I have taken Stephen Semmes‘s topics course for each of the past six semesters, and think he has a fantastic teaching style, partly because I could never see myself using the same one. During a typical class, he will make a definition or prove a theorem, and then ask “Does anyone have anything to say about this?” The path the class takes is then largely controlled by answers to this question. I am constantly impressed by how many answers to this question (often early in the morning!) Dr. Semmes and the students in the class have. More specifically, I am interested in the process of formulating such answers.

This is, of course, a huge topic (that is to say, how are scientific ideas/good research questions generated). Just to put down a few observations of good questions to ask:

**Statement of a definition (“***this is an X*“):

- An example of the object/phenomena being described (“
*x is an X”)*
- A characterization of the object/phenomena (“
*x is an X if and only if x is a Y,Z,…”*
- The object/phenomena’s relationship to previously defined objects (“
*every object that is X is also Y*” or “*every object that is Y is also X*“)

**Statement of a theorem (“***If A, then B*“):

- Is the converse true? (“
*If B, then A*“)
- What is true with weaker hypotheses? (“
*If a, then b*“)
- What is true with stronger hypotheses? (“
*If ***A**, then **B**“)
- Is the converse true with stronger hypotheses? (“
*If B and C, then A*“)
- Is there a stereotypical example of the situation described? (“
*X is A, so X is also B*“)
- Why is this surprising or interesting? (“
*If *almost *A, then not B*“)

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