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Abstract thinking

Tim_Tex wrote:

Given my difficulties in my calculus class, and now struggling in structural geology, I definitely need to do something that deals primarily with abstract things.

Those are nice subjects. I've done the math stuff, but I've always been curious about studying geology. Sadly, I've only done some modest reading on the subject and don't know much. I've never been much of a collector, though. Mostly I've enjoyed reading about the grand sweep of changes that have taken place over time. In the Pacific Northwest, there is some interesting history I've read going back about 300 million years and I liked what bits I learned from that. One particular area that came up in my lifetime was in 1987 when the first relatively solid case began being made for 9.0-sized earthquakes off the shore of Oregon, every 300-800 years for the past 15,000 years, at least. That was curious to discover about the area.
Anyway, is there a part of geology that might rely more upon the abstract reasoning part and less on calculus kinds of stuff?

Jon

_________________

Say what you will about the sweet mystery of unquestioning faith. I consider a capacity for it terrifying. [Kurt Vonnegut, Jr.]

There is only one correct solution to your calculus problems, and only one reasonable way to find it.

When a person first learns to play the piano, first they just play scales, the same predictable way, over and over.

But 'real' math (and music) is about invention, not calculation. You don't know how to proceed. Different people will take different approaches. Some will find 'elegant' solutions, others will be ugly. Different personal styles emerge.

If you wish to test your mathematical creativity, you might want to try some of the (very, very!) hard Putnam Exam problems you can look up online.

VMSnith wrote:

There is only one correct solution to your calculus problems, and only one reasonable way to find it.

When a person first learns to play the piano, first they just play scales, the same predictable way, over and over.

But 'real' math (and music) is about invention, not calculation. You don't know how to proceed. Different people will take different approaches. Some will find 'elegant' solutions, others will be ugly. Different personal styles emerge.

If you wish to test your mathematical creativity, you might want to try some of the (very, very!) hard Putnam Exam problems you can look up online.

I'm guessing from his posts here that he's lost all confidence about it. I don't think that means anything about him, personally. He may be quite able to get past all this, but has become his own worst enemy and needs someone to show him that he really does have what it takes by helping him to see his own abilities shine through. That takes a very careful kind of hand holding, so to speak. One that doesn't talk down, but helps him find a barrier or two and then to provide just the right kinds of suggestions about how to approach getting through them, letting him find them for himself though so that he can gain confidence from some small and very real successes that he has earned for himself.

These times are difficult and need a personal touch, often. If he's already fully competent, then succeeding at Putnam questions might give him the objective feelings he needs. But in most cases, I've found students have some real but sometimes very narrow 'notches' in their understandings, things that are just enough to prevent them from getting across enough of the hurdles to feel "good at" math, but not really in hindsight all that much.

I remember one student, after three days of constant struggle both on my part in class as well as his own finally came up to me after a class and said, "I got it!!" I brought him over to the chalk board and gave him some new problems. He got every one of them right! I said, "What did I say?? Was it something in particular?" And he said, "Nothing in particular. It just finally dawned on me. And I feel so stupid. It's so easy, now. You were teaching all this perfectly well. Not too fast, not too slow. It was me. I had a mental block. I just couldn't let myself get it. And then.. suddenly, it slipped through."

I've found that I can get through these more quickly, 1:1, because I can listen to the "music" of their math and the way they think when I let them explain their struggles to me and then it's not so hard to conjur up the right problem or two that makes the issue stand out for them or to suggest a new avenue for insight. (There are so many ways to "see" in math, and students just learning to solve some problem need only one of them to start, but what exact one it should be at that moment for a particular student varies and needs to be customized, where possible.)

Jon

_________________

Say what you will about the sweet mystery of unquestioning faith. I consider a capacity for it terrifying. [Kurt Vonnegut, Jr.]

Tim_Tex wrote:

As I was struggling through my calculus homework tonight, I came to a possibly self-defining realization:

I do much better in situations where there could be many answers, depending on how one perceives something--rather than situations where one has to dig deeply and use up tons of energy to find only one correct answer.

I do much better in situations where there could be many answers, depending on how one perceives something--rather than situations where one has to dig deeply and use up tons of energy to find only one correct answer.

I'm the same way.

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