Thinking Mathematically

This book really makes you think mathematically. It gives you a systematic approach to solving mathematically and analytical problems. It gives you tips for keeping your concentration on working problems involving numbers. If you make the effort to really go through this book bit by bit it will really help you think mathematically in all areas of your life.

This book helped me quite a great deal with my foray into university mathematics, which are quite different from the algorithmic problems one is often dealt in highschool. Before reading this book, I would often read a problem and just be /stuck/. If it were a test, I would put a question mark in the answer blank and just move along. This is because I didn't have a sense of where to begin with novel problems. After reading this book, though, I learned the tricks of specializing and generalizing. Much of the advice given in the book might seem obvious ("start with small cases," "draw a picture," etc.) but doesn't really get thought of during a stressful exam. By working through this book (and you have to *work* through it, don't expect to read it like a novel trying to glean advice), any sufficiently mathematically-minded person can deserve to call themself a mathematician, for they will truly begin to think like one. After it, they should check out Velleman's "How to Prove It" and R.P. Burn's "Numbers and Functions."

I used this book for a course I taught at the University of Connecticut. It has a lot to offer, especially for the price. Sample problem: draw a bunch of lines across a piece of paper. Can the resulting picture be colored only in black and white, with no adjoining regions sharing the same color? Works through examples like this one in excruciating detail, encourages the reader to sweat through problems, the payoff coming when you start to see patterns not in the problems themselves, but in how you approach them. Last chapter consists entirely of problems, with suggestions on how to attack, and then extend them.