Would David Cameron, the Conservative leader - with Boris Johnson, Latin scholar, at his elbow - both Old Etonians, or Nick Clegg, the Liberal Democrat leader and a product of Westminster School, do a better job than the two Scottish-educated leaders of the past decade? That might depend on how well they used the experience of their own schooldays to clear away the clutter of the past 20 years.

I can think of seven reasons why they might do better.

1. A secondary school lacking a “specialism” has no discernibly adverse effect on its educational performance. Eton and Westminster, like other high-performing schools, have always been wise enough to concentrate on all-round excellence. The belated recognition of this may be why the supporters of the “flagship” specialist schools programme have fallen strangely silent recently.

2.The present preoccupation with school governance is misplaced. The governing bodies of Eton and Westminster are no doubt efficient, but that is not why these schools succeed. They do well mainly because the teachers in them teach well; so those responsible for the country’s education system should concentrate on that. To that end, the parent-as-consumer, teacher-as-producer model that still infects the relationship between government and the teaching profession, needs to be discarded. How would a teacher at Eton or Westminster respond to being called a “producer” by a pupil? With whatever is the modern equivalent of a clip round the ear, I imagine.

3.The most important component of academically successful schools is the quality of their sixth forms. The years a pupil spends beforehand are seldom crucial to that pupil’s later success. Here the Westminster School alumnus might invite the Old Etonian to note that girls who enter the Westminster boys’ school sixth form from other schools do not seem to suffer academically. Both might then notice that Hampshire, where children are not selected at 11 and leave their secondaries at 16, has about eight times as many pupils in institutions with an average A-level points score of over 1,000 than Kent, which still maintains 11 to 18 grammar schools.

4.Successful sixth forms need a concentration of well qualified teachers able to teach at the required level. The necessary concentration of teachers, across a wide range of subjects, is achieved at leading independent schools and elsewhere. But far more of these high quality post-16 groupings of teachers need to be developed. This sometimes requires structural changes: fewer, better, sixth forms, not more. It is the responsibility of government to see that those changes occur. Meanwhile, the “presumption”, promoted by Lord Adonis, that every secondary school wanting one should have its own sixth form, is precisely what is not needed.

5.The curriculum, nationalised 20 years ago, is proving unreasonably expensive to maintain. Recently proposed changes are to cost pound;92.5 million, and require amendment to 13 statutory instruments. This is absurd. Both Eton and Westminster flourish without a legally enforced curriculum and its associated testing regime. Does anybody really believe, were the national curriculum now to be transformed into statutory guidance, that teachers would thereupon teach less well? Teachers would remain accountable for their pupils’ performance in public examinations and still have Ofsted breathing down their necks. Besides, most teachers are committed to doing their best for children without the array of tick-box accountabilities that surround them.

6.The present government is to be commended for building new secondary schools in areas that need them, but the academy programme is not the way to do this. One objection to academies is that, as they are directly dependent for their funding on the Secretary of State, they are government schools. As such, their governing bodies are contractually subject to systematic intrusion by civil servants. But the fundamental objection to academies was pointed out six centuries ago by William of Ockham. “Entia,” he then pronounced, “non sunt multiplicanda praeter necessitatem” - avoid doing complex and expensive things that can as well be done simply and cheaply. If there is something of educational importance that a new academy can do which cannot be done as well and at minimal cost by a new voluntary aided school, what does anyone believe that something is? One hopes taxpayers will not have to wait for further millions to be added to what has already been spent on inflated administrative and other costs of the academies before any politician insists on receiving an answer to that simple question.

7.There are important lessons to be learnt from the way Eton and Westminster are managed. Their headmasters do not, I trust, attempt to run these schools themselves. That is the job of heads of department and housemasters. The job of the headmaster is to see that they do theirs. The problem those in charge of education in England now have is that successive governments have destroyed the ability, even the will, of some local authorities to take responsibility for post-primary education. This has left ministers and civil servants trying to run the whole education system themselves with mounting inefficiency. What any new administration has to do is what Arthur Balfour did in 1902 and Rab Butler in 1944: pull together elements of what exists locally into some form of democratically accountable administration to which major responsibilities can safely be devolved. This will not be easy to do.

For political leaders to have been educated at Eton or Westminster does not disqualify them from assuming responsibility for England’s education service. If such people rely on their own experience of how good schools work, it must be possible they could do the job rather well.

Sir Peter Newsam, Former chief schools adjudicator, 1999-2002, now retired.