I know that some people don't like these kind of questions, but it's really important for me to get this straight.
A statement from a friend of mine, top-notch math student, always makes me uncertain about how 'many' of my thoughts I should write down with pen and paper. He said: "Most of the students I know tend to write everything down. I don't understand that. Although notes can be extremely useful at one time or another, one should always try to build the mathematics inside his head."
Up to this day, I still tend to shift between both extremes: approaching a problem and writing everything down that comes to my mind and write nothing down, but construct everything inside of my head.
The problem with the first extreme is that it might consume a lot of time, the problem with the second extreme is that I often reach boundaries that I cannot overcome without taking at least some notes.
When I write things down, I often work in an algorithmic way. For example, when I tackle an excercise where I need to start with refreshing a lot of definitions, I ask myself (and write down): "What does this mean? What does that mean? Why is this relevant to the problem? How is this object connected with this one? Why am I stuck right know? What do I really want to show here?" Sometimes, I am simply not able to wrap my head around a certain concept or step, but the moment I write a specific question down, the gears inside my head start producing answers, building connections. I even use a specific scheme for it and always try order objects depending on their properties and connections. For example: "What is object X? Okay, so, now, what is object X with property Y then? Okay, and what does this mean in hindsight of the excercise? ..."
Yet, I have found that this is the most effective way for me, but of course I always want to look out for ways to improve myself in terms of my learning technique. So, I always try to follow the advice of my friend, but I'm simply not able to work that much inside of my head, and I don't know why.
Formulating a question inside my head and actually writing it down is a huge difference for me, but as mentioned above, it may consume time that could be rather invested in developing the solution further.
So, I would like to know if there are certain techniques or strategies to actually train this "building mathematics inside ones head", anything that helps me to be able to think more and write less.
Personally,
For studying: after reading, I write down (in my own style) every non-trivial proof and everything that I need to memorize (definitions, statements of theorems, etc.). Moreover, I write down examples/counterexamples accompanied with detailed proofs.
For solving problems: I write nothing down in the process. I either "figure it out" then write down some little details as a sanity check, or keep torturing my brain until I lose interest in the problem or ask someone for help (the second case usually just happens when I'm solving a really tough problem, beyond what is required to understand the relevant material).