On the whole, America has become a much less racially stratified society than it was in the 1940s, '50s, or '60s. You can see indications of this all over the place, starting with the fact that we have a black man working in the Oval Office. At the same time, pure income inequality has become much bigger and on a variety of different fronts income-linked stratification has become a bigger deal. One way in which this reflects itself is that the "achievement gap" in school between white kids and black kids used to be bigger than the gap between rich and poor. An excellent Sabrina Tavernise article (featuring the chart I've poached) touting research from Stanford's Sean Reardon makes the point very clearly.
Racial education gap narrowing, income education gap growing
February 10, 2012
Slate
By:
Matthew Yglesias
Income achievement gap now nearly twice as large as black-white achievement gap, says Reardon.
Race and class in the United States are obviously linked phenomena, but also to some extent separate ones. They've both mattered throughout American history, but over time race has tended to be very important and in the middle of the 20th century class was at something of a historic low point in importance. But over the past generation the basic forecasts of William Julius Wilson's The Declining Significance of Race have tended to be vindicated as you see here.

