Poorer children, on the whole, DO achieve less in school. I'm not sure if this is because they CAN'T do as well. Most of them are from single-parent homes, often the parents are very young and don't have a clue, many of the parents don't have a lot of time for child-rearing (whether it's because they are working or partying), and on the whole there isn't as much emphasis put on education as there would be in middle-class homes.
On the other hand, if you expect less of these children, you get what President Bush expressed so well as "the soft bigotry of low expectations" -- the children end up undereducated, with no job skills to speak of, and continue the cycle of poverty.
What they need is something along the lines of parochial schools for at least the elementary grades, IMO.