I have to say, I actually loved this article. Especially “Understanding doesn’t build fluency; instead, fluency builds understanding.”
I love math and majored in it in college. The rest of my family is all scientifically inclined, but I think found/find math itself opaque and somewhat intimidating. I remember my brother asking me at one point how one would ever find, for example, the Pythagorean theorem intuitive. The author’s quote is the response I wish I had. The Pythagorean theorem becomes intuitively true not when you have some deep insight about Euclidean space, but when, on seeing a right triangle, three proofs of it spring instantly to mind. Which happens after a lot of practice.
FWIW I think it’s appropriate that the author talks about herself a lot. She’s trying to explain the subjective, cognitive experience of going from math-phobia to math mastery over her career. She can’t explain that without talking about her background and her perception of the process from inside her head.
That talk is about something superficially different—ego in math—but on reflection, I think the desire to look smart actually really does set one up for success in math in the particular way that the OP article describes.
When you just want to look smart, you don’t care whether you know something because you thought of it or because you read it in a book. You just care that you can show off what you know and solve problems easily. So you voraciously read and memorize and try to accumulate a massive mental database of facts to show off. Then at the end you find you’re actually good at the thing.
> When you just want to look smart, you don’t care whether you know something because you thought of it or because you read it in a book. You just care that you can show off what you know and solve problems easily. So you voraciously read and memorize and try to accumulate a massive mental database of facts to show off. Then at the end you find you’re actually good at the thing.
What should one do instead, in order to avoid merely “looking”/“sounding” smart?
Just do math. A student driven merely by the pleasure of doing math without concern for external validation is lucky. But if external validation is a driver, that's lucky too. In both cases, math gets learned.
>>> The Pythagorean theorem becomes intuitively true not when you have some deep insight about Euclidean space, but when, on seeing a right triangle, three proofs of it spring instantly to mind.
To be honest, this sounds like orienting one's self in the 'space of mathematics'. Is it not possible that, just like one can navigate by landmarks (proofs) or by the space itself (deep understanding), that there are in fact two roads to intuition in mathematics, of which ones is practice and fluency, and the other is deep insight and understanding?
I'm not sure I understand the question. Yes, it is often attributed to him. If you mean "is it not true that he said it?", I don't know, but my suspicion, as with many such famous quotes, is that he probably at best said something like, or reminiscent of, it.
I love math and majored in it in college. The rest of my family is all scientifically inclined, but I think found/find math itself opaque and somewhat intimidating. I remember my brother asking me at one point how one would ever find, for example, the Pythagorean theorem intuitive. The author’s quote is the response I wish I had. The Pythagorean theorem becomes intuitively true not when you have some deep insight about Euclidean space, but when, on seeing a right triangle, three proofs of it spring instantly to mind. Which happens after a lot of practice.
FWIW I think it’s appropriate that the author talks about herself a lot. She’s trying to explain the subjective, cognitive experience of going from math-phobia to math mastery over her career. She can’t explain that without talking about her background and her perception of the process from inside her head.