Since my appointment as the new vice-chancellor of the University of Buckingham was announced, I have received some wonderfully supportive and encouraging messages from people in the university sector, including vice-chancellors. But I have also received some less pleasant responses, such as “It will be very different to running a school, you know”, and “The private sector has no place in British higher education. Get lost.”

It is perfectly fair to ask whether a school headmaster has the experience to run a university. I have much to learn about this new world – not least the acronyms – and it all seems quite daunting. But I am already visiting universities the length and breadth of the country and am trying to learn fast.

It is also true that much seems familiar. Wellington College is not so far removed from a university as all that. It is a large residential school that employs 500 staff, which also has campuses in China and two state-funded schools in the UK. Like a vice-chancellor, I galvanise the organisation, decide its goals with the support of my board and drive them through to realisation. Those goals include attracting and retaining the best academic staff and students, maximising the student learning experience, marketing the institution internationally and overseeing a substantial budget (of some £40 million, in Wellington’s case).

My primary task at Wellington is to maximise the academic performance of students and to ensure that we have sufficient finance to allow us to innovate. In the past 10 years, we have been the fastest improving independent school in the country at A level, and student numbers have almost doubled. We have also attracted outstanding staff, many with first-class degrees from top universities and some with research degrees on top.

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The UK has a small but disproportionately powerful private school sector, accounting for only 7 per cent of school places but producing some 35 per cent of the students who go on to top universities. The private sector acts as a spur, as well as an irritant, to the state sector, which would be much more complacent and less successful without it.

The UK’s private higher education sector is considerably smaller and weaker. Buckingham was the UK’s first private university, starting life in 1976. The furore that greeted A. C. Grayling’s announcement in 2011 that he was creating the New College of the Humanities in London showed how raw feelings are, and how suspicious the higher education world is of private providers. The outcry was unpleasant and counterproductive. What exactly were the protesters so afraid of? Why is the very idea of private provision anathema to them?

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But we must distinguish between two kinds of private higher education institutions: those that are not for profit, and those that are for profit. Buckingham is most definitely in the former camp. The importance of the distinction is best illustrated by looking at the US, where the private sector dominates both the top and the bottom of university rankings. But while the 24 private institutions packed into are non-profit, the bottom positions in the US rankings are held by for-profit wholesale providers, a number of which were investigated by a US Senate committee for poor standards.

Universities motivated primarily by profit find it very hard to resist maximising quantity over quality, and engaging in academic cost-cutting for the sake of quick returns. It is also much harder for them to justify resources going towards pure research, except where it is of direct interest and benefit to a funder. There may well be a place in the UK for for‑profit higher education institutions, but Buckingham under me will most certainly not be one of them.

The UK needs a virile private not-for-profit university sector that can use its freedom to challenge, innovate and indeed irritate the state-funded sector. The competition they provide is good for everyone, for students as well as for staff. Arthur Seldon, my father and one of the intellectual forces behind the creation of Buckingham, wrote about the “common fallacy” that only the state can run universities and that the public interest and national need will be served only if the state is the monopoly provider.

As a head for 25 years, and as a father of three children who have attended different UK universities, I can see why many find the experience of securely funded public universities disappointing. I hear regular complaints about lack of engagement from teaching staff, low interaction time with academics, poor feedback on written work, undemanding assignments, lack of intellectual stimulus and poor standards of pastoral oversight and care. So I can sympathise with increasingly large numbers who want to go off to the US to study.

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At Buckingham, I will be building on 40 years of fine work, pioneered by its first principal, Max Beloff, and continued so capably by my predecessor, Terence Kealey. I want Buckingham to become a distinguished academic institution, providing high-quality degrees and conducting world-class research in a variety of fields. Because its funding does not come from the state, it should be free to devise innovative courses that appeal to the students and that truly educate them. Driving this forward is an opportunity I relish.

Sir Anthony Seldon is headmaster of Wellington College. He will take over as vice-chancellor of the University of Buckingham on 1 September 2015.