How to Study for a Math Exam: Practise, Do Not Reread

To study for a math exam, spend most of your time solving problems without looking at the solution, keep a log of every mistake and what caused it, and practise under exam conditions before the real day. Rereading your notes and watching someone solve a problem both feel like progress, but neither trains the thing the exam measures: starting a problem yourself and carrying it through.

Maths is different from most subjects you study. In history or biology, a lot of the work is remembering facts. In maths, remembering matters, but the main skill is doing: choosing a method, setting the problem up and working it out without slipping. A skill is built by practice, in the same way that you cannot learn to play the piano by watching videos.

This guide gives you a practice-first plan. It works for school maths, calculus, statistics, algebra and most university courses with problem sets.

Why reading a solution feels like understanding

Take a worked example from your textbook. You read it line by line and every step makes sense. You close the book feeling that you have got it. Then you meet a similar problem in the exam and cannot begin.

What happened is that following a solution is a recognition task: each step is checked against what you can see. Solving a problem is a production task: you have to decide the first step with nothing in front of you. The first one is easy to feel good about, and the second one is what the exam grades. Psychologists call the mismatch an illusion of knowing.

The research agrees on which habits help. In their review of study techniques, Dunlosky and colleagues (2013) found that practice testing and distributed practice were among the most effective, while rereading and highlighting were not. For maths, practice testing mostly means this: solving problems you have not seen the solution to. Rohrer and Taylor (2007) also found that students who mixed different kinds of problems in practice did far better on a later test than students who practised one kind at a time.

So the plan is built on one rule. Whenever you can, produce instead of read.

Step 1: Map what the exam covers

Start by writing a list of every topic that could be on the paper. Use the syllabus, your teacher's review sheet, your class notes and the chapter titles in your textbook. Then rate each topic honestly.

• Green: I can do problems on this without help.
• Orange: I can do the easy ones, but I get stuck on harder ones.
• Red: I do not know where to start.

Most students spend their time on green topics because it feels good. Do the opposite. Your time should go mostly to orange and red, with short reviews of green ones. Check the rating again after a few days, and move topics between the groups as they improve. If you are not sure how much time you have, build a plan with our guide to making a revision timetable.

Step 2: Fill the gaps with the smallest amount of reading

For a red topic, you do need to learn something before you can practise. Keep that stage short.

Read the relevant section of the book or your notes, or watch one explanation, and then go straight to problems. When you meet a worked example, use it actively. Read the problem, then cover the solution and try the first step. Check it. Try the next step. If you get stuck, uncover only a line, then continue yourself. Tutors call this the cover-and-try method, and it turns a passive example into a mini test.

If a topic does not make sense, go back one level. A student stuck on integration by parts may really be stuck on differentiation rules. Fix the earlier layer first, because the new topic will keep failing otherwise.

Step 3: Do problems, and choose them carefully

Practice is the centre of the plan, and the quality of the problems matters more than the quantity. You do not need to do all of the assigned exercises. You need the right ones.

• Start from the exam. Past papers and teacher-provided samples show the format, the difficulty and the way questions are worded. Our guide to using past papers explains the order to follow.
• Pick a spread of difficulty. For each topic, do a couple of easy problems to confirm the method, then several medium and a few hard ones.
• Do problems from scratch. Do not look at the solution until you have made a serious attempt. A struggle of a few minutes is not wasted. The effort of trying is part of how the learning sticks.
• Write full solutions. Write out the steps as you would in the exam, including the formula you start from and the units. Skipping steps in practice leads to skipping them under pressure.
• Mix topics. Once you have worked through each topic, shuffle problems from several chapters. It feels harder because you have to decide which method fits, but that decision is half of the exam. See our guide to interleaving for how.

A good session is short and focused. Forty-five minutes of full attention on six to ten problems beats three hours of drifting. If you struggle to start, our advice on how to stop procrastinating while studying can help.

Step 4: Keep an error log

This is the habit that separates students who plateau from students who keep improving. Whenever you get a problem wrong, or only right after peeking, write an entry in a notebook or document. Each entry has four parts.

• The problem (or a reference to it, so you can find it again).
• What went wrong. Be specific: "forgot to flip the inequality when dividing by a negative".
• The type of error. A concept gap, a wrong method, an algebra slip, a misread question or a careless arithmetic mistake.
• The fix. The rule or step you need to remember next time.

After a week, read the log. Patterns appear quickly. Maybe most of your errors are sign errors, which you can cut by writing each line more slowly. Maybe they are all in one topic, which tells you where to study. Or maybe they come from misreading the question, which is solved by underlining what is asked before starting.

Redo the problems in the log a few days later, with no peeking. If you get them right, they are learned. If not, they go back into the rotation. Our guide to checking your own answers covers how to mark yourself honestly.

Step 5: Memorise only what you must, and test it

Even in a problem-based subject, some things must be known by heart: key formulas, definitions, standard values, theorems and the shape of common graphs. Check what your exam provides. If you get a formula sheet, learn which formula fits which type of problem. If you do not, you need to know them.

Keep this part small and efficient:

• Write each formula on a card with a prompt ("area of a sector", "derivative of a quotient") and test yourself from memory.
• Learn what the symbols mean and when the formula applies before you memorise it.
• Review in short sessions spaced across several days.

Our guide to memorising formulas gives the full routine. Do not let this stage grow. Knowing the formula is only worth something if you can use it, so most of your week still goes to problems.

Doing it with Orbily

Gathering the formulas from a long chapter is one of the slower parts of preparation. In Orbily, you can import your notes or a file, and the Formulas tool collects the formulas from the course into a list, each with a short explanation of what it means and when to use it. If you open it from a chapter, that chapter becomes the source and there is nothing to re-enter. You can then test yourself on that list. The tool does not solve problems or do the practice for you, so the problem solving, the error log and the mock exam still have to be done by you. You can read more about the Formulas tool.

Step 6: Rehearse under exam conditions

At least once, and ideally two or three times, take a full practice paper as if it were the real thing. Use a timer, put your notes away, and use only what is allowed in the exam. Do it in a quiet place, at the time of day the exam will be held if you can.

This does several things at once. It shows whether your pace is right. It reveals gaps that problem-by-problem practice hides. It lets you try out your strategy for hard questions, and it makes the real exam feel familiar. Mark it using the official mark scheme if you have one, and add every error to the log.

Step 7: Plan the schedule backwards from the exam date

A realistic plan for most exams looks like this.

• Two to three weeks out: map the topics, learn the red ones, and begin daily problem sets. Start the error log.
• One to two weeks out: shift to mixed problem sets and past-paper questions. Review the log every few days.
• The last week: take one or two full timed papers, fix the patterns they reveal, and run through your formula cards.
• The day before: light work only. Redo a few log problems, check your equipment, and sleep. Do not start new topics.

If you have less time, the same structure applies on a smaller scale. Our plan for revising for an exam in a week shows how to cut it down.

During the exam

• Skim the whole paper first and start with questions you can do, to build confidence.
• Underline what is asked and write down what is given. Many errors come from solving a slightly different question.
• Write your steps clearly. Method marks are often awarded even when the final answer is wrong.
• Check with a quick sanity test. Does the sign make sense? Is the size plausible? Do the units match?
• Do not get stuck. If a question takes too long, write what you have, mark it, and come back.
• Leave time at the end to re-read the questions you were unsure about and to check the arithmetic on the ones worth many marks.

Common mistakes when studying for a math exam

• Rereading notes and solutions instead of solving problems.
• Practising only what you already know, because it feels good.
• Looking at the solution too early, which cancels the benefit of the struggle.
• Skipping the error log, and repeating the same slips on the day.
• Cramming the night before. Maths needs several days of practice for the methods to become automatic.
• Never working under time pressure. Knowing how to solve a problem is not the same as solving it in four minutes.
• Memorising worked examples instead of understanding the method. The exam will change the numbers and the wording.

Final thoughts

You do not need to be naturally good at maths to do well in a maths exam. You need a method that makes you practise the right thing: producing solutions, learning from each mistake, and rehearsing the real conditions. Start today by rating your topics, picking the weakest one, and solving five problems on it without looking at the answers. Write down what went wrong, and try again in two days.