Anecdotally I know this to be true. I live near a major urban high school in Denver. It's an interesting school as the surrounding neighborhoods have been undergoing rapid gentrification. The student population is half upper-middle class and the other half traditionally poor. It's pretty easy to pick out which group is which based on clothing, habits, and (sadly) ethnicity.
On any given day you'll see dozens and dozens of kids from the poorer demographics skipping class and getting into trouble. The bus into Downtown is filled with these kids literally blowing-off their education.
Talking to teachers at the school reveals what you'd expect. The cultural bias against education among the poorer students means most of those students do really poorly. The kids from more wealthy backgrounds take the same classes, with the same teachers, and use the same curriculum. They actually tend to do quite well.
Point being: I don't think the education system is broken at all. I think we have MAJOR cultural issues at play that heavily bias certain demographics from achieving in school.
I have absolutely no idea how you fix that.