The lack of social mobility will be a familiar refrain during the general election campaign. The seven per cent of children who are privately educated exert an inordinate dominance over British life, as The New Statesman has frequently highlighted: they provide 70 per cent of high court judges, a majority of FTSE100 chief executives and journalists and over 40 per cent of students at Oxford and Cambridge.

The implication seems clear: Britain is becoming a society ever more rigged in favour of those born into family wealth and connections, especially in an age of annual university tuition fees of £9,000. If it is a depressing picture, it should not be mistaken for the complete one. University education has never been more accessible.

There are more disadvantaged pupils at university in England today than ever before – including at the most prestigious universities. Since 2006, entry rates for disadvantaged pupils to university have soared by 61 per cent, and disadvantaged pupils have gone from being four times less likely to go to university than advantaged ones to only 2.5 times less likely. Entry rates for disadvantaged pupils at the universities with the highest entry requirements have soared a quietly staggering 40 per cent since 2011.

In two of the last three Parliaments, governing parties have reneged on their promises not to increase tuition fees. But this has not stopped university applications rising from people of all background. Indeed, as Les Ebdon, the Director of the Office for Fair Access, reckons: “It’s possible that the tuition fees debate has galvanised some universities into upping their game” and opening their institutions up to disadvantaged students.

In January 2004, Labour faced losing a vote on increasing tuition fees, which Tony Blair and Charles Clarke believed was imperative to secure the long-term funding of universities. To get the support of recalcitrant MPs, the bill contained a provision to create an independent body, the Office for Fair Access, that would monitor university access agreements – and, crucially, had the power to stop them charging higher fees if they failed to do enough to open up their institutions. This month marks a decade since the first set of university access agreements were approved by OFFA.

“The fact that OFFA has to approve access policies in return for universities being allowed to charge full fees gives it significant leverage and it has been effective at promoting outreach programmes in particular,” says Rick Muir, a specialist in education at the think-tank IPPR. Emran Mian, the Director of the Social Market Foundation, is less effusive, doubting how much universities fear the scrutiny of their access agreements, but credits OFFA with “a useful role in sharing best practice around universities.”

“If they don't come up with sensible plans, they're not ambitious in their own targets and if they're not delivering on what they say they're going to deliver, then of course I’ve got a very powerful weapon - the nuclear option,” says Ebdon. “I’m always tempted to go down that route.” The “nuclear option” – a phrase which attracted uproar when Ebdon used it before he took over in 2012 – has never been used by OFFA, but Ebdon credits its mere existence with spurring universities on to be more proactive. Three years ago, OFFA had a tempestuous meeting with Cambridge University over whether its colleges’ access plans were adequate.

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Because of fears that poorer children would be priced out of university, OFFA initially focused upon putting pressure on universities to ramp up their bursary provision: a mistake. “It was seen as above all about money in those early years but it was very clear, very quickly that wasn't the main issue,” says Martin Harris, the first Director. By ranking universities’ inclusivity on the basis of their bursaries, OFFA was “set up to tackle some of the wrong questions.”

It turned out that generosity of bursary provision was almost irrelevant: students rarely researched the specific bursaries offered by institutions, and it was not uncommon for one-third of eligible students to fail to claim their bursaries. “As we now know bursaries don't influence what universities pupils go to,” Harris says. Rick Muir from IPPR explains: “Research has shown that cash bursaries have a limited impact on the behaviour of students from disadvantaged areas, and that those students’ applications do not change in favour of universities that offer higher bursaries.” While there is limited evidence that bursaries help ensure disadvantaged students remain at university, “it's still a very untargeted way to do that,” notes Nick Hillman, the Director of the Higher Education Policy Institute, who cites Oxford University as a particularly bad example of bursary funds being “spent inefficiently.”

“It was always clear to me that money was not the decisive thing - a whole range of social factors were going to underlie whether any given group went to university or not,” reflects Harris. “You can get a much better return on the money you invest by effective forms of outreach and working very closely with schools than you will ever get by getting individual students who have already decided to go to university a sum of £600 or £900.” He “encouraged universities to move away the bursary road and down the road of outreach and summer schools.”

For far too long, university access schemes were woefully inadequate. “Sending the senior tutor once every ten years to a school in Truro was pointless - you might feel morally virtuous,” Harris says. “What you do need is regular contact between selective universities and schools which will have at least a few candidates.” He believes that universities allowed their commitment to access to dwindle after grammar schools were closed en masse in the 1960s (though he does not support them). “The gap between state school high flyers and the selective universities got wider because less effort was made to keep in touch with young people who were harder to find because they were scattered more.”

Greater outreach efforts from universities in the last decade have gone in tandem with improvements in state schools. “Until fairly recently, there were still teachers in some schools who would say these posh universities are not for you. This is hopeless,” Harris says. Early in his reign, OFFA analysed a North London state school that was very successful at getting students to university, but not at getting those with top grades into elite institutions. OFFA interviewed a cohort and heard an array of reasons why children with the potential to go to UCL or KCL opted for South Bank instead. They said the alternatives were more expensive (the opposite was true as they would have received far more generous bursaries), or even too inconvenient to get to on the tune. “All their mates were going to South Bank,” Harris says. “What you've got to do is get KCL or UCL to have events in schools like that, and regularly.”

While there is a long way to come, standards in state schools have been dragged up in the last 20 years. “The biggest single factor is likely to be improvements in achievement in school”, especially in London, says Emran Mian, the Director of the Social Market Foundation. More than anything else, the quality of state education determines the socioeconomic make-up of elite universities.

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For all the remarkable improvements in university access, the challenge ahead is no less daunting. In 2006, the most advantaged students were 9.2 times more likely to go to elite universities than the most disadvantaged students; the gap has fallen by a quarter, which is testament to the effectiveness of OFFA and the improvements in access schemes. But the most advantaged students are still 6.8 times more likely to go to the best universities than the most disadvantaged students, and 2.5 times more likely to go to university at all. The gains of the last decade have not been spread equally, either: in 2006 white children were more likely to go to university than black children, but they are now seven per cent less likely, and comfortably the worse performing ethnic group.

And politicians and universities have been guilty of neglecting vast swathes of students: part-time and mature students. Ebdon accuses politicians of thinking of “vanilla students” – school leavers going to university – at the expense of others.

Neglect has brought a severe cost for part-time and mature students. The number of part-time students in England has halved, from 248,000 to 129,000, since 2010, a decline that cannot merely be attributed to a reduced pool of potential students in an age when far more people go to university at 18. “That’s a very real concern to me,” Ebdon says. “More than average, part-time students come from those disadvantaged groups that we're most concerned about.”

As I have noted before, part-time and mature students are the forgotten victims of the tuition fee rise: the decrease in part-time students in England has not been mirrored in Northern Ireland and Scotland, which have lower tuition fees (and none at all in the case of Scotland). “The new fee system has militated against flexibility of provision and that’s very important for part-timers and quite possibly mature students as well,” Ebdon believes. “Whereas young people seem to worry a lot less about carrying the burden of debt, mature people seem to worry more. It may well be if you've already got a large mortgage debt that you’re not looking for another debt.”

“You say to yourself: will I get a nine per cent pay rise for finishing this course? And the answer is probably no so the whole balance of reward has been changed,” Ebdon says. “There are very strong reasons for saying that the government should encourage part-timers - because while they're studying they're also paying tax, they aren't collecting student maintenance, they're being useful to society, they're upskilling in a way that we know that in a high skill economy you need to do.” He implores the government “to be a much better employer”, by doing more to support employees to pursue part-time degrees.

Ebdon says that mature or part-time students (the two groups often overlap) “often do have very real financial pressures and walk out suddenly”. But too little has been put in place to prevent them from doing so. He advocates making loans for mature and part-time students available “on a much more flexible basis” to make them more attractive.

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Much more could also be done to make university more accessible for vanilla students. Both Ebdon and Harris are ardent supporters of contextual data. This attracts the opprobrium of the right-wing press, for seeming to give preferential treatment to those from less advantaged backgrounds, but the evidence in its favour is overwhelming. A Higher Education Funding Council for England report last year found that state school pupils with BBB were as likely to gain a 2:1 or first as private school applicants with ABB at A level. While 70 per cent of English universities make some use of contextual data, including Cambridge and Oxford, none yet regard a slightly lower set of grades from a poor school as superior to slightly better ones from an elite school. “Some universities worry a lot about the legal ramifications of that,” Ebdon says. “I’m not worried, I think the evidence base is strong enough to sustain an argument that what universities are looking for is potential.”

The balance of spending between bursaries and other access spending has become much more efficient – in 2009/10, 90 per cent of total access agreement expenditure was spent on bursaries, compared with 69 per cent today. Yet it might still be overly tilted in favour of bursaries. “We think it’s smarter to spend it on activity. I’ve been saying to universities I’m not after greater expenditure, I’d like to see a more strategic spend. We think that money spent on activity is more impactful than money on bursaries,” Ebdon says, advocating, “more evidence-based expenditure” and “more collaboration between institutions.”

“Vice-chancellors tell me the reason why they spend so much on financial support is that they feel that’s what expected of them - that the newspapers, students and certainly donors and alumni all expect that. It’s up to us to get out the message that says actually they are some very impactful ways of increasing numbers of disadvantaged students in universities and they don’t all involve handing out bursaries.”

If too much is spent on bursaries, insufficient amounts are spent on maintenance grants. “Relatively little attention has been paid to them compared to fees - I think they're incredibly important psychologically,” Ebdon says. “The advantage of maintenance grants is they conform to the principles of good student funding: they’re predictable, you’ve got a clear line of sight and you get the money when you need it.”

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Yet serious attempts to improve access need to confront the limits of what universities themselves can achieve. As Harris says, “Universities are doing more than ever but I don't think that could ever be the entire solution.”

One way of improving access would be for all students to apply after their A-levels: state school pupils suffer because their teachers are more likely to under-predict their grades than is true of privately educated students. Both Harris and Ebdon support the government introducing post-qualification admissions.

Just as the NHS and social care should not be viewed in isolation and require an integrated solution, the same is true for schools and universities. Take Sure Start, a scheme set up in 1998 to provide early learning for deprived students, which could be critical in helping more disadvantaged children go to elite institutions. Harris reflects: “Someone said ‘shouldn't universities start running Sure Start’? That’s nonsense - someone should run Sure Start, but it’s not the universities.”

Boosted by Vince Cable and David Willets, OFFA today has more clout and resources than ever before – 22 staff and a budget of £1.6 million, compared with four staff and a budget of £500,000 when it was created. Neither Ebdon nor Harris advocates more power for OFFA over university admissions policy – they are both self-professed believers in the autonomy of universities. “They set their own targets, they tell us what they’re going to do to achieve it,” Ebdon explains. “We've harnessed the creativity and innovation of the universities to come up with their own ways of doing it. With universities that works so much better than telling them what to do, because if you tell them what to do they use all their innovation and creativity to get around it.”

But Harris suggests that OFFA needs a broader mandate to be truly effective. “An ideal replacement for OFFA would not be only related to universities - it would be a joint endeavour between schools, further education colleges and universities. It would be looking at kids from 11 onwards,” he says. “The next step for access - not necessarily for OFFA - is to encourage schools to work with the university sector and offer outreach efforts, to have inreach efforts by identifying people at schools, absolutely no later than 14, and help to make the right transition from GCSEs to A-levels.”

Universities have made sterling - and largely unacknowledged - progress in opening up their gates to disadvantaged students in the last decade. But if they are to become truly representative of society today, universities need much more help from the wider education system.

These interviews took place before the beginning of the purdah period.