Amelia S. Malone

Fraction Intervention for Students With Mathematics Difficulties: Lessons Learned From Five Randomized Controlled Trials

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.

Error Patterns in Ordering Fractions among At-Risk Fourth-Grade Students

The three purposes of this study were to (a) describe fraction ordering errors among at-risk fourth grade students, (b) assess the effect of part-whole understanding and accuracy of fraction magnitude estimation on the probability of committing errors, and (c) examine the effect of students’ ability to explain comparing problems on the probability of committing errors. Students (N = 227) completed a nine-item ordering test. A high proportion (81%) of problems were completed incorrectly. Most (65%) errors were due to students misapplying whole number logic to fractions. Fraction-magnitude estimation skill, but not part-whole understanding, significantly predicted the probability of committing this type of error. Implications for practice are discussed.

Multilevel Response-to-Intervention Prevention Systems: Mathematics Intervention at Tier 2

In this chapter, the focus is on mathematics intervention at Tier 2 of multilevel response-to-intervention prevention systems. The authors begin by providing an overview of the principles involved in effective Tier 2 intervention and illustrating their application in a validated tutoring program for addressing students’ difficulty with word problems. Then, the authors discuss more recent innovations in Tier 2 intervention by focusing on early arithmetic skill at first grade and on conceptual understanding and procedural skill with fractions at fourth grade are discussed. The chapter concludes with a discussion of the limitations of Tier 2 intervention research in mathematics and identifies areas for future research.

Child-Level Predictors of Responsiveness to Evidence-Based Mathematics Intervention

We identified child-level predictors of responsiveness to 2 types of mathematics intervention (calculation and word problem) among second-grade children with mathematics difficulty. Participants were 250 children in 107 classrooms in 23 schools pretested on mathematics and general cognitive measures and posttested on mathematics measures. We randomly assigned classrooms to calculation intervention, word-problem intervention, or business-as-usual control. Intervention lasted 17 weeks. Path analyses indicated that scores on working memory and language comprehension assessments moderated responsiveness to calculation intervention. No moderators were identified for responsiveness to word-problem intervention. Across both intervention groups and the control group, attentive behavior predicted both outcomes. Initial calculation skill predicted the calculation outcome, and initial language comprehension predicted word-problem outcomes. These results indicate that screening for calculation intervention should include a focus on working memory, language comprehension, attentive behavior, and calculations. Screening for word-problem intervention should focus on attentive behavior and word problems.

Error Patterns with Fraction Calculations at Fourth Grade as a Function of Students’ Mathematics Achievement Status

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.

A Research-Validated Program for Improving At-Risk Students’ Fraction Magnitude Understanding, Word-Problem Solving, and Explanations

In this chapter, we describe a research-validated Tier II intervention program designed to prevent at-risk students from developing significant difficulty with fractions. We provide an overview of the instructional strategies used to improve at-risk students’ fraction magnitude understanding (i.e., comparing fractions, ordering fractions, and placing fractions on the number line), word-problem solving (i.e., multiplicative reasoning), and explanations (i.e., verbally explaining how to compare fraction magnitudes). We discuss each instructional strategy incorporated within the program to enhance these outcomes and explain how they can be effectively implemented to improve students’ understanding of fractions. We conclude by discussing results from 4 years of intervention research testing the efficacy of the program. At-risk students who participated in the intervention program outperformed at-risk classmates receiving fraction instruction in the general education classroom on assessments of fraction magnitude understanding (i.e., number line estimation), calculations, released fraction items from the National Assessment of Educational Progress, multiplicative-reasoning word problems, and verbal explanations of magnitude comparisons.

Comparing the Contribution of Teacher Versus Tutor Ratings of Inattentive Behavior in Predicting Mathematics Achievement

The purpose of this study was to assess the relative contribution of teacher and tutor ratings of inattentive behavior in two different instructional settings in predicting students’ performance on fraction concepts and whole-number calculations. Classroom teachers rated each student’s attentive behavior in a whole-class setting and tutors rated each student’s attentive behavior in an intensive small-group setting. The sample was 131 fourth-grade students identified as at risk for developing mathematics difficulties (MD) and randomly assigned to fraction tutoring. Teachers and tutors rated each student’s inattentive behavior in the 10th week of the 12-week intervention period. On average, tutors rated students as more attentive than teachers. Also, tutor ratings had greater predictive power than teacher ratings on fraction concepts, but neither set of ratings uniquely predicted performance on whole-number calculations. Possible explanations for findings and directions for future research are discussed.