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I'm an undergrad math major and I'm taking a fairly advanced math class this semester which has pretty long and difficult problem sets. I've noticed that several students (probably most) use online resources (such as Math SE) for get help with the problems. A lot of the problems are difficult, but standard, results in the area, and so it's not hard finding proofs online. I will say that people don't blindly "copy" these proofs: they just look at them, understand them, set their computer aside, and reproduce the argument in their own words and understanding. I do this often too.

But I always feel a bit weird doing this. Isn't the purpose of homework to be able to come up with arguments on your own? Even though - and pretty much the entire class - perfectly understand the arguments we find online, we probably wouldn't be able to come up with the tricks ourselves. Is this a productive way of doing math? If so, why? If not, what is the alternative approach, considering that pretty much the entire class uses these external resources and probably cannot come up with the more elaborate arguments on its own?

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  • $\begingroup$ Help in solving problems? Yes, Help in doing your homework? Ask the instructor if that is allowed. And (if you do get help from the Internet) you will of course cite your sources, since otherwise it may be considered plagiarism. $\endgroup$
    – GEdgar
    Nov 17, 2019 at 23:33
  • $\begingroup$ @GEdgar Thanks, but my question is less about ethicality and more about whether it is a sensible way to learn math. $\endgroup$
    – gtoques
    Nov 17, 2019 at 23:34
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    $\begingroup$ The purpose of homework is not to come up with argument yourself. Homework is just extra work. It can be to work yourself, to cover content not covered in class, to motivate research. It is perfectly fine when that research takes you to find the answer somewhere, be it online or anywhere. $\endgroup$
    – conditionalMethod
    Nov 17, 2019 at 23:44
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    $\begingroup$ Any instructor using homework for evaluation expecting exclusively individual work simply doesn't understand what is homework and is misusing it. $\endgroup$
    – conditionalMethod
    Nov 17, 2019 at 23:46
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    $\begingroup$ A large part of learning is experience. An idea that you have seen before, you can apply later. There is little value in a homework that almost no one solves because they worked exclusively individually. Many ideas in mathematics have taken years and many people trying many other unsuccessful ones. Acquiring that experience from others, faster, is crucial to be able to go further and solve new problems with them or to adapt those ideas to new situations. $\endgroup$
    – conditionalMethod
    Nov 17, 2019 at 23:55

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There's a balance to find, and it depends on the context. So I have a few levels of answer to your question.

$(1):$ From the instructor's perspective.

If you are seeking help on homework to turn in, then it is entirely up to the instructor to decide what is appropriate. I have had professors who explicitly state that they consider looking on stack exchange to be cheating; others are totally fine with collaboration as long as you type your own solutions. I also had a professor one time who would allow collaborating on regular homework, but also do "special assignments" where you were required to work alone.

$(2):$ From the student's perspective.

The next question is whether seeing others' solutions will actually benefit you, the learner. This is where you discretion and intuition come in. Personally, I always attempt problems by myself, and might look online if I am stuck or have no clue how to begin. Some people vehemently believe that looking at solutions ruins the learning experience, but I disagree. Particularly when entering a new field, the proof methods are foreign, and not necessarily worth re-discovering on your own.

$(3):$ Other thoughts.

  • It is easy, given the opportunity, to become overly reliant on others' solutions. This is obviously no way to learn, and it might be considered cheating, which is really something you do not want to mess with.

  • To reiterate, it is very important to always attempt problems before seeking help.

  • You will develop must better sense of how much of a solution came from yourself versus something you found online. Personally, I have a sense of whether I used someone else's argument/proof as a resource, or whether I really just copied their work.

  • If you use an online source, and you feel weird about it, email the professor and ask if it is ok. That might sound weird, but I actually have done this, and your professor will might even appreciate your honesty / concern.

  • An example of using stack exchange I think is appropriate: you are trying to prove that a series converges, and none of the methods you try are working. After an hour of this, you google the question and see that someone suggests using the Weierstrass $M$-test. With this hint, you return to the problem and it becomes straight-forward to complete.

  • An example of using stack exchange I think is not appropriate: you find a similar problem with an accepted answer. You outline the proof based on the answer, and type up a solution that you didn't put any work into.

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Doing homework is a learning process and it could be done through text books, library books or websites.

Students' job is learning and teacher's job is providing guidance and help to ease the learning process.

If the instructor allows using the internet on homework then so be it.

I would not give more than $20%$% weight to homework any way and we can assign homework problems that require research so students have to use resources available to them.

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