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Next year, I will be teaching a very challenging second year "multivariable calculus and calculus on manifolds" course, and will assign a large number of difficult problem sets. I am thinking of assigning students to go back to problems where they had good but not perfect solutions and rewrite an elegant, well presented solution. There are two goals here:

  • Pedagogically, to force students to really think through and present their ideas well, and to avoid the feeling that each problem set is passed through and ignored.

  • Logistically, to save myself the time of writing lengthy solutions.

Have some of you tried this? How did you make the task fair, when some problems are much harder to write than others? What were the logistical practicalities: How long did students have to produce solutions, and how were they evaluated?

To be clear about the sort of student population we are talking about, this is the top 20 or so kids at Michigan, who have already done a similarly challenging single variable course and should be anticipating a high workload. For those who know Michigan, this is 395-396; I believe the Chicago and Harvard equivalents are 20700-20800-20900 and 55b (but in a year, not the term that 55b tries to squeeze into.)

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    $\begingroup$ I am not quite sure I understand the question. Are you saying that you will (a) assign a large number of difficult problems (b) grade them with comments (c) ask the students to incorporate those comments to rewrite their answers so it is clean and well-presented? $\endgroup$ – Willie Wong May 28 '14 at 8:03
  • $\begingroup$ I had a similar experience with writing course notes for a course in electromagnetism I took in graduate school. We had a partner and had a couple days assigned. If memory serves me correctly, we voted or graded each as a class and somehow my group won. I learned about complexion in that task and a bit of LaTeX which was then new to me. It was useful. I wonder if some sort of wiki format with a sort of muted-Moore method grading would work for this... $\endgroup$ – James S. Cook Jun 4 '14 at 22:11
  • $\begingroup$ As someone who took 295-396 in the early 2000's, I can tell you that I was one of the top students in the class, but you would not ever have wanted to trust me (at that time) to write up your official solutions! For example, I loved proving things by contradiction, because why not? You get an extra assumption to work with. I found the grader's displeasure for my tactic to be unreasonable! What I'm saying is that your students, although excellent, may still not be mathematically mature enough to really write official solutions! $\endgroup$ – Chris Cunningham Jun 5 '14 at 3:54
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As a person who still has a fresh memory of being assigned problems, I would like to point out several possible problems/reasons why this might not be the best approach. I'm not sure to what extent this answers the question.

  1. The most rewarding thing about solving problems is the thrill of figuring out a solution to a problem you previously did not know how to solve. The rest, in my experience, is mostly mundane (writing things down, doing explicit computation, etc.). What you will be asking them is to do the boring work without the exciting bit. This probably won't be very motivating. [Maybe consider just giving a new assignment, somehow related to one that a lot of people got wrong?]

  2. This opens an opportunity for strategically making mistakes (depending on your student's ethics, it may be fair from their point of view.). They will have an incentive to take a perfect solution, introduce artificially some mistakes, hand it in, then have you ask them to produce an improved version, and hand in the original. (This saves them the bother of producing one extra assignment, and lets them avoid a situation when they have to improve a problem which they find difficult).

  3. As you yourself mention, this is inherently 'unfair', since the students with few/easy mistakes will have an easier assignment than the rest. (Unless you're willing to balance it on a case-by-case basis, which will involve a lot of work on your part.) Even when you balance things out (somehow), the class will likely not feel that the system is fair, because the system won't be transparent/simple enough. [Maybe consider using another scheme with similar effect, but which feels like it is just working in the student's favour? E.g. allow them to resubmit a solution to a problem, but only under the conditions that (a) new solution is very well written (b) old solution was at least OK; the new mark then replaces the old one. Make the grading scheme such that it makes sense for them to fight for those few extra points. Or a prize (extra credit) to whoever produces the best solution.]

  4. If you give them comments on their previous work, they may end up parroting what you told them. This is bad because they can hand in correctly looking things without really understanding what is going on. Even worse, they may easily produce results which clearly show that they do not understand the problem, but which are still technically correct/difficult to grade down. Also, you'd be doing part of their work for them.

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    $\begingroup$ +1 I see this happen a lot with my colleagues in humanities who allow students to revise and resubmit their essays. Eventually you end up grading your own essay, especially if you allow multiple rewrites. $\endgroup$ – ncr Jun 4 '14 at 23:57
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I see a few other ways to accomplish your stated goals:

  1. If time permits, have students present solutions. This is an even more transparent way to see what they were thinking and how well they understand their own solution.
  2. Having students present at least a few problems and maybe asking a scribe to write down what the presenters say will save you from having to write solutions to those problems.
  3. I don't know that students want elegant and streamlined solutions -- at least I know that as a student I'd rather have seen the big ideas and insights of the argument rather than the details. The details I would try to figure out on my own. If you were to write solutions that were a little more skeletal, I think the students would be satisfied. Something you could do if you're not comfortable with only a skeletal solution is to have a skeletal solution, put it on a computer and then record your voice talking over it. That way you could say out loud all the elegance that takes a long time to write.
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