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From: Geary, D. C., & Hoard, M. K. (2002). Learning disabilities in basic mathematics: Deficits in memory and cognition. In J. M. Royer (Ed.), Mathematical cognition (pp. 93-115). Greenwich, CT: Information Age Publishing.
LEARNING DISABILITIES IN BASIC MATHEMATICS
Deficits in Memory and Cognition
David C. Geary and Mary K. Hoard
ABSTRACT Recent findings regarding the understanding of number concepts and counting principles, and the simple arithmetic competencies of normal and mathematically disabled (MD) children are reviewed. Differences across these groups are evident in the understanding of counting concepts and for several features of arithmetic skill development. In comparison to normal children, many children with MD show a developmental delay in the un-derstanding of counting concepts, and use immature counting-based proce- dures and commit more procedural errors during the solving of arithmetic problems. Unlike normal children, children with MD do not transition as easily from use of counting-based procedures to memory-based processes, such as arithmetic fact retrieval. Use of immature counting procedures to solve arithmetic problems may be due to their poor understanding of count- ing concepts or poor skill at detecting and correcting counting-based errors. Deficit in arithmetic fact retrieval may be due to difficulties in lexical access
Mathematical Cognition, pages 93-115 Copyright 2002 by Information Age Publishing All rights of reproduction in any form reserved.
D.C. GEARY and M.K. HOARD
(and storage) and/or inability to inhibit irrelevant associations from enter- ing working memory during problem solving.
During the past 30 years, considerable progress has been made in identify- ing the genetic, neural, and cognitive mechanisms that contribute to read- ing disability (RD), and in the ability to diagnose and remediate this form of learning disorder (LD; Lyon, Alexander, Yaffe, 1997; Morris et al., 1998; Smith, Kelley, & Brower, 1998; Torgesen, Wagner, Rashotte, Alexander, & Conway, 1997). Research on learning disabilities in mathematics (MD) has progressed more slowly than has research on RD, in part because of the complexity of the field of mathematics. In theory, MD can result from defi- cits in the ability to represent or process information used in one or all of the many areas of mathematics (e.g., arithmetic, geometry), or in one or a set of individual domains (e.g., theorems vs. graphing) within each of these areas (Geary, 1993; Russell & Ginsburg, 1984). To make the study of MD tractable, scientists have focused primarily on mathematical domains for which competency development in academically-normal children is well understood (Geary, Hamson, & Hoard, 2000; Jordan & Montani, 1997; Ostad, 1998a, 2000). These domains include number, counting, and arith- metic (e.g., Fuson, 1988; Geary, 1994; Gelman & Gallistel, 1978). The sec- ond section below provides an overview of normal development in these domains and the performance characteristics of children with MD. The section closes with an overview of research on the memory and cognitive deficits that contribute to these performance characteristics (McLean & Hitch, 1999). The first section provides the requisite background informa- tion on the etiology and prevalence of MD.
BACKGROUND CHARACTERISTICS OF CHILDREN WITH MD
Diagnosis Measures that are specifically designed to diagnose MD are not avail- able. As a result, most researchers rely on standardized achievement tests, often in combination with measures of intelligence (IQ). A score lower than the 20th or 25th percentile on a mathematics achievement test com- bined with a low average or higher IQ score are typical criteria for diagnos- ing MD (e.g., Geary et al., 2000; Gross-Tsur, Manor, & Shalev, 1996). However, lower than expected performance (based on IQ) on a mathemat- ics achievement test does not, in and of itself, indicate the presence of MD.
Many children who score poorly on achievement tests one academic year score average or better during successive academic years. These children do not appear to have any of the underlying memory and (or) cognitive deficits described in the next section, and thus they should not be diag- nosed as having MD (Geary, 1990; Geary, Brown, & Samaranayake, 1991; Geary et al., 2000). Children who have lower than expected achievement scores across successive academic years, in contrast, often have some form of memory and (or) cognitive deficit and thus a diagnosis of MD is often warranted. It should also be noted that the cutoff of the 25th percentile on a math- ematics achievement test does not fit with the estimation, described below, that between 5% and 8% of children have some form of MD. The discrep- ancy results from the nature of standardized achievement tests and the often rather specific memory and (or) cognitive deficits of children with MD. Standardized achievement tests sample a broad range of arithmetical and mathematical topics, whereas children with MD often have severe de-ficits in some of these areas and average or better competencies in others. The result of averaging across these topics is a level of performance (e.g., at the 20th percentile) that overestimates the competencies of children with MD in some areas and underestimates them in others. Prevalence and Etiology Measures that are more sensitive to MD than are standard 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 dis- trict) in the United States (Badian, 1983), Europe (Kosc, 1974; Ostad, 1998b), and Israel (Gross-Tsur et al., 1996; Shalev, Manor, Kerem, Ayali, Badichi, Friedlander, & Gross-Tsur, 2001). These measures have largely assessed competencies in number and arithmetic and have been informed by neuropsychological studies of mathematical deficits following brain injury, that is, dyscalculia (for a discussion, see Geary & Hoard, 2001; Sha- lev, Manor, & Gross-Tsur, 1993). Performance that deviates from age- related norms and is similar to that associated with dyscalculia is often con- sidered an indication of MD. The use of these methods suggests that 5% to 8% of school-age children exhibit some form of MD. As with other forms of LD, twin and familial studies suggest both genetic and environmental contributions to MD (Light & DeFries, 1995; Shalev et al., 2001). For instance, Shalev and her colleagues studied familial patterns of MD, specifically learning disabilities in number and arithmetic. The results showed that family members (e.g., parents and siblings) of children
D.C. GEARY and M.K. HOARD
with MD are 10 times more likely to be diagnosed with MD than are mem- bers of the general population.
MODELS OF MATHEMATICAL DEVELOPMENT AND DISABILITIES
One framework for systematically approaching the study of MD involves applying the models and methods used to study mathematical develop- ment in academically-normal child