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Mix kinds of problem that are easy to confuse, name each problem's kind before solving it, and build cumulative homework sets.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will tell blocked from interleaved practice, decide when mixing is worth it, and turn a week of blocked homework into mixed sets that ask you to choose the method.
The last two lessons were about when to come back to material. This one is about what goes together in one session. Most homework comes blocked: twelve problems on the method taught today. You already know that a test is not blocked — it mixes everything from the unit — and this lesson is about practicing the way the test asks.
| Term | What it means |
|---|---|
| Blocked practice | All the problems of one kind together: A A A A, then B B B B. |
| Interleaved practice | Kinds mixed so the next problem's kind cannot be predicted: A C B A B C. |
| Kind | A family of problems solved the same way, such as cone volumes or preterite sentences. |
| Discrimination | Telling which kind a problem is, before solving it. |
| Cumulative set | A practice set that mixes today's problems with problems from earlier lessons. |
On a blocked page, the heading tells you the method. Problem 7 of Section 5.3: Volume of a Cone is a cone problem before you read it. So the page practices carrying out the method and skips the step a test depends on: looking at a problem and deciding which method it needs.
Interleaved practice mixes the kinds, so every problem starts with that decision. It feels slower and you get more wrong during the session. In Rohrer and Taylor's 2007 study, college students practicing volume problems in blocks got far more right during practice than students who practiced them mixed. A week later the order reversed: about 63 percent correct after mixed practice, about 20 percent after blocked.
Interleaving is not shuffling anything with anything. It pays for kinds that look alike and are easy to confuse — the three area rules, two Spanish past tenses, similar bird species. Learn a brand-new method in a short block first; mix it in as soon as you can do one on your own.
Another way: picture
Think of a batting cage that throws only fastballs. You hit well, because you know what is coming. A game mixes fastballs and curveballs, and the skill it asks for first is telling which one this is.
Another way: steps
Rohrer and Taylor (2007) taught college students to find the volume of four unusual solids. During practice, the blocked group got about 89 percent right and the mixed group about 60 percent. On the test a week later: about 20 percent for blocked, about 63 percent for mixed.
Rohrer, Dedrick and Stershic (2015) ran the idea in real classrooms: 126 seventh graders did the same practice problems over about three months, arranged mostly blocked or mostly mixed. On a surprise test a day after the review, the interleaved students scored higher, and the difference was larger on a test a month later. Kornell and Bjork (2008) found the same pattern outside mathematics: people who saw paintings by different artists mixed together were better at naming the artist of new paintings — yet most of them believed the blocked order had helped more.
Dunlosky and colleagues (2013) rate interleaving as moderate utility: the results are strong where tested, but most tests are in mathematics and in learning categories. The claim here stays inside that.
You rarely need new problems. Take the ones you already have and change their order. Cover the section headings with a strip of paper, or copy the problem numbers onto a list in mixed order. Deal problems from different chapters like cards. Keep a few problems from each earlier lesson in every homework session — a cumulative set mixes the kinds and spaces the earlier lessons at the same time.
Before solving each problem, write the kind in the margin: cone, preterite, systems. When you check, check the kind too. A right answer reached by the wrong method is luck, and a wrong kind is the error to fix first.
A fictional ninth grader, Maria, has an algebra unit taught over four days: Monday linear equations, Tuesday systems, Wednesday factoring quadratics, Thursday the quadratic formula. Her teacher assigns ten problems a night from that day's section.
Maria keeps the number of problems and changes the mix. Monday: all ten linear, because the method is brand new. Tuesday: six systems, four linear. Wednesday: four factoring, three systems, three linear. Thursday: four quadratic formula, three factoring, two systems, one linear, shuffled. She covers the section headings and writes the kind in the margin before each problem.
Thursday is slower than it would have been and she gets two wrong by picking factoring for a quadratic that does not factor. That mistake is the one the unit test will tempt, and she has now made it at home.
"I got them all right, so blocked practice works." You got them right while the heading told you the method. The test will not.
"Mixed practice is harder, so it is worse." It is harder because it asks for the decision. Rohrer and Taylor's mixed group did worse in practice and better a week later.
"Interleave everything: history, then chemistry, then French." Switching subjects is fine for a schedule, but the benefit found in these studies comes from mixing kinds that could be confused.
"Mix from the very first problem." A new method needs a few blocked problems first. Mix it in once you can do one on your own.
List the problems by kind.
Prisms: 1–4. Cylinders: 5–8. Cones: 9–12.
The sheet is three blocks of four; the headings give each kind away.
Deal them in turn, one of each kind.
Order: 1, 5, 9, 2, 6, 10, 3, 7, 11, 4, 8, 12
Dealing guarantees no two of one kind sit together.
Cover the headings and add a kind column.
Margin: kind? ___ before each problem
Writing the kind first turns every problem into a decision.
Check the kind as well as the answer.
Problem 10: wrote cylinder, it was a cone → kind error
A kind error is the one to fix first; the arithmetic may have been fine.
Keep the number of problems the same.
10 problems, as assigned
Interleaving changes the mix, not the workload.
Give today's lesson the largest share.
4 from the quadratic formula
The newest method still needs the most practice.
Fill the rest from earlier lessons.
3 factoring, 2 systems, 1 linear
Each earlier kind comes back, which also spaces it.
Shuffle and label before solving.
Q-formula, factoring, systems, Q-formula, linear, factoring, …
The order no longer tells her what comes next.
Take ten sentences from the preterite page and ten from the imperfect page.
20 sentences, 2 kinds
Two tenses that are easy to confuse: a case for mixing.
Shuffle them and hide which page each came from.
Mixed list: sentence 1 to 20, no page labels
Without the label, the tense has to be chosen.
Before each sentence, write the tense you will use.
A fictional student has nine geometry problems: three on prisms (P), three on cylinders (C) and three on cones (K). Which order is interleaved practice?
A fictional worksheet has 3 kinds of problem with 5 of each. Complete the worked count of how many times the student must decide which kind a problem is, blocked and mixed.
Count the problems on the sheet.
3 kinds × 5 each = a problems
Blocking and mixing use exactly the same problems.
Count the decisions when the sheet is blocked.
One per block: 3 decisions
Inside a block the kind is already known, so only the first problem of each block asks for a choice.
Count the extra decisions mixing adds.
Mixed: one per problem. Extra: problems − 3 = b
Every one of those extra decisions is practice at the thing a blocked page never asks for.
A mixed set of four volume problems, like the ones in Rohrer and Taylor's study. Before solving anything, name the kind of solid each problem is about.
| Kind of solid | |
|---|---|
| Cereal box 30 × 20 × 7 cm | |
| Soup can, radius 4 cm, height 11 cm | |
| Party hat, radius 6 cm, height 15 cm | |
| Basketball, radius 12 cm |
A fictional student turns tonight's algebra homework into a mixed set of 24 problems: a third from today's lesson on the quadratic formula, and the rest split equally between the two earlier lessons, on factoring and on systems of equations. How many problems come from today's lesson, and how many from each earlier one?
From today's lesson: a
From each earlier lesson: b
A fictional student has two problems each from chapters 4, 5 and 6 of a statistics book, and chapter 6 is the newest. She deals them like cards: newest chapter first, then 5, then 4, and round again. Fill in the chapter for each position.
| Chapter | |
|---|---|
| Position 1 | |
| Position 2 | |
| Position 3 | |
| Position 4 | |
| Position 5 | |
| Position 6 |
Four fictional study situations. For each, decide whether to **interleave** the practice or whether there is **no need**.
| Decision | |
|---|---|
| Spanish preterite and imperfect | |
| Sparrow and finch photos | |
| The fifty state capitals | |
| Speed, acceleration and force problems |
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A mixed set of three area problems. For each, name the shape and then the area rule to use.
| Shape | Area rule | |
|---|---|---|
| Sail: base 4 m, height 6 m, to a point | ||
| Tabletop: radius 0.6 m | ||
| Garden bed: parallel sides 3 m and 5 m, 2 m apart |
You can build a mixed practice set. Take your next homework, add a few problems from earlier lessons, cover the headings, and write the kind before each problem. Next: why the strategies that work feel worse. You can stop in the middle of any of this. Closing the page is the whole of stopping — your place and anything you had typed are already saved, and the lesson opens again where you left it.
11. Your turn: Spanish past tenses, step 3
1: imperfect (was raining, background). 2: preterite (one finished action).
Writing the choice first makes it the thing you practice and check.