Working Memory 27
Running head: WORKING MEMORY IN EARLY ARITHMETIC
Representation and Working Memory in Early Arithmetic
Carmen Rasmussen and Jeffrey Bisanz
University of Alberta
Abstract
Working memory has been implicated in the early acquisition of arithmetic skill, but the relations among different components of working memory, performance on different types of arithmetic problems, and development have not been explored. Preschool and Grade 1 children completed measures of phonological, visual-spatial, and central executive working memory, as well as nonverbal and verbal arithmetic problems, some of which included irrelevant information. For preschool children, accuracy was higher on nonverbal problems than on verbal problems, and the best and only unique predictor of performance on the standard nonverbal problems was visual-spatial working memory. This finding is consistent with the view that most preschoolers use a mental model for arithmetic that requires visual-spatial working memory. For Grade 1 children, performance was equivalent on nonverbal and verbal problems, and phonological working memory was the best predictor of performance on standard verbal problems. For both age groups, problems with added irrelevant information were substantially more difficult than standard problems and, in some cases, measures of the central executive predicted performance. Assessing performance on different components of working memory in conjunction with different types of arithmetic problems provided new insights into the developing relations between working memory and how children do arithmetic.
Representation and Working Memory in Early Arithmetic
Working memory is an important factor in the mathematical development of children, but the relation between different components of working memory and different types of arithmetic problems is unclear. The focus of the present research was to explore the relation between working memory and arithmetic and to test hypotheses about changes in mental representation and processing. To do so, we examined young children’s performance on simple arithmetic problems, presented verbally and nonverbally, and measures of working memory.
Working Memory and Development
In 1974, Baddeley and Hitch proposed a multiple component model of working memory. Baddeley (1992) defined working memory as the system used for short-term storage and manipulation of information required for diverse cognitive tasks. Working memory comprises three main subsystems: the visuospatial sketch pad, for holding and manipulating visual-spatial information; the phonological loop, for maintaining and rehearsing verbal information (Baddeley, 1992); and the central executive, an attentional controlling system involved in the coordination of performance on separate tasks, selective attention, set shifting, and inhibition (Baddeley, 1996). More recently, Baddeley (2000) added a fourth component to his model, the episodic buffer, which is a limited-capacity system that integrates and provides temporary storage of information from the two subsystems and long-term memory. Developmental research related to this fourth component is very limited, so we focus on the first three components.
The three-component model of working memory proposed by Baddeley and colleagues has been supported by numerous brain imaging and neuropsychological studies. In adults, functions of the central executive component of working memory has been found to be mediated by the frontal lobe as well as parietal areas (Collette & Van der Linden, 2002; D’Esposito, Detre, Alsop, & Shin, 1995). Phonological working memory is associated with left hemisphere functioning, whereas visual-spatial working memory is associated with right hemisphere functioning (Smith & Jonides, 1999; Smith, Jonides, & Koeppe, 1996). Studies showing selective interference on dual-task paradigms and specific sparing and impairments in individuals with brain damage support the separability of the different components of working memory (see Baddeley & Logie, 1999, for a review). Moreover, in a large scale study on the structure of working memory in children, Gathercole, Pickering, Ambridge, and Wearing (2004) found that the three-component model of working memory best fit the data and was supported in children aged 6-7, 8-9, 10-12, and 13-15 years.
In terms of development, performance on tasks measuring the three components of working memory increases from preschool through adolescence (Gathercole et al., 2004). Children may rely more heavily on different types of working memory as a function of age. Hitch, Halliday, Schaafstal, and Schraagen (1988) examined visual-spatial working memory in preschool (5-year-old) and school-age (10-year-old) children. On a picture memory task, the 5-year-olds were disrupted by effects of visual similarity, indicating that they relied on visual working memory to remember the pictures. To the contrary, the 10-year-olds were not disrupted by visual similarity but showed a word length effect for the names of the pictures, indicating that they rehearsed the names of the pictures. Only the older children showed a primacy effect, indicative of rehearsal, when remembering pictures on cards. Furthermore, when doing a visual task the 5-year-olds were disrupted by visual but not verbal retroactive interference, whereas the 10-year-olds were more disrupted by verbal than visual retroactive interference.
Hitch et al. (1988) concluded that preschool children rely on visual-spatial working memory more than older children, who tended to use a phonological or verbal approach. Older children appeared to recode visual tasks to a verbal code in the phonological loop. The authors suggested that younger children may not have the information-processing capacities to recode visual information to a verbal code and that they may lack sufficient metamemory. The results of this study indicate developmental changes in the types of working memory that children rely on and also support the multiple-component model of working memory proposed by Baddeley and Hitch (1974). The ability to recode from a visual to a phonological code may also be related to reading and an increased use of in phonological processing, as well as the ability of the central executive to integrate complex tasks (see Pickering, 2001, for a review).
Working Memory and Mathematics
Working memory is implicated in academic performances including reading comprehension and mathematics in both children and adults (Swanson, 1994). In adults, working memory is related to arithmetic performance (Furst & Hitch, 2000; Heathcote, 1994; Kyoung-Min & So-Young, 2000; Logie, Gilhooly, & Wynn, 1994), mathematics problem solving (Cary & Carlson, 2001), and even mathematics anxiety (Ashcraft & Kirk, 2001). Working memory is also related to arithmetic performance in children (Adams & Hitch, 1997, Hitch & McAuley, 1991). For example, deficits in arithmetic achievement have been linked to poor performance on measures of the visual-spatial working memory and the central executive (Gathercole & Pickering, 2000a; Geary, Hoard, & Hamson, 1999; McLean & Hitch, 1999; Siegal & Ryan, 1989) but not on phonological working memory tasks (McLean & Hitch, 1999). Furthermore, working memory for visual-spatial detail is related to the arithmetic deficits that are common in girls with Tuner syndrome (Buchanan, Pavlovic, & Rovert, 1998).
Bull and Scerif (2001) demonstrated that many different measures of executive functioning predict mathematics performance in 7-year-old children, with counting span being the best predictor. Similarly, 7-year-olds with low mathematics performance were also impaired on the Wisconsin Card Sorting Task (WCST), a test of executive functioning, but not on the Corsi blocks task, a measure of visual-spatial working memory (Bull, Johnson, & Roy, 1999). In both studies Bull et al. concluded that children of low mathematical skill have significant difficulty inhibiting prepotent information and learned strategies, and they also have difficulty maintaining information in working memory.
Working memory is also important for mathematical problem solving on arithmetic word problems. Passolunghi and Siegel (2001) concluded that fourth-grade children who are poor at problem solving have significant difficulty with central executive tests of working memory and with digit span, which is often used as a measure of the phonological loop (e.g., Gathercole et al., 2004). Passolunghi, Cornoldi, and De Liberto (1999) further found that fourth-grade children who are poor at solving arithmetic problems have particular difficulty on working memory tasks where irrelevant information must be inhibited. Furthermore, measures of visual-spatial and phonological working memory predict accuracy in mathematics problem solving in children with learning disabilities (Swanson & Sachse-Lee, 2001).
These findings highlight the link between mathematics performance and working memory in school age children, but this relation has largely been ignored in preschool children. Klein and Bisanz (2000), however, found compelling evidence for the relation between working memory and arithmetic in 4-year-olds. They demonstrated that the working-memory demand of nonverbal arithmetic problems accounted for differential problem difficulty better than other variables related to problem size, attentional or counting requirements, and operation (addition or subtraction).
Based on research to date, it is evident that working memory is related in some way to mathematical development. Discrepancies in findings may be due to the fact that researchers have used many different tasks to measure the three components of working memory, as well as many different tests of mathematical ability. In most of the studies with children, researchers have only looked at relations between different measures of working memory and a measure of general mathematical ability, often by dividing children into high and low mathematical ability groups based on standardized tests. Mathematical problems vary considerably in type and complexity, however, and this diversity may provide a means for exploring developmental relations between specific types of mathematical performance and specific components of working memory. To date, researchers have not examined children’s performance on specific mathematical problems that vary in demand on phonological, visual-spatial, and central executive components of working memory. Dual-task and interference paradigms that have proved successful in distinguishing specific relations between components of working memory and types of mathematical problems with adults (Logie et al., 1994; Furst & Hitch, 2000) are difficult to use with young children.
Mental Model of Arithmetic
Performance on simple arithmetic problems varies depending on how the problems are presented, perhaps because of different types of working memory that different presentations require. For example, Levine, Jordan, and Huttenlocher (1992) examined performance of children aged 4 to 6 years on arithmetic problems presented nonverbally (using disks) and problems presented verbally (story problems and number-fact problems).1 The 4-year-old children performed substantially higher on nonverbal problems than verbal problems, but this difference was negligible for the 6-year-old children. In a related study, Jordan, Huttenlocher, and Levine (1992) further demonstrated that nonverbal problems are easier for preschool children than verbal problems, and that low-income children performed less well than middle-income children on verbal problems but not on nonverbal problems. In fact, even some children between the ages of 2 and 3 years are able to solve nonverbal problems with small operands (Huttenlocher, Jordan, & Levine, 1994).
Huttenlocher et al. (1994) proposed that toddlers and preschool children solve nonverbal problems using a mental model (Halford, 1993; Johnson-Laird, 1983). How such a mental model might work for nonverbal arithmetic problems has not been proposed in detail, but at a minimum three steps might be expected. First, an initial quantity that is presented nonverbally would be represented internally in terms of tokens that map one-to-one with the external objects. Second, as objects are added or subtracted to the external display, the child adds or subtracts tokens from the internal representation. Finally, when an answer is required, the child generates an answer that maps onto the internal representation. This process can be contrasted with, for example, retrieval of number facts, which presumably would operate on the verbal/symbolic representations of number. Huttenlocher et al. suggested that these mental-model representations and processes are readily available to preschool children. Verbal problems, in contrast, may require quantitative skills and symbols that are not so readily available in young children. Alternatively, verbal problems may require translation of information presented verbally to a nonverbal, mental-model format, and this translation process may be difficult. Thus, the ready use of mental models to represent nonverbal information and transformations, combined with difficulty in representing quantitative information presented verbally, may account for why nonverbal problems can be solved at such a young age and are typically easier than verbal problems for preschool children.
Issues To Be Explored
We hypothesize that this proposed mental model requires visual-spatial working memory on the assumption that the child must store and manipulate internal tokens that are analogous to external objects. Nonverbal problems involve external objects, making them easier to encode and represent in a mental model than verbal problems. If children use a mental model, then performance on nonverbal problems should depend more on visual-spatial working memory than performance on verbal problems. Preschool children may generally depend heavily on visual-spatial working memory (Hitch et al., 1988) and perhaps this reliance on visual-spatial working memory among preschool children corresponds to frequent use of mental models of representation by young children. To the extent that the use of mental models on arithmetic problems also requires planning and selective attention, the central executive would be implicated.
In contrast, older children who have entered school perform as well on verbal as nonverbal problems (Levine et al., 1992), presumably because they are able to represent and code both kinds of problems verbally and solve them without resorting to a mental model. This assertion is in concordance with the verbal recoding among school-aged children found in Hitch et al. (1988). School-age children learn to manipulate quantitative symbols with verbal labels, and this verbal approach would demand phonological working memory because children must store verbal information. Presumably, they would rely even more on phonological working memory as they learn to solve larger problems, which would be very difficult to represent with a mental model. School-age children also learn strategies, such as counting on their fingers, that would reduce the need to rely on a mental model. If children use their fingers on verbal problems, however, phonological working memory would still be required to remember the numbers in the problem. Hence, children who have started school are less likely to use a mental model for arithmetic because they have learned conventional skills and methods based on verbal representations and they tend to use phonological coding. Again, the central executive would be involved to the extent that solving arithmetic problems requires planning and selective attention.
The relation between mental model and verbal solution processes is illustrated in Figure 1. Our conjectures about changes in mathematical representations and process, and about the relations between solution processes and working memory, lead to two key predictions. First, preschool children should perform better on nonverbal than on verbal problems, whereas children in Grade 1 should perform equally well on both problems. This outcome would replicate the findings of Levine et al. (1992), and it also would be consistent with the view that preschool children use mental models to solve nonverbal problems but have difficulty in using an appropriate model to solve verbal problems. Second, to the extent that individual differences in working memory influence performance on arithmetic problems, the nature of that influence should conform to the relations in Figure 1. More specifically, for preschool children accuracy on nonverbal arithmetic problems should be related primarily to performance on measures of visual-spatial and possibly central executive components of working memory, but less so or not at all to performance on phonological working memory. In contrast, if children in Grade 1 use verbal representations and processes, then accuracy on arithmetic problems might be related primarily to performance on phonological and possibly central executive working memory, but less so or not at all to visual-spatial working memory. If this pattern is found for verbal but not nonverbal problems, then the results would be consistent with the view that Grade 1 children might use both the mental model and verbal approaches on nonverbal problems.