Adventures in Mathematical Reasoning

Sherman Stein's Adventures in Mathematical Reasoning (which also goes by the title How the Other Half Thinks) is a book which can teach you how a mathematician thinks. I'm not claim that it will teach you, just that it can teach you, if you remain open-minded. For that reason, I think it might me one of the most directly useful math book's I've read.

Adventures in Mathematical Reasoning includes eight chapters, each of which pose a different question at the beginning. All of the chapters involves strings of letters of some sort, although I didn't actually notice until I was almost done with the book. The chapters cover a wide range of topics, from statistics to combinatorics (which is like mixing things up). All of the topics are interesting examples of mathematical ideas, made relatively simple.

The great strength in the book is its ability to show hoe mathematical reasoning works, when actually creating new mathematical theorems. The tricks which Stein uses in these chapters—abstraction, generalization, simplification, or just plain getting data manually—are tricks which can be used in math much more advanced than this. And just about any branch of math would be more advanced. Stein chose topics which can be understood with nothing more than arithmetic, basic geometry (like, "what is pi?" basic), and an open mind.

If you've ever seen a mathematical theorem or idea and thought, "how could they possibly have thought of that?" then you might want to read this book. As long as you take your time to understand everything, and keep trying to guess where things are headed, you should finish the book with a better understanding of how all the work is done. If this sounds good to you, give Adventures in Mathematical Reasoning a read.

P.S.: He usually goes about a similar strategy as is shown in this Numberphile video, so check that out if you want to see an example.