Free resources

Things worth doing between sessions.

No sign-up, no email capture. Just the handful of things I would actually tell a student to do, and the places I would send them.

The core technique

How to vary a problem

This is the single most useful study habit in mathematics, and almost nobody is taught it. When you finish a question, do not move on. Break it.

Start with something ordinary

Take a question you have just answered. It does not need to be hard. Say:

Solve x² − 5x + 6 = 0

You factorise, get x = 2 and x = 3, and normally that is the end of it. Instead, run it through the six moves opposite. Each one takes a couple of minutes and tests something the original question did not.

If any of them stops you, you have found something real — and it was hiding underneath a question you thought you had finished.

  • 1 · NudgeChange one number. Make the 6 a 7. Why can you no longer factorise neatly, and what does that tell you about the discriminant?
  • 2 · ReverseI give you the answers, you build the question. Roots of 2 and 3 — now write a different equation with those roots.
  • 3 · GeneraliseReplace a number with a letter. For which c does x² − 5x + c = 0 have two roots? One? None?
  • 4 · Re-dressPut it in a context. If y is the height of a ball, when is it on the ground and when is it highest?
  • 5 · MixCombine it with an older topic. Where does this curve meet y = 2x? Where is its gradient zero?
  • 6 · BreakSomeone answers x = 2 or x = −3. Find their error without solving it yourself.
Why this works

Answering the same question type twenty times trains recognition of that question type. Varying one question six ways trains the understanding underneath — which is what an unfamiliar exam question is designed to test. Ten varied minutes beat an hour of repetition.

Study habits

Four things that actually move the needle

Interleave your practice

Do not do twenty questions on one topic. Do five each from four topics, shuffled. It feels worse and works considerably better, because choosing the method is the skill the exam is testing — and a topic-labelled exercise removes that choice entirely.

Test yourself, do not re-read

Re-reading notes produces a feeling of familiarity that is easily mistaken for knowledge. Close the book and attempt the problem cold. Getting it wrong from memory teaches you more than getting it right with the page open.

Keep an error log

One page. Every mistake, and one line on why — not “careless”, but what actually happened. Sign errors when expanding brackets. Forgetting to check the domain. Patterns emerge within a fortnight, and they are usually fixable in an afternoon.

Explain it to somebody

A parent, a friend, an empty room. The moment your explanation gets vague is the moment your understanding does. It is the cheapest diagnostic there is and it costs five minutes.

Where to go next

Genuinely good places on the internet

All free. I have no relationship with any of them.

For enrichment and problem solving

  • NRICHnrich.maths.org — Cambridge’s problem collection, from primary to sixth form. The best free resource in British mathematics education.
  • Undergroundundergroundmathematics.org — A level material presented as connected ideas rather than a checklist. Especially good for the GCSE-to-A-level jump.
  • Project Eulerprojecteuler.net — mathematical problems that need a bit of programming. Excellent if a student likes computers as much as maths.

For admissions preparation

  • TMUAOxford’s TMUA page — the authoritative source on the test that replaced the MAT, with past papers and current dates.
  • STEPmaths.org/step — the STEP Support Programme from Cambridge. Free, structured, and genuinely the best preparation available at any price.
  • MAT archiveOld MAT papers are still worth doing as problem practice even though the test is retired — just do not treat them as a guide to the current format.

For watching and reading

  • 3Blue1Brown3blue1brown.com — the finest mathematical visualisation being made. The linear algebra and calculus series are the ones to start with.
  • Quantaquantamagazine.org — serious mathematics journalism that a sixth-former can follow. Where a lot of my news feed starts.
  • Numberphilenumberphile.com — short films with real mathematicians. Reliably makes people curious.
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