Work problems with the book shut
Most students revise by reading their notes again. When Karpicke, Butler and Roediger (2009) asked 177 college students how they studied, 84% listed rereading and 55% named it as their main method. Only 11% said they practised recalling the material, testing themselves, though 43% did mention practice problems.
Rereading feels productive because the page gets more familiar each time. In a study by Roediger and Karpicke (2006), students who reread a passage did better on a test five minutes later, but students who had been tested on it instead remembered more after two days (68% against 54%) and after a week (56% against 42%). In a second experiment, the students who only reread were the most confident they would remember the passage a week later, and remembered the least. Those studies used prose passages, not maths. In maths, the equivalent of being tested is solving a problem you have not just watched someone else solve.
Worked examples still have a place. Read one through, cover it, and write out the solution yourself. Then do a problem of the same kind with different numbers. The line where you get stuck is the step you were following without being able to produce, and that is the step to go back over.
Mix the problem types once each one makes sense
Textbook exercises usually come in blocks: twenty problems on the chain rule, then twenty on the product rule. In a block you already know which method to use, so the practice never asks you to choose one. An exam does. Mixing types together, called interleaving, makes you pick the method each time.
Rohrer and Taylor (2007) taught college students to find the volumes of four unusual solids. Students who practised in blocks scored 89% during practice and 20% on a test a week later; students who practised the same problems mixed scored 60% during practice and 63% on the test. That study had only 18 students. A larger one in real classrooms, Rohrer, Dedrick and Stershic (2015), gave 126 seventh-graders the same problems over three months, either blocked or mixed. On a surprise test a month after the final review session, the mixed group averaged 74% and the blocked group 42%.
Two things follow. Mixed practice feels worse while you are doing it, as the practice scores above show, so a rough session is not a sign it is failing. And mixing only works once you can do each type on its own: learn a new method in a short block, then fold it into mixed sets with the topics before it. The review by Dunlosky and colleagues (2013) rates interleaving as moderately useful, below testing and spacing, because the evidence for it was still limited: its benefits had “just begun to be systematically explored”.
Spread the same practice over weeks
The same number of problems, spread across days and weeks, is remembered for longer than the same problems done in one sitting. The review by Cepeda and colleagues (2006) analysed 317 experiments and found that the longer you need to keep something, the wider the best gap between sessions. On Consiliae the first review of a concept comes 2 days after you pass its lesson, and the gaps widen from there. To run a schedule like that on paper, see spaced repetition for maths, which sets one out day by day.
When a topic keeps going wrong, test the step under it
Here is an example of a stall that is not about the topic it appears in. Solve x/3 + 2 = x/4. The standard move is to multiply every term by 12, the lowest common multiple of 3 and 4. That gives 4x + 24 = 3x, so x = −24. Check: −24/3 + 2 = −6, and −24/4 = −6.
A student can stall at 12 × x/3. That is multiplying a fraction by a whole number, not solving an equation, and more equation practice will not fix it. Our prerequisite map records the same dependency. The concept equations with fractions sits directly on multi-step equations and multiplying fractions, and on 49 earlier concepts in all. Multiplying fractions is underneath 90 concepts on the map, so a gap there shows up in many places that look unrelated.
The practical rule: when you have failed the same kind of problem three or four times, stop and write out each step on its own. Find the first one you cannot do without help, and practise that one. On Consiliae, each review of this topic is a newly generated problem, for example x/6 − 3 = x/2 (where x = −9), so you cannot pass it by remembering an earlier answer.
Proof-based maths needs a different routine
Everything above assumes the problems have answers you can check. Once a course moves to proofs, the work changes: definitions become the thing you memorise and test yourself on, and a proof can be wrong in ways no answer key will catch. That needs its own approach, set out in how to learn proofs.
Studies checked on 27 October 2026. If you are returning to maths after a long break rather than studying a current course, learning maths from scratch covers where to begin and in what order.
