Ungentrified schools in gentrified neighborhoods

This articles describes a study that found that schools in gentrifying, increasingly white areas of New York City often remain predominantly poor and non-white.  This is a trend that I think most people who have grown up in gentrified or gentrifying cities are well aware of.  The white, wealthier families in such neighborhoods often don't send their kids to neighborhood public schools, opting instead for private schools or non-neighborhood special public schools.  I am a product of this trend, as I went to an out-of-neighborhood special program within the Chicago public schools.  During this time, my neighborhood became almost totally white and wealthy, while the neighborhood school half a block from my house remained almost totally nonwhite and poor. 

This is not a good thing, because it means that one of the few possibilities that exists for gentrification to also improve life for the poor in a city is not being realized.  Namely, if the higher income and education levels of the new arrivals do not bleed over as their children attend school with poorer kids, then there really is no upside for the poor families; they see their cost of living rise, but receive no corresponding improvement in their economic situation or the prospects for their children.