How to study math: do problems, mix them up, and learn from mistakes

Watching your teacher solve a problem, or reading a worked solution, makes a method feel easy. The real test is whether you can do it from a blank page.

• Cover the solution and try the problem yourself first.
• If you're stuck, look at one step of a worked example, then cover it again and carry on.
• After finishing, try a similar problem without help to check the method has stuck.

Textbook exercises often give 20 problems of the same type in a row. That's useful when you first learn a method, but it hides the hardest part of a test: deciding which method a question needs.

In the Rohrer et al. (2020) trial, 787 students practised either in blocks (one type at a time) or interleaved (mixed types). On the unannounced test, the mixed-practice group scored 61% against 38%. Mixing feels harder while you practise, and that difficulty is part of why it works.

Once you can do each type, build mixed sets: a fraction problem, then an equation, then a percentage question, then an area problem.

Every mistake is information. Keep a page (or note) with three columns:

A few days later, redo the problems in your log without looking. The mistakes you keep making are the best guide to what to practise.

Doing 30 minutes of maths on four different days works better than two hours in one go. Each time you come back to a topic it's slightly harder to remember, and that effort of recalling is what makes it last. Start each session with two or three quick questions on older topics before moving to new work.

As you solve a problem, say or write why each step is allowed: "I can divide both sides by 4 because it keeps the equation balanced." If you can't explain a step, that's the part to ask about. Self-explanation was rated a moderately useful technique in the Dunlosky review, and it's especially valuable in maths, where understanding why lets you tackle unfamiliar questions.

Many maths questions are lost to wording rather than method. Make sure you know what words like integer, factor, product, expression, equation and simplify mean, and what notation such as $f(x)$ or $\leq$ is asking for. A short list of definitions you test yourself on goes a long way.

• Estimate first: roughly what size should the answer be?
• Substitute back: if you solved an equation, put your answer back in and see if it works.
• Check units and signs: a negative length or a speed of 2,000 m/s for a cyclist means something went wrong.
• Show your working: it makes mistakes easier to find, and in exams it can earn marks even if the final answer is wrong.

Being stuck for a few minutes is useful: that struggle helps learning. Being stuck for half an hour usually isn't. Set yourself a limit, then ask your teacher, a friend or a tutor about the specific step you can't do. An AI tutor such as TutorAura's AI math tutor is useful here: it explains one step at a time and asks you to do the next one, rather than handing over the answer.

For a student with a test next week, about 30 minutes a day:

For general advice on preparing for any test, see how to study for a test. For specific topics, try our free fractions and linear equations guides, which include practice questions with instant feedback.