How to Revise Maths Without Just Rereading Worked Examples

When you read a worked solution, the hard part has been done for you: deciding which method applies. The steps then look inevitable, because they are being shown in the right order with the right technique already chosen. What the exam tests is exactly the part the solution skipped, the moment of looking at an unfamiliar question and recognising which tool fits.

Psychologists call the feeling of understanding after reading a solution the illusion of competence, and it is stronger in maths than almost anywhere, because a correct proof is so persuasive. The fix is not to avoid worked examples but to change when you look at them.

Cover the solution and try the problem cold, even if you are fairly sure you cannot do it. Write down what the question is asking, what you know, and any method that might apply, and push as far as you can. Then uncover the solution and compare. The attempt, even a failed one, does two things: it tells you precisely where your knowledge stops, and it makes the solution's key step stick, because you now have a question that the step answers.

Research on what is called the generation effect finds that a failed attempt followed by the answer produces better retention than reading the answer straight away. The struggle is not wasted time before the learning. It is the learning.

Textbooks group problems by method: twenty quadratic equations, then twenty simultaneous equations. Working through a block like that trains you to apply one method twenty times, and never to choose between methods, which is what the exam asks. The alternative is interleaving: shuffle problems from several topics together so that each one has to be identified before it can be solved.

Interleaved practice feels worse while you do it. Scores during the session are lower and the sense of fluency is gone. The same studies find that a week later the interleaved group remembers and transfers far more. If your textbook will not mix for you, take one problem from each of five chapters and do them in random order, or work from past papers, which are interleaved by nature.

Every mistake belongs to a type, and most people make the same few types over and over. Dropped minus signs. Forgetting to square both terms in a bracket. Misreading which quantity the question wants. Keep a page where each wrong answer is recorded as a category rather than a problem, and read it before the next session. After a few weeks the page is a personal list of the traps you fall into, which no textbook can provide, and the act of naming an error makes it much easier to catch the next time.

The same page should record the questions you could not start, with a note of what the first step turned out to be. Those first steps are the real content of revision: not the algebra, which you can probably do, but the recognition that this question is a disguised version of that method.

Exam maths is done in silence, against a clock, without notes, and with the pressure of a mark. None of those conditions are present at a desk with a textbook open, and skills do not transfer perfectly between conditions. In the last weeks before an exam, some sessions should be whole past papers, timed, marked honestly against the scheme, and reviewed with the error log. That is where you discover that you know the method but take too long, or that you always skip the last part of the question. A weekly routine that works:

• •Attempt each worked example cold before reading its solution
• •Do a mixed set of ten problems from five topics, in random order, every session
• •Log each error by type and reread the log before starting
• •Redo yesterday's failed problems before starting new ones
• •One timed past paper a week, marked against the scheme, errors logged

A technique called self-explanation turns a worked example from something you read into something you do. After covering and attempting a problem, go through the solution one line at a time and say, aloud or on paper, why each step is allowed and why it was chosen. Not just that the equation was divided by two, but that dividing by two isolates the quantity the question asks for. Studies of students learning from examples have repeatedly found that those who explain the steps to themselves learn more than those who simply read the same material.

Suppose a solution factorises x squared minus 5x plus 6 as (x minus 2)(x minus 3). The explanation might run: I need two numbers that multiply to 6 and add to minus 5, so both must be negative, and minus 2 and minus 3 work. Wherever you cannot give a reason, you have found a gap, and that is worth more than a page of lines you only nodded at. Explaining to another person works even better, because a listener will ask exactly the why question you were hoping to skip.

Many lost marks come from answers that could have been caught in seconds, so build checking into practice rather than saving it for the end of the exam. The quickest check is substitution: if you solved an equation and got x equals 4, put 4 back into the original and see whether both sides match. For a pair of simultaneous equations, test the answer in the equation you did not use in the final step. A second check is estimation. If you are working out 19.8 multiplied by 5.1, you know the answer should be close to 100, so 1,009.8 is obviously wrong.

Other checks suit particular topics. A quick sketch shows whether a triangle's missing side should be longer or shorter than the others, and whether a probability above 1 or a negative length has crept in. Units are a check in themselves: an area must come out in square units and a speed in distance per time. Reading the question once more to see what it actually asks catches the common slip of finding a radius when the diameter was wanted. Practise these checks during revision until they are automatic, because under exam pressure you only do what has become habit.

The Hungarian mathematician George Polya spent much of his career on the question of how people find solutions, and his short book How to Solve It, published in 1945, is still a classic guide to being stuck. He divided problem solving into four stages: understand the problem, devise a plan, carry out the plan, and look back. The stage students most often skip is the last. Looking back means asking whether there was a quicker route and whether the method could solve other problems, which is exactly how one question becomes a tool for the next.

For the moment of being stuck, Polya suggested questions to ask yourself, and one of the most useful is whether you can solve a simpler version. Suppose ten people each shake hands once with everyone else. How many handshakes are there? Try two people: one. Three: three. Four: six. Each newcomer shakes hands with everyone already there, so adding the tenth person adds nine, and the total is 1 plus 2 and so on up to 9, which is 45. Other prompts work the same way: draw a diagram, work backwards from the goal, or recall a problem with the same unknown.

Most maths exams in Britain are marked with schemes that split each answer into different kinds of mark, and knowing them changes how you write. Method marks, labelled M, are given for a correct approach even if the arithmetic later goes wrong. Accuracy marks, labelled A, are given for correct answers and usually depend on the method mark that leads to them. B marks are independent, awarded for a correct result however it was reached. Many schemes also allow follow-through: if you use your own wrong answer from part a correctly in part b, the later marks can still be earned.

Picture a four-mark question on simultaneous equations whose scheme gives one method mark for a correct elimination, one for substituting back, and two accuracy marks for the values of x and y. A student who writes only a wrong final answer scores nothing. A student who shows a correct elimination but drops a minus sign may still collect both method marks and lose only the accuracy marks. So practise the habits this rewards: write each step on its own line, never cross out working you have not replaced, and when marking a past paper, use the scheme so you see exactly which kind of mark you are dropping.

Maths revision works when it makes you choose and produce methods rather than follow them: attempt problems before looking at solutions, mix topics so that every question has to be recognised, keep a log of your own error types, and practise under timed conditions. Reading worked examples is the last step of learning a method, not the first.