Working Memory and Cognitive Modification

Kerry Lee and Swee-Fong Ng

Recent studies on individual differences in mathematical abilities show that aspects of working memory contribute to early arithmetic performance. In this chapter, we provide a brief overview on working memory: from its inception, as a replacement or supplement to the multistore model, to more recent studies that address its relationship with other cognitive functions. In particular, we focus on the applied literature and describe several studies that provide insight on the relationship between working memory and mathematical problem solving. In our own work, we found working memory played a more prominent role in mathematical problem solving than previously thought. We also found working-memory predicted performance in school examinations. To provide additional insight on “why working memory works,” we describe our initial efforts to tease out the relationship between working memory and different components of mathematical problem solving.

WORKING MEMORY AND COGNITIVE FUNCTIONS

In the last two decades, research on working memory found reliable associations between working-memory span and several other measures of cognitive functions such as intelligence (for a review, see Conway, Kane, & Engle, 2003), reading comprehension (for a review, see Daneman & Merikle, 1996), and arithmetic abilities (e.g., Bull & Scerif, 2001; Lee, Ng, Ng, & Lim, 2004). Just what is working memory and why is it correlated with such a wide range of cognitive functions? In this chapter, we present a brief review of perspectives on working memory. This is followed by the presentation of some empirical findings that provide insight into the nature of the relationship between working memory and mathematical problem solving.

WHAT IS WORKING MEMORY?

The notion of working memory stems from earlier theories that posit a structural view of memory with a separation of long-term and short-term memories. Such a view is not new; as early as 1892, William James wrote of primary and secondary memory, with the latter being “the knowledge of a former state of mind after it has already once dropped from consciousness” (p. 287). This notion of memory being separable into two components survived and influenced thinking well into the 20th century. One key modification occurred in 1968. Atkinson and Shiffrin (1968) proposed a multistore model with three components: sensory memory, short-term memory, and long-term memory. Of importance was that they specified control processes that allowed for the transfer of information from short-term to long-term memory. Their theory broke away from a perspective that memory was strictly a structure of the mind to one that emphasized the importance of cognitive functions. Subsequent work went further and focused on rehearsal processes. Craik and Lockhart (1972), for example, showed that the manner in which information was processed influenced the likelihood it was retrieved. Semantic processing was generally associated with higher likelihood of retrieval than shallower forms of processing that focused on the physical characteristics of to-be-remembered information.

Although a great deal of evidence was consistent with the multistore model, two key findings showed that it could not be sustained. First, the model assumed that the longer information was maintained in short-term memory, the more likely it was to be transferred to long-term memory (Baddeley, 2000b). Findings from Craik and Lockhart (1972) showed that transferral was more strongly affected by the type rather than the length of processing. Second, a prediction of the multi-store model was that patients with impaired short-term memory should have corresponding difficulties in long-term memory and in reasoning abilities. Yet, clinical observations provided examples of patients with short-term but not long-term memory or reasoning deficits (Baddeley, 2000b).

To account for these discrepancies, Baddeley and Hitch (1974) proposed a working-memory model in which short-term memory was transformed into a multiprocess memory system. Working memory, according to Baddeley and Hitch, is a vital part of our information processing system that allows for the representation and manipulation of several pieces of information at a time. It is responsible for short-term memory storage, reasoning, problem solving, and other cognitive functions that require a consideration of the past and present. Although the short-term memory store in Atkinson and Shiffrin's model also performed higher cognitive processes, Baddeley and Hitch's model specified a tripartite system that separated the storage and processing components. This separation overcame problems encountered in the multistore model when patients presented with short-term memory deficits with intact reasoning abilities.

The latest version of the working-memory model, shown in Figure 2.1, consisted of four components: central executive, phonological loop, visual-spatial sketchpad, and an episodic buffer (Baddeley, 2000a).

Both the phonological loop and the visual-spatial sketchpad are short-term storage systems. The former is responsible for storing and rehearsing auditory and articulatory information. The latter maintains and manipulates images. The characteristics of the phonological loop have been examined in numerous studies. Apart from being of limited capacity, it is divided into two subsystems: one is responsible for the temporary storage of phonological information and the other acts as a rehearsal mechanism that helps maintain information in the storage system. Traditional tests of short-term memory capacities, e.g., digit span, are believed to measure the storage capacity of the phonological loop. In the last several years, a number of studies have focused on the visual-spatial sketchpad. Some authors argued that it can also be sub-divided into two functional components: one responsible for storage and the other responsible for maintenance (Baddeley & Logie, 1999). Others argued for finer subdivisions with systems responsible for spatial versus visual identification (Logie, 1986).

The central executive is a resource manager and controls the allocation of attentional resources. Like most cognitive accounts of thinking processes, Baddeley's working-memory model postulates an upper limit in attention capacity. Performance deteriorates when task demands exceed this capacity. In a recent work, Baddeley (1996) suggested that in addition to resource allocation, the central executive is likely to be involved in the execution of other higher cognitive functions (e.g., inhibition, switching between problem-solving strategies, and activating information stored in long-term memory). The characteristics of the central executive and the way in which it interacts with other components of working and long-term memory are questions that continue to attract attention. One outcome of this effort is a proposal that the original tripartite system cannot fully account for several phenomena that seem to involve active interaction between working and long-term memory. Baddeley (2000a) cited as example the phenomenon of chunking. When given a list of unrelated words, adults generally remember five to nine words in immediate recall tests. This number increases to 16 or more when the words form a meaningful sentence. Presumably, long-term knowledge is used to increase word span. What is unclear is the mechanism by which the to-be-remembered information, stored in the phonological loop, is integrated with long-term knowledge. By reviewing clinical data, Baddeley (2000a) argued that neither the phonological loop nor the long-term store is likely storage sites for integration. Baddeley (2000a) proposed that this function is served by a domain-free, limited capacity temporary storage system: the episodic buffer. In addition to serving as a bridge across the short and long memory systems, it is assumed to serve as an interface across different types of representational codes, e.g., visual-spatial and phonological. Although the need for processes or a structure similar to that proposed by Baddeley (2000a) is clear, at present, empirical validation of its proposed characteristics is still at an early stage.

In addition to Baddeley's model, there have been other proposals that do not posit a strict structural division between working and long-term memory. In a collection of works on working-memory theories, Miyake and Shah (1999) brought together ten perspectives. Two of the more divergent views proposed that working memory is an activated portion of long-term memory. Cowan (1999) focused on functional characteristics and argued that working memory was the collection of cognitive processes that “retain(ed) information in an unusually accessible state” (p. 62). From a structural perspective, Cowan's theory maps onto a long-term memory system that contains information in different levels of activation: dormant, activated, and activated but in the focus of attention. Engle, Kane, and Tuholski's (1999) model bears similarity to Cowan's. They argued that working memory consists of the activated portion of long-term memory together with the system that effects and maintains long-term memory activation. These views represent a major departure from Baddeley's model. At present, there are concerted efforts by different laboratories to test the various theories.

WORKING MEMORY AND OTHER COGNITIVE ABILITIES

Turning to more applied issues, a number of studies examined the relationship between working memory, reading comprehension, and mathematical performance. In a review of 77 studies, Daneman and Merikle (1996) found measures of working memory accounted for 17 percent and 27 percent of variance in global and specific comprehension. In the remainder of this chapter, we will focus on the relationship between working memory and mathematical performance.

Most studies found reliable relationships between aspects of working memory and mathematical performance. The magnitude of this relationship varies depending on the component of working memory under consideration and the nature of the mathematical task. The literature on mental arithmetic typically shows a strong correlation between phonological storage and performance. Fürst and Hitch (2000), for example, showed that attenuating participants' access to the phonological store reduced their ability to perform mental addition. In contrast, suppression of access had little effect when the numbers to be added remained visible during the addition process. In a second study, they asked participants to perform a central executive task while they were performing mental addition. Performance on the addition task deteriorated even when the numbers to be added remained visible. The magnitude of deterioration increased when the number of carry operations required in the task increased. Fürst and Hitch interpreted these findings as suggesting different roles for the phonological loop and the central executive in mental addition. The former simply retains problem information. The central executive, on the other hand, is required for computation.

Findings from studies that utilized school examination data or standardized mathematics tests are more varied. In Bull and Johnston (1997), for example, 7 year olds were administered the Group Mathematics Test (Young, 1980) and a number of processing speed, phonological span, and long-term memory measures. Phonological span correlated reliably with performance on the Group Mathematics Test. However, when the contributions of other measures were taken into account, its contribution was found to be marginal. Bull and Johnston concluded that once reading ability was controlled, only processing speed contributed additional information to the prediction of arithmetic ability. Similar findings on the role of the phonological loop were reported by Gathercole and Pickering (2000a). In their study, 6 and 7 year olds were administered an earlier version of the Working Memory Test Battery for Children (Pickering & Gathercole, 2001) and a number of vocabulary, literacy, and arithmetic measures. The WMTB-C was based on Baddeley and Hitch's (1974) model and contained a number of subtests designed to index phonological, visual-spatial, and central executive spans. The results showed that phonological span was correlated with performance on the Group Mathematics Test. However, its contribution was rendered non-reliable when performance on the central executive measure was taken into account. The importance of central executive span to arithmetic performance is further illustrated by a study by Lehto (1995). Similar to Gathercole and Pickering's findings, central executive span predicted and accounted for much of the correlation between phonological span and arithmetic performance.

As mentioned earlier, the central executive is believed to be multi-functional. In addition to being an attentional resource manager, the literature suggests that it is involved in other executive functions such as inhibition, switching between mental strategies, and planning. Some of these functions may contribute to mathematical performance in different ways. Bull and Scerif (2001) examined this issue by using a battery of executive function measures to predict the performance of 7 year olds in the Group Mathematics Test. The results showed that measures of mental flexibility, inhibition efficiency, and working-memory span were all correlated reliably with mathematics performance. Each measure accounted for an additional 2 percent to 3 percent of variance in mathematical performance when differences due to reading abilities and IQ were controlled. Others found similar patterns amongst subgroups of children and adolescents. Passolunghi and her colleagues (Passolunghi, Cornoldi, & De Liberto, 1999; Passolunghi & Siegel, 2001) found poorer problem solvers to have lower inhibitory abilities than the better problem solvers. This relationship remained reliable even after differences in vocabulary were controlled. Sikora, Haley, Edwards, and Butler (2002) found children with arithmetic difficulties exhibited poorer planning or inhibitory abilities—as indexed by performances on the Tower of London Test—than their peers with either reading difficulties or no difficulties.

Regarding the contributions of the other two components of working memory to mathematical performance, little information is available. Studies performed on visual-spatial memory suggest that the relationship varies across visual-spatial tasks and subgroups. Gathercole and Pickering (2000b), for example, found differences in one of four visual-spatial memory span tasks between 7 year olds with normal achievement versus those poor in mathematics, but not between those with normal achievement versus those poor in both English and mathematics. McLean and Hitch (1999) found that children with poorer arithmetic abilities had lower spatial memory span compared to their age matched peers, but not when compared to their ability matched peers. In a recent study, McKenzie, Bull, and Gray (2003) found the mental arithmetic performances of both 6 and 9 years olds to be affected by visual-spatial suppression. These findings suggest that being able to store information in a visual-spatial fashion is an important component of mental arithmetic. The authors suggested that children used the visual-spatial sketchpad as they would a physical sketchpad and as a place for active manipulation.

This brief summary shows that measures of working memory, particularly the central executive, are reliably correlated with mathematical performance. Several questions remain unanswered. First, why is working memory related to mathematical performance? At what stage is working memory needed in mathematical problem solving and computation? Fürst and Hitch's (2000) and McKenzie, Bull, and Gray's (2003) studies go some way in answering this question by showing that the availability of the phonological loop and the visual-spatial sketchpad are particularly important in mental computations. Second, previous studies had concentrated on numeracy skills such as counting, number knowledge, and basic arithmetic such as addition and subtraction. Very little is known about the contributions of working memory to the performance of more complex mathematical problems.

In our work, we extended the investigation to algebraic word problems. How do algebraic word problems differ from arithmetic problems? Compared to simple arithmetic problems in which operands and operators are explicitly stated, information in algebraic word problems are typically embedded in a more extensive linguistics context. However, compared to arithmetic word problems, the linguistic demands can be similar. The most salient difference between the arithmetic algebraic word problems is the complexity of relationships specified in them. Take, for example, the two problems presented below.

Dunearn Primary school has 280 pupils. Sunshine Primary school has 89 pupils more than Dunearn Primary. Excellent Primary has 62 pupils more than Dunearn Primary. How many pupils are there altogether? (Arithmetic)

A cow weighs 150 kg more than a dog. A goat weighs 130 kg less than the cow. Altogether the three animals weigh 410 kg. What is the mass of the cow? (Algebraic)

For the arithmetic problem, the number of pupils in Dunearn Primary is a known state (Bednarz & Janvier, 1996). Enrolments at Sunshine Primary and Excellent Primary are unknowns. For this type of problems, the known state allows easy entry into the problem as operating on this known state allows the unknown to be found, simply by using the stated relationships to link the unknowns with the known.

In algebraic problems, the known state is a combination of the unknowns. In the example above, the total weight of the three animals is the known state. The individual weights of the three animals—related to each other by comparative relationships—are the unknowns. In this problem, two types of relationships must be processed simultaneously: addition and subtraction. A fundamental difference with arithmetic problems is that entry into the problem is via an unknown. Although any of the unknown states can be chosen as the entry point into such problems, there is a usually one, identified as the “generator” (Bednarz & Janvier, 1996) that enables easier entry into the problem.

In our algebraic example, there are three unknowns; any one can be chosen as the generator. An important task faced by problem solvers is to decide which is the most appropriate generator. If the cow is taken as the generator, then the weights of the dog and the goat are 150 kg and 130 kg less than that of the cow's respectively. The sum of these weights, which are all stated as unknowns, is equivalent to the total weight of the three animals. Solving the resulting equation leads to the cow's weight. If the dog is taken as the generator, the equation consists of the dog's weight, the cow's weight, which is 150 kg more than the dog's, and the goat's weight which is 20 kg more than the dog's. The total is also equivalent to the total weight of the three animals. However, solving this equation gives the dog's weight but not the cow's. The cow's weight can only be found by further operating on the dog's weight.

One reason for algebraic word problems being more challenging than arithmetic questions is that problem solvers face the additional task of having to decide on the most appropriate generator. The need to decide on an appropriate generator does not arise in arithmetic word problems (for these problems, the generator is known: the enrolment of Dunearn Primary is given, and the subsequent unknowns can be found by carrying out the necessary operations on Dunearn Primary).

In previous studies, working-memory span and other central executive functions have been found to account for 2 percent to 3 percent of variation in mathematical performance (Bull & Scerif, 2001). Given their added complexity, algebraic word problems should impose more cognitive demands on the working-memory system than simple arithmetic problems. In our study (Lee et al., 2004), we focused on the relationship between working memory, literacy, non-verbal IQ, and performance on algebraic word problems. Children from five primary schools (n = 151, average age = 10.7 years) located in the central and western zones of Singapore participated in the study. All pupils were functionally bilingual with the majority of children having received 7 years of schooling in English.

All children were administered (1) the Working Memory Test Battery for Children (WMTB-C, Pickering & Gathercole, 2001), (2) a literacy battery, consisting of subtests from the Wechsler Objective Reading and Language Dimensions (Rust, 2000) and the vocabulary subtest of the Wechsler Intelligence Scale for Children—Third Edition (WISC III, Wechsler, 1991), (3) a non-verbal IQ test (block design from the WISC-III, and (4) a mathematical test consisting of ten algebraic word problems. The WMTB-C utilizes a span approach to index capacities in each of three working-memory components: central executive, phonological loop, and visual-spatial sketchpad. The phonological loop, for example, was measured using the digit span, word recall, and several other tasks. The central executive component was measured using dual tasks requiring both storage and processing similar to those pioneered by Daneman and Carpenter (1980).

The results showed that all three working-memory measures were correlated reliably with mathematical performance. Literacy and non-verbal IQ also correlated strongly with mathematical performance. When the intercorrelations between the various predictor variables were taken into consideration, the predictors were found to account for 49 percent of variation in mathematical performance. Measures of the central executive, literacy, and non-verbal IQ all contributed uniquely to the prediction model. Of particular interest was the contribution from the central executive. Similar to previous studies conducted with younger children that focused on elementary mathematical skills, the central executive accounted for 2.6 percent of variation in our study.

One explanation for this lower than expected contribution is that some of the effects of the central executive may be indirect. Previous studies showed that both vocabulary development and language processing might be dependent on the phonological loop and the central executive. Thus, it is possible that in addition to its direct effect on mathematical performance, working memory affects it indirectly via literacy. We examined this possibility by conducting a path analysis. As illustrated in Figure 2.2, the data suggest that the best-fitting model contains both direct and indirect effects from the central executive to mathematical performance. For the phonological and visual-spatial components,

only indirect effects were reliable. The main contribution of these findings is that they show that the influence of working memory is greater than previously thought. When both direct and indirect effects were taken into account, the standardized total effect of the central executive (.42) was larger than that of both literacy (total effect = .36) and performance IQ (total effect = .33).

To examine the extent to which these findings extend to school examination results, we collected data on children's performance in Mathematics, English, and Mother Tongue at two time points: at the same time as the working-memory data were collected and eight months later. Both sets of examinations were conducted as part of normal school assessment conducted at the end of the fourth term and at the end of the second term in the following year. Compared to the algebraic word problem test, the school-based assessment was more comprehensive and contained all the material covered in the two terms prior to examination.

Data from the Mathematics and English examinations showed reliable correlations with all three working-memory components, as well as the non-verbal IQ and literacy measures. Data from Mother Tongue revealed some differences. Only the phonological loop, central executive, and literacy measures were correlated reliably with Mother Tongue performance. To take into account intercorrelations amongst both the predictors and the examination data, we conducted some preliminary path analyses with fully identified path models. In these models, only direct paths between the predictors and examination data were included. The analyses showed that the central executive measures were the only working-memory component to contribute uniquely to school examination performance. Furthermore, its contribution was only reliable for examination data collected at the same time as the working memory measures and not for the longitudinal data collected eight months later. For the longitudinal data, the literacy measure provided a unique contribution to the prediction of all three academic subjects. The non-verbal IQ measure predicted mathematics performance.

These findings provide important information on the utility of the working-memory measures. On an applied level, they reconfirm the efficacy of traditional measures of literacy and non-verbal abilities in predicting academic performance. Working memory measures provide a modest amount of additional information on children's performances that cannot be obtained from the literacy and non-verbal measures alone. These measures may be useful for diagnostic purposes and may aid in high stake educational decision making. On a theoretical level, these findings suggest that the influence of central executive span is more fundamental and directly affects literacy, non-verbal, as well as mathematical performances. To gain additional insight into possible mechanisms responsible for the correlation between working memory and mathematics, we examined the children's performances on the mathematical instrument in more detail.

The data reported above were based on overall accuracy. Another way to examine children's competency was to examine their ability to (1) understand and translate the problem into a structural representation, (2) build a set of problem-solving procedure from the structural representation, and (3) conduct the necessary computation. This de-composition of processes involved in word problem solving was based on the work of Mayer and Hegarty (1996) and was described in more detail in Ng and Lee (2004).

In our study, pupils were asked to solve the word problems using the model method. Primary school children in Singapore are taught to solve algebraic word problems using a variety of heuristics and the model method. For the model method, students draw diagrams, usually made up of rectangles, to represent the mathematical information presented in the word problem. The rectangles represent the unknown and students are expected to solve the unknown by analysing relationships between the rectangles. In our study, children's model accuracy was scored in terms of text-to-structure accuracy (TS), structure-to-procedure accuracy (SP), and computational accuracy. TS was used as an indication of their structural understanding. In awarding TS scores, we focused on children's accuracy in depicting proportional relationships between protagonists in the problem. Ability to translate the model into problem-solving procedure, or SP, was indexed by children's ability to write a set of mathematical operations that allow them to solve the problem. Although all problems used in the study could be solved using algebraic equations, most primary children rely on the model method or other arithmetic heuristics. Both arithmetic and algebraic methods were scored as correct if they followed logically from the model drawn by the children. Computation accuracy was scored independently of the previous two indices. Children who committed errors in the model drawing or procedural stages could still obtain full computation scores if the latter were carried out accurately.

The data showed that the central executive, literacy, and non-verbal IQ measures all provided unique contributions to the prediction of the TS and SP scores. Only the central executive predicted computation scores. These findings further confirm that both literacy and non-verbal skills are important for mathematical word problem solving. Of importance is that they show that both types of abilities are particularly important during the initial stages of problem solving: when information must be decoded from a textual form to a visual, mathematical format. Although the second stage of problem solving—from structure to procedure—does not seem to have any explicit linguistic requirements, both literacy and non-verbal abilities were reliable predictors to children's performances. One explanation for this finding is that transforming the models to a set of mathematical equations involves language-like abilities. A recent finding from the functional magnetic imaging literature is consistent with this explanation. It shows that both linguistic and visual-spatial processing areas are activated during the performance of complex mathematical tasks (Zago et al., 2001).

FUTURE DIRECTIONS

There are several unanswered questions. First, why is working memory working? Although empirical demonstrations of the relationship between working memory and other cognitive functions abound, works on the processes responsible for these linkages have only just begun. There are several likely explanations. First, cognitive functions that are closely related with working memory require or draw on working-memory resources. Second, both working memory and cognitive functions to which it predicts rely on the same fundamental processes. Third, both working memory and closely related cognitive functions are mediated by the same set of cognitive developmental variables. Variables that influence the development of specific cognitive function also affect the development of working memory. Readers who have followed arguments regarding the relationship between traditional IQ measures and academic performance will be familiar with the latter two explanations.

Our work has focused on teasing out the processes involved in complex academic performance in order to examine the boundaries of the relationship between working memory and such processes. Although still at an early stage, our work suggests that the various working-memory components make different contributions to academic performance. Furthermore, the nature of these relationships differs across academic subjects.

A second issue that requires further investigation, and that is of particular relevance to the focus of this book, is the modifiability of working memory. To what extent does working-memory span change across the early years and is it susceptible to clinical or pedagogical intervention? A number of investigators have examined the developmental profile of working memory. Gathercole, Pickering, Ambridge, and Wearing (2004), for example, administered sub-tests from the WMTB-C and the Visual Patterns Tests to 4- to 15-year-olds. The results showed linear increases in span in all three working-memory components from early childhood to adolescence. They also found some evidence of asymptotic performance in the early teenage years (earlier—11 years old—for visual tasks). Furthermore, the tripartite model was shown to provide a good fit to the data from age 6 onwards. To examine the modifiability of working memory, Swanson (1992) adopted a dynamic assessment approach in administering a battery of working-memory tests. Using a graduated prompting procedure based on the work of Campione and Brown (1987), Swanson administered 11 working-memory tasks to 129 ten-year-olds. The children were given a baseline score based on their initial performance on the tasks. A series of hints were then given until the correct response was obtained. Several measures were taken including the number of hints needed to achieve the correct response and the total number of correct responses when hints were given. The results showed that use of hints produced large improvement in working memory performance. Swanson also showed that inclusion of the dynamic assessment measures improved working memory's prediction of reading achievement. One problem, acknowledged by Swanson, was that the long-term effect of such intervention was unknown.

CONCLUSION

From short-term memory to limited capacity processor to working memory and executive functions, the last 40 years had seen some major changes in our understanding of a central aspect of cognition. Although not uncontroversial, Baddeley and Hitch's (1974) working-memory model has proved productive and has generated much translational work regarding the relationship between working memory and various cognitive functions. Research conducted in our laboratory shows that central executive span plays a more prominent role in mathematical problem solving than previously thought. It was found to play a greater role than literacy and non-verbal IQ when both direct and indirect effects were taken into account. From an applied perspective, literacy and non-verbal IQ tests are still superior if only one test is to be used to predict mathematical performance. Because the central executive measure provided unique contribution even when literacy and non-verbal IQ were taken into account, it may serve as a useful adjunct in high stake examinations.

References

Atkinson, R. C., & Shiffrin, R. M. (1968). Human memory: A proposed system and its control processes. In K. W. Spence (Ed.), The Psychology of Learning and Motivation: Advances in Research and Theory (pp. 89–195). New York: Academic Press.

Baddeley, A. (1996). Exploring the central executive. The Quarterly Journal of Experimental Psychology, 49A, 5–28.

Baddeley, A. (2000a). The episodic buffer: a new component of working memory? Trends in Cognitive Sciences, 4, 417–423.

Baddeley, A. (2000b). Short term and working memory. In E. Tulving & F. I. M. Craik (Eds.), The Oxford Handbook of Memory (pp. 77–92). New York: Oxford University Press.

Baddeley, A., & Hitch, G. J. (1974). Working memory. In G. A. Bower (Ed.), Recent Advances in Learning and Motivation (pp. 47–90). New York: Academic Press.

Baddeley, A. D., & Logie, R. H. (1999). Working memory: The multiple-component model. In A. Miyake & P. Shah (Eds.), Models of Working Memory: Mechanism of Active Maintenance and Executive Control (pp. 28–61). Cambridge, UK: Cambridge University Press.

Bednarz, N., & Janvier, B. (1996). Emergence and development of algebra as a problem-solving tool: Continuities and discontinuities with arithmetic. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra (pp. 115–136). Dordretch, The Netherlands: Kluwer.

Bull, R., & Johnston, R. S. (1997). Children's arithmetical difficulties: Contributions from processing speed, item identification, and short-term memory. Journal of Experimental Child Psychology, 65, 1–24.

Bull, R., & Scerif, G. (2001). Executive functioning as a predictor of children's mathematics ability: Inhibition, switching, and working memory. Developmental Neuropsychology, 19, 273–293.

Campione, J., & Brown, A. (1987). Linking dynamic assessment with school achievement. In C. Lidz (Ed.), Dynamic Assessment (pp. 82–115). New York: Guildford Press.

Conway, A. R., Kane, M. J., & Engle, R. W. (2003). Working memory capacity and its relation to general intelligence. Trends in Cognitive Science, 7, 547–552.

Cowan, N. (1999). An embedded-processes model of working memory. In A. Miyake & P. Shah (Eds.), Models of Working Memory: Mechanisms of Active Maintenance and Executive Control (pp. 62–101). New York: Cambridge University Press.

Craik, F. I. M., & Lockhart, R. S. (1972). Levels of processing: A framework for memory research. Journal of Verbal Learning and Verbal Behavior, 11, 671–684.

Daneman, M., & Carpenter, P. A. (1980). Individual differences in working memory and reading. Journal of Verbal Learning and Verbal Behavior, 19, 450–466.

Daneman, M., & Merikle, P. M. (1996). Working memory and language comprehension: A meta-analysis. Psychonomic Bulletin and Review, 3, 422–433.

Engle, R. W., Kane, M. J., & Tuholski, S. W. (1999). Individual differences in working memory capacity and what they tell us about controlled attention, general fluid intelligence, and functions of the prefrontal cortex. In A. Miyake & P. Shah (Eds.), Models of Working Memory: Mechanisms of Active Maintenance and Executive Control (pp. 102–134). New York, NY: Cambridge University Press.

Fürst, A. J., & Hitch, G. J. (2000). Separate roles for executive and phonological components of working memory in mental arithmetic. Memory and Cognition, 28, 774–782.

Gathercole, S. E., & Pickering, S. J. (2000a). Assessment of working memory in six- and seven-year-old children. Journal of Educational Psychology, 92, 377–390.

Gathercole, S. E., & Pickering, S. J. (2000b). Working memory deficits in children with low achievements in the national curriculum at 7 years of age. British Journal of Educational Psychology, 70, 177–194.

Gathercole, S. E., Pickering, S. J., Ambridge, B., & Wearing, H. (2004). The structure of working memory from 4 to 15 years of age. Developmental Psychology, 40, 177–190.

James, W. (1892). Psychology. London: Macmillan.

Lee, K., Ng, S. F., Ng, E. L., & Lim, Z. Y. (2004) Working memory and literacy as predictors of performance on algebraic word problems. Journal of Experimental Child Psychology, 89, 140–158.

Lehto, J. (1995). Working memory and school achievement in the ninth form. Educational Psychology, 15, 271–281.

Logie, R.H. (1986). Visuo-spatial processing in working memory. Quarterly Journal of Experimental Psychology: Human Experimental Psychology, 38A, 229–247.

Mayer, R. E., & Hegarty, M. (1996). The process of understanding mathematical problems. In R. J. Sternberg & T. Ben-Zeev (Eds.), The Nature of Mathematical Thinking (pp. 29–53). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

McKenzie, B., Bull, R., & Gray, C. (2003). The effects of phonological and visual-spatial interference on children's arithmetical performance. Educational and Child Psychology, 20, 93–108.

McLean, J. F., & Hitch, G. J. (1999). Working memory impairments in children with specific arithmetic learning difficulties. Journal of Experimental Child Psychology, 74, 240–260.

Miyake, A., & Shah, P. (1999). Models of Working Memory: Mechanisms of Active Maintenance and Executive Control. Cambridge, UK: Cambridge University Press.

Ng, S. F., & Lee, K. (2004). Model method—A window into pupils' mental worlds. Manuscript in preparation.

Passolunghi, M. C., & Siegel, L. S. (2001). Short-term memory, working memory, and inhibitory control in children with difficulties in arithmetic problem solving. Journal of Experimental Child Psychology, 80, 44–57.

Passolunghi, M. C., Cornoldi, C., & De Liberto, S. (1999). Working memory and intrusions of irrelevant information in a group of specific poor problem solvers. Memory and Cognition, 27, 779–790.

Pickering, S. J., & Gathercole, S. E. (2001). Working Memory Test Battery for Children. Kent: The Psychological Corporation.

Rust, J. (2000). Wechsler Objective Reading and Language Dimensions (Singapore). London: The Psychological Corporation.

Sikora, D. M., Haley, P., Edwards, J., & Butler, R. W. (2002). Tower of London test performance in children with poor arithmetic skills. Developmental Neuropsychology, 21, 243–254.

Swanson, H. L. (1992). Generality and Modifiability of Working Memory Among Skilled and Less Skilled Readers. Journal of Educational Psychology, 84, 473–488.

Wechsler, D. (1991). Wechsler Intelligence Scale for Children (3rd ed.). San Antonio, TX: Psychological Corporation.

Young, D. (1980). Group Mathematics Test. Kent: Hodder and Stoughton.

Zago, L., Pesenti, M., Mellet, E., Crivello, F., Mazoyer, B., & Tzourio-Mazoyer, N. (2001). Neural correlates of simple and complex mental calculation. Neuroimage, 13, 314–327.