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In very advanced courses, typically there are only a few students attending and sometimes they are not willing to put so much efford into the course to participate or to pass an exam.

So I made the experience that hardly anyone will do the exercises. Of course, the students will not learn that much if they do not exercise during the course and at some point it will be very hard to follow.

Are there some strategies to get as much content and methods as possible to the students? What are the advantages of e.g. repeating proofs, applying theorems, etc.? Should the tutor enforfe the students to do it themselves or should the tutor repeat it for them?

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  • $\begingroup$ Which kind of course are we talking about? How many students are there? Is there some kind of test at the end of the semester? Are the excecises being graded? $\endgroup$ – Roland Mar 14 '14 at 8:47
  • $\begingroup$ I talk about about the "last" course of some cycle (e.g. courses like partial differential equations 3, differtial geometry 5, recent trends in [put in some field] etc.). Students often do this course due to their interest (but don't need to take an exam and don't need a grade), but are normally not willing to put enough time to re-work the course since they are writing their master thesis. $\endgroup$ – Markus Klein Mar 14 '14 at 8:53
  • $\begingroup$ It sounds like there are not too many students involved, i.e. the tutoring sessions (and maybe even the lectures) would allow a high level of interactivity to actively involve the students. If most of the students are there for the interest, and not for the credit, I do not see many means to actually enforce more participation. $\endgroup$ – Roland Mar 14 '14 at 8:59
  • $\begingroup$ Yes, there are only a few people there. $\endgroup$ – Markus Klein Mar 14 '14 at 9:09
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    $\begingroup$ This seems to be the first question on educating graduate students. I have therefore added a new "graduate-school" tag to this question. $\endgroup$ – Jim Belk Mar 14 '14 at 21:59
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The exact situation you are in is not completely clear to me, I thus give a two-part answer assuming two different scenarios.

Scenario 1. The exercises are given by somebody else (the instructor of the course), and you run only the excercices-session.

The middle ground between 'enforce the students to do it themselves' and 'the tutor repeat it for them' seems like a good solution to me. The students in such a course already should have attained a certain level of mathematical maturity and are likley also somehow good students. So, if they do not prepare the exercises try to solve them on-the-fly together instead of just presenting them to them. Ask a lot of small questions, or ask for ideas they had/have. Also, do not shy away from trying ideas they have of which you know they will not work. To see the failing of the idea can be very instructive and lead to finding a better one.

Scenario 2. You have full control.

Something that can work is to run it almost like a (students') seminar. Instead of excercises hand out to each student (or small groups) a couple of pages from a book (or paper) of closely related material and let them prepare and present this material in class. Some points in favor:

  1. They might find this more interesting as it adds something new and is perhaps different in style from other courses.

  2. It might better teach them skills (understanding new material and presenting it) they need at that point of their developpment than the classical excercices-setup.

  3. They do not have to work each week a bit, but at some point a lot. This could be a better fit with their time-management at that point.

The approach in scenario 1 is not really limited to advanced students but I think it is more feasible than, and at least it tends to work better with small groups. So far I never found an occassion to try the thing I describe in scenario 2 myself but know some people that do this (they sometimes use the time-slot for excercises-seesion at the beginning of the term for addition courses to have some weeks towards the end for the student presentations).

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