Let me begin by putting some biographical cards on the table: I was the first member of my family to do A levels or go to university, thanks largely to the education I received in my local comprehensive school; and I am now the head of an independent school that used to be a grammar school. As a result, I have some pretty firm views on the importance of social mobility and the need to ensure that the poor are not excluded by academic selection.

Amid the howls from across the education sector in response to the return of grammar schools, I’m certain I could also hear the sound of champagne corks popping in tutoring agencies around the country. The scale of the problem is outlined by the Sutton Trust in their report Shadow Schooling: private tuition and social mobility in the UK (2016).

The prevalence of tutoring, which does not seem to have played a factor in entry to the grammar schools of the 1960s, has the potential to distort any 11-plus assessment process. This is because attempts at designing a “tutor-proof” test are doomed to failure, in my view, as tutors and parents will always find ways to practise, and any such assessment benefits from a training effect.

This is an issue for schools like mine, where places are oversubscribed by a factor of ten to one. So how do we level the playing field for 11-plus candidates from diverse backgrounds?

Well, in 2014, we removed the dreaded verbal and non-verbal reasoning tests, which were driving candidates from the primary state sector into the waiting arms of tutors. (Incidentally, our internal analysis showed the verbal reasoning and non-verbal reasoning tests were less useful predictors of future success with us than maths and English.) So our entrance exam now consists of key stage 2 maths and English questions.

But we also consider it essential to work closely with local state primary schools, who typically provide 50 per cent of our successful candidates, to offer advice and reassurance about the test and to get feedback on the accessibility of our papers to non-tutored candidates.

We also attempt to take into account contextual factors. So if a candidate applies from a school in special measures that hasn’t had a maths teacher for two years, it might be that their maths scores will not be stellar but they are still worth meeting for interview. Interviews give us the opportunity to sniff out candidates who have been over-coached and to spot those with more potential than their current attainment might indicate.

Our aim, of course, is to ensure that, as far as possible, we offer places on genuine merit. Successful candidates from poor backgrounds have their fees paid by our bursary programme, as well as funding for ‘extras’ such as uniform,sports kit,school trips, or music lessons. Currently 10 per cent of our pupils benefit in this way, and we aim to increase our provision to 25 per cent by 2024 (our 400th birthday).

There is, of course, a wider question in this debate: even if the 11-plus could be made fairer, is expanding academic selection a good thing, per se?

Whether one considers the greatest good for the greatest number, like the philosopher Jeremy Bentham; or thinks from behind a “veil of ignorance”, akin to philosopher John Rawls, I suspect the answer is the same as the picture painted by an analysis of the data for selective areas: no.

In my view, if the prime minister wants to improve schools so that they work for everybody, a bigger issue is the underfunding of budgets and the crisis in teacher recruitment and retention.

Although I understand the enthusiasm for governments to tackle educational problems with big levers such as exam reform, national curricula or school structures, I believe that the single most important factor in improving outcomes for children from all backgrounds is the quality of teaching.

David Goodhew is a headteacher at Latymer Upper School in Hammersmith, London