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When talking with students it frequently happens that they misunderstand what you meant. The common example is the amount of rigor that one would consider "a proof", but there are other things, like

  • smooth meaning $C^2$ or $C^\infty$,
  • ring meaning a ring with or without multiplicative identity,
  • representing points as row or column vectors,
  • defining Fourier transform with different constants,
  • zero being a natural number or not,
  • using term "graph" for undirected graphs only,
  • tree meaning rooted binary tree,

or for more soft examples

  • participation of the students in lectures,
  • checking homework or some specific way of grading it,
  • relation between exam problems and the problems covered in the class,
  • the way the instructor answers questions,

and others. Of course, it's best to address all the uncertain points (e.g. at the first lecture or in the notes), however, the problem is it's not always possible, to name a few reasons:

  • there is not enough resources (time/space/attention span),
  • some things seem too obvious,
  • one my find him/herself unprepared (for whatever reason),
  • one doesn't know the audience,
  • one doesn't have enough experience,
  • some point was addressed, but the student missed or forgot it.

Naturally, it doesn't apply only to students, but this is frequent and may have some specifics (e.g., I'm not saying it's good, but it is possible to force some convention).

General question: How to best address these issues?

More specific questions:

  • What approach to use, when there is no enough time or space to explain? For example, if I want to fit all the homework problems on one sheet (for example to avoid tl;dr reaction).
  • How to notice misunderstanding when it happens? E.g. it's hard to distinguish between non-understanding and misunderstanding.
  • How to fix the problem when spotted? For example, I may have noticed that the student uses a different definition, but we are in the middle of reasoning and making a digression may be undesirable. Another example, the student solved an effectively different (perhaps easier) problem.
  • How can I make the students be aware of it and careful in this aspect? E.g. to make them ask for clarification instead of assuming some particular version.

Edit: There's a very nice post about definitions by Jeff Stuart here, thanks to @Dave L Renfro for sharing it.

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    $\begingroup$ I think this is mostly not a problem when the teacher appropriately anticipates possible problems and takes preemptive measures to avoid such problems. For example, I would always (at the beginning, and if it came up) specify whether $0$ is a natural number or not, whether "increasing" means non-decreasing or strictly increasing (in fact, my preference in this last case is to avoid the problem altogether by never using a bald "increasing"), whether limit exists disallows infinite limits, etc. $\endgroup$
    – Dave L Renfro
    Commented May 29, 2014 at 19:58
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    $\begingroup$ This 10 October 2007 AP-calculus post by Jeff Stuart (archived at Math Forum) explains how what we think are reasonable assumptions can sometimes not be reasonable assumptions to those first learning a subject. $\endgroup$
    – Dave L Renfro
    Commented May 29, 2014 at 19:59
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    $\begingroup$ What! Who on Earth uses "smooth" to mean $C^2$? This greatly offends me :) $\endgroup$
    – Santiago Canez
    Commented May 29, 2014 at 23:05
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    $\begingroup$ For a related post, see my answer to "Is using different notation in one course a good idea?" here: matheducators.stackexchange.com/a/1477/262 $\endgroup$
    – Benjamin Dickman
    Commented May 30, 2014 at 2:15
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    $\begingroup$ @SantiagoCanez: I think that some authors use "smooth" as shorthand, possibly informal, for "sufficiently differentiable for the purpose/problem at hand." $\endgroup$
    – J W
    Commented May 30, 2014 at 5:22

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When I teach some basic graph theory, I explicitly mention (and include in the notes) that certain definitions and terminology vary from author to author, in some cases giving the alternatives. In fact, the field is notorious for this. Of course, there is a risk that students will use one of the alternatives instead of my preferred definition, but my experience so far is that this doesn't happen extremely often. I try to be alert to this possibility and point it out when it occurs, such as with the question: Which definition did you use?

In general, I try to develop my own awareness of clarity issues that may arise and inform students about them, preferably early on in a course or when the relevant topic is introduced, but sometimes on a need-to-know basis.

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My website lists a good many math words with two widely used meanings. It is at [dead link]. Search the index for "two meanings" and you will find a bunch of them. --Charles Wells

Edit many years later: It may still be possible to find some things at an archive of this site; try perhaps:

https://web.archive.org/web/20180831121859/http://www.abstractmath.org/MM/MMGlossary.htm

https://web.archive.org/web/20180831121859/http://www.abstractmath.org/MM/MMIndex.htm

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Define your terms, if needed point out there are alternative definitions/notations (and perhaps point some of them out). When using them, be careful to go back to first definitions whenever it makes sense.

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    $\begingroup$ It is not always the case that you are able to do it all. The question is about the cases where something goes wrong, how to spot it (hopefully in time) and how to react. $\endgroup$
    – dtldarek
    Commented May 30, 2014 at 21:02

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