Interleaved Practice Improves Learning

A typical mathematics assignment consists primarily of practice problems requiring the strategy introduced in the immediately preceding lesson (e.g., a dozen problems that are solved by using the Pythagorean theorem). This means that students know which strategy is needed to solve each problem before they read the problem. In an alternative approach known as interleaved practice, problems from the course are rearranged so that a portion of each assignment includes different kinds of problems in an interleaved order. Interleaved practice requires students to choose a strategy on the basis of the problem itself, as they must do when they encounter a problem during a comprehensive examination or subsequent course. In the experiment reported here, 126 seventh-grade students received the same practice problems over a 3-month period, but the problems were arranged so that skills were learned by interleaved practice or by the usual blocked approach. The practice phase concluded with a review session, followed 1 or 30 days later by an unannounced test. Compared with blocked practice, interleaved practice produced higher scores on both the immediate and delayed tests (Cohen’s ds 0.42 and 0.79, respectively).

In contrast, most math assignments are blockedmeaning that they consist of problems that can be solved by the same strategy. For example, a blocked assignment might be devoted entirely to the Pythagorean theorem. Although the problems might demand slightly different tasksfor instance, finding the length of the hypotenuse vs. finding the length of a sideevery problem is nonetheless about the Pythagorean theorem:

The intervention
Although most mathematics students typically receive some interleaved practice, often in the form of mixed review assignments before exams, we suggest that a portion of the practice problems in a course be rearranged so that students receive a higher dose of interleaved practice than usual. To be clear, we are not suggesting that students work more problems than they ordinarily do. In other words, the intervention requires only a rearrangement of practice problemsnot an increase in the number of practice problems. Here's a simplified illustration:

Evidence for interleaving
Giving larger doses of interleaved practice is supported by scientific evidence. In several randomized control trialsthe gold standard for testing an interventionstudents who received mostly interleaved practice scored higher on a final test than students who received mostly blocked practice, even though every student saw the same practice problems.
For instance, we recently conducted a research study in 54 seventh-grade mathematics classes. Each class was randomly assigned to either the mostly-blocked practice group or the mostlyinterleaved practice group. Students received practice assignments nearly every week for several months. All students saw the same practice problems, but the problems were either arranged in blocks devoted to a particular concept or interleaved with other problems. After each assignment, teachers presented solutions and required that students correct any mistakes in their work. The practice phase was followed by a review assignment and, one month later, an unannounced test: Even though every student practiced the same problems, the test results showed that the mostlyinterleaved group outscored the mostly-blocked group by a large margin: Finally, an anonymous survey completed by the teachers after the study (but before they knew the results of the study or its purpose) showed that nearly all the teachers endorsed interleaved practice.

1) Learning when to use a strategy
To solve a problem, students must first choose an appropriate strategyoften the hardest step. Consider the following problem and the strategies a student might consider: One reason it's often hard to choose a strategy is that problems that look alike cannot always be solved by the same strategy. Students face this challenge in every math course from Arithmetic to Calculus: With a block of problems devoted to a single strategy, such as the Pythagorean theorem, students do not practice choosing an appropriate strategy. Rather, they know the strategy before they read the problem. Thus, blocked practice provides students with a crutch. Students can often zip through blocked practice without understanding why a particular strategy is appropriate, perhaps leading some students (and their teachers) to falsely believe they have mastered the material. This illusion of mastery is often shattered when students encounter a problem on an exam, without the usual cues indicating which strategy is appropriate. In other words, if students don't learn to solve problems without the crutch of blocked practice, they will struggle during a test when their crutch is taken away.
When practice problems are instead interleaved, students cannot infer the strategy for a problem before they read it. Because consecutive problems require different strategies, students must choose a strategy on the basis of the problem itselfjust as they must do when they encounter a problem during a high-stakes test or a subsequent course. Thus, with interleaved practice, students practice choosing a strategynot just executing a strategywhich helps them understand when to apply the strategy. Furthermore, interleaved practice is a better indicator of whether students have indeed mastered a strategy.

2) Retaining knowledge across time
If you want students to retain a skill they've learned, practice of the skill should be distributed, or spaced, across assignments. Indeed, decades of research have demonstrated that spaced practice supports long-term retention. Importantly, a greater degree of interleaved practice ensures a greater degree of spaced practice. Consider a lesson on circumference: Several circumference problems might immediately follow the lesson (blocked practice), but other circumference problems can be mixed into future assignments following other lessons (interleaved practice). Interleaved practice ensures that circumference problems are seen on more days and spread across more assignments than is typically the casewithout changing the total number of practice problems. Although students ordinarily receive some degree of spaced practice because of review assignments, replacing some blocked practice with interleaved practice ensures a greater degree of spacing.

The scarcity of interleaved practice in textbooks
Despite its effectiveness, interleaving is underutilized. In most math textbooks, lessons are followed by blocks of problems devoted to that lesson. Even review assignments typically include small blocks of related problems. For example, a chapter review might include a few problems on Lesson 5-1, followed by a few problems on Lesson 5-2, and so forth. These "miniblocks" again provide a crutch for students.
In one study, we classified every practice problem in six popular U.S. math textbooks for seventh graders, and we found that most problems in each textbook were blocked:

Implementing interleaved practice
Until textbooks, workbooks, and other teaching materials provide more interleaved practice, teachers can create their own interleaved assignments or draw interleaved assignments from other resources. For example: 1. Create an interleaved assignment by choosing one problem from each of several blocked assignments in a typical textbook. For instance, the assignment might include p. 37 #19, p. 117 #21, p. 156 #3, and so forth. 2. Use the review assignments in a textbook or workbook. Although review assignments often include mini-blocks, these assignments provide at least some interleaved practice. 3. Find interleaved worksheets and practice tests on the Internet. For instance, a web search for "mixed review mathematics worksheets" returns a list of websites with freely downloadable interleaved assignments. Also, interleaved practice tests can be found on the websites of companies and organizations that create mathematics assessments.
Some caveats should be kept in mind, however: 1. Some blocked practice is probably useful when students first encounter a new concept or skill. Thus, we are not recommending that all practice be interleaved but, rather, that more practice be interleaved. How much interleaved practice is enough?
The ideal amount depends on the student and the material, but studies suggest that at least a third of practice problems should be interleaved. 2. An interleaved assignment consists of a variety of problem types, including ones not seen recently, so students should have the opportunity to seek help if they're stuck. This might include asking a teacher for help or having access to lessons or tutorials that explain the concept and provide examples. 3. Interleaved practice should be followed by informative feedback. For example, after students have attempted every problem in an assignment, they can be shown the solutions, asked to correct the errors in their work, and given a chance to ask questions. The key point is that students should be encouraged to learn from their mistakes. 4. Interleaved practice probably won't improve performance on a quiz that covers only the last couple days of learning. But short-term learning isn't the goal of education, anyway. The benefit of interleaved practice is most evident for long-term learning and on cumulative tests.

Takeaway
The bottom line is that interleaved practice boosts test scores without requiring any extra practice. It can be added to any mathematics curriculum by merely rearranging some practice problems. Interleaved practice also makes sense: Students must learn how to both choose and use a strategy. Students must also retain the knowledge across time. Interleaved practice helps students retain the strategies they learn and know when to use them.
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