Differential contribution of specific working memory components to mathematics achievement in 2nd and 3rd graders

Abstract

The contribution of the three core components of working memory (WM) to the development of mathematical skills in young children is poorly understood. The relation between specific WM components and Numerical Operations, which emphasize computation and fact retrieval, and Mathematical Reasoning, which emphasizes verbal problem solving abilities in 48 2nd and 50 3rd graders was assessed using standardized WM and mathematical achievement measures. For 2nd graders, the central executive and phonological components predicted Mathematical Reasoning skills; whereas the visuo-spatial component predicted both Mathematical Reasoning and Numerical Operations skills in 3rd graders. This pattern suggests that the central executive and phonological loop facilitate performance during early stages of mathematical learning whereas visuo-spatial representations play an increasingly important role during later stages. We propose that these changes reflect a shift from prefrontal to parietal cortical functions during mathematical skill acquisition. Implications for learning and individual differences are discussed.

Introduction

Although the basics of mathematics are among the more important competencies that children need to master for successful living in modern societies, our understanding of the cognitive mechanisms that support mathematics learning is limited (Mazzocco, 2008). What is known suggests that working memory (WM) is pivotal to many aspects of learning mathematics (Bull et al., 2008, Geary, 1990, Geary and Brown, 1991, Geary et al., 2000, Geary et al., 2004, Hitch and McAuley, 1991, Passolunghi and Siegel, 2001, Passolunghi and Siegel, 2004, Siegel and Ryan, 1989, Swanson, 1993, Swanson, 1994, Swanson and Sachse-Lee, 2001, van der Sluis et al., 2005, Wilson and Swanson, 2001). However, the relations between the different components of WM and mathematical competence are not as well established in children compared to adults (Ashcraft, 1992, Furst and Hitch, 2000, Hecht, 2002, Lemaire et al., 1996, Logie et al., 1994).

Many children begin school with an implicit understanding of aspects of number, counting, and arithmetic and WM may contribute to their ability to build on this informal knowledge during schooling (Geary and Brown, 1991, Geary and Burlingham-Dubree, 1989, Siegler and Jenkins, 1989). Children who excel in early mathematics learning tend to have high WM capacity (Hoard et al., 2008, Passolunghi et al., 2008), and mathematically gifted adolescents tend to have enhanced visuo-spatial WM (Dark & Benbow, 1990). WM has also been reported to mediate the relationship between IQ and mathematical performance as early as the 1st grade (Passolunghi et al., 2008). Children's learning of the mathematical number line is influenced by a combination of intelligence, the central executive, and visuo-spatial WM. However, the relative contributions of these WM components to learning changes from 1st to 2nd grade as the central executive increases in importance, whereas the roles of intelligence and visuo-spatial WM decline (Geary, Hoard, Nugent & Byrd-Craven, 2008). In short, we know that WM is critical for mathematics learning but we do not fully understand how the different components of WM contribute to learning in different areas of mathematics and we do not know whether the importance of one or more WM components changes as learning progresses. We begin with a brief review of the components of WM and their relation to mathematics learning, and then outline how the current study addresses the issue of whether the relative importance of the different WM systems changes from one grade to the next.

WM is a cognitive system specialized for storage and manipulation of information (Baddeley, Hitch & Bower, 1974). Although different theoretical models of WM have been proposed (for a review see Miyake & Shah, 1999), Baddeley and Hitch's model has been the most influential. In this model, WM is composed of a central executive and two subsystems for temporary storage and rehearsal of auditory-verbal and visuo-spatial information, the phonological loop and the visuo-spatial sketchpad, respectively (Baddeley, 1986, Baddeley, 1996, Baddeley et al., 1974, Miyake and Shah, 1999).

The central executive plays an important role in sequencing operations, coordinating the flow of information, and guiding decision-making (Baddeley, 1996, Baddeley et al., 1998), particularly when problems are more complex and facts cannot be easily retrieved from memory. The central executive is important for many aspects of mathematical performance, including use of complex arithmetic procedures that involve carrying and borrowing operations (Ashcraft, 1992, De Rammelaere et al., 1999, De Rammelaere et al., 2001, Frensch and Geary, 1993, Geary et al., 1993, Hecht, 2002, Lemaire et al., 1996). The two other components of WM are specialized for storage of domain-specific information. The phonological loop is involved in encoding and maintaining arithmetical operands (Furst and Hitch, 2000, Logie et al., 1994, Noel et al., 2001) and maintaining intermediate results (Heathcote, 1994), but not specifically in calculation of answers (De Rammelaere et al., 1999, Furst and Hitch, 2000, Hecht, 2002, Lemaire et al., 1996). Furst and Hitch showed that the phonological loop is involved in retaining and storing information about complex problems, but it is not critically involved in retrieving factual mathematical knowledge (Furst & Hitch, 2000). Consistent with this, the relationship between phonological loop and adults' mathematical performance has been relatively weak, except in dual-task paradigms when the phonological loop is excessively taxed (Heathcote, 1994, Lehto, 1995, Logie and Baddeley, 1987, Logie et al., 1994). In adults, the visuo-spatial sketchpad has been implicated in solving multi-digit operations (Heathcote, 1994) and in more complex algebraic and geometric problem solving (Reuhkala, 2001). Notwithstanding these findings, the role of the visuo-spatial sketchpad in mathematical cognition remains poorly understood.

Each WM component has a specialized role in mathematical cognition that varies with expertise and development. Different levels of experience with numbers and mathematical concepts, familiarity of the stimuli and the strength of representations can lead to changes in the types of strategies applied to solve a mathematical task; this in turn calls upon different WM components (Gathercole and Adams, 1994, Henry and Miller, 1991).

Children under the age of seven tend to rely more on visual memory to remember material such as pictures of familiar and nameable objects rather than coding visual items to verbal labels (Hitch, Halliday, Schaafstal & Schraagen, 1988). Some researchers have suggested that preschoolers tend to perform better on nonverbal rather than verbal arithmetic tasks and that the visuo-spatial sketchpad capacity is the best predictor of these abilities in this age group (Levine et al., 1992, McKenzie et al., 2003, Rasmussen and Bisanz, 2005, Simmons et al., 2008). From the age of seven onwards, however, children increasingly rely on verbal rehearsal to maintain information in memory, thus recruiting the phonological loop (Hitch et al., 1988). Consistent with this, Rasmussen and Bisanz found that by the 1st grade, performance becomes equivalent on nonverbal and verbal mathematical tasks, and that the phonological loop becomes the best predictor of performance on verbal mathematics problems (Rasmussen & Bisanz, 2005). WM also influences math performance in elementary school: in a large sample of 1st, 2nd and 3rd graders, Swanson found that younger children and children who were poor mathematical problem solvers performed less well on WM tasks than older children or children who were good problem solvers (Swanson & Beebe-Frankenberger, 2004). However, the specific contributions of each WM component across grades were not examined.

In 7- to 8-year-old children, one study found that mathematics performance is most strongly correlated with the central executive, followed by the phonological loop (L. Henry & MacLean, 2003). As the supervisory system, the central executive facilitates children's problem solving by aiding in selection of appropriate strategies (Barrouillet and Lepine, 2005, Bull et al., 1999, Geary et al., 2004) and by allocating attention resources to implement the strategy execution. Using a longitudinal design, Gathercole and Pickering found that central executive measures shared significant and unique links with children's standardized test scores in mental arithmetic at 7 years of age and again at 8 years of age (Gathercole & Pickering, 2000). On the other hand, in a large sample of 8- to 11-year-old children, Adams and Hitch found that articulatory suppression significantly disrupted children's ability to solve arithmetic problems (Adams, Hitch & Donlan, 1998), suggesting an important role for the phonological loop. Other studies have suggested that the phonological loop is engaged when children transform symbol and number strings into verbal code when using verbally mediated counting strategies during basic arithmetic problem solving (Baddeley and Logie, 1987, Geary et al., 1996, Geary et al., 1993, Logie et al., 1994, Miura et al., 1999). More recently, Holmes and Adams found that in a group of typically developing 8- and 9-year-olds, the central executive and the visuo-spatial sketchpad, but not the phonological loop scores, predicted overall curriculum-based mathematics achievement (Holmes & Adams, 2006). Interestingly, this study also found that for 8-year-olds, the visuo-spatial sketchpad was a stronger predictor of mathematics performance than the central executive. Similarly, Gathercole and Pickering (Gathercole & Pickering, 2000) found that 6- and 7-year-old children's performance on national curriculum mathematics assessments correlate with performance on measures of visuo-spatial WM.

Taken together, these studies suggest that WM plays an important role in both mathematical performance and skill development in 7- to 11-year-old children. Current data also hint at the changing role of different WM components in relation to performance and skill development. However, the research to date has been contradictory; some studies implicate the phonological loop, others the visuo-spatial sketchpad, and still others the central executive (Henry and MacLean, 2003, Holmes and Adams, 2006) There are several reasons for such inconsistencies. The first is related to the large age-range across studies (Adams et al., 1998, Andersson, 2007, Durand et al., 2005, Holmes and Adams, 2006, Swanson, 2006), resulting in high variability in the level of the participants' mathematical competence and in the curricular content of the mathematical tasks. To address these issues, we focus on two groups of children in the 2nd and 3rd grades who are at an important stage in formal mathematical skill development.

A second reason for the inconsistencies in findings is the large variability in the types of tasks used to assess mathematical performance. For example, some studies have focused either on individual arithmetic operations, such as addition (Adams & Hitch, 1997), subtraction (Barrouillet, Mignon & Thevenot, 2008) or other more complex arithmetic problems (Henry & MacLean, 2003). The use of a single measure of mathematical ability is useful for studying particular cognitive questions, but is less useful for revealing more general links between WM and developmentally relevant mathematical abilities. To obtain a more complete and ecologically valid profile of mathematics competence, we administered two standardized mathematics achievement measures — the Numerical Operations and Mathematical Reasoning subtests of the Wechsler Individual Achievement Test (WIAT-II; (Wechsler, 2001)). A key distinction between the two measures is that Numerical Operations plays a greater emphasis on counting and computation whereas Mathematical Reasoning emphasizes word problems.

A third reason for inconsistencies across studies is the use of non-standardized instruments to assess WM. We use the Working Memory Test Battery for Children (WMTB-C), a comprehensive, standardized assessment of three core WM components (Pickering & Gathercole, 2001). Importantly, large-scale studies have found that the three-component model of WM best fits empirical data on the structure and development of WM in 6- to 16-year-old children (Gathercole, Pickering, Ambridge & Wearing, 2004). Additionally, WM capacity can be expressed both in terms of raw and standardized scores on each of the three WM components. Raw scores reflect age-related differences, whereas age-normed scores are useful in assessing performance differences after controlling for normative development. Accordingly, we used both raw and age-normed scores from the standardized WMTB-C measure to examine how each WM component influences mathematical abilities assessed by the Numerical Operations and Mathematics Reasoning. Based on previous research on 2nd and 3rd graders' strategy use (Geary et al., 2004, Wu et al., 2008) we hypothesized that 2nd graders would rely more on the central executive compared to the 3rd graders because of the greater use of counting and other algorithmic strategies, whereas 3rd graders would rely more on automated retrieval processes that do not require the central executive to the same extent as 2nd graders.