Migrated from Mathoverflow.net

This is a questions about teaching/research in math academia.

During the pandemic, many things have been moved online: courses, seminars, informal gatherings, etc. As a student, I've had a positive experience with online courses: they allowed me to work at my own schedule (because lectures were recorded), and I was not tied down to a geographic location.

There seems to be a strong argument to be made in favor of continuing this post-pandemic which may bring a host of benefits to all levels of math academia. (I've mentioned some of the benefits on the student side.)

On the teaching side, I could see how remote teaching can eliminate many of the inefficiencies that were present pre-pandemic. For instance, why do we need to give three identical lectures on Calculus I, three times a week, every year (many of which are taught very poorly or unenthusiastically)? I think everybody would benefit if one records a set of lectures taught by a very good teacher, once every few years. (Imagine how much time this would free up for everyone.)

I could see how remote work can be beneficial for math academics outside of teaching. For instance, I hear about a lot of people leaving math academia because they cannot get an academic position in a city they want to live in (or, similar problems tied to geographic location). Can we not resolve this remote work? Of course, there are benefits to interacting with people face-to-face, but I still don't see why all academic positions have to be in-person.

My Question. Is there any movement within math academia (either in the U.S. or elsewhere) to make the following changes:

  1. permanently move some (or all) teaching online (either undergraduate or graduate level)?
  2. Have "remote" academic positions? (i.e. be
    affiliated with a research institution, but not be required to be
    physically present at a certain location)

Subquestion. What are arguments against implementing either of the above (if any)? (So far, I haven't heard any convincing arguments about the above issues from anyone. My conclusion is that things are the way they are largely because of inertia. If people have strong arguments in favor of doing in-person work all the time, I would be interested in hearing about them.)

Note: I considered posting this on academia.stackexchange, but I realized a lot of the question was math-specific. (For instance, the content of Freshman Calculus or Linear Algebra will not change 20 years from now, whereas in some other field of natural science, there might be a groundbreaking discovery that forces people to reevaluate the fundamentals of the field.)

Note 2: On MathOverflow, the question faced many oppositions. (In short, people thought a.) most people preferred in person instruction to remote, and b.) there is enough variation in presentation that warrants the in person instruction. Someone also implied that in person instruction was the justification for their salary. Please see the link above for details and for the exact phrasing.) I still am not convinced with the above arguments. Is there a definitive argument in favor of in person instructions?

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    $\begingroup$ 1. we learn by teaching, even Calculus 1. 2. administrators are not going to pay for people to use all that free time for intellectual pursuits, they'll just hire that one dude to teach all courses on a 3 year cycle. 3. it robs the whole enterprise of teaching the process of teaching, conversation born of the natural give and take of lecture is replaced with what ? Email chains ? Sure, for the best students it can still work, but... having taught a few semesters online, I've yet to find a genuinely curious online students Now, for free uncredited instruction, that's another story.. .more later $\endgroup$ Commented Aug 14, 2021 at 2:34
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    $\begingroup$ In my experience the main benefit of in-person instruction is providing students with a source of accountability. On the other hand, there are some real accessibility benefits to online instruction, particularly for those with unpredictable chronic illnesses. Best to offer both options, and that's exactly what we saw even pre-pandemic with online courses. Check out SNHU for one example of a mainly online program: snhu.edu/online-degrees $\endgroup$
    – TomKern
    Commented Aug 14, 2021 at 4:09
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    $\begingroup$ Useful question, although I should point out that the same arguments you're giving for taped lectures have been used for many decades. Indeed, these arguments were being made even in the 1950s when TV was beginning. (Source, that I can look up later if anyone is truly interested: various articles I've read in old volumes of Proceedings of the Iowa Academy of Science.) $\endgroup$ Commented Aug 14, 2021 at 16:49
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    $\begingroup$ This summer I taught an online math class which I already taught online last summer. For sure, I recycled some videos. But I also updated a lot of them, where I found a better presentation, a better example, learned last year which parts seem to come easy to students and which I should stress more / earlier; occasionally I use examples tied to ongoing news, these had to be updated etc. What I'm saying is: If the teaching (even just the lecturing part of teaching) is meant to be good and stay good, you cannot just re-use a recorded lecture once and for all. $\endgroup$ Commented Aug 14, 2021 at 17:20
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    $\begingroup$ Also, and this is just anecdotal evidence from my little sample: In remote versus in-person classes, a few students ask more questions, but many students ask less questions. (And I think students asking questions is a good thing; because I think the amount of unclarity they have is overall the same, but now it goes un-verbalised and un-addressed.) $\endgroup$ Commented Aug 14, 2021 at 17:26

4 Answers 4


Context: This answer is based an experiences from maths programs at German universities. Things might be considerably different in other countries.

This is mainly an answer to the teaching part of your subquestion, but first a few preliminary comments:

  • It is definitely possible to set up an entire maths program at a university in a purely remote way, and this has already been done long before Covid 19 occurred.

    For instance, the largest university in Germany (in terms of students) is (and was, also before the pandemic) Fernuniversität Hagen, which offers exclusively remote programs. Thy offer both a Bachelor's and a Master's program in maths. You don't have to (and I guess, you probably can't) show up in person, except for the exams at the end of each semester.

    A similar concept is offered by Open University in the UK.

  • You won't get any argument from me if you claim that there is a lot of "inertia" in how mathematics is taught at universities, and that many things should and could be improved.

    I am also under the impression that the disruptions in teaching caused by the pandemic took off some of the inertia and opened various options for improvements, which at least some people are happy to use.

However, I do not see any way how such improvements could or should result in a large fraction of the teaching being done online, let alone by means of mostly standardised online teaching materials.

You ask for concrete arguments against implementation of online teaching on a large scale - so here we go.

Disclaimer: The following comments are exclusively about mathematics; I do not make any claims about different academic fields.

Remote teaching

While remote teaching can be done and has been done for a long time by some institutions, it can have some severe negative effects which make it, from my point of view, hardly suitable for a large fraction of students:

  • Learning mathematics becomes much easier if students work together.

    When I began studying mathematics (in 2007 - damn, I'm getting old...), one of the very first things we were told by various lecturers and tutors was that it is extremely hard to be successful if you only work on your own. We were thus encourage to work on the homework problems together with our peers. We were also encouraged (and sometimes even required) to write down and submit the solutions to all our homework problems in pairs. To most students this was extremely helpful. You learn and understand much more if you discuss the problems you're trying to solve with other students. You also learn to communicate about mathematics, which is an extremely important skill.

    Of course, remote teaching does not make cooperation between students impossible, but it makes it much harder. In particular, if remote teaching becomes somekind of standard, many students probably won't move any longer to the city where their university is located. Thus, they can't simply meet their peers in person, unless they are lucky and a few of their peers happen to live close to them (and they happen to get along well with precisely those peers). So in many cases, the only available option will then be to work with their peers remotely, too. This might work for a few students, but for most of them it will most likely come close to a disaster: it's more than difficult enough to get used to understand and discuss mathematics as a freshman. Doing this remotely will add considerably to these difficulties.

    Technical tools such as video calls and tablets are useful for some purposes, but there is simply no way to make an online meeting between, say, three first year students who try to solve their homework in Linear Algebra nearly as efficient as the same meeting when they all sit at the same table.

  • Students are social beings.

    Of course, you can go to a bar at your hometown, with people completely unrelated to your maths program. However, when it comes to motivation for their studies, many students will feel much more motivated if they have a close and positive relationship to some students from the same program (with whom they can, for instance, discuss the course materials, discuss their homework, learn for exams, and so on.) Positive social relationships are much (much, much) easier to establish and to maintain if people meet in person and in high frequency.

  • Physical presence provides daily routine.

    Yes, I know, students are adults, and all this. But that's besides the point. Most adults work more efficiently with daily routine, too. It is, obviously, possible to establish daily routine without physical presence, too. But it is usually harder.

  • Asking questions is much easier when you are physically present.

    This refers to lectures, tutorials, and also office hours. If a student without much mathematical experience yet, asks me to explain something, this will be much easier if we are both in the same room. During the last few weeks, infection numbers in Germany have been quite low, so I was able to meet some students in office instead of remotely. Whenever I suggested this to a student, they were quite happy (or even relieved) to do this - apparently, they were all under the impression that things are much easier to discuss when everybody is in the same room.

    Again, it is far from impossible to ask and answer questions remotely. It is just harder and less efficient in most cases.

  • Lecturers are social beings.

    Personally, I like teaching quite a lot. I love mathematics, and I like the idea to give students the opportunity to become passionate about mathematics, too. Well, most of them don't become really passionate about it, of course, but I still like it to see how they learn and understand something that as new to them.

    In order to experience this, I need to meet my students - in real life, not just on some computer screen. Not giving me this opportunity will, in the long run, most likely drag down my motivation. Dragging down the motivation of people who are passionate about teaching is not a particularly wise course of action if you're interested in high quality teaching.

  • Re-usage of materials results in lower quality, unless the materials are standardized.

    One of the main advantageous of teaching a course several times is that it gives the lecturer the option to improve each time. But this is only possible if the lecturer does not re-use, for instance, the same videos every year. At first glance, one might argue that changing only parts of the materials still saves times, but in my experience this is unlikely to work.

    For instance, a year ago I taught a course in point set topology. During the course I took a lot of notes about things that I would like to improve in case that I'm going to teach the course again. These note will most likely result in re-organizing all the contents of the course. I did the class as "inverted classroom", with a manuscript and videos. But I will be essentially unable to reuse any video at all when I teach the course again.

    Hence, online course will, in general, only save time if most of the materials are somehow standardized, and the same materials are used by many lecturers. This brings us to the next point.

Usage of standardized materials

  • A preliminary remark about textbooks.

    There seems to be a misconception in the original post, where it is claimed that materials are already highly standardized for undergraduate mathematics. It is worthwhile mentioning that the idea to base an undergraduate course on a textbook seems to be very much a US thing. (When I started to regularly read posts here and on Academia StackExchange I was first quite confused how naturally many people spoke about "the" textbook in their course.)

    In fact, I can't remember a single course or lecture I've attended which was based on a textbook. Instead, we were typically given a list of books, along with the remark "These books treat the same topic as the course, but of course there are various differences. We advise you to have a look at these books, choose one that suites your preferences, and get it from the library so that you have a reference which gives you a second perspective on the topic."

    The lecturer would then give the lecture based on their own course design. (Obviously, many lecturers use textbooks as a source of reference when designing and preparing a course; but in Germany - and I think the same is true for various further European countries - they typically don't follow one single textbook. Exceptions do exist, obviously.)

So using standardized materials is far from universal. But also in places where undergraduate course are often based on textbooks, the suggestion in the original posts amounts to more, namely to also use standardized videos for online teaching.

So, to continue the list, here are some objections against using standardized materials in order to "save time":

  • Mathematics is done by mathematicians.

    This is in extremely important point, I think (and I find it a bit surprising how rarely it is mentioned in discussions about this topic).

    It might be somewhat related to the academic system - but in my experience, first year students in mathematics find themselves in a very strange situation: they have been dealing with all these nice computations of derivates and integrals, and minima and maxima of functions, in school but have hardly had any significant exposure to what mathematics really means. Then they enter university and are essentially blown away both by the level of abstraction and by the pace of the courses.

    (A nice example: In my very first week aa a university student, I attended my first Linear Algebra lecture ever. The lecturer talked a few minutes about organizational matters; then the mathematics started like this: "Alright, you also attend a course on analysis, and there you'll learn some basics about propositional logics and elementary set theory during the next two weeks; you will also learn there what a field is. That's important, because we need this now. Definition 1.1: Vector spaces.")

    Now, this is tough for most students (of course, one might start on a less heavy note than this Linear Algebra course, but this will only briefly delay the shock), and there is a high risk that many students will very quickly come to the conclusion that mathematics at universities is just abstract non-sense which mortal beings like them cannot understand.

    Now if all this stuff is presented to them in videos that have been recorded by a different person they never met, and their lecturer will only answer their questions on the material rather than teaching the material, the "not comprehensible by mortal beings"-effect will probably even amplify by an order of magnitude or so.

    So to put it differently: I strongly believe that most students need to see a real and alive human being in front of them who presents and explains the contents of the course. They need this in order to understand that mathematics is not some completely remote mumbo jumbo done by weird professors who are sitting somewhere in basement offices without windows, in cities at the other end of the country, and recording videos - but by mathematicians who exist in the same universe as the students do, and are standing there right in front of the classroom and communicate with the students in person.

  • Lecturers learn a lot by designing a course.

    This is an argument not only against standardized videos, but also a reason why I don't mainly following a single textbook when designing a course. Whenever I prepare a course, it turns out that I learn a lot (this is an understatement, actually) about the things I'm going to teach. This is true both at the level of single results (I might, e.g., include a theorem in a lecture that I haven't been aware of before, or might find a proof of the theorem somewhere that I haven't known before and use it in the course, or I might already know the theorem but have never bothered to study its proof before) and on the level of the entire course (which definitions and results do I need to treat first; which organization of the material keeps the proofs as smooth and efficient as possible; which organization of the material is likely to provide proper motivation for the students, and so on).

    The understanding of such things that I get from designing the entire course on my own is much higher than the understanding I would gain from just following a textbook. Consider for instance the following line of reasoning:

    "In Chapter 2, I include a Lemma 2.4 which is instrumental in the proof of Theorem 2.5; in fact, there would be a shorter and more direct proof of Theorem 2.5, but Lemma 2.4 combined with Example 4.9 (which we will treat later) will allow me to give a very efficient and easy-to-follow proof of a major result later on in Chapter 5."

    These dependencies between various parts of a theory are extremely difficult to grasp of you don't organize the material on your own. The major advantage of a good textbook - that the material is already very well-organized - is also a major drawback here: In a really well-thought book, the material will be so smoothly organized that you often don't even notice how much thought went into it and why the organization is precisely done the way it is.

    So, to sum up: I learn much more of I design my course materials on my own.

  • Lecturers who give the lecture themselves will be better prepared for questions.

    Our human minds tend to be lazy. If somebody provides my with a video and a chapter of the lecture notes for a course, and I'm supposed to answer students' questions on this material, I will most likely briefly go through the lecture notes before class, and then see what happens.

    Of course, I will be able to answer many questions of the students with this kind of preparation, and I'm confident that my answers will be of reasonably high quality on average.

    However, there is no chance that my answers will be as good as they would be if I had prepared and given the lecture myself. When I have to give a lecture, I will take a sheet of paper before the lecture, and write down most of the content of the lecture for me, once again. In particular, I will write down the proofs that I'm going to discuss. Obviously this will make me much more familiar with the contents than just reading briefly through the lecture notes.

    Of course, I "could" do them same thorough preparation even if there's the pre-recorded video and I'm just supposed to answer questions. But I'm quite sure that I won't, and I'm also quite sure that many colleagues won't, either. You could argue that this is stupid behaviour, that we should overcome the underlying psychological bias, and that we should thus prepare in the same way as if we were to give the lecture ourselves. But this is besides the point: Good management decisions are not about what people should do, they are about what people will do.

  • It is important to keep the contents diverse.

    This point is not so much about the students' or faculties' perspective, but rather about society (or at least, mathematics) as a whole. Where I live, university are almost exclusively funded by taxpayers' money - so there's a legitimate requirement that teaching at universities should not only benefit the students, but also society on a larger scale.

    I argue that it is beneficial to cover a very diverse range of ways to teach undergraduate maths courses. It is true that, for instance, the basics of Linear Algebra are unlikely to change in the next few decades, and it is also true that the contents of a course on, say, Linear Algebra are to some extent pre-defined by the syllabus (which will, in most cases, be designed by the math department).

    But even under these boundary conditions, it is possible to vary the approach to a course on Linear Algebra considerably. If we were to use a set of pre-defined videos to teach this course, numerous students would learn Linear Algebra in precisely the same way, in many other perspectives on Linear Algebra would likely be hardly recognized. Such a loss of diversity is not a particular good foundation for scientific progress (to put it mildly).

  • Lecturers can be somewhat idiosyncratic.

    This is one more point about the motivation of the lecturer. Many mathematicians I know (including myself) tend to have quite a few preferences (not too say: strong opinions) on how certain things should be done in teaching in general, and in certain courses in particular. Urging (or even forcing them) to use pre-defined material, with quite restricted options to adjust it to their own preference, is quite likely to have a negative impact on their motivation.

    Personally, I try to follow the leitmotiv "If I do something, I try to do it right." So if you gave my three different textbooks with associated video courses and asked me to choose one of the three and to give a course by providing the students with these materials and answering questions, chances are that I would find that none of the three sets of materials "does it right", and consequentially I would care much less about the course. (Important note: Obviously, "do it right" is completely subjective here, and there will, in general, not be any objective reason to back up my feeling that the materials should be presented differently. However, this doesn't change the fact that I would be much less motivated to teach from materials which do not fit my preferences.)

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    $\begingroup$ I have never read an answer here that I agreed more with. Fantastic. I wish educators would listen more to master teachers like yourself: "Dragging down the motivation of people who are passionate about teaching is not a particularly wise course of action if you're interested in high quality teaching." All the times I've heard about this or that pedagogical technique or technological gimmick, hardly ever a concern about the motivation of the passionate teacher. I am giving this +500 if I can. $\endgroup$ Commented Aug 15, 2021 at 3:06
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    $\begingroup$ Joined this community to upvote this answer. It's a very nice summary of why the last three semesters (online teaching) were … so little fun - for me and my students. $\endgroup$
    – CL.
    Commented Aug 17, 2021 at 8:57

The question presents teaching as lecturing; that is, as presenting content in an organized manner. This is not without value; it is easier to understand well-organized content than non-organized content. Hopefully the level of the presentation is at least somewhat tailored to the audience, too.

This is essentially a behaviourist view of learning and knowledge: the teacher presents the knowledge at a suitable pace and suitable bits, which the students absorb. Thus there is learning.

We might contrast this with social-constructive view of learning. Here, learning happens when people make sense of the material, first together and then in the process move to personal understanding. This is a lot more challenging to do via a pre-recorded video lecture.

This also happens in any kind of learning situation where students are discovering something and the teacher is taking the role of helping them along.

One might say that these approaches are very challenging in the context of traditional lectures. This is a problem with traditional lectures.

  • $\begingroup$ Same comment applies here that I just now gave to "guest". $\endgroup$ Commented Aug 14, 2021 at 16:43

To start with, look up the well-written-about phenomenon of MOOC attrition rates.

Other than that, this is a very long "question" based on extrapolating a personal (sample of one) experience/belief, with little/no basic research by yourself, and asking for a broad discussion ('list all the counter reasons'), rather than a focused question.

Other than that, you emphasize the lecture too much (video or in person) as a part of the learning process. What trains you is working problems, getting tested on them, and asking questions about the parts you don't get. NOT listening to lectures.

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    $\begingroup$ (+1) for "NOT listening to lectures." While important (not just for content, but also for soft-skill things hard to measure such as inspiring students, which I've even seen teachers with poor mathematical background able to do well), it seems to me that many of the stack exchange comments/discussions I've read (not just here, but also in Mathematics Stack Exchange) tend to place the lectures (YouTube videos, etc.) on too high a pedestal. $\endgroup$ Commented Aug 14, 2021 at 16:41

On the teaching side, I could see how remote teaching can eliminate many of the inefficiencies that were present pre-pandemic. For instance, why do we need to give three identical lectures on Calculus I, three times a week, every year (many of which are taught very poorly or unenthusiastically)? I think everybody would benefit if one records a set of lectures taught by a very good teacher, once every few years. (Imagine how much time this would free up for everyone.)

Personally, I think there's a huge difference between "giving a lecture" and "teaching". The first one focuses on the speaker; the second one, on the listener/student. I agree with you that pre-recorded sessions will save hours for the speakers, so they will have more time to research. But, if you want pre-recorded sessions, why don't we just use Youtube? There are thounsand or millions of youtubers doing that for a long time ago, at a really high level, in all languages, and using the best tools to catch their audience. The reason is quite simple, and it's because the main ingredient in the teaching process are the students, and the feedback they provide in class. In a fancy language, the evolution of a lecture is uniquely determined by the initial conditions of the system, which are distinct provided two different group of students.

I have always believed that teaching requires more social skills than academic ones, and it's so weird because we were forced to develop digital skills, and to be constantly using new tools and softwares. So, academically speaking, we are now giving better lectures than before. Nevertheless, this social side is the one we miss the most due to pandemic.

Regarding the research issue, actually I don't see any "good" reason to be forced to do an exclusive research job at one place or another. Maybe, it could be argued some financial issues, such as fraud, taxes or whatever (maybe too much Netflix at this point). However, and to the best of my knowledge, there is no University that will hire you exclusively for research.

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    $\begingroup$ Same comment applies here that I just now gave to "guest". (For some reason, guest's answer was highest when I began reading, but now it's lowest. Anyway, that's why he/she got the full comment.) $\endgroup$ Commented Aug 14, 2021 at 16:44
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    $\begingroup$ There exist research universities where you can talk (in person) to other people doing the same sort of research as yourself. This is a great advantage over the prospect of communicating at a distance. Well, it depends..., if the person you're working with understands you completely, if you are kindred souls of math, then sure, the at a distance isn't too bad. But, what if you're not aligned in spirit, what if there are pernicious motivational gaps in your math ? And, in addition you don't speak the same language ? See, then in person would be hugely advantageous. $\endgroup$ Commented Aug 15, 2021 at 2:57
  • $\begingroup$ @JamesS.Cook I agree with you. Actually, I was thinking in a lonely man doing his research in the woods when I wrote that part of my comment. When feedback becomes an important part of the topic (research/teaching) then face-to-face communication will be better than an online mode. $\endgroup$
    – FormerMath
    Commented Aug 15, 2021 at 4:36
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    $\begingroup$ @FormerMath I have rather mixed emotions on the whole of it. On the one hand, I want to promote online learning for poor remote students who have no other options. However, I have really only had good success with students online who I have previous residentially formed relationships. How to create mathematical culture in an online setting ? How to do it residentially in the age of COVID ? So much of it seems forced to me. The community of peers I found was not a program of the university, it was a natural outgrowth of our shared interest. I don't see how that happens online... $\endgroup$ Commented Aug 16, 2021 at 3:30

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