Pamela M. Seethaler

The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems

The purpose of this study was to examine the cognitive correlates of 3rd-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Third graders (N = 312) were measured on language, nonverbal problem solving, concept formation, processing speed, long-term memory, working memory, phonological decoding, and sight word efficiency as well as on arithmetic, algorithmic computation, and arithmetic word problems. Teacher ratings of inattentive behavior also were collected. Path analysis indicated that arithmetic was linked to algorithmic computation and to arithmetic word problems and that inattentive behavior independently predicted all 3 aspects of mathematics performance. Other independent predictors of arithmetic were phonological decoding and processing speed. Other independent predictors of arithmetic word problems were nonverbal problem solving, concept formation, sight word efficiency, and language.

Do different types of school mathematics development depend on different constellations of numerical versus general cognitive abilities

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; mean age = 5.77 years) were assessed on 2 types of basic numerical cognition, 8 domain-general abilities, procedural calculations, and word problems in fall and then reassessed on procedural calculations and word problems in spring. Development was indexed by latent change scores, and the interplay between numerical and domain-general abilities was analyzed by 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 procedural calculations and word problems development. Yet, for procedural calculations 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 word problems 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. Language, attentive behavior, nonverbal problem solving, and listening span were uniquely predictive.

Mathematics Screening and Progress Monitoring at First Grade: Implications for Responsiveness to Intervention:

The predictive utility of screening measures for forecasting math disability (MD) at the end of 2nd grade and the predictive and discriminant validity of math progress-monitoring tools were assessed. Participants were 225 students who entered the study in 1st grade and completed data collection at the end of 2nd grade. Screening measures were Number Identification/Counting, Fact Retrieval, Curriculum-Based Measurement (CBM) Computation, and CBM Concepts/Applications. For Number Identification/Counting and CBM Computation, 27 weekly assessments were also collected. MD was defined as below the 10th percentile at the end of 2nd grade on calculations and word problems. Logistic regression showed that the 4-variable screening model produced good and similar fits in accounting for MD—calculation and MD—word problems. Classification accuracy was driven primarily by CBM Concepts/Applications and CBM Computation; CBM Concepts/Applications was the better of these predictors. CBM Computation, but not Number Identification/Counting, demonstrated validity for progress monitoring.

Effects of Preventative Tutoring on the Mathematical Problem Solving of Third-Grade Students with Math and Reading Difficulties:

This study assessed the effects of preventative tutoring on the math problem solving of third-grade students with math and reading difficulties. Students (n = 35) were assigned randomly to continue in their general education math program or to receive secondary preventative tutoring 3 times per week, 30 min per session, for 12 weeks. Schema-broadening tutoring taught students to (a) focus on the mathematical structure of 3 problem types; (b) recognize problems as belonging to those 3 problem-type schemas; (c) solve the 3 word-problem types; and (d) transfer solution methods to problems that include irrelevant information, 2-digit operands, missing information in the first or second positions in the algebraic equation, or relevant information in charts, graphs, and pictures. Also, students were taught to perform the calculation and algebraic skills foundational for problem solving. Analyses of variance revealed statistically significant effects on a wide range of word problems, with large effect sizes. Findings support the efficacy of the tutoring protocol for preventing word-problem deficits among third-grade students with math and reading deficits.

Intensive Intervention for Students with Mathematics Disabilities: Seven Principles of Effective Practice

The focus of this article is intervention for third-grade students with serious mathematics deficits at third grade. In third grade, such deficits are clearly established, and identification of mathematics disabilities typically begins. We provide background information on two aspects of mathematical cognition that present major challenges for students in the primary grades: number combinations and story problems. We then focus on seven principles of effective intervention. First, we describe a validated, intensive remedial intervention for number combinations and another for story problems. Then, we use these interventions to illustrate the first six principles for designing intensive tutoring protocols for students with mathematics disabilities. Next, using the same validated interventions, we report the percentage of students whose learning outcomes were inadequate despite the overall efficacy of the interventions and explain how ongoing progress monitoring represents a seventh, and perhaps the most essential, principle of intensive intervention. We conclude by identifying issues and directions for future research in the primary and later grades.

The Effects of Computer-Assisted Instruction on Number Combination Skill in At-Risk First Graders

The purpose of this pilot study was to assess the potential for computer-assisted instruction (CAI) to enhance number combination skill among children with concurrent risk for math disability and reading disability. A secondary purpose was to examine the effects of CAI on spelling. At-risk students were assigned randomly to math or spelling CAI, which they received in 50 sessions over 18 weeks. Acquisition and transfer effects were assessed. The results indicated that math CAI was effective in promoting addition but not subtraction number combination skill and that transfer to arithmetic story problems did not occur. Spelling CAI effects were reliable on acquisition and transfer spelling measures, with small to moderate effect sizes on transfer to reading measures. These results provide the basis for additional work with larger samples.

The Effects of Schema-Broadening Instruction on Second Graders' Word-Problem Performance and Their Ability to Represent Word Problems with Algebraic Equations: A Randomized Control Study

The purpose of this study was to assess the effects of schema-broadening instruction (SBI) on second graders word-problem-solving skills and their ability to represent the structure of word problems using algebraic equations. Teachers (n = 18) were randomly assigned to conventional word-problem instruction or SBI word-problem instruction, which taught students to represent the structural, defining features of word problems with overarching equations. Intervention lasted 16 weeks. We pretested and posttested 270 students on measures of word-problem skill; analyses that accounted for the nested structure of the data indicated superior word-problem learning for SBI students. Descriptive analyses of students word-problem work indicated that SBI helped students represent the structure of word problems with algebraic equations, suggesting that SBI promoted this aspect of students emerging algebraic reasoning.

The Predictive Utility of Kindergarten Screening for Math Difficulty

This study examined the reliability, validity, and predictive utility of kindergarten screening for risk for math difficulty (MD). Three screening measures, administered in September and May of kindergarten to 196 students, assessed number sense and computational fluency. Conceptual and procedural outcomes were measured at end of first grade, with MD operationalized as below the 16th percentile. The authors compared single-versus multiple-skill screeners, fall versus spring kindergarten screening, and conceptual versus procedural outcomes. Reliability and validity coefficients were adequate. Logistic regression and receiver operating characteristics analyses indicated that the single- and multiple-skill screeners produced good and similar classification accuracy at the fall and spring screening occasions in forecasting conceptual outcome. To forecast procedural outcome, the screeners produced similar but less accurate fits.

A Framework for Remediating Number Combination Deficits

This article introduces a framework for the remediation of number combination (NC) deficits. Research on the remediation of NC deficits is summarized, and research program studies are used to illustrate the 3 approaches to remediation. The Framework comprises a 2-stage system of remediation. The less intensive stage implementing 1 of 3 intervention approaches hypothesized to be most productive for a student uses a validated protocol while monitoring student response. The more intensive stage, which is reserved for nonresponders, involves integrating the 3 intervention approaches within a skills-based diagnostic-prescriptive scheme for individualizing intervention.

Two-Stage Screening for Math Problem-Solving Difficulty Using Dynamic Assessment of Algebraic Learning

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.