![]() In any case, the fundamental core of early number sense includes an implicit and potentially inherent understanding of the exact quantity of small collections of actions or objects and of symbols (e.g., Arabic numerals) that represent them (e.g., `3' = ■■■), and of the approximate magnitude of larger quantities ( Butterworth & Reigosa, 2007 Dehaene, Piazza, Pinel, & Cohen, 2003 Gallistel & Gelman, 1992 Geary, 1995). Although there were early studies of infants' and preschooler's number sense before 1993 and similar studies of adults ( Mandler & Shebo, 1982 Starkey & Cooper, 1980), the corresponding methods were not typically applied to the study of individual differences in children's mathematical achievement and thus were not included in my original review. Subsequent studies have confirmed the fact retrieval and procedural deficits ( Jordan, Hanich, & Kaplan, 2003 Raghubar, Cirino, Barnes, Ewing-Cobbs, Fletcher, & Fuchs, 2009) and have extended these to a potentially more fundamental deficit in number sense. ![]() These manifested as deficits in the retrieval of arithmetic facts from long-term semantic memory, in the execution of procedures for solving arithmetic problems, and in the ability to represent and interpret visuospatial representations of mathematical information. On the basis of studies available by 1993, I concluded there are three cognitive components to MLD whose expression and development were influenced in part by underlying working memory deficits. The benefit was readymade empirical and theoretical contexts for interpreting the performance of children with MLD, but with the cost of being constrained by what was known and how this was studied. With the early cognitively-motivated studies, the methods and theories used in the study of typical development were adopted to better understand the achievement deficits of children with MLD. Unless otherwise noted, hereafter MLD refers to both MLD and LA children, as these groups were conflated in much of the earlier research. ![]() I reflect on some of these gains following the organization of the 1993 review specifically, cognitive, neuropsychological, and genetic components of MLD. Substantial progress has been made in all of these areas, and the set of articles in this issue provides a fine illustration of how far our understanding of MLD and mathematical difficulties associated with LA has come since 1993. In fact, there was no agreed upon criterion for diagnosing MLD. The revolution in brain imaging technology was just underway and thus there were few such studies on mathematical processing and even fewer randomized control studies of cognitively-motivated interventions for MLD. At the time of my 1993 review, there were less than a score of cognitively-motivated studies of MLD and LA (low achievement), a somewhat richer literature in neuropsychology, and only a few behavioral genetic analyses of individual differences in mathematics achievement. Cirino and Berch asked me to reflect on the articles in this special issue with respect to my 1993 review of what was then known about mathematical learning disabilities (MLD Geary, 1993).
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