Previous Award Number: R305A150262
Previous Awardee: Carnegie Mellon University

Purpose: The purpose of this project is to identify optimal instructional approaches for improving elementary and middle school students' understanding of fraction magnitudes and middle school students' and pre-service teachers' understanding of fraction arithmetic. Fraction understanding is critical for success in algebra, trigonometry, chemistry, physics, and other mathematics and science subject areas, yet many students struggle to acquire adequate proficiency with fractions. In this project, the research team will build and refine the theory that emphasizing fraction magnitudes and conceptual understanding of fraction arithmetic can improve students' mathematics outcomes.

Project Activities: The research team will conduct two studies per year during Years 1, 2, and 4 and three studies during Year 3 aimed at identifying optimal instructional approaches for improving students' understanding of fraction magnitudes and fraction arithmetic. Each year, the research team will conduct at least one study that focuses on improving knowledge of fraction magnitude and one study that focuses on improving understanding of fraction arithmetic. Studies focused on improving knowledge of fraction magnitude will involve elementary and middle school students, while those focused on improving understanding of fraction arithmetic will involve middle school students and pre-service teachers.

Products: This project will generate preliminary evidence of potentially promising instructional approaches for improving elementary and middle school students' understanding of fraction magnitudes and middle school students' understanding of fraction arithmetic. The researchers intend to publish their findings in peer-reviewed publications.

Structured Abstract

Setting: Participating elementary and middle schools will be located in urban and suburban areas in Pennsylvania and New York. Studies with pre-service teachers will take place at undergraduate institutions located in urban areas in West Virginia, New York, and Canada.

Sample: Across all studies, participants will include approximately 270 fourth and fifth grade students, 566 sixth and eighth grade students, and 138 pre-service teachers. Among the elementary and middle school students, about half will come from low-income backgrounds, and about half from middle or upper-middle income backgrounds.

Intervention: The research team will use two interventions to provide different types of instructional approaches to students and assess their impact on student learning: the Catch the Monster game and the Understanding Fraction Arithmetic intervention. Catch the Monster combines a 3-minute conceptual explanation of fraction magnitudes and a computer game to provide practice and feedback on magnitude estimates. Understanding Fraction Arithmetic includes explanations and practice regarding the meaning of multiplication and division in the context of whole numbers and fractions.

Research Design and Methods: The research team will conduct two studies per year during Years 1, 2, and 4 and three studies during Year 3 aimed at identifying optimal instructional approaches for improving students' understanding of fraction magnitudes and fraction arithmetic. Each year will include at least one study that focuses on improving knowledge of fraction magnitude and one study that focuses on improving understanding of fraction arithmetic. Studies focused on improving knowledge of fraction magnitude will involve elementary and middle school students while those focused on improving understanding of fraction arithmetic will involve middle school students and pre-service teachers. For all but one study, researchers will use a between-subjects experimental design where students will be randomly assigned to instructional conditions. All participants will be given pre-tests and post-tests of fraction knowledge. The other study will be a correlational study of relations across fractions and decimals. In this study, participants will complete the directions of effects task, which involves judging whether multiplying or dividing by fractions yields an answer above or below the larger multiplicand or the dividend as well as other tests of their fraction magnitude knowledge. Participants' strategy use and confidence in their responses on the direction of effects task will be obtained.

Control Condition: In the studies that use a between-subjects experimental design, researchers will implement one or multiple comparison conditions. The particulars of these conditions vary across experiments depending on the research question being asked. The comparison conditions are similar to the experimental conditions, except on the manipulation of interest in the study.

Key Measures: For all studies, primary measures will include participants' performance on five researcher-developed tasks: number line estimation (estimating a fraction's position on a number line), number line strategy explanation (locate a fraction on a number line and explain how they did so), magnitude comparison (identify whether a fraction is less than or greater than 3/5), fraction arithmetic, and fraction recall (recall fractions and non-fraction information from a short vignette). For the studies focused on fraction arithmetic, an additional primary measure will be participants' performance on a researcher-developed direction of effects task, which involves judging whether multiplying or dividing by fractions yields an answer above or below the larger multiplicand or the dividend as well as other tests of their fraction magnitude knowledge.

Data Analytic Strategy: In the studies that use a between-subjects experimental design, the research team will use ordinary least square regression, analysis of variance (ANOVA), multivariate analysis of variance (MANOVA), Pearson correlation, and non-parametric statistics. The correlational study will use the same statistics except for ANOVA and MANOVA.

Products and Publications

Journal article, monograph, or newsletter

Braithwaite D.W. and Siegler R.S. (2017). Developmental Changes in the Whole Number Bias. Developmental Science.

Braithwaite, D., Tian, J., & Siegler, R. S. (in press). Do Children Understand Fraction Addition?. Developmental Science.

Fazio, L.K., DeWolf, M., and Siegler, R.S. (2016). Strategy Use and Strategy Choice in Fraction Magnitude Comparison. Journal of Experimental Psychology: Human Perception and Performance, 42(1): 1–16.

Fazio, L.K., Kennedy, C.A., and Siegler, R.S. (2016). Improving Children's Knowledge of Fraction Magnitudes. PLOS ONE.

Lortie-Forgues, H., and Siegler, R. S. (2017). Conceptual Knowledge of Decimal Arithmetic. Journal of Educational Psychology, 109(3): 374–386.

Siegler, R. S. and Lortie-Forgues, H. (2017). Hard Lessons: Why Rational Number Arithmetic Is so Difficult for so Many People. Current Directions in Psychological Science, 26(4): 346–351.

Siegler, R. S., and Braithwaite, D. W. (2017). Numerical Development. Annual Review of Psychology , 68: 187–213.

Siegler, R.S. (2016). Magnitude Knowledge: The Common Core of Numerical Development. Developmental Science, 19(3): 341–361.

Tian, J. and Siegler, R.S. (2016). Fractions Learning in Children with Mathematics Difficulties. Journal of Learning Disabilities: 1–7.

Tian, J. and Siegler, R.S. (in press). Which Type of Rational Numbers Should Students Learn First?. Educational Psychology Review: 1–22.