As someone both studying and teaching math: there should be two different ways to teach math - for other mathematicians and for non-mathematicians.

For mathematicians you want to use all the formal proofs and sharp definitions and so on. But we have so much fun teaching that way, we forget when we switch classes that engineers don’t like/care/are motivated to think the same way. We should pivot towards application-based, result-oriented teaching but we often just don’t. And students have to deal with it because the other class managed (pure mathematicians).

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