I had explained to a class of 11-year-olds what they had to do, and they were working silently on their exercise. Suddenly one of the brighter boys announced that he had finished. Marking his book, I wondered how to best keep him occupied for the time being. "Why don't you try to write down how you think about these problems to get them right," I suggested.

     His alarm at the thought jerked a reply from him. "No fear!" he blurted. "If I try to interfere like that with my thinking, I'll start to get them wrong."

     Now it was my turn to be alarmed - and astonished. This kind of pupil is always likely to succeed, yet he was unable to explain the line of thinking that gave him the right answer. What hope was there, then, for the many youngsters who fail repeatedly in mathematics and can never say why, even to themselves? Too much mathematics is taught without open discussion of what it means.

     This is a serious failing, given how vital mathematics is to us. The German poet Hans Magnus Enzenberger has observed that there has never been a civilisation quite like ours today, which is so inescapably dependent on it. Mathematics is uniquely capable of providing us with a common language of influence and respect, and of risk, value, probability and certainty.

     Without its discipline, politicians can speak like lawyers with words intended only to convince, not bind. And democracy inevitably becomes a realm of fudged agreements, endless obscurity and spin. Between increasingly frustrated citizens and their increasingly unresponsive representatives is only an ever-increasing bureaucracy, answerable to neither.

     Sp democracy and mathematics are inextricably intertwined. Two thousand years ago the Greeks devised the logic- and evidence-based arguments that we use in today's mathematics. And they did so, as the Cambridge philosopher Geoffrey Lloyd said, not to create mathematics per se, but to give ordinary citizens the means and confidence to participate in democracy. Deprived of these skills, people are so much political putty.

     No country can claim to be democratic if too few of its people are ready and able to participate in its debates, to take on their responsibilities, to vote or think and speak for themselves. As the Oxford historian Theodore Zeldin says, to be a true member of a society, what matters is whether you are willing to think for yourself and to say what you think.

     Today, too many people are unwilling to do so, either because they've been told too often they are just ordinary and thus assume they have nothing of importance to say, or because they have received too many knocks in life. And where do we first feel the harsh rasp of such judgement, or ridicule, most acutely for daring to voice our uncertainties or doubts, or question given ideas? It's among our peers - and especially in the mathematics classroom.

     So mathematics teaching can hardly be said to be politically neutral. Taught sensitively and thoughtfully, and mainly through dialogue, as the ancient Greeks intended, it can help to unify societies. It shows the most privileged and disadvantaged that they can share a common discourse, can use a common language and can arrive at an agreement. But taught in an authoritarian, dogmatic, insensitive way, mathematics can have a very different result and highlight fundamental differences between people.

     Such ideas can subdue millions. Just as my pupil is likely to become convinced of his right to be a member of a technical elite, so many others will become convinced that they are members of an underclass. This is where the greatest damage is now being done to democracy.

     These are worrying thoughts in the face of what is happening to the European Union's bureaucracy, which, like that of the former Soviet Union, just keeps on growing. I have long argued that democracy in the EU is not supported as it should be by its mathematics teaching. Three years ago, the EU agreed to fund a major study, led by the German mathematician Harmut Köhler into a link between mathematics teaching and democratic education.* One of its basic tenets is that mathematics teaching should never be used, as the French mathematician Didier Norden has protested, "to stab young people through the heart". He claimed that no other discipline was ever so perverted from its original path of social harmony and cohesion to become a tool of social division and calibration.

     In mathematics lessons young people should be taught models of simple, complete, logical argument. Through constructive dialogue they should be encouraged to express their doubts and fears, and taught how to explain their line of thinking clearly and logically. Everyone can take part in arriving at decisions, and no question should ever be met with ridicule. This is the foundation of democracy. The classroom is where it must be built.

Colin Hannaford is the director of the Institute for Democracy from Mathematics, Oxford

*Mathematics Teaching and Democratic Education is available from Harmut Köhler at the Stuttgart Institute for Education and Teaching, D-70174 Stuttgart

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