The title is self-explanatory, but what is the appropriate response to "motivate your answer"? Should you prove your answer? Should you just explain it, or give the intuition for it?

For example, suppose students are given a true/false question, and "motivation for the answer" is required. A student could provide a formal, rigorous proof of the answer, which is obviously sufficient, but a student could also give, say, provide a sketch of a graph with an explanation that demonstrates the validity of the answer. Although the latter is not a formal proof, it does point out -albeit, less rigorously - why the answer is correct. Thoughts?

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    $\begingroup$ It depends. Can you give an example? $\endgroup$ – Jasper Jan 8 at 16:54
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    $\begingroup$ Please provide more context. Is your question related to mathematics education? $\endgroup$ – Joel Reyes Noche Jan 9 at 0:11
  • $\begingroup$ Are you the student in this scenario, or the instructor? If you are the student, I would suggest that you ask your instructor what that instruction means. If you are the instructor, I would suggest that you rewrite the question to unambiguously state what you want your students to do. I suspect that the question should be phrased as "Justify your answer," then explain that "justification" could be a complete proof, a counter example, a proof sketch, a well-drawn figure, a computation, or anything else which convincingly explains why the answer given is correct. $\endgroup$ – Xander Henderson Jan 9 at 15:11
  • $\begingroup$ @DavidJSilverberg Welcome! To make this more on-topic, I would suggest changing this question to a question about how to set the expectations on students. So I would change the title to "How do I communicate my expectations of the level of rigor for student justifications?" and explain that when I say "justify your answer," I sometimes mean one thing and sometimes mean another. Then ask the community here how to concisely explain what I want to the students. --- If you think this would be an improvement, you can edit your question, or simply tell me that you agree and I will edit it myself! $\endgroup$ – Chris Cunningham Jan 11 at 20:28
  • $\begingroup$ "Motivate your answer" is clearly a sneaky way to keep instructor's options open regarding grading. If overall class grades are too low, the instructor can boost scores for the answers that are not proper proofs. Or, conversely, the instructor does not know a correct proof himself, and accepts any half-decent justification for an answer. $\endgroup$ – Rusty Core Jan 15 at 19:28

The word "motivate" isn't sufficiently specific to make clear what's meant by such an instruction without some further context. (It might suffice a short hand on an exam, but only if students had seen examples in class and in practice problems.)

I would argue that the same is true of the word "prove" - more seriously, since mathematicians often underestimate how much variability there is in what arguments count as proofs.


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