Studying maths: the techniques that actually work

Re-reading your notes feels productive and does almost nothing. Here is what the evidence says about revising maths, and how to build a week of practice that actually moves your grade.

Almost every student who is disappointed on results day revised hard. The problem is rarely effort — it is that the most comfortable ways to revise are the least effective ones. Reading through worked solutions, highlighting notes and watching someone else solve a problem all produce a strong feeling of understanding without producing the ability to do it yourself under exam conditions.

1. Retrieval, not recognition

Recognising a method when you see it is a much weaker skill than recalling it when the page is empty. Close the book, write down everything you know about a topic, then check what you missed. The gaps you find are the whole point — they are the questions you would have dropped marks on.

• Cover the solution before you start, always. Every time.
• Write the method from memory first, then check it against your notes.
• Anything you could not recall goes on a list to attempt again in two days.

2. Space it out

Four thirty-minute sessions across a fortnight beat one two-hour session, even though the total time is identical. Forgetting a little between sessions is what makes the next recall effortful, and effortful recall is what sticks. Plan to revisit a topic three times: a few days after you learn it, a fortnight later, then again the month after.

3. Mix topics in one session

Doing twenty differentiation questions in a row teaches you to differentiate. It does not teach you to notice that a question needs differentiating — and in the exam nothing is labelled. Once you can do a method reliably, start mixing: a few questions on integration, then vectors, then trigonometry, shuffled together.

4. Work from examples, then take the scaffolding away

When a topic is genuinely new, studying a fully worked example is more efficient than struggling from nothing. The mistake is staying there. Work through one example in full, then do the next one with the solution covered, then do a third with no support at all.

Take differentiating from first principles. Read this once, then reproduce it with the page covered: $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.$$ For $f(x)=x^2$ this gives $$\frac{(x+h)^2-x^2}{h}=\frac{2xh+h^2}{h}=2x+h,$$ and letting $h\to 0$ leaves $f'(x)=2x$. If you cannot rebuild those three lines without looking, you do not know it yet — and that is a question the examiner is entitled to ask.

5. Keep an error log

This is the highest-value habit on the list and almost nobody does it. Every time you drop a mark, write one line: what the question was, and what specifically went wrong. Not "careless" — that is not a diagnosis.

• Sign error when expanding a bracket — a method you can drill.
• Did not know the identity — a knowledge gap you can close in ten minutes.
• Knew it but ran out of time — a pacing problem, not a maths problem.
• Misread the question — usually a habit, and habits respond to practice.

After a fortnight the log tells you exactly where your marks are going, and the three or four patterns it exposes are worth more than another past paper.

6. Practise under real conditions

A paper done over an afternoon with your notes open measures something, but not the thing you are graded on. At least once a fortnight, sit a full paper to time, in one go, with only the equipment you get in the exam. Speed is a skill of its own and it only improves under pressure.

Putting it together

• Every session starts with three questions from a topic you studied last week, cold.
• New material: one worked example, then two questions with no support.
• Every dropped mark goes in the error log with a specific reason.
• One timed paper a fortnight, marked honestly against the mark scheme.