Customer Reviews

Thinking Mathematically

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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.

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.

For anyone who might be interested, this book is available in Spanish as "Pensar Matemáticamente", from at least two publishers: Anagrama (Argentina) and Labor (Spain). Its ISBN is 84 335 5139 6 .

This is my favorite math book because it advises us to find problems that interest us, especially ones that have to do with things that we observe in our surroundings, and explore them without turning them into a competition. Of course, the authors also show us how to investigate such problems effectively! For such reasons, it's an effective accompaniment to books like Writing to Learn Mathematics: Strategies That Work, K-12 and Mathematical Enculturation: A Cultural Perspective on Mathematics Education (Mathematics Education Library) (In Spanish, Enculturacion Matematica: LA Educacion Matematica Desde Una Perspectiva Cultural (Spanish Edition).

This book has made me more aware of my own weaknesses, and not only those that have to do with math. Its suggestions, and the atmosphere they create in the classroom, really encourage students. Especially important are its themes that

1. Becoming "stuck" is an honorable condition, and essential to improving thinking. It happens to every mathematician. (E.g. how long did it take to prove Fermat's "Last Theorem"?)

2. For thinking to grow, the important measure is not the number of questions solved, but the quality of thought put into tackling the question and reviewing the effort.

3. As many educators have shown, practice alone does not improve us. Instead, the key to becoming adept in any field is to "accumulate a rich store of useful and accessible experience, making analogy-seeking more fruitful.[Therefore] success, with Review of why it was successful, does breed success." (p. 109)

For more details on how this book is useful to the teacher, please see also my review (March 18, 2010)of Engel's , Problem-Solving Strategies (Problem Books in Mathematics).