Mathematics learning disabilities (MLD)

The breadth and complexity of the field of mathematics make the identification and study of the cognitive phenotypes that define mathematics learning disabilities (MLD) a formidable endeavor. In theory, a learning disability can result from deficits in the ability to represent or process information in one or all of the many mathematical domains (e.g., geometry) or in one or a set of individual competencies within each domain. The goal is further complicated by the task of distinguishing poor achievement due to inadequate instruction from poor achievement due to an actual cognitive disability (Geary, Brown, & Samaranayake, 1991). Yet another complication arises from contention regarding instructional goals and approaches (Loveless, 2001), which in turn may influence whether a particular deficit would be considered a learning disability at all. Instruction that focuses on mathematics as an applied domain tends to de-emphasize the learning of procedures and mathematical facts and to emphasize conceptual understanding (National Council of Teachers of Mathematics, 2000), whereas procedures and facts are more heavily emphasized in instruction that approaches mathematics as a scientific field to be mastered (California Department of Education, 1999).

With the former approach, the deficit in arithmetic fact retrieval described later in this article may not be considered a serious learning disability because of the de-emphasis on this memory-based knowledge, whereas in the latter approach it would be considered a serious disability. One strategy that is not dependent

on instructional issues involves applying the theories and methods used by cognitive psychologists to study mathematical competencies in typically achieving children to the study of children with MLD (Bull & Johnston, 1997; Garnett & Fleischner, 1983; Geary & Brown, 1991; Jordan, Levine, & Huttenlocher, 1995; Jordan & Montani, 1997; Ostad, 1997, 1998b; Russell & Ginsburg, 1984; Svenson & Broquist, 1975). When this approach is combined with studies of dyscalculia—that is, numerical and arithmetical deficits following overt brain injury (e.g., Shalev, Manor, & Gross-Tsur, 1993; Temple, 1991)—and brain imaging studies of mathematical processing (e.g., Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999), a picture of the cognitive and brain systems that can contribute to MLD begins to emerge. The combination of approaches has been primarily applied to the study of numerical and arithmetical competencies and is, thus, only a first step to fully understanding the cognitive and brain systems that support mathematical competency

and any associated learning disabilities. It is, nonetheless, a start, and the following sections provide an overview of what this research strategy has revealed about MLD. The first section provides a discussion of diagnostic and etiological issues, and the second provides a description of some of the performance and cognitive patterns that distinguish children with MLD from their peers. The final section presents a framework for guiding future research on mathematics and learning disabilities (LD) and reviews the basic cognitive and neural mechanisms and deficits that may underlie the performance and cognitive patterns described in the second section.

Unfortunately, measures that are specifically designed to diagnose MLD are not available; thus, most researchers rely on standardized achievement tests, often in combination with measures of intelligence (IQ). Ascore lower than the 20th or 25th percentile on a mathematics achievement test combined with a low average or higher IQ score are typical criteria for diagnosing MLD (e.g., Geary, Hamson, & Hoard, 2000; Gross-Tsur, Manor, & Shalev, 1996). However, a lower than expected (based on IQ) mathematics achievement score does not in and of itself indicate the presence of MLD. Many children who score low on achievement tests one academic year score average or better in subsequent years. These children do not appear to have any of the underlying memory or cognitive deficits described in the next section, and thus a diagnosis of MLD is not appropriate (Geary, 1990; Geary et al., 1991; Geary et al., 2000). In contrast, children who have lower than expected achievement scores across successive academic years often have some form of memory or cognitive deficit, and a diagnosis of MLD is often warranted. It should be noted that the cutoff of the 25th percentile on a mathematics achievement test does not fit with the estimation, described later, that between 5% and 8% of children have some form of MLD. This discrepancy results from the nature of standardized achievement tests and the often rather specific memory or cognitive deficits of children with MLD. Standardized achievement tests sample a broad range of arithmetical and mathematical topics, whereas children with MLD often have severe deficits in some of these areas and average or better competencies in others. The result of averaging across items that assess these different competencies is a level of performance (e.g., at the 20th percentile) that overestimates the competencies of children with MLD in some areas and underestimates them in others.

In addition to the development of diagnostic instruments, another issue that needs to be explored is whether treatment resistance can be used as one diagnostic criterion for MLD. As described later, many children with MLD have difficulties retrieving basic arithmetic facts from long-term memory, and these difficulties often persist despite intensive instruction on basic facts (e.g., Howell, Sidorenko, & Jurica, 1987). Although the instructional research is preliminary, it does suggest that a retrieval deficit resistant to instructional intervention might be a useful diagnostic indicator of arithmetical forms of MLD.

Experimental measures that are more sensitive to MLD than are standardized achievement tests have been administered to samples of more than 300 children from well defined populations (e.g., all fourth graders in an urban school district) in the United States (Badian, 1983), Europe (Kosc, 1974; Ostad, 1998a), and Israel (Gross- Tsur et al., 1996; Shalev et al., 2001). These measures have largely assessed number and arithmetic competencies and have been constructed based on neuropsychological deficits associated with dyscalculia (for discussion, see Geary & Hoard, 2002; Shalev et al., 1993). Performance that deviates from age-related norms and is similar to that associated with dyscalculia has been used in these studies as an indication of MLD and suggests that 5% to 8% of school-age children exhibit some form of MLD. Many of these children have comorbid disorders, including reading disabilities (RD) and attention-deficit/ hyperactivity disorder (ADHD; Gross- Tsur et al., 1996). As with other forms of LD, twin and familial studies, although preliminary, suggest both genetic and environmental contributions to MLD (Light & De- Fries, 1995; Shalev et al., 2001). For instance, Shalev et al. studied familial patterns of MLD, specifically, learning disabilities in number and arithmetic. The results showed that family members (e.g., parents and siblings) of children with MLD are 10 times more likely to be diagnosed with MLD than are members of the general population.

As noted earlier, the use of cognitive theory and its associated methodology to study children with MLD has

yielded a number of insights regarding the potential sources of their learning disability. These studies have primarily focused on the number, counting, and arithmetic competencies of children with MLD (e.g., Ackerman & Dykman, 1995; Barrouillet, Fayol, & Lathulière, 1997; Bull, Johnston, & Roy, 1999; Geary, 1993; Geary, Widaman, Little, & Cormier, 1987; Hanich, Jordan, Kaplan, & Dick, 2001; Ostad, 2000; Räsänen & Ahonen, 1995; Rourke, 1993). The results have suggested that the basic numerical competencies (e.g., identifying arabic numerals, comparing the magnitudes of numbers) of most children with MLD, though often delayed, are largely intact, at least for the processing of simple numbers (e.g., 8, 12; Badian, 1983; Geary, 1993; Geary, Hoard, & Hamson, 1999; Gross-Tsur et al., 1996). Based on these findings, the numerical competencies of children with MLD are not discussed further (for discussion, see Geary & Hoard, 2002).

Unless otherwise noted, MLD refers to children with low achievement scores— relative to IQ in many of the studies— in mathematics. When studies have only focused on children with low mathematics achievement scores but average or better reading achievement scores, participants will be referred to as children with MLD only. If the study assessed children with low achievement in both mathematics and reading, participants are identified as children with MLD/RD.