Executive functions underlying multiplicative reasoning: Problem type matters

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Abstract

We investigated the extent to which inhibition, updating, shifting, and mental-attentional capacity (M-capacity) contribute to children’s ability to solve multiplication word problems. A total of 155 children in Grades 3–6 (8- to 13-year-olds) completed a set of multiplication word problems at two levels of difficulty: one-step and multiple-step problems. They also received a reading comprehension test and a battery of inhibition, updating, shifting, and M-capacity measures. Structural equation modeling showed that updating mediated the relationship between multiplication performance (controlling for reading comprehension score) and latent attentional factors M-capacity and inhibition. Updating played a more important role in predicting performance on multiple-step problems than did age, whereas age and updating were equally important predictors on one-step problems. Shifting was not a significant predictor in either model. Implications of proposing executive function updating as a mediator between mathematical cognition and chronological age and attention resources are discussed.

Introduction

Studies on individual differences reveal that many school-age children struggle with mathematics (e.g., Mazzocco & Myers, 2003). Research on cognitive processes underlying mathematical cognition may lead to better understanding of developmental and individual differences in math performance. However, mathematics is a complex subject, encompassing a host of abilities and skills (Bull & Espy, 2006). Thus, the identified cognitive processes may depend on the type of mathematical skill or ability under investigation.

From a young age, children are taught a variety of computation procedures (e.g., addition, multiplication) and are expected to memorize basic number facts (Hecht et al., 2001, LeFevre et al., 1996, LeFevre et al., 2004, Lemaire et al., 1996). Although computational knowledge is important, research indicates that students’ errors on mathematical word problems are due mainly to problem misrepresentation (De Corte and Verschaffel, 1987, Jordon and Oettinger, 1997, Verschaffel et al., 2000). Regardless of computational skill, inability to represent the nature of the problem correctly and to identify the required operations will undermine performance (Goldman, 1989, Verschaffel et al., 1992). To solve a word problem, children must understand the nature of the problem, plan a method or strategy, execute the plan, and verify the solution (Goldman, 1989). However, children often adopt “short-cut” strategies as opposed to more meaningful approaches (Hegarty, Mayer, & Monk, 1995).

In short-cut approaches, the solver selects the numbers in the problem and the key relational terms (e.g., “times”) and develops a simple solution plan combining the numbers using arithmetic operations that are primed by key terms (e.g., multiply) (Greer, 1997, Hegarty et al., 1995, Verschaffel et al., 2000). Short-cut strategies may lead to errors when the keyed operation does not suit the problem. However, this approach often can be successful when applied in routine problem solving. In the classroom, routine problems typically are one-step situations requiring a simple computational procedure to be applied (Greer, 1997). In contrast, multiple-step word problems are more complex, and their solutions require children to formulate a plan and coordinate procedures rather than simply apply an overlearned operation. This requires that they monitor their thinking and transfer and apply previously learned strategies to new problems. Thus, different underlying cognitive processes may be identified depending on whether children use short-cut or more meaningful strategies to solve a set of mathematical word problems.

We examined children’s performance on word problems tapping different multiplicative concepts: scalar, proportion, and Cartesian problems (i.e., array and combinatorial). Scalar multiplication is applied in equal groups situations (Greer, 1992) in which there are a number of groups each holding the same number of objects. Proportion problems also fall into the equal groups category because their solutions eventually involve a multiplier and a multiplicand. However, in proportion problems, one of the multipliers is not given and a ratio calculation (e.g., division) must first be carried out. Array and combinatorial problems are categorized as Cartesian because their solutions are a direct product of sets. Array problems involve units that are the same and discrete and that produce one entity when arranged in two dimensions (rows and columns). Combinatorial problems involve construction of a new entity using two different entities. We assessed children’s ability to reason about such concepts in the context of one-step and multiple-step word problems. The problems did not contain keywords such as “times.” Nevertheless, we expected that previous experience with multiplicative concepts (indexed indirectly by age) would be relatively more important in predicting performance on one-step word problems. In contrast, executive processes should be more important in multiple-step items.

Executive function refers to a broad range of activities that control and regulate cognition. Executive functioning plays an important role in math reasoning (Bull and Johnston, 1997, Bull and Scerif, 2001, Lee et al., 2009, Passolunghi and Siegel, 2001). Researchers investigating the relationship between cognitive control processes and mathematical cognition often have adopted Baddeley’s working memory model as a framework. This model has four main components: central executive, phonological loop, visuospatial sketchpad, and episodic buffer (Baddeley, 1986, Baddeley, 1996, Baddeley, 2000, Baddeley and Hitch, 1974). The phonological loop and visuospatial sketchpad are short-term storage, or slave, systems dedicated to a content domain (verbal or visual–spatial). The episodic buffer holds representations that integrate phonological, visual, and spatial information. The central executive is a multifunctional construct that acts as a supervisory system controlling the flow of information from and to its slave systems, mediates the relation between short-term storage and retrieval from long-term memory, and is responsible for control and regulation of cognitive processes (Baddeley, 1986, Baddeley, 1996). Within this framework, working memory typically is assessed with dual tasks requiring one to perform two tasks simultaneously (e.g., store words while doing simple arithmetic). Dual tasks are assumed to tax the central executive; therefore, they tap mental resources not relied on in short-term memory tasks (Daneman and Carpenter, 1980, Engle et al., 1999, Turner and Engle, 1989).

Researchers have suggested that individuals with poor mathematical skills (Bull and Espy, 2006, Bull and Scerif, 2001) and children identified as math disabled have a deficit in some aspect of the central executive (Holmes and Adams, 2006, Swanson and Kim, 2007, Swanson and Sachse-Lee, 2001), and numerous studies have obtained correlations between working memory measures and performance in math (see LeFevre, DeStefano, Coleman, & Shanahan, 2005, for a review). For example, Mabbott and Bisanz (2003) found a relationship between working memory measures (backward digit span and operation span) and children’s multiplication competency. However, the use of number-based working memory measures makes it is difficult to determine the extent to which the obtained relationship between working memory and mathematical competency was due to control processes of the central executive or domain-specific knowledge. Children with math disabilities may have domain-specific deficits, showing reduced ability to process information that is specifically numerical (Bull & Johnston, 1997).

Although the central executive is possibly the most important of the four components in Baddeley’s model of working memory, it is the least well defined (Bull & Espy, 2006). Other researchers have begun to dissect component features of executive function (Miyake, Friedman, Emerson, Witzki, Howerter, & Wager, 2000). Miyake et al. (2000) found that executive functions can be divided into at least three independent but related factors: updating, shifting, and inhibition. Updating refers to the ability to monitor and code information in working memory. Shifting, in a multiple tasks scenario, refers to the ability to change between mental sets or tasks. Inhibition refers to the ability to deliberately inhibit dominant, automatic, or prepotent responses.

Researchers have found a link between poor mathematical reasoning skills and inhibition. Bull and Johnston (1997) found that children with arithmetical difficulties did not differ from higher ability children in word span, digit span, or counting span after controlling for reading comprehension. However, Bull and colleagues found that children of lower mathematical ability had difficulty in inhibiting prepotent responses (Bull, Johnston, & Roy, 1999; Bull & Scerif, 2001). Passolunghi and Siegel (2001) compared good versus poor arithmetic problem solvers on measures of working memory and inhibition. There were no group differences on short-term memory tasks; but poor problem solvers performed worse on working memory measures. Poor problem solvers also made more intrusion errors on the working memory task than did good problem solvers. Passolunghi and Siegel hypothesized that inhibition may mediate the relationship between working memory and math performance. Mathematical problem solving requires relevant information conveyed by the problem to be held in mind. To do this effectively, the person must reduce accessibility of less relevant information that could overload working memory and interfere during processing. Note that Passolunghi and Siegel used intrusion errors and Bull and Scerif (2001) used perseveration responses on the Wisconsin Card Sorting Task to measure inhibition ability. These are rather complex and indirect measures of inhibition. In the current study, we assessed inhibition with three relatively pure measures.

Updating and shifting also have been found to be good predictors of math cognition (Bull and Scerif, 2001, DeStefano and LeFevre, 2004, Furst and Hitch, 2000, Kalaman and LeFevre, 2007, Kroesbergen et al., 2003, Passolunghi and Pazzaglia, 2004, Passolunghi and Pazzaglia, 2005). For example, Passolunghi and Pazzaglia (2005) found that children with high scores on a memory updating task were better at solving word problems and recalling relevant information than were those with low updating scores. The authors concluded that updating ability is involved in math problem solving and is connected to the ability to inhibit irrelevant information.

St Clair-Thompson and Gathercole’s (2006) investigated the extent to which inhibition, updating, shifting, and working memory contribute to children’s mathematical competencies (indexed by a math scholastic attainment test). They found an inhibition factor and an updating factor but failed to identify a distinct shifting factor. Because Miyake et al. (2000) had reported a strong association between updating and working memory, the authors conducted a second analysis to test the hypothesis of a common working memory factor underlying updating and working memory. Analyses showed the updating and working memory measures to load onto a single factor. To examine relationships among mathematics, working memory (i.e., working memory and updating), and inhibition, they computed a series of partial correlations using factor scores. After controlling for working memory, inhibition correlated moderately with math score. A slightly stronger relationship was found between working memory and mathematics while controlling inhibition. Inhibition and working memory predicted unique variance in math score.

St Clair-Thompson and Gathercole’s (2006) results are consistent with previous findings that inhibition is dissociable from other executive functions in children; however, the authors failed to identify a shifting factor. This is consistent with previous research suggesting that mental flexibility (i.e., shifting) may be less differentiated from working memory in young children than in adults (Senn, Espy, & Kaufmann, 2004). Alternatively, this disparity may be due in part to the shifting measures used. Note that Lehto, Juujarvi, Kooistra, and Pulkkinen (2003) identified shifting as an executive function distinct from updating and inhibition in child samples.

St Clair-Thompson and Gathercole (2006) concluded that inhibition and working memory play important roles in children’s learning. Although the working memory score correlated with math achievement, it is unclear which cognitive processes (updating, storage capacity, or control processes occupying the central executive) are more important in math performance. In addition, the outcome measure included mental arithmetic problems (which generally involve recalling learned math facts) and word problems (which may involve a host of cognitive abilities). As the word problem solving literature suggests, problem type (e.g., one-step vs. multiple-step problems) must be considered when exploring processes underlying mathematical reasoning.

Recently, Berch (2008) pointed out limitations of Baddeley’s multicomponent model and suggested that math researchers examine other approaches to working memory such as those of Cowan (2005) and Engle (2002). The latter models do not see working memory in terms of multiple stores; rather, they see it in terms of a highly activated state of long-term memory. Both relate working memory to capacity for focusing or controlling attention. Pascual-Leone, 1970, Pascual-Leone, 1987 theory of constructive operators (TCO) describes mental attention in terms that are closer to those of Cowan and Engle than to those of Baddeley (Pascual-Leone, 2000). However, unlike Cowan and Engle, the TCO makes explicit developmental predictions about the capacity of mental attention, a maturational component of working memory (Pascual-Leone and Johnson, 2005, Pascual-Leone and Johnson, in press). We adopt the TCO framework in the current study. This cognitive developmental theory considers participants’ processing capacity, executive function abilities, and previous experiences when attempting to understand cognitive development and performance such as mathematical reasoning (Pascual-Leone and Johnson, 2005, Pascual-Leone et al., in press).

The TCO views cognitive development as being due partially to maturation of a general-purpose central processing resource or mental-attentional capacity (M-capacity). When assessed behaviorally, M-capacity is proposed to increase by one symbolic unit every other year from 3 to 15 years of age (e.g., 7- and 8-year-olds have an M-capacity of 3, 9- and 10-year-olds have an M-capacity of 4) (Pascual-Leone, 1970). M-capacity serves to boost activation of mental schemes (i.e., functionally unitized pieces of information) that are relevant for the task at hand but are not salient or automatically activated by the situation. Inhibition is hypothesized to actively interrupt (i.e., decrease the activation level of) task-irrelevant schemes (Pascual-Leone, 1987). In the TCO, the ability to inhibit is not viewed as an executive function (as characterized by others, e.g., Miyake et al., 2000). Rather, capacities to effortfully activate (M-capacity) and inhibit schemes are seen as cognitive resources that are in dynamic interaction (Pascual-Leone, 1984).

According to the TCO, executive control processes monitor allocation of M-capacity and inhibition when problem solving. The extent to which executive processes are needed to control allocation of M-capacity and inhibition is dependent on both the misleadingness of the problem-solving situation and the person’s previous experience. For instance, one-step problems may carry less demand for executive control than do multiple-step problems. Executive control processes do not contain content-specific knowledge. Instead they carry know-how, abstracted through experience within or across domains, regarding application and coordination of core cognitive abilities and strategies to reach desired goals (Pascual-Leone, Goodman, Ammon, & Subelman, 1978). In the TCO, shifting and updating refer to control executives for decentration (i.e., ability to change levels of analysis from one problem space to a different one) and recentration (i.e., ability to change the set of activated schemes without switching the level of analysis), respectively (Pascual-Leone & Johnson, 2005).

Pascual-Leone and colleagues have developed tasks to measure M-capacity across a range of content domains (e.g., Pascual-Leone and Johnson, 2005, Pascual-Leone and Johnson, in press). These mental-attentional measures (M-measures) are not traditional working memory tasks. They contain classes of items that vary parametrically in demand for M-capacity such that the highest item class passed serves as the index of the child’s M-capacity. M-measures have been found to exhibit high reliability and validity (Im-Bolter et al., 2006, Johnson et al., 2003, Pascual-Leone and Baillargeon, 1994) and to be good predictors of multiplication reasoning (Pascual-Leone et al., in press).

M-capacity constrains children’s ability to conceptualize various mathematical problems. A child should not be able to solve a problem if the mental-attentional demand (M-demand) of the task exceeds the M-capacity of the child. The M-demand of a task, estimated using metasubjective task analysis (MTA), corresponds to the minimum number of nonsalient schemes the child must hold in mind at once to solve the task (Pascual-Leone and Johnson, 1991, Pascual-Leone and Johnson, 2005, Pascual-Leone and Johnson, in press). We used MTA to estimate the M-demands of the various problem types used in the current study. Our conclusion was that our one-step word problems did not differ from the multiple-step problems in terms of M-demand but that they did differ in terms of the need for executive control. Sample task analyses are presented in Appendix A.

Pascual-Leone et al. (in press) used MTA to estimate the M-demands of various types of multiplication word problems and investigated the effect of training on the different problem types. They found that training did not improve performance when a child’s measured M-capacity was below the predicted M-demand of the type of multiplication problem. Adequate M-capacity is a necessary condition for understanding word problems, but it is not a sufficient condition. In this study we examined the combined power of attentional resources and executive functions in predicting performance on one-step and multiple-step multiplication word problems.

We constructed a test composed of word problems that tapped various multiplicative concepts at two levels of difficulty: one-step and multiple-step problems. Metasubjective task analyses suggested that one-step problems typically did not differ from multiple-step problems in M-demand, that is, processing load (see Agostino, 2008, for additional task analyses). Rather, the two levels of items differed in executive demand. Multiple-step problems required a computation to find one of the multipliers for the final solutions. Thus, multiple-step solution strategies required more coordination and sequencing of steps, suggesting a need for skill in updating.

Our study had three main goals. The first goal was to investigate the extent to which inhibition, updating, shifting, and M-capacity are unitary or separable in children. The second goal was to assess the extent to which these cognitive processes contribute to children’s multiplication reasoning abilities. This was examined by testing a theoretical model stipulating that multiplication performance would be predicted by a model in which the effect of age was partially mediated through M-capacity and inhibition, whose effect in turn was mediated through the executive function updating. We compared this model with one in which latent factors for activation and inhibition capacities and executive functions all were aligned in a single level, analogous to subcomponents of the central executive. The third goal was to assess the prediction that multiple-step word problems carry higher updating executive demand relative to one-step problems.

Section snippets

Participants

Participants were 155 children in Grade 3 (n = 34, mean age = 8;6 [years; months]), Grade 4 (n = 40, mean age = 9;6), Grade 5 (n = 44, mean age = 10;6), and Grade 6 (n = 37, mean age = 11;9) from two public schools in suburban areas just outside a large metropolitan Canadian city. The mean age of the sample was 10;1 (SD = 1;3), ranging from 8 to 13 years. Boys constituted 44% of the sample.

Multiplication test

A written multiplication test was administered individually to participants. The tester also read the problems aloud. The

Results

Prior to analyses, data distributions were examined for normality and sphericity. Data points that were more than 3 standard deviations from the mean were modified to be equal to the most extreme score within 3 standard deviations. Four data points were modified: two Trails scores, one number Stroop score, and one color Stroop score. For path analysis, bivariate, residual, and influence plots were examined. All variables appeared to have a linear relationship with the math measure. Skew and

Discussion

We studied children’s ability to solve word problems reflecting various multiplicative concepts (scalar, array, combinatorial, and proportion). For each multiplication type, we constructed one-step and multiple-step problems. As predicted, children had more difficulties when problems required coordination of multiple steps. Simple correlations indicated that measures of M-capacity, inhibition, updating, and shifting were related to developmental performance on both kinds of word problems.

Conclusions

This study demonstrates the importance of examining mathematical cognition in relation to different cognitive processes. It also shows that a theoretical developmental model can be used as a tool to explicate processes that underlie children’s mathematical reasoning abilities. The relative impact of executive functions in mathematical reasoning depends on the problem type. The ability to update information in working memory is particularly important in word problems that require coordination of

Acknowledgments

This article is based on Alba Agostino’s doctoral dissertation submitted to the Department of Psychology at York University. Portions of this research were presented at the 2007 meeting of the Society for Research in Child Development in Boston. The research was supported by a Social Sciences and Humanities Research Council of Canada operating grant (410-2006-2325) and a York University Faculty of Arts research grant. We thank M. Arsalidou, C. Lee, and M. Nuyens for assistance with data

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