- Title: Mixing problems increases performance of all students but especially of low-achieving ones in college calculus
- Authors: Steve Bennoun, Veronica X. Yan & Alice Xu
- Access the original paper here
- Watch a video overview:
Paper summary
This study tested whether reorganising homework could raise exam scores in a large undergraduate calculus course at the University of California Los Angeles, with 585 students. Two sections of the course, taught by the same instructor, completed identical homework problems arranged in one of two ways. Blocked homework grouped problems by topic, while interleaved homework mixed topics together and spread each one across several weekly assignments. Halfway through the ten-week term, the sections swapped. After accounting for students’ backgrounds and midterm scores, the interleaved homework was linked to higher final exam scores for all students. It helped most those who had scored below average or only slightly above average on the midterm, and it narrowed gaps for students from underrepresented minority groups, first-generation students, students whose mothers had less education, and those with weaker school maths preparation.
If teachers remember one thing from this study, it should be…
Mixing up homework questions, so each assignment covers several topics and each topic returns over the following weeks, was linked to higher final exam scores for all students, with the biggest gains for those who were struggling. The only change was the order of the same questions.
*** PAPER DEEP DIVE ***
What are the key technical terms used in the paper?
- Blocked practice: questions grouped by topic, one topic at a time.
- Interleaved practice: questions on different topics mixed together.
- Spaced practice: returning to a topic across several sessions, instead of all at once.
- First-generation students: the first in their family to go to university.
What are the characteristics of the participants in the study?
585 undergraduates (563 in the final exam analyses), nearly all first-years, across two sections of a UCLA calculus for life sciences course taught by one instructor in autumn 2023. Students weren’t randomly assigned to sections, which differed from the start (29% versus 48% from underrepresented minority groups).
What does this paper add to the current field of research?
Interleaving’s benefits are well established, notably in Doug Rohrer’s school maths trials. Less clear was who gains most, and earlier studies gave mixed answers. This study shows benefits on high-stakes university exams, with the largest gains for lower-attaining students, easing fears that mixed practice overwhelms those who struggle.
What are the key implications for teachers in the classroom?
- Build homework so that every topic keeps coming back. In the study, once a topic had been taught, a question on it appeared on each of the next three weekly homeworks, and two questions on the same topic never sat next to each other. You can do this with questions you already have. When you set this week’s homework, keep a few questions on the new topic and add one or two from each of the last few weeks’ topics, spread through the set. The authors point out that mixing topics that sit next to each other in the scheme of work was enough to see a benefit, so you don’t need to hunt for cleverly contrasting topics.
- Give lower-attaining students mixed practice too. A common worry is that mixed-up questions will overwhelm students who find maths hard. In this study, those students gained the most. Students who had done less well on the midterm, and those with weaker school maths, got the biggest lift from interleaved homework. If your instinct is to give a struggling group a diet of one-topic-at-a-time worksheets, try including some mixed review questions in their practice as well.
- Give students practice at working out which method a question needs. The authors’ explanation (which they flag as a hypothesis they didn’t directly test) is that blocked homework tells students the method before they’ve read the question, whereas exams mix everything up. Mixed practice makes students decide for themselves. You can target this directly. Give a short set of mixed questions and ask students to write down which method they would use for each one, and why, before they solve any of them. Then discuss the ones where they disagreed.
- Tell students why their homework looks different. Research elsewhere has found that learners tend to feel mixed practice is going worse than blocked practice, even when they are learning more from it. If you switch, explain to the class that mixed questions are meant to feel harder, and that the struggle to recall which method fits is part of how it helps. That makes it less likely students (or parents) read the extra difficulty as a sign that something has gone wrong.
Why might teachers exercise caution before applying these findings in their classroom?
Students weren’t randomly assigned, and the sections differed from the start. The study covered a single US university course, and exam questions resembled the homework, so performance on unfamiliar question types is untested. Some of the gain for lower scorers may reflect top students having little room to improve.
What is a single quote that summarises the key findings from the paper?
“Our results suggest that spaced and interleaved practice is a promising intervention to both enhance all students’ performance and to help reduce achievement gaps among students through a simple reordering of homework practice questions.“
