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There are a lot of undergraduate courses out there and most of them agree on certain things, with regard to the subjects covered.

Courses that include mathematics (engineering, physics, etc) are preparing you for a particular job, and so the content reflects this. You learn the methods that will suit your profession.

But it's not so simple to catalog the techniques required to learn, when there's no specific application at the end of it. The goal of a math degree is to broadly cover as much as possible. Then, you specialise later, if you want to.

Given those open-ended terms, how do you construct an undergraduate curriculum? How do you decide which topic should be undergraduate instead of graduate?

I think the solution is to assume that an undergraduate course's goal is to prepare a student for the most likely next step. Courses in differential equations, linear algebra, etc, are taught because they're immediately useful in many fields.

Being a good mathematician isn't really a goal. You can't solve this with a curriculum.

Well I'm rambling. The question is, if you could design the perfect course of mathematics, how would you do it? (I'll let you decide what perfect means.) Perhaps you wish they'd included more of something in your undergraduate degree. Maybe, less of something else. Maybe you'd use different learning methods to what you were exposed to.

EDIT: also, I think this is important too: what standard would set for your course? Consider this: should a student be great at every topic you can think of? Should they be very good at one particular thing? I'm sure there are universities out there that give degrees to students who are fine at many things, but a master of none. Should that be acceptable?

How would you gauge dexterity in any subject, and how dexterous should a student be at a given subject?

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Are you going on to be a teacher? Or, are you going on to be a researcher in a university? Or, are you going on to do physics? Or, are you going on to do biology? Or, are you going on to work in industry? Or, are you going on to work in an insurance company? It obviously matters what you're going to do with it if you want to decide what is most important. –  Graphth Oct 2 '12 at 18:22
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OOOOOOOOOOOO,I wish I had the time to give a proper answer to this question right now.........grrrrrrrrrrr............ –  Mathemagician1234 Oct 2 '12 at 18:23
    
@Graphth I already covered that. If you assume that the course goes on to something else like physics, biology, then the question is trivial. Try to have fun with the question. –  Korgan Rivera Oct 2 '12 at 18:35
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Barry Mazur. –  Ragib Zaman Oct 2 '12 at 18:58
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@Graphth If you have an answer, please post it. –  Korgan Rivera Oct 2 '12 at 20:22
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There was a discussion this morning on BBC Radio 4 by Jim Al-Khalili in a programme called The life scientific, and the education of scientists was discussed with the next government chief scientific advisor, Sir Mark Walport, who stressed the importance of writing, presenting, and communicating, as well as getting to grip with scientific issues. Tim Porter and I wrote an article ``What should be the context of an adequate specialist undergraduate education in mathematics?'', The De Morgan Journal 2 no. 1, (2012) 411--67, which discusses many issues such as the aims of an undergraduate course. A notable feature of a course we ran at Bangor on "Mathematics in context" was the enthusiasm of the students: a free discussion with them on the ideas of the course was like opening the floodgates!

Part of the reason for the course was to provide students with some background and language so as to be able to have a view and talk about the value of the subject in general. This is important for the subject so that the students can be able to speak for the subject in their future careers.

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Another take: whatever the concrete (well, or abstract!) content is, the most important point is that you should know what you know very well. Depth is more important than breadth! you should know not only the theorems and definitions, but their use (well, some of it ...), understand proofs, being able to do proofs, understand counteraxamples, understand why a concept is defined in some way and not in some other way (why does the other, appealing way brake down?). As Halmos said, you should fight it: not only do exercises, make your own exercises, proofs should be worked until you understand every little detail better than the author ... and so on.

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Studying everything to a certain level of depth is definitely important. But I would argue that for an undergraduate education breadth is very important as well! An undergraduate should be exposed to a wide variety of mathematics - specialization is more for grad school IMO. If a student is preparing for grad school, they will need to experience many different topics so that they can decide which holds the most interest for him/her. –  Jair Taylor Oct 3 '12 at 2:37
    
well, yes, I am not really in disagreement with that, the point is more that bread without depth do not have much value. What is common to the many digfferent topics lay in the depth! so learning something in depth is a very good preparation for almost anything. –  kjetil b halvorsen Oct 3 '12 at 2:45
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One criterion that could be used is to ask if this or that subject (and the manner in which it is taught) help the student to learn to learn mathematics. We cannot determine all the needs for the future mathematics or statistics graduate but if they are trained to learn mathematics (and not by just sitting in a lecture watching `us' write on the board or click a button to go to the next slide), then they are set up for future learning.

... and having learnt to learn, as Ronnie said in his reply, they should learn to write, and communicate the mathematics (and any conclusions that it gives) in an intelligible way, (and a Maths in Context course is an excellent way to do that).

So do not just look at the content, look at the context!

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