Thinking Mathematically is about mathematical processes, and not about any particular branch of mathematics. Our aim is to show how to make a start on any question, how to attack it effectively and how to learn from the experience. Time and effort spent studying these processes of enquiry are wisely invested because doing so can bring you closer to realizing your full potential for mathematical thinking.
Experience in working with students of all ages has convinced us that mathematical thinking can be improved by
● tackling questions conscientiously;
● reflecting on this experience;
● linking feelings with action;
● studying the process of resolving problems; and
● noticing how what you learn fits in with your own experience.
Consequently while encouraging you to tackle questions, we show you how to reflect on that experience by drawing your attention to important features of the process of thinking mathematically.
How to use this book effectively!
Thinking Mathematically is a book to be used rather than read, so its value depends on how energetically the reader works through the questions posed throughout the text. Their purpose is to provide recent, vivid experience which will connect with the comments that are made. Failure to tackle the questions seriously will render the comments meaningless and empty, and it will be hard to use our advice when it is needed. Three kinds of involvement are required: physical, emotional and intellectual.
Probably the single most important lesson to be learned is that being stuck is an honourable state and an essential part of improving thinking. However, to get the most out of being stuck, it is not enough to think for a few minutes and then read on. Take time to ponder the question, and continue reading only when you are convinced that you have tried all possible alleys. Time taken to ponder the question and to try several approaches is time well spent. Each question is followed by suggestions under the heading STUCK? to provide signposts when progress seems blocked. Because different resolutions follow different paths, some of the suggestions might be mutually contradictory, or irrelevant to your approach, so do not expect every suggestion to provide immediate insight!
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