Firstly I'll mention the difference between proofs using natural deduction systems or formal proofs and "standard" mathematical proofs.
The former is an actual deduction system, you're proving statements in a formal way by using certain inference rules.
The latter is completly different! A more correct term for the latter would be plausibility argument. It's just an informal reasoning to get from the hypothesys to your goal, that's what people usually mean by proof (though an actual proof must be formal).
Secondly, as I have little information about what exactly you're having trouble with, I can only guess what the problem is, but here's my guess anyway: you're having a hard time following standard mathematical proofs because your only training is in formal proofs, you're too used to the clarity that comes with them. Also you probably are really unused to what I named above as plausibility arguments or informal proofs. From here on when I mention the term proof I mean a proof in the latter sense.
Thirdly, I'll address how to solve this. I'm assuming that what you mean by an algebraic proof is actually an arithmetical proof, i.e., proofs in which you're basically doing some calculations.
Mathematics is built on blocks. Some blocks are tougher than others, but all of them can be divided into atoms. In an atomic state everything is simple (or at least as simple as it can get). If you're having trouble with a proof, break the block into smaller blocks, if you still can't handle it, break the smaller blocks into even smaller blocks and keep doing that until you understand the proof.
This is easier said than done, but nothing is more important than you trying it yourself, I believe. Also if you can't understand a proof, there's a good chance you can't identify which block to break or how to break it. To handle this issue I suggest that every time you infer something (or something is infered in the proof), you ask yourself "why is this true?", when you can't answer that question, you've found a block you want to break. Obviously it also helps that you have someone you can ask what you don't understand and break blocks with you.
I also suggest that you try going through some simpler proofs. For instance like the ones one would find in an elementary number theory course. It's a nice bridge between arithmetical proofs and the other proofs, while still mantaining a lot of logical simplicity. Finally, don't think for a minute that my next (and last) suggestion is any less worthy than the previous. If I was forced to give you an answer in ten words or less, this would be it: How to Prove It: A Structured Approach, by D.J. Velleman. Go through it.
I hope I understood your issues correctly and wish you luck.