I think most guys on this board keep score and play it down. I keep score by how I am in relation to par. I know where I am all the time. I only use a card and pencil if I am playing a new course.
My wife, on the other hand, plays 18 different games. Every hole is a new game, and she does not connect them after the round. Previous holes are forgotten. She plots each hole and tries to play it according to her plan. Par is irrelevant. She does not keep a handicap.
I have a friend who keeps score by where he is in relation to five. Par is irrelevant. For instance, one over fives or two under fives. He never writes down scores.
Others I play with write down every hole. Sometimes they know how they are scoring, and sometimes not.
I love it that golf can accommodate many types of scoring.