Educational philosophy

Principles, not procedures.

A procedure is a set of steps that works on the questions it was designed for. A principle is an understanding of why those steps work — which means you can rebuild them, adapt them, or spot when they do not apply. Only one of those two things survives contact with an unfamiliar question.

The problem

Why able students hit a wall

A student can score well for years on procedural fluency alone. They have memorised the recipe for each question type, and the questions arrive labelled by chapter, so the right recipe is always obvious.

Then the questions stop being labelled. GCSE gets to its harder end, A level starts mixing topics, an admissions test asks something genuinely novel — and suddenly there is no recipe to reach for. The student has not got worse. The scaffolding was always missing; it just was not load-bearing until now.

The usual response is to do more practice papers. If the underlying understanding is not there, that mostly builds a bigger library of recipes. It is exhausting, and it does not generalise.

The method

How a session actually runs

  1. We start with a real problem from your course

    Not a warm-up, not a worksheet I brought along. Something from the actual specification you are being examined on, ideally something that went wrong recently. It keeps the work anchored to what you need, and it tells me quickly where the real gap is.

  2. I break it down until it is obvious

    This is the part I care most about. If an explanation has not landed, my assumption is that it was not broken down far enough — not that you are missing something. So we go smaller. And smaller again, if needed. There is no point at which I will suggest you should already understand this.

  3. Then we vary it

    Once the original problem is solved, it becomes raw material. Change a sign. Remove a constraint. Ask for the general case. Ask what would have to be true for the answer to be zero. Each variation probes whether the understanding is real or whether a procedure has just been reproduced.

  4. And we mix it with older work

    Topics are taught in blocks and then examined all together. So we deliberately interleave: today’s quadratic meets last term’s trigonometry. This is uncomfortable at first and it is the single highest-value thing you can do. It is also well supported by the research on how memory actually works.

  5. You explain it back

    The last few minutes are you teaching me the idea. It is the fastest way to find the parts that are still fuzzy, and it is what turns “I followed that” into “I own that”.

Why it works for exams too

Understanding is not the slow route

There is a suspicion that teaching for understanding is a luxury — lovely in principle, but that with an exam in ten weeks you should just drill past papers.

I do not think that trade-off is real. Examiners deliberately write questions that look unfamiliar, precisely to separate students who understand from students who have memorised. A student who knows why a method works can reconstruct it under pressure. A student who has only memorised it has nothing to fall back on when the question is dressed differently.

So we do practice papers — plenty of them, under timed conditions, close to the exam. But we do them on top of a foundation, not instead of one.

The test I use

Can you meet a question you have never seen before and know what to reach for?

If yes, the topic is done. If no, we have not finished, whatever the mark scheme says.

And beyond the exam

I spent fourteen years using this maths commercially — forecasting, pricing, risk, optimisation. Almost none of it arrived labelled with which method to use. The ability to look at an unfamiliar situation and recognise the structure underneath is the whole job. It is worth building early.

Plainly

What I will and will not do

I will

  • Go at whatever pace the understanding actually needs.
  • Tell you honestly where you are, including when that is uncomfortable.
  • Send you the session notes and a short set of varied problems afterwards.
  • Say so if I think you would be better served by someone else.
  • Teach adults with exactly the same patience as fifteen-year-olds.

I will not

  • Imply that you should already have understood something.
  • Sell a fixed package of hours you may not need.
  • Promise a grade. Nobody honest can.
  • Do your coursework for you.
  • Drill past papers at a student whose foundations are not yet there.
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