When doing analysis, I have difficulties to recite the proofs given in the lectures in oral examinations. I am able to do calculations with the theorems that I encountered. (Note that in central europe, there is no separation of calculus and analysis.)
My way of learning focuses on finding out the ideas, to be able to state which other theorems are involved and how they interact at which state of the proof. I am mostly unable to formalize the ideas behind the proof when it becomes technical, even though the outline of the proof itself is understood. I can convince myself why the corollaries are meaningful and how they relate to the preceding theorem.
Example: When proving the inverse function theorem (we did this by using the implicit function theorem), I know how to define the function that transfers the problem to the requirements of the implicit function theorem, and I know how I arrive at all given objects that are used for instance in the picture drawn to visualise the problem (the open sets involved in the assertion of the inverse function theorem).
The question is the following, as I find it particularly hard to get rid of my aversion to study technical details:
What are possible ways to master making the step from talking about the connections to really show the connections? When studying a proposition and its proof, I am always working with a sheet of paper, writing down the things and the core aspects, though I am having a particularly hard time to write down more than the vital aspects. Also, I am looking for good reasons (might be philosophical ones) why it is important to not exclude technical details in the studies and overcome studying for the big picture.
When checking out related questions, many answers suggest to learn the ideas and getting a big picture. Here, I am having troubles converting proof sketches/skeletons to formalised mathematics and becoming rigorous in reasoning.