Why mixing up your maths practice beats doing twenty of the same problem

In a laboratory experiment with 18 undergraduates at the University of South Florida, students who practised similar maths problems in blocks scored 89% during practice, while those using mixed problems scored only 60%.

This discrepancy is not a failure of the mixed-practice group. It is a measurement of how traditional learning methods mask the actual difficulty of a subject. The high scores in the blocked group do not represent mastery. They represent the ability to repeat a single, known motion.

Blocked practice masks the real challenge

Most maths assignments are designed around the principle of repetition. A textbook chapter introduces a single concept, and the subsequent exercises require the student to apply that same concept repeatedly. Rohrer, Dedrick, and Stershic (2015) note that in most maths assignments, each set of practice problems is devoted to the same skill or concept.

This structure creates a significant cognitive shortcut. When a student sees the fifth problem in a set of ten, they do not need to think about which formula to use. They already know. The strategy has been decided by the context of the previous four problems. The student is no longer practising problem-solving. They are merely practising execution.

The challenge of mathematics is not just knowing how to calculate. The challenge is knowing which calculation to perform. By grouping similar problems together, assignments remove the most critical step of the learning process: the selection of a strategy.

Initial practice scores can be misleading

The University of South Florida study highlights a dangerous illusion. The blocked-practice group, which worked on one type of geometric solid at a time, looked successful because its scores were high. The mixed-practice group, which had to switch between different types of solids, looked as if it was struggling. Then came the test, one week after the second practice session: the mixed-practice students scored 63%, the blocked-practice students 20%.

This creates a false sense of competence. A student can breeze through a worksheet of twenty identical quadratic equations and leave the desk feeling prepared. In reality, they have only practised the mechanical steps of the solution. They have not practised the mental leap required to identify a quadratic equation in a sea of different mathematical structures.

The lower scores in the mixed-practice group are not a sign of poor learning. They are a sign of active engagement. Every time a student switches problem types, they must retrieve a different rule from their memory. This friction is where actual learning happens.

Long-term retention improves with mixed problems

The true value of interleaved practice only becomes visible when the safety net of the practice session is removed. When students are tested on what they have actually retained, the results shift dramatically.

In a study of 140 seventh-grade students, material learned by interleaved practice scored 72% versus 38% for material learned by blocked practice on an unannounced test two weeks later. Once the practice context was gone, mixing won by a wide margin.

The likeliest explanation is that the blocked-practice students had never practised telling problem types apart, so once the context was gone they could not pick the right strategy. They had memorised a sequence of steps rather than a library of methods.

Students often underestimate the value of harder methods

There is a psychological disconnect between how effective a method is and how effective it feels. Learners tend to prefer methods that feel easy, even when those methods fail to produce results.

A study of UCLA undergraduates demonstrated this bias. Participants were asked to identify the styles of 12 different artists. 78% of participants did better when the paintings were mixed together. Yet 78% said that massing the paintings was as good as or better than spacing them. The learners were entirely unaware that the harder-feeling method had actually worked better.

This is a common trap. When learning feels effortless, we assume we are progressing. When learning feels difficult, we assume we are failing. In the context of mathematics, the difficulty of interleaving is a desirable difficulty. It is the effort of switching gears that makes the learning last.

The method works in real classroom settings

The benefits of mixing problem types are not limited to controlled laboratory environments or small groups of undergraduates. The effect holds up even in the messy, large-scale reality of public education.

A randomised controlled trial of 54 seventh-grade mathematics classes found that the interleaved group outscored the blocked group 61% to 38% on an unannounced test one month later. This result was consistent across all 15 teachers involved, even though the teachers received no specific training on how to implement the method.

This suggests that interleaving is a robust cognitive principle. It does not require a specialised curriculum or high-tech tools. It simply requires a change in how problems are organised. It is not a magic fix. The trial’s authors note that important caveats remain, and a major review of 10 study techniques rated interleaving “moderate utility”, below practice testing and spaced practice, because more research was needed on how broadly it helps.

Practice should mimic the conditions of an exam

To learn effectively, students must stop practising for the worksheet and start practising for the exam. An exam does not present problems in neat, thematic blocks. An exam presents a variety of challenges that require the student to identify the correct strategy on sight.

The most effective way to study is to rearrange practice sets so that no two consecutive problems require the same strategy. If a student is studying algebra, they should not do ten problems on factoring followed by ten on expansion. They should mix them. They should force themselves to look at a problem and ask: What is this, and how do I solve it?

This approach is harder. It is slower. It will lead to lower scores during the initial study sessions. But that is the point. The goal of practice is not to feel successful in the moment. The goal is to be prepared for the moment when the context is gone and the only thing left is the problem itself.