If you haven't started watching the series "Breaking Math" then you should. It's a new Youtube TV series where the main goal is to:
- Find a major flaw in math that mathematicians can't solve.
- Understand the problem.
- Look for a real world math problem that allows us to experiment.
- Look for ways to solve the flaw.
http://www.youtube.com/view_play_list?p=C0A040EDA1C10909