In the comments to last post, reader math gradstudent asked, “I would be very interested in reading a professional mathematician’s (a math professor’s, to be precise) version of this post!”
No math profs have chimed in thus far, so this question is still open. Math profs among readers, please tell us your thoughts!
I am not a mathematician, but I think I am reasonably well acquainted with the field and some other closely related ones.
I think a key question is how different the type of work done as PI is from that done by a grad student and postdoc. In most physical and biological science fields, the PI is really the manager of the group and doesn’t do the technical work themselves any more. Students and postdocs are supported by grants, so they have less freedom to choose what they work on; that their labor is expected to be in the service of the project that pays them. They are also constrained in terms of available funds, equipment, and expertise in the group. Of course, grad students and postdocs are expected to be creative and innovative, but they don’t have absolute freedom to do whatever they like because there is always the question “and who’s going to pay for that?”The PI is the group’s manager, who organizes the work of junior folks, and provides direction and advising, as well as the resources for the work to be done.
Most of math operates differently. The work is not done by groups of trainees overseen by a senior person, a professor, but by individuals or small, deeply enmeshed teams. (Applied math is much closer to computer science, statistics, and some fields of engineering in that people have groups of students whom they advise, more grant money is expected and possible to raise, etc.) In pure math, it seems to me that young mathematicians are meant to be independent since a very early age. This independence is much more absolute than the in other sciences, where independence is “within the constraints of the grant and the equipment we have and the techniques we are experts in.” Young mathematicians are expected to develop their own ideas and execute them since graduate school. If there is advising from senior faculty, it’s much less intrusive than in other sciences and engineering. In part, it’s because young mathematicians are largely paid as teaching assistants, so there isn’t the aspect of “I pay you as RA, so you have to do this project that the grant is for” that we have with group science in STEM. But mostly, it’s the culture of the field.
It seems to me that a young, creative, original mathematician who is able to both come up with ideas and execute them on his or her own is a desired professorial profile. There are collaborations, but they seem much closer and deeper, with intellectual contributions inextricably interwoven, than what is typical for collaborations in other STEM fields, which are based on complementary expertise. Therefore, the research work is done by the mathematician largely alone both before and after taking a professorship.
In contrast, the skills necessary to be successful as a professor in the sciences are not necessarily the same ones that bring success during grad school and postdoc. For instance, someone could have mediocre lab skills as a grad student and postdoc, but turn out to be a great visionary, grant writer, and mentor; this person will be very successful as a PI if he or she knows how to pick technically strong students. Conversely, we all know people who were excellent “doers” in the lab and had “great hands”, but always had to be told what to do, and were not capable or interested in developing the big-picture skills needed to be a PI.
Where am I going with this? Some of the great PIs in STEM could have conceivably not been great in grad school or during postdoc, but they have other skills more important for the PI job, which may not have been obvious to their advisors during training if they weren’t given much freedom. Some people find they are superb teachers, enjoy lecturing, and have really great rapport with students, whereas they didn’t teach very much in grad school, so that aspect would not have been on their advisor’s radar. Thus, I can see how even well-meaning advisors in most STEM fields can honestly think that a student or postdoc is nothing special because he or she was not a super productive group member, yet these junior folks fluorish when given the chance. That’s why I advocate for leading with the facts and then lending as much support as needed rather than trying to dissuade the student from pursuing a chosen career path in STEM fields.
In pure math, it seems that the skills needed to be a successful grownup mathematician are much the same skills that are needed to be a successful grad student or postdoc: individual creativity and technical prowess. That’s why I would say the correlation between grad school success and professorial success is probably higher in math than in other STEM fields, and as such the recommendation of future success based on traineeship days is probably a better idea in math than in other areas. However, I don’t think mathematicians are any less biased toward women and minorities than other scientists, so they are just as much in danger of erroneously dismissing worthy candidates as anybody else. Long story short, math is different, but is populated by people, and people are biased. So, yeah.
xykademiqz 
Thank you so very much for this response. The comparison to other fields is particularly insightful. As a math grad student, this reader is only aware of the scenario inside Math departments and could only make guesses about how faculty in other subjects work…
What I’ve been told is that it helps is you are willing to move around a lot for leave and for post docs. Spending 6 mo in France, that sort of thing. For people who want to be at R1 eventually. It must be different for SLAC profs though.
I’m a pure math professor married to an experimental physicist and I agree with this account (I’m in Canada, so my view may be a little bit different from the US, the system is not exactly the same at the level of trainees). It is generally considered that the main predictor for success as a faculty member in R1 institutions is research. So for example, hiring committees for tenure-track positions in R1 institutions will mostly look at the research record of postdocs (and very specially, the papers published after PhD) pretty much ignoring anything else, unless there is something bad that stands out. Now, I’d argue that, at least in Canada, you need some additional skills to be a successful professor such as being good at applying for grants and being very strong at mentoring students (my understanding is that being a good mentor in pure math doesn’t count as much in the US).
The student mentoring is different from experimental sciences, as the post properly describes. But it has its own challenges. Basically, a student that is not independent enough needs to be given a problem. It;s a big art to find a problem that is challenging enough, but solvable, and also that no one else solves it in between (students tend to be slow at working in the problems since they are new to the subject). So far this is more or less like any other science. But, if everything goes well, the professor doesn’t even get the chance to write their name as an author in the paper. So, basically, you need to spare a problem that you could be solving yourself, to give it to a student, and get zero credit for its resolution. The professor only gets to write their name in their own problems, not the students’. What I’m trying to say is that being a good mentor is somehow in conflict with being a good researcher. Of course strong people manage to be both, because they have many questions and don’t have the time to work on them, so they give some of them to their students. But even deciding which questions you keep to yourself and which ones you give to your students is a big issue in my opinion
My observations are also similar to xykademiqz’s. (I’m a computer scientist working in an ECE department and many of my friends are mathematicians, since I did a CS/math undergrad). One comment is that math PhD students tend to have not so many publications during grad school, only 1 or 2 from my observation.
Thanks for this post, though. It made me realize that I actually work a lot more like a mathematician than a computer scientist (not so many students, still do a bunch of technical work) although I’m pretty applied as these things go. Not systems, but definitely not theory. It’s worked for me, although I’m in Canada and not the US, so I feel less pressure (but more than 0) to have a huge group and lots of money.