .31 .06 .07 .42* .33 .52*** .47** .25 .60*** .54**2. PRT Score 1.00 .20 .16 .16 .11 .15 .12 .06 .07 .12 .12 .263. Calculation Test 1 1.00 .36* .30 .05 .30 .34 .53** .54** .22 .60*** .51**4. Digit Span 1.00 .46** .22 .31 .23 .07 .30 .06 .31 .265. Nonword Repetition 1.00 .34* .08 .02 .16 .08 .09 .11 .126. Matrix Span 1.00 .04 .24 .02 .12 .04 .18 .077. Corsi Span 1.00 .40* .42* .44** .26 .40* .34*8. Trails Written (s) 1.00 .26 .53** .40* .38* .209. Trails Verbal (s) 1.00 .37* .21 .34* .38*10. Trails Color (s) 1.00 .30 .47** .2211. Crossing Out (s) 1.00 .42* .1312. Missing Item (s) 1.00 .39*13. Arithmetic Span 1.00Note. Correlation coefficients: df 33 in all cases.* p .05.** p .01.*** p .001.

254 MCLEAN AND HITCHrates were low, difficulty with the idea of equivalence could have reduced thespeed with which appropriate procedures could be carried out. However, further(as yet unpublished) data argue against both of these interpretations. A somewhatdifferent perspective is provided by Ericsson and Kintsch’s (1995) concept oflong-term working memory as activated knowledge structures. These authorsproposed that expertise within a domain is associated with more developedknowledge structures and an increase in the functional capacity of long-termworking memory. In this framework, therefore, poor performance on the MissingItem task could be interpreted as reflecting an impairment in long-term workingmemory relative to children’s level of expertise.More generally, a number of researchers have suggested ways in whichlearning difficulties in arithmetic might reflect links between working memoryand long-term memory (e.g., Geary et al., 1991; Hitch & McAuley, 1991; Siegler& Shrager, 1984). To discuss all of these would be beyond the scope of thepresent study. However, the present results do speak to one particular suggestion,namely that some children fail to develop long-term memory for basic numberfacts because information in working memory decays too quickly for relevantassociations to be formed (Geary et al., 1991). To the extent that simple andcomplex spans for verbal materials are influenced by decay (see, e.g., Baddeley,1986; Towse & Hitch, 1995), this account is supported by the lower digit spansand addition spans for the Poor Arithmetic group compared with their Age-Matched controls. However, faster decay of information would not explain theabsence of deficits in other retention tasks such as Nonword Repetition andMatrix Span. Furthermore, if faster decay were a causal factor, children withPoor Arithmetic might have been expected to perform significantly worse thanAbility-Matched controls on all the above tasks, and this was not the case.Overall, therefore, the present results provide little support for the hypothesis thatfaster decay in working memory is a major factor in poor arithmetical attainment(Geary et al., 1991).Other studies have attempted to explain the deficits of children with arithmeticdifficulties in terms of speed of calculation (Garnett & Fleischner, 1983; Geary,Widaman, Little, & Cormier, 1987; Geary et al., 1991). These studies have foundthat arithmetic disabled children are slower than their normal-achieving peers insolving arithmetic problems (Garnett & Fleischner, 1983; Geary & Brown, 1991)and that speed of addition fact retrieval is related to performance on arithmetictests (Geary & Burlingham-Dubree, 1989). However, in the present study thereis no evidence to suggest that the Poor Arithmetic group were slower than theirAbility-Matched peers in solving simple arithmetic problems. For example, therewas no difference in the number of arithmetic problems they could do in 1 minon the Calculation Test (see Table 1). Thus, inclusion of Ability-Matchedcontrols is valuable in showing that children with poor arithmetic do not have anunusually slow calculation speed relative to their level of attainment.The second major feature of the results is that the Poor Arithmetic group were

WORKING MEMORY AND ARITHMETIC DIFFICULTIES255significantly impaired relative to the Age-Matched controls on 6 of the 10working memory measures: Corsi Span, Trails Written, Trails Verbal, TrailsColor, Missing Item task, and Arithmetic Span (Digit Span was also nearsignificance). Given that the Corsi task has no obvious arithmetical or numericalcontent, the first observation raises the possibility that a spatiotemporal workingmemory deficit contributes to poor arithmetic attainment. Interestingly, incidentalobservation of children’s behavior suggested that none of the Poor Arithmeticgroup had unusually poor handwriting or misaligned their arithmetic problems,implying that they did not have problems with spatial coordination. Neuropsychologicalstudies of dyscalculia by Rourke and colleagues (e.g., Rourke &Findlayson, 1978; Rourke & Strang, 1978) had found visual–spatial deficits intasks such as Picture Completion, Picture Arrangement, Block Design, andPicture Assembly from the WISC-R (Wechsler, 1974). Given that the PoorArithmetic group were impaired on a test of spatial but not of visual workingmemory in the present study, Rourke and colleagues’ conclusions appear togeneralize only partly to children with problems within the normal schoolpopulation.A second interesting set of differences between the Poor Arithmetic andAge-Matched groups are those on the Making Trails tasks. The visual versions,Trails Written and Trails Color, have been administered in previous studies oflearning disabled children (e.g., McIntosh, Dunham, Dean, & Kundert, 1995;Share, Moffitt, & Silva, 1988; White et al., 1992; Williams et al., 1995). Whiteet al. (1992) concluded that arithmetic disabled children were impaired on thesetasks relative to normal and reading disabled children of the same age andinterpreted this as reflecting a problem of visuomotor integration. However,although their basic finding was replicated, the observation that children werealso impaired on the verbal Trails task argues against a visuomotor explanation.The idea of a deficit in switching between retrieval plans provides a moreparsimonious interpretation. Moreover, given that the arithmetical content of theTrails tasks was minimal, it seems unlikely that children’s poor performance wasa simple consequence their low arithmetic ability.It is interesting that the Poor Arithmetic group were impaired on Digit Spanrelative to the Age-Matched controls, though the difference just missed significance.Other studies where groups with poor and normal arithmetic werematched on reading ability have shown small impairments in digit span (e.g.,Hitch & McAuley, 1991). However, the picture is not entirely clear, as Bull andJohnston (1997) and Swanson (1994) found no evidence for lowered digit spanin their groups of arithmetic disabled children when reading was controlled forstatistically. As noted above, the deficit in digit span does not appear to reflect aproblem associated with the phonological loop, as Nonword Repetition wasunaffected. An alternative possibility is that the deficit arises from a specificdifficulty in remembering numerical stimuli.Results from the partial correlation matrix are useful in showing that all the

256 MCLEAN AND HITCHtasks which gave significant differences between the groups were correlated withGraded Arithmetic–Mathematics Test score. None of the measures correlatedwith reading ability. This was surprising with regard to Digit Span and NonwordRepetition given evidence for an association between verbal short-term memoryand reading (Rugel, 1974). However, it should be noted that the way the groupswere selected resulted in the sampling of a restricted range of reading abilities,reducing sensitivity of this aspect of the correlational analyses.CONCLUSIONSChildren with specific difficulties in arithmetic were impaired relative toAge-Matched controls on a number of tasks tapping different aspects of workingmemory. Two of these, Corsi Span and Making Trails, suggest possible underlyingdeficits in spatiotemporal working memory and executive processes forswitching retrieval plans. When compared with younger Ability-Matched controls,children with specific difficulties in arithmetic were significantly impairedon only one measure, a Missing Item task. This deficit is tentatively interpretedas reflecting a deficit in executive processes controlling interactions with longtermmemory (Baddeley, 1996) or in long-term working memory (Ericsson &Kintsch, 1995).REFERENCESAdams, J. W., & Hitch, G. J. (1997). Working memory in children’s mental arithmetic. Journal ofExperimental Child Psychology, 67, 21–38.Ashcraft, M. H. (1982). The development of mental arithmetic: A chronometric approach. DevelopmentalReview, 2, 213–236.Ashcraft, M. H. (1987). Children’s knowledge of simple arithmetic: A developmental model andsimulation. In J. Bisanz, C. J. Brainerd, & R. Kail (Eds.), Formal methods in developmentalpsychology: Progress in cognitive development research (pp. 302–338). New York: Springer-Verlag.Ashcraft, M. H. (1995). Cognitive psychology and simple arithmetic: A review and summary of newdirections. Mathematical Cognition, 1, 3–34.Ashcraft, M. H., & Faust, M. W. (1994). Mathematics anxiety and mental arithmetic performance: Anexploratory investigation. Cognition and Emotion, 8, 97–125.Ashcraft, M. H., Donley, R. D., Halas, M. A., & Vakali, M. (1992). Working memory, automaticity,and problem difficulty. In J. I. D. Campbell (Ed.), The nature and origins of mathematical skills.Amsterdam: Elsevier.Atkinson, R. C., & Shiffrin, R. M. (1968). Human memory: A proposed system and its controlprocesses. In K. W. Spence (Ed.), The psychology of learning and motivation: Advances inresearch and theory (Vol. 2, pp. 89–195). New York: Academic Press.Backman, J. E., Mamen, M., & Ferguson, H. B. (1984). Reading level design: Conceptual andmethodological issues in reading research. Psychological Bulletin, 96, 560–568.Baddeley, A. D. (1986). Working memory. Oxford: Oxford Univ. Press.Baddeley, A. D. (1992a). Working memory. Science, 255, 556–559.Baddeley, A. D. (1992b). Is working memory working? The fifteenth Bartlett lecture. QuarterlyJournal of Experimental Psychology, 44A, 1–31.Baddeley, A. D. (1996). Exploring the central executive. Quarterly Journal of Experimental Psychology,49A, 5–28.Baddeley, A. D., & Hitch, G. J. (1974). Working memory. In G. H. Bower (Ed.), The psychology of

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