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At a course at the university, the students have to do homeworks every week which will be graded and discussed in exercise groups.

Is it a good idea to put "official" solutions of the homework on the homepage (or give it to the students in some other way)? Is there some study that this has some advantage?

I personally think that most people will not go through their solution since they have some solution and maybe will never more look at the solution. It would be nice to have a reference for my statement.

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    $\begingroup$ In some courses, a relevant amount of students will only attend the exercise classes to copy the solution from the blackboard (and usually insists on every detail being written down). In such a case, it might give the tutors more time for actual teaching, if the solutions are published in a neat format. $\endgroup$
    – Wrzlprmft
    Commented Mar 14, 2014 at 12:44
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    $\begingroup$ Once when I was TAing for an undergraduate class, I realized that some number of my students were copying down solutions from a solution set (for the same textbook) that an instructor at a different university had placed on their website. I would recommend, instead of placing such solutions in the public domain, giving students access in some more restricted way (online course management system? actual hardcopy? email?) $\endgroup$
    – Aru Ray
    Commented Mar 14, 2014 at 13:12
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    $\begingroup$ Pedagogy aside, there's the argument that publishing your homework solutions helps make sure that you didn't make a mistake. I've seen instances (myself being culpable a few times) where the person assigning the homework "solves" the problem making use of some implicit assumptions (small things like "the variable $n$ is a positive integer"). It is only when the students ask questions about the published solutions that such small mistakes are caught. $\endgroup$
    – Willie Wong
    Commented Mar 14, 2014 at 13:52
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    $\begingroup$ Regarding your final paragraph, a professor I knew in graduate school was a big believer that physical copies of solutions should be handed to students immediately after they turn in their homework, since this is the only time that they would actually be likely to look at them. There's likely much truth to this, at least for most students. $\endgroup$
    – Santiago Canez
    Commented Mar 14, 2014 at 14:29

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I think a great reason to post homework solutions is so that students have a way of reviewing their homework before exams (it would be nice if they were to review homeworks whenever they get their graded work back, but this seems to be a rare practice). While a well-graded problem set should have comments pointing out any mistakes the student may have made, they might not make it clear what a correct solution would look like.

Providing homework solutions also gives me more license to ask somewhat harder questions on exams if they are related to homework questions. It is true that some (perhaps significant) number of students will not actually look at the solutions and will struggle, but this seems to me to be a good way to distinguish between the students who are actively preparing for class/exams, and those who are not.

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    $\begingroup$ Not to mention producing "a well-graded problem set" takes a (sometimes prohibitive) time commitment. $\endgroup$
    – user614
    Commented Apr 3, 2014 at 18:53
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    $\begingroup$ That's a great point @Shay. In addition, if one has TAs/graders, one might have little control over how well the problem sets are being graded. $\endgroup$
    – Aru Ray
    Commented Apr 3, 2014 at 19:46
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Homework (and exam) solutions should be published, at the very least to enable students to request regrading.

Yes, many students never look at the solutions (I even have a stack of homework and exams from last term that they didn't even bother to pick up). But that some don't bother isn't reason enough to deprive those who are interested of being able to look at solutions and try to learn from them.

In the end, to be able to grade you need to have complete solutions, publishing them is not much more work.

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I think it is important to make the solutions available for the homeworks and exams administered in any course, perhaps after you think that the students have had sufficient time to think about the questions. A student who has spent enough time thinking about the question will have somewhere to look for; and those who do not think about will not bother any way but this is not a reason against having some solutions written.

In principle, a good homework will entertain student's creativity. In this case, you have somewhere to make additional remarks about the material covered in the class and seen on the homework. Almost always, you can present introductions to other "areas" of Mathematics/Sciences where these ideas pop up.

To ensure that the homeworks are taken sufficiently seriously, a strategy that one could follow is to have them prove some results whose proof is a slight extension of something they have seen on the homework. This will ensure that students actually spend time thinking about the assigned homework and also develop some familiarity with techniques and ideas they would be expected to know.

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Yes. The idea is not only to give them well thought-out answers and solutions to the problems presented but also (ideally) to show how these were constructed. Enlightening solutions make explicit the intricacies of the problem and how to deal with them.

For maximum effect, the problem set should have been designed possibly with some open-ended questions requiring concept explanations (can you explain why these steps are true?) I remember fondly of a test where one of the questions consisted of a long proof and you had to (i) complete several steps, (ii) explain difficult ones (such as motivating the construction but not doing it). It led to a greater appreciation of the material, understanding of the necessity of the hypotheses and comprehension in general.

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I was a bit surprised to not see anyone taking issue with the "collected and graded" aspect of the "homeworks". Research done by the SFU mathematics department (Gray & Mulholland) showed that the concept of grading 'practice' led to the occurrence of copying solutions. Students thought these homeworks were for the professors and not for students to learn. Therefore they were being completed by Wolfram Alpha, a tutor, online sites, friends etc just to have it done and assigned marks. They actually were not learning from it.

Creating a culture of learning where students are assigned opportunities to develop the concepts of the course, do some intentional practice around these concepts in order to have a complete understanding of these for summative assessments. Provided answers provide feedback to guide students in this development and to indicate readiness or review needs. SFU changed to a series of quizzes based on the assignments that were graded instead of the actual assignments quickly led students to understand that the purpose was to learn.

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While writing and releasing solutions is in general a really good idea, because it helps the students see their errors, one major problem with doing this is that it increases cheating in the future. One has to be careful to balance releasing the solutions with the ability to re-use good problems in future instances of the course.

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  • $\begingroup$ Do you have an idea how to archieve that balance? Only give a part of the solutions away? $\endgroup$
    – Markus Klein
    Commented Mar 15, 2014 at 6:22
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    $\begingroup$ I've tried various things, and nothing really works completely. One thing that does at least help is "re-theming" the question or re-naming terms as was pointed out above. I do my best to "cycle" through questions. In other words, I have 2-3 questions for each topic that are all about equally instructive, and I try to use ones that haven't been used in a while as much as possible. $\endgroup$
    – adamblan
    Commented Mar 15, 2014 at 15:28
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    $\begingroup$ @MarkusKlein, often you can "turn around" a question, i.e., given the result derive the data, or change data enough to make the solution process the same, but the details different. $\endgroup$
    – vonbrand
    Commented Mar 17, 2014 at 16:13
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    $\begingroup$ @MarkusKlein: One thing I do to mitigate this issue is to distribute solutions on paper only, not in any electronic form. Paper solutions are much less likely to be saved long-term and passed on. $\endgroup$
    – Mark Meckes
    Commented Mar 25, 2014 at 8:43
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I like sharing solutions for a few reasons, many already mentioned by others: they are good study aids before exams, students will catch your mistakes/assumptions, students can see what is considered "ideal", etc.

However, I've noticed (through pageview tracking on course management sites) that few students actually look at them! So, I think it behooves us to do something with solutions. Pick one problem per assignment and discuss solutions in class. This can be the most difficult problem, the one with the most valid approaches, the one that tripped up students the most, the one that provides a segue to the next topic, etc. Or, have a followup assignment where students must pick one problem and write about how their solution differs from yours. Or, offer extra credit for students who notice their solutions are significantly different and who write them up and send to you to be included in future versions.

Your mileage will vary depending on your course's difficulty and size and how much class time you're willing to devote to this. Personally, I've started using activities like this in a 15ish student intro to proofs course, including class discussions of actual student solutions (anonymously) that I type up and distribute. They have really enjoyed this so far.

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Usually I give the odd-numbered problems as homework, so the students can check their answers immediately. If I am giving problems that they must think longer about, I would want to eventually provide good answers.

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The two mathematicians below would answer YES! They tout the free promulgation to students of detailed answers and solutions to EVERY exercise and problem in a book.

  1. Robert Ash (1935-2015) touted in the Preface to Real Variables with Basic Metric Space Topology,

I rely especially on one of the most useful of all learning devices: the inclusion of detailed solutions to exercises. Solutions to problems are commonplace in elementary texts but quite rare (although equally valuable) at the upper division undergraduate and graduate level. This feature makes the book suitable for independent study, and further widens the audience.

  1. David Patrick touts on page v, in Introduction to Counting and Probability (2005)

However, if you are using this book on your own to learn independently, then you probably have a copy of the solution book, in which case there are some very important things to keep in mind:

  1. Make sure that you make a serious attempt at the problem before looking at the solution. Don't use the solution book as a crutch to avoid really thinking about a problem first. You should think hard about a problem before deciding to give up and look at the solution.

  2. After you solve a problem, it's usually a good idea to read the solution, even if you think you know how to solve the problem. The solution that's in the solution book might show you a quicker or more concise way to solve the problem, or it might have a completely different solution method that you might not have thought of. [emboldening mine]

  3. If you have to look at the solution in order to solve a problem, make sure that you make a note of that problem. Come back to it in a week or two to make sure that you are able to solve it on your own, without resorting to the solution.

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