A short and dirty answer: they should read Imre Lakatos' Proofs and refutations (Google books), to get an idea of how mathematics develops/is built/is discovered. It is little use delving into the thornier issues of (in)completeness, for example, without knowing how mathematicians think about and develop their subject.

The book uses the simple example of the Euler characteristic of a (convex) polyhedron, usually expressed by the formula $V - E + F = 2$, and proceeds by dialogues. I don't believe any huge leaps of mathematical understanding are needed.