The cognitive profile of Chinese children with mathematics difficulties

Abstract

This study examined how four domain-specific skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed) may account for Chinese children’s mathematics learning difficulties. Children with mathematics difficulties (MD) of two age groups (7–8 and 9–11 years) were compared with age-matched typically achieving children. For both age groups, children with MD performed significantly worse than their age-matched controls on all of the domain-specific and domain-general measures. Further analyses revealed that the MD children with literacy difficulties (MD/RD group) performed the worst on all of the measures, whereas the MD-only group was significantly outperformed by the controls on the four domain-specific measures and verbal working memory. Stepwise discriminant analyses showed that both number fact retrieval and place value concept were significant factors differentiating the MD and non-MD children. To conclude, deficits in domain-specific skills, especially those of number fact retrieval and place value understanding, characterize the profile of Chinese children with MD.

Introduction

Children with mathematics disabilities appear to constitute a heterogeneous group, with different children showing different profiles of knowledge, learning strengths, and learning deficits (Geary, 2002). This may be the reason why different terminologies have been used. In this study, we set out to explore the cognitive profile of Chinese children with marked difficulties in mathematics. We adopted relatively lenient criteria to obtain a reasonably large sample in an exploratory study with little known findings about MD in Chinese. We use the term mathematics difficulties (MD), instead of mathematics disorder or mathematics disabilities, to include children having a broader range of difficulties.

Previous research studies have suggested that mathematics disabilities or difficulties tend to be associated with deficits in four domain-specific numerical skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed).

A deficit in arithmetic procedural skills typically means difficulties in executing arithmetic procedures (e.g., carrying or trading in complex addition problems) or in executing counting procedures to solve simple addition problems (Geary, 1996). For multistep arithmetic problems, the procedural errors may include misalignment of numbers while writing down partial answers or errors in carrying or borrowing from one column to the next (Russell & Ginsburg, 1984). Children with mathematics disabilities or difficulties, in comparison with typically achieving children, tend to use less sophisticated strategies and commit more errors in solving addition problems (Geary et al., 2000, Geary et al., 2004, Ostad, 1997). However, many of these children improve by the middle of the elementary school years (Geary, 2000, Torbeyns et al., 2004). Thus, their error-prone use of immature procedures seems to represent a developmental delay instead of a long-term cognitive deficit. Such delay seems to be domain specific and does not characterize children with other academic-related learning deficits. For example, children with reading disabilities (RD) only do not seem to differ from typically achieving children in their strategies for solving simple addition problems in the accuracy of strategy use (Geary & Hoard, 2002) or in multistep calculation (Reikeras, 2006).

A deficit in number fact retrieval means difficulties in accessing arithmetic facts from long-term memory (Geary, 2004). According to Geary’s conceptual framework, the difficulties of storing arithmetic facts are related to the limitations of MD children in retaining information in working memory while performing other operations. For example, in doing a simple addition task of 5 + 2, children with MD may favor the “counting all” strategy that refers to counting from 1, 2, 3, … instead of counting up or on from the higher number: 5, 6, 7. The counting procedure may influence formation of long-term memory representations of basic facts if activation of numerical representations in working memory decays more quickly for children with MD. In this situation, the representation of the problem addends may decay before the count is completed; thus, an association between the problem stem (e.g., 5 + 2) and the answer generated by means of counting might not be formed in long-term memory. A second form of retrieval deficit may need to do with difficulties in inhibiting the retrieval of irrelevant associations. For example, children with learning disorders more often incorrectly answer the question 6 + 2 = ? with 7 or 3, each of which is a number following an addend in the counting string, than their typically achieving peers (Geary et al., 2000). Indeed, deficits of number fact mastery among MD children seem to be rather persistent and are quite independent of reading and language abilities (Jordan, Hanich, & Kaplan, 2003a). Number fact retrieval deficits increasingly emerge as a central characteristic of mathematics disabilities, at least with respect to addition and subtraction operations (Geary, 2004, Gersten et al., 2005, Jordan et al., 2003b, Ostad, 1998, Robinson et al., 2002).

Although arithmetic procedural deficit in children with MD may prove to be a developmental delay, number fact retrieval deficit seems to persist. In the current study, we compared younger and older children with MD to examine the persistence of these and other cognitive deficits.

A key insight about numbers is that the value of a digit depends on its place in a group of digits (Chinn & Ashcroft, 1999). For example, although most second-graders have some trouble with positional knowledge and digit correspondence, typically achieving children outperform children with reading and/or mathematics difficulties on this (Hanich, Jordan, Kaplan, & Dick, 2001). A reliable connection between place value understanding and addition and subtraction skills among Chinese children has also been documented. Training in place value concept also effectively improves Chinese children’s place value understanding as well as addition skills (Ho & Cheng, 1997).

Number sense has been referred to as “a child’s fluidity and flexibility with numbers, the sense of what numbers mean, and an ability to perform mental mathematics and to look at the world and make comparisons” (Gersten & Chard, 1999, p. 19). It has also been considered as important to mathematics learning as phonemic awareness is to reading. According to Berch (2005), it has generally been agreed that number sense involves abilities related to counting, number patterns, magnitude comparisons, estimation, and number transformation. Often acquired informally, it is basic and important for early formal arithmetic learning. It was found that kindergarteners’ number sense performance and growth together accounted for 66% of the variance in first-grade mathematics achievement in a study by Jordan, Kaplan, Locuniak, and Ramineni (2007). Similarly, results of longitudinal studies suggest that screening early weakness in number sense could help to identify children who might develop mathematics disabilities or difficulties later (Gersten et al., 2005, Jordan et al., 2006, Jordan et al., 2007). For instance, Locuniak and Jordan (2008) found that number sense screening in kindergarten successfully ruled out 84% of children who were not at risk for fluency difficulties and positively identified 52% of those who later showed fluency difficulties. Because number sense is related to the meaning of numbers stored in long-term memory, Robinson et al. (2002) speculated that weak number sense might contribute to number fact retrieval deficits in children with MD. Baroody (2008) later suggested that the best way to help students master efficient number fact retrieval was to develop their general number sense by facilitating them to discover patterns and relations among numbers. The proposed close connection between number sense and mathematics achievement in young children is supported by the recent findings by Jordan, Kaplan, Ramineni, and Locuniak (2009), who reported that kindergarten number competence (which is similar to the concept of number sense) predicted the rate of growth in mathematics achievement between first and third grades and the achievement level through third grade. Poor number sense may, therefore, underlie children’s weaknesses in fact retrieval that contribute to MD.

The mental capacity for temporary processing and storage of information (i.e., working memory) has been a focus in research on mathematics performance (Rosselli, Matute, Pinto, & Ardila, 2006). Although many would agree that children with MD seem to have some working memory deficit (e.g., Bull et al., 1999, Geary et al., 2004, McLean and Hitch, 1999, Rosselli et al., 2006, Wilson and Swanson, 2001), the contribution of specific working memory components to MD remains unclear.

DeStefano and LeFevre (2004) suggested that all three components proposed by Baddeley and Logie (1999)—the phonological loop, the visuospatial sketchpad, and the central executive of the working memory system—could play a role in mental arithmetic. Yet some studies have suggested that the relation between working memory and mathematics performance varies as a function of age and ability (Andersson & Lyxell, 2007), with a stronger role for the visuospatial sketchpad in younger children’s mathematics performance (Holmes & Adams, 2006). Other studies have found that mainly the central executive was impaired in children with MD (Andersson and Lyxell, 2007, Bull et al., 1999, McLean and Hitch, 1999, Passolunghi and Siegel, 2004, Wilson and Swanson, 2001). The central executive seems to be a core deficit for children with MD, whereas the phonological loop and the visuospatial sketchpad may contribute to more specific math-related cognitive deficits (Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007). Still other studies have concluded that children with MD have problems with working memory tasks involving numerical information and not those involving non-numerical verbal information (Geary et al., 1991, McLean and Hitch, 1999, Passolunghi and Siegel, 2001, Siegel and Ryan, 1989). In other words, no conclusive findings regarding the relations between MD and various working memory components have been reported so far. In the current study, Chinese children with MD were compared with typically achieving children on both verbal and visual memory.

Children with MD are generally slow in solving arithmetic problems (e.g., Ostad, 2000). For instance, Jordan and Montani (1997) found that children with MD performed worse than the control group in both story and number fact problems in timed conditions but not in untimed conditions. Perhaps children with MD simply take more time than their typically achieving peers to execute all basic numerical processes, or perhaps they favor slower counting strategies (e.g., counting all) rather than the faster number fact retrieval strategies (Geary, 1996). With little increases in fact retrieval speed with age, many MD children lag further and further behind their peers. Because arithmetic performance seems to be strongly predicted by processing speed (e.g., Bull and Johnston, 1997, Fuchs et al., 2006), arithmetic difficulties persist, perhaps due to persistent processing speed deficit, hampering automatic basic arithmetic fact retrieval (e.g., Geary, Hoard, Byrd-Craven, et al., 2007). Slow processing speed as a general obstacle for efficient numerical operation and as slowing the fact retrieval process both are plausible but not mutually exclusive. The close connection between processing speed and storage capacity of the working memory may help to explain its contribution to arithmetic difficulties (Case, Kurland, & Goldberg, 1982). The current study examined processing speed as part of the cognitive profile of Chinese children with or without MD and tested whether processing speed contributed to fact retrieval efficiency.

The superiority of Chinese students’ mathematics performance in comparison with their Western counterparts in cross-cultural studies has been well documented (Aunio et al., 2004, Geary et al., 1996, Ho and Fuson, 1998, Miller et al., 2005, Miura and Okamoto, 2003, Yang and Cobb, 1995). Contributing factors identified for the cross-cultural discrepancy in mathematics performance include numerical language characteristics such as short pronunciation duration of Chinese numbers that would increase efficiency of mathematics operations and the regularity of a Chinese number-naming system that enhances cognitive representation of numbers and understanding of place value concepts (e.g., Ho and Cheng, 1997, Miura and Okamoto, 2003). In addition, the cross-cultural differences may be related to school curriculum, teaching practices, Chinese valuation of mathematics, parental expectations of children’s mathematics performance, and the amount of time parents spend on home teaching (Miller et al., 2005, Stevenson and Stigler, 1992). There are very few research studies on Chinese children’s mathematics abilities and learning, and those on MD of Chinese children are even more rare. The findings of the current study would provide insights into the manifestations of MD among Chinese children.

Understanding the cognitive profile of MD children is an essential first step for developing appropriate identification tools and intervention methods. Given the different numerical language characteristics of Chinese, Chinese children with MD may exhibit different numerical and cognitive difficulties that could not be informed by research studies with American or European students. The aim of the current study was to examine the cognitive profile of Chinese children with MD. Specifically, we examined whether and how children with MD would be different from their typically achieving peers in four domain-specific numerical skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed). We focused on two age groups in primary school: 7- and 8-year-olds (because mathematics difficulties typically become quite noticeable by this age) and 9- to 11-year-olds (because mathematics difficulties are not outgrown by this age and are probably rather persistent). The main hypotheses were as follows.

Hypothesis 1 stated that both younger and older MD groups were expected to perform significantly worse than the typically achieving controls on all of the domain-specific and domain-general measures. Because domain-specific numerical skills may contribute more directly to children’s arithmetic performance, it is expected that performance differences between the MD and control groups on the domain-specific measures would be greater than those on the domain-general measures.

Previous research findings have suggested that arithmetic procedural deficit in children with MD may prove to be a developmental delay, whereas number fact retrieval deficit seems to persist. Based on such previous findings and our preliminary findings (Chan, 2008), Hypothesis 2 stated that all four basic number skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) were expected to be useful for differentiating the younger MD and non-MD groups, whereas number fact retrieval and place value concept were expected to be especially useful for differentiating the older MD and non-MD groups because children with MD are likely to have outgrown their earlier deficits in arithmetic procedural skills and basic number sense by fourth grade.

Hypothesis 3 stated that because children with both mathematics and literacy difficulties (MD/RD) might suffer from a more severe general impairment than those with MD only, the MD/RD group was expected to perform worse than both the MD-only and control groups on all of the measures.

For testing Hypotheses 1 and 2, four groups of children (younger MD, older MD, and their corresponding control groups) were formed for comparison and analyses. Another three groups of children (MD-only, MD/RD, and control groups) were formed to test Hypothesis 3.