In the first few chapters, I noticed the side of the wall next to his bed is filled with sticky notes. Maybe he does write them down hence he doesn't forget after he wakes up, but probably still forgets most of them during the day when he is working(he doesn't seem to be bringing a note and pen or isnt allowed to)
Who knows maybe those aren't notes and are pictures, and I don't think they'll prohibit him to bring a booknote along because they depend on each other's work to keep the house running, so everytime he's behind on a chore, they're behind as well because they need him in other places, that's why they get mad at him when he doesn't finish kr do things on time.

If he has memory issues why doesn't he write things down? I got some memory issues as well, so I write almost everything down and put the notes somewhere I know will always be on my sight or I'll forget where I put the notes as well lol. If I'm moving around then I put the paper in a pocket with a part of the paper a bit outside so that I don't forget that I have notes on me and in which pocket.