I don’t think I understand a new piece of physics just because I can calculate it. I think conceptual understanding is more important! I and use calculations as a test of my conceptual understanding.
I think I understand what you’re saying now. You’re saying in practice, the only objective standard I’m using for judging whether or not I’ve actually been successful at understanding concepts are these indirect proxies like my ability to calculate things, or non-objective criteria like how I compare to my peers. In contrast, it sounds like you are saying that you—as a philosopher—know of a way to objectively evaluate explanations of math concepts.
Okay sure. That definitely seems plausible to me, and I see how schools are not testing for that stuff at all.
I agree with this.
I think we had a misunderstanding, because I had a very different thing in mind when you asked me to explain induction and trig. I wasn’t even considering trying to fully write something out as if I was trying teaching it to you, I was trying to produce an extremely brief summary to quickly prove that I know it to someone who also knows it. Maybe it’s a problem in and of itself that I interpreted your request in the way that I did, but idk.
In my defense though, I think I could have made something way more like the purplemath or mathisfun article if I realized that was what you wanted me to do. E.g. I am well aware that what I was saying about trig was in the context of right triangles; it took me like 20 seconds to write what I wrote, so I left out a lot of stuff.
If you want to get a better idea of my real conceptual understanding, I think you should ask me to explain a 3rd math concept (or let me re-write what I wrote about induction or trig), given my new understanding of what sort of thing you’re requesting that I write.