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I am teaching a 3rd year undergraduate algebra course, and I have told my students that I would not be giving out solutions to the weekly exercises that I set. Instead,

  • one question per week will be graded and detailed written feedback will be given by way of continuous assessment, and
  • there are weekly tutorials where they can ask questions if they struggled to solve something or if they are unsure if their solution is correct.

Many students are extremely unhappy about it, so I would like to get 2nd/3rd/etc opinions from other experienced educators here. I will provide my reasoning below, but first the question:

Do you give out model solutions to your exercises, say one or two weeks after they were set? This is primarily aimed at educators in science subjects, where there is a correct solution to each exercise.


My reason for not giving out solutions

My main purpose in teaching a mathematics course is to teach students to solve problems; to be stuck and to persevere; to seek creative approaches. I am pretty sure that if the model solution is one click away, or even if they just know that it will arrive in a few days' time, they will, on average, spend less time on the exercises, and some of them will just give up when they cannot solve something within 10 minutes. One student explicitly told me that they like to use model solutions to "work backwards" to complete their understanding of the course material. This is simply not the intended use of the exercises.

A little more background

  • I was student at Oxford UK, I taught at Cambridge UK, Warwick UK, and Postech Korea. At none of these institutions did students expect to be handed out model solutions. Now I am at Glasgow, where students' expectations are wildly different. However, due to a research grant I have not taught for a few years, and I do not know how much of this difference is not just due to geographical variation, but also to a temporal gap. I can certainly see infantilisation and bureaucratisation of university education on a wide spectrum of issues, I just don't know whether this is one of them, so one answer could be "wake up, you are stuck in 2015 with your ideas about university mathematics education; these days we are all expected to give out model solutions".

  • I did check what the School policy is on solutions to exercises. There is no need to go into details, but suffice it to say that both decisions, to give out full solutions and not to give out almost any, would be compatible with the official policy.

Frequent arguments for giving out solutions and my response to them

  • These are responsible adults, don't treat them like kids. They know that they are supposed to first attempt the exercises themselves. The solutions are there for when they get truly stuck or to check the correctness of their solution at the end.

Contrary to popular opinion and superficial appearance, this is not really an argument, but a rhetorical device dressed up as an irrefutable argument. The fact that they are of legal age is irrelevant here. Firstly, they simply have little experience at independent learning. We do not say about a patient "They are an adult, they can choose their therapy themselves", but leave that choice to experts; the age or legal status of the patient is irrelevant, only their experience in that particular domain is. Secondly, even adults can have a hard time overcoming temptation. I am sure I do not need to elaborate on this last point.

  • Everybody studies differently. It is unfair to impose your personal choices on others.

Actually, this is precisely what a pedagogue is paid to do: to impose certain choices on their students. We do that through selection of the material, of the order in which to present it, of the exercises that we set and don't set to our students, and yes, through the mode of delivery and the resources that we make available or choose not to make available. Of course what distinguishes a good pedagogue from a bad one is how good those choices are, hence this question.

  • How can the students know if they have solved an exercise correctly?

I have to confess that I underestimated this one. I always thought that in mathematics one knows when one has proven something, but many students obviously don't. However, that is what the tutorials are for. It might be relevant to add here that the tutorials are happening via zoom, and the engagement, so far, has been pretty lacklustre. Many students don't switch on their mic or camera, and about 1/3 of them show no signs of life through the entire tutorial. Certainly, the percentage of students that say "I would like to see how this question is done" is much lower than of those complaining about the lack of model solutions.

Anyway, I could say more on this, but I would like to hear your experiences and thoughts!

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    Not turning on camera/mic is not necessarily a lack of engagement. There are excellent reasons never to turn on the camera such as poor broadband or working from a space that you don't want others to see. For the mic, it's simple courtesy to have it off unless you have a question/comment.
    – JenB
    Oct 6 '20 at 15:16
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    When you don't provide model solutions, most students will simply ask the solutions to the best of them. Moreover, frequently, along the years, students produce "unofficial" solutions manual to most of the given exercises, especially if they are recurring. So you may not end up with what you want to achieve anyway.
    – Massimo Ortolano
    Oct 6 '20 at 15:46
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    @JenB: the kind of lack of engagement that I am talking about is: if I ask whether the students have done exercise xyz, and ask them to press a button in the zoom call to signify "yes" or "no", then about 1/3 of them remain in a quantum state of having neither done it nor not done. For that 1/3, I cannot tell whether they are even in front of the computer during the tutorial. The tutorials are not in large groups, there are about 20 students per tutor in one breakout room.
    – Alex B.
    Oct 6 '20 at 16:12
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    [deleted a couple of comments which were based on misremembering things] As another data point: when I started my current job in the UK (Lancaster, 2014) model solutions seemed to be expected, for both the practice questions and the assessed ones. I have a vague memory from my TA work as a PhD student (Newcastle-upon-Tyne, 2002-06) that students did expect to see some form of solutions but whether these were full model solutions, or outline solutions provided at the lecturer's discretion, I can no longer recall with clarity.
    – Yemon Choi
    Oct 6 '20 at 21:08
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    @CaptainEmacs In my experience, if the instructor then reuses the exercises through the years, the students of the first few years will solve all the exercises which will be then handed to those of the future years. When I was a student, one could find in various copy shops booklets with the solutions to the exercises provided by students, when the professors wouldn't. It was also a way for the students providing the solutions to earn some money. You could also find collections of all the questions asked in certain oral exams etc.
    – Massimo Ortolano
    Oct 18 '20 at 12:24
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If you don't provide model solutions, it is fairly likely that one of the more advanced students will end up providing their answers to the other students. It doesn't count toward the grade, so it wouldn't be helping someone to cheat. And most of these students will be friends from being in the same courses many times. So a different way to frame it would be, would you prefer: (1) your solutions - that you know are correct and you can highlight the key conceptual steps or (2) whatever the student writes.

Can you provide a skeleton of the steps? This may also help with the tutorial engagement because you could ask the group which steps they were able to complete, for example.

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    Also sharing solutions would more equitable your students who aren't friends with one of the students sharing their answers
    – GageMartin
    Oct 8 '20 at 1:38
  • I'd strongly second this answer. Further, "the internet" may provide dubious solutions, as well, so the question is whether the instructor would want to have control over what the students perceive as "good solutions". (I always make full solutions...) May 6 at 15:19
  • I disagree with this futility argument. There is a strong difference between an officially endorsed model solution and solutions that students share with each other. Officially endorsed solutions have a much stronger nudging effect on learning behaviors, and in this case, they might encourage cramming (particularly for the weaker students).
    – Holger
    Jul 26 at 16:45
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How can the students know if they have solved an exercise correctly?

Eventually your students are going to leave university and apply what they have learned in your class in their new jobs. When that happens, there will be no solution manual. Better to learn now how to convince themselves that the solution is correct. They are being trained to become the experts, to become the ones that write the solution manual.

Now, your course should give them the tools they need to convince themselves that the solution is correct. You should also clearly communicate that practicing these tools and learning how to deal with uncertainty is an important learning goal of these exercises.

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    The point of a education is to learn. But we shouldn't confuse learning with being able to do it as soon as they start. By definition is this is what they've learnt when they finish, they will not be capable of it when they start. Oct 6 '20 at 11:36
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    You are right that you need to give them the tools first. But you can't give the students the tools and throw them in the real world. They have to practice it first. I would say that practicing this skill, solving problems without knowing the solution, is much more important then anything else they will learn in university. Oct 6 '20 at 11:41
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    sure, so you'd expect the students to be comfortable doing these things by the end of their degree, not at the beginning, and their confidence to grow gradually over the time they were studying. Oct 6 '20 at 12:01
  • @IanSudbery Yes, and to achieve that goal, I expect them to start practicing those skills from the very beginning, so at the end they are comfortable and capable. You need to communicate clearly what your expectations are, and why you do this. Oct 6 '20 at 13:16
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How can the students know if they have solved an exercise correctly?

I have to confess that I underestimated this one. I always thought that in mathematics one knows when one has proven something, but many students obviously don't. However, that is what the tutorials are for. It might be relevant to add here that the tutorials are happening via zoom, and the engagement, so far, has been pretty lacklustre. Many students don't switch on their mic or camera, and about 1/3 of them show no signs of life through the entire tutorial. Certainly, the percentage of students that say "I would like to see how this question is done" is much lower than of those complaining about the lack of model solutions.

I am currently a Master student in mathematics and have a slightly different take on it. I do agree that students have to learn how to deal with a scenario where there is no solution given to you, especially if they want to go into academia. However not everyone wants to do so. A lot of them will end up in industry in insurances, banks, whatsoever. These approaches are not as necessary there.

Further, it depends strongly on how advanced they are. I remember that it took me quite a long time to get a good intuition on whether my argumentation/proof is right or whether it lacks precision. This can be learned far more efficiently if you have model solutions at hand. If this is the case also the tutorials will be only of little help because students barely know where their problem is.

Last but not least, a lot of teachers expect their students to spend a lot of time on trying to solve exercises. The students on the other hand have several subjects and therefore only limited time and energy resources which sometimes cannot be spend this way - at least not by everyone. Unfortunately not everyone is on the level of Oxford students, nevertheless one should have the opportunity to learn something. Just imagine, some people have to work besides going to university in order to finance the latter. Their life gets much harder.

You said you consider yourself a pedagogue. However one could also consider you a service provider - this identity heavily depends on the question whether students pay fees for the university. If they pay a lot of money, I think they are right to expect a certain service, no matter whether you think this is pedagogically irresponsible.

After all, why don't you find a compromise? Give out model solutions for some basic tasks, and let some advanced exercises open. Or provide the model solutions only every second week. I think a black/white solution is certainly not the best and a good compromise could be the best way for all interests.

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    This is not necessarily a skill just for academia. If you work in insurance and you don't know how to check your own calculations, you are going to loose your company a couple of million dollars/euros or more. If one of the students ends up involved in the vaccine for COVID-30, (s)he could hurt or kill a couple of million people with a single mistake... So I would say this skill is even more important outside academia. Oct 6 '20 at 13:21
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    The argument that students need solutions because they have no time to solve all the exercises is also one that comes up a lot, and I should have pre-empted it in my question. To me this is like saying to your driving instructor who tells you to go from A to B "I will try, but if I don't manage, could you just drive us to B?". The only purpose of my exercises is to get students to practice solving exercises. Them knowing the answer is not even of secondary importance, it is of no importance whatsoever.
    – Alex B.
    Oct 6 '20 at 13:35
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    @AlexB.Do you mind telling us the textbook you are using? Lower than D&F, D&F? Harder than D&F like Lang? Or your own lecture notes? I have the feeling that Max and you are talking about different thing. They are talking about Calc level, you are talking about Abstract algebra? Am I totally wrong?
    – scaaahu
    Oct 6 '20 at 14:10
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    @scaaahu: The recommended text book is Fraleigh, which is comparable to D&F. Max: the alternative to doing the exercises now is not to do nothing, but for example to do them when they revise before the exam. Also, if we are honest, for almost everybody the limiting factor in how much work they put in is not the physical amount of time they have but their perseverance and capacity for hard work, and I am treating this as a variable, not an immutable quantity. My claim is that if solutions are provided, then students will spend less time on the exercises.
    – Alex B.
    Oct 6 '20 at 16:09
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    @MaartenBuis: Your argument with COVID-30 seems quite absurd. An equally absurd argument: What if the student makes one mistake in the COVID-30 vaccine because they did not see a model solution? Maybe they would than know that the learn the right way to do their calculation instead of doing the wrong thing which they do because they don't know anything else. (Anyways, I think both our arguments show that we don't have the slightest clue of how vaccine development works.)
    – user111388
    Oct 7 '20 at 16:15
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From what I understand, you

(1) don't write model solutions, and

(2) only grade a small part of their homework.

In my book, either of (1) and (2) is well defensible, but (1)+(2) together hurt your pedagogy. Teaching (particularly at the undergraduate level) is not just about conveying concepts but also about destroying misconceptions. If your students are getting something wrong, how are they going to realize it? Normally, this is done either by them getting their homework back graded, or by them double-checking it against the model solutions (of course, they may be too lazy for that -- but that's their own problem). If both of these feedback channels are reduced to a minimum, misconceptions will grow and fester. If time constraints are making this feedback impossible, there is a third option: give students access to a pool of "training" exercises with solutions available. (The internet nowadays isn't bad at this.) This should work if you can reasonably expect the possible misconceptions to be resolved by those training exercises; still it will hardly beat the personalized feedback of actual grading.

In my experience teaching higher-level undergraduate mathematics classes, misconceptions are commonplace. Wrong ideas about what an induction proof is tend to stick around until one gets into advanced territory. Commutativity is used (through muscle memory) far beyond its legitimate domain. Polynomials and polynomial functions are merrily lumped together until one is set straight by absurd conclusions in finite field theory. "It's all proofs, so you should be able to check it yourself" doesn't work in practice when the students' familiarity with proofs goes back 1-2 years only (no one learns proofs in school any more) and when undergrad degrees have become grab bags of random classes chosen by accident or bureaucratic requirements.

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Perhaps what you have discovered is that different institutions, and different professors within them have different practices around this. I was an undergraduate more than 50 years ago and some professors at the time posted (in a locked display cabinet outside their office) the solutions to the latest assignments. This made it harder for them to propagate to future classes, of course, but some fraternities would copy them down and file them away for future use by members.

However, this question relies mostly on opinion, I think. My own opinion, which doesn't scale very well, is to give minimal hints on assignments to those who request them after they explain to me their thinking. This is fine in a class of 30, but not so much in a class of 300.

But the idea is that I want to focus on learning, not grading. So, sending a student back to the "drawing board" on an assignment is a good thing. I may need to re-steer them a bit, but when they come to the office (actual or virtual) with a question, I sometimes need to dispel them of some misconception that is blocking their understanding and progress. Posting answers on my door (actual or virtual) may give some students insight, but it is much less certain to do so, especially for those who need a bit of guidance.

So, my preference is not to publish solutions, especially for meaningful questions, but might have to do something like that if the scale was impossible.

But, you might also consider an intermediate case, if you are clever enough to figure out how to do it. You might publish, instead of the solution, a set of "hints" or "things to think about when doing this exercise".

Like many other things, mathematics is learned through practice and feedback. The practice leads to insight (we hope) and the feedback helps suppress misconceptions. But the feedback needs to be individualized to accomplish this.

It is one reason that (in computer science, though), I let students resubmit work after getting personal feedback on earlier attempts. They could "earn back" a portion of the points lost earlier, but not achieve full marks except on the first version. This re-doing of work was, again, an attempt to guide the student to "reinforcement" of good ideas based on the feedback. Alas, it doesn't scale very well and too many institutions are forcing impossible scale on courses.

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    Concerning re-submitting work: I've actually started using a system where students have to submit an initial attempt, and then have to correct their own work using the solutions I provide. I then grade the corrected versions; some portion of the grade is based on the initial attempt, and some on completing the corrections. This scales much more nicely on my end, since it's not much more work to grade a "corrected" version of a student's work than it is to correct to correct their original work. Oct 6 '20 at 20:47
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You need to give solutions to some (a few) of the questions. You can not expect students to rediscover in a semester every technique which took (the greatest minds) many years to discover. I like to tackle the problems and learn how to solve as much as I can by myself. Fortunately I often can and I know I got it right. But sometimes I get stuck, and I can not spend a whole month with a single problem.

(image from the wonderful SMBC: https://www.smbc-comics.com/?id=3947)

"How math works" (https://www.smbc-comics.com/?id=3947)

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This is from the perspective of a graduate student:

I understand that you want students to persevere, resilience is a valuable character trait that takes time and a bit of pain to develop. However, by not giving students solutions to problems, you might do your objective a disservice because students will be discouraged. But also, providing all of the solutions creates an incentives for students to convince themselves that they now understand what they're doing. One must remember that policies with the best intentions do not often produce the best results. So I think what you need is a compromise, and this is what I suggest.

Instead of giving out full and complete solutions for every problem, give your students incomplete solutions that allow them to fill in the steps. This way, you are guiding your students through the solutions and helping them to develop the intuition that they need to tackle problems on their own. You could, for example, provide a prompting question to engage the student in the solution, such as, "What is the definition of {some concept}?" and then without going through the motions, provide the solution to that particular step.

The students can then still struggle to make that connection themselves, but you are being far more constructive in guiding their thinking than just offering the solutions or throwing them to the sharks. Perhaps you give one complete solution, 3 semi-complete solutions, and then for a problem you think is the most important, you give them no solution (maybe give it to them a bit later on, your call) so that they can tackle it fresh after going through the exercise of getting themselves in the right mode of thought.

The incomplete solutions takes work on your part, but I think it's a fair compromise for your students and the objectives you have in the success of students.

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  • Agree. Hints, or leading questions, is a good way for the "semi-complete" questions. I recently did topology (also quite theoretical, much like algebra), some qood questions where on the form: 1) define concept X. (which is mostly a simple copy from the text book) 2) prove Y (which uses X) 3) define Z 4) Prove W (which uses Z, and maybe Y, maybe X) and so on. Nov 10 '20 at 18:51

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