I am a third year (starting fourth year in the fall) PhD student in mathematics. I've passed all qualifying exams and am currently doing research. As far as I can tell, I am not doing poorly. I have the good fortune of having a great advisor, being in a very supportive department, and having friends and family who genuinely care about my success.

The fact is research is hard. It appears to consist primarily of staring at a problem for days and days and days without getting anywhere. Sometimes, rarely, I do figure something out and that feels wonderful, but the overwhelming majority of my time appears to be spent banging my head against a mostly figurative wall. I am not complaining about the material being hard, and I am not averse to putting in hard work, but I get frequently discouraged when I realize the vast volumes of mathematics that I yet know nothing about (and probably never will). It's very hard to quantify progress - in particular, there are too few tangible returns after too many hours worked. I find myself thinking along the lines of "Oh, if only someone actually smart were thinking about this problem they would have solved it in moments" and so on.

I've talked about this to some number of people; here is some advice I have received:

  • Take a day off. Putting in hours upon hours of trying things doesn't magically lead to a solution, particularly if the brain is tired and just wants to sleep.
  • Have a hobby. Since math research doesn't exactly provide instant gratification, a hobby might provide some instead.
  • Talk to other graduate students. Realize that many graduate students go through this.

I'm interested to know how other folks have dealt with being discouraged as a graduate student. Does it get better with time and experience? Is this a sign that research is not for me and that I should seriously consider a life outside academia?

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    @aru I think you should watch PhD Movie ( phdmovie.com ). It shows to you that your situation in not unique... Commented Jun 29, 2012 at 8:49
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    I'd say on the contrary it's a sign that research is for you :) If you were thinking that you're the smartest, that you could find the best solution with just a few minutes of reflexion, and never question your decisions, then you probably would be working for a bank ;)
    – user102
    Commented Jun 29, 2012 at 13:06
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    I would like to impart on you some very good advice that my professor gave to me when I was researching under him: "In the thermodynamic limit, the ratio of time spent understanding things to time spent being confused needs to go to zero. If it does, then that means you're learning something new. Otherwise, you're just doing the same thing over and over." Commented Jun 29, 2012 at 14:47
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    I share a good amount of your feelings. When I compare to others, I feel I am always less good. I attribute most of this disparity to the fact that usually when other people explain me their achievements they only tell me the happy ending, the things that worked. And the explanation itself takes very little time. Instead, I know all the time and frustration I experienced in my research and study. So it is very easy for my brain to misjudge and think that it required me much more time and effort to do comparable things as them. But it is, of course, a completely deceiving perspective. Commented Oct 3, 2014 at 15:44
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    +1 for have a hobby to fulfill instant gratification. I'm not in research, but in development, and successful steps are similarly drawn out without any visible progress in between. Commented Dec 12, 2014 at 7:25

14 Answers 14


The fact is research is hard. It appears to consist primarily of staring at a problem for days and days and days without getting anywhere. Sometimes, rarely, I do figure something out and that feels wonderful, but the overwhelming majority of my time appears to be spent banging my head against a mostly figurative wall.

Yes. This. And it wouldn't be so damn tempting if those bricks didn't wiggle just a little bit every time I slammed my forehead into them. Sometimes I think my eyes must be playing tricks on me, what with the repeated cranial trauma and all. But then I remember how good it felt the last time my head actually went through the wall, and so I keep plugging away.

I've found it extremely useful to have two or three walls to bang my head against at any given time. Surprisingly, sometimes banging my head against one wall actually makes one of the other walls weaker. But most walls prove considerably stronger than my head; so it's helpful to have options, so I don't feel so bad about walking away with some scalp intact.

If you're very lucky, one good smack on the bricks will actually bring the ceiling crashing around your ears. That takes a long time to clean up, but sometimes the debris will knock down other walls for you. And then you have a whole new set of even bigger walls to bang your head against!

I find myself thinking along the lines of "Oh, if only someone actually smart were thinking about this problem they would have solved it in moments" and so on.

Do not listen to the Impostor Syndrome. Everyone "actually smart" is hearing exactly the same voice in their head saying "Oh, if only someone who actually knew how to hit walls with their forehead hit this wall, it would come down like a stack of cards." when in fact the wall really is made of brick.

Does it get better with time and experience?

Yes. Eventually, you'll move from hoping that you'll be able to knock down a wall with your head someday, to being surprised at how often the walls you hit with your head actually fall, to finally believing that you really can knock down walls with your head sometimes. (For me, the second transition happened some time after tenure.)

But your head will still hurt.

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    @DavideChicco.it: en.wikipedia.org/wiki/Impostor_syndrome
    – JeffE
    Commented Jun 29, 2012 at 11:44
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    +1 thanks for a great answer, and also introducing me to the name of my worst quality/habit.
    – posdef
    Commented Jan 22, 2013 at 6:38
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    Over the past year, I've been coming back here to re-read this answer because it speaks to me so much. In particular, 'But your head will still hurt' :) .
    – Aru Ray
    Commented Jun 7, 2013 at 23:03
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    @vzn: But research is a masochistic endeavor; you really do have to enjoy feeling stupid almost all the time. See also: jcs.biologists.org/content/121/11/1771.full
    – JeffE
    Commented Mar 29, 2014 at 21:03
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    @vzn The joy of discovery is in no way inconsistent with the frustration of hard work.
    – JeffE
    Commented Mar 29, 2014 at 21:17

It definitely does not mean research isn't for you. Research is hard, and it takes some getting used to. Your experiences sound normal, and it will indeed get better.

Part of the problem is that it's tempting to focus too much on the destination: proving theorems, writing papers. These things happen only occasionally, and thinking about them (or their absence) too much is an easy way to become depressed. Instead, you want to reach the point of enjoying the journey itself. This takes some perspective and confidence, but it will come with time.

For example, imagine that one day you mention a difficulty you're having to your office mate, who tells you about a wonderful theorem that's relevant. If you're feeling insecure, this is awful: you think about the time you wasted not knowing about this theorem, and you worry that your office mate knew it and therefore you should have. On the other hand, if you're confident in yourself, then it feels great: you learned something beautiful that will help your research, and how can it be a bad day when you learned something like that? This confidence can take time to develop, but as you feel more relaxed and bolder, everything will become more enjoyable.

Another thing to keep in mind is that progress is difficult to measure when you don't know where you're going (which is what research is!). Grad students sometimes feel bad because they don't think they're en route to solving their thesis problems. Often they're right, but that's not a problem. If you knew in advance that you were going to solve it, then it wouldn't be research. The goal isn't to solve the problem you started with, and indeed you often won't. Instead, the goal is to find something exciting along the way. Once you're used to this, you can say to yourself, "OK, probably I'm not going to solve this problem, but it's worth a try, and in any case I'm sure that if I think hard enough about it, something interesting and worthwhile will come out of that work."

Basically, I think of this as a phase transition that happens in a certain point in one's development as a researcher. Before the transition, you think "Oh no, there's so much to learn. How can I ever learn enough to be a good researcher?" Afterwards, you think "Well, I don't know that much in the overall scheme of things, but I seem to be doing research anyway. And I'm so glad there's a lifetime supply of great mathematics to learn, so I'll never be bored." The key is to relax and trust that everything will work out, even when it feels overwhelming.

I know this is easier said than done, and I struggled with it myself. For years, I felt like I wasn't a real mathematician, and I would tell myself I would be one if only I could accomplish some goal: publishing a paper, learning some difficult topic, publishing a paper I was proud of, getting a job, etc. However, it was never enough. I thought the end game was deciding I was a real mathematician, but it turns out it's developing the confidence not to worry about this, and I've been much happier since that point.

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    Developing confidence is so huge. And you won't get it all at once. Some days I have great confidence and research is grand. Other days (or even just other times in those same days) my confidence wavers and it's easy to get depressed for a while. I try to monitor my own thoughts (self talk) and correct it when it becomes less confident. As Anonymous Mathematician said, staying confident is so much easier when you step back from the goal of proving that next theorem or publishing that next paper, and simply enjoy all the beautiful math you get to learn.
    – Dan C
    Commented Jun 29, 2012 at 6:13
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    Since originally asking this question, whenever I feel discouraged about research I come back to this thread to read the different answers, and in particular this one. What you mention in your last paragraph (about not feeling like a 'real' mathematician despite milestones which sure sound like success on paper) has always worried me, and the fact that it can get better keeps me going. Thank you so much for this insightful answer.
    – Aru Ray
    Commented Jun 7, 2013 at 23:14
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    @AruRay, I continually do likewise. This thread has been one of my life-savers. Commented Oct 3, 2013 at 21:03

Does it get better with time and experience?

sort of, in that you become smart enough to realize that there ISN'T anyone smarter who would have figured it out in a couple of minutes :)

Is this a sign that research is not for me and that I should seriously consider a life outside academia?

Certainly not !

Research is hard work. You're on the cutting edge, charting territory no one has explored before. It takes courage, persistence, energy and a VERY THICK SKIN for rejection. After all, (and this pertains to CS), probably 95% of your job applications will be turned down, 75% of your papers will be rejected the first time, a grant proposal has a 1 in 10 change of succeeding.

But it's the small sublime moments of joy when you realize that you've discovered something that no one else knows that make it fun. And the feeling, as time goes on, that you're immersed in a wonderful lake of , with beautiful new ideas around you as far as you can see.

p.s the advice you were given is very sound. Take breaks, find fulfilling things to do outside of work, and realize that everyone (even seasoned researchers) feel the same frustrations and highs that you do.

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    Well, there are people who are smarter that could have figured it out in a couple of minutes, but since they are not working on your project it wouldn't matter much anyway :) Commented May 19, 2016 at 23:19
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    @MichaelLai and conversely, those people that could figure out your problem in minutes might themselves be working on a project for a long time that YOU could have figured out much faster. Everyone has different strengths, and whether or not you already have the knowledge to solve a problem might just boil down to luck. Commented Feb 11, 2022 at 15:37

I love the answers above, but here's another possible bit of advice: find ways to work with others.

Research on your own can be isolating. Working with other graduate students can make the process much more enjoyable. Staring at a problem on your own is both less fun and generally less productive than trying to work through it with a colleague. Synergy happens. Two brains isn't just better than one; it's better than two brains working separately.

In graduate school there's an artificial sense that you should be working on your own to get "your" PhD for "your" work on "your" problem. This mindset is counterproductive, but it may be unavoidable depending on your field and school. If you have to keep some problems to work on on your own, but then find one or two problems to work on with your officemates or others. Once you're done with graduate school, the artificial "work on your own" constraints will start to disappear.


Your situation was once mine. I suffered through the challenge of having a project that just wasn't working, while at the same time both of my advisors happened to be on sabbatical. Adding to the indignity, it was a theoretical/modeling project, which meant that the failure wasn't because some experimental device wasn't working, but because I simply wasn't clever enough. If I were, of course I'd see exactly what's wrong, and figure out what's going on! And, to make the pressure worse, I found out that the next semester, I'd be responsible for giving the very first departmental seminar among the graduate students in my class. So, yeah, it was a bit of a perfect storm brewing there.

Ultimately, though, the "Eureka!" moment did come—I was literally walking around campus when the idea struck. And, the next time my advisors were back (a few weeks later), I had a working prototype simulation to show them!

What am I trying to say? Well, a few things:

  • Don't give up. The course of true research never did run smooth.
  • Failure is normal—and even to be expected. Just about nothing works exactly as you predicted it would. More importantly, if something doesn't go wrong, then your project has been badly designed, and in fact, I would argue that you're only doing development, not research!
  • Don't be afraid to fail! Failure teaches you lessons that you will never learn from success. I needed a few really abysmal grades in college to get me on the right track—the proverbial kick in the pants that allowed me to realize I couldn't coast through college the way I did through high school.
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    I was literally walking around campus when the idea struck. -- Carry around a small notepad and a pencil in your pocket, for just such occasions. Commented May 22, 2013 at 17:43
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    That last bullet point defines my transition from under-grad to post-grad life. Commented Jun 20, 2014 at 10:38
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    @JonathanLandrum How about recording yourself with the phone?
    – Ooker
    Commented Mar 7, 2015 at 11:26

I have a perhaps different view, I guess, from most people here...hopefully my experience can be helpful.

Like Louiqa said, it is better to think about it now than later. And it is not about whether you are good in research, of whether you are smart enough (don’t underestimate yourself!). It could come down to what you see yourself doing in the future, perhaps in the next 5-10 years.

I used to be very sure that I wanted to be a researcher for the rest of my life. I did quite well, and actually went straight into a PhD program after undergrad. But I am a very project/task-oriented person (like to “complete” things) and I really enjoy talking to people about science - the basic science research work I did didn’t give me many of such opportunities (long hours at the bench not talking to anyone else). I also don't see myself becoming a post-doc and do more research work. It took me a while to realize that it is not just another PhD student day (and this happened after I passed my PhD qualifying exam with flying colours). I decided to wrap up my project and apply to graduate with a MSc instead.

After everyone went into shock, I freaked out for a week, and then started to look into my past experiences, trying to combine what I liked doing with what I had the skills for. I now work as a Communications Coordinator at a physics dept and I LOVE my job (despite occasionally hating the fact that I don’t have a PhD and cannot lead my own research project) I cannot be happier that I decided to do something else.

To be honest, everyone is different. Another friend of mine just finished PhD, and became a research scientist for a hospital and loves what she does now (she said she also had some really bad moments). In the end it comes down to you. My advice is to start looking at your plan for the next 5-10 years. Do you want to stay in research in academia (post-doc, faculty position, etc). Do you like teaching and inspiring students (teaching only positions?)? Do you want to go into industry? Perhaps you have other skills (a lot of what I do now depends on the soft skills I acquired during grad school, so still time well spent) that might lead to something that you want to do? What are the qualifications required for what you want to do? These are questions that you can ask yourself now, instead of later.

Good luck and I wish you the best! (btw, a lot of the other advices you got here are also very good, and were found useful by my other PhD friends)

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    Can I ask you what exactly a Communications Coordinator do? I have googled but the results are not informative.
    – Ooker
    Commented Mar 7, 2015 at 11:46

The fact is research is hard. It appears to consist primarily of staring at a problem for days and days and days without getting anywhere.

Here is what I found helpful in this regard: consider switching to a different style of research. Instead of studying problems, study techniques. Avoid focusing on questions like "Is X true?" Instead, focus on questions like

"No one seems to have observed that object X is as an instance of object of type Y. Does the available theory about objects of type Y say anything useful about X?"

"There seems to be a parallel between techniques used to prove statements of type U and statements of type V. How deep does the parallel go?"

"Objects A and B appear to both satisfy property Z. Can we prove a general theorem about when property Z is satisfied? What are the really important parts in the proofs that A and B satisfy Z? "

I don't mean my advice to apply generally - this is only my personal experience. I found working on questions of the type "Is X true?" to be very frustrating - immensely rewarding if I succeeded in resolving them, but they felt like banging my head against the wall most of the time. When I changed my research style to study techniques, there was a lot less blank staring involved and research became more fun.


I love the answers here, and I just want to add that I find the following things helpful.

  • Read the literature. You can get a lot of good ideas from seeing how other people have solved similar problems before. (It's also rather frustrating to have your manuscript rejected by a journal because you didn't do enough reading. It's better for you to find these things out on your own.)
  • A related point is that learning the vocabulary in another discipline may show you that your problem was actually already solved, but other researchers just called it something else.
  • Describe your problems to a colleague. Just restating the problem to a third party can help you to see something new. ("Confessional debugging").
  • Find something bigger to procrastinate on. You can fool yourself into working on an unpleasant task A if you feel like you're avoiding a harder task B.

YMMV because my research is engineering, not math. Good luck!

Edit: I forgot to say that for a short term boost of morale, consider reading all of the Phdcomics. They're funny, cathartic, and painfully true. If you like posting to a forum to complain about the problems in academia, The Chronicle of Higher Education will show you that you're not alone at all. Finally, if you want proof that your problems have occurred before, The NIH Catalyst goes all the way back to 1994 with hilarious comics about the types of people in academia.


The research experience varies considerably across disciplines. I hold appointments in a computer science department and a business school and also collaborate with biologists and medical researchers. In some disciplines you are expected to work solo but in other domains, collaboration and working with a team is encouraged. The depth of the problem being addressed also varies widely. Plant biologists may spend years working on a specific gene and I believe this is true for mathematicians as well. The pace is much faster in computer science and in business schools breadth and applicability to the real world is often favored over deep thoughts.

It is good that you are asking these questions now. I have seen many faculty asking these questions long after they have received tenure and realizing that research is not for them.

Keep in mind that a doctoral degree does open many doors beyond research and that the skills that you have required (logical thinking, formal reasoning) apply widely.


All researchers-in-training must constantly grapple with:

Uncertainty – You have no idea whether the hard work you are putting into your project even matters.

Isolation – Nobody around you understands or empathizes with what you are doing, since they either lack technical context or are too busy with their own creative struggles.

If you can properly manage these two emotions and make consistent forward progress every single day, get private feedback from a mentor every week or two, and get external feedback from paper submissions a few times per year, then you can successfully finish your Ph.D.

The bad news is that it's impossible to fully eliminate uncertainty and isolation when doing research. But if it's any consolation, recognize that these feelings are completely normal; all of your fellow grad students are facing them as well.

Additionally, you have to deal with discouragement. The very fact that there are others like you can be encouraging.


If at any time you feel

  • You are not understanding what supervisor is saying.
  • You are not on the right path.
  • Your colleagues are performing better than you.
  • Only you don't understand a thing and lagging behind in deadlines.
  • Your supervisor is not happy with your work.
  • Your advisor is rude to you but you saw him laughing with another student.
  • Your work seems easy but others' work is very impressive.
  • You just want to quit.
  • You can't quit because of social pressure.
  • You just wanna go with flow.

Any symptoms like above mean you are doing PhD and almost every student get such mood swings any time. I almost gone through all but when I talked with my colleagues they were thinking exactly like me, some even saying my work is very nice and their work is easy etc... So don't worry just work and read.


Obviously research is difficult but the gratification, though not instant, is worth the effort in the end. Problem solving requires you to bang your head on the proverbial wall. You have to experience failure because working through an obstacle builds resilience. Resiliency stems from your beliefs & attitudes about yourself and your experiences.

Does it get better with time and experience?

Yes, it does. You learn how to cope with not finding the answer. You will also stretch your brain to find other ways of arriving to the answer. As you progress in your career & continue to build your knowledge base you will have more information to pull from when you get "stuck."

Is this a sign that research is not for me and that I should seriously consider a life outside academia?

I would advise exploring other options not because "research is not for you" but because it is necessary to see what life is like in the industry. Then after having experienced both you can make an educated decision. Furthermore, nothing is set in stone. You could start in academia and then move to industry or the reverse.


A duplicate question directed me here. My answer adds another piece of evidence that doing research (in mathematics) can be a drag and it is normal. I've recently read the preface to the textbook "All the Mathematics You Missed: But Need to Know for Graduate School" by Professor Thomas A. Garrity. I recommend reading the entire thing, but in particular, these few sentences made it clear to me that struggling is part of the journey for everyone:

Math is Hard. Unfortunately, people are just not that good at mathematics. While intensely enjoyable, it also requires hard work and self-discipline. I know of no serious mathematician who finds math easy. In fact, most, after a few beers, will confess how stupid and slow they are. This is one of the personal hurdles that a beginning graduate student must face, namely how to deal with the profundity of mathematics in stark comparison to our own shallow understandings of mathematics.


Mathematics is exciting, though. The frustrations should more than be compensated for by the thrills of learning and eventually creating (or discovering) new mathematics. That is, after all, the main goal for attending graduate school, to become a research mathematician. As with all creative endeavors, there will be emotional highs and lows. Only jobs that are routine and boring will not have these peaks and valleys. Part of the difficulty of graduate school is learning how to deal with the low times.

Thomas A. Garrity - All the Mathematics You Missed: But Need to Know for Graduate School

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    Does the book provide strategies to deal with the low times? It would be useful to outline them in the answer if that is the case.
    – Anyon
    Commented Jul 3, 2022 at 17:11

It's very hard to quantify progress - in particular, there are too few tangible returns after too many hours worked.

This comes down to something called Delayed Gratification. Certain jobs, schools, activies, or frankly anything else, require different levels of Delayed Gratification. For example, at the lower levels, there is the Marshmellow Test. In the middle there are things like studying instead of having fun with your friends. And on top there is college and graduate school. I once heard the following saying, "The Greatest Test Of Delayed Gratification Is Graduate School" or something similar. You clearly have a bunch of it, as you are most likely in a good graduate school and have survived high school and college with good grades. Researching can be one of the things that takes the most. This is a life skill, so try to work on that too. Research will get easier because you'll get better at this - but it will never be easy. If you go into mathematical research then easy discoveries are not what's in store for you - ever.

As to the quantifying progress - Don't try. Also, instead of looking at it as a step function, even if it may be, instead assume you are making progress all the time. And I think you really are. Every path you go down is progress - You've now ruled something out or discovered something. This a tangable return.

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