Working memory and arithmetic calculation in children: The contributory roles of processing speed, short-term memory, and reading

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Abstract

The cognitive underpinnings of arithmetic calculation in children are noted to involve working memory; however, cognitive processes related to arithmetic calculation and working memory suggest that this relationship is more complex than stated previously. The purpose of this investigation was to examine the relative contributions of processing speed, short-term memory, working memory, and reading to arithmetic calculation in children. Results suggested four important findings. First, processing speed emerged as a significant contributor of arithmetic calculation only in relation to age-related differences in the general sample. Second, processing speed and short-term memory did not eliminate the contribution of working memory to arithmetic calculation. Third, individual working memory components—verbal working memory and visual–spatial working memory—each contributed unique variance to arithmetic calculation in the presence of all other variables. Fourth, a full model indicated that chronological age remained a significant contributor to arithmetic calculation in the presence of significant contributions from all other variables. Results are discussed in terms of directions for future research on working memory in arithmetic calculation.

Introduction

Arithmetic calculation is one of the primary academic skills children learn in school. Indeed, next to learning to read and write, the development of arithmetic calculation skills takes a place near the top of the curricular pyramid. Given the value of these skills, difficulties in these areas would pose substantial problems for children’s educational development and for their daily experiences throughout life. For instance, the importance of addition as a primary skill in schools rests in its place as a building block for the development of increasingly advanced mathematical skills (Ashcraft, 1992, Jensen and Whang, 1994, National Council of Teachers of Mathematics, 2000). Indeed, the National Council of Teachers of Mathematics (2006) recently underscored the importance of calculation abilities early in school-based curriculum and instruction. Proficiency in simple addition leads to the development of more complex addition skills (Geary & Brown, 1991), and the development of multiplication skills stems from proficiency in addition (Cooney, Swanson, & Ladd, 1988).

Across the field of mathematics, research has focused on describing factors that influence the development of arithmetic performance (e.g., problem-solving strategies [Fuchs & Fuchs, 2002]), on isolating specific areas of arithmetic difficulty (e.g., level of problem difficulty [Ostad, 1998]), and on developing curricular and instructional interventions to respond to poor arithmetic performance (Fleischner, Nuzum, & Marzola, 1987). However, compared with the substantial body of research focused on identifying the cognitive processes that undergird reading (e.g., phonological processing [Snow et al., 1998, Stanovich, 1982]), little research has been directed at identifying the cognitive processes that are involved in arithmetic calculation. Extant theoretical and empirical research suggests that working memory, a cognitive system that is in part responsible for the construction of information and the transfer of this information into long-term memory, is likely to play an integral role in arithmetic calculation (Baddeley, 1996, Baddeley, 2001).

A generalized definition of working memory defines it as a limited capacity information processing resource. It has two principal processes: the preservation of information and the concurrent processing of the same or other information (e.g., Engle et al., 1992, Just and Carpenter, 1992). The latter element is important because it serves to distinguish working memory from other forms of memory such as short- and long-term memory.

The most widely accepted and researched model of working memory was formulated by Baddeley and Hitch (1974). Consistent with 30 years of research, this model supports a three-component structure consisting of the central executive interacting with two subsystems: the phonological loop and the visuospatial sketchpad. The central executive is viewed as a limited capacity component responsible for directing information toward the two subsystems, coordinating activity within the memory system, and retrieving information from long-term memory (e.g., Baddeley, 1986, Baddeley, 1996). The phonological loop (analogous to verbal working memory) is responsible for the temporary storage of speech-based information and is vulnerable to rapid decay. Decay can be interrupted by vocal and subvocal rehearsal through articulation (Vallar & Baddeley, 1984). The visuospatial sketchpad (analogous to visual–spatial working memory) is responsible for the temporary storage of visual–spatial information and is involved in the generation and manipulation of mental images.

In the field of mathematics, the cognitive processes involved in performing arithmetic calculations are embedded within the working memory system in that they require a combination of temporary information storage while performing other mental operations. For instance, to solve the problem 18 + 7, one must concurrently retain two or more pieces of information (phonological codes representing the numbers 18 and 7) and then employ one or more procedures (e.g., counting) to combine the numbers to produce an answer. Alternatively, employing carrying or regrouping involves maintaining recently processed information while conducting a related operation. To solve 18 + 7, one must retain the 5 from adding 8 + 7 while adding the 1 from the tens column of the 18 to the 1 from the tens column of the 15 produced from adding the 8 + 7.

Results from the limited number of studies conducted to date, largely with adults, suggest that working memory resources are required to solve multistep problems such as 125 + 97 (Fürst and Hitch, 2000, Geary and Widaman, 1992, Logie et al., 1994). In these problems, working memory appears to be related to retaining intermediate values during calculation and to carrying partial results across columns. Adams and Hitch (1997) explored this relationship in 8- to 11-year-olds using simple mental addition (e.g., 8 + 1) and complex mental addition (e.g., 231 + 16) problems. Results suggested that proficiency in solving problems presented verbally was associated with high working memory functioning, whereas proficiency in solving problems presented visually was related to low working memory functioning. Findings were interpreted as indicating that problems presented visually require more working memory resources than do problems presented verbally. In addition, level of problem difficulty was related to working memory contributions, with higher working memory spans corresponding with more complex problems.

Studies have also indicated that individual working memory components have specialized roles in arithmetic calculation (e.g., Ashcraft, 1995, Geary et al., 2007). Verbal working memory has been associated with exact calculation in addition (Lemaire, Abdi, & Fayol, 1996), subtraction (Seyler, Kirk, & Ashcraft, 2003), and multiplication (Seitz & Schumann-Hengsteler, 2000). In addition, relationships have been reported between verbal working memory and calculation procedures such as counting (Logie & Baddeley, 1987), retaining problem information (Fürst & Hitch, 2000), and holding interim results during counting (Logie & Baddeley, 1987). Hecht (2002) attempted to clarify the role of working memory by contrasting verbal working memory with general working memory. In examining simple arithmetic calculation in undergraduate students, Hecht found that verbal working memory was used to store intermediate values, whereas general working memory resources were involved in coordinating procedures to be used in calculations. Application of these results to describe the role of working memory in arithmetic calculation should be made with caution given that Hecht’s study focused on adults and did not include a measure for visual–spatial working memory.

The contribution of working memory to arithmetic calculation in children has not received much attention. Since the early work of Hitch and colleagues (Adams and Hitch, 1997, Hitch, 1978), only a few studies have examined this relationship. Geary, Hoard, Byrd-Craven, and DeSoto (2004) examined the involvement of working memory in the strategy choice of elementary school children (Grades 1, 3, and 5) when solving simple and complex addition problems. Results indicated that, similar to earlier studies, higher working memory capacity was associated with higher accuracy in solving complex arithmetic problems (Adams & Hitch, 1997). Geary and colleagues (2004), however, found that this relationship was related to specific strategies. For instance, higher working memory facilitated the use of more sophisticated strategies such as decomposition, whereas lower working memory was related to less sophisticated strategies such as finger counting. In a related study, Fuchs and colleagues (2006) investigated the cognitive predictors of children’s arithmetic calculation. Their results suggested that working memory was not a significant predictor in third graders’ performance on simple mental addition (e.g., 3 + 2), complex mental addition (e.g., 35 + 29), or arithmetic word problems; rather, phonological decoding, attention, and processing speed emerged as unique predictors. However, a narrow selection of working memory measures might have overemphasized the contribution of verbal working memory. Similar to most studies on arithmetic calculation in children, both Geary et al., 2004, Fuchs et al., 2006 did not include measures of visual–spatial working memory; rather, these studies used verbal working memory tasks as the sole measure of working memory.

Largely absent from research on arithmetic calculation is consideration for the role of visual–spatial working memory. Limited studies with adults suggest that the involvement of visual–spatial working memory in arithmetic calculation is important during the initial stages of arithmetic calculation for encoding problems presented visually (Heathcote, 1994, Logie et al., 1994). Heathcote (1994) examined the specific roles of verbal working memory and visual–spatial working memory in pairs of three-digit addition problems presented visually and verbally. Whereas visual–spatial working memory was involved in retaining carries, verbal working memory was involved in retaining interim results through articulatory rehearsal. Furthermore, verbal working memory and visual–spatial working memory both were involved during the initial stages of calculation through encoding of the problem in working memory, with visual–spatial working memory involved in problems presented visually and verbal working memory involved in problems presented verbally. Geary and colleagues’ (2007) study offered insight into the role of visual–spatial working memory in arithmetic calculation in children. Specifically, Geary and colleagues found that visual–spatial working memory was related to the use of counting procedures (i.e., min strategy) when performing complex addition (e.g., 16 + 7) but not when performing simple addition (e.g., 4 + 3).

Although research suggests a relationship between working memory and arithmetic calculation, with potential contributory influences of individual components related to verbal and visual–spatial processing, the influence of other cognitive processes that inform working memory has received little attention. Specifically, arithmetic calculation proficiency is supported by two cognitive processes that interact with working memory: processing speed (Case, 1985) and short-term memory (Baddeley & Hitch, 1974).

The speed at which information is introduced into working memory and used within working memory is critical to the efficiency of the working memory system. Case (1985) highlighted the importance of processing speed by implicating its role in increasing the available short-term storage space. Based on research with children, Case and colleagues found a linear relationship between processing speed and storage capacity of the working memory (Case, Kurland, & Goldberg, 1982). In a study of 6- to 11-year-olds, they reported that faster counting speed predicted higher counting spans. Case interpreted this and similar findings as indicative of a speed–capacity relationship whereby the faster a child processes relevant information, the more information the child can retain over a short period of time. In essence, faster operations require less workspace, leaving more available for information storage.

Although processing speed and short-term memory have been examined in relation to various arithmetic activities such as problem solving (e.g., Swanson & Beebe-Frankenberger, 2004) and to strategy use in arithmetic calculation (Geary and Brown, 1991, Jordan and Montani, 1997), their relative importance in the complex of memory processes that have been associated with arithmetic calculation is unclear. Bull and Johnston (1997) examined the role of processing speed and short-term memory in children’s arithmetic calculation. They reported that when a group of 7-year-olds were assessed on measures of short-term memory, long-term memory, processing speed, and sequencing ability, the strongest predictor of arithmetic calculation was processing speed. Their study, however, used a set of cognitive tasks that did not include measures of working memory.

The most telling evidence for the potential contribution of processing speed and short-term memory to arithmetic calculation is available from recent studies by Swanson and colleagues (Swanson, 2006, Swanson and Beebe-Frankenberger, 2004). Swanson and Beebe-Frankenberger (2004) examined the contributions of working memory to arithmetic problem solving and to arithmetic calculation in children in Grades 1 to 3. Results revealed that processing speed contributed unique variance to each of these arithmetic areas in the presence of unique contributions of working memory, short-term memory, phonological processing, and age. Although this study is notable in its inclusion of several cognitive processing domains, it treated working memory as a general domain (creating a composite score of all working memory tasks) and did not differentiate between verbal working memory and visual–spatial working memory in predictive models. Swanson’s (2006) study investigated the cognitive processes that contribute to arithmetic calculation in children with advanced mathematical skills in Grades 1 to 3. This study differentiated between individual components of working memory (verbal and visual–spatial) and included measures of the executive system, processing speed, and inhibition (generation and fluency). In the presence of significant contributions of age and reading, processing speed did not emerge as a significant contributor, yet the executive system and generation (numbers and letters) both proved to be significant contributors. It should be noted, however, that the measures used to represent executive functioning in Swanson’s study (e.g., auditory digit sequence) have been viewed as representative of verbal working memory in other studies (e.g., Swanson, 2004, Swanson and Sachse-Lee, 2001). Thus, an alternative interpretation of the contribution of executive functioning might be that verbal working memory was a significant contributor.

In review, since early work by Hitch (1978), only a few studies have directly investigated the role of working memory in children’s arithmetic calculation performance. Subsequent research has been equivocal on the role of working memory. Three challenges in particular underscore our current difficulty in reaching consensus on the role of working memory in arithmetic calculation. First, many studies treat working memory as a unidimensional memory system (e.g., Hecht, 2002). Second, there has been a lack of consideration for the role that processing speed plays in the complex of cognitive processes that are involved in children’s arithmetic calculation. Third, the role of short-term memory in the complex of cognitive processes that relate both to working memory and to arithmetic calculation has received little attention. The current study was designed to address these issues with a central purpose of examining the relative contributions of processing speed, short-term memory, verbal working memory, and visual–spatial working memory to arithmetic calculation in children.

Research suggests that, when examining the cognitive underpinnings of performance in academic areas, it is important to consider similar contributory processes that relate to each academic area. With respect to mathematical performance, reading has been associated with word problem solving (Swanson & Beebe-Frankenberger, 2004) and with arithmetic calculation (Hecht, Torgesen, Wagner, & Rashotte, 2001). Ostensibly, although reading and arithmetic use different knowledge structures (e.g., letter sound knowledge vs. counting knowledge), performance in these areas draws on similar cognitive processes. For instance, one will use his or her phonological processing ability when identifying words through a sounding-out procedure (Bradley & Bryant, 1985). To solve arithmetic calculations, implementing a counting-based procedure also engages the phonological system (Buchner et al., 1998, Geary, 1993). Consistent with these relationships, the current study sought to understand the influence of the target variables after accounting for the interaction between reading and arithmetic calculation.

Section snippets

Participants

A total of 90 children (44 boys and 46 girls) in Grades 3 to 6 from three schools in central Canada participated in this study. Children’s ages ranged from 98 to 145 months. The socioeconomic status of individual students was not assessed; however, each of the schools that participated in the study was located in a predominantly middle-class neighborhood. All children spoke English as their first language. No child had been identified as having a neurological disorder (e.g., learning

Descriptive statistics

Means, standard deviations, and score ranges for all measures are reported in Table 1. Due to participant selection criteria, large ranges in chronological age were expected. As a result, large ranges were also expected between the academic achievement variables. Reading raw scores ranged from 28 to 46, and arithmetic calculation raw scores ranged from 20 to 41. Reading standard scores ranged from 79 to 132 and arithmetic standard scores ranged from 69 to 128.

Correlations among the measures are

Discussion

Results of the current study provide further evidence of the role of working memory and related cognitive processes in arithmetic calculation (e.g., Adams and Hitch, 1997, Swanson and Beebe-Frankenberger, 2004). Building on a dearth of research on arithmetic calculation in children, this study sought to examine the relative contributions of processing speed, short-term memory, and working memory in children’s arithmetic calculation. Results suggested four important findings. First, processing

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