How to Learn Maths

If maths is not working, more practice at your current level is usually wasted. The problem is nearly always further back, and practice on top of a gap reinforces the workaround rather than fixing the cause.

The way to find it is to work backwards. Take something you cannot do and ask what it assumes. Then ask what that assumes. Keep going until you reach something you are completely solid on, and start one step above it. This is uncomfortable because it means doing work that feels beneath you, and it is the fastest route by a wide margin.

In practice the gap is fractions surprisingly often, and negative numbers second. Both are where the rules learned for whole numbers stop applying, and both are usually taught quickly.

Most people first meet the equals sign as an instruction meaning here comes the answer. That reading works throughout arithmetic and breaks completely in algebra.

An equation asserts that two things are the same. Everything you are allowed to do to it follows from that: whatever you do to one side you must do to the other, because otherwise they stop being the same. Students who hold the instruction reading experience algebra as a set of arbitrary moves to memorise, and it is the single most common reason algebra feels impossible.

If that description sounds familiar, spend an hour on what an equation is before touching another equation.

Working through a worked example and following every step feels like learning and largely is not. You are checking comprehension, which is easy, rather than retrieval, which is what an exam demands.

The version that works is closing the book and attempting the problem cold, getting stuck, and only then looking. Being stuck is the mechanism, not a sign the method is failing. It is uncomfortable, it feels less productive, and it produces substantially more learning per hour, which is one of the best-established findings in the psychology of learning.

Mixing problem types matters too. Ten questions of the same kind lets you stop reading the question after the first two. The same ten shuffled among other types forces you to work out what kind of problem it is, which is most of the difficulty in any real assessment.

The habit that catches the most errors is deciding roughly what the answer should be before working it out. It costs a few seconds and turns a wrong answer from something you hand in into something you notice.

This is particularly valuable with percentages, where slips produce answers off by a factor of ten that look perfectly plausible on paper. If you know a 15 per cent tip on 80 is somewhere near 12, an answer of 120 stops being a possibility.

If you will only master one area of school mathematics for practical use, make it this one. Recipes, maps, exchange rates, discounts, scale models, unit pricing, mixing anything with anything - all one idea.

It is also the area where small errors are most expensive in daily life. A percentage increase followed by the same percentage decrease does not return you to where you started: up twenty then down twenty leaves you at ninety-six per cent. That single fact is worth more than a term of geometry to most people.

Probability is the branch where trained intuition is most reliably incorrect, and this is not a failure of intelligence. Human intuition about chance evolved for a different set of problems.

A coin that has landed heads nine times running is exactly as likely to land heads again. Feeling that tails is due is universal and wrong. Accepting that your instinct is unreliable here, and calculating instead, is the whole skill.

The claim worth taking seriously is that the difference between people who find maths easy and people who find it impossible is almost entirely a difference in whether their gaps got caught. Not a difference in capacity.

That is good news, because gaps are findable and fixable and capacity would not be. It is also inconvenient news, because it means the fix is going backwards, which nobody wants to do.