Physics Questions Solver
Marsilius Boa
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We investigated whether continuously alternating between topics during practice, or interleaved practice, improves memory and the ability to solve problems in undergraduate physics. Over 8 weeks, students in two lecture sections of a university-level introductory physics course completed thrice-weekly homework assignments, each containing problems that were interleaved (i.e., alternating topics) or conventionally arranged (i.e., one topic practiced at a time). On two surprise criterial tests containing novel and more challenging problems, students recalled more relevant information and more frequently produced correct solutions after having engaged in interleaved practice (with observed median improvements of 50% on test 1 and 125% on test 2). Despite benefiting more from interleaved practice, students tended to rate the technique as more difficult and incorrectly believed that they learned less from it. Thus, in a domain that entails considerable amounts of problem-solving, replacing conventionally arranged with interleaved homework can (despite perceptions to the contrary) foster longer lasting and more generalizable learning.
In each of the two stages of the course, students completed 84 practice problems across 10 homework assignments. Blocked assignments typically featured three successive problems for each of three topics, whereas interleaved assignments typically featured only one problem per topic. In the figure, letters represent topics and subscripts represent the problem number for a given topic (1, 2, or 3). Different topics are also assigned different colors so that it is easier to visually tell them apart. Reflecting the relative simplicity of practicing one topic at a time, topics in each row of the blocked condition correspond perfectly to the assignment subject labeling that row, but this is not the case for the interleaved condition. Topics addressed on the criterial tests are also listed. Due to course time constraints, the last two blocked assignments of each stage include only two problems per topic instead of three. Topics from these assignments were not included in criterial tests.
During the course, each of the three weekly lectures was accompanied by a homework assignment. With blocked assignments, each topic was repeatedly practiced in succession with no intervening topics, whereas with interleaved assignments, each successive problem involved a change in the topic (for a list of topics, see Table 1). Of the nine problems per assignment, blocked assignments had three successive isomorphic problems per topic (i.e., having the same underlying problem-solving structure with contrasting surface features), which resembles the arrangement of practice exercises that occurs in many educational contexts1, whereas interleaved assignments had only one problem per topic, thus requiring students to engage in switching between topics (with the second and third problems per topic appearing on subsequent assignments). Crucially, within each stage, all students completed the same 84 total problems, with only the arrangement of those problems differing.
To measure the potential effects of interleaving, we administered an in-class surprise criterial test at the conclusion of each stage. These tests followed the approach taken in recent studies of interleaving and mathematics31,33 and avoided contaminating effects of cramming, study group activities, and other events that can occur with increasing frequency in the period leading up to pre-announced exams. Both tests featured three novel problems that were more difficult than those included in the homework assignments. The first two problems required integrating concepts and procedures from two separate topics, whereas the third problem required applying a single topic in a new scenario. All three problems required recall and application of factual content conveyed in formulas (see Fig. 1). To derive answers, students had to correctly recognize the topics involved, all of which were last encountered more than 1 week prior; recall relevant formulas, rules, and principles; and in two of three problems, integrate and apply that information to devise a new solution strategy37 (which could be viewed as requiring higher-order reasoning, integration, and constructive thought processes as opposed to simply recalling and repeating previously learned information)38,39.
Across both lecture sections, 290 students in stage 1 (83% of the total enrolled) and 286 students in Stage 2 (82% of total enrolled) experienced the experimental manipulation in its entirety by completing and turning in all of the homework assignments. Per our preregistered inclusion criteria, only data from those students were analyzed. Although that analysis revealed disparities between interleaving and blocking in terms of student performance, judgments of difficulty, and judgments of pedagogical effectiveness, there was no advance indication of any interleaving benefit.
With respect to overall performance, students correctly solved more blocked than interleaved homework problems (Table 2), with a mean deficit on interleaved assignments of 0.05 and 0.09 proportion correct in Stages 1 and 2, respectively. When interpreting these results, it is important to consider that there were nine different problem types on most interleaved assignments, with each type requiring a different problem-solving strategy, whereas, with most blocked assignments, there were only three problem types. Hence, the expectation that the blocked assignments would be easier was confirmed by student performance.
Each histogram displays the distributions of criterial test scores in a given stage, with green representing performance in the interleaved condition and purple representing performance in the blocked condition. The median score in each condition is included as a vertical bar of the corresponding color. Histograms are normalized so that in each condition, the sums of values of all bins equals 1. Mean performance in Stages 1 and 2, respectively, was 0.43 and 0.27 in the blocked condition and 0.54 and 0.47 in the interleaved condition.
Results at the level of individual problems (Table 3) also showed the advantages of interleaving. These advantages were the most consistent (i.e., across both sub-measures) for the easiest problem in each stage (which addressed one as opposed to two topics). Overall, interleaving yielded at least a numerical advantage on both sub-measures for all three problems on both criterial tests.
Several theoretical mechanisms may account for the observed benefits of interleaving. Here, we summarize five candidates. These explanatory accounts are not necessarily mutually exclusive and have been largely drawn from the literature on interleaving, with some adaptations to problem-solving in introductory physics.
First, interleaving may have facilitated inductive learning of problem categories defined by specific physical concepts or principles. These categories, whose correct identification was necessary to solve criterial test problems, are often easily confusable to novice physics learners, who tend to base their problem representations on literal features instead of abstract principles47. The course progressed in a hierarchical manner whereby problems across topics commonly shared literal features, but problem classification was never explicitly discussed; hence, any inductive learning of problem categories would most likely have occurred during practice on homework sets. As has been repeatedly demonstrated in the literature (e.g., the attention bias and discriminative contrast framework), inductive learning of confusable perceptual categories is a context wherein interleaving can excel relative to blocking12,13,23,24. It is plausible that the interleaved homework sets, which provided more opportunities to compare non-isomorphic problem categories than the blocked homework sets, yielded similar benefits. However, it is important to note that the criterial tests required additional problem-solving steps, including memory retrieval of formulas. As such, inductive learning of problem categories alone might not be sufficient to explain the observed results.
Second, as previously noted, interleaving incorporates distributed practice (i.e., learning spread out over multiple sessions), which is known to improve long-term memory10. According to the study-phase retrieval account of the spacing effect, distributed practice during homework sets may have forced students to engage in repeated long-term memory retrieval processes, which are known to enhance the durability and accessibility of memories3. In contrast, with blocking, every successive set of three homework problems involved the same topic, thus allowing students to bypass memory retrieval in favor of knowledge temporarily held in working memory (i.e., repeatedly reusing the same solutions). Hence, productive memory retrieval processes may have been attenuated in the blocked condition, potentially reducing the rate of successfully recalling correct formulas on criterial tests, even in the case that the problem solver had achieved a correct conceptual classification of the problem. Other cognitive processes that distributed practice may engage, such as increased encoding of varied contextual cues, may have also had a facilitative effect on learning48.
Relatedly, there is evidence in the interleaving literature to support both minimal and major roles of distributed practice depending on the learning context. In the case of perceptual category learning, conditions that feature extensive amounts of distributed practice in the absence of interleaving have failed to yield similar learning benefits13,15, which suggests a minimal role, whereas, in studies involving mathematics or second language learning, interleaving schedules that incorporate substantial amounts of distributed practice have yielded larger benefits, which suggests a major role2,24. It is important to note, however, that differences in experimental and task design across studies may have also been factors.
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