When I read a theorem and read its proof and fully understand it, am I supposed to know the proof even after a long time or is it natural to forget the it?
I ask this question as I'm a self learner who forget many many proofs of the theorems I study even after reading it and understanding it (in many cases I rewrite the proof with full details entirely in my own words but even here I forget it after some time). Forgetting proofs annoys me really and I don't know if this is the natural thing or this is a something I've to deal with .
I also wonder, What should math student do with proofs? What is she expected to do with it? read it understand it and forget it or read it understand it and keep it forever? or just keeping the idea of the proof and reconstruct the details herself when needed?
To be concrete, in the last few weeks, I was studying Boolean algebra, ordinals and Godel first incompleteness theorem, Is it natural that I forget most of the proofs after sometime only keeping the main ideas and the main results from those topics? Is that what math student are supposed to do or am I facing a problem?
I'm going to major in math next September and I really want to know what I'm expected to do with proofs while learning for the coming years.
As a wider question, What are the outcomes math students should gain from studying some mathematical topic?