Which Mathematics to Teach?

In 2002, Paul Lockhart wrote an article about the state of mathematics education called A Mathematician’s Lament.  Lockhart argues passionately that what children are typically taught is neither useful nor interesting.  He believes children would get far more from emulating what mathematicians actually do, unstructured mathematical exploration and proofs.  One of the problems, Lockhart acknowledges, is that many math teachers don’t understand or appreciate what he calls “real math,” leaving them unable to teach it.  While I agree we need to improve our math teaching, and I agree that exposing many more children to “real math” is a good idea, I think Lockhart is at once too optimistic and too pessimistic.