I think the problem is that not enough time is spent explaining these very common techniques - what the rationale behind them is, hot to intuit that it could be applied to a specific problem, and how to get from that intuition to the actual answer / application.

Often, there's a very large skill gap between the student and the teacher (especially at college level, where many of the career mathematicians live and breathe maths), and these things are hand-waved away as obvious. Even worse is when the teacher doesn't actually know, and is just presenting the material straight from a guide. The way most course material seems to be set up is to skip over the middle part, and as a result the best short term strategy is to learn it as a bag of tricks.

I love maths, but I think its universal applicability and beauty get lost due to the way it's generally taught.

You know that from study and experience. If that underlying concept wasn't the lesson, like the parent indicated, it would I deep end up a trick in the bag.

The toughest part of teaching, I think, must be really knowing that your students are internalizing the core rules/principles/concepts behind the examples you teach with.

I remember a similar problem in Calc 2. I forget the specifics now, but I think it was an integral of some combination of sin/cos that ended up being circular. You had to recognize an opportunity to swap one of the steps for an equivalent, which would lead you to the final solution.

Probably the second example here [1] for those curious (I think the integral of sin(x)*e^x dx is the only place I've seen this used, would love to know if there are other examples).

[1] https://en.m.wikipedia.org/wiki/Integration_by_parts#Tabular...

This becomes much more transparent if you realize you're integrating Im(e^x * e^{ix}). And it's no longer a trick but a technique.

It would be great if that were how it was taught, but when I took the class it was taught as a trick. No theory behind it, just the prof on the board saying, "But look! :swap: And now you can integrate it."

It becomes a technique once you realise it is a specific case of change of bases. Even just getting to basic theorem of algebra does not provide this insight.

The common techniques ARE a bag of tricks. Feynman was famous for being really good at integrals, because he had memorized the huge bag of tricks. Today, we have Mathematica for that, you don't need to be Feynman.

This makes me wonder. Is Mathematica also applying a bag of tricks (I suppose in a breadth first search), or does it have a more structural approach?

As I recall, Mathematica embodies a (perhaps incomplete) implementation of the Risch algorithm.

I don't remember it being presented that way at all. If it really is a core technique, I would expect it to have been emphasized strongly and the test to have a problem with a different expansion instead of the exact same 2 -> 1/2 + 3/2 problem that he had done in class.