# But where's the proof?

Margaret Brown, in her inaugural lecture as professor of mathematics education at King's College, London, locates debate about standards in the wider cultural nexus between conservative anxiety and liberal passion for egalitarianism. Conservatives fear lest the achievements of the elite be diluted. On the other hand, as one educationist declared, "I don't give a sod about the top 10 per cent, what about the rest of the country ? They need maths."

Well, but what do we need maths for? Should the curriculum be "top-down", dictated to the schools by the needs of universities, or "bottom-up", led by the desire to have numerate citizens? There is no doubt, says Professor Geoffrey Howson, professor of mathematics at Southampton University and one of the five signatories to the Guardian letter, that now the average student at 16 knows more maths than 20 years ago. But what about the top 6 or 7 per cent of the cohort who do maths A level? Professor Christopher Robson of Leeds University asks: "Does the country want decent mathematicians, engineers, physicists, technologists, at high level? These are the students for whom the national curriculum is just not working."

Robson has a "feeling" - but one which he shares with his fellow signatories, particularly Professor David Crighton of Cambridge University - that the past three or four years have shown a marked decline in two new ways. Students using maths at university are not technically proficient in manipulative algebra. Yet, says Crighton, "secure technique is absolutely indispensable". Worse, they do not have a firm grasp of the conceptual base of mathematics - the centrality of proof. Is this the fault of the national curriculum or are other factors at work?

Mathematical surveys over the decades show a decline in maths ability in many countries. The effects of instant entertainment - television and video games - may have made students generally less keen on the sheer slog of studying. "General laziness," says Crighton.

Algebraic competence is, says Robson, like scales in music: it may not be much fun, but it is necessary to develop a grasp of the subject. And calculators, which are not accurate for decimal points and irrational numbers, reinforce an already hazy notion of proof.

In primary schools, says Sue Burns of the faculty of mathematical education at King's, the = sign is used, not as it should be in the language of maths, to mean "both sides equate", but to mean "Fill this in - Do It". Is this the beginning of an idea that "an intelligent guess will do"? As Howson remarks, "it seems students dislike accuracy".

Against this, many educationists will urge that people approach maths with more confidence. The Cockcroft Report encouraged maths teachers to involve children more in maths, to apply it in everyday situations. But has this approach somehow gone wrong? Sir William Cockcroft himself, also one of the signatories to the letter, feels that it has. He agrees with Robson that from valid ideas about experimenting seems to have come a disregard for technique and an unwillingness to learn technique before application. As Robson points out, it is actually easier, as well as more accurate, to learn a technique before attempting to apply it: in the messy world of experiment, proof may appear to fail when actually the experiment has gone wrong.

Howson laughs at suggestions that it is problem-solving which is amiss. "Mathematics is problem-solving. That's what the whole game is about," he says. "But perhaps we need to think more clearly what is an appropriate kind of problem-solving for what stage of learning." Robson adds: "I want students to guess, to experiment, but I want them to understand there must be proof at the end." They decry the notion that technique can't be taught without "relevance". How will students solve the "multi-task" problems for which maths is uniquely suited, Crighton wonders, when they only learned "each technique as a quick fix to a problem, not a useful tool for many problems." One maths teacher declares that "these open-ended tasks are rubbish".

Crighton adds : "What kids need to find out about is what giants thought about for 300 years: how are they going to pick it up for themselves in a few lessons a week?" He points to a recent Engineering Council report charting a decline in manipulative algebra among students and, more passionately, to his experience "here, in this room, talking in twos" to 250 of the brightest, "most highly motivated" maths students in the country. "Far too many have no recall of elementary facts and no grasp of the critical discipline."

If maths in schools is misfiring, when, why and how? According to Margaret Brown, blaming the national curriculum is mistaken. Not only is the timing wrong, but the national curriculum itself springs out of these perceived concerns. Attainment target 1, reasoning and logic, is supposed to inculcate just that "morality of maths" which Robson misses in his students. Unfortunately proof, which was present in the national curriculum at key stage 4 level 10, has been ditched. GCSEs rule Years 10 and 11. Is GCSE, which began in 1988, and which has a "stranglehold" on teaching at ages 14-16, to blame?

There is no doubt, says one head of maths at a north London comprehensive, that "GCSE is only a shadow of the old O level". One key worry at GCSE is the competition between boards for most candidates, which has a knock-on as schools are driven to compete for better results by league tables and parents pressing for their children to do better in exams. It is well known that schools can improve their performance by entering children for different boards. One teacher, writing pseudonymously in the Daily Telegraph, detailed his pilgrimage around the syllabuses, ditching SMP because it had irrational numbers and Oxford and Cambridge because it used the Simpson curve method. He settled for NEAB, which offered the easier (but less accurate) method of step-calculation. How can children imbibe accuracy in such a climate?

Another worry, and perhaps the biggest, is the softening of the middle tier at GCSE. Candidates who take the middle tier need do hardly any algebra - it is not aimed at those going on to A-level maths and the top grade used to be C. Last year there were more Bs in GCSE maths awarded at the middle tier than at the top. A student gaining B in GCSE may well go on to do maths A-level and then a maths or maths-related subject at university - having done little or no algebra. Without algebra there is no understanding the maths used for engineering, physics, technology, As Robson says, a physicist takes a problem and feeds it into some maths. If the maths isn't in place, the physics can't be looked at.

Clever students may pick up algebra in the sixth form, but the less able, those for instance who took the middle tier - and now that 30 per cent of the age cohort go on to university, the students must perforce include some who are less naturally able - will founder. Misplaced, confidence is a liability.

Margaret Brown has a different set of analyses. Teachers, hard pressed to push children through exams, may not feel they have time for the core values of the subject. Many are not trained mathematicians. She "would defend to the hilt" Cockcroft's "movement to get children actively to think". She sees its fruition in the national curriculum. With "curriculum backwash" national testing can drive up standards. Administered nationally, the tests compare like with like, cost less to run than competing boards and maintain fixed standards. So far from being a diluter, the national curriculum, she believes, will be a tonic.

There are other issues. Professor Peter Mortimore of the Institute of Education has called for more specialised maths teachers at junior level to offset the average primary-trained teacher's "maths-panic". Assessment of modular A levels is controversial.

And is this just another lot of boring old chaps fantasising about how much better it was when the world and they were young? Arthur Rowe of Leicester University conducted an experiment. He gave 1967's maths entrance test to two lots of contemporary students. Against the "gut feeling" of Rowe himself and every one of his colleagues, they scored significantly better than the intake of 1967. So is it time to stop whingeing?

"What we want is a genuine debate," reply the academics, who are setting up a working party with members from the London Mathematical Society, the Institute of Mathematics and its Applications, the Royal Statistical Society, the Royal Society, the Institute of Physics, the Engineering Council and the Royal Society of Chemistry. They hope to bring this party's report to a discussion with scientific employers, the Department for Education, the School Curriculum and Assessment Authority and classroom teachers.

Perhaps they will consider testimony like Cassandra's. Cassandra is a recent science graduate: "I got an A at GCSE. I was bored - maths was so easy. Then I did double maths A level, and I really liked it. But it was all of a sudden so difficult. Can't they make the GCSE harder?"