Interleaving: a blocked practice sheet skips the step the exam tests¶

Level: 101 · for anyone who did every problem at the end of the chapter and still froze on the exam

One line: A sheet of problems all of one kind lets you skip choosing the method, since it is the same as last time; an exam mixes the kinds, so choosing is most of what it tests. Interleaved practice rehearses the choice, which is why it feels harder, scores worse in practice, and does better on the exam.

The advice¶

Todd Zakrajsek's The New Science of Learning (third edition, 2022), chapter 4, "Improving the Learning Process", lists four ways to strengthen memory, and one of them goes against how courses are built: interleaving. Instead of studying in blocks, chapter 1 and then chapter 2, weave them together. Textbooks are written in blocks, the problems at the end of each section use that section's method, and exams often follow the order of the book. Life does not, and neither does a final exam.

The book reports two findings that seem to contradict each other. Interleaving keeps material in memory much longer, most of all when the things mixed are similar (Brunmair and Richter, 2019). And students shown course plans rate the interleaved one as the least desirable (Hartwig and colleagues, 2022). This page shows why both are true, by counting.

The sheets¶

The program uses four volume formulas from area and volume formulas ↗: cylinder πr²h, cone ⅓πr²h, sphere ⁴⁄₃πr³, hemisphere ⅔πr³. Twelve practice problems, three of each, are ordered two ways: blocked, three cylinders, then three cones, and so on; and interleaved, cycling through the four. The exam is eight problems in mixed order, as exams are.

Verified output of interleaving.py — regenerated by tools/run_examples.py, never hand-typed.

1. The two practice sheets and the exam

   blocked:     cylinder, cylinder, cylinder, cone, cone, cone, sphere, sphere, sphere, hemisphere, hemisphere, hemisphere

   interleaved: cylinder, cone, sphere, hemisphere, cone, sphere, hemisphere, cylinder, sphere, hemisphere, cylinder, cone

   exam:        cone, sphere, sphere, cylinder, hemisphere, cone, cylinder, hemisphere

2. How often each sheet makes you choose the formula

   sheet          problems   a new kind of problem, so a choice to make
   blocked             12      4  (33%)
   interleaved         12     12  (100%)
   exam                 8      7  (88%)

   On the blocked sheet the formula is chosen 4 times in 12 problems; the
   other 8 are the same as the problem before. The exam asks for a choice
   on every problem but one. Interleaved practice rehearses what the exam
   tests; blocked practice rehearses using a formula already chosen.

3. A student who never chooses: 'use the formula from the last problem'

   They work out the first problem properly and then stop choosing.

   sheet          right   of   score
   blocked           9   12     75%
   interleaved       1   12      8%
   exam              2    8     25%

   On the blocked sheet the shortcut scores 9 of 12 without ever telling
   a cone from a cylinder, so the sheet feels easy and the score looks good.
   On the interleaved sheet it gets only the first problem, and on the exam
   2 of 8. A blocked sheet's score predicts the exam badly: it measures
   using a formula, and the exam measures choosing one. The interleaved
   sheet's low score is the honest one, and it is low in practice, where
   it costs nothing.

4. The answers, exact

   kind         r   h   volume
   cone         5   3   25 pi
   sphere       3   -   36 pi
   sphere       1   -   4/3 pi
   cylinder     1   7   7 pi
   hemisphere   6   -   144 pi
   cone         2   9   12 pi
   cylinder     4   2   32 pi
   hemisphere   3   -   18 pi

What each sheet makes you do¶

A problem has two steps: decide which formula it needs, then use it. Section 2 counts how often each sheet asks for the first step, that is, how often a problem is a different kind from the one before. The blocked sheet asks 4 times in 12; after the first cone, the next two are cones because the last one was. The interleaved sheet asks 12 times in 12. The exam asks 7 times in 8.

So the blocked sheet gives twelve rehearsals of using a formula and four of choosing one, and the exam is mostly choosing. Every blocked sheet has this shape, whatever the subject: the heading of the section chose the method for you.

The student who never chooses¶

Section 3 makes that concrete with a student who works out the first problem properly and then stops choosing: every later problem gets the formula of the one before. On the blocked sheet this scores 9 of 12, 75%, without ever telling a cone from a cylinder. On the interleaved sheet it gets the first problem and nothing else, and on the exam 2 of 8.

No real student is that extreme, but every student leans on the shortcut a little, because it works on the sheet in front of them. That is why blocked practice feels better and why students rate interleaving least desirable: the blocked sheet's score is higher. It is also why the score misleads. It measures the step the exam barely tests and hides the one it does, and a student stopping on a good practice score, the rule from metacognition, stops too early. The interleaved sheet's low score is the honest one, and it arrives in practice, where it costs nothing.

What the experiments found¶

Doug Rohrer and Kelli Taylor (2007) ran this with real students and real volume problems (wedges, spheroids, spherical cones and half cones). During practice the blocked group was ahead, about 89% correct against 60%. On a test a week later the order reversed: about 63% for the interleaved group and 20% for the blocked group. A review of many such studies (Brunmair and Richter, 2019) found a benefit on average, largest when the kinds mixed are easy to confuse, like cones and cylinders, and not for every kind of material. The program does not reproduce those numbers, and it does not try to; it shows the one mechanism that makes them unsurprising.

Try it¶

Shuffle your homework. Do the odd-numbered problems from the last three sections in one mixed list, not section by section.
Name the method before you start. For each problem, write which formula or technique it needs before solving. That is the step a blocked sheet skips.
Mix in something old. Put two problems from an earlier chapter into every study session. It is interleaving and spaced retrieval at once.
Change the exam. Make the exam blocked in the program, and see how the shortcut does. Would a blocked exam be measuring what you will need later?

Flashcards¶

The page as a deck of Anki cards: interleaving.txt. Import with File → Import. Tags: definition, mechanism, evidence, trap.

Po polsku, w skrócie¶

Przeplatanie (interleaving) to mieszanie w czasie nauki różnych typów zadań, zamiast robienia najpierw wszystkich zadań jednego typu, a potem drugiego. Zadanie ma dwa kroki: rozpoznać, jakiej metody wymaga, i ją zastosować. Arkusz blokowy pomija pierwszy krok, bo metoda jest taka sama jak w poprzednim zadaniu. Egzamin miesza typy, więc sprawdza głównie właśnie rozpoznawanie.

Program liczy to na czterech wzorach na objętość (walec, stożek, kula, półkula). W arkuszu blokowym trzeba wybierać wzór 4 razy na 12 zadań, w przeplatanym 12 razy na 12, na egzaminie 7 razy na 8. Uczeń, który bierze zawsze wzór z poprzedniego zadania, ma w arkuszu blokowym 75%, a na egzaminie 25%. Dlatego nauka blokowa wydaje się łatwiejsza i daje lepszy wynik na ćwiczeniach, ale ten wynik wprowadza w błąd. W doświadczeniu Rohrera i Taylor (2007) grupa blokowa miała na ćwiczeniach 89% wobec 60%, a tydzień później na teście 20% wobec 63%.