The article resonated with me on some level, because it does take a long time to learn how to actually do math. If you are at the point of just doing algebraic

The article resonated with me on some level, because it does take a long time to learn how to actually do math. If you are at the point of just doing algebraic manipulations on equations to try to figure something out, you've lost the battle (as opposed to using algebraic manipulations to encode your thoughts, and work out the details). On the other hand, I think everybody really did see the beauty in geometry. Yes, the initial manipulation to prove something about symmetric angles is a bit silly, but you aren't being taught symmetric angles here, you are being taught how to do a geometric proof. Which is not easy to learn, so you start with something super simple and obvious. I didn't observe anyone in my classes (well this was back in 1982, so memory is a challenge here) confused about that point. And, as soon as we learned it, the requirement for formalism was dropped. It was the same in algebra. In the first weeks you weren't allowed to go directly from x + 3 = 1 to x = -2. You had to do something like x + 3 - 3 = 1 - 3; x + 0 = 1 - 3; x = 1 - 3; x = -2; with all the rules that you are using written out. Annoying yes, once you grasp it, but once you proved you grasped it that was the end of that, and we never had to do it again. But, I had good teachers that always tried to explain the 'why' of what we were doing, and did not make us engage in pointless formalisms. But you need to understand terminology like ABC in geometry; when you get to tough problems you'll be using it. So, learn it with the easy problems.