Mathematics is, in itself, a perfectly polished diamond.  But what about the messy, imperfect human endeavor of doing mathematics?  How can we understand—and how can we help our students understand—this duality?

In this talk, we will explore some of the near misses of mathematics: the botched calculations, the pseudo-patterns, the almost-theorems, and the specious proofs.  By reflecting on how mathematical reasoning goes wrong, we hope to gain insight into the nature of mathematics and into the human struggle of learning it.