By breaking them up, do you mean dividing the clump? You can if you're careful and make sure each sprout/division has it's own tuber for a food source. I recently bought a package of 2 clumps with multiple short sprouts. Just for fun I wanted to see how many divisions I could get and ended up with 10, though a few were very small. I divide my clumps when I dig them in the fall. Since they are individual tubers. I seldom end up with multiple sprouts.
I've never done cuttings so I can't offer any help with that. There are lots of YouTube videos about taking cuttings.
