Sir Peter Lampl wants schools to do more to improve social mobility; here's some research that suggests some possibilities.
But I'm not sure how much education can do to improve mobility.
Put it this way. Rich parents tend to transmit advantages to their children in all sorts of ways outside school: home tutoring; good genes; investment in activities and books; social networks; a culture that values learning; and positive attitudes.
To offset these advantages - that is, to create genuine equality of opportunity - poor children must get better schooling than rich ones, perhaps much better.
But this is not feasible. For one thing, it's expensive to improve schools; the link between inputs and outputs in education is weak. And there'd be huge hostility to this; just look at the outrage prompted merely by cheap ways to equalize children's chances of going to good schools. And even if poor areas did have better schools, richer parents would game the system to get their children into them.
Instead, a better solution might be to increase equality of outcome - not just through more progressive taxation, but by flattening organizational hierarchies. International evidence suggests countries with less inequality of income have greater social mobility.
One reason for this is that greater equality amongst parents would tend to reduce inequalities in investments in children. Another reason is purely mathematical. The narrower is the gap between the top and bottom quartiles, the more likely are people to move between them.
Also, less inequality of income would reduce the problem of social immobility; if the bottom quartile is a tolerable place to be, it's less of a problem that people stay there.
Now, one might object that progressive taxation is costly. True. But so too is spending on the education of the poor. I'm not at all sure that equality of opportunity is a cheaper form of equality, still less a desireable one.