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Lack of proof is evidence of decline;Secondary

An ability to demonstrate proof lies at the heart of rigorous mathematical thinking. But a new survey suggests it is beyond many GCSE students, reports Victoria Neumark

Pupils in England are not learning the core principle of mathematics, according to a new study from the Economic and Social Research Council.

Celia Hoyles, professor of mathematics at the Institute of Education, University of London, and Lulu Healy, a research officer at the institute, surveyed 2,500 high-attaining Year 10 pupils to see if they could demonstrate a grasp of mathematical proof. Despite expecting good GCSE maths results, most of them failed to provide convincing proofs for algebra and geometry problems. The results of the survey, released this week, are "quite sad", says Professor Hoyles.

A report by the London Mathematical Survey in 1995 deplored the removal from the national curriculum of the requirement to deal with proof. The institute's study seems to confirm that if proof is not formally taught pupils will not pick it up for themselves.

Professor Hoyles says proof is the rigorous heart of mathematical thinking. Students learn not just a formal language and how to analyse, but also how mathematics is founded on rules that are verifiable within a chosen domain, and on accuracy. They learn how thinking is tested by example and counter-example, about induction and deduction. Proof is the skeleton of maths, not an added-on extra, and pupils must learn it in order to see and reflect on their own thinking at any level above calculation and computation.

The pupils in the study were asked a series of multiple choice and problem-solving questions in both algebra and geometry. The results in geometry were "quite awful", says Professor Hoyles. The students could not apply logical deduction to geometrical problems, and their ability to analyse was hamstrung by having no formal language in which to display their working. They did better in algebra, Professor Hoyles believes, because they are accustomed to number investigations in which they have to back up extrapolations.

The study is "not entirely a disaster story", she says, but only because students are good at investigating and reporting, in line with the national curriculum target, "Using and Applying Mathematics". Pupils are, however, "extremely bad" at building on this to support a formal argument.

A sophisticated breakdown of responses to problem-solving questions revealed no variation according to teacher qualification or training, class, location or type of school (though no independent schools were included). Girls were significantly better at algebraic proof. Some pupils did, however, score much better: those from schools which had a majority of pupils in the set entered for the higher tier at GCSE, andor which gave a greater share of the timetable to maths. The question this begs, says Professor Hoyles, is how "a better mathematical atmosphere" affects pupils' ability to think.

The bigger question for maths educators is whether, by removing proof from the national curriculum before 16, too much of the ambiguity and challenge on which the more able pupils thrive - and which they need to progress to further maths - has been removed.

Many maths educators share Professor Hoyles's view that mathematics at GCSE is simply "not rigorous and challenging enough" and that in following the recommendations of the Cockroft report (1982) and raising the standard of the average pupil, too little has been done to develop the potential of the above-average.

Justifying and Proving in School Mathematics by Celia Hoyles and Lulu Healy is available from the Institute of Education, University of London, 20 Bedford Way, London WC1H 0AL. Tel: 0171 612 6651

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