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I am an undergraduate student in computer science, and I have been working as an undergraduate researcher in theoretical computer science for 7 months. There was this problem that my supervisor mentioned, and I worked on it for a while. But I only managed to solve a sub-case of the problem, which didn't interest my supervisor at that time. However, he wanted me to write a manuscript about it anyway. I put a lot of effort into writing it, but he didn't bother to read it. And we have moved on to other things.

A week ago, my supervisor called me, and he said that he found an interpretation of the sub-case of I've solved, that economists are interested in. And he said that we can publish it in a journal. So, I was really hyped. We met the next day, to review my proof. But when I arrived, I learned that a friend of my supervisor found an implicit assumption at the beginning of my proof which was clearly wrong.

As you can imagine, I was devastated. I've tried to find a workaround, and fix my solution, but I've just found another thing that I've overlooked. So I've lost my enthusiasm and self-confidence.

However, my supervisor claims he found a completely different idea to solve the problem. He explained it to me, however it has some parts to be completed. So he wants me to write a manuscript about it and fill the gaps. However, I cannot find any motivation to do so. I am overwhelmed by the feelings of failure and anger.

I know that I should be working on my supervisor's idea, but I'm so distracted with the mistake I've made. How can I overcome my mistake, and start to work again?

UPDATE: A month later, we solved the problem :)

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    Errorist, you are in a fantastic spot for personal growth. I am honestly excited for you right now. Encountering these sorts of issues and overcoming them will be an AMAZING life event for you. The fact that you can identify what is going on in your head and write them down like this is a big step. Do some more thinking about why you feel the way you do, and it may help you identify ways to overcome those feelings. – Jeff.Clark Sep 9 '16 at 16:25
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    I also really like @user37208's answer, except for the last two paragraphs. Sometimes avoiding situations like this are impossible, and powering through emotions like those is nearly impossible. They are a part of life, and what you should learn is how to deal with them when they do happen. Because they will. When I get overwhelmed and discouraged, I have found that writing everything down on paper, and then creating a step-by-step list of what I should do to solve the problem helps immensely. – Jeff.Clark Sep 9 '16 at 16:32
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    Possible duplicate of How should I deal with discouragement as a graduate student? – EnergyNumbers Sep 9 '16 at 18:02
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    If science were easy enough that we never made any mistakes, there would be no sense of satisfaction or achievement when you succeed. Take it as evidence that your research is non-trivial and hence probably worthwhile. You have experienced one of the lows of research, when you get back to the high's you will appreciate them all the more for the contrast. – Dikran Marsupial Sep 10 '16 at 12:25
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    Congrats on solving the problem, thanks for the update :) – ff524 Oct 10 '16 at 0:42
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As an early-career mathematician, I have had this experience, more or less exactly, several times. I think I've solved a problem (or made good progress toward doing so) and I become proud and emotionally invested, only to show it to a mentor (first my PhD advisor, then my postdoc supervisor) who quickly points out a fatal flaw in my reasoning. Sometimes, he will propose a different and much better method. I completely get how demotivating this can be. After all, I would think, why am I even working on this project if my mentor could do it himself more quickly?

But you have to overcome these feelings. Here are some points of view that helped me do so:

  • Trying things and failing is how we learn. There's a mathematical lesson here about why my techniques failed and my mentor's succeeded. There's also a lesson about how to do research: how to do sanity checks, how to avoid "premature optimization," etc.

  • My mentor is a professional researcher with decades of experience. Of course he can solve things more quickly than me. Even if it feels like my role is to work out the details of his ideas, I'm learning a lot by doing that. And these "brilliant ideas" are usually adapted from previously existing techniques anyway--it's more an experience gap than an intelligence gap.

  • Experiences like this keep me humble, which comes in very handy when teaching people who understand less than I do.

I would encourage you to try to power through these feelings of inadequacy. Part of being a mature researcher is not being afraid of being wrong.

Having said all that, a good way to avoid situations like this is to keep the person you're working under more in the loop while you're doing the work, so that they can steer you away from dead ends. In your case, it wasn't necessarily your fault that your supervisor didn't devote much time to your project, but in the future you should try to get high-level feedback earlier in the process.

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    ..."Part of being a mature researcher is not being afraid of being wrong." The OP is an undergraduate. In my experience, when undergraduates lose motivation to do X, it's healthy to allow them to do Y, while making clear that they can certainly return to X later on if they wish. If someone is just temporarily discouraged, I think that strategy also works well (and was suggested in my answer). Also: my feelings here are directly informed by my experiences mentoring actual undergraduates, whereas your answer seems to take the perspective of a junior mathematician. These are not the same. – Pete L. Clark Sep 9 '16 at 0:05
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    @errorist: I am trying to be helpful to both you in particular and undergraduates in general. I have mentored several undergraduates over the years and am currently the Graduate Coordinator in my mathematics department. I have a stake in the future of our profession and guiding younger people through it. If you continue in the profession, by all means feel free to contact me and let me know how my advice played out for you. – Pete L. Clark Sep 9 '16 at 0:30
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    @PeteL.Clark I'm sorry, it was a little bit rude. I appreciate your help. – errorist Sep 9 '16 at 0:47
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    @PeteL.Clark The vibe I get from OP is more "smart student who hasn't failed much in life, getting a first taste of something truly hard" than "someone who doesn't really enjoy math research, but keeps going out of pride." In other words, I don't think math research is the problem, it's never having failed before. I was in those shoes about 10 years ago, and my answer is what I wish I'd been told at the time. But OP isn't me, and I admit this is extrapolation, based mainly on the fact that OP describes the situation in much the way I would have described mine. – user37208 Sep 9 '16 at 1:04
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    @PeteL.Clark - I didn't read user37208's answer as telling the OP what to do. Rather, s/he was sharing what has been helpful for him/her in dealing with the problem the OP described. – aparente001 Sep 9 '16 at 1:21
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Perhaps I'm taking the OP too literally at his word, but my first inclination is to see it like this: an undergraduate tried his hand at research, had a predominantly negative experience, and now is not motivated to continue even though his advisor gave him something else to do. Well, there is one obvious answer: don't continue with your research if you don't want to. Whenever an undergraduate is doing mathematical research (TCS research counts), they are doing it for the purpose of getting a sense of whether they like it and want to continue with it more than any other purpose. So if you feel strongly enough that you don't like it, it may well be that in fact you achieved what you were supposed to achieve in a fundamentally deeper sense than the one you are feeling bad about. Most undergraduates doing mathematical research decide at some point that it is not really for them, at which point they stop.

To be honest, my experience with undergraduates doing mathematical research is that they are often rather fickle and flighty in their approach when viewed through the lens of a professional researcher. I have spent hours, days or more trying to give undergraduates the training they need to work on a problem that I've chosen for them, only to hear from them next week that they are now interested in something entirely different, e.g. some pattern they saw in the numerical output of some code that they wrote. Whether that is disappointing depends entirely on my expectations. I now think that treating an undergraduate as a "non-positive year graduate student" is the wrong approach: in particular they should not be assigned a problem that one would expect an older student or researcher to spend several months working on. Certainly they should not be given a problem such that a failure to solve it -- or even a complete neglect of it -- will have a significant negative impact on their advisor or anyone else.

Circling back to the situation: I worry that the OP has been placed in a situation which is "too serious" for an undergraduate student. A little context is missing; e.g., the OP writes

I have been working as an undergraduate researcher in theoretical computer science for 7 months.

Does that just mean that you've been thinking about TCS for 7 months? Are you getting paid to do it? (Most undergraduates in the US are not paid for academic work they do during the academic year...but the OP may well not be in the US.) Are you getting paid as a sort of undergraduate research assistant to your advisor?

There are some nuances in the situation that make it hard for me to read. On the one hand, it's possible that your supervisor is doing a bang-up job: he gave you a problem that you were interested enough to work on for several months (already a successful undergraduate research experience, I claim). You claimed to solve a piece of it, which is nice for you but not necessarily something that needed to be written up. Then your advisor discovered that there actually was interest in what you had done, and was involved in the process of discovering a mistake in your writeup. After that happened twice, he stepped in by sketching an approach of his own. That sounds really supportive to me. (I'm sorry that you put a lot of work into a writeup that he didn't comment on in detail, but: I've been on both sides of that situation. It happens, too often, to good people.) On the other hand, if your advisor actually expects you to function as someone who can mostly carry him through achieving his own research goals: when it comes to mathematical research, at least, that's very bad. At the most you have a short internship as a mathematical researcher: it's his career, and the burden of responsibility needs to fall on him in every way. If you are feeling bad because you think that you've derailed his own research plans or otherwise disappointed him: well, I think that's very poor behavior on his part.

To end, perhaps you should embrace the fickleness of undergraduate research. Not feeling excited about the project? Don't work on it...for now. If there is something else that interests you, anything else, look at that instead, and talk to your supervisor about that if you choose. It may well be that you could regain your positive feelings about TCS research by thinking about -- well, anything -- and when that occurs and/or some time has passed, you may actually enjoy returning to the problem and proposed solution your supervisor gave you. (A big secret: I am a real life research mathematician, and giving up on something that isn't going well for a little while in order to regain enjoyment and confidence is something that I do very often. Sometimes there is exactly one hard problem that I want to chip away at for days, weeks and months at a time...but more often, there isn't, and sometimes I wonder a bit at how hard people who only work on one thing are actually working.) Or not, which (according to me!) is really okay.

Final word: don't do mathematical research if you don't enjoy it. Seriously.

Added: I am concerned that my "final word" is not being understood as I meant it. First of all, an equivalent version is "If you don't enjoy mathematical research, then don't do mathematical research." Maybe this stops people from reading this as "don't do mathematical research [plus other words]"! Second, the statement has to be construed in a reasonable way, and in fact anyone who is doing mathematical research has to construe it in a reasonable way for them. Certainly I don't mean "If you wake up one (or two, or three...) days and have negative feelings about mathematical research -- or if you have one (or two, or three...) setbacks in your mathematical research -- then you're not cut out for it; just stop now"! What I mean is "The main reason that people do mathematical research is because of the joy and sense of personal fulfillment it gives them -- not that it always makes them happy and fulfilled, but that in choosing which of the many things to do with their life, this choice seems to make them more happy and fulfilled than the alternatives." There are extrinsic rewards to mathematical research as well -- if you are successful enough at it, you make more than enough money to survive, you have more control of your daily schedule than many other professionals, there are many opportunities for travel, and so forth -- but if you remove the intrinsic rewards, the extrinsic rewards don't stack up against other careers that a similarly talented, hard-working person could probably achieve. Also not enjoying mathematics is not a failure and is not the same as being bad at it. It almost seems silly to have to say this out loud, but: the people who do mathematical research as a career are not the ones who in all the world are the most talented or most successful at mathematics -- they are the ones who are talented and successful enough to get paid to do it and who value the activity enough to devote their lives to it. Most undergraduates who do some mathematical research do not want to devote their lives to it. That's not a character flaw or an intellectual weakness on their parts: there are just a lot of other valuable things to do. Finally, I'm not saying "Look, most undergraduates quit, so you'll probably quit." I'm saying "Most undergraduates quit, so the idea that you might quit should not be so horrific that you can't even contemplate it." All things in moderation: if you have a bad experience with something, taking a small break from it, reassessing your commitment to it, adjusting your practice of it to rekindle your joy are all reasonable, healthy responses. So is powering through the pain, sometimes. These are all good things for us to have in our repertoire.

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    Whether you are getting paid for something definitely makes a difference, e.g. in what might happen if you stop doing the thing you are getting paid for. Are you in the US? (If not, even "undergraduate" can mean something very different!) What are the duties of an undergraduate research assistant? Can you just quit if you decide you don't like it? (And a more difficult, but still pertinent question: do you know why your supervisor assigned you the task of fleshing out his argument rather than just doing it himself?) – Pete L. Clark Sep 8 '16 at 23:25
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    "However, your [sic] want me to quit if I don't like it." Well, I certainly wanted to suggest the option. Why are you doing something for which by far the largest inherent reward is the satisfaction you get from doing it if you don't get satisfaction from doing it? I think you'll agree that's a fair question. "I enjoy mathematical research." That's the first time you've said that! As you say, it's very important, and the fact that my answer motivated you to say that is, in truth, not so "weird". – Pete L. Clark Sep 8 '16 at 23:54
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    @aparente001: I see nothing in my answer that reinforces self-doubt. Is the idea that one might be happier doing Y rather than X really "self-doubt"? That seems ridiculous to me. We don't help students by denying that they have any options. We help them by giving the options and conveying that it is okay for them to explore them. – Pete L. Clark Sep 9 '16 at 3:14
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    But I want to be honest: yes, most research mathematicians have experiences like those described by the OP. But how they respond to them is extremely subjective and individual. I am a "successful" research mathematician, and when I find a mistake in work I've spent months on (or worse, published years ago), it still counts as a negative experience, to say the least! If I made five times as many mistakes as I do or each mistake was five times as painful, I would have a terrible job. So "Do you enjoy it, overall?" is not a euphemism for something else; it's the real, key question. – Pete L. Clark Sep 9 '16 at 6:29
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    @PeteL.Clark - The accepted answer states "you have to overcome these feelings."* Feelings can get you tied up in knots! OP was overwhelmed by feelings of failure and [self-]anger, ... distracted by thinking about the mistake made. The ideal would be for the OP to learn to distinguish between the soundness of an argument and his or her self-worth, regardless of how long this particular project and working relationship continues, and regardless of what field or type of project OP chooses to work on in future. – aparente001 Sep 9 '16 at 12:55
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Okay. I was in a well respected PhD program for two years, doing research, when I got sick, and now I've been sick for 3 years unable to do math. I'm won't say what your experiencing isn't real or unpleasant, but I'd like to suggest I have some perspective to offer you. A lot of it comes from realizing unhealthy emotional habits doing math that contributed to the stress that brought me to where I am now, unable to do math.

Fundamentally, I believe the healthiest way to approach mathematical research is to truly and deeply appreciate your ability to exercise your mental faculties, your ability to ponder interesting ideas, and your abilities to think of interesting questions. This is something like, "Neat, summation by parts is a neat tool and it's neat how it can be used in all these sorts of problems and, wow, maybe it is useful in this problem in such-and-such a way," vs, "Gosh, my attempt to use summation by parts fails to solve the problem. I suck."

This is different from successfully solving problems. That is a fine goal, but if you allow it to dominate how you relate to doing math research, you will be stressed out all the time unless you happen to be wildly and luckily successful.

I'm suggesting an attitude of humility. With this sort of an attitude, you can more easily accept what everyone else tells you, that failing and making mistakes is a big part of research. Of course it will happen to you. It happens to others too, though you don't get the inside scoop into their research lives. Certainly some people appear to never fail, and maybe some people really are that successful, but envy will only stress you out and make you feel depressed.

You need to practice this by expressing wonder or thankfulness for what you do understand. Here are some examples.

"Wow. I actually do understand the problem statement!"

"Neat! People actually care about this problem, it impacts the world in some small way, and I get to participate in the community by thinking about it."

"It's neat that I was able to use [this clever integration by parts trick] in an attempted argument."

"I am thankful I have an opportunity to participate in the mathematical community and potentially write a paper."

"[Anything you think it positive: i.e., something about your experience, your paper, etc., you'd rather have than not have, no matter how small or trivial.]"

Fake it until you make it. If you think there is a modicum of merit in adopting the humble attitude, then you are being honest when you practice such thoughts.

The point is, if you approach things from a humbler perspective, the experience wasn't wasted even if you don't achieve your goal. And it really is a privilege to think about mathematics. If you find you cannot, not matter what you do, enjoy the process of thinking of mathematics and working through ideas on your own regardless of whether it is fruitful or not, then maybe it is true that research isn't your path.

I miss my days of research and have realized I love thinking about math on my own. Hope there's something useful here.

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Focusing too much on your personal experience may have something to do with it - the world is bigger than your own self, connect to it.

Maybe find other motivations: e.g. "some economists out there just can't wait to use the methods you derive" or "theoretical CS is beautiful no matter how treacherous"

It also may be a case of burn-out. Usually those recedes with Rest-and-Recreation.
Or is it 're creation'? In the sense of trying to create something in an unrelated area, succeeding and regaining self-confidence by proving yourself you still have value (despite being just human and making mistakes).

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