Do mathematicians write on paper all the time?

This is an older question but I'll answer yes for me.

I come at it from the approach that most of my teachers and professors were so anti computers or just not good with computers, the math ones that is. So when I was in class, they didn't want me coding. I could be on my calculator but they would yell at me. After years of arguing with them, I asked myself what I could do instead and the thought came to me to take notes (what a revolutionary idea). So at the time everything the teacher said or wrote went into my notes. They were very detailed notes. So good that other students wanted to borrow them thermal regularly.

Then I started reading math books. I don't mean really reading cover to cover. But I'd look for the highlights, the lemmas, theorem's, corollaries and the main things, the proofs. I'd copy those into a separate set of notes to make sure I understood them.

At that same time I was doing things like flash cards for other classes but I found that while I was doing those same things for math classes, the very act of writing these proofs and lemmas out was so helpful to me. Not reading it but writing it and seeing each step slowly and being able to sit and think about it for a second before moving on.

Since then I've learned a lot more things, multiple programming languages, gotten my PhD, machine learning, some religious studies and now some chess stuff, and I'm still doing this same method. It's really helps me to focus.