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I am in math and I am fairly new in supervising PhD students, so I need some advice. I have a fairly good student who has made very good progress initially. But after a promising start, he's been stuck at the same problem for over one year, without making any progress. I have at a few occasions suggested that he should try another problem. He would then spend one or two weeks on another problem, but then he would always get back to the original problem with some new approaches/ideas (which still haven't turned out to work). He seems to have grown attached to this problem and feels that it would be a waste to abandon on something he's been working on for so long. I obviously admire his persistance and I always feel bad to discourage him from working on something he feels passionate about. But at the same time, I am worried that the longer he spends on this problem, the less likely he would be willing to move on and eventually give up altogether. What would you do in this situation?

Thanks in advance.

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    You could always work more closely with him on the problem. Do you think there's a reasonably accessible solution to be found? – user37208 Aug 25 '16 at 21:40
  • What would you say is working closely? At the moment, we have regular meeting about once a week and we would discuss for maybe one hour. It's very hard to say whether an accessible solution can be found. I have suggested a few possible approach but we haven't been able to get anywhere... – user119481 Aug 25 '16 at 23:20
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    I think it would be better if you approach another ideas and problems, and also not giving up this problem. Perhaps after a few months working on another problems you came back to this particular problem fresh. – Nikey Mike Aug 26 '16 at 8:58
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    Are there ways of weakening the problem, such as instead of proving something for all integers, it's proved for infinitely many integers or for a set of integers whose limiting density is bounded above zero? Without knowing anything specific about the problem, I don't know what to suggest, of course, but I find it hard to imagine a problem that can't be weakened in various ways, and perhaps some of these weakenings have already been proved by your student, and if these proofs are sufficiently nontrivial, perhaps they could form a contribution to the final dissertation. – Dave L Renfro Aug 26 '16 at 14:30
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    To flesh out user37208's suggestion, you could try thinking about the problem yourself for a few days (outside of meetings). After that you may either have a better suggestion or a more convincing argument that the problem is definitely too hard. – Noah Snyder Aug 26 '16 at 23:53
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This is one of the hardest questions a math PhD advisor can face. (Perhaps it is the hardest question. It is certainly the one whose failure to adequately solve has caused me to feel most, um, inadequate as an advisor over the years.) Here are some ideas:

1) You say the student is fairly good. I'm not sure exactly what that means. For me, a good student can make substantial progress on problems of interest if the advisor sets things up very well: by choosing the problems well and by providing help when needed. You don't want to watch a good student waste much more than a year working on something that is not going to come to anything. Within this category, a better student will be faster, more successful and independent, and a worse student will need so much help from the advisor that although the student does solve his own problems, the advisor must put at least as much thought and energy into the situation as would be needed to solve the problems herself.

So for me, "fairly good" would mean a student who can be quite successful with a very hands-on advisor. I would not want such a student to spend a substantial amount of time on a problem that I didn't have some idea how to solve. That's a strategy for a higher grade of students (a really excellent student can carry the risk of failure of working on truly difficult problems and can also learn from the experience of choosing the wrong problem).

2) One year is too long for almost any mathematician to work on a problem without making any progress. Certainly I have never done anything close to this (although I have spent many years writing certain papers: progress was being made, just slowly and there was always more to do).

I obviously admire his persistance and I always feel bad to discourage him from working on something he feels passionate about.

To be honest, I don't really feel the same way. Working on something for a year without any payoff sounds irrational, maybe a bit obsessive to me.

3)

But at the same time, I am worried that the longer he spends on this problem, the less likely he would be willing to move on and eventually give up altogether.

To be honest, you don't sound quite worried enough. The fact that he has nothing to show for a full year of work is already a problem for him and his future career. A student can no longer graduate with "one theorem" and expect to be competitive in the academic job market: increasingly, successful graduates already have a body of work. Moreover, most successful graduates draw on the knowledge and skills of their advisors in very deep, substantial ways: without enough of this access, they may not be competitive.

For a student who has been spinning his wheels for a year, I think you need to step in and give him a new problem in which you know exactly how to solve some initial segment of it. Starting in a place where you are essentially telling them what to do seems like a good remedy for the situation: even implementing exactly what your advisor is telling you is more than one or two weeks work for a good student. If you make clear that you do expect him to work and make progress on this new problem -- with your help -- he is likely to do so. If you have a meeting with him in which he tells you that he worked on the other problem instead, reiterate what you want him to do and schedule another meeting only a couple of days later. If you can keep on him closely enough, he'll get the idea: you fully expect him to make progress on this new problem. If he wants to work on his own problem in his spare time, great, but you are making sure that doesn't interfere too much with his new day job.

I hope it doesn't sound like I am scolding you. As I said, advising is such a hard job that most people I know don't feel they are doing as well as they should. Anyway, good luck.

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    Great answer - I also feel much more focused and am making more substantial progress, after my advisor decided that we communicate almost every day via email. But wouldn't you say that this student's obsession with the problem is a good trait - and welcomed / admired, even expected by thesis advisors? A least initially, before he learns his lesson that one can obsess over a problem and still make no progress on it and must move on. – user58865 Aug 31 '16 at 20:50
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    Very nice answer from Pete as usual. I would add however that if the student doesnt want to stay in Academia after the PhD, then actually it might be better for him to stick with it to the end, even if it takes another year. Another consideration, which i find to be true for programmers but maybe less so for mathematicians, is that the more you re-write your work from scratch and discover alternative/better ways to do something, the more you develop as a programmer. In a sense, it's always good to dive deep into 1 thing. From a learning perspective at least. – Wetlab Walter Aug 31 '16 at 21:20
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    @WetlabWalter: "I would add however that if the student doesnt want to stay in Academia after the PhD, then actually it might be better for him to stick with it to the end, even if it takes another year." What makes you think the end will take only one more year? What makes you think he will ever solve the problem? – Pete L. Clark Aug 31 '16 at 22:51
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    @user58865: "But wouldn't you say that this student's obsession with the problem is a good trait - and welcomed / admired, even expected by thesis advisors?" No, I am a thesis advisor and don't expect students to be obsessed. Very committed to a problem is great...if progress is being made. "A least initially, before he learns his lesson that one can obsess over a problem and still make no progress on it and must move on." Hang on -- we're not at initially anymore; we're one year later with no progress. You are really changing the question. But yes, I agree with what you end up saying. – Pete L. Clark Aug 31 '16 at 22:55
  • Yes, good point - thanks again, @PeteL.Clark. – user58865 Sep 1 '16 at 6:35
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I have a professor who was very good friends with Paul Erdos; Erdos had a room in my professor's house and actually flooded the basement one morning.

The first time they met, my professor was in graduate school and presented a graph theory problem to Erdos and asked for his input. My professor wanted to solve this problem for his dissertation. Erdos stroked his chin and thought for a moment and finally said: "young man, you should consider a different problem."

My professor went back to work and eventually wrote his dissertation on a different problem. About ten years later he reminded Erdos about that and Erdos asked, "has anyone solved that problem yet?"

"No."

"Well, see, I saved you ten years."

I know my professor continued to work on the problem, I don't know if he ever solved it, but he got a few papers from the work that came out of that problem.

Tell your grad student they're lucky they found a good, hard problem. They do not have to abandon it; they can certainly spend a couple hours a week on it during graduate school, if they manage their time well. But grad school is about efficiency. It's about learning what problems can be solved quickly and what takes more effort and time. Don't tell them to abandon their problem, tell them to keep at it, but spend most of their time on something more reasonable. Balancing short term and long term goals, that is a skill worth learning.

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Is it something he can set aside and come back to later, when he's completed his PhD?

It might help to remind him that he doesn't need to accomplish everything he wants to research as a graduate student. If there are areas more likely to bear fruit, he may be well-served by focusing on what will get him to his degree and into a research position first, then coming back to his unsolvable problem, when it's not holding up his career. This way the work he did actually isn't wasted time.

You might also tie that into issues of funding and such, although you didn't say anything about whether that's a constraint.

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From the student perspective, and also a teacher. It would be good to have an informal meeting along with tea and discuss the matter. You can tell your honest opinion about the problem, the timeline and the progress. Ask him that instead of working in this problem it will be better if we go for that problem so that we both have interest as this probkem is taking too much time and i feel bad that you are working alone on that problem and my supervision is of nonuse in this case.

Dont tell him that you wasted time with no result/progress on this problem as he will try to prove his self and work on the same problem just to prove that he did it or will give up considerin him self a low performing student.

The more friendly environment you keep the better results you can expect and control as well.

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Is the problem solvable? Maybe it isn't, and the student just needs to stop working on it.

Is the problem divisible? Taking any problem and breaking it down into multiple steps can help.

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I think that in every work we should consider the deadline. You can set a deadline and if your student won't progress, change the thesis topic very quickly. Based on "fail fast" strategy, you should always have an alternative plan and if the primary plan didn't work, fail fast and go to alternative plan. It could save you a lot of time.

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