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.