Mathematicians love to play with ideas. They experiment with puzzles. They tinker with the connections between shapes and numbers, patterns and logic, growth and change. To a mathematician, the fun of the game is in experimenting, in trying new things and discovering what will happen. Many modern strategy games were invented primarily for the fun puzzle of analyzing who would win.
Wise mathematicians are never satisfied with merely finding the answer to a problem. If they decide to put effort into solving a math puzzle, then they are determined to milk every drop of knowledge they can get from that problem. When mathematicians find an answer, they always go back and think about the problem again.
Our childhood struggles with schoolwork gave most of us a warped view of mathematics. We learned to manipulate numbers and symbols according to what seemed like arbitrary rules. We may have understood a bit here and a bit there, but we never saw how the framework fit together. We stumbled from one class to the next, packing more and more information into our strained memory, until the whole structure threatened to collapse. Finally we crashed in a blaze of confusion, some of us in high school algebra, others in college calculus.
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Denise Gaskins' Let's Play Math 