The Value of Failure

In his latest MOOC blog posts Keith Devlin insightfully discusses expectations, real math, mathematical thinking, AND, the 'power of failure.'
According to Keith high school math and college math are "in many ways completely distinct subjects," and the "mathematical thinking" needed at the college level cannot be "taught" but must be "learned," which includes "learning by failing":

"...it is only when we fail that we actually learn something. The more we fail, the better we learn; the more often we fail, the faster we learn. A person who tries to avoid failure will neither learn nor succeed."

Read his post (part 2) if you need clarification on all that -- heck read both posts (much food for thought) even if you need no clarification! -- at the risk of sounding like a broken record, these ought not be missed if you're an educator:

http://mooctalk.org/2013/12/14/maththink-mooc-v4-part-1/

http://mooctalk.org/2013/12/15/maththink-mooc-v4-part-2/

ADDENDUM (12/17): now part 3 of this series from Keith is up: 
http://mooctalk.org/2013/12/17/maththink-mooc-v4-part-3/

By the way, elsewhere, Dr. Devlin cites J.K. Rowling's 2008 commencement speech at Harvard (having to do with "failure") as possibly the best commencement address ever. If you've never seen it, enjoy: