We are three teachers from three very different countries and academic backgrounds. We hope you will find it remarkable that we have arrived at a remarkably similar conclusion. We have discovered, independently, that learning from discussion is far more effective than learning from instruction.

We can explain why this is more effective at all ages and all levels. Our aim, however, will be more. We can explain why, if your own teachers adopt this method progressively - not, please, all at once - your government agencies can stop attempting to find enough mathematics graduates for your civil service, your industry, commerce, and science, whilst also persuading hundreds of others to teach in mathematics classrooms. We will explain why this need not be entirely necessary.

But we aim to do more. We can explain why teaching by instruction fails badly to develop any really positive understanding of mathematics in the majority of young people - but why it has far greater negative success in producing serious social and moral divisions in the same majority: divisions which will be increasingly marked, as they grow older, by selfishness , dishonesty and belief in the value of dishonesty, confusion, rejection, anger, disruption, and eventually, crime.

Our aim, therefore, is to do more than tell you how we believe education can be improved. We want also to show you how mathematics teachers can be prevented from destroying the coherence of your society that education should help to achieve.

Of course, it can be argued that these effects can all be produced by other teachers. Why is mathematics teaching so important?

First - probably - because ability in mathematics is widely and wrongly believed to be an objective measure of intelligence. Frightened by this belief, children are more likely to pretend to understand mathematics, to conceal misunderstanding - and finally to copy, to lie, and to cheat - even while these foolish, panicky, defensive strategies destroy their confidence in themselves, in one another, in their teachers, and eventually in the value of truth itself.