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And I like this one! I agree completely that the process of learning mathematics is probably highly multidimensional for... pretty much everyone ever. In parti

by tel 16y ago

And I like this one! I agree completely that the process of learning mathematics is probably highly multidimensional for... pretty much everyone ever. In particular, it's easy to see how formal descriptions can push mathematical generalization forward far before we have a suitable concept of the mathematical object we're describing. I think we're all (incl. the op) in some kind of agreement here about the didacticism of math. The op didn't disregard the power and utility of mathematical languages — he came from being trained pretty heavily in engineering math, at least up to playing with higher-order differential equations — but instead was, perhaps not directly, arguing for increased metaphorical/physical descriptions in taught mathematics. He's just responding to the rather eye-opening feeling one gets when one starts to realize that math is so interpretable! I think that's a perfectly fair argument to have. I know that in my own experience, I never understood the joy of math until the day that linear algebra took an interpretation as linear space transformation. --- So we're just sort of all oscillating here in strong rebuttals of whatever interpretation of the "heart" of math the prior author champions for a while. Which is fun but unproductive. --- I like the challenge of taking someone from physical concepts and metaphors directly to a mathematical object. I think it'd be possible, and maybe even useful when someone first starts to learn real math, but certainly it's not the most efficient way to become well-read. It'd be a lot like explaining the meaning of, I dunno, Día de los Muertos without immersing someone in Mexican language and culture. A single point of contact can be forced, but you lose so much context and fluency.

scott_s 16y ago
I would be more comfortable with the original author's goals if he wanted to augment conventional math education with simulation. Perhaps "augment" is too weak a word, perhaps they should be done equally. But I certainly don't want it to replace manipulating abstract symbols. I think that understanding one helps understanding the other, but both are much less meaningful in isolation. In particular, I think that symbolic manipulation is important because it is both simplified and precise; simulation and visualization is important because it appeals to our intuition.

by tel 16y ago

scott_s 16y ago

I would be more comfortable with the original author's goals if he wanted to augment conventional math education with simulation. Perhaps "augment" is too weak a word, perhaps they should be done equally. But I certainly don't want it to replace manipulating abstract symbols. I think that understanding one helps understanding the other, but both are much less meaningful in isolation. In particular, I think that symbolic manipulation is important because it is both simplified and precise; simulation and visualization is important because it appeals to our intuition.