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For background, I'm a first year PhD student in a mathematics program. Basically, I can't tell if I'm "where I should be".

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.

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    Yes this is standard. Commented Feb 16, 2021 at 3:23
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    To be brutally honest with you, you will know where you need to be once you take the prelim exams. The purpose of taking prelims is so that you are qualified to do research and can then stay in the program. Until then, the answer to your question is highly speculative. Commented Feb 16, 2021 at 4:51
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    However, that shouldn't stop you from asking for feedback. I suggest you start making mentor-mentee relationships with one or two of your professors. Go to them for guidance and advise, bring them (interesting) math questions. Once you've shown them your potential, they will be more open with you and give you specific feedback. Note that this is very different than showing up to office hours to ask for help with homework. Commented Feb 16, 2021 at 5:03
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    Do you have a PhD advisor?
    – Jeroen
    Commented Feb 16, 2021 at 10:31
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2 Answers 2

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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.

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First, let's eliminate a couple of ways that aren't helpful to tell if you're on-track:

  • Don't evaluate yourself by comparing your knowledge/ability to that of your professor (e.g., worrying that you spend two hours understanding something and then he is able to breeze over it in a small fraction of that time). Most academic staff were high achieving even by the standards of their PhD cohort, and additionally, they have several decades of additional experience since then. It is completely normal for PhD candidates to have difficulties with problems that academic staff can solve easily (for an example, see this related question).

  • Don't expect to understand more from a talk than what is reasonable: By design, good talks do not usually go into heavy levels of technical detail. Going into deep technical details requires more time, and it requires an audience having time to "digest" the material. This is usually not possible in the timeframe allocated for a talk, so you will generally find that good speakers find ways to gloss over the technical details and present their topic in a way that gives the audience a good overarching understanding, but without the minutae. By design, this is sometimes a bit "hand-wavey". Usually a talk topic is backed up by an academic paper (already published, submitted, or under contruction) so if you want more technical details you can follow-up with a request for any paper the speaker has written on the topic.

  • Don't impose made-up impediments on asking for feedback: It may feel awkward to you, but there is nothing at all unprofessional about asking your teachers for feedback on your progress and standing. In fact, this is literally professional --- i.e., within the direct scope of the professional duties of the academics you are dealing with. For the lecturers of your courses, they are unlikely to be able to give much feedback beyond what your assessment results in the course tell them, but they might have some more insight on your strengths/weaknesses. If you have a supervisor or panel at this stage of your candidature, you can ask them how you are going.

Now, here are some things that are helpful in evaluating whether you're on-track:

  • Other students are in the same boat: You mention that from talking to your classmates (I am assuming these are other PhD students) you are not unusual. That suggests that the difficulties you are having are standard difficulties at this stage of the program, and are not an indication of being off-track.

  • Your grades are good: You mention that your coursework is progressing well and your grades are good. That means you are learning the course material up to the required standard, which also indicates that you are on-track. The early part of the PhD program is often devoted to coursework and "upskilling" so that you come into the research part with a solid base of technical skills. The grades you get in your coursework are a reasonable indication of how you are doing in that part.

  • You are having trouble understanding papers beyond a superficial level: This is the only part of what you write that is a concern, and it is not particularly unusual at your level. For many students, first-year of a PhD program might be the first time they ever read academic papers (as opposed to textbooks, course notes, etc.), and it takes some practice to learn how to digest this material. Unlike textbooks or course notes, academic papers are less "self-contained" and they operate in a context of a wider literature on a topic. These papers often rely on context and referencing of other work to point to parts of the topic, so it is not unusual that you will miss details or subtleties when you are new to reading them. The main skill to develop here is to learn to "dig in" to the literature, read some of the papers cited by the paper you are reading, and so on, until you feel you understand the material in the paper under examination. I recommend you dig deeper in your readings until you feel that you have understood a paper at a reasonably deep level. That is a skill you will need to develop for your later research work.

  • You haven't failed any of your candidature reviews: You PhD candidature has a number of review steps where you will be formally evaluated, and these are points at which the department can identify weaker candidates. You mention that you have some kind of preliminary examination coming up later, prior to progressing to the research phase of your candidature. That will obviously give you some feedback on your progress. Presumably that preliminary exam has been held before, so perhaps you can have a look at some past exams and see how hard they seem to you. In any case, as things stand, you have not failed any candidature reviews, which means that there is no indication of being off-track.

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