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This question is possibly ill-advised. (If it is not right for this site I will delete it.)

I, suddenly, have students.

It is very clear to me that there is nothing in my education that has prepared me for the task of training graduate students.

Yes, I know that graduate school is the place where one finally assumes full responsibility for one's own mathematical progress. It is also equally clear to me that there are innumerable things that an advisor might do, unwittingly, to irrevocably damage the career of their own student. This is keeping me up at night. And unlike searching for advice on, say, parenting, it appears that most people keep their opinions on the process to themselves, especially with respect to issues specific to training mathematicians. The more senior people I have approached have generally told me that "things work themselves out".

I see people I know, not so much younger than me, for whom the job market is not working itself out.

I was very lucky, and as a result I have many questions about things I didn't deal with myself. I don't know how to strike the balance between a doable research project and a significant one. I don't know how to help students move from reading background into exploring on their own. I don't know when and how much to help when they are struggling, or what to say when they become unhappy about their progress.

And I don't know where to find resources to do so. As I've said, sometimes I don't know that people take my concerns seriously... my own mentors deal with students at an n'th rate university, rather than an 8n'th.

Any direction would be appreciated.

(This question is anonymous, but not for my own sake.)

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This question is very important. –  SNd Apr 25 '11 at 21:29
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I expect this question to generate some "appropriateness" discussion. Please direct it to meta: tea.mathoverflow.net/discussion/1025 . Please also up-vote this comment so that it stays "above the fold". –  Theo Johnson-Freyd Apr 25 '11 at 21:45
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I vote against closing. As usual, if you want to vote to close, leave a comment canceling this vote rather than clicking the "close" button. Please vote this up so that it stays visible. –  Andy Putman Apr 25 '11 at 21:49
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I suspect that one of the reasons why people are reluctant to share their opinions/stories is the conviction that whatever they do is applicable to themselves only and does not translate well to other cases. That would be a good point: even within math, cultures are very varied. Also, unlike child-rearing, the person whose destiny lies somewhat in your hands comes a lot more fully formed, so I imagine that dedicated advisors attempt to adapt to their students. –  Thierry Zell Apr 25 '11 at 22:47
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To those considering answering this question: please note that it asks for references. I think that anecdotes are not useful here, and will only serve to swamp the useful stuff (ie references). –  Loop Space Apr 26 '11 at 7:14
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8 Answers

One important thing is to make sure your students talk enough to other mathematicians, by introducing them to people at conferences or visitors to your university, encouraging them to talk regularly with other faculty, making sure they get to know some of your friends and collaborators, trying to help them find other mentors, etc. Ideally, they should have substantive interactions with a mixture of other specialists in their area and mathematicians in other areas.

Aside from the obvious intellectual benefits (learning from many people and developing one's own identity as a researcher) and career benefits (getting good letters of recommendation), this directly addresses one of the biggest advising issues, namely the Rumsfeldian unknown unknowns. You may not know what your blind spots are as an advisor, or how to fix them even if you can identity them, but talking to other mathematicians will help your students fill in any gaps.

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+1 for pointing out the Rumsfeldian unknown unknows (some of which I know very well...). –  Dirk Apr 26 '11 at 11:03
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The chapter on advice to a young mathematician from the Princeton companion to Mathematics by Atiyah, Bollobas, Connes, McDuff and Sarnak are advices for young mathematicians but one can infer from them some advices also for young supervisors. This is also the case for Tao online advices for graduate students.(Tao's page has links to various other good resources.) I did not see an advice meant directly for the supervisor and it will be nice to have one.

I did find one nice source - a paper by Oded Goldreich: Demystifying the Master Thesis and Research in General: The Story of Some Master Theses. It is rather specific and get into technical details but maybe it can help.

"I don't know how to strike the balance between a doable research project and a significant one. I don't know how to help students move from reading background into exploring on their own. I don't know when and how much to help when they are struggling, or what to say when they become unhappy about their progress."

What I can say is that this is very individual and there are various possible styles of supervision. I dont have clear answers for these questions even after supervising a number of graduate students over the years. Having a lot of research-level questions that you did not deal with yourself is certainly a very good basis for good supervising. A general good advice is to try to be realistic in problems you suggest to your students and not to aim too high. Like other duties, give it a lot of thinking and don't hesitate to consult with others. (Also dont hesitate to co-supervise with a colleague.)

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I also found a document (not specific to math) which seems nice: ppss.ph.unimelb.edu.au/2520/node33.html –  Gil Kalai Apr 25 '11 at 22:37
    
Gil's link: ppss.ph.unimelb.edu.au/2520/node33.html –  Did Jun 5 '11 at 19:52
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Since the OP is asking for resources, I'll give two references that I've found to be useful.

  • The Survival of a Mathematician, by Steven Krantz
  • I Want to Be a Mathematician: An Automathography, by Paul Halmos

Both books have something to say on the subject.

And it can't hurt to add the AMS ethical guidelines:

http://www.ams.org/about-us/governance/policy-statements/sec-ethics

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I see that you are worried about the way the job market will greet your students, which is a perfectly legitimate concern. The one thing I would impress on you is to make sure that your students are aware of their non-academic career options.

If they are at all interested in non-academic options, they need to prepare for them, things don't just happen. I know people for whom this just meant taking a couple of well-chosen classes here and there, it does not have to drastically affect the course of their graduate studies. But intentionality is key here.

More generally, there is a lot you can do to improve the way the job market will welcome your students, and none of it has to do with math. Impress on them the importance of professionalism, and of researching their future career. Too many people assume that simply getting a phd is enough to get you ready for academia, but in these difficult times, anything that makes you look better prepared is a huge bonus.

So sure: be a cheerleader for their math, help them out through the rough patches, but ultimately the impact that you will have on their research might not be so important as the impact you will have on their career preparedness. Depending on where you work, your institution might already have a lot in place to help you with that aspect. Otherwise, do your homework, and impress on your students the importance to do theirs.

I've just led my first job search this year, and I can tell you that the mathematical level of applicants was fairly even with few outliers (as far as one can judge such things from a cv). On the other hand, the range in the level of preparation for the job was stunning.

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"On the other hand, the range in the level of preparation for the job was stunning." Do you mind giving a few more details, explanations or examples for this? –  Arend Bayer Apr 26 '11 at 14:47
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@ABayer: I cannot go into too many details for obvious reasons, but there were basic issues like CV writing and good cover letters. But then, I've seen graduate students who are already looking ahead at the next step rather than just their thesis and graduating. For research profiles, they might be writing grant proposals or at least thinking about it, and have a clear program. For the teaching profiles (our candidates), many had taken part in programs such as preparing for the professoriate or had other significant experiences. To me, these differences were really significant. –  Thierry Zell Apr 26 '11 at 15:46
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Allen Knutson has written this on the subject.

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Ravi Vakil has a similar thing: math.stanford.edu/~vakil/potentialstudents.html –  Graham Leuschke Apr 26 '11 at 11:00
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In 1997 the (US) National Academies' Committee on Science, Engineering, and Public Policy issued a report called Adviser, Teacher, Role Model, Friend: On Being a Mentor to Students in Science and Engineering.

The report is freely available (click on the link above), here is its table of contents: 1 What is a Mentor? (1-16) 2 The Mentor as Faculty Adviser (17-42) 3 The Mentor as Career Adviser (43-52) 4 The Mentor as Skills Consultant (53-60) 5 The Mentor as Role Model (61-64) 6 Recommendation: Improving the Quality of Mentoring (65-68) 7 Resources (69-78) Report Brief: Reshaping the Graduate Education of Scientists and Engineers (79-84).

Maybe this can help. However, one should keep in mind that the focus of the authors is not specifically on mathematics but, as the title says, on science and engineering in general.

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I hope Matthew Valeriote will not mind my sharing this experience. I was talking with him and his wife at a (math-induced) social function when the topic of family raising came up. One of them mentioned that they had a role model or mentor family for guidance. I vastly underappreciated that remark at the time.

I recommend you supplement your mentor pool, even if you must do it a la Internet, if your current mentors don't satisfy you. You still have to do the tough choices yourself, but if you can't find the good advice within yourself, keep looking, and cut yourself a break for the occasional (well-inentioned) mistake.

Gerhard "Ask Me About System Design" Paseman, 2011.04.25

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As for direction, guidance counselors could probably tell you where to look. (Yes, I am talking about those found in American high schools and elsewhere.) Gerhard "Ask Me About System Design" Paseman, 2011.04.25 –  Gerhard Paseman Apr 25 '11 at 22:01
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+1 For "and cut yourself a break for the occasional (well-inetntioned) mistake." I just hope your students do you the same courtesy. –  BSteinhurst Apr 25 '11 at 22:05
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Section 2B of Indiana University's How to be a Good Grad Student is Advice for Advisors. The first half of this is very generic, but there are good specific bits of advice in the second part. Also, I think giving your student booklets like How to be a Good Grad Student would probably also help them.

Since the OP mentions that he's new to advising, perhaps The Assistant Professor's Guide to the Galaxy would be helpful. Despite being a guide to the galaxy, it's only 6 pages long, and only section 7 has to do with advising. This section focuses more on what grad students can do for you, but does have one concrete suggestion:

In my personal work with students, I set goals for them and insist that they document their progress with draft manuscripts. My work with them on these drafts often leads to conference papers. My students always publish before they finish, sometimes jointly with me and sometimes on their own, depending on the degree of my own involvement.

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The first source seems excellent! The second link is broken.. –  Gil Kalai Oct 18 '12 at 22:29
    
Thanks for pointing that out. It worked when I posted it. Anyway, here are two more (I don't want to edit because I don't want to bump this old thread): dcslab.snu.ac.kr/~ilhwan/work/grad/survival/section1.9.0.6.html and www.cs.iastate.edu/~honavar/assistantprofgalaxy.pdf –  David White Oct 18 '12 at 22:43
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