Earlier this month, Katharine Birbalsingh announced her resignation as Chair of the Social Mobility Commission. Defending her decision, she wrote that her “propensity to voice opinions that are considered controversial” was putting the commission “in jeopardy”.

She is right. From arguing that children shouldn’t be allowed smartphones to suggesting girls were put off taking Physics A-level as it contained too much “hard maths”, Birbalsingh regularly attracted public backlash for the opinions she expressed.

Her views on Oxbridge, however, appear to have hit the most sensitive of social nerves. In her inaugural speech in June, she spoke of the need to radically rethink what social mobility means, calling for a move away from the fixation with “rags to riches tales” of “caretaker’s daughter goes to Oxbridge and becomes a top surgeon”. Instead, she appealed for “small steps up the ladder” rather than “giant leaps”, in a bid to promote social mobility for a wider range of people, not just by creating “elite pathways for the few”.

Her views sparked outrage when they were misrepresented in some parts of the media. As pointed out in her resignation letter, the press painted her as believing that “working class people should stay in their lane”, when she merely called for a broader conceptualisation of the mechanisms of social mobility.

“Deep down, we want to believe that an elite education will cue the shattering of the class ceiling”

Her comments also require us to question the idea that Oxbridge is a golden ticket to a successful and fulfilling life. Deep down, we want to believe that an elite education will cue the shattering of the class ceiling. We are seduced into believing we will be on equal footing with our peers who went to private schools. This myth of meritocracy is in part perpetuated by the same narrow notions of social mobility that Birbalsingh sought to dispel.

I should know – I too have been blinded by academic glitter and gold. When I received my offer from Cambridge, as an 18-year-old at a state school in Swansea, I was told by those around me that I had “made it”. Family and friends joked about how one day I would be the Prime Minister. My foot was in the door, and the sky was my limit.

It is high time we assess more realistically what an Oxbridge degree alone feasibly can and cannot do for social mobility in our country. A diploma from one of Britain’s two best universities undoubtedly opens doors. But an Oxbridge education might not always be the “giant leap” we expect or want it to be, nor does it give you the means to undertake a six-month long unpaid internship in central London. Opportunities like these are out of the question for most people who don’t have a home in London nor parents with the extra cash to support them.

It also will never serve as a substitute for the “right” accent or the “right” mannerisms. It doesn’t tell you to wear black shoes to a job interview. A senior leader at a FTSE 100 firm anonymously admitted: “We all know that people with the right accent, the mannerisms, you know… sound much more believable. Equally, I want to say we can see through that, but the truth is we can’t”.

“You need to pretend to be posh to get ahead in life”

Amol Rajan’s recent two-part BBC documentary, “How to Crack the Class Ceiling” tells a similarly bleak story. The programme follows a group of young working-class graduates from elite universities trying to find jobs in media, law and banking – professions still largely dominated by the privately educated. All struggle to find a way “in”, continually thwarted by intractable factors that even a top degree cannot reverse. Rajan’s tragic conclusion is that you need to pretend to be posh to get ahead in life.

Birbalsingh is right about “small steps”. Progress has to start somewhere, and lasting, systemic change will take generations. Oxbridge itself certainly has a role to play in supporting its students. My own college, Pembroke, has been taking steps in the right direction. It recently launched its LEAP Scheme, a programme offering support for internship applications, mentoring sessions and opportunities to network with industry professionals.

Ultimately, simply working to increase the share of state school students at Oxbridge won’t shatter the class ceiling. To start seeing cracks at its seams, we must start by rethinking what degrees from elite universities can and cannot do for social mobility more broadly.