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I just finished my first year as a graduate student and I feel like I learned nothing

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It depends on what your goal is. If your plan is to become an academic, the real goal is to publish a lot of papers and get grants. Depending on your exact mathematical specialization, "proof-writing ability" is only loosely coupled to those career incentives.

If your plan is to compete in olympiads, then it's a different kind of goal. Unless these competitions require "showing your work", I think they incentivize remembering a lot of different tricks more than they do being able to present clear and rigorous arguments.

And if you're thinking about bailing out of academia and earning a living in government or industry, again the incentives are different and rarely really require proof.

So is it an XY problem based on unfairly comparing yourself to some ideal that doesn't exist, or are you seriously envisioning a likely future where your current abilities are going to be a major barrier to progress?
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"It's strange because I can sit in lecture and follow most of the professor's proof. I can even follow the proof in the texts with no major problems either. It is only when it comes down to me having to produce a proof that I fall short."

That's a trap.  Following a proof that someone else produces is useful, but it's not the same as producing your own at all.  The key for getting the concepts to come alive in your mind is practice: *Lots of practice.*
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I rarely could perfectly remember everything after I finished a class. When I really learned a topic, that usually happened when I was studying for my qualifiers. I would sit down for three to five ours a day for months, and practice, chunk, and repeat theorems over and over again. I would go through the book forwards and then backwards until I knew everything I could.

Then a year after the exam, I’d forget most of it again, but I can always go back when in need to relearn a topic.

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