My favorite current mystery novels are the Brodie Farrell mysteries by Jo Bannister. One character is a high school math teacher (as Americans would call him) named Daniel Hood. Here are some things he says about math in Breaking Faith:

“Numbers are different. Numbers I can do. There are no nuances, no room for debate. They always and only mean one thing: you can’t twist them to mean something else… (page 63).

“Look, I’m a mathematician. I’ve never had an original thought in my life. I can’t imagine how you create something entirely new…” (page 116).

Daniel is expressing a widely-held attitude, but he is wrong. Mathematical research does have a place for applying a known method to a problem that obviously can be solved in that way. But math usually requires more creativity than that.

1. To solve a hard problem, you may have to recognize a non-obvious pattern in the problem that reveals its amenability to a known method…
2. …or you may have to reformulate a known method in a non-obvious way to solve the problem.
3. Sometimes you have to invent a genuinely new method to solve a problem.
4. Sometimes you accomplishment is coming up with a new kind of problem and solving it, or part of it.
5. Sometimes in trying to understand a type of mathematical object you have a creative insight that seems to come out of nowhere, an “entirely new” way to think of or describe a mathematical phenomenon.

Most research papers fit (1) or (2). They both involve what I would call minor-league creativity and can get your paper published. (3), (4) and (5) can get you wide recognition in your field. In exceptional cases you may receive one of the several prizes awarded to mathematicians (Fields medal, Wolf prize). Then many mathematicians outside your field will have heard of you.

When you do serious math research you always find yourself thinking of the objects involved in many ways. (For example, if you “twist” the meaning of complex numbers to be vectors in 2-space then you get the Argand representation, which makes some baffling things about complex numbers obvious.) You cannot do serious math without having several metaphors for each object in your head at once.

Jo Bannister may have been putting her understanding of math in Daniel’s mouth. Or she might think that many high-school math teachers think that way even though she knows better. For all I know, many high-school math teachers do think that way.

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