This study examined the efficacy of preventive 1st-grade tutoring in mathematics, estimated the prevalence and severity of mathematics disability, and explored pretreatment cognitive characteristics associated with mathematics development. Participants were 564 first graders, 127 of whom were designated at risk (AR) for mathematics difficulty and randomly assigned to tutoring or control conditions. Before treatment, all participants were assessed on cognitive and academic measures. Tutoring occurred 3 times weekly for 16 weeks; treatment fidelity was documented; and math outcomes were assessed. Tutoring efficacy was supported on computation and concepts/applications, but not on fact fluency. Tutoring decreased the prevalence of math disability, with prevalence and severity varying as a function of identification method and math domain. Attention accounted for unique variance in predicting each aspect of end-of-year math performance. Other predictors, depending on the aspect of math performance, were nonverbal problem solving, working memory, and phonological processing.

The purpose of this study was to examine the interplay between basic numerical cognition and domain-general abilities (such as working memory) in explaining school mathematics learning. First graders (n=280; 5.77 years) were assessed on 2 types of basic numerical cognition, 8 domain-general abilities, procedural calculations (PCs), and word problems (WPs) in fall and then reassessed on PCs and WPs in spring. Development was indexed via latent change scores, and the interplay between numerical and domain-general abilities was analyzed via multiple regression. Results suggest that the development of different types of formal school mathematics depends on different constellations of numerical versus general cognitive abilities. When controlling for 8 domain-general abilities, both aspects of basic numerical cognition were uniquely predictive of PC and WP development. Yet, for PC development, the additional amount of variance explained by the set of domain-general abilities was not significant, and only counting span was uniquely predictive. By contrast, for WP development, the set of domain-general abilities did provide additional explanatory value, accounting for about the same amount of variance as the basic numerical cognition variables.Inquiries should be sent to Lynn S. Fuchs, 228 Peabody, Vanderbilt University, Nashville, TN 37203. Publisher's Disclaimer: The following manuscript is the final accepted manuscript. It has not been subjected to the final copyediting, fact-checking, and proofreading required for formal publication. It is not the definitive, publisher-authenticated version. The American Psychological Association and its Council of Editors disclaim any responsibility or liabilities for errors or omissions of this manuscript version, any version derived from this manuscript by NIH, or other third parties. The published version is available at www.apa.org/pubs/journals/dev NIH Public Access Author ManuscriptDev Psychol. Author manuscript; available in PMC 2011 November 1. Published in final edited form as:Dev Psychol. 2010 November ; 46(6): 1731-1746. doi:10.1037/a0020662. NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author ManuscriptLanguage, attentive behavior, nonverbal problem solving, and listening span were uniquely predictive.Keywords mathematics development; procedural calculations; word problems; basic numerical cognition; domain-general abilitiesAchieving mathematics competence in its many forms during the elementary school years provides the foundation for learning algebra and other higher forms of mathematics and eventually for success in the labor market and a society that increasingly depends on quantitative skills (National Mathematics Advisory Panel, 2008). Yet, the cognitive mechanisms that support learning of formal mathematics during elementary school are not well understood: specifically, the relative contributions of children's basic numerical cognition that emerges without formal schooling (e.g., competence in number, counting, and simple arithmetic) as contra...

Response to intervention (RTI) models for identifying learning disabilities rely on the accurate identification of children who, without Tier 2 tutoring, would develop reading disability (RD). This study examined 2 questions concerning the use of 1st-grade data to predict future RD: (a) Does adding initial word identification fluency (WIF) and 5 weeks of WIF progress-monitoring data (WIF-Level and WIF-Slope) to a typical 1st-grade prediction battery improve RD prediction? and (b) Can classification tree analysis improve the prediction accuracy compared to logistic regression? Four classification models based on 206 1st-grade children followed through the end of 2nd grade were evaluated. A combination of initial WIF, WIF-Level, and WIF-Slope and classification tree analysis improved prediction sufficiently to recommend their use with RTI.

Contributions of domain-general and domain-specific numerical competencies were assessed on 1 st graders' number combination skill (NC) and word problem skill (WP). Students (n=205) between 5-7 years of age were assessed on 2 aspects of numerosity, 8 domain-general abilities, NC, and WP. Both aspects of numerosity predicted NC when controlling for domain-general abilities, but domain-general abilities did not account for significant additional variance. By contrast, when controlling for domain-general abilities in predicting WP, only precise representation of small quantities was uniquely predictive, and domain-general measures accounted for significant additional variance; central executive component of working memory and concept formation were uniquely predictive. Results suggest that development of NC and WP depends on different constellations of numerical versus more general cognitive abilities.In an analysis of six large-scale longitudinal studies, Duncan et al. (2008) demonstrated that mathematical competence at school entry predicts mathematics achievement throughout the elementary-school years, above and beyond general cognitive ability, classroom attention, social skills, or socioeconomic background. In fact, performance on early mathematical achievement tests was by far the single best predictor of later mathematics achievement. They suggested "it may be beneficial to add domain-specific early skills to the definition of school readiness" (Duncan et al., p. 1429), but their analysis did not allow for the assessment of which specific mathematical competencies may be the best target for such programs. Candidates center on children's early number sense, including the ability to quickly apprehend the quantities of small sets of items, use counting to determine quantity, estimate the value of large quantities, and intuitively understand the effects of addition and subtraction on quantity (National Mathematics Advisory Panel, 2008).Inquiries should be sent to Lynn S. Fuchs, 228 Peabody, Vanderbilt University, Nashville, TN 37203. NIH Public Access Author ManuscriptChild Dev. Author manuscript; available in PMC 2011 September 1. NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author ManuscriptAt the same time, Duncan et al. (2008) showed that classroom attention, a domain-general factor, also predicts later mathematics achievement, above and beyond early mathematical competence. Other studies have revealed that general cognitive ability is also a strong predictor of achievement across academic domains (e.g., Walberg, 1984). General cognitive ability includes working memory capacity, speed of information processing, and logical reasoning (Embretson, 1995;Engle, Tuholski, Laughlin, & Conway, 1999;Kail, 1991), although the relative importance of these domain-general abilities is debated (e.g., Ackerman, Beier, & Boyle, 2005).In the present study, we focused on the relation between two measures of children's early number sense as well as domain-general cognitive and attentional measures with perf...