OK, I'm a Cambridge humanities academic. State school background myself. Have been in the ecosystem here for twenty years. I have lots to say about the 'indy schools are full of tim-nice-but-dims who have been heavily overtutored while sink comprehensives are full of sharp-as-nails poor kids who just need to be given the chance'. Yes, this is definitely true - but it doesn't really impact the Oxbridge scene that much because we spot the over-tutored kids a mile off. A few will be admitted, but not overwhelmingly. Cambridge is not flooded by hundreds of stupid private school children who were only let in because their parents gave large donations to the college. The indy kids who are admitted, are bright kids. The question is just whether there are lots of state schools kids who, given the same training, could do as well.
Lots of people know this I know but some don't, so I shall explain the setup here. The Oxbridge system is built around what are called supervisions (Cambridge) or tutorials (Oxford). Students have one to one (or one to two) meetings with academic supervisors once a week to discuss essays that they have written specifically. These essays are designed to be more than 'informative' - students are being taught to be deliberately argumentative, provocative, interesting. To not just answer the question, but challenge the assumptions of the question. Early on, I get some of my students to write essays that argue something that they themselves don't believe. You get the idea. Now you might say that this only prepares people (men?!) to go into Westminster with its heavily bombastic, rhetorical MO - rather than producing genuinely curious, intelligent graduates who know a lot of 'facts'. In which case, Oxbridge might want to redesign what it's here to do. But I would say it actually does both.
Two observations:
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Overall, indy students have always presented as much better at all this - their essays have always been miles more sophisticated from the start. And I mean miles. I have had Westminster, Winchester, St Pauls and Eton (yes - it's now pretty academic there) undergrads who from the very first supervision of Michaelmas in their first year have hit the ground running with extraordinarily creative, detailed, clever arguments. The sort of stuff I wasn't turning out until my third year at best. They have been taught the art of rhetoric, they know how to digest material quickly and effectively and produce something genuinely interesting. They can reference stuff from the classics, the Bible, politics, the arts - stuff that isn't covered in the National Curriculum. Meanwhile my state-educated students, including the grammar ones, have always had a much slower start. In the past, some managed to catch up, some not. Of course, I have also had a whole lot of less-intelligent indy students who just coasted, and some state educated students who were writing essays that were 'fine' early on, but it's true to say that 99% of the time, my very top undergraduates were always from indy schools and 99% of the time, my struggling ones were always from state schools. I know this sounds like hyperbole; it's genuinely not. The top ones are not just 'tutored' - they have been completely immersed in a world of curiosity and intertextuality that doesn't exist in the state sector where a class of 30 needs to be regularly evacuated if one kid is having a meltdown, or even just in a place where there aren't the resources to take the time for this stuff. So even if they get into Cambrdige, they're already turning up at a disadvantage.
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Over the last 10/15/20 years, the decline in the standard of undergraduates that come up to Cambridge over that time has been HUGE. I mean, seriously seriously worrying. I now have students turning up who don't really know how to write essays. Some don't write in proper sentences. And this is Cambridge! These students can't structure a simple argument, let alone write a 2500 essay every week that involves reading several books and journal articles, and finding something vaguely 'original' to say. And this is most seen in the state students - so the advantage gap I described above between the indy and the state students who get in is widening.
This is entirely the fault of lack of funding, and the National Curriculum which no longer has very high expectations, which indy schools are able to mitigate, but state schools don't always have the resources to do. It's also the case that many state schools are not even offering some of these subjects or opportunities. Music is a case in point - you are just simply not going to produce so many good musicians if you 1) don't actually offer the subject, but also 2) don't have a chamber orchestra or chapel choir. Nobody expects the Royal College of Music to accept someone who can't play the violin as well but possibly has the 'potential' just because they didn't go to the Yehudi Menuhin School, so why are we expecting Cambridge to do that? Likewise, nobody expects the Arsenal junior team to accept someone who only has the 'potential' over someone who is already showing that potential fulfilled. Kings College Cambridge has had a reputation now for accepting a high percentage of state school students - the upshot now is that nobody (and I mean not one) in the back row of Kings College Choir is actually at Kings College. Because the best singers will be coming from places that nurture the choral tradition, and that can only be in a place with a chapel choir. I think that's a little sad.
I don't think anyone really believes that indy school kids are innately cleverer (aside from the small percentage benefiting from genetic advantage of generations of success). There will be plenty of intelligent state kids who had they been to indy schools, would have turned out to produce even more sophisticated arguments. The problem is that 18 is just too late to start this process. Lady Margaret Hall in Oxford spearheaded a foundation course for students, and now an increasing number of colleges at both institutions are running it. But by this age, a different style, approach, habits have set in, and it's very difficult to reverse and build on. It would simply require extremely intense and expensive 1-to-1 supervision over a long period to bring out the true potential of all the 'brilliant but failed' state kids. Which really isn't the job of Oxbridge colleges - it would be a much cheaper, easier, more effective job if it were done much earlier. I suppose you could make a case that Oxbridge could start an intensive programme for state school students much earlier - but really this stuff needs to be lived and breathed. It can't be a few weekend courses here and there. Most supervisors are academics whose 'real' work is research. So when they supervise undergrads, they want to talk about actual content and ideas, not have to go through their essays finding typos, looking for where they've just used ChatGPT. That's not a good use of academics' time or college's money.
I don't know what the answer is for Oxbridge, but it's not to just let more state school children in and try to 'top them up'. The result of that is 1) the state school kids are failed - they never turn out as bright, and they drop out more than the indy ones; 2) the standards of these institutions just keep dropping, because not only are the standards of graduates lower, but the universities are spending lots of money and resources on trying to bring up the standards of a few by miniscule amounts. The cost/benefits is woeful.
Like I say, I don't know the answer for Oxbridge. But I entirely understand why Trinity Hall has done this, and said it publicly, in a hope to attract some of the sorts of kids I've described above.