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...

The purpose of this study was to explore patterns of difficulty in 2 domains of mathematical cognition: computation and problem solving. Third graders (n = 924; 47.3% male) were representatively sampled from 89 classrooms; assessed on computation and problem solving; classified as having difficulty with computation, problem solving, both domains, or neither domain; and measured on 9 cognitive dimensions. Difficulty occurred across domains with the same prevalence as difficulty with a single domain; specific difficulty was distributed similarly across domains. Multivariate profile analysis on cognitive dimensions and chi-square tests on demographics showed that specific computational difficulty was associated with strength in language and weaknesses in attentive behavior and processing speed; problem-solving difficulty was associated with deficient language as well as race and poverty. Implications for understanding mathematics competence and for the identification and treatment of mathematics difficulties are discussed. Keywordscalculations; word problems; cognitive predictors; mathematics Mathematics, which involves the study of quantities as expressed in numbers or symbols, comprises a variety of related branches. In elementary school, for example, mathematics is conceptualized in strands such as concepts, numeration, measurement, arithmetic, algorithmic computation, and problem solving. In high school, curriculum offerings include algebra, geometry, trigonometry, and calculus. Little is understood, however, about how different aspects of mathematical cognition relate to one another (i.e., which aspects of performance are shared or distinct, or how difficulty in one domain corresponds with difficulty in another). SuchCorrespondence concerning this article should be addressed to Lynn S. Fuchs, Peabody College, Box 228, Vanderbilt University, Nashville, TN 37203. lynn.fuchs@vanderbilt.edu. NIH Public AccessAuthor Manuscript J Educ Psychol. Author manuscript; available in PMC 2010 January 6. NIH-PA Author ManuscriptNIH-PA Author Manuscript NIH-PA Author Manuscript understanding would provide theoretical insight into the nature of mathematics competence and practical guidance about the identification and treatment of mathematics difficulties.The purpose of the present study was to explore the overlap of difficulty with two aspects of primary-grade mathematical cognition and to examine how characteristics differ among subgroups with difficulty in one, the other, both, or neither. The first aspect of performance was computation, including skill with number combinations (e.g., 2 + 5; 8 − 3) and procedural computation (e.g., 25 + 38; 74 − 22). The second aspect of performance was problem solving, including one-step, contextually straightforward word problems (e.g., John had 9 pennies. He spent 3 pennies at the store. How many pennies did he have left?) and multistep, contextually more complex problems (e.g., Fred went to the ballgame with 2 friends. He left his house with $42. While at the game, he bought 5 h...

Metropolitan-Nashville Public Schools, Nashville, TennesseeThe purposes of this study were to investigate the effects of an intervention designed to improve at-risk 4th graders' understanding of fractions and to examine the processes by which effects occurred. The intervention focused more on the measurement interpretation of fractions; the control condition focused more on the part-whole interpretation of fractions and on procedures. Intervention was also designed to compensate for at-risk students' limitations in the domain-general abilities associated with fraction leaming. At-risk students (n = 259) were randomly assigned to intervention and control. Whole-number calculation skill, domaingeneral abilities (working memory, attentive behavior, processing speed, listening comprehension), and fraction proficiency were pretested. Intervention occurred for 12 weeks, 3 times per week, 30 min per session, and then fraction performance was reassessed. On each conceptual and procedural fraction outcome, effects favored intervention over control (effect sizes = 0.29 to 2.50), and the gap between at-risk and low-risk students narTowed for the intervention group but not the control group. Improvement in the accuracy of children's measurement interpretation of fractions mediated intervention effects. Also, intervention effects were moderated by domain-general abilities, but not whole-number calculation skill.

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...