I think that the biggest mistake in the pedagogy of high-school geometry is how we teach the trigonometric functions (sine, cosine, etc). We tell people that these functions are about triangles, and spend a lot of time in classes finding the missing sides and angles in triangles. But they these functions aren’t about triangles at all! They’re about circles, and it just happens that triangles are about circles as well, since just about everything is about circles.This is why π appears in so many unexpected places — circles are really fundamental. But instead of taking the opportunity to teach people how fundamental circles are, and building the skill of figuring out missing lengths and angles of triangles from that,This is actually a very useful skill, if you want to build physical things, and I think it’s a good thing to teach people! It just shouldn’t be the only thing we teach people about trigonometric functions. people are given busywork and told to memorize SOHCAHTOA, as if trigonometric functions are just made of magic. Remembering what sine and cosine do in relation to a circle isn’t just easier, it’s also more generally useful — there are lots of circle-related tasks that come up that are much easier to solve if you know the fundamental definitions of sine and cosine, rather than the triangle based ones.
I think there’s a more general point here about the tendency to teach specific knowledge, rather than fundamental knowledge along with the skills to generalize it, but I’m too tired to come to any real conclusion right now.