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...

The primary focus of this study was to determine the effectiveness of a classwide peer tutoring program in reading for three learner types: low achievers with and without disabilities and average achievers. Twelve schools, stratified on student achievement and family income, were assigned randomly to experimental and control groups. Twenty teachers implemented the peer tutoring program for 15 weeks; 20 did not implement it. In each of the 40 classrooms, data were collected systematically on three students representing the three learner types. Pre- and posttreatment reading achievement data were collected on three measures of the Comprehensive Reading Assessment Battery. Findings indicated that, irrespective of type of measure and type of learner, students in peer tutoring classrooms demonstrated greater reading progress. Implications for policymaking are discussed.

M any school administrators, policy makers, researchers, and advocates recognize the variation that exists between states (and between districts within states) in how RTI is implemented, a fact recently documented by Berkeley, Bender, Peaster, and Saunders (2009). Despite such variation, it is our impression that most stakeholders assume there is consensus on basic questions about the nature or essence of RTI and its general purpose. This isn't the case. Rather, there are two large, loosely configured campswhich we'll call an Individuals With Disabilities Education Act (IDEA) group and a No Child Left Behind (NCLB) group-that hold starkly different answers to questions about the nature and purpose of RTI . . . and special education. In claiming the existence of these two camps, we're not saying people refer to themselves as belonging to one or the other. Nor are we trying to divide stakeholders-or, maybe more accurately, to deepen a divide that exists. Our discussion of the IDEA and NCLB groups is more exploratory than confirmatory. It is a means of probing multiple meanings of RTI; helping practitioners and policy makers think more clearly about what they want to accomplish in its name; and suggesting a different, distinctive, and important role for special education. In short, our basic purpose and 301 ABSTRACT: For nearly 10 years, the response-to-intervention (RTI) policy initiative has engendered enthusiasm at federal, state, and local levels and among various stakeholders. Nevertheless, there are basic and important disagreements about its nature and purpose. The authors describe two groups with contrasting perspectives on RTI in an effort to examine its multiple meanings, to argue that neither group has a credible plan to educate children and youth with severe learning needs, and to encourage all interested parties to think productively about what they want to accomplish in the name of RTI.hope is to offer analysis and a heuristic that leads to stronger RTI frameworks.We have organized the article in four parts. We first provide an overview of how the IDEA and NCLB groups think about the purpose and nature of RTI, and where they agree and disagree. Second, we explore the groups' respective perceptions of and plans for special education in an RTI framework. Next, we describe the IDEA group's "standard protocol" approach to general education and evaluate its likelihood of success in comparison to the NCLB camp's "problem-solving" approach. In these second and third sections of the article, we develop several points. The most important is that, although much of the two groups' thinking on RTI is well formed, neither one has yet developed a persuasive plan to meet the academic needs of our nation's most difficultto-teach children. We speak here and in the remainder of this article of subgroups of children with learning disabilities, behavior disorders, speech and language impairments, and intellectual disabilities. Finally, in part four, we describe what special education might look like as RTI's most intensiv...

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...

The purposes of this study were (a) to identify measures that when added to a base 1st-grade screening battery help eliminate false positives and (b) to investigate gains in efficiency associated with a 2-stage gated screening procedure. We tested 355 children in the fall of 1st grade, and assessed for reading difficulty at the end of 2nd grade. The base screening model, included measures of phonemic awareness, rapid naming skill, oral vocabulary, and initial word identification fluency (WIF). Short-term WIF progress monitoring (intercept and slope), dynamic assessment, running records, and oral reading fluency were each considered as an additional screening measure in contrasting models. Results indicated that the addition of WIF progress monitoring and dynamic assessment, but not running records or oral reading fluency, significantly decreased false positives. The 2-stage gated screening process using phonemic decoding efficiency in the first stage significantly reduced the number of children requiring the full screening battery.

Objective To validate a diagnostic instrument for pediatric delirium in critically ill children, both ventilated and nonventilated, that uses standardized, developmentally appropriate measurements. Design and Setting A prospective observational cohort study investigating the Pediatric Confusion Assessment Method for Intensive Care Unit (pCAM-ICU) patients in the pediatric medical, surgical, and cardiac intensive care unit of a university-based medical center. Patients A total of 68 pediatric critically ill patients, at least 5 years of age, were enrolled from July 1, 2008, to March 30, 2009. Interventions None. Measurements Criterion validity including sensitivity and specificity and interrater reliability were determined using daily delirium assessments with the pCAM-ICU by two critical care clinicians compared with delirium diagnosis by pediatric psychiatrists using Diagnostic and Statistical Manual, 4th Edition, Text Revision criteria. Results A total of 146 paired assessments were completed among 68 enrolled patients with a mean age of 12.2 yrs. Compared with the reference standard for diagnosing delirium, the pCAM-ICU demonstrated a sensitivity of 83% (95% confidence interval, 66–93%), a specificity of 99% (95% confidence interval, 95–100%), and a high interrater reliability (κ = 0.96; 95% confidence interval, 0.74–1.0). Conclusions The pCAM-ICU is a highly valid reliable instrument for the diagnosis of pediatric delirium in critically ill children chronologically and developmentally at least 5 yrs of age. Use of the pCAM-ICU may expedite diagnosis and consultation with neuropsychiatry specialists for treatment of pediatric delirium. In addition, the pCAM-ICU may provide a means for delirium monitoring in future epidemiologic and interventional studies in critically ill children. (Crit Care Med 2011; 39:150–157)

First, we describe two types of assessment (problem solving and standard treatment protocol) within a “responsiveness-to-instruction” framework to identify learning disabilities. We then specify two necessary components (measures and classification criteria) to assess responsiveness-to-instruction, and present pertinent findings from two related studies. These studies involve databases at grades 1 and 2, which were analyzed to compare the soundness of alternative methods of assessing instructional responsiveness to identify reading disabilities. Finally, conclusions are drawn and future research is outlined to prospectively and longitudinally explore classification issues that emerged from our analyses.