Mathematicians do not know in advance if their assertions are true, so they need to verify assertions with rigorous proofs or produce counterexamples, then attempt to salvage the assertions by transforming them into theorems.
Extending the Frontiers of Mathematics: Inquiries into proof and argumentation
invites students to experience this progression of mathematical discovery.  Burger's informal, humorous writing style and intriguing puzzles engage students throughout the learning process.  The innovative text offers a discovery-based approach that is ideally suited for use in a proofs course, a discrete math course, or any bridge course to more abstract mathematics.