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I've encountered the following misunderstanding. I pose a question (to undergraduates in the U.S.), for example:

Let $P$ be a polygon of $n$ vertices. Is it true that every triangulation of $P$ has the same number of triangles?

This question depends on what constitutes a "triangulation," but assume the student know that. The answer is Yes: every triangulation of $P$ consists of $n-2$ triangles.

Here is the problem I encounter. The students apparently don't understand that "Let $P$ be a polygon" means, let your mind run over all possible polygons, so $P$ is an "arbitrary" polygon, in that it can be anything that fits the definition of a polygon (which the students also know). They wonder, well, maybe $P$ is a convex polygon, and should I answer specific to convex polygons?

This example doesn't quite illustrate the problem because the answer is always Yes. But when the answer is sometimes Yes, sometimes No, they seem to get confused over the quantifier. I think it may come down to the meaning of the phrase: "Let $A$ be a $B$." Let $p$ be a point in the plane $\mathbb{R}^2$—meaning any point in the plane, an "arbitrary" point in the plane. Let $P$ be a polygon, meaning any polygon. Let $T$ be a triangulation of a set of $n$ points. Does every triangulation of $n$ points have the same number of triangles? (No, not always.)

Q. Have you encountered this confusion in your teaching? If so, how do you circumvent it?

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    $\begingroup$ Reminds me of this question I asked: matheducators.stackexchange.com/questions/13594/… Perhaps we need to be more explicit/verbose with "let"! Would students be less inclined to this confusion if we said, for instance, "Suppose P is any polygon, with no other assumptions about its properties"? $\endgroup$ – Brendan W. Sullivan Feb 24 at 7:02
  • $\begingroup$ Isn't that exactly what "Let P be a polygon" means outside our classroom? @BrendanW.Sullivan $\endgroup$ – Chris Cunningham Feb 24 at 13:36
  • $\begingroup$ I should say that this confusion arose in written questions, not verbal explanations, where it is easy to elaborate as @BrendanW.Sullivan suggests. $\endgroup$ – Joseph O'Rourke Feb 24 at 15:05
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    $\begingroup$ @ChrisCunningham Yes, exactly. But it seems like maybe the problem is students don't realize that the single word "let" for us means all of that other stuff. I'm recommending that we spell it out for them a lot. Emphasize that idea in class so that, on a written problem like in OP, part of the question is assessing their understanding that "Let P be a polygon" means to consider all possible polygons. $\endgroup$ – Brendan W. Sullivan Feb 24 at 16:17
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    $\begingroup$ My initial thought was that "convex" was intended, because omissions such as this are common (sometimes on purpose, sometimes just overlooked), and in my opinion to avoid having a gotcha question you'd want to say something like "polygon (convex or not)", unless prior context or reader exposure was that non-convex polygons arose often enough that your intended audience should be thinking about them (which I can't judge based only on what you've posted). So regarding "how do you circumvent it", I would say by anticipating common/expected assumptions unless you're specifically testing that. $\endgroup$ – Dave L Renfro Feb 24 at 16:55

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