Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability

Abstract

Groups of first-grade (mean age=82 months), third-grade (mean age=107 months), and fifth-grade (mean age=131 months) children with a learning disability in mathematics (MD, n=58) and their normally achieving peers (n=91) were administered tasks that assessed their knowledge of counting principles, working memory, and the strategies used to solve simple (4 + 3) and complex (16 + 8) addition problems. In all grades, the children with MD showed a working memory deficit, and in first grade, the children with MD used less sophisticated strategies and committed more errors while solving simple and complex addition problems. The group differences in strategy usage and accuracy were related, in part, to the group difference in working memory and to group and individual differences in counting knowledge. Across grade-level and group, the switch from simple to complex addition problems resulted in a shift in the mix of problem-solving strategies. Individual differences in the strategy mix and in the strategy shift were related, in part, to individual differences in working memory capacity and counting knowledge.

Introduction

Between 5 and 8% of children suffer from some form of learning disability (LD) in mathematics (MD; Badian, 1983; Gross-Tsur, Manor, & Shalev, 1996; Kosc, 1974; Ostad, 1997; Shalev et al., 2001). Yet, with the exception of some aspects of competencies in the domains of number and arithmetic, little is known about the phenotypic expression of this form of LD or the underlying brain and cognitive deficits (Geary, 1993; Geary and Hoard, 2001, Geary and Hoard, in press). Even in the comparatively well studied domain of arithmetic, most of the research has focused on how young children with MD solve simple problems (e.g., 4 + 5; Garnett & Fleischner, 1983; Geary, 1990; Jordan & Montani, 1997), and less commonly on the cognitive mechanisms that may underlie these performance characteristics (Bull & Johnston, 1997; Geary, Bow-Thomas, & Yao, 1992). The goals of this study were to fill some of these knowledge gaps. Specifically, we extended the assessment of arithmetic competencies to children with MD who are older than those typically studied and assessed performance for solving simple (e.g., 3 + 5) as well as more complex (e.g., 16 + 7) problems. We also assessed working memory capacity and knowledge of counting principles and tested predictions regarding how group differences in these domains might contribute to group differences in arithmetic performance. In the following sections, we review normal development and the characteristics of children with MD for arithmetic development, working memory, and counting knowledge, respectively. Specific predictions of this study are presented in the final section.

Normal development. The most thoroughly studied developmental and schooling-based improvement in arithmetical competency is change in the distribution of strategies children use during problem solving (Ashcraft, 1982; Carpenter & Moser, 1984; Geary, 1994; Siegler, 1996; Siegler & Shrager, 1984). When first learning to solve simple addition problems (e.g., 5 + 3), children initially count both addends. These counting procedures are sometimes executed with the aid of fingers, the finger counting strategy, and sometimes without them, the verbal counting strategy (Siegler & Shrager, 1984). The two most commonly used counting procedures, whether children use their fingers or not, are called min (or counting on) and sum (or counting all; Fuson, 1982; Groen & Parkman, 1972). The min procedure involves stating the larger-valued addend and then counting a number of times equal to the value of the smaller addend, such as counting 5, 6, 7, 8 to solve 5 + 3. Sometimes children will start with the smaller addend and count the larger addend, which is the max procedure. The sum procedure involves counting both addends starting from 1. The development of procedural competencies is related, in part, to improvements in children’s conceptual understanding of counting and is reflected in a gradual shift from frequent use of the sum procedure to the min procedure (Geary et al., 1992; Siegler, 1987).

The use of counting also results in the development of memory representations of basic facts (Siegler & Shrager, 1984). Once formed, these long-term memory representations support the use of memory-based problem-solving processes. The most common of these are direct retrieval of arithmetic facts and decomposition. With direct retrieval, children state an answer that is associated in long-term memory with the presented problem, such as stating “eight” when asked to solve 5 + 3; direct retrieval is typically used for problems for which both addends are less than 10. Decomposition involves reconstructing the answer based on the retrieval of a partial sum. For instance, the problem 6 + 7 might be solved by retrieving the answer to 6 + 6 (i.e., 12) and then adding 1 to this partial sum. The use of retrieval-based processes is moderated by a confidence criterion that represents an internal standard against which the child gauges confidence in the correctness of the retrieved answer. Children with a rigorous criterion only state answers that they are certain are correct, whereas children with a lenient criterion state any retrieved answer, correct or not (Siegler, 1988). The transition to memory-based processes results in the quick solution of individual problems and reductions in the working memory demands that appear to accompany the use of counting procedures (Delaney, Reder, Staszewski, & Ritter, 1998; Geary, Bow-Thomas, Liu, & Siegler, 1996; Lemaire & Siegler, 1995).

Children use similar strategies when solving more complex problems, such as 17 + 6 (Fuson, Stigler, & Bartsch, 1988; Geary, 1994). With initial learning, the most common of these involves use of the min procedure, as in counting 17, 18 … 23, or decomposition. As an example, decomposition would involve breaking the 6 into two 3s, and then adding these in succession, 17 + 3=20 + 3=23. With formal schooling and especially with complex problems, such as 27 + 38, children will use the commonly taught columnar strategy (i.e., summing the ones-column integers and then summing the tens-column integers).

Children with MD. During the solving of simple arithmetic problems (e.g., 4 + 3), children with MD use the same types of strategies (e.g., verbal counting) as their normally achieving peers, but differ in the strategy mix, strategy accuracy, and in the pattern of developmental change (Geary, 1990; Hanich, Jordan, Kaplan, & Dick, 2001; Jordan & Hanich, 2000). These differences have been found in the United States (Geary & Brown, 1991; Jordan & Montani, 1997; Jordan, Hanich, & Kaplan, 2003a), Europe (Barrouillet, Fayol, & Lathulière, 1997; Ostad, 1997, Ostad, 1999, Ostad, 2000; Svenson & Broquist, 1975), and Israel (Gross-Tsur et al., 1996). As an example, Geary and colleagues (Geary, Hamson, & Hoard, 2000; Geary, Hoard, & Hamson, 1999) found that in first and second grade, children with MD committed more counting errors and used the developmentally immature sum procedure more frequently than did their normally achieving peers or children with reading disability (RD). In keeping with models of normal arithmetical development, in first and second grade, the children with RD and the normally achieving children shifted from heavy reliance on finger counting to verbal counting and retrieval, and committed fewer errors. The children with MD, in contrast, did not show this shift, but instead relied heavily on finger counting in both grades, and continued to commit finger and verbal counting errors. The most consistent finding in this literature is that children with MD have difficulty retrieving basic arithmetic facts from long-term memory (Barrouillet et al., 1997; Garnett & Fleischner, 1983; Geary, 1990, Geary, 1993; Jordan & Montani, 1997; Ostad, 1997, Ostad, 2000). Unlike the use of counting strategies, the ability to retrieve basic facts apparently does not substantively improve across the elementary-school years for most children with MD, suggesting the retrieval difficulties result from a persistent cognitive deficit, as contrasted with delayed development (Geary, 1993).

Russell and Ginsburg (1984) compared fourth-grade children with MD to normally achieving children in third and fourth grade on a variety of number and arithmetic tasks. Included among these were tasks that assessed the ability to solve simple (e.g., 3 + 9) arithmetic problems mentally and to solve more complex (e.g., 17 + 34) arithmetic problems mentally and with paper and pencil. For the mental solving of simple problems, the children with MD correctly retrieved fewer answers than did the normally achieving children in third or fourth grade, in keeping with the research already described. For the solving of more complex problems, children in all of the groups tended to count, or use decomposition or regrouping (e.g., adding the units column values and then the tens column values). For the complex problems, the children with MD did not differ from the normally achieving third graders, but they did not perform as well as the normally achieving fourth graders. The children with MD used the same types of counting and decomposition strategies as their normally achieving fourth-grade peers, but committed more execution and working memory errors, in keeping with the results described in the next section.

Working memory is the ability to maintain explicitly a mental representation of some amount of information, while being engaged simultaneously in other mental processes. According to Baddeley, 1986, Baddeley, 2000, working memory is dependent on a central executive that is expressed as attention-driven control of information represented in three slave systems, a language-related phonetic system, a visuospatial sketch pad, and an episodic buffer. Debates regarding the nature of these components of working memory are discussed elsewhere (Miyake & Shah, 1999). The issues here concern developmental change in the overall capacity of working memory in normally achieving children, and the working memory of same-age children with MD.

Normal development. The capacity of working memory increases from preschool through the elementary school years. As an example, preschool children can hold three to four items of some forms of information, such as numbers, in working memory, whereas a typical fourth grader can hold five to six items (Kail, 1990). The mechanisms underlying these developmental changes appear to include an improved ability to use strategies, such as rehearsal, to keep the information active in working memory (e.g., Kreutzer, Leonard, & Flavell, 1975), and changes in more fundamental components that support age-related improvements in working memory capacity. The latter include one or some combination of an improved ability to control the focus of attention, increased speed of processing information represented in the slave systems, or slower decay of information represented in the slave systems (e.g., Cowan, Saults, & Elliott, 2002; Kail, 1991). In a recent review, Cowan et al. determined that all of these fundamental components improve as part of normal development in childhood, and thus each contributes to the observed increase in working memory capacity.

Children with MD. Children with MD do not perform as well as their same-age peers on a variety of working memory tasks (Geary, Brown, & Samaranayake, 1991; Geary et al., 1999; Hitch & McAuley, 1991; McLean & Hitch, 1999; Siegel & Ryan, 1989; Swanson, 1993). One often-used task, counting span, is highly relevant in terms of the working memory processes involved in the use of counting procedures (Hitch & McAuley, 1991; Siegel & Ryan, 1989); thus, we used this task in this study. Here, children must maintain one or a series of integers in working memory while engaged in the act of counting; counting span is the number of integers that can be accurately held in working memory. However, the mechanisms underlying group differences in counting span and other working memory tasks are unclear. Some studies suggest that the differences reside in fundamental differences in speed of processing, as in speed of articulating number words (Bull & Johnston, 1997; Hitch & McAuley, 1991), whereas other studies suggest that the differences reside in poor information representation in the slave systems (Geary, 1993), or in executive/attentional control (Bull, Johnston, & Roy, 1999; McLean & Hitch, 1999).

Whatever the causes, the relation between the working memory deficit of children with MD and the group differences in the developmental maturity and mix of the strategies used to solve arithmetic problems has not been explored systematically. Geary, 1990, Geary, 1993 hypothesized that children with MD rely heavily on finger counting and commit more counting errors as a result of poor working memory resources. If so, then measures of working memory, such as counting span, should be related to individual differences in use of finger counting and frequency of finger and verbal counting errors, and should contribute to differences in these strategy variables comparing children with MD to their normally achieving peers. The importance of working memory may also increase with increases in task novelty and complexity, and thus may be more important during the solving of complex compared to simple arithmetic problems, and during the initial stages of learning (Ackerman, 1988).

Normal development. Children’s early counting knowledge and counting behavior can be represented by Gelman and Gallistel’s (1978) five implicit and perhaps inherent principles. These principles include one–one correspondence (one and only one word tag, such as “one,” “two,” is assigned to each counted object); stable order (the order of the word tags must be invariant across counted sets); cardinality (the value of the final word tag represents the quantity of items in the set); abstraction (objects of any kind can be collected together and counted); and, order-irrelevance (items within a given set can be tagged in any sequence). The principles of one–one correspondence, stable order, and cardinality define the “how to count” rules, which, in turn, appear to constrain the nature of preschool children’s counting behavior and to provide the skeletal structure for children’s emerging knowledge of counting (Gelman & Meck, 1983).

In addition, children make inductions about the basic characteristics of counting by observing standard counting behavior and the associated outcomes (Briars & Siegler, 1984; Fuson, 1988). These inductions may elaborate Gelman and Gallistel’s counting rules (1978) and result in a belief that certain unessential features of counting are essential (Briars & Siegler, 1984). In particular, young children often induce that the unessential features of adjacency (items must be counted contiguously) and start at an end (counting must start from the leftmost item) are in fact essential. The latter beliefs indicate that young children’s conceptual understanding of counting is rigid and influenced by the observation of standard counting procedures.

Children with MD. Geary et al. (1992) contrasted the performance of first-grade children with MD and their normally achieving peers for tasks that assessed all of Gelman and Gallistel’s (1978) basic principles and most of Briars and Siegler’s (1984) unessential features of counting. The procedure involves asking children to help a puppet learn how to count. The child watches the puppet count a series of objects. The puppet sometimes counts correctly and sometimes violates one of Gelman and Gallistel’s counting principles or Briars and Siegler’s unessential features of counting. The child’s task is to determine if the puppet’s count was “OK” or “not OK and wrong.” In this way, the puppet performs the procedural aspect of counting (i.e., pointing at and tagging items with a number word), leaving the child’s responses to be based on her conceptual understanding of counting.

The results revealed that children with MD differed from normally achieving children on two types of counting trials, pseudoerror and error. Pseudoerror trials involved counting, for instance, the first, third, fifth, and seventh items and then returning to the left-hand side of the array and counting the second, fourth, and sixth items. Technically the count is correct, but violates the adjacency rule and assesses the child’s understanding of the order-irrelevance principle. Error trials involved double counting either the first or the last item. Children with MD correctly identified these counts as errors when the last item was double counted, suggesting that they understood the one–one correspondence principle. Double counts were often labeled as correct when the first item was counted, suggesting that many children with MD have difficulties holding information in working memory—in this case noting that the first item was double counted—while monitoring the act of counting. A follow-up study that controlled for group differences in intelligence (IQ) confirmed these findings (Geary et al., 1999; Geary et al., 2000). Children with MD, regardless of their reading achievement levels, performed poorly on pseudoerror trials in first and second grade and on error trials (double counting the first item in a series) in first grade. The pattern suggests that even in second grade, many children with MD do not fully understand counting concepts, and in first grade, many children with MD may have difficulty holding information in working memory while monitoring the counting process (Hoard, Geary, & Hamson, 1999).

Geary et al. (1992) found that performance on pseudoerror trials was correlated (r=.47) with use of the min procedure when finger counting or verbal counting was used to solve simple addition problems, and explained the difference in use of min counting comparing children with MD to their normally achieving peers. Ohlsson and Rees (1991) predicted that children’s counting knowledge and skill at detecting counting errors would enable them to correct these miscounts and thus eventually result in fewer counting errors. In support of this prediction, Geary et al. found that a combination of pseudo and error-trial scores from the counting knowledge task was significantly related to the frequency of finger and verbal counting errors while solving simple addition problems (r=−.44), and explained the group difference in the frequency of these errors.

This study was the first to simultaneously assess working memory capacity (i.e., counting span) and counting knowledge as they contribute to individual and group differences in the pattern of strategy usage and strategy accuracy during arithmetical problem solving. This study also adds to the literature by extending the assessment across the elementary school years and with the inclusion of both simple and more complex addition problems. First, performance on the counting knowledge task was predicted to correlate with individual and group differences in counting errors and use of the min procedure, based on the findings just described (Geary et al., 1992, Geary et al., 2000; Ohlsson & Rees, 1991). Second, we predicted that working memory/counting span would correlate with the use of finger counting as a problem-solving strategy, would correlate with the frequency of counting errors, and would contribute to group differences on both of these dimensions. These predictions follow from earlier suggestions that the use of fingers during counting is a working-memory aid (Geary, 1990), and that the poor working memory skills of children with MD (Hitch & McAuley, 1991) may contribute to their tendency to rely heavily on finger counting and to commit finger and verbal counting errors frequently (Geary et al., 2000; Jordan & Montani, 1997). The predictions do not preclude working memory contributions to the use of other procedures, such as verbal counting, but rather were specifically derived as an attempt to understand the mechanisms that contribute to group differences in the use of finger counting and counting errors.

In contrast, we predicted working memory capacity would be a less important contributor to individual or group differences in the execution of more automatized memory-based processes, particularly direct retrieval. This prediction was based on Ackerman, 1988, Ackerman and Cianciolo, 2000 findings that working memory is most important during the initial phases of skill acquisition and becomes less important with learning, and Siegler, 1996, Siegler and Shrager, 1984 model of strategy choice and patterns of normal arithmetical development. As noted earlier, use of counting during problem solving appears to result in the formation of associations between the problem and the answer generated by means of counting. Eventually, the associations result in the automatic retrieval of the answer, and at this point, working memory is predicted to be less important in terms of understanding group and individual differences. Working memory capacity may still be correlated with retrieval frequency, because children with poor working memory resources may execute counting procedures more slowly and less accurately than other children and thus not easily form the associations needed to support direct retrieval (Geary et al., 1996). In short, poor working memory may result in slow acquisition of basic facts, but should not be as important for the dynamics of retrieving those facts that do become committed to long-term memory.