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I'm a first year PhD student in mathematics and I'm in a pretty bad spot.

Over the last year or so of undergrad and first semester of grad school, I've completely atrophied my problem solving skill. At some point I became more comfortable with looking up a solution than trying to solve it myself. At this point my first instinct is to google something instead of trying to solve something. I need to fix this; it's already been affecting my performance and well being across the board. I'm almost instinctually aversive to trying to solve a problem by myself at this point. I feel like I've lost the ability to actually do math. I initially justified it by saying that my interest in math stems from my interest in the theory, and that I'm not particularly interested in problem solving. It's clear now that that was just cognitive dissonance. I need to fix this.

I know what the obvious answer is- don't look stuff up. And try to do problems on my own. Practice, practice, practice.

But I feel like it's not so simple either. I'm doing graduate level math after all. I managed to get into a fairly top level, rigorous program. I have performed well enough in the past that I managed to place ahead of my peers, and am doing relatively advanced courses (after all, I wouldn't have resorted to looking up stuff if it wasn't working well for me, until recently). As such, it seems like I already need a solid, strong problem solving capability in order to deal with my classes, which are quite demanding. So when I'm faced with HW or other problems, I'm unable to solve most of the problems even if I try really hard, because my problem solving skill is just so bad at this point, and I have to resort to looking things up once more. This further worsens my skill and on and on. It's a negative feedback loop. And I'm struggling to break out of it. I wanted advice on how to escape this loop especially. The idea of simply not looking stuff up is sound, but it's hard to follow through when I have only a finite amount of time before I have to stop thinking and submit my answers, or when I simply don't possess the capability anymore to try and solve the problem.

These days the idea of solving a graduate level HW set seems impossible to me, and I'm just incredibly lucky graduate level courses tend not to have exams. It's reached the point where it's threatening my future in my PhD program so I really do need to fix it. Googling my way through life isn't possible (or desirable either). I really am desperate now. I feel like a lost cause at this point, like the damage has already been done to me, and I can't really fix it without going back to undergrad or something.

I was just hoping for concrete advice and from people who have had similar experiences, and what I should do.

Clarifications:

  • I am referring to problems/solutions in my graduate courses. The research portion of my PhD has not started yet.
  • My advisor and I are not particularly close (only interacted a few times so far due to the pandemic), so my awkward attempt to bring it up with him didn't really go anywhere.
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    Some clarifications have been edited into the post; the rest of this conversation has been moved to chat. Please do not post answers in the comments. Hint: if you find yourself adding five consecutive comments, it probably shouldn't be a comment. – cag51 Feb 12 at 6:35

12 Answers 12

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I think it’s not a coincidence that the timing of this decline matches up with the pandemic. A lot of people are really struggling right now, I feel like I can’t do math as well as I could a year ago. In this post it sounds like you’re being really hard on yourself. Counterintuitively I think a big part of what might help is being more compassionate with yourself. It’s ok that you’re struggling, and it’s ok to just not solve some of the problems. Research math mostly involves failing to solve problems most of the time, so it’s really ok to just not answer some of the questions and get used to that feeling and be kind to yourself because they’re hard problems. You can always come back to those problems next year. I guess what I’m saying is that you also want to develop the skill of giving up on a problems as well as the skill of solving them. The former is perhaps more valuable in research than the latter.

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    This answer cannot be emphasized more. The pandemic is, among others, also a large-scale mental health crisis. Many universities have started to recognize that and give students additional retake options for failed classes. – lighthouse keeper Feb 9 at 14:39
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    "you also want to develop the skill of giving up on a problems as well as the skill of solving them. The former is perhaps more valuable in research than the latter." - excellent point, +1. – Captain Emacs Feb 9 at 14:44
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    I appreciate the value in what you're saying and you're probably right to some degree. But I feel obliged to mention that all of this is with respect to regular coursework; I haven't gotten to research yet. While research math involves a lot of failing I imagine regular coursework is designed to be solvable. But still, thank you for your advice. – Atrophied_Skills Feb 9 at 14:54
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    The point of these first few years of grad school is preparing you for research. Part of that preparation is learning to be ok with not solving problems (even solvable ones!). I don’t design graduate homework with the expectation that students should be able to solve every problem. (I want students to be able to solve most of them, and I don’t want to assign problems no one gets, but it’s normal for a problem or two to stump each student each week.) – Noah Snyder Feb 9 at 15:02
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    That's fair. Thanks for your advice! – Atrophied_Skills Feb 9 at 15:04
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I experienced a similar deterioration in those skills during my time as a Ph.D. student. A key element in enabling me to recover those problem-solving skills was TAing (working as a teaching assistant) on undergraduate maths courses, which gave me a good excuse to schedule some time to practise those skills, on entry-level problems, in order to help students with then.

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    Yes I have also found this helpful and have been seeking out courses to TA that would help my fundamentals. – Atrophied_Skills Feb 9 at 14:14
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    +1 for TAing. I actively used TA hours to keep refreshing basic maths skills, and by extension problem solving skills. In few distinct occasions this actually helped by giving me ideas for my actual research work, too! – AnalysisStudent0414 Feb 10 at 0:03
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    Problem: You can't immediately think of ideas that may lead directly to solving a HW question. Answer: Just start doing something in algebra/topology/whatever and after a while your subconscious mind will loosen out and ideas/intuitions will come. – Trunk Feb 10 at 15:18
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    @JKHA Not stupid at all. TAing=working as a teaching assistant. – Daniel Hatton Feb 12 at 0:39
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    ... similar to being a Doctorant contractuel chargé d'enseignement, but you don't actually have to be a doctoral student to do it. – Daniel Hatton Feb 12 at 0:46
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Let me add my perspective on the first two points in the answer of Spark.

  1. Some students, and I was one, reach the limit of their natural "ability" in math early on, and others later. Before you reach that limit things seem fairly obvious and you can do math without a lot of hard work. But once you reach that limit only really hard work will get you to the required insights. It is possible to do that, as I did, but as soon as I started undergraduate studies I had to work hard to advance. My sister, on the other hand, hit the wall later and didn't really recover from it. But in her early life it was all pretty easy.

The solution for me was to do more exercises, even if I had to buy workbooks in order to find them. I once graphed hundreds of rational functions (by hand) using derivative information and developed deep insight into the behavior of real functions in doing so. But I had to learn not to depend on "ability" to get the job done.

  1. You may have reached a point of burnout as I did early on in grad school. This is pretty natural if everything up to now has been very intense. Your brain, and maybe your body generally, need a break of some kind. I didn't get over it until I changed institutions and found a more supportive environment. But I also learned to take breaks - especially aerobic breaks, so that my mind was fresh when I attacked a new problem.

I don't know if either of these actually resonates with you or not. But it is probably worth taking a bit of time (a time-out) to look at where you are what underlying causes there might be for your block. A professional counsellor (perhaps at the university) might be able to give you advice fairly quickly. Small changes might be all that is needed.

I don't have more to add to Spark's last points, but they resonate with me also. Good advice there.

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    Your #1 sounds like the introduction Isaac Asimov's essay collection View From a Height (I think this is the correct book), but I have to leave now and thus don't have time to look for my copy for more specifics. But I recall it was about how Asimov found all math easy and trivial and obvious until differential calculus, when the cracks began, and then with integral calculus (2nd semester U.S. based calculus) he hit a wall that he never passed, and from then on he regarded math as an arena in which he could only roam around as long as he stayed on the lower side of integral calculus. – Dave L Renfro Feb 9 at 16:40
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    I got back home a few minutes ago, looked for the Asimov comment I was thinking of, and found it as the italicized introductory comments to Part 3. Mathematics in his 1969 book Opus 100 (pp. 89-91 in my 1969 Dell Publishing paperback version; I think pp. 87-90 in the 1969 Houghton Mifflin hardback version). Unfortunately, the passage does not seem to be on the internet anywhere. – Dave L Renfro Feb 9 at 18:31
  • +1 for your first point. The later you hit that wall, the more it hurts and the harder it is to recover. – JS Lavertu Feb 9 at 19:01
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    @JS Lavertu: Because I apparently had nothing better to do, I've posted the Asimov comments, which I previously mentioned, as an answer to Success in maths (soft question). – Dave L Renfro Feb 9 at 20:58
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    The wall thing needs to be bought up more with students who find math easy. I remember hitting it in first-year university calculus. I had no idea the wall was coming and it sucked. I wish someone had told me. – stanri Feb 10 at 10:48
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There are a few things that might be happening:

  1. You are in a grad program, which is probably presenting you with more challenging problems. So - problems you are encountering might be actually more difficult than problems you encountered in your undergrad studies.

  2. Your brain is tired. Stress of starting a PhD program; from a global pandemic; from knowing you are in a top program and feeling you're not up to snuff; from failing. Let go of these. This is clearly easier said than done, but perhaps finding comfort with friends and family, or talking to a professional (no shame in that at all! This is a stressful time).

  3. You are giving yourself a crutch of Googling stuff, which is super tempting. Try to avoid that as much as you can, and even start by solving the basics first as was suggested by others here.

  4. One of the big differences between grad research and undergrad studies is that you don't know whether an answer exists. In an undergrad assignment, you know that there's a solution, which is giving you some comfort that if you dig enough you'll find the solution. This is definitely not the case for research problems, where your problem may have no solution (or one that's way beyond your skill level).

Good luck!

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  • +1 for the uncertainty on the existence of an answer. What's worse, sometimes one isn't even sure about what the question actually is, and whether it is relevant. – Dohn Joe Feb 11 at 15:04
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I think your skills might not have atrophied as much as you think and you simply haven't adjusted your expectations!

It's difficult to diagnose the problem from afar but there might be the following vicious cycle at play: You start working on a problem with the attitude of "I am good at math. And I can do any math problem thrown at me!". But now because those problems are much harder you struggle much more compared to your times as an undergraduate. This generates enormous frustration ("I should be able to do this!! Why can't I do this??") which in turn makes it much harder to solve the actual problem.

Because solving hard math problem requires creativity and very high focus you need to be in a somewhat relaxed and positive state of mind. In order to achieve that I'd advise you to give yourself much more credit for partial progress. As an undergraduate you probably didn't give yourself any credit for just doing run of the mill homework. Maybe if you got 100/100 on every single problem set you felt a little bit proud. But below that you probably just didn't consider solving homework problems as an achievement. Now in grad school game is a different one!

The way to break our of your current state, is not to beat yourself up and try harder but give yourself positive feedback for much smaller incremental progress than you used to. Getting started with even understanding exactly what the problem is used to be "worthless" to you ("everyone can do this") whereas now it would equate to a very productive afternoon that you should feel proud and accomplished.

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Here are some suggestions for both your courses and your future life as a researcher. These will reflect my own taste and approach, so may or may not be relevant to you. In particular, I think this applies to some areas of mathematics much more than to others. Some of the points below will be corollaries of the more general points preceding them:

  • Instead of thinking of your job as solving problems, think of it as understanding "what is going on". In many cases at the level of graduate courses this will, in fact, be "all" it will take to solve the homework exercises (which is not to say that this is a small step). In research this is also often the most important part. Rediscover that understanding what is going on is something only you can do, and that looking up can only help so much with this. When we write papers, we often create an illusion: namely that at some point (or at several points) we had a brilliant flash of problem solving genius. In reality what happens much more often is that we grope our way to the final polished solution by way of many concrete examples, special cases, iterations of abstraction, etc., ultimately just understanding better and better "what is really going on". The next points are just going to be concrete elaborations on this general attitude.
  • Compute concrete examples, unprompted. Don't delay this until you have been given concrete exercises to solve, start doing it when you learn the theory. As soon as you see a definition, come up with examples and non-examples for the concept. As soon as you read a theorem statement, try to disprove it before continuing to read the proof; work out and verify the statement in concrete examples; if you start out immediately believing the statement, try dropping some of the hypotheses and see if you still believe it; now find counterexamples to the versions with hypotheses omitted; now actually try to prove the result. You will discover (to your huge delight, I promise) that by the time you have done all of the above, you will often actually just be able to prove the theorem. Even better, you might get a sense of how one would actually discover the statement one one's own. That is always the ultimate goal in learning a new theory. And you will be able to transfer all these strategies to your homework.
  • Try, whenever possible, to ignore time pressure, and actually to ignore time. Just set out to get to the bottom of things.
  • If you can find a suitable colleague, try explaining what you have learned to them. When you prepare such a lesson, anticipate the questions, starting with the most naive ones. If you don't have a suitable colleague for this, prepare such a talk anyway. What this will do is it will make obvious to yourself the various gaps that you still have in your understanding that you were not aware of. Often when trying to get "to the bottom of things", one of the big obstacles is seeing where "the bottom" is. You get used, so to speak, to floating at a particular depth, and just by force of habit start thinking that this is where you are supposed to be.
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    Excellent advice! Especially the first point that many people claw their way forward incrementally, but then publish a "revised" history that shows how to get there with smooth, clean leaps. – Cole May 23 at 6:09
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If you were able to intuitively solve problems up until now you then probably never developed a systematic approach to problem solving. Now when your intuitive approach fails, you have nothing to fall back on. The good news is that you can learn a systematic approach to problem solving. Then you just have to remember to apply your systematic approach instead of panicking (this can actually be quite hard, from my experience).

You may even recall other people using this kind of strategy while you "just solved" things.

  1. Read over the whole question once.
  2. Organize what information you're given--I recommend copying this off onto a second piece of paper, it really helps focus you.
  3. Make sure you understand what you're being asked to find or show. If it's a multi-part question, focus on one part at a time.
  4. Write down any definitions or ideas that might connect what you're given to what you're asked to find or show. Re-familiarizing yourself with earlier (undergraduate) classes will be helpful here!
  5. Note any techniques that might help you (perhaps it seems like a proof by induction may be helpful, perhaps what you're being asked for is a change over time so some calculus might be useful, etc.).
  6. If you still have no ideas, try either... a) Forming a simpler version of the problem. b) Trying a few sample cases to try and understand what is happening. c) Drawing a picture. d) Pretending that you are explaining the problem to someone else, using full sentences, and adding in more detail until the problem is clear.

This previous question about how to get better at proofs might also be useful for you. There are several very good answers that offer structured approaches to proofs.

Also, if you can find a bit of time (it's hard, I know), it can be useful to go back and find the boundary where your intuition was no longer enough, and start working at that boundary. Find a few problems that were difficult but doable and practice solving them using deliberate techniques...then see if you can move that boundary a little bit.

Oh, and one other thing. You may have to change your study skills, too. Or at least, I did. I never memorized things because I found it easier to learn the general ideas and then piece the specifics back together from principles when needed. But at the point where my intuition started failing I wasn't able to do that, I had to start actually studying the hard way using note cards and spaced repetition software and that kind of thing (it was awful, haha).

(Side note, I agree with others re: pandemic is hard and probably making things worse.)

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A few things that have helped me in similar situations:

  1. Write down several problems to try (on paper), turn off your phone and computer, and work for an hour. If you're staring at a blank page for an hour, pick easier problems from earlier in the course. The goal is to strengthen your Sitzfleisch, a mathematician's most important muscle.
  2. If you're still stuck on some problems after the hour is up, talk to someone! Your professors have office hours -- tell them what you've tried and where you're stuck. Nowadays, you'll have to turn on your computer to go to office hours, but don't look up anything about the problems before going to office hours. The point is that you want to practice formulating and asking questions on your own.
  3. Reach out to other grad students and work on homework with them. Be clear about your goals to not look things up right away. Working with others is a good way to keep yourself accountable. This is also good practice for collaborating on projects once you start research.

Regarding the last point -- it can be intimidating to ask to work with others. You may worry that they all know what they're doing, and that they will all see how hopelessly lost you are. However, being brave enough to feel stupid is an important part of doing research. And who knows if your peers actually know what they're doing? If they do, you can learn a lot from them, and if they don't, then you can all figure it out together!

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Just to add to the great advice already given.

Please consider consulting a psychiatrist to make sure if there is a physiological component of your struggle, it is dealt with appropriately. Sometimes we forget that the brain is an organ and it needs proper nourishment and irrigation. A good psychiatrist has the right tools to assess whether someone needs treatment, and it could really help. I write from personal experience - I am not a mathematician.

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  • This was my first reaction from reading the "tone" of the OP. – Leonardo Herrera Feb 10 at 14:09
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How much sleep are you getting? What other stress are you under? Financial, relationship, etc.? Are you doing anything that is sapping your intellectual strength? Drinking, video game binging, porn, etc.? Are you keeping up some measure of exercise? Eating reasonably healthy?

Mental sharpness is affected by a lot of things. These are a few basics to rule out before looking deeper.

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  • I would also state the converse, that if you struggle with many of these surface behaviours, it is likely indicative of a deeper psychological issue that needs to be addressed. (ie. they are coping mechanisms, what are you trying to cope with?) – Cole May 23 at 6:14
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Hah, I had a different situation, but it may be useful to you. Several years ago, I was able to consistently solve 4/6 problems at any IMO problemset.

Then happened my undergraduate studies and this horrible phase of mathematics education where I didn't really need to solve math problems. You learn the method for the exam problems, solve it, forget it and go on. I never did much homework, nor did I take any advanced courses.

I switched universities owing to the Bologna Accord, and then started taking serious courses. And I really found myself worse than I remembered, not able to really solve the homework in advanced graduate courses.

Then it was painful for a while, to be honest. I felt like my mental skills have declined. But, I kind of slowly got my brain back once I started spending >10h/week just trying to solve problems, just like in high school. Ideas just started appearing more often after the first semester.

The point: it takes consistent practice to be OK at anything. And you need grit; you won't get any returns in a few weeks only.

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Ask your self:

Do I like doing math?

There's no rule that a person will always enjoy doing what they are good at. It's possible that you have mathematical skills, but you don't actually enjoy doing math.

Ask yourself why you enrolled in this program; was it to pursue passion, or to pursue a specific career, or was it simply the path of least resistance?

Other answers point to several effects that could cause a temporary, reversible lack of interest/motivation (e.g. pandemic, undetected depression...) and these would likely have a broad impact over many aspects of life. So if your ability to enjoy anything has atrophied, then look for a temporary, reversible effect and if possible seek some counseling or professional assistance.

However, if it seems to be specifically mathematics that's lost it's shine, then consider some significant courses of action sooner, rather than later:

  1. Look into a more applied field of mathematics like engineering, physics, some aspects of molecular biology or social sciences and see if these spark a new interest in wanting to solve problems. Maybe it's simply that you are more of a goal oriented person and when the solution has a practical use it feels more compelling.

  2. If that's not effective, then consider that you've discovered that Math just isn't interesting to you. Maybe you simply don't like it any more. In that case you may have some decisions to make.

For comparison, here's how some people feel about mathematical problem solving:

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