LXXXVIII.

"Mathematical thinking is more than being able to do arithmetic or solve algebra problems. In fact, it is possible to think like a mathematician and do fairly poorly when it comes to balancing your checkbook. Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns. Moreover, it involves adopting the identity of a mathematical thinker.

For instance like most people, when I am doing something routine, I rarely reflect on my actions. But if I’m doing mathematics and I step back for a moment and think about it, I see myself not just as someone who can do math, but as a mathematician.
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In many cases, the real value of being a mathematical thinker, both to the individual and to society, lies in the things the individual does automatically, without conscious thought or effort. The things they take for granted – because they have become part of who they are.
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That then, is mathematical thinking. How do you teach it? Well, you can’t teach it; in fact there is very little anyone can teach anyone. People have to learn things for themselves; the best a “teacher” can do is help them to learn."

© Keith Devlin [link]