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):
- Have them prove a statement similar to one in class so that they at least get the main ideas.
- 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.
- Have them work through toy problems first before attacking a general statement.
- 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.