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For example, I gave an exam earlier today with a problem that ended in the sentence

Use the chain rule to find $(f\circ g)'(3)$.

During the exam, one of the students asked me what the circle between the $f$ and $g$ means, and I answered that it represents the composition of functions.

Later on, another student was working on a question that contained the phrase

. . . where $V$ represents the volume of water in the tank.

and also the phrase

Given that water initially drains from the tank at a rate of $2.0\;\mathrm{L}/\mathrm{min}$ . . .

The student asked whether $2.0\;\mathrm{L}/\mathrm{min}$ is the value of $\dfrac{dV}{dt}$, and I said I couldn't help with that.

Would you have answered these students' questions? How do you decide what to do in these situations? I usually handle these on a case-by-case basis, but I'm curious if anyone has any good general rules for deciding what information to give.

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  • $\begingroup$ I remember a (very profound) teacher in classical languages walking around during the highschool finals (which are very important in my country), silently walking around, looking over students' shoulders for a couple of seconds, and, while frowning, pointing out some mistakes on the sheets. Not a very fair approach, but it still makes for a funny story. $\endgroup$ – Ruben May 5 '14 at 4:39
  • $\begingroup$ It depends on how the exam is structured. For any exam with more than one proctor, I have a no answering questions policy, unless it is 100% clear that the question is due to not understanding the definition of a regular English word (not a math word). This helps prevent the unfair case where some proctors tend to interpret what constitutes help more strictly than others. Here, none of these questions are answerable. The question "What does the word drains mean?" is answerable. $\endgroup$ – WetlabStudent May 6 '14 at 4:37
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In some cases (e.g., during an open-book exam), you can opt for no-questions policy.

However, I think that case-by-case is the best option. Each time a student asks a question, what I ask myself is (with some stretch, you could call this a “general rule”),

$$\text{Would the answer give the student unfair advantage?}$$

Such an approach might mean that

  • some clarifications would be told to the whole group if spotted early;
  • even sensible questions could be left unanswered if asked late.

To consider your examples, in my opinion explaining notation would not give the student an unfair advantage (i.e., the content of the tasks should be widely accessible). On the other hand, acknowledging the connection between $2.0 \mathrm{L/min}$ and $\frac{dV}{dt}$ would (this is already beyond the basic, literal understanding of the problem).

I hope this helps $\ddot\smile$

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    $\begingroup$ Do not forget that questions may also draw your attention to a real problem with a question (such as a mistake in the question) early, that you can then amend just in time. I therefore would consider a no-questions policy a really bad idea. $\endgroup$ – Wrzlprmft Mar 22 '14 at 0:04
  • $\begingroup$ @Wrzlprmft Indeed, I consider no-question policy a bad idea. Nevertheless, I would answer your comment with "it depends". It is possible that to preserve fairness you will have to keep quiet and scrap the results afterwards. Sometimes it might be possible to salvage some of it, and in rare cases dealing with ambiguous/unclear problem statement could be a part of the challenge (but it would be clearly stated at the beginning of the semester, and then reminded before the exam). $\endgroup$ – dtldarek Mar 22 '14 at 0:12
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If you are worried about fairness, I'd offer the answer to any students' questions to the whole class, whether announced to the class verbally, or written down on the board.

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    $\begingroup$ This is the approach taken by almost all the math exams I've been in. If it can't be announced to all students, it's likely an unfair advantage. $\endgroup$ – Muz Apr 13 '14 at 11:18
  • $\begingroup$ Most important answer IMO. $\endgroup$ – Torsten Schoeneberg Oct 31 '17 at 18:26
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Since dtldarek's answer addresses well the issues of fairness to students, I'll mention another consideration. When writing exam questions, I try to make sure each question has a certain intent, that it probes the student's knowledge of a certain concept/definition/technique (or some combination). When a student asks a question during the exam, I have these goals in mind, and I will not reveal any information about what is supposed to be probed by the relevant question.

For instance, consider your example with $(f\circ g)'(3)$. Are you probing the student's ability to recognize the applicability of, and subsequently use, the chain rule? If that is your goal of the question, then reminding them that $f\circ g$ means composition is totally fine. Or, have you been stressing the importance of notation in class? Is this question more about probing their ability to understand various ways of writing functions and compositions? If so, I would rather not tell them what the circle means and instead say, "I cannot answer that. That's the point of the question."

So, ultimately, I'm advocating a second part to dtldarek's general rule: Would my answer adversely affect the intent of the exam question?

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  • $\begingroup$ I would rather avoid this approach, students frequently side-step examiner intents and giving away some hints but not the others might, if not handled properly, make the test unfair. $\endgroup$ – dtldarek Mar 23 '14 at 14:31
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In a course I taught a while back I made it a rule: "I ask you 4 questions, each of you gets to ask me at most 4 questions." Did wonders to cut down on dumb questions. And obviously, the only questions that get answered (but all count as asked) are questions about what is being asked. "Can this be done by..." gets no answer, counts as a question. This only works for relatively small groups (I had 15 to 20 then).

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    $\begingroup$ Implicit in your answer is that before you made the rule, it was a too-common occurrence that students would ask you more than four questions each during the course of an exam. I find this curious: the average number of questions that I get from the entire class on an exam is probably 3-4. I wonder why your students are asking more than a full order of magnitude more questions than mine? $\endgroup$ – Pete L. Clark Apr 12 '14 at 23:34
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    $\begingroup$ Also, I have to say that I don't really like the rule: the exam time is for the students to use as they see fit. Unless the instructor is giving the game away in her answers to questions (which it sounds like you are certainly not), in most situations it cannot be to the student's advantage to ask too many questions: rather they should get on with the business of solving the problems. But if, for instance, it turns out there are several mistakes or ambiguously worded questions on the exam, then all of a sudden you may get lots of questions, and it seems unfair to limit them. $\endgroup$ – Pete L. Clark Apr 12 '14 at 23:35
  • $\begingroup$ @PeteL.Clark, it was a smallish course (20 to 30) for non-specialists, which somehow had gotten into the bad habit to ask, ask, ask in the hope that the game was given away in answers. The rule was announced beforehand, and in the (rare) cases where the question was about some real mistake or ambiguity in the exam it did not count as a question, and it was clarified to all. $\endgroup$ – vonbrand Apr 12 '14 at 23:57
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For questions like "what does the circle between $f$ and $g$ mean" in a calculus course, I would tell the student and then write on the board (for everyone else) something like:

In problem 5, $(f \circ g)(x)$ stands for $f(g(x))$ (composition of functions).

I wouldn't do this in a precalculus course, however, where this notation is one of the things being tested.

As for questions that I don't think are appropriate to answer, I usually make an assessment of how much I think knowing the answer is worth for the problem at hand. For example, for a 10-point applied related rates problem in which your second question came up, I might tell the student that I'll provide the answer for a 3-point penalty, which they can then choose to accept or reject. I explain this sort of bargaining in advance, when reviewing the syllabus the first day of class and/or when reviewing for the test before the test is given. Personally, I'd much rather do this (and mark on their test, in ink and in my handwriting, "$-3$ for hint") than to later deal with the much more time consuming task of deciding how to grade a problem that I think they could have worked if they knew such-and-such, but it's difficult to tell for sure because what they don't know and what they do know is mixed together in their solution.

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Don't neglect the students in your class with unasked questions!

If one student in your class asks you a question, then it's safe to assume that at least one or two also have the question somewhere in mind. If this is the case, then I would argue that it is unfair to answer a question for only the students who are willing to ask. If we really want to avoid giving "unfair advantages" then surely it is unfair to bias against those students without the courage or volition to ask a question even when their question isn't really what the the test is "about."

The school I teach in currently has a "no questions allowed" policy during exams, including when there is a flagrant or misleading mistake in the question paper. At first I thought this was harsh, but I've come to think that this really is the most fair way of handling things for everyone. Additionally, this really motivates teachers to moderate their exams and carefully check for wording and clarity.

In my class tests, if a student asks a question I will respond in only two ways:

1) I provide general encourage along the lines of "You can do it!" This (I hope) is a way of calming the nerves of those with test anxiety or a memory block.

2) If I find that I have made a really bad mistake in writing a class test, then I'll write the correct question on the board and explain the mistake to the entire class.

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