Question

How do people come up with mathematical arguments on the fly?

Answer 1

I think for just about anybody, successfully answering "why is the fundamental group of the projective plane Z_2" is a matter of recall, not a matter of producing something on the fly. Someone who answers that question with fluency has answered that very same question (with less fluency) several times before. If someone really did come up with an answer to that question the first time, rather than being shown by somebody else, then they probably came up with it over the course of hours of work.

Based on your negative self-talk, what you struggled with is as much the social skill of keeping a cool head in an interview as the mathematical skill of producing mathematical arguments on short notice.

Different people have different comfort levels with being "on the spot" this way. I'm lucky in that while I am prone to getting overwhelmed and shutting down and wanting to run away in other situations, this one-on-one technical "interview" kind of format is one where I tend to plug in and focus on what's at hand. This isn't true for everybody by default, but I think it is a learnable skill.

During the interview itself is exactly the wrong time to be worrying about whether it's going well. (But of course, putting that into practice is easier said than done.) Having been on the "interviewer" end of things like this a few times, there's not that much relationship between how well the interviewee seems to think it's going and how well it's "actually going". Whether someone is nervous and panicky isn't that closely related with whether they're actually demonstrating the knowledge and ability I'm looking for.

There's some degree of live-action tracking-of-what-impression-you're-giving-off and gaming-the-process involved in this kind of thing. But that's dressing on top of a baseline that is *not* that at all. Ideally you're mostly running on autopilot, being yourself, and letting whatever impressions form form. If you want pen and paper to work through something, say "Let me try to work through that on pen and paper" and try to work through that on pen and paper. You are who you are, and you know what you know. You are a human being talking to another human being about mathematics.

Answer 2

A mix of general mathematical maturity, knowledge in the particular field, knowledge of the precise question, state of mind, luck, occasional epiphanies, blood sugar levels,...
General advice, stop comparing yourself to others, start comparing yourself to yourself a year ago.

Answer 3

if he offered to supervise you, this is really good. The important thing is to just *try* to answer his questions with as much courage as you can. Try to get rid of that self consciousness. I really grew a lot from having an advisor that would grill me with questions i couldn't answer. I got better at thinking on my feet. Some light "shaming" shouldn't hurt you. Just go back to your room and find the answers for next time. You can't truly be creative if you're too self conscious so you just have to find some way to push that part down. You will probably find that if you weren't under pressure, you could think about projective spaces more clearly and at the very least have some ideas about the answer. You gotta tap into that even under pressure. It's not easy.

Answer 4

Most of us went through similar experiences. I don't think you should be too hard on yourself. It sounds like the professor wasn't being particularly helpful. He was effectively giving you an oral exam without you having a fair chance to prepare.

Did he at least try to help you when you struggled? Any experienced professor knows how discombobulated a student can be when put on the spot like this. He should have tried to guide you, maybe by asking easy leading questions. If that doesn't work, then he should have told you what he expects of you and ask you to return later when you're better prepared.

I also have no idea why he asked you about the Putnam. I don't see why that was relevant at all.

Sounds like he wanted a student who, as you say, can think fast. My suggestion is to find a more helpful professor.

But one lesson I did learn in grad school was to never tell a professor you "did" or "know" something unless you're ready to provide basic details on the spot.

Answer 5

You've already received a lot of good responses, but let me see if I can fill in some gaps that I didn't see in the other replies.

In a situation like this, it's okay to let the other person know you're feeling nervous and that's making it harder to recall things. I'm sure that professor knows what it's like to be in that situation, but unless you remind him in the moment he may not realize you feel that way right then. Some people will even try to be accommodating. And if the person decides to be a dick about it, then you know that you probably don't want to work with them. So it's typically a win-win to mention it when you're in these one on one sort of situations.

So much of mathematics is about knowing the definitions and the major results. I had the good fortune early in undergrad of having a math professor tell us how to do our homework. He said, for each problem you sit down to do write out each definition that the problem statement uses. Try to write it from memory, but it's okay to look it up if you get stuck and you should double check that you recall it correctly. Doing this will build your ability to recall the definitions, even if it seems boring. Actually, boredom in this sense is a good signal that you're starting to master recalling it.

The next thing he said to do, as it becomes less effort to recall the definition, is to start thinking about what the definition means. What ideas does it capture? What is something that fits the definition and what is something that doesn't fit the definition?

He also said, often times with undergraduate text book problems, they can be solved quite readily when the definitions are sitting there on the page next to each other.

In many ways, understanding a subject of math can be reduced to the combination of knowing the definitions and their implications (the theorems/lemmas about them). Once you've studied mathematical reasoning for a while, many proofs are sort of rote in the sense that they follow from the definitions. However, some theorems tend to embody some sort of insight or cleverness. Those tend to be named theorems. For those, it's fine to just memorize the cleverness. You should at least know their implications and the major insight behind their proof. The rest of the details are hopefully the small rote reasoning steps that you can reconstruct as needed.

Another tip, is to take the things you felt you blanked on or wished you had known and study up on those before you talk to him again. It's a "better late than never" situation. And you could then tell him you reviewed it and you'd like a second chance at answer some of his questions. See what he says. Ideally, he'll see it a sign of maturity and willingness to improve yourself.

Answer 6

If i have a talk with other students, no problem.
If i am to present something i prepared,  works fine.
If i'm in a room with a professor or any Supervisor really, i feel like i got a lobotomy.
I have to admit that it depends on the prof i talk to, when i talk to someone who can run through the lecture all semester long without looking at his notes once i tend to feel very intimidated.


I guess it's just about getting used to it. At one point it will become natural. Unfortunately i cannot tell you when this is about to happen because i am still waiting for that moment myself.

Edit: continuing on what others wrote: i learned the hard way that just referencing a result from a lecture without being able to explain further, even though the direction is correct, when being asked does not bring you any further other than having traded a question you couldn't answer against one that is even more detailed that can't answer either.

Answer 7

Sorry, do you go to UofT? If you don't mind me asking, which prof was it?

Answer 8

Honestly, being able to have a conversation about math with someone who is more experienced than you is something that just takes practice. Talking about math is different than writing it on a homework set. And you probably would have felt a lot more comfortable and therefore done a lot better if you were talking to a peer.

I’d be willing to bet that many students who’ve put themselves out there have a similar experience as you. I know I did many times. I would recommend trying to get out of your comfort zone by regularly putting yourself in situations where you have to talk about math. Are there any math clubs where people give talks? Any math seminars for undergrads where you can give talks? Also, accepting this professor’s offer to go through some graph theory could be helpful as well. I became much more comfortable talking about math through weekly meetings with my advisor. I felt very nervous and under qualified when I started my masters, but got a lot more confident after a few months of meetings.

EDIT: Also to answer your title question. Many people can’t make a lot of arguments on the fly if they aren’t familiar with them already. I can’t count the number of times my advisor tried to do something on the fly in one of our meetings and it ended up not working out. And he’s brilliant. I think being confident and comfortable enough to just try things and think out loud even if your thoughts don’t end up leading anywhere is the key. This confidence just comes with time.

Answer 9

Comes with more practice. Basically, reading and writing lots of proofs in the field you’re interested in.

Answer 10

The more intuitive you can make it, the easier it is for you to recall it. If you learn a proof, figure out how could someone had came up with the proof, why is the derivation reasonable. When you do exercises, apply those ideas. Eventually you will get used to it and have intuitions for the topic.

The more you can intuitively understand something the faster you will be able to get those argument; so this will keep happening no matter how far you go, because there are always harder topics. I have no troubles explaining many undergraduate topics on the fly, but when my professor talk he jumps so fast through concepts that nothing makes sense to me.