Interleaving: why mixing problem types beats blocked practice (and how to shuffle a textbook that is organized in blocks)

Interleaving, in studying, means mixing different kinds of problems in one practice set: a chain-rule problem, then a product-rule one, then a related-rates one, instead of doing twenty of the same kind in a row, which is called blocked practice. Mixed practice feels harder and scores lower during the session, and higher on the test, because the test makes you choose the method and blocked practice never does. Here is the evidence, a way to shuffle a textbook that is organized in blocks, and a six-problem set to try.

What interleaving is (in studying, not in computing)

The word has other meanings. In computing, interleaving is a way of ordering memory or data. In textiles it is a way of weaving. In learning, it means one thing: the order of your practice problems.

Take the same twelve problems, four of each of three kinds. Blocked practice does them as AAAA BBBB CCCC. Interleaved practice does them as ABCBCACABACB, or any shuffle. Same problems, same time, different order. That is the whole method.

Why would the order matter? Because a test question does not tell you which method it wants. On a textbook page headed "The chain rule", every problem is a chain-rule problem, so you never have to ask. On the test, the first thing you have to do with every question is decide what kind it is. Blocked practice never practises that decision. Interleaved practice practises it on every problem.

The evidence: interleaving vs blocked practice

Rohrer and Taylor (2007) taught college students to find the volumes of four unfamiliar solids. One group practised the problems blocked by type; the other practised the same problems mixed. During the practice session the blocked group was the more accurate, getting about 89 per cent right against about 60 per cent for the mixed group. On a test one week later the order reversed, and not by a little: the mixed group scored about 63 per cent and the blocked group about 20 per cent (Rohrer and Taylor 2007). The blocked students had looked better while practising and knew less a week on.

Taylor and Rohrer (2010) ran the same comparison with schoolchildren learning four kinds of problem about prisms, and found the same shape: blocked practice looked better during the lesson, interleaved practice tested better the next day. Rohrer, Dedrick and Stershic (2015) took it into real classrooms, with 126 seventh-grade students whose regular assignments were arranged so that some skills were practised interleaved and others blocked. On an unannounced test given one day after the final review, the interleaved skills scored higher; on a test given thirty days after, they scored higher by a wider margin (Rohrer, Dedrick and Stershic 2015). The gap grew with the delay, which is what you want from a method you are using for an exam weeks away.

The effect is not only for maths. Kornell and Bjork (2008) had people learn to recognize the styles of twelve painters, either seeing each painter's works in a block or mixed with the others'. The mixed group was better at naming the painter of a painting they had never seen. Then came the part that matters for you: asked which schedule had helped them more, most of the learners said the blocked one (Kornell and Bjork 2008). The method that worked felt worse.

Where does this sit among the study methods? Dunlosky and colleagues (2013) reviewed ten common techniques and rated interleaved practice as moderate utility, with the strongest evidence in mathematics and fewer studies elsewhere. The ranking of study methods puts it below practice testing and spacing, and above rereading and highlighting. Moderate is an honest rating: the maths result is solid and repeated; for other subjects, the evidence is thinner.

Why mixed practice feels worse and tests better

Every problem has two parts: working out what kind of problem it is, and then solving it. Researchers call the first part discrimination. A mixed set forces you to discriminate on every problem. A blocked set hands you the answer to the first part by position: you are on the chain-rule page, so it is a chain-rule problem.

That is why blocked practice feels smooth. You are only doing half the task, and the half you skip is the half that makes the test hard. It is also why the mixed set scores lower during practice: you are doing both halves, and getting the first half wrong costs you the problem.

The confidence trap follows from the same fact. Blocked practice feels fluent, and fluency feels like learning. Kornell and Bjork's learners judged by how it felt and were wrong. If a practice set feels easy, ask whether the page is making your decisions for you.

How to interleave a textbook that is organized in blocks: the interleaving study method

Textbooks are blocked, and you cannot reprint yours. You can reorder the problems.

1. Pick three or four sections you have already met. Not brand-new material; sections you can do slowly but have not had to choose between.
2. Write the exercise numbers from each section on slips of paper or in a list. Section and number: "3.4 #7", "3.6 #12". Six to ten problems a session is plenty.
3. Shuffle. Slips in a cup, or a list in random order. The point is that you do not know what is coming.
4. Do them in that order without looking at which section each is from. Cover the section label if it helps.
5. For every problem, write the method's name before you solve it. "Product rule." "Implicit." This is the discrimination step made visible, and it is the part you are training.
6. Mark, and log the errors. A wrong method name is a different error from a wrong line of algebra; log both, and note which it was.

The homework version: do the odd-numbered problems from today's section mixed with the even-numbered problems from two sections you did earlier this month. Your homework still gets done, and the older sections come back. Spacing comes free with interleaving, because every mixed set revisits material from days or weeks ago.

Worked example: a mixed set from MCV4U

Six problems from Grade 12 calculus (MCV4U), one from each of six ideas on the course map: the power rule, the product rule, the quotient rule, the chain rule, implicit differentiation and related rates. Here they are in a shuffled order, with the method named first and the answer after. Try each before you read the right-hand columns.

#	Problem	Name the method first	Answer
1	Find dy/dx for x² + y² = 25 at the point (3, 4).	Implicit differentiation	2x + 2y·y′ = 0, so y′ = −x/y = −3/4
2	Differentiate (x + 1)/(x − 1).	Quotient rule	[(x − 1) − (x + 1)] / (x − 1)² = −2/(x − 1)²
3	Differentiate 4x³ − x.	Power rule	12x² − 1
4	A 5 m ladder leans against a wall. Its foot slides away from the wall at 1 m/s. How fast is the top sliding down when the foot is 3 m from the wall?	Related rates	x² + y² = 25; 2x·x′ + 2y·y′ = 0; at x = 3, y = 4, x′ = 1: y′ = −(3)(1)/4 = −0.75 m/s, so the top falls at 0.75 m/s
5	Differentiate √(3x + 2).	Chain rule	3 / (2√(3x + 2))
6	Differentiate x²eˣ.	Product rule	2x·eˣ + x²·eˣ = x·eˣ(x + 2)

Check the related-rates answer yourself: when the foot is 3 m out, the top is 4 m up (because 3² + 4² = 25), and differentiating x² + y² = 25 with respect to time gives 2x·x′ + 2y·y′ = 0. Put in x = 3, y = 4 and x′ = 1 and solve: y′ = −3/4 m/s. The sign says the top is going down.

Now look at problems 1 and 4. They share an equation, x² + y² = 25, and ask different things: one wants a slope at a point, the other a rate of change in time. On a blocked page you would never meet them side by side, so you would never have to tell them apart. In this set you do, and that act of telling apart is the discrimination the test will demand. If you wrote "implicit" for problem 4 and then wondered where the time went, that is a method-name error, and it is the most useful kind to find at your desk rather than in the exam room.

How to study maths has the error log and the worked-example cycle that go with a set like this; the MCV4U course map lists the six ideas and where they sit in the course.

Interleaving in biology and chemistry

In the sciences the kinds to mix are kinds of question, not only kinds of calculation. A review session for a biology unit mixes a definition ("what is a competitive inhibitor?"), a mechanism ("why does raising the substrate concentration undo it?") and a calculation or a graph reading, in a shuffled order, so you practise switching the kind of thinking each question wants. Mixing units in one session does the same for the whole course.

In chemistry the discrimination is the set-up. A pH problem, an equilibrium problem and a stoichiometry problem each start from a different first line, and a test gives you no heading. Put three of each in a cup and the first thing you practise on every draw is "which set-up is this?", which is exactly what the test grades first. Mixing set-ups in chemistry and physics has the detail.

When not to interleave

Two cases.

The first hour with a brand-new procedure. You cannot choose between methods you cannot yet do. When a new rule arrives, do three or four blocked problems to get the steps into your hands. Then mix it in. The research we have read suggests that interleaving earns its gain after initial learning, not instead of it, and that reading is a reading of the studies, not a quotation from them.

Unrelated subjects in one set. A calculus problem followed by a history question followed by a chemistry calculation is not interleaving. There is no decision to practise, because no one would ever confuse them. That is task switching, and it costs attention for nothing. Interleave things that could be confused: methods within a course, set-ups within a unit, question types within a topic.

One more honest line. Interleaving helps you choose the method. It does not teach you a method you do not know, and it does not replace recall. It is one of the four moves, and the other three still have to happen.

The method on one page

• Interleaving is mixing problem types in one practice set; blocked practice is twenty of a kind in a row.
• It scores lower during practice and higher on the test, because the test makes you choose the method.
• Name the method first. Write it before you solve, every problem.
• The shuffle: three or four sections you have met, exercise numbers on slips, shuffle, solve blind to the section, log method errors apart from algebra errors.
• Homework version: today's odds mixed with two earlier sections' evens.
• Do not interleave a brand-new procedure in its first hour, or unrelated subjects.
• Chapter 1 of the free playbook is the attempt-first rule that every mixed set depends on: answer before you look.