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:
- In class meetings, the vast majority of time is spent with the professor lecturing.
- Nearly all the assessment is done via problem sets and quizzes/exams.
- 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.
Best Answer
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.)