The 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 learning. At-risk students (n = 259) were randomly assigned to intervention and control. Whole-number calculation skill, domain-general 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 narrowed 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.
This study investigated whether individual differences in working memory (WM) moderate effects of 2 variations of intervention designed to improve at-risk 4th graders' fraction knowledge. We also ...examined the effects of each intervention condition against a business-as-usual control group and assessed whether children's measurement interpretation of fractions mediated those effects. At-risk students (n = 243) were randomly assigned to control and 2 intervention conditions. The interventions each lasted 12 weeks, with three 30-min sessions per week. The major focus of both intervention conditions was the measurement interpretation of fractions. Across the 2 conditions, only 5 min of each 30-min session differed. One condition completed activities to build fluency with 4 measurement interpretation topics; in the other, activities were completed to consolidate understanding on the same 4 topics. Results revealed a significant aptitude-treatment interaction, in which students with very weak WM learned better with conceptual activities but children with more adequate (but still low) WM learned better with fluency activities. Both intervention conditions outperformed the control group on all outcomes, and improvement in the measurement interpretation of fractions mediated those effects.
The purpose of this study was to compare subgroups of students with various forms of learning difficulties (< 25th percentile) on cognitive and mathematics characteristics. Students with mathematics ...difficulty (MD, n = 105), reading difficulty (RD, n = 65), both (MDRD, n = 87), or neither (NoLD, n = 403) were evaluated on an array of cognitive measures (e.g., working memory and language) and on mathematics measures of foundational numerical competencies, computation, and problem solving. Results revealed expected level differences among groups in both domains: NoLD outperformed RD, and MD outperformed MDRD. Profile differences were noted among pairs of subgroups on cognitive measures. On mathematics measures, profile differences were noted between RD and other subgroups, but not between MD and MDRD subgroups. The most discriminating cognitive measures were processing speed and language; the most discriminating mathematics measures depended on the subgroups being compared. Results were further evaluated according to more severe (< 10th percentile) criteria for MD and RD, which generally affected level differences more than the profile patterns. Results have implications for understanding comorbid MD and RD and for conceptualizing core deficits in MD.
Using a randomized controlled trial, we examined the effect of a fractions intervention for students experiencing mathematical difficulties in Grade 5. Students who were eligible for the study (n = ...205) were randomly assigned to intervention and comparison conditions, blocked by teacher. The intervention used systematic, explicit instruction and relied on linear representations (e.g., Cuisenaire Rods and number lines) to demonstrate key fractions concepts. Enhancing students’ mathematical explanations was also a focus. Results indicated that intervention students significantly outperformed students from the comparison condition on measures of fractions proficiency and understanding (g = 0.66–0.78), number line estimation (g = 0.80–1.08), fractions procedures (g = 1.07), and explanation tasks (g = 0.68–1.23). Findings suggest that interventions designed to include explicit instruction, along with consistent use of the number line and opportunities to explain reasoning, can promote students’ proficiency and understanding of fractions.
In this article, the authors summarize results from 5 randomized controlled trials assessing the effects of intervention to improve the fraction performance of fourth-grade students at risk for ...difficulty in learning about fractions. The authors begin by explaining the importance of competence with fractions and why an instructional focus on fractions magnitude understanding may improve learning. They then describe an intervention that relies strongly on this type of understanding about fractions instruction, and they provide an overview of the intervention’s overall effects. This is followed by an overview of 5 intervention components for which the authors isolated effects. They conclude by discussing some of the lessons learned from this research program.
The purpose of this study was to investigate the contributions of domain-general cognitive resources and different forms of arithmetic development to individual differences in pre-algebraic ...knowledge. Children (n = 279, mean age = 7.59 years) were assessed on 7 domain-general cognitive resources as well as arithmetic calculations and word problems at start of 2nd grade and on calculations, word problems, and pre-algebraic knowledge at end of 3rd grade. Multilevel path analysis, controlling for instructional effects associated with the sequence of classrooms in which students were nested across Grades 2-3, indicated arithmetic calculations and word problems are foundational to pre-algebraic knowledge. Also, results revealed direct contributions of nonverbal reasoning and oral language to pre-algebraic knowledge, beyond indirect effects that are mediated via arithmetic calculations and word problems. By contrast, attentive behavior, phonological processing, and processing speed contributed to pre-algebraic knowledge only indirectly via arithmetic calculations and word problems.
Magnitude understanding is critical for students to develop a deep understanding of fractions and more advanced mathematics curriculum. The research reports in this special issue underscore magnitude ...understanding for fractions and emphasize number lines as both an assessment and an instructional tool. In this commentary, we discuss how number lines broaden the concept of fractions for students who are tied to the more general part–whole representations of area models. We also discuss how number lines, compared to other representations, are a superior and more mathematically correct way to explain fraction concepts.
The main purposes of this study were to test the effects of teaching at-risk 4th graders to provide explanations for their mathematics work and examine whether those effects occur by compensating for ...limitations in cognitive processes. We randomly assigned 212 children to 3 conditions: a control group and 2 variants of a multicomponent fraction intervention. Both intervention conditions included 36 sessions, each lasting 35 min. All but 7 min of each session were identical. In the 7-min component, students were taught to provide high quality explanations when comparing fraction magnitudes or to solve fraction word problems. Children were pretested on cognitive variables and pre/posttested on fraction knowledge. On accuracy of magnitude comparisons and quality of explanations, children who received the explaining intervention outperformed those in the word-problem condition. On word problems, children who received the word-problem intervention outperformed those in the explaining condition. Moderator analyses indicated that the explaining intervention was more effective for students with weaker working memory, while the word-problem intervention was more effective for students with stronger reasoning ability.
The focus of this study was connections among 3 aspects of mathematical cognition at 2nd grade: calculations, word problems, and prealgebraic knowledge. We extended the literature, which is dominated ...by correlational work, by examining whether intervention conducted on calculations or word problems contributes to improved performance in the other domain and whether intervention in either or both domains contributes to prealgebraic knowledge. Participants were 1,102 children in 127 second-grade classrooms in 25 schools. Teachers were randomly assigned to 3 conditions: calculation intervention, word-problem intervention, and business-as-usual control. Intervention, which lasted 17 weeks, was designed to provide research-based linkages between arithmetic calculations or arithmetic word problems (depending on condition) and prealgebraic knowledge. Multilevel modeling suggested calculation intervention improved calculation but not word-problem outcomes; word-problem intervention enhanced word-problem but not calculation outcomes; and word-problem intervention provided a stronger route than calculation intervention to prealgebraic knowledge.
The goal of this study was to describe fraction-calculation errors among fourth-grade students and to determine whether error patterns differed as a function of problem type (addition vs. ...subtraction; like vs. unlike denominators), orientation (horizontal vs. vertical), or mathematics-achievement status (low-, average-, or high-achieving). We specifically addressed whether mathematics-achievement status was related to students’ tendency to operate with whole-number bias. We extended this focus by comparing low-performing students’ errors in 2 instructional settings that focused on 2 different types of fraction understandings: core instruction that focused on part-whole understanding versus small-group tutoring that focused on magnitude understanding. Results showed students across the sample were more likely to operate with whole-number bias on problems with unlike denominators. Students with low or average achievement (who participated only in core instruction) were more likely to operate with whole-number bias than students with low achievement who participated in small-group tutoring. We suggest instruction should emphasize magnitude understanding to sufficiently increase fraction understanding for all students in the upper elementary grades.
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