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Academic Guidance Academic Guidance - Academic guidance for those pursuing a college degree... what college? Grad school? PhD help?


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June 1st, 2010, 01:51 AM   #1
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What you wish you knew - designing a course

I have recently been asked to help design a new introductory course for entering freshmen math majors. In general, we expect almost all of the freshmen to have taken a calculus course in high school, but about half will be repeating Calc I at the university. This is a small course, only one hour per week.

But I think that with one hour per week, many great things can be accomplished. So I wondered- what do you wish that you knew early on? How do you explain what math really is?

For example, when I learnt Galois Theory, I was actually studying in Budapest for a semester. And it turns out that everyone around me knew things about symmetric polynomials and complex numbers that I simply wasn't familiar with. In fact, I thought that I should have been familiar with them. But I think that symmetric polynomials, complex numbers, and many other ideas from classical algebra are easily accessible to an entering freshmen math major, and so they merit consideration for this course. I also want to give a sense that math can be more than just arithmetic, and perhaps even start looking into the idea of proofs (although there is a stronger proofs/vector spaces course further on).

Math education is something that interests me greatly, and I'd be thrilled to hear your ideas and discuss what we think. I also know that it's difficult to say what you would have wanted to know, as this school has its own background, and people will come in with a variety of skills as it is. But nonetheless, this is a start.

What do you think?
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June 1st, 2010, 11:55 AM   #2
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Re: What you wish you knew - designing a course

That sounds like fun! Another idea that I think is worth studying is cyclotomic polynomials. There are some rather accessible approaches to the subject, and they come back quite a bit.
I really think you should at least show limits from a very rigorous perspective, and describe (at least briefly) how easy it is to extend this idea. I very likely would have been more open to calculus (and analysis) if it had been better discussed when I was first learning... of course, maybe it was discussed well, and I was too naive and blase to notice.

There are other things, certainly, but I cannot think of them off-hand. I'll definitely give this some thought.

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Originally Posted by jason.spade
For example, when I learnt Galois Theory, I was actually studying in Budapest for a semester.
When were you in Budapest? Was Csaba teaching it?
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June 2nd, 2010, 07:10 AM   #3
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Re: What you wish you knew - designing a course

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Originally Posted by cknapp
When were you in Budapest? Was Csaba teaching it?
I was in Budapest last semester, and yes, Csaba was teaching it! When were you there?
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June 2nd, 2010, 07:54 AM   #4
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Re: What you wish you knew - designing a course

Last semester, as in spring or fall? I was there last fall.

I didn't actually take Galois theory... as a subject, it scares me for some reason. I know Csaba because I spent a lot of time hanging out in Anna's office, and Csaba would always give me shit about drinking the coffee he made... So eventually I started making it if I showed up early. At the risk of derailing the thread, what classes did you take?

Back on subject:
I think the "crash course" that they have at BSM has a lot of the ideas that you should be trying to show-- Most of them I hadn't seen until more advanced courses, but they are low level, approachable things which really show the deeper structural relationships that get mathematematicians excitied.
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June 3rd, 2010, 08:51 AM   #5
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Re: What you wish you knew - designing a course

And I must admit, the classical algebra course that Cliff taught in the beginning was great, and he introduced me to the book Polynomials, by Barbeau. I suspect this will play a big role in the course.
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