I don't believe this is true. I attended a total of 7 different schools in 4 different cities in 2 different states during my primary/secondary education. Only one of these schools was notably non-diverse. I was very often exposed to children of other cultures/attitudes/backgrounds, as well as other ages. It is possible that my experience is exceptionally different, but it doesn't seem like it.

I tried to find some research on this, but unfortunately everything I find is an aggregate across all schools, rather than diversity within each school.