[thanatropism]

What's best about black/whiteboards for me is precisely the nonlinearity. Numerous times I've seen professors realize near step 10 of the proof that something they said in step 3 isn't quite right; then a sidebar is opened and arrows start to be drawn to piece the puzzle together.

The point of going to classes versus studying alone at home is that you get to watch mathematicians doing mathematics. It's apprenticeship learning, the same as carpentry. Watching the professor reason and make mistakes and second-guess himself is invaluable. The typed class notes where everything is neat and theorems are numbered gives the impression of an immaculate birth that the student can't fathom arriving at. The best maths classes are the ones that teach you how to be a mathematician.

[Bartweiss]

This is a much better expression of what's great about chalkboards than just the order of steps, thank you. It's not so much seeing a problem worked out in sequence as seeing it worked out for real. Even if the professor is working from lecture notes with all the steps, writing them out live engages with the content in a way that pressing "next bullet" doesn't. And as you said, what's missing is the messiness - catching mistakes in the notes, making mistakes on the board and finding them, puzzling over what belonged at a missing step.

Everyone jokes about the frustration of copying a full blackboard worth of notes and then having the professor go "wait no, that wasn't right" (myself included) but it's a valuable experience. That moment of "hm, hold on a second" is where the math is happening, and a chalkboard helps professors and students notice that. (And of course, when something "isn't quite right" on the Powerpoint, it's a tremendous pain to fix...)