I looked up at the boy in astonishment. I had only asked him to try and write down how he had arrived at the answers to an exercise he had just completed correctly. I had not asked him for any kind of logical analysis - that would have been beyond him.

“No fear,” he said. “If I do that, I’ll start getting them wrong.”

Suddenly, I realised that rather than helping my pupils to use the potential of their young minds, to enjoy more and more their years ahead in mathematics, I was helping them to dig their intellectual graves.

His remark revealed a horrifying fact: he did not want to know what he was doing because he feared reducing his own efficiency. He knew that he was working like an automaton, and was satisfied with this. It was, after all, his success at working like an automaton that I had hitherto been rewarding.

He was not an especially clever little boy, but he was diligent. I was not an especially clever teacher, I suppose, but I was also diligent. I was always ready to explain and demonstrate, to mark and correct. Every evening I dragged home a bag of mathematics books to mark. Every week I handed them back. I believed that my energetic corrections in red ink were my most important contribution to their lives.

But every year I would become aware of a puzzle I had never solved. Every year fewer and fewer pupils would really understand what I was teaching them - or even what they themselves were doing. By the senior years the fact could no longer be denied. Often quite a respectable fraction would understand, and within this fraction there would be some who could even work creatively at mathematics.

They were not the problem. The problem was; no, the problem is - for it exists in every school in every year in every country - the unhappy rump of pupils, not necessarily unintelligent, often still persevering, but often bewildered, who have done the same work with the same teachers and even passed the same exams, and yet understand virtually nothing. Ten to twelve years of effort; and nothing. Despite their well-trained responses, they can be as innocent of understanding as a pocket calculator.

The comparison is apt. And let us not imagine that this is a peculiar kind of British dumbness. It is widespread. Turning children into extensions of their pocket calculators is a world-wide problem.

A distinguished German poet, Hans-Magnus Enzensberger, wrote to the 50th Mathematicians Congress in 1998: “It’s as though one were to acquaint people with music by having them practice only scales year in and year out. The result would undoubtedly be a lifelong hatred of this art.”

We teach too many to hate this science. To hate and to fear it.

Earlier this year I was in Budapest to hear the formidable Professor Eva Vasarhelyi explain the Hungarian Method. Encouraging teachers and pupils to engage in friendly dialogue improves enjoyment and creativity, not only in elite mathematics classes, but increasingly in every Hungarian school.

The Hungarian Method, essentially a method practised by the early Greeks, even demonstrated personally by Socrates, is now being rediscovered as the most successful method of maths teaching.

In my own classroom, an 11-year-old begins to read aloud: “Adding and subtracting whole numbers...”. “Wait a bit,” I say. “What do ‘adding’, ‘whole’ and ‘numbers’ mean? What is a whole number?” After 10 minutes spent in discussion the class realise that whole numbers are names we give to collections. Adding means finding the name of a collection of collections. “So, now, what do we mean by subtracting?” In this way, children gain confident possession of the logical bases of mathematics. The method is just as applicable to the higher reaches of maths, be it trigonometry or calculus. The point is to make pupils use maths to think for themselves.

There is another, even more important aspect to this. The Greeks did not develop the kind of argument used in mathematics for maths alone. Their purpose was to give ordinary citizens more confidence to argue democratically. In an age saturated with information, and misinformation, such confidence is essential for true democracy.

Colin Hannaford is senior mathematics and ethics teacher at the European School in Culham, Oxfordshire, one of nine EU created and governed schools in Europe and officially described as the most successful for the last five years. His book, Learn Mathematics in Ten Minutes a Day can be downloaded from www.gardenofdemocracy.org. In return, donations are requested for the Central and Eastern Europe Democracy Appeal. Details on the website or from the European School, Culham OX14 3DZ