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In a perfect world, yes, we want our students to learn math (substitute physics, or astronomy, or basketweaving, or whatever else), but we also want them to internalize how to think. I agree that learning about integrating factors makes the light go out in students' eyes, but it's possible to both motivate those "tricks" and use them.

I ran into this when I was a TA for intro astronomy, as well: it was challenging to formulate well-motivated examples, in a way that illustrated that solving a word problem was not just about trying different hammers and pounding away until an answer emerged. Rather, we want students to (a) develop a correct physical intuition; and (b) convince them that mathematics holds the answers. The issues come from getting mired in counter-productive bookkeeping, of the kind we so often see in our "mathematical methods/intro" classes.