What is the fairness and usefulness of writing proofs with help?

Question

As an undergraduate student taking upper division math, I naturally have to prove many theorems, identities, etc. However, sometimes I simply cannot work out a valid proof, even after many, many hours of work. So, I end up back here, find a question someone else asked, and suddenly the proof is obvious. Now the problem is that I didn't write the proof, and by some metrics that means I plagiarized my proof, and "didn't answer".

Obviously the ideal course is that I would stubbornly spend as many waking hours as is necessary parsing them through in my head, but sometimes I just don't know enough, or I don't have time to spend 10 hours on it when I have other work to finish, too.

Additionally, what is the "proper" way to get help? I believe that the widespread approach is to search online, but this doesn't help if your proof isn't standard coursework. What is a student to do if they need help with a complicated or obscure proof?

The real question here is:

What is the value of writing proofs yourself? How much do you lose by struggling with it, realizing you can't accomplish it, and getting help? To what extent is it reasonable to do this, and does it indicate that a student is under qualified for that course?

Answer 1

The simple fact that you spend hours working on a proof is useful. It's not a waste of time at all. Even if you don't finally find the proof, this work will make you progress, because it makes you explore ideas and try many different things. And this will pay off later, because whether you are aware of it or not, these ideas will remain in some place of your brain, and you'll be able to use them later, for another (similar) problem. Or at least, you'll learn to know a little better if an idea has a chance to work or not. That's how you develop intuition in math: spend hours trying to find proof and read other's proofs.

By the way, not being able to find all proofs and needing help is perfectly normal, this is part of the learning process in maths.

Answer 2

I believe you should be able to get help at your university in person, then you could also ask for hints instead as well.

I don't think it implies that you are under qualified, keep in mind the entire point of mathematics is to stand on the shoulders of others who took decades to get there, it would be silly to assume you should do all of that yourself. I think if something grips you it's important to keep thinking about it. There is no cheating in this sense if you understand, you might only cheat your own sense of accomplishment which you might need to adapt a bit =)

Answer 3

The idea behind learning to write proofs is precisely to learn how to write proofs. That does not mean that you should be able to find proofs of all the theorems you encounter during course by yourself but you should spend time trying to understand everything in the proof, and when you do, be able to reproduce it. You will often encounter useful ideas, in a sense that you can reuse them in other proofs, or to solve some exercises.

At later point, when you've firmly grasped proof writing, extracting ideas from proofs is all that matters. In more advanced books you will often find proofs that are more sketches than proofs, as reader maturity is expected. Also, to cite my professor: "Not all theorems have proofs that should be learned."

Exercises (of theoretical nature) is another story. You should try to solve them by yourself, and if you struggle, seek for hints before solution. Of course, some exercises will be easy and others hard, and you shouldn't be bothered too much if you can't solve everything by yourself. This is why MSE exists after all, after you put effort but still can't find solution yourself, there are others to help.