Effective Note-Taking for Maths

Most maths notes are useless, and the students who write them are usually the conscientious ones. That is what makes this worth writing about.

A typical maths copy contains worked examples copied from the board, neatly, in full. Come revision time, the student reads back through them, follows every line, feels reassured, and learns almost nothing. The reading felt productive. Nothing was retrieved from memory, so nothing got stronger.

The problem is not effort. It is that maths notes are usually a transcript of a lesson, when what you need is a record of what you did not already know.

Why copying the board fails

Two reasons, and both matter.

Following is not producing. When you read a worked solution, each line follows sensibly from the last, and that generates a strong feeling of understanding. That feeling is largely unrelated to whether you could produce the work on a blank page. This is the most reliable illusion in maths study, and it catches hard-working students hardest.

Everything gets recorded equally. The step you found obvious and the step you did not are written down in the same handwriting with the same emphasis. Six weeks later, nothing tells you which was which, so revision means re-reading everything, which is slow and mostly wasted on material you never had trouble with.

Good maths notes are selective and asymmetric. They record the difficulty, not the procedure.

What to actually write during a lesson

The shift is from transcription to annotation.

Write the method as steps in your own words, once. Not the example, but the method. "Get everything to one side, factorise, set each factor to zero." Four lines that generalise beat two pages that do not.

Write down why a step happens, when you find out. The line you would not have thought of yourself is the valuable one. Why did we multiply by the denominator there? Why that formula and not the other? If a teacher explains a choice, that explanation is worth more than the arithmetic that follows it.

Mark the moment you got lost. A star in the margin at the exact line where you stopped following. This takes half a second and it is the single most valuable thing in the copy, because it is the only record you will ever have of what you actually need. Without it you will remember only that the topic "was confusing".

Note the trap. When a teacher says "this is where people go wrong", write it down and box it. Those sentences are distilled from years of corrected papers.

Do not copy anything you already understand. It feels wrong. It is the whole point.

If you are working from recorded lessons, this is easier, because you can pause, attempt the next step yourself, and only write down where you diverged from what the tutor did. That divergence is your note.

The two pages worth more than the copybook

Two documents do most of the work of revision, and they take about ten minutes a week to maintain.

The error log

One page per subject area. Every time you lose a mark, one line: what the question wanted, what you did instead.

Not "algebra mistakes". Something specific enough to act on: "dropped the sign on the second term inside the bracket".

Within a fortnight you will have something no textbook can give you: a list of your recurring errors, which is usually four or five items rather than the dozens students expect. That list is the highest-value revision material you own, and in the final week before an exam it is the only thing worth working from.

The method sheet

One A4 page per topic, and the constraint matters: it must fit on one page. For each topic, the steps of each method in your own words, the traps, and the location of the relevant page in Formulae and Tables.

Because the formulas are provided in the exam, this page should not be a formula list. It should be a decision guide: when do I use the cosine rule rather than the sine rule, what tells me this is a conditional probability question, which test shows this is a maximum.

Making it is the exercise. Compressing a topic to one page forces you to decide what actually matters, which is a far better use of an evening than re-reading a chapter.

Notes are not revision

Worth stating plainly, because a lot of study time disappears here.

Making notes is preparation for revision. Revision is attempting questions without the solution visible. A student who spends six evenings producing beautiful summaries and two attempting problems has the ratio exactly backwards.

The test for any note you have written: does this help me when I am stuck on a question? If yes, keep it. If it is a copied example you can follow but not reproduce, it is not doing anything.

Practical matters

Handwrite the maths. Typing mathematical notation is slow and awkward, and the physical act of writing appears to help retention. Type the prose if you prefer, but the working belongs on paper.

One copy for classwork, one for the method sheets and error log. The second one is short, and short is what makes it get opened.

Date everything. When a topic reappears in a later year, and it will, you want to find what you wrote the first time.

Leave margins. You will want to add the thing you did not understand until three weeks later, and it belongs beside the original, not forty pages further on.

The test that settles it

Take a topic you covered a month ago. Close everything. On a blank page, write the method from memory, then attempt a question on it.

Then open your notes and see whether they contained what you actually needed.

For most students the first time, the answer is no. The notes contained the example, and what was needed was the decision behind it. Fixing that changes what you write for the rest of school.

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Every Summit lesson is recorded and indexed by topic, and our notes are written as method-and-trap summaries rather than transcripts, so students can annotate rather than copy. Our guides to balancing maths with other subjects and setting exam goals cover the rest of the study picture.

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