The purpose of this study was to assess the effects of small-group tutoring with and without validated classroom instruction on at-risk (AR) students' math problem solving. Stratifying within schools, 119 3 rd -grade classes were randomly assigned to conventional or validated problem-solving instruction (Hot Math [schema-broadening instruction]). Students identified as AR (n = 243) were randomly assigned, within classroom conditions, to receive Hot Math tutoring or not. Students were tested on problem-solving and math applications measures before and after 16 weeks of intervention. Analyses of variance, which accounted for the nested structure of the data, revealed the tutored students who received validated classroom instruction achieved better than tutored students who received conventional classroom instruction (ES = 1.34). However, the advantage for tutoring over no tutoring was similar whether or not students received validated or conventional classroom instruction (ESs = 1.18 and 1.13). Tutoring, not validated classroom instruction reduced the prevalence of math difficulty. Implications for responsiveness-to-intervention prevention models and for enhancing math problem-solving instruction are discussed.Mathematics word problems require the transfer of knowledge to novel situations, and this form of transfer can be difficult to effect (cf. Bransford & Schwartz, 1999;Mayer, Quilici, & Moreno, 1999) especially for primary-grade children (Durnin, Perrone, & MacKay, 1997; Foxman, Ruddock, McCallum, & Schagen, 1991, cited in Boaler, 1993. Whereas a calculations problem is already set up for solution, a word problem requires students to use text to determine what information is unknown and to construct and solve a number sentence for finding that unknown information. So, although calculation skill is foundational to word problems (e.g., Fuchs et al., 2006), it seems likely that for some students, word-problem skill will not develop automatically from instruction on math calculations, but rather that instruction needs to be designed specifically to prevent difficulty with word problems.The context for preventing academic difficulty in the schools has changed over the past 5 years with the introduction of multi-tiered prevention systems. 1 Adapted from the health care system,

Dynamic assessment (DA) involves helping students learn a task and indexing responsiveness to that instruction as a measure of learning potential. The purpose of this study was to explore the utility of a DA of algebraic learning in predicting 3 rd graders' development of mathematics problem solving. In the fall, 122 3 rd -grade students were assessed on language, nonverbal reasoning, attentive behavior, calculations, word-problem skill, and DA. On the basis of random assignment, students received 16 weeks of validated instruction on word problems or received 16 weeks of conventional instruction on word problems. Then, students were assessed on word-problem measures proximal and distal to instruction. Structural equation measurement models showed that DA measured a distinct dimension of pretreatment ability and that proximal and distal word-problem measures were needed to account for outcome. Structural equation modeling showed that instruction (conventional vs. validated) was sufficient to account for math word-problem outcome proximal to instruction; by contrast, language, pretreatment math skill, and DA were needed to forecast learning on wordproblem outcomes more distal to instruction. Findings are discussed in terms of responsiveness-tointervention models for preventing and identifying learning disabilities.A major purpose of educational assessment is to forecast academic achievement. The goal is early identification of students who are at risk for poor learning outcomes so that intervention can be initiated before the development of severe academic deficits, which can be intractable and can create life-long difficulty in and out of school (e.g., Rivera-Batiz, 1992). The predominant approach for forecasting academic achievement is traditional testing with a general measure of intelligence (e.g., Raven Progressive Matrices [Raven, 1960]) or a test of specific ability or skill presumed to underlie future academic performance (e.g., phonological processing for development of word-reading performance or calculations skill for development of mathematics word-problem performance). In these conventional testing situations, examinees respond without examiner assistance, and a body of work demonstrates that assessments of intelligence or precursor abilities/skills capture varying amounts of variance in forecasting academic development. For example, a measure of quantity discrimination, when used as a screener near the beginning of first grade, accounts for 25% to 63% of the variance in end-of-year math outcomes depending on the study (e.g., Chard et al. 2005;Clarke & Shinn, 2004; Lembke & Foegen, 2006). Because these conventional assessments are imperfect predictors of academic learning (e.g., Sternberg, 1996), they have long been the target of scrutiny and criticism (e.g., Tzuriel & Haywood, 1992). A major concern is that these "static" estimates of performance reveal only two states: unaided success or failure. By contrast, as Vygotsky (e.g., 1934Vygotsky (e.g., /1962 proposed, children may function somew...

Responsiveness to intervention (RTI) is an innovative approach to the identification of learning disabilities (LD). The central assumption is that RTI can differentiate between two explanations for low achievement: poor instruction versus disability. If the child responds poorly to validated instruction, then the assessment eliminates instructional quality as a viable explanation for poor academic growth and instead provides evidence of a disability. For children who do respond nicely, RTI serves a critical prevention function. Most of RTI research has been focused on early reading. In this article, we describe two ongoing programs of research on RTI in the area of mathematics: one on a comprehensive mathematics curriculum at first grade and the other focused on word problems at third grade. For each research program, we describe the sample, explain how students are identified as at risk for mathematics disability, provide an overview of the interventions to which responsiveness is gauged, and describe some results to date.

The purpose of this study was to explore the utility of a dynamic assessment (DA) of algebraic learning in predicting third graders’ development of mathematics word-problem difficulty. In the fall, 122 third-grade students were assessed on a test of math word-problem skill and DA of algebraic learning. In the spring, they were assessed on word-problem performance. Logistic regression was conducted to contrast two models. One relied exclusively on the fall test of math word-problem skill to predict word-problem difficulty on the spring outcome (less than the 25th percentile). The second model relied on a combination of the fall test of math word-problem skill and the fall DA to predict the same outcome. Holding sensitivity at 87.5%, the universal screener alone resulted in a high proportion of false positives, which was practically reduced when DA was included in the prediction model. Findings are discussed in terms of a two-stage process for screening students within a responsiveness-to-intervention prevention model.

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