Mixing Up Practice Problems Boosts Learning for All Calculus Students—And Helps Strugglers Most

A new study published in NPJ Science of Learning offers some of the strongest classroom-based evidence yet for a deceptively simple change in how mathematics is taught: rather than grouping practice problems by type, instructors should interleave them. The research, conducted in the demanding environment of college calculus, found that when students were given mixed sets of problems instead of blocked sets, every group of students improved—but the gains were largest for the students who needed help the most. Low-achieving students, who traditionally fall further behind in gateway mathematics courses, closed a measurable portion of the gap separating them from their higher-performing peers.

The finding challenges a practice so entrenched in mathematics education that most students and teachers never question it. Open virtually any calculus textbook and you will find chapters organized so that every derivative rule, every integration technique, and every limit-evaluation procedure is practiced in a dedicated block of near-identical exercises. This blocked arrangement feels efficient. Students appear to master a technique quickly, teachers can confirm comprehension at a glance, and homework sessions proceed with a satisfying sense of momentum. But decades of cognitive science have argued that this fluency is largely an illusion, a phenomenon researchers call the illusion of competence: because students know in advance which strategy each problem requires, they never practice the most difficult and most important step—deciding which strategy to use.

Interleaved practice removes that crutch. When problems drawn from different topics appear in mixed order, students must first diagnose the problem—recognizing, for example, whether a given integral calls for substitution, integration by parts, or a trigonometric identity—before they can execute the solution. This diagnostic step, sometimes described as discriminative contrast, forces learners to compare and contrast problem categories rather than repeatedly applying a single memorized template. Laboratory studies dating back to the mid-twentieth century, and more recent classroom experiments in algebra and geometry, have consistently shown that this added difficulty during practice produces substantially better retention and transfer, a counterintuitive pattern known as a desirable difficulty.

What makes the new study consequential is its setting. Calculus is not a laboratory task but a high-stakes, credit-bearing college course that serves as a gateway to degrees in engineering, the physical sciences, economics, and medicine. It is also a course with a well-documented attrition problem: students who arrive with weaker preparation are disproportionately likely to fail or withdraw, and those failures ripple outward, discouraging students from pursuing scientific careers altogether. Demonstrating that a low-cost, curriculum-neutral adjustment to homework design can improve outcomes in this environment matters far beyond the psychology of memory. It suggests that part of the achievement gap in STEM may be an artifact of instructional convention rather than an inevitability of prior preparation.

The study’s headline result is that mixing problems raised performance across the entire distribution of student ability. Higher-achieving students, who might have been expected to gain the least from a change in practice structure, still benefited from the interleaved format, consistent with the broad laboratory literature on spaced and varied retrieval. But the effect was not uniform. Students at the lower end of the achievement spectrum showed the largest improvements, a pattern with significant implications for equity in mathematics education. Interventions that lift the whole class while disproportionately lifting struggling students are rare in educational research, where the most common outcome is that advantage compounds: students who start ahead pull further ahead.

Why would low achievers gain the most? The authors’ explanation, grounded in established learning theory, centers on what blocked practice conceals. Under blocked conditions, struggling students can complete an entire assignment by mechanically repeating the worked example from the top of the page, without ever engaging in genuine problem-solving. The feedback signal is delayed until the examination, when the support of topical grouping disappears and the weakness is exposed too late. Interleaving converts that hidden failure into immediate, low-stakes feedback: students discover early which distinctions they cannot yet make, and instructors can see and address misconceptions while there is still time to correct them. In effect, mixed practice functions as a continuous diagnostic instrument woven into ordinary homework.

The mechanics of the improvement are worth spelling out, because they illuminate why the effect appears in calculus specifically. Calculus is a subject of bewildering surface variety concealing a relatively small set of underlying procedures. Two problems that look nothing alike—a related-rates word problem and an implicit differentiation exercise—may rely on the identical chain-rule computation, while two problems that look nearly identical may demand entirely different tools. Blocked practice teaches students to classify by surface features, which fails the moment an exam mixes contexts. Interleaved practice compels classification by mathematical structure, which is precisely the skill that expert mathematicians deploy automatically. The mixed format thus trains the categorization process itself, not merely the execution of procedures within a category.

The practical barriers to adoption are modest, which adds to the study’s policy relevance. Interleaving does not require new technology, smaller classes, additional instructional hours, or retraining in novel pedagogy. It requires reordering existing problem sets so that review of earlier material is distributed throughout the course rather than concentrated in a single pre-exam scramble—a change that also delivers the well-established benefits of spaced repetition as a side effect. Textbook publishers and online homework platforms could implement the restructuring at scale, and instructors can begin immediately by pulling a handful of problems from prior weeks into each week’s assignment. The chief obstacle, the literature suggests, is perceptual: interleaved practice feels harder and slower to students, and performance during practice sessions often looks worse, which can dissuade teachers who rely on short-term performance as evidence of learning.

That perceptual hurdle is also why studies conducted in real courses, with real grades and real students, carry more weight than laboratory demonstrations. Laboratory experiments on interleaving typically use artificial materials and short retention intervals, and skeptics have reasonably asked whether the effects survive contact with the messy realities of motivation, attendance, and competing coursework. By showing the effect in an authentic college calculus setting—and by showing that it operates most powerfully for the students whom standard instruction serves least well—the new research strengthens the case that desirable difficulties are not merely a laboratory curiosity but a practical lever for improving learning in the courses where the stakes are highest.

The broader message for students, teachers, and curriculum designers is a lesson in intellectual humility about what learning feels like. Performance during study is a poor proxy for durable knowledge, and the teaching practices that feel smoothest often produce the shallowest results. Mixing problem types makes practice harder, slower, and less comfortable—and that discomfort is the signature of the brain doing the comparative, structural work that long-term mathematical competence requires. If the findings generalize across institutions and course levels, as the underlying cognitive theory predicts, then one of the cheapest reforms available to mathematics education may also be one of the most equitable: stop telling students which tool to use before asking them to solve the problem, and let the mixed problem set do the teaching.

Subject of Research: The effect of interleaved versus blocked practice problems on student performance in college calculus

Article Title: Mixing problems increases performance of all students but especially of low-achieving ones in college calculus

Article References: Bennoun, S., Yan, V. X., & Xu, A. (2026). Mixing problems increases performance of all students but especially of low-achieving ones in college calculus. npj Science of Learning. https://doi.org/10.1038/s41539-026-00450-6

Image Credits: AI Generated

DOI: 10.1038/s41539-026-00450-6

Keywords: interleaved practice, blocked practice, college calculus, desirable difficulties, mathematics education, STEM equity, low-achieving students, retrieval practice, spaced repetition, cognitive science of learning, problem-solving, achievement gap