Interleaving: why mixed practice feels harder and teaches more

A page of similar problems can supply the method for you

Imagine working through a page headed “Adding fractions”, with an example showing how to find a common denominator. Once you have understood the example, the heading supplies a decision that every later problem requires. You can practise carrying out the calculation without deciding whether addition is appropriate. An examination removes that assistance when questions requiring different operations appear together, so a learner who completed the practice page smoothly may hesitate over the first unfamiliar arrangement.

Interleaving makes that decision part of practice by placing different kinds of problem among one another. A fraction addition can follow a multiplication, with a subtraction appearing later. Before calculating, you must examine the signs and quantities to select a method, rather than carry the previous procedure forward. This arrangement gives you opportunities to practise matching a problem to its solution, including the occasions when an apparently familiar question needs a different approach.

Doug Rohrer and Kelli Taylor tested mixed mathematics practice in their 2007 paper, The shuffling of mathematics problems improves learning. Their second experiment is the relevant one here. College students learned to calculate the volumes of several unfamiliar geometric solids and then practised problems that were either blocked by type of solid or randomly mixed, with no increase in the total number of practice problems. A first experiment in the same paper examined the timing of practice separately, so the mixing comparison is reported on its own.

On a test one week later, the authors describe performance as vastly superior after mixed practice. Blocked practice usually yields more correct answers during practice itself, which is why a delayed test matters when judging a routine: easier progress through a worksheet can reflect help supplied by its arrangement, even when learners receive the same problems. The study was a laboratory experiment with college students on one mathematical task, so its results cannot establish the size of a benefit for every school subject.

Comparing neighbouring examples can reveal a useful difference

Mixing can also help when learning to recognise categories, even when there is no calculation to perform. In Learning concepts and categories: Is spacing the “enemy of induction”?, Nate Kornell and Robert Bjork asked learners to study paintings labelled with their artists' names. Paintings by an artist appeared together or among works by other artists. On subsequent tests using previously unseen paintings, interleaving helped learners identify the artist, although participants who compared the arrangements tended to believe that grouped paintings had taught them more.

One plausible explanation is that placing different styles near each other helps learners discriminate between them, although the experiments did not measure what any participant noticed. Grouped examples let you look for a style's common features, while alternating styles invites attention to their differences. The painting tests followed the study period within the experiments; this paper therefore supports recognising new examples without establishing how well those distinctions survive months later.

For a small exercise in observation, compare tree leaves. Look at an oak leaf beside a maple leaf and describe the feature you would use to distinguish another specimen. Then examine different examples of each, because a single memorable image may conceal variation within the category. This is a suggested exercise in observation, with the usefulness depending on whether the selected examples expose the distinction you need to learn.

Practice fluency and delayed performance can point in different directions

The diagram uses letters for kinds of question, so each arrangement contains the same amount of each kind. Its delayed scores are deliberately illustrative. They show how a practice arrangement that feels orderly can produce less successful answers later, without assigning an expected improvement to you or to either research paper.

Schematic, not data: both sequences contain nine questions, with three each of types A, B and C. The illustrative delayed results of 40 and 80 correct answers out of 100 describe no participants and predict no particular learner's score.

The schematic sequences and result values appear here.
Arrangement	Practice sequence	Questions per type	Total questions	Illustrative delayed result
Blocked	A A A B B B C C C	A: 3; B: 3; C: 3	9	40 correct answers out of 100
Interleaved	A B C A B C A B C	A: 3; B: 3; C: 3	9	80 correct answers out of 100
Result scale	Lower endpoint: 0 correct answers out of 100; upper endpoint: 100 correct answers out of 100

A slower answer during mixed practice need not indicate that you have forgotten how to calculate. You may be spending time deciding which calculation fits, a decision the blocked page previously made easy. Keep track of that distinction when checking a solution. Choosing multiplication when addition was required needs a different correction from choosing addition correctly and then mishandling the denominator; the first calls for examining the problem's structure, while the second calls for repairing the procedure.

An unfamiliar method may need a short block first

Mixing is demanding when you have no usable method to choose. If a formula and its notation are unfamiliar, following a worked example and trying related problems can give you something coherent to compare with other methods. A short block is a sensible first exposure in that situation. Treat this as practical guidance about managing difficulty, rather than a research result establishing a fixed number of blocked questions before switching.

The wider evidence also gives reasons to be selective. Matthias Brunmair and Tobias Richter's 2019 review, Similarity matters: A meta-analysis of interleaved learning and its moderators, pooled 59 studies and found a moderate benefit overall, a small one for mathematical tasks and a clearer one for paintings. Results for expository texts were ambiguous, while studies using words favoured blocking on average. Differences between the materials mattered, which makes a universal rule to mix every subject difficult to defend.

Those findings suggest asking what your practice arrangement makes possible. Related problems that are easy to confuse offer a reason to compare their distinguishing features. Unrelated subjects may offer little such comparison, even though switching between them creates effort. Reading a paragraph about rivers between every algebra problem adds variety, yet it does not necessarily help you distinguish mathematical procedures. For your next practice set, choose related questions whose differences you can explain after checking the solutions.

Mix familiar questions and explain the choice before solving

To revise mathematics, collect previously studied questions that require different methods and remove headings that announce the operation. Attempt them in a mixed order, stating why your chosen method fits before doing the calculation. Keep the solutions available for checking afterwards. When an answer is wrong, explain the relevant cue in the question before trying a fresh example, so that practice addresses the choice as well as the arithmetic.

Chart interpretation offers another way to apply this idea. Compare a graph of a total with a graph of a rate, then describe what each vertical axis measures before making a claim about growth. The distinction changes what an upward line means. Return later to unfamiliar examples with their explanations covered and assess whether you can identify the quantity correctly, because recognising a page you have just read supplies weaker evidence of understanding.

A mixed session is a place to practise changing approaches

Funga Wega applies this reasoning in sessions of about eight minutes that mix pattern puzzles, estimation, comprehension and reasoning with a working-memory task, a field note and review. A change of question gives you a reason to reconsider your approach rather than repeat whatever helped previously. The different tasks exercise different practised skills; evidence about mixing mathematical procedures does not establish that this particular session improves intelligence or provides the same benefit across every task.

Use a change of task as a cue to pause long enough to identify what the question requires. An estimation needs a defensible range, while a comprehension question may require distinguishing a stated fact from an inference. If you are stuck, the layered hints can direct attention towards a cue or an approach. On the next fresh question, try naming the relevant feature before requesting that assistance again.

Does interleaving mean doing several things at once?

Interleaving usually means completing different kinds of question in sequence, giving each your attention before moving on. Responding to messages while attempting a problem adds interruptions, which is a different arrangement from selecting the appropriate method for the next complete question.

How should I choose the topics to mix?

Begin with methods you have already encountered, especially ones whose questions you sometimes confuse. Keep their answers available for feedback after your attempt. If you cannot understand an example even with its solution, spend time on that method before adding it to the mixed set.

Can I interleave practice and space it across days?

You can mix different kinds of question within a session and revisit those kinds on later days. Mixing changes the neighbours of each question, while spacing changes the time between encounters. These arrangements can coexist, although neither supplies a universal schedule for every material or learner.