I have stack-exchanged through my undergrad math program. Am I likely to succeed in mathematics PhD programs?

Question

I'm currently a third-year mathematics student at a top-20 private research university in the US (though mostly known for the humanities). We have a small undergraduate mathematics department, but we were offered the option to take many graduate courses. I’m always passionate about studying pure math, specifically algebra, and I’m very seriously considering pursuing a PhD degree in mathematics. Nonetheless, I’ve not been able to participate in mathematical research of any kind so far – getting undergraduate research experience is impossible due to personal status issues, and my department has no resource committed to undergraduate research.

Indeed, as of my course load, I’ve finished the three-semester algebra sequence by my second year (i.e., group/representation, ring, field/Galois, category theory, commutative algebra, and homological algebra). Though I started late, I'm also working my way through the analysis and topology/geometry sequence, taking differentiable manifold and complex analysis now. I maintained all As in the graduate courses I’ve taken.

Regarding this, I do have one important confession to make, and it is in fact the reason why I’m asking this question. Math SE is a very robust community with respect to algebra. This worked very conveniently for me as there are very few peers at my school to discuss math with. As a combined consequence I actively seek ideas on SE whenever I get stuck on homework. (Please note that I'm not in violation of any collaboration rules set by my department: I understand and then proceed to write every proof myself.) This happens in about 30% of the assignments. I have no problem with exams since they are usually much easier than assignments.

Only after having recently talked to graduate students and professors at a conference, did I realize this is a terrible approach. I vividly remember one said something like: “unless you went through a textbook and attempted to prove every theorem yourself first you won’t truly understand the subject”, which is, the exact contrary of what I’ve been doing. I’m seriously in doubt about my aptitude over these subjects, fearing that I will be subpar on the level of understanding as well as the ability to conduct research, to approach open questions when I reached graduate school. I fear I never try/explore “hard enough” to come up with proofs like others have suggested. I managed to do most just by familiarity of common methods/tricks and theorems, but those things can be forgotten over time.

So here comes some specifics of my question:

• Is searching SE for homework problems common for math students?
• How will doing so affect a student’s understanding of the material?
• In what ways does doing so tie to one’s ability to do research?
• What are some possible ways to remedy this, besides completely re-learning the material?
• How much do I have to pay in the future for stack-exchanging through my courses?

Answer 1

First of all, I believe this is extremely common these days. More and more I notice students neglecting to develop important problem solving skills and instead developing great “google-fu” and “stack-exchange-fu” skills to achieve the same goals.

Now, don’t get me wrong, SE-fu is a terrific skill to have. Just like you are worried about using the internet too much and coming to rely on it as a crutch, some people genuinely ought to worry about having the opposite problem of obstinately trying to figure everything out themselves even if it takes them weeks or months and refusing to ask for help. This is just hugely inefficient. These people may in fact become excellent problem-solvers given enough time, but it’s just not a practical approach to covering the large amount of material a modern mathematical education requires.

So what I’m saying is, there is a right amount of stack-exchange usage that can be really good for you. Someone who makes the right use of math.SE and other great online resources can really boost their ability to master complex topics and speed up the learning process compared to their peers who don’t use those resources. And then... there is definitely also a wrong amount of stack-exchange usage. It is certainly possible to rely on it too much, or more generally to rely too much on asking other people to help you figure things out when you get stuck (before SE was around, people with such tendencies also existed, they would just nag their friends and class mates with lots of questions instead of using google/SE).

So, is this a fatal flaw or an indication you’re unlikely to succeed? Absolutely not. From your description it sounds like a slightly bad habit at worst, but one that you likely share with a lot of other students (I mean probably something like 70% of them, if we interpret your habit to include heavy use of google and not just SE).

I do advise you however to actively work on shaking this habit and investing more time and effort in trying to solve problems by yourself before you give up and ask for help. Getting yourself unstuck when you get stuck is a skill in and of itself, and involves important sub-skills like learning how to identify when you have a serious misconception about a problem, learning to believe in your ability to solve problems by yourself, learning to be attentive to small details, and probably other things that are equally important but that I would have a hard time articulating in words. When you look up the answer or ask for help on SE, you end up solving a specific short-term problem (figuring out the answer to the specific question you need solved), but deprive yourself of broader opportunities to acquire these very valuable problem-solving skills. It’s certainly not too late to start though, and your tendency to over-rely on help sounds fairly mild in any case, if it even exists. Good luck!

Answer 2

Yes! In fact, I think you're well on your way to doing better than your peers! Taking longer to understand something isn't something to be proud of! There's no need to reinvent the wheel. If someone can help you understand something, you would be well-advised to make use of them. In the same way, you would be well-advised to attend the lectures, thereby getting help from the professor, instead of staying at home with the ZFC axioms and attempting to derive the whole of mathematics from scratch.

My experience asking questions on Stack Exchange also indicates that simply writing the problem in a form which others can understand is a great help in clarifying my own understanding of the problem. Indeed, I've solved some of my problems simply by beginning to write a (never asked) question.

Of course, this doesn't mean you should outsource your understanding to others. You should make it so that you can explain the concept to next year's students without help (answer some other SE questions while you're at it!). But even then you should feel free to ask for help with next year's problems.

Answer 3

I wouldn't worry too much. It sounds like you are making excellent progress. You are still an undergraduate, you have tons of time ahead of you!

There are lots of good ways to learn mathematics. Talking with others (including over the Internet) is one. Allowing yourself to get stuck, and trying hard to come up with your own proofs is another.

If you feel that your study habits have skewed too much towards the former, I'd recommend trying out the latter approach. (Which does not mean you have to change to it permanently.) For example, choose a course or book and try to get through it without SE. See what happens!

Answer 4

"... unless you went through a textbook and attempted to prove every theorem yourself first you won't truly understand the subject"

Nonsense! This is anxious, perfectionistic thinking, and internalizing thoughts like this ultimately caused me to leave academia. I felt that I couldn't pursue my research unless I fully understood everything from first principles. I would get stuck for weeks trying and failing to get my head around a particular tiny nuance of probability theory. Needless to say, it was exhausting, and I barely managed to finish my master's thesis.

The thing is that learning is a continuous process, and you don't have to understand everything perfectly to move forward. You can learn as you go.

Answer 5

No-one really knows the educational effects of reliance on SE yet, mostly because you, and your generation, are the canaries down the coal-mine. Most of the experts who answer questions on the technical SE sites are people who completed their graduate education before SE existed, and some before the Internet was even in regular use. Those of us who answer questions on these technical SE sites rely on the students who use this facility to have good judgment about when reliance on this assistance is helping them learn, as opposed to being a substitute to learning. From our perspective, we certainly hope this assistance will help your understanding of the material, and we try to frame our answers in this way, but we are relying on you to let us know how it all works out, not the other way around.

So here comes some specifics of my question:

• Is searching SE for homework problem common for Math students?

You are probably in a much better position to judge this than most of us. Ask around the students in your faculty and see if they use this resource, or if it is just you. There are certainly a reasonable number of student questions on SE.math, but the volume is still far below the number of mathematics undergraduates presently studying. This suggests that only a small proportion of math students are using SE to ask questions, though many may be looking up answers to existing questions on that facility.

• How will doing so affect a student's understanding of the material?

This really depends on the exercise of good judgment by the student, so you tell us! Has SE helped you or hindered you? Have SE answers functioned as a tool to assist your learning, or as a substitute for learning? Have there been any instances in which you used an SE answer to substitute having to learn to understand a piece of material? Was this common?

• In what ways does doing so tie to one's ability to do research?

Unless the use of SE answers has hampered your previous learning, there is no reason to think it will impose a limit on your ability to learn to do research. It takes a long time to learn to do research, which is why we have PhD programs, but if you are able to get into one of those programs, there is no inherent reason that you should lack the ability to learn the material and ultimately succeed in this area. You mention your professor's comment that you won't truly understand a subject unless you attempt to prove every theorem in the relevant textbook yourself. I would say that is rather aspirational, and it assumes that the textbook is some kind of golden-tablet that perfectly delineates all necessary knowledge in the subject. In practice, this level of engagement wouldn't necessarily occur, and it is not a necessary condition for expertise. As you acquire expertise in a subject, you are naturally going to be curious to check and re-check all the foundations, so you are probably going to find that at some point you will learn to prove all or most of the relevant theorems yourself. That might occur slowly, as you learn more about your subject and relate it to other relevant areas of mathematics.

In regard to research trainnig, it is worth noting that the major difference between undergraduate mathematics, versus research mathematics, with respect to SE, is that the latter is going to involve problems that are sufficiently difficult that it might be hard to get answers on SE. At the level of graduate research, you are expected to be developing into someone who is becoming an expert in your topic, so at that point the pool of people that can help you diminished substantially. Some researchers do ask questions on SE.math pertaining to research topics, and sometimes they get useful answers, but often the complexity/obscurity of the topic is such that it is difficult to get assistance through this medium.

• What are some possible ways to remedy this besides completely re-learning the material?

You haven't really told us exactly what deficiencies you feel you have that you're trying to remedy. Presumably you have learned something in the courses you passed, so even if there are gaps in your knowledge, plugging those gaps would require only a partial relearning of the material. Certainly, if you feel that there are areas of your mathematics education where you did not acquire the relevant knowledge, go back and have another go at them, and try out some practice problems until you gain the understanding that you missed.

• How much do I have to pay in the future for stack-exchanging through my courses?

As I said at the start, you are the canary in the coalmine here, so we are waiting to see the results, and then you can tell us.

Answer 6

Instead of me looking through your questions on Math SE like Ethan Bolker suggests, let me tell you what I would look for and evaluate instead, and then let yourself do the self-evaluation (which is also an important skill to develop as a researcher!).

Basically, I would look at if your questions are well received. If you hit the hallmarks of a "good question" (as defined on many SE sites), you are probably benefiting from your interaction with Math SE. To list some of these characteristics of good questions:

• shows that you have put thought in the problem,
• shows that you have attempted to solve the problem yourself,
• shows that you understand the material.

If you had good guidance and supervision, you would probably be able to discuss these things with your advisor. If you had many peers with similar interests, you would be able to discuss these things with your peers. Neither your advisor, nor your peers, nor the community at Math SE would value if you came asking for help while showing you had put no effort into understanding the problems yourself.

So if your questions are not well received, SE might be pulling your weight, and the same way the community does not appreciate questions showing low research effort, a prospective adviser would not develop confidence in your understanding and dedication if you expected them to serve you with all the answers.

Additionally, and excellent way to deepen your understanding of mathematics (or any topic, really) is to start answering others' questions. You need a deeper understanding of the material in order to explain something clearly and concisely. This would be similar to explaining a topic to one of your peers asking for help.

Finally, I'd like to note that not everybody has the same learning style: some people benefit more from sole studying with a textbook, some people benefit from using visual aids or drawing, and some people (me included) work and learn (and come up with new ideas!) best when interacting with other people.

To summarise: if you are asking well-received questions and providing well-received answers on Math SE (or a similar technical SE) you are probably benefiting from it similar to what you could get out of a good adviser and peers with similar interests. This is especially true if you have an interactive learning style; you were simply looking for the most effective learning method fitting your learning style which was possible in your situation.

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_{^{On a personal note: I used to be much more active on Stack Overflow during my BSc and MSc. My activity and reputation graph pretty much follows my project and seminar schedules: they had required me to think about more than we had to do for class, or to combine the course material in new ways. Being able to get feedback on my ideas and approaches was invaluable for my learning and understanding.}}

Answer 7

There are a lot of aspects of training to be a mathemtician. For example, you want to learn:

• Mathematical theory (definitions, key theorems, key constructions)
• Literature search (figuring out what's in which papers, finding results in books, using google to find relevant ideas)
• Mathematical tricks and proof techniques
• How to struggle with a difficult problem that takes hours or days to solve. What kind of techniques do you use to get unstuck? How do you deal psychologically with being stuck? How do you decide when to give up on a problem or approach?
• Writing math papers (dealing with writer's block, making diagrams, organization, clarity of proofs)
• Collaboration (learning how to solve problems together with other people, figuring out how to learn from and explain to your peers, etc.)

The bad news is it sounds like you've focused exclusively on the first two bullet points at the detriment of the other ones. Getting stuck on 30% of problems suggests that you're pretty far behind where you should be in terms of mathematical tricks and proof techniques. If you go to a graduate school that has a preliminary exam you're likely to struggle with it. Not having the experience of really struggling with questions you're stuck on, means you're underprepared for the psychological experience of research, and for figuring out how to work on problems to which no one knows the answer. It's important to try to learn the difference between being stuck and close to an answer, and being stuck and far from an answer. By turning to stackexchange exclusively instead of talking with your fellow students and your teachers in person, you've fallen behind on learning how to collaborate.

The good news is that you have time to learn these things during the rest of this year and during your first year or two of graduate school before you move on to research. But you really need to start thinking about the aspects of your mathematical training beyond just learning mathematical theory. You need to stop giving up on problems early, and you need to talk to people in person about problems you're stuck on rather than just finding answers online.

Answer 8

Really, the answer to this is very simple.

Everything that you can find on the internet has already been done by someone else.

What you are required to do in PhD research is something which has not been done by anyone else yet.

Of course, the web may still give you good ideas about techniques, etc. But SE or (any other web forum) isn't actually going to "do your PhD for you" in the same way it can "do your homework for you" - even if you have been avoiding straightforward "copying" of what you discovered from web sites.

You will also eventually hit the problem that there are very few people in the world who are working on anything very similar to what you are doing - and that handful of people might not hang out on SE, or anywhere else you are looking on the web for advice!

Answer 9

You can just carry on with this approach through graduate school and well into your career as a professional mathematician.

For instance I have at least one series of papers that started because of answers I got to a mathoverflow post when I was confused about something.

Answer 10

I imagine that this is one of those situations in which far too many people have this "not invented here" mentality. It is something every one of goes through at some point in our lives. For example, a beginner programmer might insist that he or she has to write their own code for any and all functions that they wish to implement, including shunning the use of open-source, freely-available or otherwise acceptable to use libraries, functions, plug-ins or other assists. By the same token (and this is something I have personal experience with) a 3D beginner might insist on refusing to use any texture art, mesh data or keyframe information that they had not explicitly created themselves, even if it served the purpose to flesh out certain areas of a concept they might have been going for in the first place.

There is nothing wrong with properly-attributed work being used to support your case, and examining other people's work (and collaborating with other people) is often a fantastic and accelerated way to learn how to do things. I greatly enjoy 3D modeling, animation and rendering, but I absolutely suck at mathematics. I learned just enough to be able to work on my 3D stuff, and I use computational tools that simply didn't exist 30 years ago to provide me the assistance I need any time I work on anything requiring calculations.

In my experience, you will not be treated poorly by the academic community for using other people's work to support your own, just as long as you are open and honest about doing so. That's the key issue here. Academic dishonesty is severely frowned upon for very good reason, but so long as you do not engage in such shenanigans, I think you'll be fine.

Answer 11

I intended to answer after looking at the kinds of questions and answers you posted on Math SE. I was surprised to find none linked from your profile.

Stack Exchange can be a good place to "discuss mathematics", but if all you did was lurk, reading other people's entries, you have not discussed much, and may not be as ready for further study as others here suggest.

Answer 12

If you are looking up the solutions to the exact (or trivially similar) questions you are being asked in your homeworks, then that is cheating. It is bad for all the reasons that cheating is bad:

1. you are not doing the assignment the professor wants in the manner they want you to do it; you are side-stepping the challenge.
2. Your performance in the assignment is incorrect feedback to the professor that you understand the problem, potentially causing them to accelerate or at least not bother reviewing and circling back as much. At the very least you should cite the online sources you used so they will know. This provides a good opportunity to test your theory that it is perfectly kosher to use those sources.
3. Your grade will not represent your abilities as it was meant to, and potential grad schools and employers will be misled into thinking you are better at math than you really are. This is also blatantly unfair to everyone who did not look up solutions.

As for rewriting the theorems after you understand them, this is better than nothing sure, but couldn't you claim the same after "collaborating" to get the answers? That doesn't make cheating ok. Many other answers here focus on the question of learning, but you could also learn wonderfully while plagiarizing fellow students' work, hacking the profs laptop for the solutions, and performing other forms of overt cheating, as long as you did it "properly". Hey we can't expect you to solve every problem in the book yourself right?

When I made this point in a comment it was met with a surprising amount of alarm and the comment was even deleted. To everyone with such mindsets I'd suggest you consider the problem viewed from a higher level of moral development. See here: Kohlberg levels of moral development. What matters is the universal principal underlying the rules against cheating, which I listed above, not what you can get away with due to less-savvy profs and outdated student handbooks.

As for re-doing every proof in a textbook, this is creating a false dichotomy. There are more alternatives between the extremes of cheating through all problems versus re-writing a content-packed book with your eyes closed. That middle ground is the job of your instructor to find for you. You don't really understand math unless you can redo it yourself without first knowing the answer. Yes, that's a fact. You may or may not need to understand every theorem in a particular book. Depends on the book, most aren't even peer-reviewed and have some mix of key topics and the author's personal preferences. Your instructor's job is to feed you material and challenging practice work a little at a time so that you can follow along and keep up.

As for your specific questions, yes it is very common as is all cheating. The modern view of students seems to be that a college degree is worth millions as some kind of job ticket, while merit is practically irrelevant. It will hurt you far more than you realize because your tendency will be to seek help at precisely the times you shouldn't; while you are motivated to do problems yourself when they are easy and you aren't challenged. Expertise requires Deliberate Practice which means increasing challenge, not just working easy problems or reading how others solved problems. This also provides the framework for answering the rest of your questions. Yes working through an entire book is one way to get this increasing challenge. But a better way is to do just certain key parts in a good order from a really good book. Use stackexchange to find these superior resources. Another way of course is to take a class with a good instructor that leads you through the material and provides assistance when you are stuck, rather than you looking to the internet for assistance.

Answer 13

As students read textbooks and solve exercises, but do not have access to them while writing the actual exam. So is the case with StackExchange. You don't have access to it while writing the actual exam. Whatever case you follow, as long as you did well on the exams, why should it matter?

BTW, if you do join PHD, please use mathoverflow.