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«Working Memory Impairments in Children with Specific Arithmetic Learning Difficulties Janet F. McLean and Graham J. Hitch Lancaster University, ...»

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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 on

Working Memory Impairments in Children with

Specific Arithmetic Learning Difficulties

Janet F. McLean and Graham J. Hitch

Lancaster University, Lancaster, United Kingdom

Working memory impairments in children with difficulties in arithmetic have previously been investigated using questionable selection techniques and control groups,

leading to problems concluding where deficits may occur. The present study attempted to overcome these criticisms by assessing 9-year-old children with difficulties specific to arithmetic, as indicated by normal reading, and comparing them with both age-matched and ability-matched controls. A battery of 10 tasks was used to assess different aspects of working memory, including subtypes of executive function. Relative to age-matched controls, children with poor arithmetic had normal phonological working memory but were impaired on spatial working memory and some aspects of executive processing.

Compared to ability-matched controls, they were impaired only on one task designed to assess executive processes for holding and manipulating information in long-term memory. These deficits in executive and spatial aspects of working memory seem likely to be important factors in poor arithmetical attainment. © 1999 Academic Press Key Words: working memory; executive processes; arithmetic; children; learning difficulties.

There are many reasons children may fail to learn arithmetic. Examples include 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 difficulties can 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 cognitive deficits in a subgroup of children within the general population who have specific difficulties in arithmetic but normal reading.

This research was conducted in partial fulfillment of a Ph.D. by the first author. The cooperation of schools in the Lancaster area and the financial support of the Economic and Social Research Council are gratefully acknowledged.

Address correspondence and reprint requests to Janet McLean, Department of Psychology, Florentine House, University of Glasgow, Glasgow G12 8BQ, United Kingdom.

0022-0965/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.


Describing the cognitive deficits of this subgroup can be problematic. Studies based on clinical samples have suggested three different types of cognitive deficit in arithmetic (see Badian, 1983, and Geary, 1993, for reviews). These are characterized as visuospatial deficits (Hartje, 1987; Rourke, 1993; Rourke & Findlayson, 1978; Rourke & Strang, 1978), difficulties with reading and writing of numbers (Kosc, 1974; Temple, 1989), and problems of retrieving arithmetic facts from long-term memory (Badian, 1983; Benson & Weir, 1972; Jackson & Warrington, 1986). White, Moffitt, and Silva (1992) found that arithmetic disabled children performed poorly not only on the visuospatial tasks used by Rourke and colleagues, but also on the Making Trails tasks (Reitan, 1958), the Grooved 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 are associated with a variety of deficits. However, it is difficult to extrapolate from these observations to the general population.

In a comprehensive review of cognitive deficits in arithmetic disabled children which included nonclinical studies, Geary (1993) identified two basic categories with different developmental trajectories. The first is manifested by the use of developmentally immature arithmetic procedures which usually regain normal levels after 2 or 3 years of schooling. This category has been associated with deficits in counting, computational skill, and working memory (Geary, 1990;

Geary, Bow-Thomas, & Yao, 1992). The second category involves a more persistent difficulty with the representation and retrieval of arithmetic facts from long-term semantic memory. Fact-retrieval difficulties have been linked to deficits in counting speed and working memory (Garnett & Fleischner, 1983; Geary, 1990; Geary et al., 1991). It is interesting to note that both of these categories include working memory, the limited capacity system responsible for transforming and maintaining temporary information (see, e.g., Baddeley & Hitch, 1974).

A number of previous studies have compared measures of working memory in normal children, children with arithmetic difficulties, and children with enhanced arithmetical 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, they agree in suggesting that working memory is related to differences in the ability to perform arithmetic. For example, Geary et al. (1991) investigated arithmetically disabled and normal children over 1 year. The disabled group received remedial education in mathematics and were below the 46th percentile on national achievement tests. Both forward and backward digit span were significantly reduced for the disabled group, consistent with a working memory deficit.

Geary et al. (1991; see also Geary, 1990, 1993) proposed that although skills such as counting knowledge and strategy choice are the primary areas of deficit in the mathematically disabled child, working memory deficits may lead to a failure to develop long-term memory representations of basic facts. Geary et al. (1991) 242 MCLEAN AND HITCH suggest that if the representation of a problem’s integers is lost more rapidly from working memory it is less likely to be associated with the answer. However, their arithmetic disabled children also had poor reading skills. Their children’s average percentile ranking for reading was actually below that for arithmetic, and was much lower than the reading ranking for the control group. This is important because a significant amount of language is used in mathematics, and problems in reading and mathematics often co-occur (Muth, 1984; Richman, 1983; see Geary, 1993, for a review). Therefore, children with more general academic difficulties have been included in Geary and colleagues’ studies, and this leads to problems interpreting their findings with respect to arithmetic per se.

Evidence that working memory deficits differ as a function of the specificity of learning difficulties is presented by Siegel and Ryan (1989). They gave children two complex span tasks designed to assess working memory capacity.

These were a counting span task (Case, Kurland, & Goldberg, 1982) and a sentence span task (adapted from Daneman & Carpenter, 1980). Children with specific arithmetic difficulties (i.e., WRAT Arithmetic 25th percentile and WRAT Reading 30th percentile) were impaired on counting span but not on sentence span. In contrast, children who were impaired on reading as well as on arithmetic had significantly lower counting and sentence spans. Siegel and Ryan suggested that the reading-and-arithmetic disabled group had a general working memory impairment, whereas the arithmetic disabled group had a domain-specific working memory deficit. In a related study, Hitch and McAuley (1991) confirmed that children with specific arithmetic difficulties were impaired on counting span but not on other complex span tasks. They also showed that these children had significantly lower digit spans than did age-matched controls.

A general criticism of the studies described above is that they have employed a design in which children with arithmetic difficulties are compared to normal achievers of the same chronological age. This type of design cannot determine the cause of the disability, as any cognitive deficits may be a consequence rather than a cause of low arithmetic attainment. Vellutino, Pruzek, Steger, and Meshoulam (1973) attempted to resolve this problem in the area of reading difficulties by suggesting that a younger ability-matched control group should be included. The logic is that if the poor achievers perform worse on some cognitive task than do younger, ability-matched controls, the deficit is unlikely to be a simple consequence of their low level of academic achievement. Given poor achievers’ greater age and experience, such a deficit would suggest they are arriving at similar levels of ability via different routes. However, Backman, Mamen, and Ferguson (1984) noted that age-matched controls remain useful for identifying differences in cognitive function at the same level of exposure to the skill in question.

A second concern about previous work is the sparsity of attempts to assess the various facets of working memory in children with poor arithmetic. If this system


is 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 used here, as it explains a wide range of experimental and neuropsychological evidence (see also Baddeley, 1986) and tasks have been developed for assessing some of its components. According to this model, working memory consists of a central executive processor which interacts with two slave subsystems, the phonological loop and the visuospatial sketch pad. The phonological loop is specialized for the storage and rehearsal of speech-based verbal information (Baddeley, 1992a, 1992b), whereas the sketch pad is specialized for holding visual and spatial material (Logie, 1986; Quinn & McConnell, 1996). In a recent development, Baddeley (1996) proposed a fractionation of the central executive into separate but overlapping functions of coordinating concurrent activities, switching retrieval plans, attending to inputs, and holding and manipulating information in long-term memory.

Evidence from studies of the normal population suggests that different components of 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 Furst & Hitch, in press). The ¨ visuospatial sketch pad appears to be involved in multidigit problems where visual 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), but without linking them to working memory. The role of the central executive has been noted several times (see, e.g., Ashcraft, Donley, Halas, & Vakali, 1992;

Lemaire, Abdi, & Fayol, 1996; Logie et al., 1994). Ashcraft suggested that the executive 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 the role of different executive functions in arithmetic.

The first executive function identified by Baddeley (1996) is the capacity to coordinate performance on two or more separate tasks. Arithmetic can be considered a multiple task when it involves subtasks such as calculating partial totals and keeping track of other information. However, given that the subtasks are parts of an integrated skill, the requirement for coordination is presumably low relative to performing multiple independent tasks. The second executive function is switching retrieval strategies. This is clearly necessary for problems such as multidigit multiplication, which typically involves both multiplying and adding. Experimental evidence suggests that switching is also required for carrying operations (Furst & Hitch, in press). The third executive function is ¨ attending selectively to different inputs. This is clearly a feature of multidigit 244 MCLEAN AND HITCH problems carried out as a series of subtasks, where attention is paid to selected parts of a problem at different times. The fourth executive function is activating and manipulating information in long-term memory. This seems likely to be involved 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, nearly all the components of working memory seem likely to be involved in arithmetical calculation, each playing a somewhat different role.

In light of the foregoing considerations, the present study had two aims. The first was to gain a more complete picture of working memory deficits in children with specific arithmetic difficulties. This was achieved by using a battery of tasks assessing different components of working memory. The second aim was to consider whether working memory deficits are responsible for children’s difficulties in arithmetic. This was investigated by applying the age-matched and ability-matched design described above. To ensure the selectivity of arithmetical deficits, all three groups were matched on reading. That is, in all three groups, children’s reading was at normal, age-appropriate levels. An incidental advantage of using reading in this way is that it avoids controversial issues associated with matching groups on general intelligence (Siegel, 1988).

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