Polya's How to Solve It is all about heuristics and strategies for problem solving, but it's mainly written for problems which require students to find values, not to prove theorems. While Polya's methodology works even for theorem proving (and even theorem finding), I suppose, but at the same time it would be nice to have a similar book especially for the kind of problems students encounter later in their life.
Is there such a book? In particular:
- In How to Solve It, Polya wrote (if I recall correctly) he was going to write a similar book especially for theorem proving. Did this actually happen?
- Many people here have recommended Velleman's How to Prove It. I have only skimmed at its table of contents, and I felt that this book was more about what a valid mathematical proof was and about individual tactics that appear in proofs, rather than about heuristics and strategies about combining the tactics organically. Can I use the book to learn heuristics and strategy for theorem proving, just as I could with Polya's book for elementary problems?
Sorry for the plug but this is precisely what my book Proof Patterns is about. The idea is to learn and name the patterns that come up commonly in undergraduate mathematics proofs.