They did not understand the meaning of proof and they failed to grasp that maths was a subject requiring precision. They were unable to do numerical algebraic calculations accurately and incapable of solving problems involving more than one step. Most had little notion of what maths was all about.

The trouble, declared the London Mathematical Society, the Institute of Mathematics and the Royal Statistical Society, began at school. Trendy teaching methods were diverting children's and teachers' attention away from basic matters such as decimals, fractions, ratios and algebra. Instead, pupils were wasting their time on such activities as "data surveys" - gathering and organising varied kinds of information.

The report attacked the national curriculum for being too woolly and GCSE exams for being too easy. International comparisons showed us lagging behind many European and Asian countries. School maths, it concluded, was in crisis.

For anyone familiar with the world of education, these complaints have a familiar ring. For students of education history, they are depressingly predictable.

If maths is in crisis, the difficulties go back a long way. In 1910, a pamphlet from teachers and lecturers argued that new methods were needed to raise standards: "We have to get away from big figures and learn from discovery mathematics."

In 1924, His Majesty's Inspectors (HMI) of schools attacked maths teaching for being "the mere imparting of factual knowledge, passively received". In 1925, an inspectors' report on the teaching of arithmetic in elementary schools pointed out: "It has been said that accuracy in the manipulation of figures does not reach the same standards as was reached 20 years ago."

Forward to 1977, and we find a Commons committee report lamenting that witness after witness stressed the same problems about school maths. What were the problems? "These concerns are the apparent lack of basic computational skills in many children, the increasing demands made on adults, the lack of qualified maths teachers ..."

That report inspired a further thorough investigation - the Cockcroft inquiry into maths teaching, which reported in 1982. Yet now, 13 years later, if last week's verdict is correct, British schools are still struggling to teach maths properly. Why? Why have universities, schools and governments failed to solve such a long-standing and well-publicised problem?

PART OF the answer lies in the adversarial nature of educational debate in Britain. The traditionalists want more rote learning and drill. The progressives want children to discover maths through practical activities. Norman Thomas, a former senior HMI for primary schools, says: "There is a national idea that we always have to be in an argument. We swing from one extreme to another, but neither extreme is right."

As a result, the 1950s produced a generation either terrified of maths or bored by it, associating the subject with an endless drill of apparently pointless questions about needless bullets travelling at arbitrary speeds when fired for mysterious reasons out of trains. The 1970s and 1980s, by contrast, have produced young people who may enjoy maths more but who are bad at mental arithmetic and some basic arithmetical functions.

Those who lament the passing of long division and the present drive towards modern methods may have the Russians to blame. The Soviet Union's success in putting a Sputnik into space in 1957 alarmed British politicians, who attributed it, wrongly, to the Russians' education system. One HMI was given the full-time job of shifting maths away from old-fashioned lists of sums and towards practical applications such as building Sputniks. But the shift went too far.

In 1982 Sir Wilfred Cockcroft tried to correct the balance in a report commissioned by the Government. He argued that there was a place for rote learning, provided that children understood what they were doing. His careful prescription of practical work and acquisition of basic skills, however, was misinterpreted by many schools.

The reaction against the rote learning and drill of the 1950s went too far. Number work and mental arithmetic were downgraded and problem-solving was emphasised. Experts agree that teaching methods over the past 30 years must share the blame for the state of maths.

Professor Alan Smithers of the University of Manchester says: "The ethos of education in the 1960s and 1970s was that learning should be enjoyable and exploratory. In English, the emphasis went away from spelling and punctuation; in maths, from the notion that multiplication tables needed to be learnt by heart and that you needed continual practice in adding and subtracting."

Even those who oppose a return to traditional maths teaching acknowledge that schools have neglected numbers and algebra. Margaret Brown, professor of maths education at King's College, London, and chair of the Joint Mathematical Council, says: "We have not stressed enough what is important. There is a tendency for secondary teachers not to tackle key things in number and algebra."

Michael Barber, dean of the University of London's Institute of Education, says some schools have clung to mixed-ability teaching in maths out of misplaced egalitarianism. "There is a persistence with mixed-ability teaching which is quite palpably failing. 'Setting' and streaming in maths is necessary unless you have very good teachers."

The widespread use of calculators is also a mixed blessing. An inspectors' report last year showed that calculators were more prevalent in Britain's primary schools than those of almost any other country. The argument for them is that they enable children to move more rapidly on to complex mathematical operations. The argument against is that they deprive children of the practice they need in basic arithmetic.

Sensible use of the calculator is inevitable - no one would ban word- processing computers from schools because they inhibit writing - but, like many educational reforms in this country, it may have got out of hand.

Professor Sigbert Prais, of the National Institute of Economic and Social Research, says: "On the Continent you begin counting on your fingers or with bottle tops. Then the child learns to work things out in their head. They can then use their calculator. Use of the calculator too soon means that a vital phase of working out in the head is missed."

So much for teaching methods. There is plenty of evidence that the standard of maths in Britain is the result not just of how the subject is taught but of who is teaching it. Ninety per cent of entrants to English teaching have a second-class degree or better. In maths, 40 per cent have a third- class degree or worse. Why? Professor Smithers' research into maths teacher recruitment suggests that it is to do with personality: mathematicians by nature like order and patterns whereas children are messy and emotional. Other countries have recruitment problems in the subject but they are less acute because the nature of education in most places is different. Professor Smithers says: "Our teachers are supposed to be loving people. We expect them to be in loco parentis and almost to mother children as well as teach them. In other countries a teacher's job is much more restricted."

Most continental classrooms are much more orderly than ours because more formal methods are used. That, too, may make the job more appealing to mathematicians.

The difference between Britain and other countries runs much deeper, however. There is growing evidence that our relatively poor performance in maths may have cultural causes. The International School Effectiveness Research project, which compares the performance of nine-year-olds in nine countries, shows that even by that early age children in Britain lag way behind those of the Pacific Rim countries in maths. Parents and children in Pacific Rim countries are strongly motivated by the belief that education is a key to economic advancement. Here, even in homes where education is thought important, maths is not a priority.

Professor David Reynolds of the University of Newcastle, one of the study's architects, says: "In the Pacific Rim countries, parents see maths as important for young people in ways that they don't in Britain. Parents in this country see reading as important. Before Taiwanese children start school, parents would be helping them with maths as well as reading."

Professor Brown says Japan's performance in maths is partly due to motivation and partly to the long hours that Japanese society expects children to spend in mathematical activity both in and out of school.

Yet maths teaching in Britain is not an unrelieved failure. Our performance in maths in relation to Pacific Rim societies is not good, but it is not disastrous in relation to the rest of the world. In most league tables Britain comes around the middle, usually in front of the United States and Canada.

British pupils tend to do better in geometry than in arithmetic and algebra. They have traditionally scored more highly in problem-solving than in arithmetic. Mental arithmetic may be in decline but pupils now tackle a wider range of concepts than their parents. How many of the middle- aged understand probability? How many know the difference between the median and the mode? Both are skills required of today's 11-year-olds.

A 1976 HMI survey found that 14 mathematical concepts were required in 11-plus exams in 1962. By 1976 the figure had risen to 37. The national curriculum of 1988 retained nearly all 37 and added several. Today's 11- year-olds have, for example, to be able to "group data in equal class intervals, represent collected data in frequency diagrams and interpret such diagrams".

There are success stories. At Redborne Upper School in Ampthill, Bedfordshire, more than 60 per cent of the pupils get A-C grades at maths GCSE. Last summer nearly 16 per cent got As or starred As in maths. Steve Robinson, head of maths, stresses the increased range of pupils: "Standards haven't slipped. But there is a difference. Students now are being taught a much wider range of maths and, for the student, that has to be right. We may not go into the same depth always. A- and O-levels were set up for the universities - they used to write them. It is the universities that have not adjusted."

Mr Robinson says the advent of calculators means pupils now do less repetitive basic arithmetic and there is less rote practice of algebra because it is not required by the national curriculum.

"Quadratic equations used to be on CSE papers; now we only do them on higher-level GCSE papers. But then some things we do at GCSE or A-level, I didn't do until I was at university or until I was teaching. This morning my lower-sixths were looking at trigonometry functions and doing computer graphs with them. I didn't do that until I was teaching and the syllabus required it.

"Overall, I would say that our top-ability sets are better all-round mathematicians than their predecessors."

Pupils of differing abilities at Redborne talk of the sheer enjoyment of maths in a way that would have been inconceivable to most of their parents. Where they have a complaint, it is often that they encounter prejudice from their elders - grandparents or colleagues in their weekend jobs - who believe that because they use calculators and computers, they are not up to scratch in maths.

UNDOUBTEDLY, as a country, we could do better, The mathematicians want a government committee of inquiry into maths - a second Cockcroft - to remedy the defects they see both in maths teaching and the national curriculum.

Would such a committee help? It seems unlikely. The curriculum might be changed, but since it was introduced in secondary schools too late to affect most of the inadequately prepared undergraduates now showing up in their surveys, it is an odd target to choose. The signs are that the curriculum, and particularly the associated tests, is shifting the balance back in the direction they want.

A committee might also propose more formal teaching methods, more whole- class teaching and more repetitive arithmetical exercises. The dangers of that are clear. The British tendency to swing from one extreme to another would surface and we should be back in the 1950s with rows of bored children wondering what on earth was the point of logarithms and desperate to escape from maths as soon as they could.

Professor Brown says: "It is possible to raise standards and maintain children's interest, but it is a very fine balance."

Then there is the question of culture. In a society where the Establishment is still dominated by arts graduates and where professions such as engineering have a low status, it is hard for maths to make its way.

Professor Brown says we need to give maths much greater priority if we are serious about improving our performance. "We won't improve standards very much unless we increase the time spent on it to the same level as they do in places like Japan."

"In the past," says Professor Barber, "we have tolerated too much failure in maths. As a society we must raise our expectations of what children can do."

Additional reporting by Wendy Berliner.

Can you do it?

The following two questions were set in this year's UK Junior Mathematical Challenge, a competition entered by 100,000 of brighter 11-12-year-olds. 1. If five-sixths of a number is sixty, what is three-quarters of it?

A) 371/2, B) 40, C) 45, D) 54, E) 60

2. You have to make up a sum using two different numbers chosen from 6, 8 and 72 and one operation from +, -, x, and 3. For example, choosing 8, 6 and + gives you 8+6, a sum whose "answer" is 14. Which of the options below is not a possible answer? A) -2, B) 0.75, C) 1.3, D) 9, E) 432

The next two questions come from the Intermediate Challenge, taken by 14-16-year-olds:

3. The diagonal of a square has length 4cm. What is its area in cm2?

A) 2, B) 4, C) 4.2, D) 8, E) 16

4. Augustus Gloop eats x bars of chocolate every y days. How many bars does he get through each week?

A) 7x/y, B) 7y/x, C) 7xy, D) 1/7xy, E) x/7y

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