What Do We Really Know About Teaching Kids Math?

Earlier this week, I wrote about the history of progressive math education, the culture wars it has inspired over the past hundred years, and the controversy over the California Math Framework. Today, I want to start with a much broader question: What do we really know about how to teach math to children? The answer is not all that much--and what little we do know is highly contested. An American math education usually proceeds in a linear fashion, with the idea that one subject prepares you for the next. Take, for example, the typical path through mathematics for a relatively advanced student.