# Try another angle;Mathematics

It's that time of year again, with GCSEs looming and pupils spending more time drawing up timetables than actually revising. How can teachers help their pupils make best use of revision?

Many students are not sure how to revise GCSE mathematics. They may have followed a "spiral curriculum", meeting each topic many times, learning a little more each time. Such piecemeal experience of mathematics can result in a poor overview.

Take area. GCSE students have learned to find areas of various shapes at different times. They may do rectangles and right-angled triangles in key stage 2, parallelograms, other triangles and circles in key stage 3 and trapezia and sectors of circles in key stage 4.

For revision, they need to bring this information together and notice the two key area concepts: that area involves counting squares; and that complex shapes can be approximated by or broken into known shapes. They can then create methods for more difficult problems such as finding the area under a curve or the area of a circle segment.

Revision should involve practice in reconstructing these methods; rote-learning area formulae can provide some shortcuts, but is no substitute for mastering the underlying ideas.

The secret of helping students to revise is to prepare them in advance. Teach for understanding; let them think; emphasise the organisation of knowledge; and get the students to help themselves and each other.

Naturally, students find topics difficult to revise if they have never understood them in the first place. Unfortunately, they can get reasonable marks in class with only limited comprehension of the work, particularly if they are given exercises that are mostly identical to worked examples they have already met. When we reward routine practice with lots of ticks, we reinforce the common notion that only right answers matter. Practising is important, but skills are useless without a sense of when to use them. This sense develops with harder problems.

As teachers we profess to want our students to think. Unfortunately, we short-circuit their thinking processes. We intervene too soon, providing answers to questions they haven't asked. Sometimes, we do all the thinking, deconstructing a problem into a series of leading questions, each requiring a trivial answer. We need to remember that "the one who does the thinking does the learning" and that students develop much deeper understanding when they struggle with a variety of harder problems than when they sail through dozens of easy exercises.

All secondary mathematics teachers ought to have an overview of GCSE mathematical knowledge. They can help students by emphasising mathematical connections throughout the whole course. For example, proportionality is a recurring theme of GCSE mathematics - from working out the cost of 2.4kg of apples at 80p per kg to fractions and percentages - from scale drawing and enlarging shapes to trigonometry. Many GCSE questions ultimately involve multiplication and division, the operations associated with proportionality.

Students need structures that allow them to help themselves and each other. This means routinely presenting new topics in ways that challenge the students to apply what they already know. It means providing problems where the means of solution is not immediately obvious, generating discussion. It means routinely giving students time to think - it is easier and more effective to help a student who has got stuck with a problem than one who hasn't started it.

Revising a topic should not be the same as teaching it all again. Trigonometry, for example, is initially introduced gradually, perhaps beginning with work on the tangent ratio. Gradually sines, cosines and the inverse functions are brought in. For revision, however, the focus should be on using a diagram to select the right function - everything else is mere calculation or simple algebra. Mastery requires practice, but students need a variety of contexts and wordings to see that diverse problems share common methods of solution.

Steve Abbott is deputy head at Claydon High School, Ipswich

IT'S NOT TOO LATE

* Start with major topics: get pupils in groups to brainstorm what they know about the topic.

* Refuse to believe that they remember nothing.

* Share the results with the class.

* Ask them which parts they need reminding about.

* Give them collections of examination questions to be attempted alone and to a deadline.

* Then offer them time in class (or in revision clubs) to help each other with the questions.

* Minimise the time when they are all listening to the teacher.

* Create a sense of togetherness where students want each other to do well.

* Give them some timed tests, aiming for a standard above their predicted grades.

* Finally, use all the help you can get. Encourage students to buy one of the many revision guides. Welcome the support of parents and private tutors. Use colleagues who have built up resource banks.

* Remember to think about next year's revision now!

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