Why is solving the same type of problem over and over misleading? The science of interleaved practice

Your maths homework has ten problems under the heading “direct proportion”. You think a little over the first, find your rhythm by the third and by the tenth you could solve them with your eyes closed. You close your notebook feeling pleased: you have finished the topic. A few days later the exam mixes the proportion problems up. In one, the more workers there are, the shorter the job takes; in another, the more kilograms you buy, the higher the price. You pause: which one was this?

That pause comes from a skill you never practised in your homework: looking at a problem and choosing which method it needs. In this article we describe a way of studying that closes that gap: interleaved practice.

Blocked practice and interleaved practice

Most textbooks and homework sheets are “blocked”: first a run of problems of one type, then a run of the next type. In interleaved practice the types are mixed together: a direct-proportion problem, an inverse-proportion problem, a percentage problem, then a direct-proportion problem again.

You solve the same number of problems and cover the same topics either way. The only difference is the order.

Why does it make a difference?

In blocked practice, the previous problem has already given away the method for the next one. Because “this section is about direct proportion”, you do not have to think about what to apply. This makes you feel you are making progress as you work, but it lets you skip something: telling types apart. In an exam, and in real life, problems do not arrive with a heading. First you have to ask, “what kind of problem is this?”

Interleaved practice makes you ask that question every time. So you learn two things at once: the method itself and when to use it. The benefit is expected to be greatest for topics that look alike and are easy to confuse: direct and inverse proportion, area and perimeter formulas, mean and median, adding and multiplying fractions.

What does the research show?

• Rohrer and Taylor (2007) taught university students to calculate the volumes of certain solids. One group practised in blocks and the other in a mixed order. During practice the blocked group did better; but in a test a week later the interleaved group came out clearly ahead.
• Rohrer, Dedrick and Stershic (2015) ran a similar comparison in real seventh-grade maths classrooms. On topics practised in a mixed order, later test results were better.
• Brunmair and Richter (2019) carried out a meta-analysis that pulled together the studies in this area. They reported that interleaving is generally better than blocking, but that the size of the effect varies with the material being learned. So you should not expect the same size of benefit for every topic.

An honest note: in the broad review that Dunlosky and colleagues published in 2013, interleaved practice was rated as “low utility”, because at that time there was not enough evidence. Later classroom studies strengthened the evidence. In science, conclusions are updated as new studies appear.

Is feeling hard a flaw?

No; it is even expected. This is exactly what Robert and Elizabeth Bjork call a desirable difficulty: some difficulties that seem to slow learning down actually improve its durability in the long run. When you interleave, you make more mistakes and feel slower. What is more, in studies such as Kornell and Bjork (2008), participants thought blocked practice was better for them, the opposite of what was measured. So do not rely on the feeling “this is too hard, it is not for me” alone.

A small example

Consider three problems:

1. If 3 kg of apples cost 45 TL, how much do 5 kg cost?
2. If 6 workers finish a job in 12 days, how many days do 9 workers need?
3. A product priced at 200 TL is discounted by 25%; what is the new price?

All three contain numbers and all three feel a bit like “proportion”. But the first is direct proportion (more kilograms, higher price): 45 ÷ 3 = 15 TL, 15 × 5 = 75 TL. The second is inverse proportion (more workers, fewer days): 6 × 12 = 72 worker-days, 72 ÷ 9 = 8 days. The third is a percentage problem: 200 × 0.25 = 50, 200 − 50 = 150 TL.

Solving the three in a mixed order makes you choose the method for each. Solving each in its own block tells you the method in advance.

How can you add it to your studying?

1. Name the type first. Before solving a problem, write one word on paper: “inverse proportion”. If you are wrong, you see it straight away. A good clue for proportion: “As one goes up, does the other go up too, or go down?”
2. Learn first, then mix. If you are learning a topic for the first time, it is natural to solve a few problems of the same type. Once you understand it, add problems from earlier weeks’ topics to your study list.
3. Work through a question bank by moving between sections, not from start to finish. Put the types you got wrong on a separate list and fold them back into mixed problems a few days later. This works well alongside spaced repetition and retrieval practice, which we covered earlier.

There are also situations where interleaving will not help much: if you are acquiring a single skill for the first time, or the types look nothing alike, the gain may be small. There is no single magic method in studying; the aim is to pick the practice that prepares you for what you will actually face in an exam and in life.

Summary

• In blocked practice the order of the problems gives the method away; in interleaved practice you choose which method to use.
• It feels harder while you work, but research generally shows better results on delayed tests.
• The effect varies with the material; do not expect the same size of benefit for every topic.
• Learn first, then mix; name the type before each problem.

For interactive lessons where you can practise proportion, percentages and probability in a mixed order, see the Mathematics topic. For more study techniques, browse our Learning Science articles.