What does it mean to understand a certain area of mathematics completely? One possible answer is that you understand it when you can solve its problems *mechanically*. Consider, for instance, the following question. Jim is half the age of his mother, and in twelve years’ time he will be three-fifths of her age. How old is his mother now? For a child who is just old enough to understand the concept of “three-fifths,” this is likely to be an impossibly difficult problem. A bright and slightly older child may be able to solve it after some hard thought, which will probably include a certain amount of trial and error. But for anybody who has learned how to translate such problems into equations and who ...

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