For whatever reason, whenever I need to work out a problem on paper quickly, I find myself reaching for a post-it note or index card instead of my notebook. I'm not sure why I prefer the smaller format for scratch work.
I had to work to learn to use my lab notebook for this kind of thing. In school I only recorded lab setup and results. Once I got a "real job" I realized (well it was sternly told to me) that even random calculations and stupid incomplete ideas could be very important to my employer at some point in the future if something related later came out. This included crossing things out in a way that was still legible.
When we'd sign each other's notebook pages nobody ever commented on the half-assed stuff in my notebook and I saw it in others' too.
I got this lecture too, but (unfortunately?) all that really changed was that my post-it notes were stapled into my notebook.
I understand the US Secret Service has analyzed lab notebooks in some major disputes.
(To the best of my knowledge nobody has ever looked at my notebooks after I was done with them).