Lecture to visiting students from Hungarian Transylvania
to the Eötvös Loránd University Summer School in Mathematics
Budapest, Hungary, 27th August 2001

Good morning,

    I am very honoured to be asked by Professor Éva Vásárhelyi* to talk to you about my experience as a teacher.

    I suppose one reason for this invitation is that I have some unusual experiences to talk about. I only became a school teacher at the age of thirty. Before that I was a soldier. I was a engineer captain in the British Army. I have a degree in engineering, and after my own training as a soldier and officer, I became an instructor of young soldiers.

    After leaving the Army I trained to become a teacher at Cambridge University, where I was very fortunate to be accepted by Trinity College. Trinity College is one of the most important university colleges in Britain. It was the home of Isaac Newton, and the fellows of Trinity College after him have together won more Nobel prizes, in all the sciences, than the whole of France. By the way, I do not mean you to believe that I am all that clever, but to find myself in such company after the Army was a most interesting contrast.

    My last work for the Army was in the military headquarters in Northern Ireland. In Ireland at that time there was - and, indeed, some would say there still is - a kind of limited civil war between those Catholic Irish who want all Ireland to be Catholic and the Protestant Irish who want to stay in Ireland but who do not want to live in a Catholic society.

    This was therefore my introduction - thirty years ago - to a problem that is again becoming very serious in Europe. We now call it ethnic cleansing. It is really a problem that people have in living with others of different opinions. Some think that this is only about different religions. This, however, is only part of the story. In fact - as you will know very well - this particular conflict, between Catholic and Protestant is mediaeval; that between Christian and Muslim is even older; between Christian, Muslim, and Jew older still. But such conflicts can also be about any different political ideas, and even about different ideas of science.

    One of the most puzzling and terrible aspects of the past century has been that whilst people everywhere in Europe - also in Russia and Asia - were receiving more education, their ability to accept more diversity did not increase. On the contrary, many appear to have become more than ever convinced that only one system of thinking must be absolutely correct, and that all other systems, as a result, must be wrong. Nor have we today escaped from this manner of thinking. Currently the most accepted economic model is called global free-market capitalism. We are supposed to believe that global free market capitalism will eventually create liberal democracy, freedom, and prosperity for everyone on earth - and, believing this with total confidence, that we therefore should plan for no other future. But planning for no other future than the one that is confidently expected was the great mistake of all the absolute systems of political truth of the 20th century. Why can we not learn?

    I think the reason is essentially that human beings want something emotionally that their powers of inquiry will not allow them to have. Emotionally they want simplicity, certainty, continuity, and above all, permanence. They want to feel safely enclosed by the set of ideas - preferably their own set of ideas - that promises this security.

    Unfortunately, for those who have believed in its security, every time such a closed set of ideas has been constructed, reality is found to contain more to be explained than this closed set of ideas can explain - and thus the whole process of looking for another level of simplicity, of certainty, continuity, and permanence has to begin all over again. If we are therefore to try to help our pupils to avoid civil wars, like those in Ireland, or far wider conflicts, as between Islam and the West, we need to persuade our pupils that knowing the truth cannot depend on where they are born or what they are told is the truth. The truth must be more general. It must be more open. It must be less certain.

    After leaving Trinity I went to teach in a public school in Oxford called Magdalen College School, where I was very happy, but I was paid very little, at that time, in fact, less than a bus inspector. Eventually I looked for another job, and in 1979 I began teaching at one of the official European Schools of the European Union. This means that I meet teachers - especially in mathematics - from every tradition in Europe. Some years ago I began to be more interested in teaching ethics, and for some years now I have represented my school in conferences on ethics education.

    What I would like to do today is to try to tell you everything that I have learnt of importance in my life as a teacher. This is impossible. If, however, I have only time to tell you one thing, one thing that I believe will help you to become a better teacher than I have been, I think I would have to tell you this story.

    Some years ago I asked a young class to write an essay to describe what was worrying them in the world. Most remarkable, I found, was that nearly all of the class - all children between eleven and twelve years old - expected the world to end. And by that I mean they expected it to end soon. When I questioned them about this, they could not agree whether the world would be destroyed by a comet or an asteroid striking the earth, or by earthquakes, or by volcanoes, plague, nuclear or biological disaster, by the melting ice-caps disturbing the dynamic balance of the lithosphere and causing the poles to move to the equator (really: they knew about this), or by the deep sea deposits of methane hydrate exploding and turning the atmosphere overnight into a gas oven (which some knew about too).

    Such is the power of television. Teaching soldiers, even young soldiers, is very different from teaching modern young children. I was astonished by the variety of knowledge they possessed about these disasters. All of them, it must be admitted, are certainly possible, but on most we would probably set a time-scale in terms of hundreds or thousands of years. My young class, however, reckoned on one or more of these cataclysms wiping out all life within twenty years. This is something else to keep in mind when working today with the young, especially those who have access to television. Many may not seriously expect to grandparents - even to be parents.

    In another class, and in different year, one young girl surprised me in a very different way. She was already tall for her age, and was also thin, awkward, and not pretty. I had already noticed that she was very afraid of seeming stupid, and so she copied a great deal from her neighbour. She was not my favourite pupil. At the end of her essay for me, however, I found something truly unusual. She had written: "Most children would like their parents to tell them more often that they loved them."

    She was twelve years old. In her next mathematics class, when I had a moment, I asked her to join me at my desk. And then I showed her essay saying: "This is a most remarkable thought. But you know, as well as being true - for I am sure it is - most parents would also like to be told more often that they are loved by their children. And then," I went on, "there is a deeper problem. Some parents have never really learnt from their own parents what love really is. They simply do not know what it means. How could you teach a parent what a parent does not know?"

    We sat together, this thin, awkward, skinny girl and I, head to head, and discussed this problem. I told her I think it is an immense problem, probably the most important problem in all societies - and she had noticed it already!

    There will often be times, when you are a teacher, of opportunities like this. They are the most important opportunities in a child's life. For as I talked with this little girl, I could feel her becoming a young woman. Perhaps for the first time in her life an adult was talking to her as an adult whilst showing respect for her observation and ideas. She will remember this all her life. She may, of course, become, as a result, a bully and a monster. But I hope not, because at the same time that I showed respect for her ideas, I also showed her affection.

    Most of the children you will teach will need desperately to be shown respect for something that they have done or that they have tried to do - and they will need your honest, balanced, critical, non-sexual affection. If you cannot respond like this to your pupils, also to children who are being deliberately silly, stupid, disruptive, or disobedient, you had better give up teaching right away, because instead of helping you to grow as an adult and to become more useful to your pupils as a teacher, your responsibilities will only exhaust and destroy you.

    As a teacher you must be determined to be happy. This is essential. Angry teachers are useless. Of course you must correct silly, undisciplined behaviour, but instead of being angered by, you must always be ready to laugh. Children are, often, very funny, both deliberately and accidentally. If necessary - but not please, too often - you must also be ready to admit to being wrong or mistaken. And finally you must know that being the person who can most usefully direct and assist a child's learning, and who can help them to become an adult, means that occasionally you may be the most important influence in a child's life. Know as well, however, that you are never, ever, responsible for the learning itself. You can help a child to learn, you cannot learn for the child. Only the child has that responsibility.

    What I have just told you, I hope will keep you alive and happy in your classroom. Never underestimate the dangers. My successor at my first school, a clever young graduate with a wife and son, lasted only a few years after I left before committing suicide. Look for colleagues who will encourage and support you as you must look for ways to encourage and support them. Never let yourself get so tired that you lose your sense of humour. You, too, will always be learning. Listen to criticism, but do not let criticism destroy you.

    Now I want to tell you of another kind of discovery, one that I hope as well will give you a most unusual sense of purpose - especially if you are a mathematics teacher, as all of you will be.

    I had been teaching for fifteen years before I became aware of a fact that everyone knows who works in mathematics or with mathematicians, but which outside mathematics is almost completely unknown. As a matter of fact to most people outside mathematics it is impossible, unbelievable, ludicrous. This is the fact that mathematics, essentially, powerfully, and even unavoidably, is democratic.

    In the past ten years I have been able to persuade audiences of this fact in Britain, France, Germany, Spain, Turkey, the Czech Republic, and now in Hungary. With the support of the Stuttgart Institute for Education, LEU, I have helped lead a European Union project to explain the connection, and to explain its importance in teaching mathematics, in English, German, and Spanish. The French mathematics teaching association have published it in France. I hope you will help to publish it in Transylvania. Soon we hope to begin another European Union project to improve mathematics teaching and democratic education in many more countries in the world.

    Usually the disbelief of the audience lasts only for minutes if I show them this table. On the left is how I expect you will want to behave in your classroom to be most effective in teaching mathematics. On the right is how I think you would expect a democratic politician to behave most democratically. Please take a minute to examine it.

1. Political leaders should treat people as if they are their intellectual equals.

    Once this pattern of behaviour has been established in the mathematics classroom - especially because of the very powerful prestige you will have as mathematics teachers - it is socially extremely influential. When young people have been treated by their teacher with this degree of respect, and, perhaps even more importantly, with the clear understanding that they have the right to criticise, even to refuse an argument, then they are being prepared to take their place in a society as its democratic citizens.

    Conversely, of course - and this must be used as the test of any similar argument - if any detail on the left is changed, one should expect to find an appropriately similar change on the right. This is a correspondence you may like to play with yourself. I think you will find it will produce very familiar social and political patterns. Imagine the result, for example, if every teacher of mathematics that a child ever has insists that no-one may criticise any argument pronounced by authority, that they must all be accepted, without criticism, without discussion, as commands. This is actually very common in many societies today. What kind of society would you expect to result? You can find them on any map.

    Finally you should not believe that knowledge of this correspondence is new. It is as old as Western civilisation. The first society to develop democracy and systematic mathematical argument together was that of Athens around 500 BC.

    Of course you may object that Athens was a slave-owning society in which women had no voice. This is true. Slave-owning and sexual discrimination was not unusual at that time. Athenian society, however, did not at first develop this form of argument just to do mathematics. In their time their democratic assemblies were being dominated by the very rich, their lawyers, their spokesmen, and, above all, their speech-writers. All learnt a very clever form of argument called rhetoric. Teachers of rhetoric famously boasted that they could teach how to prove anything to anyone, and next day, with equal success, teach the opposite. Today we call this political spin.

    The Athenians became aware that this power of rhetoric - this continual use of spin, as we would now say - was systematically destroying people's confidence in democracy. Even Plato warned against the damage. And so a new kind of democratic argument was rapidly developed to restore people's confidence in democracy. Instead of rhetorical arguments that were often designed to confuse, people were shown - and learnt - simple but reliable arguments that followed fixed patterns that they could use again and again in many different situations. They all had the same characteristics. They all began with evidence that everyone agreed was relevant, they used connections that everyone could follow, and they arrived at their conclusion quickly. This was most important. Only the rich had time for elaborate speeches and leisurely dramas. The working citizens had businesses to run.

    These patterns are the patterns that we call logic. They are the kind we use in mathematics,

    The following is an historical fact that you can teach. Mathematics as a tradition and a system of arguments that anyone can learn and use, did not come before democracy. It came after democracy had first created such a system for political use. It was on the back of a democratic political tradition that mathematics began to be systematised, also democratically, so that everyone can understand and use it.

    And this is what every mathematics teacher should communicate to every pupil today. The power and quality of democracy in your society depends, not uniquely, but very essentially, on the way that you teach mathematics.

    Remember, therefore, that as mathematics teachers you are not only the guardians of the mathematical ideas of mankind, a great responsibility in itself, you have an even greater responsibility: you guard the democratic tradition of mankind.

    I mean this most seriously. The right to vote, the right to criticise, the right to be informed, and the right to argue, all of these are completely meaningless if citizens can still be confused, bullied, and frightened by rhetoric: by spin-doctors and by spin. Clarity of thought, clarity of argument, and fearless expression are the weapons that all tyrannies fear. They are what you must teach if your pupils are to do mathematics themselves. They are what you must teach to everyone who enters your classroom.

    Thank you for listening to these ideas. I hope you may find some of them useful - and if not useful, at least provocative. If you have any questions, I can try to answer them now, or you can write to me in England: preferably, please, in English!

* Professor for Mathematical Didactics, Department of Mathematics Education, Eötvös Loránd University

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