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Outside of your field of research / application, how much of your undergrad education have you retained?

Thought experiment: How would you fare today if handed old exams from your introductory topology / number theory / differential equations / whatever classes? No studying or preparation.

I've always wondered how much of this material one should expect to have internalized, and how much is acceptable to forget over time and need a reference aid.

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I would probably do pretty bad on an analysis exam, both real and complex, and definitely the advanced linear algebra. I know I would do well on differential equations and matrix theory. I also could get through the groups part of abstract algebra, but rings and fields were tougher and I know my retention is not there now. –  Eleven-Eleven Dec 11 '13 at 4:06
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Well, it's not exactly like anyone ever passed their college exams unprepared, without studying first before taking them, so your question is a bit odd... :-) –  Lucian Dec 11 '13 at 4:07
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why do you want to know? –  Will Jagy Dec 11 '13 at 5:07
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Fairly well, apart from differential equations and complex variables. But I took those when I was $15$ and $16$, respectively, never cared much for them, and managed to avoid teaching diff. eq. my whole career. (I did teach complex variables once, probably in the late $70$s.) Oh, and I’ve lost whatever Galois theory I learned as an undergraduate. –  Brian M. Scott Dec 11 '13 at 12:43

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Professional mathematicians are not always professors. There are many working for government-based research labs, other government agencies, private research firms, and industry.

In these cases, in my experience many of them -- because they do not teach as a regular activity -- do not retain many of the details, but they understand all the concepts.

Many of these folk might not remember things like trigonometric integrals, or they might not be able to recall the proper way to solve certain classes of ordinary differential equations, but they certainly understand the math -- they just might not be able to recall it off-the-cuff.

When you work in industry or government, you have ample access to resources. For instance (just to use an off-the-cuff example), if I see $\int \arctan x\ dx$, I don't know offhand what that is. But I know it's something, and I know how to find it, and I can fully understand all of the details on how to compute it. So if it comes up in my work, I can recognize that and go from there. Memorizing it isn't particularly useful for me. I haven't had to compute the antiderivative in 15 years. But if I needed to know how to do it, it would take me less than 5 minutes to figure out.

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Yes, I guess this is more or less what I was wondering. How much does one retain of the stuff one doesn't use often. If you teach the subject is one factor, but as you say, many go on to careers where they don't necessarily get to review those subjects through teaching. –  o_o_o-- Dec 11 '13 at 18:12

Very basic stuff, like linear algebra, basic topology, elementary number theory, elementary real analysis, finite group theory, etc comes up often enough in research, and the kind of problems you would see on undergrad exams are simple enough that most mathematicians would do fine on those tests, even if they haven't taught those classes recently (which they probably have).

I think that in more advanced undergrad and early graduate classes the situation would be a little different, the material isn't so easy and those things might not come up as often if you research outside whatever area. Plus people usually only teach advanced courses in topics that are close to their research area.

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If the pros here are honest, unless those exams are in areas of mathematics they've either done substantial research in or recently taught classes in several times, they'll say they'd fail them absymally. I know how counterintuitive that sounds to most people, but the landscape of modern mathematics is so vast that once one specializes in a tiny area of it-as all research mathematicians must do to be effective researchers-thier entire cognitive faculty has to be focused completely on that tiny drop in the ocean of mathematics. All else is forgotten-and shockily quickly. This is one of the reasons I think it's so important for research mathematicians to teach a diverse set of undergraduate courses in thier career-it's precisely to prevent the kind of amniesia that sets in with that neglected knowledge.

Unfortunately, suggesting that research mathematicians teach at most prestigious universities today is somewhat like suggesting an accomplished chef show a kindergarten class how to make perfect hard boiled eggs-it's so beneath them,they might become offended. There are MANY exceptions,of course.But in my experiences, ask the average young postdoc at an Ivy League school what his approach is to teaching linear algebra or calculus and you may get laughed out of the room.

And that's quite sad.

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To the question asker: this opinion seems to be coming with someone with very narrow experience in the mathematical community. Important research takes skilled knowledge in and many ideas from many areas of mathematics. Moreover, many professors (yes, even in Ivy Leagues, if I were to take that example) care about teaching. –  Maxim G. Dec 11 '13 at 4:17
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As an undergrad I had a chance to talk to or work on projects with several professors who are active in research. Every one of them had more than just a good understanding of undergrad course topics---even ones well outside their field. This post (esp. second paragraph) strikes me as someone's frustration over past interactions with mathematicians rather than evidence of any trend relevant to the OP. –  Daenerys Naharis Dec 11 '13 at 5:34
    
@Max G,Joseph: First of all,I specifically said there were many exceptions. It is less about individual professors then the atmosphere at research institutions,to which the first commandment is "publish or perish". Secondly,I have considerably more experience with the mathematical community then you would imagine-and I freely admit it's been less positive then I'd want.We each can only speak from our own experiences. And Joseph,if that's true and I have no reason to doubt it-I'm certain those professors teach those courses regularly. Sadly,that's not always the case of researchers. –  Mathemagician1234 Dec 15 '13 at 17:31

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