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 Posted: Mon Nov 20, 2006 7:59 pm
This isn't really math, but rather about how to think about math. Epistemology, and that sort of stuff.
So I was drinking tea with my co-CAs (it would be funny if they were the duals of CAs xd ). Anyway, they tend to snipe at one another over how they understand things, since one is an algebraic number-theorist and the other is a topologist/geometer. The TG had a triplet of interlinked rings that we were playing with; the AN couldn't visualize why you can't realize Borromean rings physically with rigid circles, so the TG made fun of him for that.
Later, we were trying to find the name of a textbook; the TG remembered what the book looked like but couldn't read the image, so the AN remarked that if the TG memorized things more abstractly, as text, then he would remember.
Last week they were arguing about inversion geometry on the plane (sending x->x/|x|^2). TG had it as projecting R^2 onto the sphere and then rotating the sphere 180 degrees, then projecting back. AN had it as applying an involution to the function representation of the circle.
What do people here do? I tend to side with the TG, being that I'm TG myself. You guys?
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Posted: Wed Nov 22, 2006 10:33 am
Hmm. Interesting question, surely since in our class, 5 out of eight also like things that are very abstract and try to think that abstract. 2 of them are even kinda extreme in that.
While the 3 others more try to visualize the things we see. I'm one of the last three and, to be honnest, the worst of them. I love math, but when something is too abstract and impossible to visualize, it's very difficult for me to fully understand it.
Off course, as a result of the above, there are sometimes conflicts, merely me (sometimes together with a few others) against the two (sometimes three) very abstract ones. For example, they hate statistics, saying it's not math while I like it. They off course love the very abstract things, sometimes trying to make it more difficult, while those things most of the time are already very difficult for me and a few others.
It's kinda clear by now what my answer is, I guess.
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Posted: Fri Nov 24, 2006 5:38 am
Thinking is hard enough. Thinking about how I think is nearly impossible.
I don't really feel like I have a set way of thinking just yet. I'm much more visual than most of the other people in my classes, but I prefer to use the visual stuff to understand, and work the problems from a more abstract level. To me, abstraction represents the more...hard-lined nature of a problem, while the visual represents intuitive understanding. You need both to really attack most problems.
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Posted: Sat Nov 25, 2006 3:48 am
I have always loved geometry and tend to prove things geometrically than algebraically. To be quite honest, I am a math student who somewhat hates numbers or algebra because I am a highly visual person. I like the being able to imagine a proof or solution without having to use long algebraic equations because I tend to be careless in writing my solutions. If I made a mistake in my long solution, it sometimes takes me some time to spot the error (and I often do more than two steps in writing a solution because I am too lazy to write a whole bunch of numbers/variables). For me, the geometric is more efficient way to solve a problem because I find it less time consuming- just imagine the situation and everything else falls into place. I like improvisions rather than the rigid technical structures
Even if the visual and abstract seems contradictory, it can sometimes help one another. In one occassion, I forgot a formula during a test so I derived it by using a picture/graph. If only I can translate the abstract into visual and the visual into abstract, I would have an easier time in all my subjects.
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Posted: Sat Nov 25, 2006 8:04 am
I find algebra easier than geometry, but I'm happiest when they're together smile
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