I keep a research notebook. The following issues have come up in conversation with other mathematicians.
To scratch or not to scratch: Do you include your scratch work in the notebook? Some people include everything in research notebooks, but I feel this might dilute their usefulness. I don't include most scratch work. It takes up a lot of paper. I include some done when I'm in a location in which I don't have access to real scratch paper (by that I mean paper that's already been printed on or used on one side). I try to include summaries of my calculations done on scratch paper. Sometimes, the details of the calculation are useful.
Paper thickness: Paper must be sturdy enough that ink does not bleed through.
Hardcover or softcover?: I used to prefer a cardboard cover but I've switched to a lighter notebook. The new kind fits better in my lap than the old kind. I don't like a terribly floppy cover, but I've decided I do not need a very rigid cover.
Wire bound or other?: I really hate spiral-bound notebooks, actually -- the wire spiral always gets caught in my bag and pulls out. I know other researchers who insist on spiral-bound.
What's in there, anyway?: Notes. Research notes -- things I am trying to prove, random ideas I had, snippets from talks with other people. Reading notes on things I'd like to know more about. Every now and then a grocery list or other to-do item. Details of some, not all, calculations. QUESTIONS I need to answer.
I refer back to them every now and then: they're useful when talking to others about old mathematical conversations I've written down, when figuring out what I did last summer or last week, when someone asks, "Did you check (blah)?" I store up some ideas for future work as well.
I don't have a very systematic way of going back over them, though, and I wish I did have a way of doing so usefully. I haven't figured out what's useful yet.
Stuck in the South of France
20 hours ago