Children with a mathematical learning disability (MLD, n = 19) and low achieving (LA, n = 43) children were identified using mathematics achievement scores below the 11th percentile and between the 11th and 25th percentiles, respectively. A control group of typically achieving (TA, n = 50) children was also identified. Number line and speed of processing tasks were administered in 1st and 2nd grade and a working memory battery in 1st grade. In both grades, the MLD children were less accurate in their number line placements and more reliant on a natural numbermagnitude representational system to make these placements than were TA children. The TA children were more reliant on the school-taught linear system in both grades. The performance of the LA children was similar to that of the MLD children in first grade and to the TA children in second. The central executive component of working memory contributed to across-grade improvements in number line performance and to group differences in this performance.Several large-scale population-based, prospective studies and a number of smaller-scale studies have consistently found that between 5% and 10% of children and adolescents will experience a substantive learning deficit-not attributable to low cognitive ability-in at least one area of mathematics before graduating from high school (Badian, 1983;Barbaresi, Katusic, Colligan, Weaver, & Jacobsen 2005;Ostad, 1998;Shalev, 2007;Shalev, Manor, & Gross-Tsur, 2005). These individuals are considered to have a mathematical learning disability (MLD), and are joined by another 5% and perhaps many more children and adolescents who experience more mild and circumscribed learning difficulties in mathematics (for recent reviews see Berch & Mazzocco, 2007). The mathematics achievement of this latter group of low achieving children (LA) is below expectations based on their cognitive ability and reading achievement, and the mechanisms contributing to their difficulties with mathematics learning may differ from those underlying MLD (Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007;Murphy, Mazzocco, Hanich, & Early, 2007). The central executive component of working memory has been implicated as a core mechanism underlying differences in the mathematical cognition of children with MLD but not their LA peers, but this remains to be confirmed. Moreover, the mathematical areas in which these groups may be similar or different are not well understood.
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Author ManuscriptChildren with MLD have deficits in a wide range of basic mathematical domains, including a delayed understanding of counting concepts (Geary, Bow-Thomas, & Yao, 1992), difficulties remembering arithmetic facts (Geary, 1993;Jordan, Hanich, & Kaplan, 2003;Jordan & Montani, 1997), and poor conceptual knowledge of rational numbers (Mazzocco & Devlin, in press). The delayed learning of LA children, in contrast, may center on basic numerical representations, including the number line (Geary et al., 20...