After six years of national curriculum maths, high-achieving students are overwhelmingly unable to engage with proof - the recognition and construction of chains of logical argument based on agreed rules and procedures.
This was the finding of a survey of 2,500 pupils from 90 secondary schools, both mixed and single sex. All the students were from high-achieving schools or sets; none scored below level 5 in their key stage 3 tests.
The survey was carried out by Professor Celia Hoyles and Lulu Healy from London University's Institute of Education .
Proof is fundamentally important for those planning careers built on mathematical literacy but, says Celia Hoyles, "If pupils have not learned proof by the time they get to Year 10, I am really sure it is too late".
Maths uses a particular language to communicate the logical sequences of proof; it is a language which needs to be learned like any other. It cannot "just be picked up".
The Economic and Social Research council-funded project began in November 1995. Some of the teaching experiments which spun off from it are still going. Celia Hoyles is convinced it is not teaching but the national curriculum that has failed these "bright, well-motivated kids". She deplores the fact that in using and applying mathematics "they are not faced with the task of making logical arguments. Not long axiomatics like Euclid, but more than one step of calculation. Calculation is not maths".
In the survey, pupils were asked to recognise and construct proofs for algebraic and geometrical proofs. They showed a crippling ignorance of geometrical terminology, most being unfamiliar with such a term as "bisector". In terms of formal language, they were tongue-tied. Not one student was able accurately to assess all 10 geometry arguments, and only 20 were able to do this in algebra.
The survey reflects the consequences of building a curriculum on practical, investigative bases. Students can make narratives and explain their work to others. What they cannot reliably do is make a formal argument. That is what Profesor Hoyles means by it being "too late".
The whole thrust of the curriculum is towards calculation and estimation. There is little geometry, in the sense of arguing from rather than simply recognising the properties of shape. Whereas 40 per cent made some correct deductions in algebra without necessarily offering a complete proof, only 10 per cent of students in the survey could cope with unfamiliar arguments in both algebra and geometry.
Because pupils are familiar with number investigations and drawing general conclusions from them, they could have a shot at algebra; since they do not do problem-solving with shape, they were at a loss to infer equal lengths given a perpendicular bisector of a triangle.
Pupils in the survey were more comfortable with empirical arguments from example than deductive arguments from known properties. They are used to answering questions but not to developing arguments. Yet, well below the university level of symbolic logic, the ability to isolate mathematical properties, marshal those properties and deduce an argument of several steps will be useful.
Maths, which is all about abstraction, seems to have been buried in the curriculum in a concrete overcoat. As Professor Hoyles says, "If you want to apply maths, for instance to the workplace, and that's all you stress and test, then that's all that will be learned. But there is so much more to maths."
That "more to maths", the aesthetic pleasure of which mathematicians speak, the training in thinking, the ability to dissect and analyse problems, has at its heart the rigorous demands of proof. Though difficult, it is not beyond a majority of those in top sets for maths - quite a large proportion of the population. Teaching proof, as the survey's continuing teaching experiments show, is highly popular with able children, who enjoy the challenge. To introduce it in the national curriculum at Year 7 would not be difficult, says Celia Hoyles: "it would need a change of emphasis to include a level of reflection on what students are doing."
From the results of the survey, which threw up a few examples of schools which had pupils who understood proof, it seems that some teachers, either unbeknown to themselves or deliberately venturing outside the curriculum, are managing to incorporate proof in their teaching. There is hope here for a mathematically richer future.
Introducing proof to the secondary school curriculum should not be a revisiting of the dreary memorising of classical proofs of yesteryear. In fact, one of the most interesting results of the survey was the response of a student from Hong Kong. She was able to give a formal rendition of a geometrical proof, but unable to give her own narrative of what she had done. Proof in her education had become a meaningless ritual - proving something "you know is true" rather than making an argument.
Our home-grown students consistently score well in their confidence to share with others their own explanation of what they have done. Students who already enjoy these challenges in key stage 3 are far from being stretched by the national curriculum as it stands. Nor are they prepared for the harder maths which awaits them at A-level.
It would not take too much rewriting of the curriculum to include formal proving. To judge from some of the responses to the survey, there are already students happy and able not just to recognise arguments but also to adapt them to their own purposes. As Celia Hoyles says, "Our kids are really good at conjecturing, discussing and attempting; they really value explanation. Can we not build on that to enable them to write more formally and connect their ideas and explanations?" '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