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IES Grant

Title: Interleaved Mathematics Practice
Center: NCER Year: 2011
Principal Investigator: Rohrer, Douglas Awardee: University of South Florida
Program: Cognition and Student Learning      [Program Details]
Award Period: 3 years Award Amount: $901,694
Goal: Development and Innovation Award Number: R305A110517

Co-Principal Investigator: Robert Dedrick

Purpose: In most U.S. mathematics textbooks, the majority of practice problems following a lesson are devoted to that same lesson. This "heavy repetition" approach gives students many problems on the same topic in immediate succession. An alternative approach is to rearrange the practice problems so that a portion of each assignment includes “interleaved" problems from previous topics. Evidence suggests that interleaving different kinds of practice problems drawn from other lessons in the textbook, without adding or changing problems, can improve test scores. One possible reason is that whereas heavy repetition allows students to infer the relevant concept or procedure for a practice problem before they read the problem, interleaved practice provides students with an opportunity to learn to select the appropriate strategy to solve the problem, just as they must do when they encounter a problem in later courses or standardized tests. The purpose of this grant is to develop and refine a simple, inexpensive practice strategy that teachers could use to interleave practice problems.

Project Activities: Researchers will develop and test different ways of interleaving problems within workbooks in order to identify the most effective ways to implement interleaving problems within homework assignments. The team will gather evidence about the feasibility of using the materials. The problems for these homework sets will be drawn from the textbooks currently being used by school districts and will be arranged in different configurations. During the development phase of the project, they will conduct two semester-long, classroom-based, fully embedded experiments, in order to identify features of the workbooks to be incorporated into the Year 3 test of the fully developed intervention. In the first experiment, the team will identify how many of the assigned practice problems should be interleaved. In the second experiment, the team will incorporate the interleaved practice proportions found to be optimal in the first experiment, and will test whether interleaving is more effective when a solved example is provided for each practice problem. In the final year of the project, the team will compare the efficacy of standard practice assignments to those that include interleaved practice. In each proposed study, researchers will compare two kinds of practice assignments using a repeated-measures, counterbalanced, crossover design, in which half of the assignments completed by students will incorporate interleaving, and the other half of which will not.

Products: Products from this study will include a fully developed workbook for teachers to use and a method that teachers can adopt to amend the presentation of practice problems. Publications in peer reviewed journals will also be produced.

Structured Abstract

Setting: These studies will take place in suburban public high schools in Hillsborough County, Florida.

Population: Participating students will be enrolled in Algebra 1. In the participating school district, 57 percent of students receive a free or reduced-price lunch, and 59 percent are racial minorities.

Intervention: The intervention will be a set of materials with interleaved practice problems drawn from textbooks already in use by teachers. These materials will be developed iteratively and will incorporate the optimal configuration based on a series of considered features. These features include different spacing intervals between the topics and number of practice problems. A second feature will be the difficulty posed by problems themselves, using an approach called “incremental development" (i.e., showing students simpler problems on a topic before giving them harder ones). A third feature pertains to the use of superficially-similar problems that actually demand different strategies, and a fourth is the use of just-in-time review problems (i.e., when the skills needed to solve a problem are actually presented in the subsequent topic).

Research Design and Methods: The researchers will use the practice problems in the textbooks already in use in the classroom. They will create workbooks with different types of orderings and gather evidence about the feasibility of using the materials and the effects of the different types of orderings. During the development process, they will also conduct three semester-long, classroom-based, fully embedded experiments. The three studies will be conducted in the spring semesters of three consecutive years. For each study, they will create a workbook with problems drawn from the students' regular textbook, but the practice problems will be rearranged in one or both conditions. Half of the practice problems completed by each student will include one type of interleaving, and the other half will include a different type (or standard assignments). Otherwise, the course is unchanged. Students will be taught only by their teachers, who will present topics as they ordinarily would. The three experiments will explore the effects of: (1) the number of practice problems interleaved, (2) the effect of providing solved examples as part of homework, and (3) the efficacy of standard practice assignments with those with interleaved practice. In each proposed study, they will compare two kinds of practice assignments by using a repeated-measures, counterbalanced, crossover design.

Control Condition: In the control condition for the Year 3 pilot study, students will receive assignments from their regular textbook in the same order as presented in the textbook.

Key Measures: The key measures will include pretest-posttest gains and posttest-posttest gains that is the difference between one posttest and the next posttest. There will be three posttests: one at the end of the school year; one on the first day of the following school year, approximately 14 weeks later; and a third posttest administered 1 year later depending on the students' subsequent mathematics course. The posttests will consist of two kinds of items: problems exactly like some of the problems appearing in their assignments (but with different numerical values) and problems requiring at least a moderate degree of transfer. The transfer problems will be drawn from standardized exams, such as the Iowa Test of Basic Skills (ITBS), Orleans-Hanna Algebra Prognosis Test, California Achievement Tests (CAT), and the Stanford Achievement Test (SAT). Other performance measures include scores on quarterly exams and students' accuracy and effort on practice problems. Finally, students and teachers will provide self-report preferences (e.g., Which kind of practice assignment did you prefer?) and metacognitive judgments (e.g., Which of the two kinds of assignments will better help you when you return for a test?).

Data Analytic Strategy: Researchers will compare both the difference in pretest-posttest gains and the difference in posttest scores between the two practice strategies. They will also conduct analyses to determine whether moderator variables, such as student ability, had an effect.

Related Grants: An Efficacy Study of Interleaved Mathematics Practice (R305A160263)
Harnessing Retrieval Practice to Enhance Learning in Diverse Domains (R305B070537)
Optimizing Resistance to Forgetting (R305H040108)
Optimizing Resistance to Forgetting (R305H020061)


Journal article, monograph, or newsletter

Carpenter, S.K., Cepeda, M.J., Rohrer D., Kang S.H., and Pashler, H. (2012). Using Spacing to Enhance Diverse Forms of Learning: Review of Recent Research and Implications for Instruction. Educational Psychology Review, 24(3): 369–378.

Rohrer, D. (2012). Interleaving Helps Students Distinguish Among Similar Concepts. Educational Psychology Review, 24(3): 355–367.

Rohrer, D. (2015). Student Instruction Should be Distributed Over Long Time Periods. Educational Psychology Review, 27(4): 635–643.

Rohrer, D., and Pashler, H. (2012). Learning Styles: Where's the Evidence?. Medical Education, 46(7): 634–635.

Rohrer, D., Dedrick, R.F., and Burgess, K. (2014). The Benefit of Interleaved Mathematics Practice is not Limited to Superficially Similar Kinds of Problems. Psychonomic Bulletin and Review, 21(5): 1323–1330.

Rohrer, D., Dedrick, R.F., and Stershic, S. (2015). Interleaved Practice Improves Mathematics Learning. Journal of Educational Psychology, 107: 900–908.