How can a student evaluate themself? For background, I'm a first year PhD student in a mathematics program. Basically, I can't tell if I'm &quot;where I should be&quot;.
As an undergrad, I read before class and lecture felt like a nice review. As a grad student, I can read before class, spend an hour or two pouring over a few pages, understanding every sentence, and the professor will breeze over it in 10 minutes. If I don't prepare, I'll be left in the dust, totally lost. Or, he will cover material with so few details it's hard to tell if I understand anything at all. Talking to classmates, I'm not unusual, but I find this very stressful because it's consistent and I'm not used to it. If it's typical to get lost and not understand the lecture, how can I tell if I understand anything at all? Is this just standard?
Reading papers or attending talks is worse. I feel like I get things only at 'handwavey' level, without details or subtlety. I know this is normal so I don't stress much, but some day I will have to be giving talks, writing papers. How can I get there if without a firm basis?
I think my fear is that I could be learning the absolute minimum and barely skating by -- which will bite me when it comes time to take prelims/do my own research -- or I could be doing perfectly fine and just stressing myself out. My question is, how can I tell? How can I gauge if I'm making typical or appropriate progress?
My grades and homework are good, but this seems meaningless. I always receive full marks, even when I find errors in my own work later, and work comes back with no comments. Talking with peers helps a lot, but is hard with distance learning. Asking a professor directly seems like it would be awkward and unprofessional, and also how could they know? They don't grade the homework and we speak for maybe 20 minutes in office hours occasionally. What else could I consider?
I've also read this post, which is relevant, but the answers are directed at the research stage of the PhD. I'm still taking courses, etc. I've also read posts on imposter syndrome and I feel like my issue is distinct. It's not that I feel like an imposter, but more that I feel like I'm untethered and in la la land as Dan writes. I'm interested in ways to feel more tethered, and in understanding how common/typical this untethered feeling is so it doesn't stress me out as much.
I think you’ve hit on a subtle point here that plagues many mathematics students up to and including the advanced graduate level: evaluating yourself without external feedback is very, very difficult.
In fact, it seems to me that the ability to evaluate your own level of understanding (in the context of studying pure math at least) and test when you understand a concept well, versus when you are just deluding yourself into thinking you understand, is a skill in its own right. And it’s a difficult skill at that, one which takes considerable time and experience to acquire. Professional mathematicians have developed all sorts of tricks and heuristics to achieve this: various sanity checks and other “unit tests” that we continuously run while working through new material to test our understanding and make sure we keep our mathematical feet planted firmly on the ground, instead of becoming untethered and flying off to some kind of la la land. At some point this becomes second nature and I would guess that most experienced mathematicians do it essentially at a subconscious level. But as I said, it takes years of practice to get to that point.
What I would suggest is that you do two things: first, put it on your todo list to develop an ability to gauge your own level of understanding and identify misconceptions about material you’re reading or hearing in lecture. It’s an extremely useful skill, and thinking of it as an explicit thing you want to work towards in a deliberate way will surely help you acquire it faster and better.
Second, until you’ve actually developed that skill and have evidence to support that, assume you don’t have it. That is, accept that there will be occasions (which become increasingly rare over time, typically) when you will absolutely have to rely on external feedback to tell you that you’re misunderstanding something, or to confirm that you’re not. I imagine people will be posting other answers here suggesting various tips and tricks for evaluating yourself. That’s all well and good, but don’t let such advice mislead you into thinking you can do without external feedback. You can’t — not yet, not at this point in your studies at least. So don’t be shy, go out and seek that feedback from your professors, peers, stack exchange or wherever else you can get it. And please get rid of the mindset that it’s “unprofessional” to seek feedback - nothing could be further from the truth in my opinion.