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I am taking a math course this semester and the problems that are given to us on a weekly basis require less than half a page to solve. The path to the proof is relatively straightforward, using the things we learned in class. However, my professor is accusing me of cheating because my proofs look very similar to online solutions. I don't know what to do because the proofs do look similar, even if they aren't word for word, but I wouldn't know any other way of solving them using the information we learned in the course.

Because the problems were relatively simple, there doesn't seem to be many places where I could "branch" off and do something completely different from what can be found online. What do you think is the best course of action in the face of this allegation?

thewayofkings
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I'll assume here that you didn't cheat, nor go outside the bounds of what is acceptable in your course.

The path, hopefully successful, is just to insist that you didn't cheat and keep on insisting on it. Don't try to "prove" that you didn't cheat. That isn't your job and will probably lead nowhere, or even to some "gotcha moment". If you didn't cheat, say that and keep saying it.

If your work process was that you solved the problem on your own using course materials and your notes, then say that.

No one can, of course, guarantee that this or anything will be successful. If offered a follow up, possibly an oral question, consider taking it.

But accusations need to be proved, not assumed.

Good luck. The world isn't necessarily fair.

Buffy
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What do you think is the best course of action in the face of this allegation?

Explain to the professor that you found the proof relatively straightforward. Perhaps mention that proving the result using taught methods leaves little room for variation, which is why your proof is similar to others.

Offer to explain the proof.

A student that can explain a proof either wrote it or is good enough to write it.

Ultimately, the burden of proof is on the professor: They must demonstrate that you cheated. You cannot definitively prove you didn't.

user2768
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because my proofs look very similar to online solutions. If the word "proof" were in the singular form, I would believe you. Otherwise I side with your professor: the probability of multiple similarities is rather small here (it is not zero, of course, but we allow new drugs to be used on people with confidence levels of 1% or so, so I would also accuse someone of cheating if the probability of random coincidence is below a certain threshold).

However, I would certainly give anybody a chance to disprove my accusations as soon as they arise. What I would do is to bring the student to my office (online mode doesn't work, sorry, so that has to wait until the next semester), give him or her two problems, one of which has a readily available solution online and the other one has not, and see what happens. If the student solves both and the student "online problem" solution is, indeed, similar to the web one, I would profusely apologize and revert all accusations and reduced points. Another way that would make me to remove all accusations is to give a student a couple of really difficult questions in the same setting. If he or she manages to solve them, I would remove all accusations too, but this time not because I am convinced that the student doesn't copy his/her solutions from the web occasionally but merely because I no longer care is he or she does. In all other cases I would stick to my original opinion. You may try to suggest something like that to your professor or ask him to invent his own verification test. Just stubbornly insisting on your innocence would prove nothing to me: the people who are best at that are usually the hardest cheaters (alas, I agree with Buffy here: the world is not fair).

fedja
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    the probability of multiple similarities is rather small here: In propositional logic (and many other settings) the probability of identical (not just similar) proofs is high, for short proofs. – user2768 Feb 24 '21 at 06:48
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    @user2768 Of course, in propositional logic, the individual probability may be 0.8. But the probability of a streak of thirty is then 0.001. One has to have sufficiently many coincidences to accuse, of course, and "sufficiently" depends on the subject. Besides, there is a question of writing style, etc. I do not disagree with you in principle, but, if the whole class submits not web-similar solution and one person submits them, then he is either a genius (so he sees the optimal route every time) or a cheater. And cheaters/genius ratio in the population is very high. – fedja Feb 24 '21 at 07:00
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    I disagree with your probabilities. Especially given that normalised solutions may exist. Nonetheless, I agree with you in principle, I'm just noting that identical proofs are reasonably likely in some scenarios (more so for class assignments). – user2768 Feb 24 '21 at 07:11
  • @user2768 Particular numbers were just for "illustrative purpose", of course. But, as I said, I also agree with you in principle. However, in my experience, even in low level algebra classes, the solutions submitted by the students differ enough from each other (not in the general way of the proof chosen, but in many small peculiar details). One has just to use one's own common sense here as to where exactly to draw the line. And proving cheating is harder than proving non-cheating. I suggested 2 ways that would convince me. How would you prove that a student cheated beyond any doubt? – fedja Feb 24 '21 at 07:26
  • How would you prove that a student cheated beyond any doubt? Ask them to explain their proof: Failure/faltering is a sign of cheating (or nervousness), digging deeper gets closer to the truth (and away from nerves), and outright asking gets admissions. – user2768 Feb 24 '21 at 07:48
  • @user2768: Many of my professors said "I don't care where you got the homework from as long as you can explain the proof well". And when somebody copied the homework, of course they spent time to understand it well, so they could explain it (we did not give our solutions to the prof, but had to present them) I would believe that (almost) every cheater does the same thing: understanding the homework well so nobody can accuse them of cheating, no? – user111388 Feb 24 '21 at 11:17
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    @user111388 I highly doubt every cheater [makes the effort of] understanding the homework well so nobody can accuse them of cheating: That's more effort than doing the homework. (Of course, fields vary.) – user2768 Feb 24 '21 at 11:41
  • @user2768: Okay, thank you! (I personally would, at least when the prof says "I think you cheated, when do you have time to discuss?") try to understand as good as possible - but I guess some.cheaters are way too stupid..) – user111388 Feb 24 '21 at 13:33
  • @user111388 Saying "I think you cheated, when do you have time to discuss?," gives too much away. Just arrange a meeting. I guess some cheaters are way too stupid: People cheat for a variety of reasons. I suspect a common reason is that they cannot do the work. (Studies have probably been conducted.) – user2768 Feb 24 '21 at 13:57