With respect to education, in this essay about inequality (seriously read that whole thing) Paul Graham makes the point that improving access to education doesn't actually decrease inequality since while to makes the poor richer, it also makes the rich richer.

There is of course a way to make the poor richer without simply shifting money from the rich. You could help the poor become more productive-- for example, by improving access to education. Instead of taking money from engineers and giving it to checkout clerks, you could enable people who would have become checkout clerks to become engineers.

This is an excellent strategy for making the poor richer. But the evidence of the last 200 years shows that it doesn't reduce economic inequality, because it makes the rich richer too. If there are more engineers, then there are more opportunities to hire them and to sell them things. Henry Ford couldn't have made a fortune building cars in a society in which most people were still subsistence farmers; he would have had neither workers nor customers.

If you want to reduce economic inequality instead of just improving the overall standard of living, it's not enough just to raise up the poor. What if one of your newly minted engineers gets ambitious and goes on to become another Bill Gates? Economic inequality will be as bad as ever.