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Tl; dr : Does a math professor get annoyed or think of it as a waste of time if he is emailed a difficult math question from an undergraduate student whom he is fairly acquianted with? Or is he ok with it?

I am currently an undergraduate at an U.S. university. I have taken two math courses with a pretty renowned (though not world-famous, though I might be wrong) and super-nice (i.e. very kind and skillful both in teaching and in math itself) professor in the past, got A's on both, and am also decently acquianted with the professor (I went to lots of office hours while taking the course). I am self-studying a math topic (set theory) he should be very familiar with (he is a tenured logic professor, and logic and set theory often go hand in hand). I recently came across a question on the math topic that's hard to answer, and wanted to email the professor about how to solve the problem. However, I am afriad that the professor might think of my question as a nuisance / waste of his time, especially since he might well be busy (he's tenured professor at a university that is ranked wihin top 5 in the world in mathematical logic, so he might have lots of conferences/research/meetings/etc). I don't want to be a bother for him in any way, but I think he can help me solve this rather difficult question. Should I email him or not? The last time I took a course with him was like February 2018...

In general, how do professors react to this? If you are a professor (who is not someone who doesn't care about students) please comment below. Are they annoyed, or are they ok with it/delighted? What if they are busy?

Please gear this toward summer! Lots of people may relax during summer so it might be different... Also I am not sure if this is relevant but he doesn't have a family (i.e. single) and is like age 42 or something...

I emailed a different professor once about a math question and his reaction seemed to be "I'm annoyed".. I actually wrote an email to another professor and he was like "I would be able to answer you if I am on campus but sorry I'm not and writing math via email is too much so I can't answer you"... Are these reactions generic or are they exceptional?

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    Professors receive an incredible amount of emails. Messages they consider a waste of time (due to the contend, sender, or other circumstances) or are currently not of sufficient priority (due to other commitments or indeed holidays) are typically simply ignored. I don't think the time of the year should be a major concern, but you could address that in the email. – Roland Jul 26 '18 at 9:20
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    Why not ask this at Mathematics Stack Exchange? I see that you have other questions there. – Dave L Renfro Jul 26 '18 at 10:15
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    @DaveLRenfro, actually, I don't think this is special to mathematics. – Buffy Jul 26 '18 at 14:04
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    Please gear this toward SUMMER!!! That’s irrelevant. It is appropriate to email the professor, at any time of the year (especially since you have a personal relationship to him, but more generally since his email is publicly listed on his department’s website). It is also appropriate for him to ignore your email if he is on vacation or reply to say he is unable to help. – Dan Romik Jul 26 '18 at 21:02
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Let me try to answer some of the questions in the body of your post:

In general, how do professors react to this? If you are a professor (who is not someone who doesn't care about students) please comment below. Are they annoyed, or are they ok with it/delighted? What if they are busy?

Of course the answers are "it depends". What does it depend on? Here are a few factors:

  1. The tone of your email: of course, people tend to respond better to polite requests than something that seems demanding or pushy. You say the professor "might well be busy". I can guarantee you that he is busy. He may or may not make the time to answer your question nevertheless, but you should make clear that you understand he is not under any obligation to do so.

  2. The difficulty of the question: you say your question is "rather difficult", but what is rather difficult for you may not be for a professor who is expert in the subject. If your question really is something complicated that will take a long time to type an answer to, people will be less inclined to spend that time just to satisfy your curiosity.

  3. The clarity of the question: if you choose to write and ask, you should be absolutely sure that your question makes sense and is clearly and unambiguously stated. Nothing is more vexing than putting in time to answer someone's question only to have them change what they are asking.

  4. Personality: some people just enjoy answering questions more than others; some people guard their time more closely than others.

Speaking just for myself, if I received such a question over the summer that was clearly written, not overly complicated, and phrased in a polite way, I would be pleased (maybe not "delighted"), and if I had time, I would try to answer.

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    +1. I would only add that such a professor is always busy. The summertime is no different than any other time. If your question is interesting you will get a reply, but it may just be a request to see him at some future date to be determined. But, as with many things in life: don't be boring. – Buffy Jul 26 '18 at 10:39
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    To be fair, if the question is too simple it may also count against your chances of receiving an answer. I frequently get questions (from students and others) that at least to me appear very boring and mundane, but where an answer takes a fair bit of time to type up. These have a much lower chance of getting answered than questions that require some thinking/research but much less time to explain or type out. – xLeitix Jul 26 '18 at 11:50
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    @AzorAhai, I think that if you ask them you will find they are still busy. When do you suppose they develop those courses? I don't know many academics who have much down time - ever. And if they also do research ... Busy busy busy. It is part of the life, but also part of the misconception about teachers. Even elementary school teachers do quite a lot during "vacation" time. – Buffy Jul 26 '18 at 20:44
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    @AzorAhai. I'll stand by what I said, from long experience. Maybe harder in the Summer. That may be their only chance to get "real" work done and think of the 500 person course as a bit of an easy job to run since they also have a staff of 40 to help them. See the total personnel list of Harvard's CS50 class, for example, compared to its class size. The actual ratio is close to one staff person for 20 students. In any case, the OP has to expect that the professor may be just as busy independent of season. – Buffy Jul 26 '18 at 20:53
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    @Buffy It would be brilliant if people here had 40 people to help them ... – Azor Ahai Jul 26 '18 at 20:54
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Tl; dr : Does a math professor get annoyed or think of it as a waste of time if he is emailed a difficult math question from an undergraduate student whom he is fairly acquianted with? Or is he ok with it?

In the worst case, he doesn't have time to read it, will never answer, will not remember having ever seen the email if you ever ask him about it, which makes it a waste of your time, not his. If this happens, please don't blame him and don't take it personally.

One of the essential skills to become a professor is excellent time management. If he doesn't have time to prioritise your email, he won't.

Getting distracted in the middle of a working day answering Stack Exchange questions means that I may lack that skill. ;-)

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    The last line - that hurts : doing the same ... :) – Solar Mike Jul 26 '18 at 14:26
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You should respect that professors may be:

1) on holiday relaxing after a busy semester

2) at a conference

3) catching up on research

4) writing a paper

Or other activities, so expecting them to be online 24 / 7 to provide you with a personal online tutorial is perhaps, a bit much. I respect your motivation to further yourself during the summer break but you should respect the professor’s break as well.

Best of luck with your studies.

I have, during the summer break, written reference letters due to special situations and provided questions with completed solutions (that students failed to download during the semester) so they could catch up.

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