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I was reading recently online Peter May's complaints (I'm a fan, you can tell, I'm sure) about teaching the third quarter of the graduate algebra sequence at the University of Chicago. This course focuses on homological algebra and attempts to be as up-to-date as possible. May's conundrum stems from the fact that homological algebra is inexorably tied to algebraic topology and as a result, it's difficult to separate the 2 out in the course completely. May questions whether or not this is in fact a good idea; however, since this an algebra course and not a topology course, he feels compelled to work hard to do this.

That being said, he raises a very good pedagogical problem in the teaching of mathematics, particularly at the graduate level where the better schools are trying to prepare students to enter research as quickly as possible. Mathematics is now a very holistic, intertwined discipline: Algebra increasingly permeates virtually all of mathematics, the study of manifolds now requires very sophisticated analytic tools from differential equations and functional analysis, probability theory now partakes of a considerable amount of harmonic analysis, mathematical physics is now a major player in the construction of new mathematical structures-I could go on and on, but you get the idea.

So here's the question: Is the old model of keeping the subdisciplines of mathematics separate in coursework for the sake of focus obsolete? I know a lot of mathematicians in recent decades have begun to draw from various disciplines in constructing the first year graduate sequences of most universities; Columbia is one local example. The question is really are they going far enough? The problem of course is that when you begin weakening those artificial barriers, you run the risk of them collapsing altogether and you ending up with a hodgepodge of theory and methods that seems to have no focus.

So anyone want to comment on what the solution here might be from their own experiences as both teachers and students? How far should courses go in being interrelated? And does this lead to better prepared graduate students for the research level?

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This seems to be completely a discussion question, so it may work better as community wiki. –  Jonas Meyer Apr 14 '10 at 17:45
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Andrew L had indented the text (which doesn't show up in posts), so I knew where he had intended to have paragraph breaks. –  Jonas Meyer Apr 14 '10 at 17:54
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About "controversial": That is surely a minority view (and possibly an offensive one as well), held I imagine only by someone with little exposure to what math logic is. Since you are here in New York, I would invite you to join my intro logic course at the CUNY Graduate Center (e.g. this semester), and I doubt afterwards that you would give any credence to such a view. –  Joel David Hamkins Apr 16 '10 at 0:54
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Well, my invitation to you stands. –  Joel David Hamkins Apr 16 '10 at 12:25

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Of course you should show students, taking into account their backgrounds, that the material they are learning in one course is relevant elsewhere. It makes it clearer to the students that topics they are studying have wide usefulness. At the same time, if you know the students don't have a background to appreciate the technicalities coming from other disciplines (not everyone in algebra has had algebraic topology), then you may have to restrict yourself only to making some broad general remarks, although maybe one or two special worked examples from the other disciplines would be accessible without a lot of machinery.

When I discussed characters in an algebra course, I explained a little about Fourier series both for context (otherwise the concept can seem rather far-out) and so they'd see that the otherwise idiosyncratic theorems on characters are related to properties of Fourier series.

I don't think such discussions in a first-year course are going to make the students better researchers, but it will make them better appreciate what they are supposed to be learning.

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I question whether mathematics is really as holistic and intertwined as some people are making it out to be.

Certainly there are a few freaks of nature out there who can understand 20% of the mathematics out there and incorporate ideas from 10 different subfields into their work. A larger group of us are capable of getting the big picture though maybe not all the specifics of 4 or 5 different subfields, at least to the extent that we know when to reach out to an expert. Many graduate students, and most of them once one leaves the world of the top ten or twenty departments, are just capable of learning one subfield well enough to write a dissertation narrowly focused on one problem in that subfield, ignoring all the wider connections if indeed there are any. Most published papers are written by people who have never done serious work outside a single narrow subfield in their entire career, even if the same is not true for the best papers.

A professor or a department may choose to aim its education at the future Fields Medalist (or, somewhat more broadly, the future NSF-or-equivalent-research-grant-receipients), but is this really fair for the other 19 students in the room?

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What a depressing answer. I'm torn between trying to argue that this is factually inaccurate (which is not so easy!) and saying that it doesn't matter if it's true or not on average, because there's an aspirational aspect to teaching: to a degree, all academic mathematicians are mathematical missionaries, and the statistical fact that we will most often fail should not dissuade us from our soul-saving attempts. –  Pete L. Clark Apr 14 '10 at 23:05
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Here's another response: it is possible that you are overestimating the level and ambition of the interconnectedness referred to above. It is, as you say, not so reasonable to expect many of us to be research experts in multiple fields, but I think it is reasonable for all of us to gain a cultural understanding of most fields of mathematics, enough to give one a perpsective of the larger picture. –  Pete L. Clark Apr 14 '10 at 23:11
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Another remark is that the separation between areas is taken to extremes in the American educational system: in most undergraduate / intro graduate textbooks on subject X [say linear algebra], references to subject Y [say, differential equations or even calculus] are expunged with alarming scrupulousness. I've seen the effects of this on ordinary students: they often choose up sides early in their education: "I hate analysis" or "I only like algebra". Some of these students grow up to be researchers, who are freaks of academic nature: incredibly limited in their interests and expertise. –  Pete L. Clark Apr 14 '10 at 23:17
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Different areas of mathematics really are deeply intertwined, and this has always been the case. But mathematics has grown so much that it is very difficult for anyone to master more than a small part of it. Although there is a place for narrowly focused courses and it is possible to be a somewhat successful narrowly focused research mathematician, I believe that students should try to learn as much as they can bear about the intertwining of different subjects. Even if you're not a Fields medalist, it greatly enriches your perspective and research of mathematics. –  Deane Yang Apr 15 '10 at 2:41
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I agree with Pete, this answer seems to start from the wrong premise. A basic understanding of most areas of mathematics shouldn't be outside the reach of most mathematicians. That's quite different to being at the cutting edge of research in many different areas. –  Kevin McGerty Apr 16 '10 at 14:53

I've almost uniformly studied the homological algebraic aspects before I got around to studying the corresponding results from algebraic topology. It did get somewhat artificial at points - specifically triangulated categories make a lot more sense once you've seen Serre fibrations than before you do.

I felt quite well motivated by the approaches I encountered though; with the study of Ext and Tor to divine interesting ring properties taking the forefront in homological algebra, with a side dish of approximating modules by things that are free everywhere that matters, but sacrifice degree concentration to achieve it.

My personal feeling is that it probably depends to a large extent on whether whoever is teaching the material wants to teach homological algebra or algebraic topology: if you're happier thinking about topology, then homological algebra will feel desolate and artificial almost no matter what you do about it; while if you are genuinely interested in homological algebra on its own, it's much easier to sprinkle in the off-ramps as you go, pointing out where certain concepts have roots outside the current area, and how to get more information about the roots.

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I am only a first year graduate student, but I am very interested in mathematics education. My own approach to teaching is very much problem based: give students interesting problems which lead to the development of the concepts you want them to have. Even if they can't come up with all of the needed concepts on their own, if you give it to them after they have wrestled with a problem they will be much more likely to be able to apply the concept in novel situations in the future. Why couldn't this approach be carried through in a math grad situation? Design a sequence of problems, varying in difficulty, which in total cover need material from most of the "first year curriculum".

Before writing this off as a crazy idea, I would like to point out Cornell's vet school. They use exactly the model given above: Every week or two there is a new case. In each of your classes (anatomy, pharmacology, radiology, ...etc) you cover general information which is pertinent to the case of the week, but it is up to you and your team to do research, come up with a diagnosis and a method of treatment. So all of the classes you take are integrated together in the context of solving some real problems. Cornell is turning out some amazing vets. Why couldn't the same model work for mathematics?

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What you're advocating is a kind of modified Moore method of teaching,Steven.That's been at the heart of much debate since Moore proposed it in the first half of the last century. My own feeling is that that leads to very isolated perspectives on mathematics despite developing strong mathematical reasoning skills.I think some courses along those lines are healthy for graduate students,but not all. –  Andrew L Apr 14 '10 at 20:56
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I think some amount of "standard instruction" would still be necessary, so this is not really in the spirit of Moore. Also I am a big believer in collaboration. This diverges from the standard curriculum mainly in that you are immediately applying the things you learn in the classroom to a novel problem which requires the input of many different fields of mathematics, so you are weaving all of the subjects together from day one. You should be using your functional analysis to put bounds on dimensions of vector spaces to help solve your number theory problem! –  Steven Gubkin Apr 14 '10 at 21:36

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