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I have recently applied to several PhD programs in math and applied math. In between undergrad and grad school, I am looking to do some summer research. During a recent interview for a research lab, I encountered a challenging situation: the interviewer posed several technical mathematics questions, and I found myself struggling to respond effectively under pressure. This experience has highlighted an area where I feel I need significant improvement: thinking on my feet in mathematical contexts, particularly under pressure or in unfamiliar situations. While I am generally confident in other subjects, I notice a marked difference in my response to on-the-spot mathematical questions.

Given this, here are my questions: Are there specific exercises, practices, or methodologies recommended for developing the ability to think quickly and accurately in mathematical contexts? How can one better prepare for high-pressure situations, such as interviews or discussions, where complex mathematical understanding needs to be demonstrated spontaneously? I often understand mathematical concepts when studied at my own pace but find applying them quickly to be a challenge. How can I bridge this gap?

Any insights, personal experiences, or resources that could guide me in enhancing these skills would be greatly appreciated.

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  • Possibly related: academia.stackexchange.com/questions/204083
    – cag51
    Commented Jan 29 at 17:08
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    Not an answer to your question, but: many of the mathematicians I respect most are terrible at thinking on their feet. Being able to apply mathematical concepts effectively is more important to a career in mathematics than applying them quickly. (Of course, sometimes there are gates to pass on the way to that career, like interviews or exams, that require some quick thinking.) Commented Jan 29 at 17:16
  • There are areas where “thinking on your feet” is not a very useful ability. Take Andrew Wiles. The ability to focus on a very hard problem for seven years until he solved it, that was important. His ability to think quick on his feet… Nobody cares. I learned that “thinking quick” actually makes you quite unsuitable in some situations.
    – gnasher729
    Commented Jan 31 at 18:22

3 Answers 3

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On-the-spot mathematical questions mostly test current knowledge, not "quick thinking"

I have seen various situations in interviews and other contexts where people are asked to solve or comment on specific mathematical/statistical problems at short notice. These cases are always highly confected situations that do not reflect actual professional practice in mathematical/statistical analysis. In practice, proper mathematical work is done slowly and carefully with significant time spent writing up the problem, attempting various avenues for solutions, re-writing the problem in revised notation, revising solutions, re-writing the problem, revising the solution, checking solutions against special cases, ancillary properties of solutions, etc. Good work in mathematics typically involves slow and careful iterations of work that usually involve (almost) as many attempts at re-writing the context and question as they do attempts at a solution.

To the extent that a mathematician/statistician is able to answer on-the-spot technical questions, this is usually less to do with "quick thinking" and more to do with having already seen and solved the question before, or at least having seen and solved similar questions in that field. For unfamiliar problems we might (at best) have a set of initial ideas of possible ways to pursue a solution, many of which probably will not work. When a mathematician is able to immediately solve a problem, or rattle off a reasonable solution method that works, that is because they have done this type of problem before and so they are familiar with the broad methodology and steps that would be most likely to yield a solution. To get a correct answer to an immediate mathematical question is likewise much more to do with the ability to impose checks on your working at the end of a solution process (e.g., checking special cases, expected limit properties, etc.) rather than relying on immediate accuracy of algebraic steps undertaken under time pressure. (Many highly competent mathematicians are people who can't do their algebra fast under pressure.)

For all these reasons, on-the-spot mathematical questions in job interviews are best seen as a test of experience rather than mathematical "quickness". Most experienced mathematicians treat questions like this with the appropriate mixture of amusement and contempt they deserve --- if they are familiar with the problem then they might have a go at it, or give an extemporaneous outline of a solution method, but otherwise the answer would be: I'll take this away and solve it in my own sweet time. I would go so far as to say that there are instances where the best response to a question is to point out that you would need time to solve it, and that it would be irresponsible for you to attempt to solve the problem under time pressure.

(As an anecdote on this point, I once attended a job interview where I was asked to give advice on the appropriate model for some data that was extemporaneously described to me. I told the panel that I would never recommend a statistical model for data without having gone through a proper process to review the sampling method and structure of the data first, and I described the process by which I would meet with them to do this. I also told them that any other applicant who answered their question and recommended a model without such a review would be doing them a disservice. For what it's worth, I didn't get the position, so make of that what you will.)

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    I'll make of it that you probably dodged a bullet avoiding that job.
    – Bryan Krause
    Commented Jan 30 at 1:18
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    Reg. your anecdote: this is the typical question you are asked at interviews&co in the finance sector. The answer that gets you high praise is along the lines of yours, i.e. "Assume we did a proper review of sampling method and structure, I will proceed with the model XZY" where XZY is whatever the cool method of the day is. That's how incompetent smart people filters themselves out from the big pool of good candidates. And this explains why financial market are behaving like short-term idiots, not like the informed invisible hand postualted by some XIX century philosophist ...
    – EarlGrey
    Commented Jan 30 at 8:32
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    As someone who at least used to be very good at such high speed thinking, I would say the ability to do such quick thinking tasks varies massively between different mathematical researchers and is essentially uncorrelated to their ability to do quality math research.
    – quarague
    Commented Jan 30 at 10:55
  • Any evidence to support the claims in this answer?
    – Kostya_I
    Commented Jan 30 at 13:26
  • @Kostya_I: I used my quick-thinking. ;)
    – Ben
    Commented Jan 30 at 21:07
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While I'm not a mathematician and won't offer any advice specific to mathematics, I think I would revise how you see these situations. I doubt these questions are intended to assess your quick thinking skills; rather, I think they're intended to assess your familiarity with the field.

You likely are not expected to spontaneously solve problems you are unfamiliar with. Rather, you're expected to demonstrate familiarity with similar problems or methodology and to relate something that is new to you to that past experience. In some cases that past experience may directly offer a likely solution, but I doubt you're being asked to produce a rigorous proof on the spot.

Besides that, practice. That means practice reading and understanding the literature and being familiar with a broad class of problems (and their solutions) in your general area of expertise.

It means practice discussing problems and their solutions with your peers in your field and adjacent areas of interest, including being able to modulate the level of discussion for the appropriate audience (roughly, I'd imagine about 3 levels of audience: someone close enough to your field that you'd always be reading each others papers, someone with PhD/professor-level expertise in your general area of study but not your very narrow field, someone who is intelligent and educated and familiar with research but otherwise unfamiliar with your field).

It means also knowing the limits of your understanding and being able to differentiate between the things you know well and where you would need to learn more. It's common to encounter students (especially those at the undergraduate level with a lot of talent) who think they know the solution to every problem (because they haven't yet encountered many that they do not) and think they have to impress others by confidently expressing a premature (and wrong) answer to something they haven't thought about in sufficient depth. Don't forget that interviews are ordinarily more of a conversation than exam and answering a question with an intelligent counter-question is likely to demonstrate your understanding far more positively than a flimsy attempt at an answer.

There is no research field I am aware of where research is performed by "quick thinking".

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You may benefit by reflecting on the emotional valence of your interview experience. For example, you might ask yourself:

  1. Was anything emotionally significant happening before or after the interview?
  2. Did you have expectations of how the interviewer would conduct the interview, against which the interviewer behaved very differently?
  3. How did you feel when the interviewer had asked their second question after you felt you had not adequately answered the first question? What about the third question?
  4. Did you feel like the interviewer was disappointed in your answers? What cues in the interviewer's spoken or unspoken language conveyed this to you?
  5. Did you feel you could have, or should have, cut the interviewer off, and said something like "I am having trouble working under pressure"? Why, or why not?
  6. For each of questions 1-4, does thinking back on the experience bring to mind memories of other emotionally significant events in your life? Do you notice any sensations of discomfort, pain, or numbed feeling in any parts of your body?

You may find these exercises useful. Research engages our fundamental curiosities, and we often find our foundational identities tied up in our results (or felt lack thereof!).

Thus, good thinking (fast or slow!) cannot happen without psychological safety, and it may be worth considering if you felt unsafe in that high-pressure scenario. It may be that the interviewer simply wasn't welcoming, and anyone would have frozen. Or some otherwise-benign feature of the experience may have reopened old past wounds and troubled you. In either case, you can then work through any psychological issues or questions that come up, whether informally with friends or formally through therapy.

(To use my own example: I find opposite tendencies in myself in that I think quickly "on my feet", which is ideal for meetings and lectures. However, I then really struggle to follow through and formally invest in pushing a quick insight all the way to a proper paper, which means my "journal output" has fallen far short of my own expectations. So it was no surprise when I recently received a diagnosis of ADHD, and medication has significantly helped me in both my personal and professional life.)

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