Thinking Mathematically

Rating:
Summary: A Students Review
Review: I used this book as a text at The Evergreen State College. It was a tremendous help in developing skills to reason through complex problems. By identifying stages of mathematical reasoning, such as being stuck, generalizing and making conjectures, I have developed an entire new approach towards solving mathematical problems. I now employ techniques in problem solving such as deciding the magnitude of the problem, collecting the information that I know, reviewing problems to find underlying principles and reflecting on the key moments in which I made large steps towards a solution. This new structured technique towards problem solving has provided me with a more thorough understanding of mathematics, better confidence in my answers and a more in-depth analysis of solutions. In addition I have increased my ability to teach others mathematics and to approach all problems requiring mathematical analysis.

Rating:
Summary: A Students Review
Review: I used this book as a text at The Evergreen State College. It was a tremendous help in developing skills to reason through complex problems. By identifying stages of mathematical reasoning, such as being stuck, generalizing and making conjectures, I have developed an entire new approach towards solving mathematical problems. I now employ techniques in problem solving such as deciding the magnitude of the problem, collecting the information that I know, reviewing problems to find underlying principles and reflecting on the key moments in which I made large steps towards a solution. This new structured technique towards problem solving has provided me with a more thorough understanding of mathematics, better confidence in my answers and a more in-depth analysis of solutions. In addition I have increased my ability to teach others mathematics and to approach all problems requiring mathematical analysis.

Rating:
Summary: you learn mathematics by doing it.
Review: 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.

Rating:
Summary: you learn mathematics by doing it.
Review: 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.

Rating:
Summary: Great for the transition to university maths
Review: 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."