When a student is first introduced to proofs in a math course, they are always presented in as cryptic a way as possible and dressed up in all the fancy math terms the textbook writers can think of. Thankfully, someone with a compassionate heart created this course and so that us poor students get a real introduction to such a fundamental can actually digest proofs and be able to write our own.
In my experience, students never understand a proof fully if it is just given to them without all the details explained. The "eureka" moment when you get to that connects the antecedent to the consequent is usually glossed over in many proofs, and that leaves me confused when I suddenly see something surprising, but the usage of it does indeed affirm the implication. It's always a good idea to explain why something is used in a proof because not everyone is smart enough to understand that extraordinary algebra in a rearrangement, or why and how the function with a and b in it is necessary in showing a certain relationship relationship. And it goes without saying that the lectures in class also tell us where we should elaborate in the proofs that we construct in the future.
Now that I am taking this course along with second year calculus, I find that I can really dig into the proofs and understand the thinking process of the person who came up with them. I can fully appreciate just how ingenious some of them are, like the proof for the Bolzano-Weierstrass theorem is a bit long written out but I can see the basic idea is sort of like searching for an element in a list and that's the point in the set that the sequence is approaching...sort of. Well, before I couldn't even find analogies to help me understand proofs so I would say that this course is definitely putting me on the right path.
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