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I am in my third year of undergraduate math, and now that classes are becoming more proof-based, many of my homework questions are proofs of relatively basic concepts that can be found with a quick Google search.

When I do my homework, I always come up with my own solution first, but then many times I will search stackexchange for other proofs and see how mine compares. Not infrequently I will come across something that's more elegant or just generally better than my proof.

I am always hesitant to replace my answer with the "better" one, since I feel as though the work that I submit should be mine. On the other hand, the point of homework is to learn, and as long as I understand the process of coming up with the "better" answer, perhaps the real goal of the assignment has been fulfilled. Most of the time I try to use as little of the "better" answer as possible and perhaps just incorporate the main idea into my own proof.

I'm curious as to whether there is a consensus on this subject, as I'm sure I'm not the first person to encounter this dilemma.

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    $\begingroup$ You should ask your teachers. Different instructors can have different policies about how much use you can make of solutions from the web. $\endgroup$ – Andreas Blass Oct 14 '16 at 5:13
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    $\begingroup$ Personally, I would be very happy you are checking your work. What you are doing is ultimately what we want our math students to do long-term... research. Of course, credit must be given where credit is due. But, some homework problems (or proofs) are so standardized and common that it is not expected to reference where you learned it. Andreas is correct, this is ultimately a relational question between you and your instructor. $\endgroup$ – James S. Cook Oct 14 '16 at 12:37
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You are paying for your own education through your tuition and your time invested. You have an obligation to yourself to make the most out of your time in school.

There is value both in working through problems alone and in studying others solutions. The former will build problem-solving and tenacity. The latter will often consolidate understanding and show optimal technique.

Many instructors will often insist that solutions be entirely your work. I think this is unrealistic to take literally. You should always be willing to try a problem on your own. But in cases where you become stuck, you can often derive more value by looking up the solution than to wallow in confusion. As long as you make an honest attempt to understand the problem and your shortcomings in trying the problem, you are working in an academically honest fashion.

Consider, if in doubt, that after reading the solution, you leave it sit for a few days. Then, write it out as best you understand it at a later date. If you can reproduce the argument from scratch, then you have made inroads towards learning the material.

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  • $\begingroup$ +1 for pausing a couple of days to investigate how much of the online solution you recall (and understand!) rather than simply copying verbatim as you read it for the first time. $\endgroup$ – kwah Oct 23 '16 at 12:32
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Of course if a course involves game-playing with (conceivably artificial) rules, ask the rule-maker about the rules.

On the other hand, many of us are aware of the existence of the internet, and its useful-though-inevitably-compromised information.

So, to my mind, it is not reasonable to tell people that they "cannot look on the internet". It's analogous to telling people they cannot go look in the library (loooong ago...).

Even long ago, the fact that many "exercises" had become completely standard, and were often proven in textbooks, suggests to me that to try to compel novices to pretend that the "model" solutions do not exist is perverse.

Thus, we come to the genuinely subtle point: since all the exercises are done and known on the internet, what should be studied? Well... I'd claim that for people looking forward to genuine mathematics (as opposed to people fulfilling requirements), the point is internalization. That is, not just "can you walk and chew gum at the same time" _in_principle_, but, ... in practice.

That is, the low friction for (not-necessarily-high-quality) information's travel is very low... and it is (in my opinion) silly to pretend that we/students should pretend things... but/and if/when people have a timed, closed-book exam to explain various things... it is infinitely better to merely be "remembering", than "problem-solving/thinking".

Yes, I do agree that it is non-trivial to convey the issue, much less convince, ... but I do think that the salient point is less simple than in the past, and some variants of the "compulsion to do " are indeed no more than filters for some other path. Which is demeaning both to mathematics and to the person... but, nevermind.

Back to the literal question: it is good to think about things oneself, and it is good to take the trouble to discover that (many) other people have thought about things before, and to learn from their travails, and ... Duh.

My most pointed comment here would be that many of the traditional mythologies of mathematics and other academic versions of things implicitly deny, or disregard, the palpable fact that most thoughts that occur to us are not at all new. The "external funding" or other administrative strictures seem ... to compel us to declare belief in a "Whig" version of history, in which everything else was just a prelude to our own ... special uniqueness. :)

In fact, very many ideas that occur to us have occured to the minds of many people before, especially in the last 230 years or so, as an approximation to "modern times". There is that ol' mythology that seems to tell us that we must/can transcend all the things that long-ago people did. But this is simply unreasonable, since we have scant advantage... apart from the accumulated scholarship, for those who track it.

So, again, ask what your "rules" are, but/and realize that it is best to learn from the hard-won (by "suffering") experiences of others, rather than repeating the same suffering.

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