In other words, the very people who always criticise schools are themselves distrusted by the public.
Heads may regard this as poetic justice; but I wonder how long it can last.
As the Telegraph poll was being carried out, the head of a Kent primary was being jailed for forging test results. The whole thrust of government policy for the past 20 years has been to change the culture of the public sector in general, and schools in particular, and to inject private-sector disciplines and working practices.
This endangers many things for which the public once valued schools - a sense that they were above sordid self-interest, that they would be honest about children's strengths and failings, that they would do their best for all pupils in their care, that they would give parents dispassionate, professional advice. The extent to which distrust is creeping over to the public sector is demonstrated in the Telegraph poll: hospital executives get a big negative score, distrusted as much as Lib Dem politicians and Daily Mail journalists.
Which leads us to another question. How highly does the public rate truthfulness? Do they really want things run by people they trust? You may think it obvious that they do. But why don't more of them vote for the Lib Dems, the politicians they least distrust?
Let us turn to (in a grossly over-simplified form) the argument between Gordon Brown and Tony Blair over public services. Number 10 believes that the public sector can be improved through breaking up producer monopolies and extending consumer choice. Number 11 believes that producer monopolies can be tolerated provided they are, in Brownian jargon, "contestable": Some authority (preferably, in Mr Brown's view, Mr Brown) should be able to boot out the producer and replace him with a better one. The way to organise this contest is to set targets for producers.
This is a profound difference but neither man promises an easier life for teachers, I'm afraid. Both want public services to strive for improvement and they believe that this striving has to be imposed through a variety of sticks and carrots. Both assume that public servants are not to be trusted.
The old public-service ethos was simple. Public servants got jobs for life, with minimal checks on how they performed. In return, they accepted lowish pay and an obligation to do their jobs conscientiously without regard to personal gain or advancement. This model gave us teachers and doctors whom the public trusted. But it also produced services that were often slow-moving, unresponsive and inefficient.
As they move away from this model, both Chancellor and PM would say that what the "consumer" wants is paramount, that "delivery" is all, and so we must press on.
Is it possible, however, that what consumers want most of all is not "choice" or even efficiency, as conventionally defined, but simply public servants whom they trust to tell the truth?