Hot take: do we really need intuitive explanations for anything? I love 3b1b videos for example, but I am more in the "mechanical application of rules" camp rather than "a-ha moment" camp.
That's why I hated integrals at the calculus course. It's all weird tricks all the way down - which a mortal couldn't come up with.
For many people, it’s deeply dissatisfying to mechanically apply magical incantations that you don’t understand and could never arrive at yourself. Once you intuitively understand something, it also makes it trivial to remember it, and also helps connect it to other information in your brain.
Intuition seems to me to be the point at which you really understand something - and it allows you to make progress and predictions about other phenomena.
For example, I don’t think anyone has an intuitive understanding of quantum mechanics yet, and it seems to be limiting progress. String theory was an attempt at an intuitive understanding that has so far been fruitless.
Spacetime is a counter example, where a difficult concept was made graspable by some intuitive concepts and analogies, and progress very quickly seemed to follow.
The way we were introduced to integration, the different rules (tricks as you put it) made sense. If you are using rote memorization and cannot derive the "tricks," I imagine all of math is harder to approach.
As a former inner-city high school math teacher, recovering, students who tried to memorize mechanical approaches could be easily spotted when they forgot or recalled incorrectly. Simple example: they may not remember X^1 and/or X^0 and they lack the understanding to figure it out.
Interesting. I disliked calculus for exactly the opposite reason. I mean, it also struck me as "all weird tricks, all the way down". But it seemed like a list of tricks one needed to memorize, lacking an intuition.
I was a TA for the EE course that all the ME's had to take. They kept trying to understand Kirchhoff's laws and do circuit analysis using intuition, using water through pipe analogies, stuff like that. The problem is, electricity just isn't intuitive, the analogies are not perfect, nothing in the mechanical world that we understand is quite like it. In fact, we don't really even know why electricity works like it does. I finally told them to stop trying to understand and just do the math and then they had a much easier time.
That's why I hated integrals at the calculus course. It's all weird tricks all the way down - which a mortal couldn't come up with.