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In some of my more advanced classes, I have students present solutions to problems on a rotating basis. Not all solutions are perfect, but for almost all students this goes okay. However, sometimes there is a student X who seems incapable of solving all but the most trivial problems (and cannot always do those correctly). I am unsure whether X is even aware of the difference between themself and the rest of the class (my impression is they are not), but the X's presentations are not pleasant for the rest of the class, and probably not for X either. They typically involve me or other students making several corrections, taking 2-3x as long as needed (and the problems X has gotten so far have pretty easy already), then me eventually cutting of X and saying "Okay, all you do is [...]. Next."

I was contemplating only giving this student trivial problems to present. Is this a good idea? Are there better ideas?

My concern about doing this is that it will probably be obvious to the rest of the class if X is routinely getting significantly easier problems to present. It may also be obvious to X, though that is less clear--if so, I can imagine it might be embarrassing. (And if X cannot successfully solve those, it would be more embarrassing.) Not letting X present would probably be the most embarrassing. Right now, my inclination is to let X's problems naturally vary between trivial and normal, based on the rotation, but make an attempt to keep hard problems away from X.

Another option is to say something to X, but I am not sure what, apart from suggesting X check their solutions with another student before presenting. One thing I definitely want to avoid is teaching X individually outside of class. Based on previous interactions, this could quickly become a time sink, and I am too busy already.

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Another option is to say something to X, but I am not sure what -- "You are not prepared for this class; I strongly suggest dropping and retaking the prerequisite class." Also, don't let X take 2-3 times as long. Use a timer, and when X's time is up, say thank you, ask X to sit down, and move on to the next problem. – JeffE Feb 20 at 14:17
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"X is routinely getting significantly easier problems to present" - whether or not this is perfectly fine depends completely on whether the goal of the class is to practice solving the given problems, or to practice presenting given solutions. – O. R. Mapper Feb 20 at 14:38
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Is X going to be able to pass the class? – Patricia Shanahan Feb 20 at 15:34
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I agree with the concerns of the three comments above, but if X is going to stay in your class, and not only embarrasses themselves during presentations, but unreasonably slows down your class for everyone, I'd consider arranging for a meeting before class (with TA or even instructor) to prep the planned presentation a bit. This isn't quite fair to others, might set a bad precedent, and is untenable if more than one need this, but it'll improve the experience in those sessions for every other student. Going forward, consider checking students' preparedness before allowing them into your class. – gnometorule Feb 20 at 16:08
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Have you tasked X with solving a problem privately where the class is not staring them down? Sort of a control to determine whether it's the material or their presentation. – CMosychuk Feb 20 at 17:33

There are at least two+ different issues: (1a) your suffering (1b) suffering of the other students, (2) the obliviousness or incompetence or low-energy of the individual student. The easiest argument that there's something needing repair is that the other students shouldn't have to suffer, in any case.

My anecdotal evidence indicates that whether it's obliviousness or incompetence or some sort of lethargy, any one of these can be an essentially fatal obstacle, and is not within anyone else's power to change in the short term. Maybe long-term. Giving easier problems will not be understood as such, and can have the effect of passively "confirming" the inadequate strategies of that individual.

So: time limits on presentations, at least. Perhaps "failure limitations", too, meaning that a certain smaller window is allowed to persuade the audience that things are going in the right direction. This criterion gives a device to get failing performances off the stage and not waste other peoples' time.

Similarly, in my experience, it is possible to spend unlimited amounts of time on some students "with problems", thinking to "lift them out of failure", but have essentially no impact on them. Don't do it. Allocate the reasonable amount of time, and then stop. This is absolutely not an argument against spending time and energy on students (a.k.a. "teaching"), but is an argument against pretending that one has infinite personal resources, and against pretending that bad allocation doesn't harm other parts of the enterprise.

Possibly time to mention the Dunning-Krueger syndrome again, too? That is, that sometimes the most problemmatic scenarios are those in which someone lacks the meta-cognition to understand that they "even have problems". If someone has already passed/failed that filter, the chances are slim that you can call on their meta-cognitive faculties to help them... Sadly.

So, you cannot deny weak students resources, but don't misallocate so as to harm others. And the cost to you yourself should be limited. One does not have unlimited responsibility (in time or energy) to students. Definite responsibility, yes, but definitely limited.

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I'd add a caution, though, (maybe not so relevant to Kimball's specific case: I'm acquainted with him... , and also he gives specifics that indicate otherwise...) that conceivably the environment in a "work a problem on the board" accidentally entails "tests/gauntlets" not really related to mathematics. E.g., hecklers? Is anyone accidentally thinking they should solve a problem in real time in front of everyone? Is the blackboard too high for a short person? I guess it's some kind of virtue to be able to cope with hecklers, but I hope that's not a prerequisite for too many things... – paul garrett Feb 20 at 22:53
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Giving easier problems just kicks the can down the road, too. If this class is a fundamental class for anything else that student will be even less equipped in the future too. – enderland Feb 21 at 22:36

Paul Garrett's answer is on spot with two recommendations: time limits on in-class presentations and limits on time and effort you spend helping X. However, there are two fast things you can do to help X.

First, inform X that he is underperforming. Sadly, he may not even realize this, and if he does, there is still that denial stage. Talk to him when you meet next time and say that he is falling severely behind with his studies and he has to put in a lot of his personal effort to catch up.

A natural request coming back at you will be, 'How do I catch up?'. Get ready. Have a list of books to read (if possible, indicate the exact chapters) and, more importantly, a list of additional problems for X to solve on his own. The only way of learning to solve problems is by solving another dozen. Tell X that solving these problems will give him a better understanding of what happens in your class. If you feel like it, tell him you'll be able to check his solutions during your next office hours. If you don't feel like it, ask a TA to check X's solutions or ask X to find a peer to check them.

Surely, you might need to give him an easy problem to present at the next class meeting, but do give him more problems to solve on his own. This way, you are giving him a chance to grow up and reach for adequate problems towards the end of the course.

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Yes, it's good to try this, but I've seen many instances where the kind of catch-up that would have salvaged the situation is even more impossible for the person with difficulties... for reasons that put them behind in the first place. In effect, "try harder" may be functionally impossible. "Allocate more time to this course material" might be plausible, but amazingly often I've been told that the student-in-trouble was already spending hours and hours every day. I grudgingly admit that it is indeed possible to spend time and accomplish very little... – paul garrett Feb 20 at 22:00
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@paulgarrett That would heavily depend on the exact reasons that put the student behind. While this may be a hard-to-tackle issue like a genuine disability / learning disorder, it may also be that the student wasn't doing his homework in the beginning of the course or hasn't found a reference book that would appeal to him. Anyway, I tried to limit my suggestions to require as little time from the OP as possible. – svavil Feb 20 at 22:04
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True, in principle such over-come-able obstacles may be the issue, but my own anecdotal experience indicates that this is relatively rare. That is, disturbingly bad outcomes are very rarely due to simple, accidental disengagement. In some sense, math is not so hard, insofar as it really doesn't require to much sense about the world, in contrast to other subjects. Just very primordial things about shapes, symmetries, and the book-keeping of algebra. Quite a few delicate linguistic considerations that we pretend are logical rather than semantic. :) – paul garrett Feb 20 at 22:24
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my own anecdotal experience indicates that this is relatively rare. — Mine, too. Hence my original suggestion: Go retake the prerequisite class. That's what prerequisite classes are for. – JeffE Feb 20 at 22:58
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I will support @svavil in that you've got to have a conversation with the student. I do find (at lower level than this case) that usually just tapping the subject at all brings an outpouring of concerns that the student has been aware of for some time. Sometimes the fact that it's not fixable in the scope of the current semester becomes clear to the student in the conversation; but you do need to express a vision of what a path to fixing it would look like. – Daniel R. Collins Feb 21 at 5:37

This question resonates with me, since I am often in what I think is a similar situation. To help fix ideas, I will describe my situation: when I teach math graduate courses -- other than qualifying courses which have a fixed syllabus and prepare students for a later written exam -- the homework issue is often a challenging one. Let me begin by setting the scene: in most math PhD programs in the United States, students continue taking coursework for their entire time in the program. For instance, my department has a regulation that a student must take at least one "real" (i.e., non-reading, non-thesis-writing) course per semester. My own graduate program (Harvard) had the enlightened practice that as soon as a student passed their written quals, their grades in courses are automatically "excused" and thus students do not even necessarily show up at all for the courses they register for (and this is not necessarily a problem for anyone). Most other programs are not like this, and students get letter grades in their courses even while they are writing up their PhD and applying / interviewing for / accepting jobs.

So one is in the position of teaching graduate courses to students, most of whom presumably are at least somewhat interested in the material (they are math PhD students, after all) but many of whom have more pressing demands on their time. On the other hand, for many if not most students attending many if not most courses, simply attending the lectures and never doing outside work is not going to get them anywhere: it would be a more efficient use of their time to simply excuse them from coming to the lectures (which is not unheard of but not guaranteed to be kosher either). So in most cases you want to at least give opportunities for the students to reinforce the material of the lecture, but if you do much in the way of homework then they may be unhappy, and perhaps rightfully so. Moreover, most professors do not get graders for these kinds of courses, and -- more teaching-focused faculty may freely roll their eyes now -- in many cases we have too many other professional responsibilities to spend too much time grading written work. Here are the ways I have navigated this myself:

(1) In all graduate courses my lectures include "exercises" that if worked will reinforce the material. (If I don't do that, then I've lost all pretense of teaching a course.) However in some courses the students are not required to solve the exercises in any way. They are -- of course? -- free to ask me about the exercises, however the last few times I used this practice I had very little in the way of such discussion, to the extent that it would be hard for me to be confident that the majority of the students were spending any significant time working the exercises.

(2) I have sometimes had students turn in written homework, however with the understanding that I cannot grade a weekly problem set in a graduate course. I have a memory of problem sets from an elliptic curves course cluttering up my office and then my study long after the end of the course. I looked at some of them but not all of them and probably not enough of them. I know a very small number of professors that do grade regular problem sets in graduate courses of this kind, and I admire them for it. I know more professors who compensate for this by asking students turn in a ridiculously small number of problems in total, e.g. less than ten for a semester course. This is not a great solution: most of the work goes unread, and they don't get enough reinforcement.

(3) My favorite solution is to have a weekly(ish) problem session in graduate courses, in which we meet -- usually for at least an hour -- and students present solutions to each other (and to me). I like this practice because:
(i) I don't have to grade written homework.
(ii) The fact that students will be presenting in front of others usually makes for more work on any given problem and improves the quality of their presentation.
(iii) When things are going well, it means that students can benefit from solutions to problems that they did not themselves work out.
(iv) It gets students in the practice of talking to and working with each other rather than just talking to me.

I am currently teaching a graduate course (commutative algebra) in which I have "flipped the classroom" by making the Monday lecture a problem session, and then in exchange I give a 60-70 minute lecture on Friday afternoons (of a more one-shot nature; for those who care, my first three lectures have been on: Swan's Theorem on vector bundles and projective modules; Galois connections; and direct and inverse limits).

This is working well -- in fact, better than any problem session I can remember. Whenever I point to a student, they go to the board and solve a problem (one of a longish list that I have given them a week or more in advance). They usually solve it correctly; when they falter, another student steps in to help them out. They do so well that both they and I often feel free to ask followup questions on the spot. Having had the experience of problem sessions that don't work as well in the past, I do not take this success for granted and am trying to figure out what's going on. One of the students is my own PhD student (which doesn't hurt!), and she told me that the majority of the students meet as a group and work out enough of the problems together. That's great! However, the one thing that really makes it work well is that the students are both strong and relatively homogeneous in their abilities and experience: they are mostly second and third year students, and though some are more interested in category theory or topology or number theory, there is no clear top and bottom to the group. There is no doubt that each of the students in the room can solve a positive number of the problems assigned every week.

In past years I've had Kimball's problem: one or two students are -- either by preparation or ability or lack of interest or lord-knows-what -- just not up to the level of the others and the requirements of the course. I have tried to compensate for this in the following ways, many of which Kimball has already mentioned:

(i) Making a wide range of difficulty in problems assigned, and allowing weaker students to solve problems that most of their classmates would regard as trivial.
(ii) Allowing weaker students to present less often than the other students.
(iii) Assigning special problems that target the weaker students' background or are more obviously related to their stated interests. (I can think of one instance where this worked well. But in retrospect I think a big part of the success was that the "weak student" was not actually weak: in fact she was a strong student, just younger and with a worse background than her classmates.)
(iv) Excusing students from presenting solutions to the problems. (I had to do this once, for a student who needed to complete other work in order to stay in the program. This student ended up with a kind of "IOU N problems in subject X" which, of course, was never properly cashed in.)

The bottom line though is that in many math PhD courses, having a student who is not that interested, is not understanding what is happening very well, and who would have to shore up his background a bit in order to engage with the problems at the level of his classmates, is best left alone, possibly after a conversation with his adviser to make sure that he is spending his time productively on other things.

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As you note, student X's presentations take up a lot of time, don't help other students learn, and don't seem to even help student X. It certainly sounds like it will be important to have a conversation with X about how to help them do better, or even whether they are prepared for class. In addition to that, having each student present the solution to a problem to the whole class may not be the best way of meeting your learning objectives; the problem is most visible with student X, but this general strategy may not be serving some other students as well.

I use jigsaw groups to meet the goals of having students work together in solving problems and of having students communicate their solutions; moreover, this is a context in which students can come to practice teamwork and see it as valuable. The way I do this is I pick a handful of problems for the class to work on, say 6 problems for a class of 36. Students work in groups of 6, with each group solving one problem (solutions can be prepared in-class or beforehand). Then, the groups re-shuffle, so that each new group has a member of each of the original groups. For each group, the student who solved the first problem presents the solution to that problem with the members of their new group, the second student shares, and so on. This can give you more useful listening to do in class: with students sharing solutions to the whole class, you have to listen all the way through problems you already know the solution to! Instead, by circulating through the room, eavesdropping on groups, I can focus on hearing the most tricky parts of solutions. I avoid intervening immediately, but this often reveals confusions multiple students have, which helps me identify topics to clarify at the end of the lesson for the whole class.

This has a handful of advantages over having each student present to the whole class. First, most students get nervous about presenting in front of a whole class; presenting for a handful of people can be much less stressful. (I grant that presentation skills are important, but practicing presenting something a student just learned may not be the best context for learning these skills.) A solution prepared by a group is more likely to be correct—this addresses the problem with X having difficulty solving many problems: X can at least practice talking through a solution, which can be a valuable way to learn. In small groups, students are more free to ask questions; of course, more freedom to ask questions is great, and, even better, these questions can even a flawed solution. For the more advanced students, teaching a median-level student, or even a student like X, can force them to communicate solutions more clearly; this is a worthwhile challenge for a bright student, and can help them exceed your expectations for them. As a student, I helped my classmates in this way in study groups outside of class, and I learned more deeply than I would have otherwise. By encouraging cooperative problem solving in class, you can help students recognize more effective ways to study together in and out of class.

In direct response to the specific question,

I was contemplating only giving this student trivial problems to present. Is this a good idea?

I do not recommend this. You want X to eventually be proficient even with difficult problems in this class. X can notice if you deliberately lower your expectations for them, and this can inhibit X's confidence that they can reach the high standards you rightly hold for your students. Developing a classroom environment in which cooperation between students is scaffolded can lead to supportive relationships between X and other students.

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Your question:

I was contemplating only giving this student trivial problems to present. Is this a good idea? Are there better ideas?

Whether this is a good idea depends on what your learning goal for the course is. If your goal is "students should be able to effectively solve an unpracticed problem in front of an audience," then helping student X get better at this skill is important. @svavil's ideas of a short meeting that makes sure the student understands the problem and provides resources is great.

If your class goal is only "students should be able to effectively solve an unpracticed problem" then my answer is different. If public presentation is not an important skill, provide other opportunities for students to work in small groups or work independently to show they can succeed without the social pressure of an audience. Mixing it up even one day a week can make a nice change and give students an opportunity to talk to each other instead of just sit and watch each other struggle.

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I'm kind of confused by this answer. What do you mean by "unpracticed"? Certainly the OP is not giving out the problems in class; the students get to work on the problems in advance -- isn't that practicing? (If you just mean problems that they haven't done before -- what other kind of problems do you ask people to solve?) Moreover, I'm not sure why you think that the presentation or the social pressure is a key aspect of the student's performance. If the student really did solve the problem correctly, then they won't have the kind of issues the OP is describing. – Pete L. Clark Feb 21 at 6:32

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