Fifteen pairs of eyes fix on the patient’s shrivelled white limb. The toes are black. “It’s gangrene,” says the surgeon cheerily to his summer school students. The patient says his leg hurts more when he is lying down.

“That’s probably because less blood can circulate when it’s on the level,” says a mullet-haired youth with a Rotherham accent. An Asian girl suggests comparing blood pressure in the arm and leg to diagnose arterial disease. Long ringlets from Somerset agrees.

Over 90 minutes they forensically diagnose the patient. No one giggles, or chats, or doodles on their notes. I did science A-levels and I can’t follow it all. These 17-year-olds, all from comprehensives and families where no one has been to university, are super-bright. They are the doctors and surgeons of the future, whose knowledge will save us when we are sick.

There has been much national soul-searching of late over Britain’s alarmingly bad - and deteriorating - social mobility. Last week, to add to the gloom, our leading universities revealed they are taking fewer students from poorer areas and the Joseph Rowntree Foundation announced that the gap between rich and poor is the largest for 40 years.

Education is the missing link; if poor bright kids don’t make it to the best universities to become the surgeons, businessmen and other professionals of the future, the engine of social mobility runs out of petrol. Oxford is the most glaring example with only 53.7% of its students coming from state schools (less than 20% from standard comprehensives). This matters because 90% of our kids go to them and, as I had rammed home to me during my day at the Oxford summer school, intelligence has nothing to do with class, income or accent.

The miracle, as I discovered as I heard more about the lives of the summer school kids, is that these teenagers have made it this far. “I kept quiet about coming here,” one lad from Lancashire told me. “Me mates would think I was daft going to school in the holidays.”

The others laughed and agreed, and one added: “At school you wouldn’t let on that you are clever. The others look down on you. You have to hide it.” The best thing about summer school is finding that “there are people like you who are on your intellectual wavelength”.

They all say they thought they would be the most stupid person on the week-long course. One girl told me she nearly got off the train because she was so sure she couldn’t cut it. To them Oxford is not just another world, it’s a different planet. Many had unemployed parents, nearly all were on EMA (education maintenance allowance) which is paid to over-16s who are still studying and whose family income is less than £31,000 a year, and nearly all did jobs - waitressing, supermarket checkouts, bar work - as well as their studies.

They all said how proud their parents were that they’d come to Oxford to be students for a week. I had expected their schools to be proud, too - that their teachers would have picked them out, encouraged them to attend (it is hard to get on the summer school, 1,500 apply for 250 places).

Not a bit of it. “My school didn’t tell me about it,” chorus a few voices. They’d found out from the local paper, posters in college, from the internet. Mostly off their own bats. Had their teachers encouraged them to apply? A few obviously had, but the majority implied that the teachers felt that Oxford was “divisive” and “elitist” - not for kids like them.

With attitudes like this it’s no surprise that we are not getting bright, poor kids into our elite universities. Harvard and the other US Ivy League institutions have teams of scouts truffle-hunting poor kids from bad schools with high SAT results.

A friend told me how he sat in on an admissions board at Harvard where they discussed a bright young black single mother from the ganglands of Los Angeles with SATs at the lowest end of their range but who they believed had the potential to be the mayor of a city, who with their help could be a catalyst for change. They wanted to create social capital. Despite the risks and the other higher qualified candidates, she was in.

At Oxford, by contrast, until 10 years ago the university ran no outreach programmes to get bright kids from unlikely schools and backgrounds into its colleges. Peter Lampl, an Oxford alumnus and himself a poor grammar school boy, was appalled to discover after spending 20 years in America that things here had gone backwards educationally.

“I realised,” he said, “that a kid like me had little to no chance of making it to Oxbridge or another Russell Group university. Something had to be done.” He started funding the Oxford Summer Schools, which have proved so successful that they now run in 10 other top universities and the government is involved in rolling the programme out further.

At the dinner on the last night of the course the sense of thrill, of widening horizons, was palpable. “I thought everyone else would be an egghead, but they’re not, they’re just like me,” said a hipster from Wales. “Oxford just seemed completely out of reach before I came here,” said another, “but now I’m going to apply.”

A week of living in college, going to seminars and hanging out with students who are already there has shown them that this could be their world, too. Half of the kids who come on the summer school apply to Oxford and about 40% of those get in. Of the rest, almost all will get a place at a top university. As one Asian boy from Birmingham put it: “I always thought I’d go to the local college with my friends. Now I’m going to apply to Oxford, Bristol and Edinburgh.”

It felt a privilege just to watch the bright young faces, chatting confidently, feeling on the cusp of great things, realising they’ve got what it takes. During the speeches when Lampl told them that they all had a great future, a black girl on the next table shouted “Yeah!” Lampl told them to work hard for their A-levels, that the next year would have more influence on the rest of their lives than any other. That anything was possible for them.

I left feeling humbled. I had expected to go to Oxford since as a child my parents (who met there) had walked me round the quads. At St Paul’s school and Westminster I was coached to get in. My time there was fantastic but not productive. I feel ashamed of my immense privilege and how arrogantly I wore it.

We need to get our brightest kids, wherever they are from, into our best universities. If we don’t, we all suffer.

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