Exactly 60 years ago, just after the greatest armada in history set off across the English Channel, a frailer vessel made a heroic attempt to cross the great British class divide.

In July 1944, on the day when General Eisenhower was reviewing his bridgehead in Normandy and the Second Army was steaming south of Caen, Lord Fleming, a Scottish judge of sound Tory views, launched his report into the public schools and the state system.

Fleming, like many people before and since, believed that the existence of the social division in education “made far more difficult the task of those who looked towards a breaking down of those hard-drawn class distinctions within society”.

In the brave new world after the war those distinctions would be relegated to the dustbin of history. The Fleming report was seen at the time as part of the war effort, like the Beveridge report or the Butler Education Act.

Winston Churchill, that most dedicated of old Harrovians, waxed lyrical when visiting his alma mater about “broadening the intake and the public schools becoming more and more based upon aspiring youth in every class of the nation”.

The way to do it, Fleming thought, was that some public schools should take 25% of their pupils from state primaries, while others would follow the model of the direct-grant schools, open to all with parents paying only what they could afford.

Trailblazing stuff. Except that almost nobody followed the trail that Fleming blazed. Hertfordshire council sent a couple of boys to Rugby. There was a West End play called The Guinea Pig about a working-class boy having a hellish time at a public school.

Otherwise privilege resumed her reign. Rab Butler, that great social reformer, sneered at Fleming as humourless and “sensationally ingenuous”. And the new Labour government wanted nothing to do with the public schools, except to abolish them, which it hadn’t the nerve to do.

Twenty years on in 1965, Harold Wilson set up another commission on public schools under Sir John Newsom, who recommended much the same solution as Fleming and oddly enough, like Fleming, died soon afterwards, uneasily aware that he had been duped by politicians who never intended to do a thing.

Far from bringing the state schools and the independent schools closer together, Labour’s only achievement was to drive 119 of the old direct-grant schools into the independent sector, depriving thousands of children from poor families of the best education in Britain. With the best will in the world, Manchester Grammar school, for example, can today afford to help only one in seven of its pupils with their fees, whereas under the old system up to half would have been helped.

So here we are, 60 years later. The cold war has come and gone. The tiny platoon of Fleming’s guinea pigs are almost as old as the veterans of Normandy. And it is as true today as it was in 1944 that, as The Times opined then, “no domestic issue excites more violent or implacable feelings”. Look at the long list of Labour notables who have been cursed by their comrades for defecting to the private sector or, more heinous still, sneaking their children across the borough boundary into a better school: Tony Blair, Harriet Harman, Diane Abbott.

Now, just in time for its diamond non-wedding anniversary, Charles Clarke declares that the war is over. He wants to see an end to the “armed neutrality” between the private and state sectors and urges them to enter into a new partnership. For the first time the government is offering to fund sixth-form places at public schools for state-school pupils to study subjects that would otherwise be unavailable.

It sounds like a splendid start. Unfortunately the government does not always sound so benign. We are told, for example, that the new Charities Bill may tighten the definition of “educational benefit”, so that independent schools will no longer be able to claim charitable status and so escape corporation tax and higher business rates, unless they can prove that they do something for the state sector — offer scholarships, sponsor inner-city academies, or at least share their sports fields.

I wonder whether this scarcely veiled menace is the right way to foster a spirit of co-operation. Already you hear fee-paying parents complain, “We’re paying twice for our children’s education as it is and saving the state thousands. Why should we be asked to pay even more to subsidise the schooling of some yob who may well destroy the atmosphere of learning and discipline?” Far from dissolving class resentments, if the government goes about it this way it is more likely to exacerbate them. The independent schools point out that they help nearly a third of their pupils with fees and the real livewires, such as Graham Able at Dulwich, are setting up new state academies themselves.

Those heads I have spoken to seem almost poignantly eager to help. They look out over their rolling grounds, Olympic rowing lakes and magnificent theatres and feel there must be more to life than shepherding bright boys from comfortable homes to their allotted berths in life. Many would like nothing better than to return to the mission statements of their medieval founders that they should instil religion and sound learning into the poor scholars of the parish.

The Tories set off on the right track with their idea of a pupil’s passport that would entitle the holder to buy an education anywhere. But then they lost their nerve and said the passport could not be used in existing independent schools, thus keeping up the old class barrier.

The government protests its enthusiasm for diversifying the system. Blair has declared that we are in the post-comprehensive era. Sir Cyril Taylor, inventor of city technology colleges under the Conservatives, is advising Clarke how to do much the same thing under a different name.

Yet neither side is brave enough to tackle the twin taboos: the taboo on charging some modest fees in state schools (means-tested, of course), and the taboo on offering poor parents help to send their children to private schools — in other words, to break down the basic difference between the two systems. And when a head like Anthony Seldon of Brighton College dares to suggest something of the sort, he is treated as something of a freak.

The financial difference between the costs of the two sectors is narrowing. The average cost per pupil at a state secondary school is more than £5,000 a year. The cost at a private day school averages something over £7,000. If we gave every pupil a ticket or voucher of, say, £5,000 a year, parents could choose between schools of whatever type they fancied.

But I fear the time is not ripe for such daring. A scheme of precisely this type was included in Forster’s Education Act of 1870. So you cannot really expect us to have got around to implementing it yet.