After school programs, ensuring all schools have equal funding, getting better at determining which kids aren't succeeding, and spending more time with them to give them the one on one support they need. I mean it's a very hard problem, and maybe it's unsolvable, but I think we have a lot more things we can try before giving up and trying to put Band-Aids on the problem.
I think if class sizes are sufficiently small though, at least in grade school, a teacher should know every student, and be able to identify those who aren't getting the support they need at home. If they are a good teacher, they can act accordingly, and maybe work with the parents to help the child succeed.