The usual explanation is that it’s too much hassle, what with all that paperwork and all those people on your back, from parents and governors to Ofsted and Whitehall. Running a private company may be stressful, but it is at least simple in one respect: you are accountable to the shareholders, and your job is to raise the value of their shares and dividends. Running a school involves serving half-a-dozen different masters, who all set their own, often conflicting, goals.

But I suspect there is more to it than that. A headship no longer carries status. This may seem a perverse thing to say when heads have power to hire, fire and promote staff, control of school budgets and freedom to repair the roof without getting permission in triplicate from the town or county hall. But power is not the same as status. Status derives from the respect accorded to your position and your institution. The Queen has high status but little power. Her status does not depend on her personal qualities or on the achievement of objectives. All she has to do is keep the show on the road.

Until recently, heads were in a similar position. Provided a school didn’t descend into obvious chaos and a head didn’t provoke an outright parents’

revolt - by, for example, ill-treating the children or returning consistently dreadful exam results - he or she kept the job. Both the school and head were treated with respect, even reverence. Longevity was a virtue, not an occasion for remarks that old so-and-so is getting a bit stale.

Now the esteem in which heads are held - by staff, parents, pupils and various superiors - depends on results. It is not enough to avoid the label “failing school”; a school must not even “mark time””. All heads know they are just one bad Ofsted report away from public odium.

Why should this matter in schools and not in, say, Vodafone or Boots? There is no shortage of people aspiring to the top in most other careers. You may say that heads aren’t paid enough given the responsibilities, and I am inclined to agree. But there is more to it. For one thing, schools have never developed a specialist management cadre as private companies do and as, within the public sector, the NHS has done. Very few doctors want to run hospitals or are expected to do so. Perhaps more important, schools are now expected to follow a management culture, imported from business, which is alien to their traditions. Schools used to depend on the quality of personal relationships, on the bonds of trust and mutual respect built up between teacher and teacher, pupil and teacher, parent and teacher. They valued continuity and stability above an eagerness to embrace the next new thing. They expected to earn respect among their communities and parents through a job well done, not through marketing. Most of this was true in private schools as well as state ones.

Many teachers become teachers precisely because they are attracted by this culture and reject the short-term, instrumental goals of modern business.

They want, as they put it, to make a difference, to work co-operatively, not competitively, with colleagues. If they become heads, they are invited, in effect, to become different sorts of people. I am not surprised that they hesitate.