Sometimes, you might miss important ideas by using known and strong theorems (which might be overkill). And it might actually be invalid! For example, using L'Hospitals rule to compute the limit $\lim_{x\to 0} \sin(x)/x$. This is technically perfectly fine, until you realize that you need this very limit in order to compute the derivative of $\sin x$!

I totally understand, but in the example you mentioned, the student would be using circular reasoning so it would actually justify taking a few marks away. However, in my case, the professor agreed after careful scrutinization that my proof was indeed correct. (I just saved around 5 steps). Would you penalize a student for getting the right answer the wrong way? That is my point. I clearly knew the theorem, and I quoted it exactly with its conditions so what is the problem of using it?!

Tests are usually given to check you understand the material you've been taught. If you use something instead of what you've been taught - even if its technically better to do it this way - you might have failed to demonstrate you understand the course material.

@Coconut The question is whether you know the proof of that theorem? Sometimes, a lot of progress is hidden under the hood. As student, I once shortened a proof considerably by writing things down in a different manner which surprised the TA/prof. But it was all self-contained, not using external theorems.

@Landric How would I demonstrate lacking the understanding of a course material if I actually showed the required proof successfully? Just because I didn't do a proof the way the course is designed doesn't mean I don't understand the content of the course, it just means I thought about the proof with all the tools I had and that theorem was one of my tools.

@CaptainEmacs The proof is actually pretty straight forward (direct proof) and yes I knew how to prove it precisely. I just didn't see the point of reinventing the wheel and reproving an already existing theorem.

Does the question forbid using theorems not taught in class? Usually in my math courses there are such restrictions.

I would say this depends a lot on whether you could reasonably be expected to understand a proof of said theorem without having to introduce much extra material. I mean, any introductory group theory exercise of the form "show that a group of order n is not simple" could be solved by writing "this follows from the classification", but that would hardly be considered OK by anyone.

@Coconut And you mentioned that to the prof? If so, then, I am afraid, there is some lack of flexibility in the way this prof expects learning to take place. I wish my students would come up with external theorems for proofs, who also knew the proofs of the external theorem :-)

@justhald No, there was no explicit mention that we couldn't do that. Not in the exam paper, and we were not informed of it anytime before the exam.

@CaptainEmacs I did, to no avail. I explained to him the 2 steps for the prove of the theorem and he kept his stance firmly. Usually I don't like discuss grade matters with professors regardless what I think, but I genuinely couldn't get why he was so firm on his stance even though he can clearly see that I can prove it and had a solid understanding of the theorem consequences!

Were the supplementary materials part of a "recommended reading list" from the professor, or did you pick them independently?

@T.Verron Chosen independently

In case anyone is interested, the professor just emailed me a few minutes ago. I've included the context in the question in case you are wondering.

Sigh... Can we please open a site arguing_on_grades.stackexchange.com separated from academia.se?

"How would someone be able to use a theorem correctly if he doesn't know its proof?" Almost anything a physicist does (barring mathematical physicists).

It is alright as long as the person can explain where the theorm is derived from. For example they should be able to write a proof that proves it, or at least show where it comes from.

@FedericoPoloni Never in my past 3 years of undergraduate studies have I ever argued about grades with any professor, except now. I felt that there was injustice happening, and I felt a need to speak out. Not any students who argues about grades are asking for marks they don't deserve! Professors are far from perfect and they can make unjust mistakes, and when this happens, I have all right to speak up! If the only thing that you got from my post is that I was arguing for grades then you missed the point of the entire post!

@FedericoPoloni "Can we please open a site arguing_on_grades.stackexchange.com separated from academia.se" As long as there is also a "how_to_run_my_classroom" stack. Have seen several questions by instructors not sure how to enforce their own rules >.<

I got marked down in a trig class for using calculus to solve the problem. But it was right..! :-)

Here's a thought: If you push your logic to the extreme, you could do ridiculous things like proving the weak law of large numbers using the strong law of large numbers, which (IMO) would in most situations not only deserve zero credit, it'd deserve negative credit if there was such a thing. So you have to realize there's a balance somewhere. So try to learn where the balance is.

@Coconut I am not putting the blame on you. As NZKshatriya correctly observes, the instructor here has many responsibilities for not being able to make and enforce their own rules. You have all the right to speak up with your professor and to escalate the matter if s/he is unjust. But I am not sure what we can do about this issue by discussing it on the internet.

@FedericoPoloni personally I think this question is very interesting and has philosophical implications that are worth discussing beyond any connection to the petty (in your view) issue of grade disputes - see my answer for additional related thoughts. Also as of now the question already has a good number of votes that is likely increase further. So, I'd say it very much belongs on academia.SE.

We've always had the policy: use any theorem you want, if you can provide a proof of it as a part of your test paper using what had been taught.

In the future, if you want to use a theorem not taught in class, just provide a proof of the theorem. That way, if the theorem has a short enough proof, you get to enjoy reproving it the short way. Alternatively, you can prove an altered form of the theorem specific to what you need. It's quite delightful

@FedericoPoloni Oh man, I would hate to see the meta for that stack exchange...