In the 1970s, it was fashionable to talk about the "hidden curriculum" - the ideas and beliefs which schools transmit to children simply by the way they go about their business. In those days, for example, schools implicitly taught that boys were more important than girls and had more need of good jobs, and that children in the bottom streams were not worth bothering about.
Today, schools still transmit certain ideas - whether they mean to or not. If a school has its own playing fields, a pool, and well-trained and enthusiastic PE staff, the pupils receive the message that their physical fitness is valued, and that outdoor activity matters. Inadequate facilities and short-sighted policies - for example, stopping swimming lessons as soon as all children can swim the minimum national curriculum requirement of 25 metres - demonstrate the opposite.
Similarly, decrepit schools with leaky roofs teach that society does not much value those schools - or the children and teachers in them. Of course, exceptional teachers can and do overcome such disadvantages; they may take their children miles to concerts in order to show them that music can have a place in their lives; or arrange trips to art galleries to demonstrate that art is for everyone. But they are working against the grain, often without support; and they usually have to take their pupils outside the school to push the message home.
The result is that one key lesson our schools teach children is about their place in society. The most influential families send their children to private schools with excellent facilities, where they imbibe the idea that they are different from - even superior to - the rest. At the same time, the very existence of the independent sector teaches many children in state schools that they are not part of that elite.
This does not mean, of course, that children in private schools necessarily have it easy. Indeed, the pressure to achieve and realise the hopes of their parents can be acute, and may be one reason for the eating disorders which currently seem to plague independent girls' schools.
But the fact remains that this educational apartheid cannot be healthy. Many children in fee-paying schools never meet those from the state sector - although over 92 per cent of the population attends maintained schools. This gives both groups a strangely distorted idea of a society divided into "snobs" and "yobs".
But could these fossilised barriers be starting to crumble? Last week, Stephen Byers, the Minister for Standards, asked independent schools to share their facilities with state schools - indeed, encouraged them to demonstrate their community credentials in order to justify their status as charities.
At last, this looks like being a step forward. But it can only work on a wide scale if such co-operative schemes avoid any sense of an independent school graciously consenting to share some of its advantages with its social inferiors down the road. After all, private schools are not always rich, and the state sector is not necessarily grossly deprived. There may be extremes of privilege and poverty, but many decently-brought-up children of all backgrounds go to well-run local schools where they are given a good education - and plenty of private schools are educating very similar children.
What is important is that both types of school - and the teachers and pupils in them - should cooperate in a spirit of mutual interest and inquiry. Educationally speaking, private and state schools are all part of the national education system, and already have more in common than they might think.
With this in mind, schools should propose to share not just facilities, but ideas, problems and expertise. In particular, pupils should be encouraged to focus on their similarities, not their differences.
What could scupper Mr Byers's plan, of course, are the petty distinctions and hostilities of the British social class system. But that's another story.