• Mathematics

Maths is a doing skill, and reading solutions feels exactly like learning

There is no subject where the gap between feeling prepared and being prepared is wider. You read the worked solution, every line follows from the last, nothing surprises you, and you close the book confident. Then the exam shows you a problem with no solution attached, and you discover that following an argument and constructing one are entirely different capacities.

6 min readSubjects

The only real measure

Can you solve the problem on a blank page, from memory, with nothing in front of you? That is the only question, and every other feeling you have about your preparation is noise.

'I understand it' is not the standard. Understanding a solution is a low bar that reading clears easily and exams do not accept.

Practise choosing, not just executing

Most maths revision is done in blocks: twenty integration-by-parts problems in a row. After the first one, you already know the method — the exercise told you. So you practise executing a technique you were handed, and never practise the thing the exam actually tests, which is *deciding which technique*.

Interleave: mix problem types so every question forces a decision before it forces a calculation. Your practice accuracy will drop and your exam performance will rise. That trade is the whole of learning science in one sentence.

The routine

  1. Attempt before you look. Always. Even a failed attempt makes the solution far more instructive — see self-explanation.
  2. When stuck, look at one line only. Then close it and continue alone. Reading the whole solution is how you learn nothing.
  3. Redo problems you got wrong, days later. Immediately afterwards you're pattern-matching on the solution you just read, which measures nothing.
  4. Automate the basics past boredom. If standard derivatives take conscious effort, they're eating the working memory you need for the actual problem — and you'll conclude you're bad at the problem.

Common questions

Why can I follow maths in class but not do the problems?

Because following and generating are different skills, and watching a solution only practises the first. The choice of method — the part that isn't written down — is what exams test.

How many practice problems should I do?

Fewer, mixed, and properly reviewed beats many, blocked, and skimmed. Twenty problems of the same type teaches you almost nothing after the first few.

Should I memorise formulae?

Automate the common ones to the point of boredom — a formula you have to reconstruct is consuming working memory you need for the problem. Spaced repetition is ideal for this.

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