Cheating on homework in a graduate course is staggeringly dumb

Bill Fulton informs me that there is a user on math.SE whose questions are almost entirely copies of homework questions from Math 592 (Algebraic Topology) and 597 (Real Analysis) here at Michigan. In the case of the analysis course, almost every question which had been assigned appeared on math.SE. These courses do not have their problem sets online, so it is extremely unlikely that this is someone self-studying the material at a different location. I am sure this is not the only case.

There is an interesting discussion to be had (and which has been had before) about how the math.SE community, and we professors, should deal with this new situation. But for tonight, I want to try persuasion. This is really extraordinarily dumb behavior on the student’s part.

I think it is safe to assume that someone who enrolls in graduate school at UMich is aiming to produce a thesis, and most likely to get a tenure track position at a research institution. Some of our students are aiming to teach at a Liberal Arts or Community College, but these positions require research too. Your future career success depends on your ability to understand the mathematics you are learning in your courses, and apply it to produce new results. There is no way to learn math without working tons of hard problems. Every bit of time you put into problem solving is a slight increase in the odds that you can get the job you are dreaming of. Why on earth would you waste that opportunity?

Moreover, you are planning to work in a closely-knit field, where personal recommendations from your senior colleagues are crucial to success. Why on earth would you take the risk of ruining your reputation in this way? And, yes, academics will view this sort of behavior very negatively — are you surprised?

Finally, you are dramatically harming the instructors ability to teach the class. When I teach advanced courses, I grade all the homework myself, and I use it to adjust my teaching. When no one solves a problem, or when the solutions all miss a basic insight, I add lectures on the relevant background. If the homework is actually being done by posters on math.SE, I lose all ability to calibrate my instruction.

ADDED: I have had a number of people raise the concern with me that I do not make clear enough that I do not know whether the student in question was a graduate student or an undergraduate. Indeed, I don’t. I believe that the professors of the effected courses know who the person is, but I don’t and I’d rather not. I assume that most students in such courses, either graduate or undergraduate, are dedicated and work tremendously hard. I’ve had several people assure me that, in particular, this is true for the students in the topology course, and I am glad to hear it.

Perhaps the student plans to work in some field very far from topology and real analysis, and thinks of the course as a waste of time. In this case, it is unfortunate that the student did not receive better advice especially because, at UMich, a student can only enroll in a limited number of courses. But, in fact, experience in any form of mathematical thinking strengthens ones overall mathematical ability. Moreover, learning fields outside one’s concentration is an excellent investment in the future. It makes you more able to participate in cross-disciplinary research (which is more and more of research). It also makes you a more valuable teacher if you can cover more topics.

Perhaps the student wants to experience listening to excellent lectures on algebraic topology, but doesn’t want to do problem sets. Maybe because the student doesn’t have enough time, or maybe he believes himself to be one of the rare people who learns best by passively listening. (I doubt that such people exist, but I’m willing to grant the possibility.) Well, good news! This is exactly what auditing a course is for! I have never known a math professor to refuse to let someone sit in on his or her course. As a grad student, you are only required to enroll in one course a term; I encourage you to audit two or three more if that is your interest. But you should be able to find one course a term where it makes sense for you to do the homework.

Perhaps the student was having a personal crises and simply couldn’t work on some particular week? Well, in this case, that isn’t true — the questions appear at a steady rate over the last two months. But, if it were true, it would make much more sense to ask for an extension, or to simply omit a problem set. In my experience, professors at the graduate level are generally very flexible. (I have some sympathy here. My senior year of undergrad, I had a major paper due the same week I started seeing an amazing lady, and I briefly considered buying a paper online to spend more time with her. What I did instead was to turn out some of the worst writing I have ever produced, collect a well deserved D+, and keep my integrity. Professor West, would you feel better to know that I married her?)

Perhaps the student was badly over his head in the course? This could be a serious problem, or just a sign that the particular teaching style/subject matter was a bad fit for him. It’s an excellent issue to bring up with the professors, or with one’s advisor. Again, faculty at this level are usually extremely helpful and flexible.

Perhaps the student is an undergrad? Well, if the student plans to go on to graduate school, all of the same issues apply. In fact, the concern about recommendation letters would be extremely pressing in this case. It’s also especially unfortunate, as I know that I, and I expect most faculty, are rather sympathetic to undergrads who take graduate courses beyond their level.

Perhaps the student is an undergrad who plans to continue in some non-academic path? In that case, it may be a good strategy to get the transcript boost of some graduate courses and not worry about learning the material. (Or it may not. I have a friend who, after majoring in Classics, went on to work at D.E. Shaw. When a job applicant mentioned ancient Greek fluency on her resume, he was called out to grill her. There are a lot of mathematicians working in hedge funds, so you might want to be careful what you claim to know.) But, if that’s your plan, I’ll let you on to a poorly kept secret. Graduate courses are usually graded very leniently. If you just want that line on your transcript and never want to think about the course again, you could probably just do a bad job on the problem sets and get your line.

Cheating is always wrong. But I understand why my calculus students cheat — they think of the course as a formal hurdle in the way of their diploma. In a graduate course, this is the knowledge you need to begin your career. It makes no, no, no, no sense.

43 thoughts on “Cheating on homework in a graduate course is staggeringly dumb

  1. It takes a nontrivial amount of effort to move from an “undergraduate” attitude to coursework (including an obsessive focus on homework as the most “controllable” sliver of one’s GPA) to a “graduate” mindset. Some graduate students take longer to make this transition than others.

  2. Perhaps the student thinks that this is how one does academic work? (“I had a go, then I got stuck, then I asked some people for solutions, now I can pass the course.”)

    But I understand why my calculus students cheat — they think of the course as a formal hurdle in the way of their diploma

    In my grumpier and more uncharitable moments, I wonder if this attitude doesn’t persist to higher levels among some people, who’ve simply never been told any better.

  3. I dunno. You see it all the time where students learn from their friends how to do a problem, or maybe they find treatments in books. I may be missing something, but I don’t see much difference in this case, except the whole thing is more public.

    Obviously these students are feeling pressure to turn in complete problem sets when it is beyond their abilities to come up with complete solutions in the traditional monastic style (deep thought -> solution -> more deep thought -> better solution & fuller understanding). So they try to skip some steps. Problem is, it’s much easier to skip to the solution than to skip to the understanding. And besides, it’s the solution that gets graded anyway.

    So there’s no surprise here, nothing vexing (or dumb) about the students’ behavior. It’s a natural outcome of their perception of the class’s expectations, and of the reward system set in place. I would seek to change those perceptions before or in addition to chastising the students.

    I suppose there could be another factor, say the student’s innate mendacious tendencies. But I think that, given the large proportion of students who write solutions they don’t truly understand (in my albeit limited experience), I think the salience of this factor can be largely discounted. In other words, your nobility, being the exception, is perhaps explained as a function of your personality; but the mendacious norm is more likely to have a social explanation.

  4. Hmm, always nice to reflect and reread after hitting the “post comment” button. I guess the fact that the student in question posted nearly all the homework questions nearly exactly as stated gives this incident a severe difference in degree from one where a student gets stuck somewhere and elicits help from a friend. I guess I just wanted to make a general point about this kind of thing, neglecting the severe degree of the case at hand. Sorry if it was off-topic.

  5. It actually differs from school to school. I do not mean to disparage University of Michigan’s academic reputation, but I do know certain graduate students (whom I would not name) from your university admitted partly because their BA coursework is based on the help of others. Perhaps the admission committee paid too attention to a polished transcript and less to the applicant’s real commitment to mathematics. I apologize if this is considered to be politically inappropriate.

    Also I doubt if these arguments is indeed reasonable from the student’s egoistic perspective. People often believe math is a time-related sport; a student who can finish the academic work quicker and hand in correct answers usually receives more attention under the system instead of a student cannot keep up with classes, fail in homework and tests, but is capable of developing very original ideas. With the system working this way, and people readily formed study groups to boost their grades, it is not easy to make the students believe that deeper understanding mathematics by one’s own efforts is the most helpful thing if there are cheaper, easier ways to achieve one’s ends. Instead, why not ask others to examine if my solution is really correct? Why not ask others’ solutions to the problems if that helps me to view them from a different perspective? Since Hilbert spaces is so difficult, why should I bother studying that by my own effort which must come with a struggle? If my experience proves that I learn better by asking others than I study alone, then why not?

    This is the common logic that a lot of students might believe. I still remember after finishing my commutative algebra midterm, a fellow students ask the classmates one by one how we approach the few difficult problems. I did not know what to say and eventually found he received an A instead of my B+ in the second day as I had been stuck as well. While this is not acceptable to me and led to my estrangement from such people, I do not see how your argument might rationally persuade them that “not cheat” in your sense is a better option. Sometimes I also ask homework questions after they were due and let the professor know I asked, but I doubt that applies to everyone or this is the appropriate attitude.

    Indeed, the line is blurred; in the old days students study to solve problems by reading studying guides or reference books, and now they have online resources readily available. If as Ryan Budney lamented it is hard to devise homework problems students cannot find by a google search online, then it makes a lot of sense to do so – especially if one can see the “essence” of a problem that way instead of muddling along for a mere solution. I remember once I told someone that it is a good idea to work everything cover to cover by oneself after finish reading a math book, she just answered: “why? The textbook gives all clear proofs!”. Unless you can show that originality is one of the most valued quality for a mathematican, not ‘speed’, ‘conformity’ or ‘popular opinion’, you will have trouble to persuade students.

    While not trying to be ironic, I doubt that your argument that their career prospect would be undermined by their ‘cheating’ effort works in real life, for from what I know while ‘honest’ students are having trouble to go to first-rank graduate schools like University of Michigan because they perform less well in comparison with these students, these students often change their career to some applied undertaking that does not require such a high level of research but enable them to enjoy a comfortable materialistic life.

  6. This is of course not the first time this has happened. A month or two ago I was informed that a graduate student at UW was doing the same thing, but for questions on a take-home final. I was informed of this by the chair of the department, who subsequently invalidated the results of the final and decided to never give such a final again. Unfortunately I think systemic incentives and human nature make it unlikely that this problem will go away.

  7. I’ve never understood the professor’s mindset in how they think that getting help online is cheating. If the student in question is using StackExchange for EVERY homework question, it shows a clear lack of commitment to learning and understanding the material. They want to make the grade with as little work or effort as possible That much is obvious. But then, isn’t it going to show up on the exams? If they were posting questions to StackExchange from the exams, I guess I could consider that cheating, provided the exam is “closed book”. But for homework?

  8. Homework is (for me at least) supposed to be educative *and* diagnostic. And I have to mark it fairly across the class. If people getting it done for them online is not “cheating”, in the POV you suggest, it is at the very least shoddy behaviour, and makes my job more frustrating.

    In my undergraduate days there was no graded homework, but a big bunch of exercises which you were meant to attempt, and on which someone would go through your attempts. So who knows, maybe I just don’t “get” the world of the “young” folk…

  9. > I’ve never understood the professor’s mindset in how they think that getting help online is cheating

    It depends on the help requested and received, doesn’t it? Am I being too antediluvian in thinking that personal interaction made between students on the course is to be encouraged, while exploiting the urge of online mathematicians to show off by doing exercises for people is not?

  10. “I think it is safe to assume that someone who enrolls in graduate school at UMich is aiming to produce a thesis, and most likely to get a tenure track position at a research institution. Some of our students are aiming to teach at a Liberal Arts or Community College, but these positions require research too.”

    I think a lot of people in grad school are just there to dodge entering the real world, and do not really understand what the point of a PhD is. Of the people who enter math grad school, what percent ever end up even just submitting a single paper to a journal?

    That said, we’re entering a 21st century world. Students have always formed small groups in person to help each other with homework, the only difference here is it’s more public (as Dustin already pointed out). Adjusting grad level math accordingly is no easy feat.

    One thing to consider is whether the student is writing the solutions in his or her own words. It’s one thing to get the answer, it’s another to understand it sufficiently well enough to explain it. If they do understand it that well, then the homework has accomplished its goal (though, yeah, you can’t very well use it to gauge how you’re teaching). As for grades, I guess that’s what in-class midterms and finals are for.

  11. Asking people who also have a stake in your education to lend a hand to your work and your learning is completely different from asking strangers to “geek out” by doing your work — then passing off their work as yours. If students in graduate mathematics programs expect to succeed just by passing, then admissions committees should learn to admit a different kind of student, and also those “get-the-grade” students should not succeed.

    A college (undergraduate) diploma is generally taken to signal that the bearer has a proven capacity for absorbing and processing information, and for critical thinking. The field of concentration is usually irrelevant for future job prospects, and may be only slightly relevant even as preparation for graduate school. One expects to complete several undergraduate classes that have little if any impact beyond adding a line to the transcript. Most students consider this acceptable because it’s the norm, because there is bankable value in demonstrated range and flexibility, and (for the idealists) because there is intrinsic value in exposure to ideas beyond one’s comfort zone, and in gaining some measure of comfort with them.

    Graduate school is designed as career preparation. This is obviously true of “professional” schools (law, medicine, &c.); the academic graduate programs are professional training for being an academic. You can’t be (or at the least shouldn’t count on being) a professional academic without highly developed critical thinking skills within your field, or without at least aspirations to occasional originality. You have to be confident you can make a contribution: a new idea or new expression or synthesis of ideas — but SOMETHING of your own.

    How can getting your homework done for you get to professionalism? Maybe you’ll still learn the answers, but you will only undermine your understanding, to say nothing of your confidence.

  12. If this student hasn’t passed quals, he won’t, and if he has passed quals, why does he have to do graded homework?

    No one who operates this way will end up doing anything creative, much less write a decent thesis. That he might end up writing a thesis and becoming a crappy professor somewhere – who cares – probably the students will like him, because he will spoonfeed them – and the dean will be happy too.

    By the way, I know a competent mathematician who never passed his quals (two tries) becauase at the time he didn’t like complex analysis (since he has written respectable papers using it). He loves math, just wasn’t very good at doing what he was told. This is just to say that in grad school too much grading and evaluation is probably counterproductive.

  13. > That he might end up writing a thesis and becoming a crappy professor somewhere – who cares – probably the students will like him, because he will spoonfeed them – and the dean will be happy too.

    Yes, let’s tolerate inadequately prepared people becoming professors, where they will raise more graduate students in their own image. How droll it will be.

    And I am not saying that grading is the be all and end all. But when you’re in the system, you play by the rules, and more importantly you *play fair*. See my comments above about the diagnostic role of assignments; or, for a more cogent take, Lark Speyer’s comment.

  14. Many things to respond to. In this comment, I want to tackle why we have homework. In particular, does working in small groups damage this purpose? Is the purpose fulfilled by finding a solution in an outside source, as long as the student can explain the solution in their own words?

    For almost all people, the least useful ways to learn material is to listen to a lecture about it and to read a text about it. (Which of these is better varies from person to person, but they are both bad.) The best is to engage actively in problems involving the material. This is my own experience of watching how students and colleagues learn and my understanding is that it is one of the most reliable findings of pedagogical research.

    In our lower level classes, this is why we try to minimize our lecturing and give our students extensive opportunities to interact with us and each other. In our upper level classes, we have too much material to get through to teach in this way, so we lecture in class and assign homework to provide students with the opportunity to learn more actively. In a well designed graduate course, students are spending somewhere between 1 and 3 times as much time on the problem sets as in class, and this is where a large portion of their learning takes place.

    In its best form, working with fellow students provides MORE active learning than solving problems alone. Every point is argued over and discussed from multiple viewpoints. “Two scholars sharpen one another”, as the Talmud says.

    Student collaboration often does not reach this height. But I think there is almost always some active thinking on both sides of a partnership. In the probably 20 or so problem set based courses I have taken, I sought out collaborators in all but two of them and I can only remember a handful of times that I felt that I was either completely providing the answers or that I was being purely swept along by a stronger student.

    I think that the help provided on math.SE encourages much more passive learning, because stackexchange users provide well formulated complete answers in a way that a fellow student is unlikely to. I don’t think that it’s useless — it is somewhere between reading a textbook and talking to an older student/postdoc/faculty member. But it makes it much more easy to copy an answer without struggling with it.

    As should be obvious from the above, I don’t think that rewriting a solution one obtains online is provides the same learning value as working the problem for oneself. I think, for many students, it will not even produce the same level of understanding of that problem. It certainly will not provide the same exercise of ones general mathematical muscles.

    Finally, I am particularly annoyed here because this case is particularly egregious. It looks like this student may have been doing no work at all. Asking for help on those some questions one can’t answer after honest attempt is, well, more grey. It is easy to give up too soon when one would have learned more by trying longer. But one will learn more from being given an answer than from never seeing an answer at all so, in a course where there are no (or poor) solution sets provided, there is a reasonable case for doing so sometimes.

    I am still thinking about what policy to announce in my combinatorics course next term. My current thought is (a) students may work together, but must credit any idea which they obtain from a fellow student (b) students should not search out solutions on line before problem sets are due. If they find such solutions, they must cite them. (c) After the problem sets are turned in, go ahead and repost the questions to math.SE! I’ll be as interested as anyone to see what solutions people find.

  15. Wow, that was way too long. Shorter responses to everyone else. Bebito and anon suggest that the solution is to move all assessment to proctored exams — either in class or quals. I really don’t like the idea of postponing all assessment to quals — that’s way too late in a graduate students career to measure what he or she knows.

    I don’t like using proctored exams in general in upper level courses, because many people have trouble working under time pressure and the sort of questions that you can fairly ask under time pressure are pretty shallow. But it may be necessary.

  16. anon suggests that a lot of graduate students are just trying to hide from the real world. I thought that was what law school was for :-).

    I can’t provide an argument against this practice. If you enjoy learning, don’t need much money to be happy, and are intelligent enough to get into a good math grad school, this does sound like a good way to pass several years having fun and not incurring any debt.

    From a faculty perspective, I want to get rid of those students. They are going to bog down their fellow student’s energy, and they will draw off attention which I could be giving to someone with a passion for math. MAJOR DISCLAIMER: I have plenty of respect for some one who wants to go to grad school and study math but not pursue an academic track. If you want to teach high school math, or do financial analysis, for example, and you think that learning advanced math is the way to pursue that, I am all in favor. I just want students who will approach their studies with passion and dedication.

  17. anon asks what proportion of graduate students ever submit a paper to a journal. I’ll try to break that into two parts:

    What fraction of entering graduate students graduate? And what fraction of graduating students have submittable work? I would guess that there are not a lot of students who produce submittable work and don’t graduate.

    What fraction graduate? I had a hard time finding this number. It should be extractable from the AMS Departmental Profile Survey, but the way that they organize their data doesn’t make it easy. The number of first year math Ph. D students in the US in 2010 was 3313. The number of students receiving a Ph. D in that year was 1632. So, if the system is in steady state, we can deduce a completion rate of roughly 50%. Interestingly, I get the same proportion at Group I schools: 538 out of 1093 (which I did not expect).

    But I’m not sure that it is a good assumption that the system is in steady state. Math departments have been expanding their enrollment, which would make the situation not as grave as those numbers suggest. What I’d really like to know is, in any given year, what proportion of those students who leave grad school are leaving with a Ph. D?

    What percentage who graduate have submittable work? I’m going to use employment as a coarse proxy for this — there are a lot of jobs you can’t get if you don’t publish. Of the 1632 Ph. D recipients in 2010, I see that 614 work at US Ph. D granting institutions and an additional 209 work at four year colleges. I think it is safe to assume that anyone who got one of those jobs has publishable work. (Possible exception — lecturers/instructors. The AMS really should break that out separately.) There are also 48 working at Research Institutes and 163 working in non-US universities. I would guess that most of these are also jobs you can’t get without publishing, but there may be more range in these sort of positions than I am aware of.

    So I am very happy with a lower bound of 50-60% here. I imagine it is higher — I know lots of people who had a good paper or two but couldn’t find a job, whereas I don’t know anyone who got one of these jobs without having something published or almost ready to publish right out of grad school.

    I’m tempted now to ask a question on :-).

  18. Dear David,

    I just read your post and it really resonated with me.

    Full Disclaimer: I am a student from Australia. I just completed a BSc at the University of Sydney majoring in pure mathematics with first class honours and the university medal. At the end of 2011 I applied to 8 of the top pure mathematics graduate programs in the US. I have been rejected from 6 of them and seem to be on the wait list for the other 2 (The reason for this seems to be my poor GRE scores). I just want to stress that I am commenting not because I am bitter, but because this post is quite relevent to the situation I am in.

    Lets assume for the sake of argument that the student in question is indeed a graduate student. The question I keep asking myself is this: Why was this student admitted to U-M in the first place? Isn’t the point of the application process to weed out the students who are not suited for graduate studies in mathematics?

    – If a student is unable to handle a real analysis course (say at the level of Royden’s Real analysis) or an algebraic topology course (say at the level of hatcher), shouldn’t this be evident from the recomendation letters?
    – If a student is unable to manage their life and their study, shouldn’t this be conveyed in their transcript? (I doubt that their is a person alive who has not had a personal crises between the age of 18 and 22)

    Both mathoverflow and mathstackexcahnge are great tools for learning mathematics, but if there is a problem at graduate schools with students abusing these sites to complete homework sets, then is it possible that the wrong students are being admitted? All I can really say is that If an honours student at USYD tried to weasel their was out of an assignment (via MSE or some other method), then they would probably be eaten alive by the other honours students. (In fact, there was someone taking algebraic topology at USYD in 2011 who tried to obtain a copy of an older PhDs written solutions. It did not go down well and they ended up droping the course).

    P.S. I hope Someone has spoken to this student in a positive way. As Terry said, they might just need a little push in the right direction!

  19. This is a thought-provoking post, and the comments are interesting as well. However, I would like to mention one thing that concerns me about the direction this discussion has taken.

    It seems that many posters here are assuming the cheater is a (first-year) graduate student. I’m a second-year math PhD student here at UM, and I find that a bit hard to swallow.

    The original post neatly outlines the reasons why grad students would have little incentive to cheat. Unlike undergraduate students, we’re not under pressure to pad our transcripts with A’s from upper-level courses; and we generally expect to use most (if not all) of the material we’re learning.

    Most graduate students are reasonably prepared for graduate-level work. The graduate program here is quite selective. On the other hand, some undergraduates do take grad courses without necessarily mastering the prerequisites.

    It is not my aim to speak ill of UM undergrads; the undergraduate students I’ve met in my courses have been incredibly talented and hard-working. However, I think it’s most likely that the “cheater” is an opportunistic undergrad.

    I find it problematic that this possibility is only raised as an afterthought in the original post. The overall effect is to cast aspersions on the competence and integrity of the PhD students at UM, with what appears to be insufficient evidence.

  20. Rachel, that’s a good point, and looking over my own comments, it’s true that I did unthinkingly make this assumption. Apologies to you and, by extension, other diligent grad students for any implied or perceived insult.

  21. Thanks, Professor Choi. It’s not an unreasonable assumption in general; it just happens that the student body for many first-year undergraduate courses at UM is about half undergrads.

    Of course, it is entirely possible that one of the grad students here is the culprit.

    It’s shoddy behavior indeed, in any case.

  22. Er…that would be first-year graduate courses in my comment above. My typing skills are failing me today.

  23. David asks what proportion of those who graduate with a PhD ever publish a paper. I’ve heard one-half, although David’s argument for (number getting academic jobs)/(number getting PhD) being a good proxy seems to imply it’s higher.

    I’ve also heard – but don’t know how to confirm early in the morning – that the median number of papers published, by those who have published any papers at all, is one. That is, more than half of the people who publish at least one paper publish exactly one.

    The latter claim is hard to check, though, without a central listing of all papers ever published. mathscinet tries to be this. But what about the physicist who happens to publish one paper in a mathscinet-indexed journal and publishes lots of papers in physics? They should either count as someone who never published (since they’re not in mathematics proper) or someone who published many times, but not as someone who published exactly once.

  24. Yemon:
    You did not catch the pitch of my sarcasm well. I don’t tolerate or like these people, but I do recognize that they are around me, and don’t believe that the energy of faculty is well expended in making the serious efforts that would be needed to bring them to even the most minimal sort of justice. My implicit question, which was perhaps too implicit is – is it worth the bother to try to check that student homework is not copied from Math.SE, or is it better simply to avoid the problem by not grading homework? Or is there a third and still better alternative? By the way – it’s not like this is a new problem – complete books of solutions for the problems in books like Rudin’s analysis book or Jackson’s Electrodynamics have long been available (one discovers this when two students hand in the same solution to some arcane exercise).

    It seems clear that no one operating this way will ever become a good mathematician or even a good teacher of mathematics. There is not much worry that someone like this will get a job at a quality university or college. On the other hand, I wish I could say that there is no way that someone like this will receive a doctorate, and I wish I could say there is no way that someone like this will become a professor somewhere, but I am acquainted with too many counterexamples. By the way – Is it really so different when the professor writes the thesis? Some cheaters survive, and they do some damage (though mostly students survive bad professors), but I think the effort required to police them is sufficiently large, and their number sufficiently small, that to some degree it is not a good use of the time of most faculty.

    Think of teaching a large lecture class of calculus. Particularly in this time of cell phones and the like, one knows that that somewhere among the hundreds there is some copying or other form of cheating going on (?always?). Most of the time it is unproductive and incompetent and it results only in a failing grade in any case – the students copying do so too obviously, or copy off of other failing papers, etc. – though some of course get through and get by (the future bankers). To catch them one has to poison the atmosphere of the class and engage in all sorts of shenanigans (colored paper, several versions of the exam, etc.) which cost one time. There is who chooses to do all these things – in my experience teaching with such people it creates a setting for the class that reminds me in an unpleasant way of the imagination-stifling school teacher who takes off points for not writing proofs in neat parallel columns. Later, the student is caught and there is the elaborate process of pursuing justice through the dean of students. Half the time the results are inconclusive, or the student fails out before the process is finished, or the dean decides to give the student a warning, and the whole business was just a waste of the professor’s time, with little benefit – the cheater has learned that it costs little. If the student shoves the cheating in your face, or it is very egregious, or clearly beneficial, then one has no option but to go through all the disciplinary process – but if it is of marginal benefit to the student and policing it is costly relative to other options, it is perhaps better not to make an extraordinary effort in that direction and instead to see if there are small changes to the evaluation process that effectively render cheating ineffective.

    I think with graduate courses one can reasonably assume that the percentage of cheaters is lower than it is in undegrad courses (why bother taking a graduate course if your goal is just to pass?) – and mostly these students will be the one or two underprepared first years whose desperation in front of their inadequate preparation leads them to cheat. For the most part, this happens little with grad students in quality programs simply because the vetting of applicants is reasonably serious, and most have genuine interest, rather than simply looking to pass, and have also the necessary background. The guy who just wants an advanced degree to later make more money does better getting a master’s or going to law school. It is more of a problem with undergrads and with students in terminal master’s programs, or in programs that, to fill TA staffing requirements, are forced to admit some riskier students, whose preparation is less clearly adequate or more difficult to assess.

    What the instructor can do to avoid these things is pose examinations in formats that make cheating more difficult, and pose problems for which cheating is less effective. With something like stack exchange effectively answering standard homework exercises at the level of standard graduate courses, maybe one has to simply stop giving graded homework – the students who care will do the homework problems anyway – at this level the other students (those who have no passion) are in any case a low priority – and the cheaters have to come take an in-class exam. At the graduate level the authoritarian motivation of getting good grades may even be counterproductive and may misdirect some students whose psychology is not independent. One could push the entire examination burden onto qualifying exams if one wanted to. Solutions along these lines seem to me both easier and more effective than trying to police how a web site gets used. If one could write software that would check whether homework matched answers on Math.SE, then one could go ahead grading homework and use the software – something akin to this is done in introductory programming courses where exactly this sort of problem has been recognized for some time. But absent something of that sort, it does not seem worth the bother to do it by hand.

  25. I don’t think it is accurate to assume that
    most math PhD students plan to become research mathematicians, especially once you get beyond the top 5 or so programs. Many want to become college professors, perhaps in a small college somewhere where
    a PhD from a good program (and perhaps with one article based on the thesis) may be enough to get tenure. But such students are not driven to become research mathematicians. So satisfying subject distribution requirements may be perceived as a roadblock even by those who are serious about earning a math PhD.

    However, I feel that it is even *more important* that these non-research career track students be made to learn in some depth as broad a swath of mathematics as possible: they are likely to end up teaching in small departments and will need to be able to teach a wide variety of courses to accomodate their students, to stay in touch with contemporary mathematical developments, etc, i.e. properly represent mathematics at their institution. If the only math they know is the tiny slice their thesis work required, they are unlikely to inspire their better students.

    And, as we know, you can only master a topic if you do the accompanying work…

    So, I may not fully agree that it is dumb on the student’s part to have others do his HW, but I think it is bad for math to allow too narrowly educated students to be college math teachers.

    I suspect that those who are driven to become research mathematicians understand the pointlessness of not doing your own homework early on.

  26. Like Rachel, an I’m a bit concerned about the underlying assumptions in the comments to this post: I have VERY good reasons to believe it was NOT a grad student at all, or a math-focused undergrad. To declare potential for bias, I’m a 1st year Michigan math PhD student myself, though I don’t take the courses concerned, but I know all the grad students taking these courses, and they are all very smart indeed – not just at math, but in terms of their attitude to the work. They ALL put a colossal amount of effort into these and other courses – as well as self-posed problems built up from them. Without just giving each other answers, they do discuss the work to positive effect – often well into the night – and prioritise it in front of other things so they can do so (in fact I’m very unused to this such a positive culture of discussion, coming from overseas). There are also quite a few extremely smart and focused undergrads in the courses – one of the courses is in fact graded by one – and it’s just as implausible they’re involved. HOWEVER, I understand there are also a few undergrads in the course from outside the math stream but are taking it to add a ‘grad course’ notch in their cane. To be fair to them too, it can only really be just one of these – but please, if the premise of much of the discussion in the comments is ‘Why would a grad student wanting to do math research do this?’, the simplest answer in this case is probably, ‘Well, they wouldn’t, and didn’t.’

    I think that part, if not all, of the problem in this case is the fact that people can sign up for any course without any adequate preparation. I see this in calc courses, and I think it’s happening here. I don’t know about the American system in general, but I think it would help if in some cases placement exams should be given and enforced – and due to supra-departmental policies this isn’t the case here.

    Edited — The person who posted this comment has asked me to remove his or her name, and I have done so. [DES]

  27. @Bobito: You wrote “I think with graduate courses one can reasonably assume that the percentage of cheaters is lower than it is in undegrad courses (why bother taking a graduate course if your goal is just to pass?) – and mostly these students will be the one or two underprepared first years whose desperation in front of their inadequate preparation leads them to cheat. ” I think undergrads might bother because grad courses tend to be effectively pass/fail and they get a transcript which says, “Woohoo me, I did a grad course!” Secondly, it is most likely NOT a first year, and definitely not in this case, where at least in analysis the über-diligent first year grad students actually form a minority of the class, if I’m not mistaken.

    Edited The person who posted this comment has asked me to remove his or her name, and I have done so. [DES]

  28. Like several others, I agree with all of the original post, except the line “I think it is safe to assume that someone who enrolls in graduate school at UMich is aiming to produce a thesis, and most likely to get a tenure track position at a research institution”, even with the qualification about people who don’t have those goals. (FWIW: the unfortunate formulation “Liberal Arts or Community College”, in my ear at least, carries a hint that the writer maybe doesn’t see much difference between the two. And community college positions generally do *not* require research.)

    In my experience, even at the “best” departments, while many math grad students do explicitly intend to be academics (and often have had this goal since they were quite young), many do not. They’re in grad school not because they are dreaming of being professors, but because they like math a hell of a lot, full stop. They are usually *interested in* the idea of being an academic, but it isn’t a goal for them, the way it is for people who genuinely dream of that. Instead, it’s sort of a half goal: maybe they will do it, and if not, they will turn the skills they have to something else. (This whole paragraph is sort of tangential to the cheating issue— I’m just pointing it out because I’m always surprised when people in math don’t seem to be aware of this. It’s quite possible to be very serious about mathematical research, at the graduate school level, without being at all committed to being a professor.)

    But of course, like everyone else, I’m just baffled. If you want to be a professor, it’s insane to have other people do all of your homework. If you’re interested in math, it’s insane to have other people do all of your homework. When is it sane to have other people do all of your homework? I can’t think of any job where having “algebraic topology” and “real analysis” as a line on your transcript would get you something, without you also having the knowledge to back it up. Either I’m the one who’s crazy, or the world just doesn’t work the way that person thinks it works.

  29. No, I don’t see Liberal Arts or Community College as interchangeable. They are two very different institutions, and I think working for either of them is an admirable goal.

  30. Anonymous 1: I think your assumptions are perhaps a bit naive. There are plenty of examples of grad students at fine institutions who have cheated in every imaginable way – from plagiarizing in their theses to turning in solutions from well known solution manuals in first year courses.

    It is an empirical reality that a substantial number of your classmates – perhaps even a majority – will not be working as research mathematicians in ten years – and also that for many of them some sort of professional success is more important than is doing creative mathematics. On the other hand, particularly in an environment in which one’s classmates are hard working and smart, there can be terrible pressure to keep up. Someone who feels himself slipping behind or unable to maintain the pace can look for something on which to support himself. Competition motivates cheating too, particulary if one is accustomed to being the best, and suddenly finds oneself second or third best.

    Generally the undergrads in a first year grad course are at least as strong as the grad students in the course (the better prepared grad students often can skip these courses). There is very little incentive to have a grad course on the transcript except that one intends to do some kind of post-graduate study – because no one but graduate programs gives a damn if you took a graduate math course – and so I don’t find so terribly obvious that there is greater incentive for an undergrad to cheat than there is for a grad student. It may be that in the case that provoked this blog post that was the case, but this kind of cheating is by no means a rare sort of thing. (Here’s a humorous example: last year I had an undergrad submit, without attribution, as part of a numerical analysis project, code that I had written!)

    It seems that one should suppose that these things happen, and the question is what should faculty do about it, if anything. What I am suggesting is that faculty should set things up so that this kind of cheating is not so easy to do, and is unlikely to be effective, and beyond that they should not worry too much about it, unless it is shoved in their faces, so to speak. The negatives associated to effective policing seem to me to outweigh the benefits. As an example: one could prohibit collaboration between students on homework problems – this has a negative effect for interested, motivated students – the ones one cares most about – and is not terribly effective in stopping the behavior of the cheaters. Not grading the homework at all achieves a similar goal without the negative side effect – the motivated students will do the problems anyway – talking about them freely as they see fit – and the cheater won’t have anything to cheat at – so won’t get any unfair benefit by cheating. Frankly, I don’t see much difference between posting to math.SE and plowing through books in the library looking for the answers (the old-fashioned way of doing the same thing – though maybe the old-fashioned way at least required having some idea where to look), and don’t think this sort of behavior can easily be prevented as long as there are people willing to answer homework-like questions on math.SE.

    Edited Changed references to Anonymous 1’s real name. [DES]

  31. @ Aaron (#7) —
    >> I’ve never understood the professor’s mindset
    >> in how they think that getting help online is cheating.

    “Getting help,” online or by any other means, is not generally considered cheating (there will be exceptions, of course). It’s also not the situation we’re discussing here. We are discussing a student who copied out almost every homework problem over 2 months of an analysis course (plus some from a topology course) and presumably sat back waiting for the answers to roll in.

    Passing off as your own someone else’s work without attribution is plagiarism. And plagiarism is condemned not just by a “professor’s mindset” but by almost universally in creative fields and beyond. Wikipedia claims to have a particular ethical and practical duty to avoid plagiarism because it does not publish original research. The Center for the Study of Ethics in the Professions has collected ethical codes prohibiting plagiarism from not just the American Mathematical Society but also the Latvian Union of Journalists, Association of Social Anthropologists of Aotearoa/New Zealand, and Market Technicians Association, Inc.

    Admittedly, Aaron, the Student Code at Weber State University, your alma mater, lists cheating and plagiarism separately, but cheek-by-jowl under “activites [that] are specifically prohibited” (IV, D, 2, a&b).

    Plagiarism is cause for retraction of research articles (written by and for those with a “professor’s mindset”). It also is cause for pulling out of circulation “hot” new young-adult books.

    And, yes, even so, plagiarism is not understood to be wrong or problematic by everyone who stands to gain or lose from it. AAUP issued a warning to members about this over a decade ago:

    But rather than feel shame at detection, some students exposed through such simple exercises [as an online antiplagiarism program] express outrage. The trend has sent up red flags for faculty. “What frightens me is that many of these students don’t even recognize their theft of prose as something that is wrong,” adds [Jill] Craven, [a professor of film studies at Millersville University of Pennsylvania]. “They show no remorse, just anger at being caught.”

    I once caught a person who had attained a terminal professional degree lifting several paragraphs out of Wikipedia for a chapter he had been invited to author for a textbook. How? Because he left in the internal hyperlinks to other Wikipedia pages. When a peer confronted him, he claimed he had not realized this was unacceptable. As incomprehensible as the story seemed, it is the most plausible explanation: had he known he was taking a frowned-upon shortcut, he could simply have removed the hyperlinks! (And, yes, putting your own name on text generated by Wikipedia is still plagiarism, even though there is no copyright to violate.)

    @ Bobito (#32) — I will argue that plagiarizing homework answers after soliciting them from an open mathematical forum is more egregious than copying from a classmate, or a student a few years ahead of you, or “plowing through books in the library looking for the answers (the old-fashioned way of doing the same thing…),” because it makes it all the easier for the next person to cheat. Any other student frequenting math.SE could happen upon actual answers when they might have been looking legitimately for help. As you say, this venue will remain open to people looking for shortcuts “as long as there are people willing to answer homework-like questions on math.SE”—and math.SE users love answering homework questions.

    @ Rachel (#20, 22, 23) and Anonymous 1 (#27, 28) — recall that David’s primary purpose as stated in the original post is persuasion. He wants readers, particularly those in graduate school such as yourselves, to conclude that this plagiarism is irrational and self-destructive. You are proof that he’s succeeding. It doesn’t matter if the student posting these homework questions is a graduate student, an undergraduate, or some sadistic auditor. That person, for the purposes of this discussion, is just an example.

    Edited Removed references to Anonymous 1’s real name. [DES]

  32. Lark: I basically agree with your assessment, but it leaves open the question – what can be done about it? I think that preventing such use of math.SE and the like is nigh impossible, and policing such activity is tiresome and time consuming – and, moreover, not very effective – so that the best remedy is for professors to structure their evaluation so that cheating in such ways is not very helpful.

    My experience is that the current generation of (university engineering) students does not understand that there is anything wrong with pasting wikipedia articles into files and handing them in – for example to provide background for a numerical analysis project for which the implementing code has been provided by an older student or a paid tutor. In many less well endowed primary institutions students are encouraged to use the web (i.e. wikipedia) as a surrogate for the non-existent library. Cut and paste is the same in principle as copy and recopy, but much easier in practice. This is a huge and widespread problem.

    I find myself falling into – those who want to learn will not do these things – and to hell with the rest.

  33. Was looking up something else (spherically closed fields) and found this very interesting discussion. One comment I didn’t notice in the posted remarks is the following. Getting a significant amount of assistance (online or from other students) with the homework may or may not be acceptable in a given course, but it is *never* acceptable to get such help and not explicitly acknowledge it. When I teach a grad course, I’m fine with students turning in HW saying “I worked on these problems with X & Y”, and I’m even okay with “I was stuck on this problem, so I asked on math.SE”. But turning in other people’s work without acknowldegement is a form of plagiarism. Many grad students don’t realize this, so must be educated, but once they’ve been told, they need to follow the proper rules of attribution. And this carrys over to writing a thesis and being a research mathematician. Most research papers include a paragraph in which the author thanks those who have assisted, with specific thanks if they’ve done something specific such as showing the author how to prove (or simplify the proof) of a lemma.

  34. Oops, sorry, I had read about 2/3 of the posts and then realized I only had a few minutes to comment before heading to a seminar. But I just noticed that Lark Speyer’s March 27 post says essentially the same things that I just posted, so sorry for the repetition.

  35. I don’t know if math.SE has a way of making sure people don’t post homework – it should be easy to pick this up with the right patrolling. It depends on the scale of the problem in general, but one hopes that it would at the very least be picked up by the prof or other students whenever it happens.

    Perhaps another part of the problem is the fact that homework is fundamentally different from research, and one way is that in a course you know the problem is possibly already solved and out there somewhere, as well as many intermediate steps towards it. In fact instead of trying to learn and understand as much as possible one has to actively not do so in case one stumbles across the solution – and one can’t ‘unlearn’ it. This doesn’t really help learning, and there’s a fairly farcical element to that. For some courses also, there is a positive culture of group discussion which is naturally restrained, and formal attribution would be quite strange here. Informal attribution though is very different.

    Reasonably mature people probably do know how they learn things best, and at the end of the day those that do have to do research will have to do research, and they probably do want to do what is going to best help them with this. If they don’t, it’s their silly choice.

    It has taken me a while to get used to the US system, too… back home graduates just attend courses and grades are even officially a thing of the past. Here grades are not so important any more, but we still do get graded at least officially in the same way… there’s a facade of an undergrad atmosphere. This confuses, and even when doing research some people can find the culture in courses of “I must do my work or I get into twubble and teacher will spank me” a bit hard to shake off.

    Edited The person who posted this comment has asked me to remove his or her name, and I have done so. [DES]

  36. To clarify, I’m only talking about the way some un-clicked graduates might see things at first, though I don’t think this sort of response to that is very common, and was doubtless not the case here.

    Edited The person who posted this comment has asked me to remove his or her name, and I have done so. [DES]

  37. I hope graduates read this blog; maintaining a strong work ethic is good practice for when you are looking for future employment after you graduate… And the only person you are cheating is yourself!

  38. As a grad student at Berkeley, I used to lend my Algebraic Geometry homework to my officemate. He returned it in a short time. But later I found him copying from a photocopy of my homework. (Apparently he took my homework to 959 Evans…)

  39. This is a good problem in “mechanism design”, namely, how to design a mechanism for homework grading, so that it is in the best interest of the homework doer to be honest?

    A partial solution would be to have a random student present his/her solution in detail, the first 15 mins of class. That would force at least some degree of understanding, so part of the reason for assigning homeworks will be taken care of—though the “cheating” would still not be cured by this simple mechanism…..

  40. I like that idea! The incentive to improve homework is a benefit, but what I like more is the opportunity to give students a chance to improve their presentation skills.

    Worries that I have:

    (1) Some students would definitely hate this. And there will be some very frustrating moments when one student presents something badly that others understand well. This will require some thought about what the correct expectations of student behavior should be, and how to create them.

    When I taught calculus last fall, I was strongly encouraged to have students spend a lot of time presenting solutions. They did it very poorly, and their fellow students got bored and frustrated. I don’t think I managed this particularly well. But I am there to teach, so maybe it would be good to teach them how to deal with this.

    (2) It is a sacrifice of lecture time. There is always an immense amount of pressure to get through material rapidly in an upper level course. And I predict that it will eat up more time than you expect, between the time of transition to get someone else up at the board, time for questions, and general disorganization.

    The general perspective of most expects on teaching is that lecturing is one of the least useful activities possible, so I should be glad to give up the time. My completely nonexpert gut reaction is that this is a concession to student laziness. It is true that going to lectures and not spending hours outside them working on problems is not useful, but I feel that, in a context where I was working problems all the time, and discussing them with my friends, outside of class, I learned a ton by interrupting this process to go listen to lectures. (B. Conrad, McMullen, Sturmfels, Haiman, Lenstra, Knutson… are a few who come to mind.) But maybe I was atypical.

  41. Suvrit-

    I’m not sure that this would actually be effective at convincing the students to do the homework; generally people respond better to certain but minor negative consequences to their actions than to serious but inconsistent ones. It’s just too easy to convince yourself that you will not be the one chosen to present in class (which assumes that the students aren’t scheduled beforehand).

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