Note, you see this effect even within the poor cohorts, I think, in the example of charter schools, which select for motivated students and kids who parents are motivated. I am skeptical about a lot of the claims that any particular method is useful due to this factor alone. Although, charter schools are a mixed bag.
Charter schools do better because they keep out the problems.
I think one thing you can do is encourage more economic integration in school systems, where kids can then at least emulate their peers. And this should go along with more economic integration of housing.
It's not going to work--this will simply trash the better schools, the parents will go private instead.