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For proof-based math courses, the gist of the learning happens in problem sets and so it is essential to design them well. We would appreciate responses containing references (eg. from active learning) and personal experiences on designing problem sets to give us ideas.

Here are some criteria and ideas we use in crafting our assignments in proof-based calculus and senior years analysis (such as epsilon-delta arguments, constructing the reals, proving properties about families of functions):

  1. Have them prove a statement similar to one in class so that they at least get the main ideas.
  2. Include many true (prove why) or false (prove why not) questions. This is the quintessential skill of modern mathematical research, where one has to build intuition to figure out which directions to go first.
  3. Have them work through toy problems first before attacking a general statement.
  4. The more all the questions tie together, the better, e.g., via the use of themes. For example, we can devote a problem set to studying a convenient application that relates to many of the class's results.
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    $\begingroup$ Are these courses where students are learning to write proofs, or are you just saying that their content is mostly proof-based? I ask because I know I certainly design problem sets differently if one of the goals of the course is to help students learn to communicate effectively, not just to teach them material. $\endgroup$
    – Brendan W. Sullivan
    Commented May 10, 2019 at 1:46
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    $\begingroup$ Could you specify more carefully what the course content is? E.g., does "proof-based calculus" mean more than delta-epsilon proofs and applications of the chain/product/quotient/[etc] rule? And is "senior years analysis" akin to a first course in real analysis (e.g., constructing the reals)? Or a second course in real analysis (e.g., measure theory)? Relatedly: Do you have sample assignments satisfying [or not] your criteria/ideas that you can share, along with how you would wish them to be different or otherwise built upon? $\endgroup$
    – Benjamin Dickman
    Commented May 10, 2019 at 2:08
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    $\begingroup$ @Sullivan We aim for both but teaching them the material comes first. $\endgroup$
    – Thomas Kojar
    Commented May 10, 2019 at 15:51
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    $\begingroup$ @Dickman We are not asking for ideas from analysis courses per se, but that is welcome. We would like to hear criteria college instructors are using in designing problem sets when their courses are heavy on proofs (both content-wise and having the students be proficient at producing them). $\endgroup$
    – Thomas Kojar
    Commented May 10, 2019 at 15:54
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    $\begingroup$ The Art of Problem Posing may be relevant. $\endgroup$
    – user5402
    Commented May 12, 2019 at 18:16

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