Tony Gardiner is a crusader. Reader in mathematics at Birmingham University, involved in local maths clubs since 1975, contributor to SMP (School Mathematics Project) since 1978, initiating and running the national mathematical Challenges since 1987, masterminding the UK Maths Foundation’s Oxford summer school for secondary pupils for the past two years whilst training the International Maths Olympiad (IMO) team, he is a man driven by the love of “hard mathematics as a spiritual challenge” and by a conviction that the current orthodoxies in maths education have got it “horribly wrong”. But what is it that is wrong and what is Dr Gardiner’s prescription for getting it right?

When he talks about being “interested in getting kids to do hard things” and about how “educationally potent” his sets of problems can be, few perhaps would dispute his argument.

But recently he has waded much further into the fray over standards of maths teaching and met with Education Secretary Gillian Shephard. Undaunted, he and others from the academic community are facing the heavy artillery of the School and Curriculum Assessment Authority, the Department for Education and Employment, OFSTED and all.

The numbers taking up Dr Gardiner’s challenge to tackle harder maths are impressive. From the 16,500 who responded in 1988, numbers have mushroomed to 220,000 entering the Junior and Intermediate Mathematical Challenges for Years 7 and 8, and 9 and 10. Stickers reward the top 40 per cent, but those who do really well get, says Dr Gardiner with a quirky grin, “more sums”. Eventually they may end up in the Olympiad team.

But it is just such exceptional students whom Dr Gardiner’s opponents cite when they call his approach elitist and out-of-date. When Dr Gardiner says that the top 50 per cent have been badly served by progressive methods, GCSE, the national curriculum and, above all, a kind of slapdash “DIY” recycling of eternal verities, many educationists would reply that the bottom 50 per cent are now much better served than before and that anyhow the brighter children always “get there”. It is this approach which infuriates Tony Gardiner. All the “guts” which make mathematics important have been abandoned, he believes, in a futile attempt to woo the punters. “If pupils can’t tackle a topic, instead of improving our teaching, we drop it. Maths content is not optional just because it’s hard.”

He lists fractions - “one of the cornerstones of maths”; learning of tables by heart - “most countries recognise you can’t make sense of number without numerical fluency”; algebra - “the development of algebra since 1600 liberated our culture, leading into the industrial revolution”; and exact constructions in geometry - “without geometry there is no mechanics, no bridges, no roads yet few pupils even learn to bisect a line segment correctly” - as crucial content upon which the national curriculum does not insist. Educationists call for pupils to use their own methods, but this, says Dr Gardiner, “wastes so much time”.

He is passionate in asserting that all pupils are being short-changed by not learning traditional methods. “These methods have evolved over centuries not just by the giants of mathematical thought but also by a kind of cumulative wisdom. Written algorithms seem unnatural; for example, the dot-and-carry method of adding and subtracting is not the most natural but it is far better for dealing with larger numbers. If I want to teach a class of 140 and they have 140 private understandings of a problem, how can I begin? We need standard methods to communicate.”

Take a problem. I buy a cabbage for 54p and an orange for 7p. How much change do I get from Pounds 1? The solution involves several steps, a knowledge of number bonds, place value, addition and subtraction. There are many ways of getting an answer, but we need the most efficient. In “real life” any reliable method would suffice. But maths, insists Dr Gardiner, is “not just about getting answers. You always have to think, to understand the structure of the problem.” This is one reason for questioning use of calculators, which emphasise the answer and make it difficult to focus on the method. Then, “if the magic number is wrong”, it is hard to sort out why. “It encourages thinking of the wrong kind.”

Yet, says Dr Gardiner, many of the instructions in the national curriculum militate against grasping mathematical thought. In key stage 3, for example, simultaneous equations are solved “by trial and improvement”. While intelligent guesswork may be fine as a start, teaching the systematic analytic solution demands the elimination of a variable. Mathematics is not an empirical science, it is a rational universe. Sadly, says Tony Gardiner, SCAA and its advisers have lost their inner confidence about a subject which has universal, permanent charcteristics.

Dr Gardiner is equally scathing about “general problem-solving skills”, so beloved of GCSE syllabuses. Time is wasted on them instead of on specific skills. Well-meant, unfocused “exploring” rouses Tony Gardiner’s ire. “At some point we have to stop pussy-foooting around and tell kids what we know. Tell them the bloody rule; let’s get people happy using what they know.”

His opponents seize on these remarks as over-rigid, denying natural creativity and diversity. He says creativity is stifled by immuring it at the lowest level of “exploring”, denying students the tools they need to solve problems involving more than two steps.

Is a “dumbing-down” process at work in the GCSE tier system? For subsequent progress in mathematics, pupils need the upper tier syllabus. But, points out Dr Gardiner, “if you can get a B in the middle tier why risk catastrophe by going in for an upper tier, even if the students could tackle the work?” He is emphatic that “market forces are having a pernicious effect. They are forcing a downwards spiral in the quality of what is being taught.”

Like the five mathematics academics who wrote a letter to The Guardian at the beginning of this year, Dr Gardiner insists that mathematicians have to be given a say in the maths curriculum. He finds it “tragic” that “many of the best kids in the country” do not understand the difference between the exactness of Pythagoras’s theorem and measuring with a ruler on a scale drawing. The proof lies not in examples of triangles - of which there are an infinite number - but in the logical structure of the theorem. The fact that there is no ambiguity in maths is something to celebrate, he says, but, alas, “there is vested interest in the woolly”.

Fractions are downplayed for the wrong reasons - “because we no longer have imperial measures”; children at key stage 3 are asked to “manipulate algebraic formulations” when simplification not manipulation is what is needed; and “a range of methods” are repeatedly advocated instead of proven standard methods.

All this to avoid upsetting those who may get it wrong. “I don’t mind them getting it wrong,” cries Dr Gardiner in exasperation. “Everyone gets things wrong, you learn by getting things wrong. What’s frightening is that they don’t seem to have any idea of what it means to get it right.”

Meanwhile, back at the maths summer school, Vin de Silva from Oxford University is teasing a bunch of teenagers with “winning ways”. As they scribble and discuss whether the paper-and-pencil game of “sprouts” can ever end or whether pairing co-primes provides the answer to a number game, Adam McBride from Strathclyde University confirms that his students too, have “a weird idea of proof. They don’t understand that maths is not proved by examples and they can’t grasp the difference between a theorem and its converse. ”

A young lad puts up his hand. Frowning over lists of numbers, he says: “We can see why, but we can’t prove it.” It looks as if the crusade is not lost yet.

Dr Gardiner will speak at a TES debate at the British Association for the Advancement of Science Festival in Newcastle on Tyne on September 13 on the theme of “Have standards in mathematics fallen over the past 10 years?” Other speakers are: Richard Brown (SCAA); Professor Carol Fitzgibbon (Newcastle University); Professor Harry Marsh (Durham University); Tom Wilson (Duchess’ High School, Alnwick). The session will be held at the Surveying Lecture Theatre, Newcastle University, at 5pm. Admission is free. Information: 0171-973 3076.