How to Revise Maths Without Just Rereading Worked Examples
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Maths revision has a trap that other subjects do not. A worked example, read line by line, makes complete sense, every step follows, and the reader closes the book certain they could do it. Then the exam asks a question that is the same idea in different clothes and the method is nowhere to be found. The gap between following a solution and producing one is the whole difficulty, and it is the thing revision has to close.
Why reading solutions fails
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.
Attempt before you look
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.
Mix the problem types up
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.
Keep an error log
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.
Rehearse the conditions, not just the content
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
The takeaway
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.