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In the fall, I am teaching one undergraduate and one graduate course, and in planning these courses I have been thinking about alternatives to the "standard math class". I have found it much easier to find different models for undergraduate classes than for the graduate classes.

There are many blogs and other online resources focusing in large part on math pedagogy, often taking "nonstandard" approaches to the math course (Moore method, inquiry based learning, flipped or inverted classroom, Eric Mazur's "Peer instruction",...). For instance, Robert Talbert's blog discusses the practicalities of the "flipped" classroom, and here Dave Richeson discusses teaching undergraduate topology using the modified Moore method. Note that although Moore originally used his method in a graduate topology course, most discussions of it and its adaptations are at the undergraduate level.

Of course, the basic principles of the methods would still apply at the graduate level, but I still find myself wanting to see specific examples of different approaches people have used to teach a graduate course.

Are there similar discussions of teaching a nonstandard math course at the graduate level? Or at least links to syllabi, course webpages, etc. for such courses?

Further background:

First, it is of course hard to pin down exactly what is meant by "standard math class"; for the purposes of this question, let us highlight the following:

  1. In class meetings, the vast majority of time is spent with the professor lecturing.
  2. Nearly all the assessment is done via problem sets and quizzes/exams.
  3. The textbook runs in parallel to the class lectures. Although lip-service my be made to reading it, there is little structural dependence on it, except perhaps as a source of problems: lectures do not depend on students having read the book, and problems are not asked on material not presented in lecture

Second, though descriptions of the form "I once took a course where..." could be interesting, It would be particularly valuable to have links to course webpages with syllabuses/etc. describing in more practical detail what happened, or discussions from the actual professors doing the teaching about what they did, their reasoning behind it, how it went it practice, etc., as opposed to just a collection of anecdotes.

Finally, though what I have discussed above tend to be rather drastic changes from the "standard math class", of course things run on a continuum, and smaller deviations are possible. For example, I have taken graduate courses where student presentations of selected material played a role, and where a part of the grade was given to an expository paper. But in the courses I took these were usually rather minor deviations from the standard structure. Examples of similar smaller variations would be valuable if they were particularly well documented, or had explanations of why these variations were particularly important and not just window dressing.

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Should we consider revising the question title to be more succinct? Something like "Non-standard structures for graduate math courses". –  user1241 Jun 13 '13 at 15:59
    
Succinct is nice. Exactly conveying the intent is better. The mods actually suggest longer, more appropriate titles. For search purposes, I suggest limiting the amount of TeX, but otherwise providing more detail. Gerhard "Brevity Does Have Its Place" Paseman, 2013.06.13 –  Gerhard Paseman Jun 13 '13 at 16:18
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Perhaps it would be better to remove the first sentence in the title and just keep the question starting with "what". –  Amir Asghari Jun 13 '13 at 19:30
    
BTW, in which country/system are you giving this course? UK-US differences, and all that. –  Yemon Choi Jun 14 '13 at 0:11
    
Yemon -- I'll be teaching in the US system. From my limited experience, UK system seems perhaps even more wed to the standard method. And the question title is perhaps a bit long -- the description "tweet and a half" as an upper limit stuck in my head. But I think Amir's suggestion is probably good. –  Paul Johnson Jun 14 '13 at 8:04
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3 Answers

It may be only a minor thing in the space of examples that you seem to be considering, but I have had a lot of success with my practice of requiring students in my graduate courses to write a substantial term paper on an original topic.

The aim is for them to undertake a simulacrum of the research experience. I definitely do not want them to just give me an account of some difficult topic on which they read elsewhere. Rather, we try to find a suitable original but manageable topic, which they will have to figure out themselves, and then write up their results in the form of a paper.

I insist that these term papers give the appearance of a standard research article, with proper title, abstract, grant or support acknowledgement, proper introduction, definitions, statement of main results and proof, with references and so on. Furthermore, I insist that the students use TeX, which I insist they learn on their own if they do not yet know it.

The most difficult part for the instructor is to find suitable topics. One rich source of topics is to take a standard topic that is well-treated elsewhere, but then make a small change in the set-up, giving the student having the task to work out how things behave in this slightly revised setting. For example, in a computability theory class, there is a standard definition of the busy beaver function, with many results known, but one can insist on a slightly different model of Turing machine (such as one-way infinite tape instead of two, or change the halt rule, or have extra symbols or extra tape), where the standard calculations are no longer relevant, but many of the ideas will have a new analogue in this new setting. But also there are usually many suitable topics if one just thinks with curiosity about some of the main ideas in the course and some relevant examples.

I always insist that the topics be pre-approved by me in advance, because I want to avoid the situation of a student just writing up something difficult they read, but rather have them really do real mathematical research on their own. Often, I meet with each student several times and we make some discoveries together, which they then work out more completely for their paper.

After students submit their final draft (I do not call it a first draft, since I want them to do several drafts on their own before showing me anything, and I don't want to look at anything that they regard as a "first draft"), then I give comments in the style of a referee report, and they make final revisions before submitting the "publication" version, which I sometimes gather into a Kinko's style bound issue Proceedings of Graduate Set Theory, Fall 2014 or whatever, and distribute to them and to the department.

Finally, on the last lecture of the course, we usually have student talks of them making presentations on their work. For example, see the student talks given for my course on infinitary computability last fall.

I think it works quite well, and gives the students some real experience of what it is like to do mathematical research. In a few exceptional cases, the terms papers have subsequently turned into actual journal publications, when the students got some strong enough and interesting enough results, and that has been really special.

The workflow for me is to assign normal problem sets in the early part of the course, and then start suggesting topics, with the students coming to me and we discuss possibilities. Then, as the work on the paper ramps up, the problem sets taper off, until they are submitted, with additional problem sets at the end of the course, except when they are making their revisions.

(And I never accept papers after the end of the course.)

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I find this technique doesn't work as well in undergraduate courses, since the students need much more hand-holding, and the outcome is generally less satisfactory. But I have tried it in a few very advanced undergraduate courses, and it did work out. –  Joel David Hamkins Jun 13 '13 at 15:10
    
Very neat suggestion. –  Andres Caicedo Jun 13 '13 at 15:50
    
Is “insist[ing] on a slightly different model of Turing machine” a typical example? Such a topic does not seem very appealing to me: The student considers an “unnatural” definition nobody is interested in and nobody uses and transfer results which are already not widely applicable (busy beaver estimates). She would probably not have learned much less if she had simply read the proof of the original result very carefully, which uses the same methods, to write it down in a different way. –  The User Jun 13 '13 at 16:08
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...My goal is to find a toy example that isn't exactly considered elsewhere, so that the student has to figure things out again anew him or herself, rather than studying someone else's work, which I don't believe is an accurate simulation of what actual research is like. I want the student to struggle with a problem entirely on their own, figuring out how to apply known ideas to the particular problem at hand. This, I believe, is what actual mathematical research is like. –  Joel David Hamkins Jun 13 '13 at 23:00
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Most (almost all?) graduate computer science courses I am familiar with follow this format; it seems usually quite successful. Often, students are encouraged to work on projects in groups of two, maybe three. A difference might be in CS that open problems are easier to identify and approach, but it seems like the approach could be very good in maths as well. –  usul Jun 14 '13 at 1:17
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To your direct question, I don't know any answer that would improve upon a standard web search, or be better than one that could be gleaned from lurking on an academia type forum. So I will change the question, and answer that.

Question: How can one design an alternative format for conducting a graduate class, presumably with the goal of improving the result, namely that the students get closer to mastery of their craft? (Of course, multiple goals could be considered, for example the convenience and satisfaction of the director of the class. However, to make things fun, I assume the director has unlimited resources.)

One answer to this question is to consider the techniques or characteristics of the goal, and tailor the course to offer repeated practice of exercises which should encourage mastery of those techniques or acquistion of those characteristics.

To demonstrate, let me take good communication as a theme. One can focus on various techniques which lead to asking a good question or giving a good presentation, but the student does not learn them without doing them, and in my worldview based on my experiences, doing them several times is required.

George Bergman had a technique which he used for both graduate and undergraduate level classes, to which I suggested a modification that he adopted. Before the class, each student was supposed to have read the material for that class, and come up with a question about some aspect of the material that they didn't understand. They wrote that question on a slip of paper (and if they understood everything, they would tag it "Pro forma" to indicate they did not need the answer), and then turned that in at the beginning of class. George would then spend some portion of the class answering those questions. I don't know if he did this, but he could have tracked a lot from these questions alone, and used it to help students ask good questions.

I could provide examples regarding other aspects of communication and ways of addressing them in a graduate mathematics class, but I instead invite you to consider designing your own format to test your ideas about effective education, and then bring those ideas and that design to a forum for critical review.

Gerhard "Really, Go Ahead And Ask" Paseman, 2013.06.13

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Gerhard, I think I may have been in Bergman's class with you. Is that right? –  Joel David Hamkins Jun 13 '13 at 15:43
    
Officially, no, as I don't recall taking any of Bergman's classes. (I recall us and Ketchersid in a model theory class by Prof H. "Lost its p-ness and can't fork anymore"), and we were likely both in McKenzie's 245. I did audit some of Bergman's 245 and I TAed for him, so odds are high we were in the same room one time with him as teacher. I don't trust my memory on such things any more. Paul, George, and I might appreciate hearing how George's technique impacted your graduate experience. Gerhard "Memory Not So Green Now" Paseman, 2013.06.13 –  Gerhard Paseman Jun 13 '13 at 15:55
    
McKenzie's universal algebra class was great! I recall Bergman as both brilliant and clear, but I think the questions-on-paper technique was not yet implemented. –  Joel David Hamkins Jun 13 '13 at 16:07
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A good strategy for a class which I endured for a couple classes as an undergraduate and to a lesser extent as a graduate student for a couple classes is when the students make presentations for at least a substantial portion of the total number of class sessions, and make most of the students grade based on presentations.

If the students make presentations for class, then there are several obvious advantages. First of all, if the students make presentations, then they be better prepared for teaching and making presentations at seminar and conferences. Being able to talk in front of people with out becoming paralyzed is a very valuable skill to have. The students may even grow comfortable enough with making presentations, that they may even become comfortable enough to tell a couple appropriate jokes during the presentations during class. Also, when making a presentation on a certain topic or result, the student should have a deeper understanding of that particular topic. A couple years ago, I had to make a presentation on a certain paper, and I can still recall from memory several details about that paper that I made in that presentation. Furthermore, if the students give presentations for class, then class can still be held when the professor is away at a conference of something. This surely beats cancelling class altogether. I once made a presentation outside (yes outside) in front of two other students while the professor was away at a conference :).

Of course, for this strategy to work, the class cannot contain too many students, and the students have to be motivated and able enough to give high quality presentations.

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