Seems to be a working memory aid for me.

If I have to manipulate equations mentally, I'll (sort of) explain the equation sub-vocally and assign chunks of it to different fingers/regions of space, and then move my fingers around or reassign to regions, as if I'm "dragging and dropping" (e.g. multiply by a denominator means dragging a finger pointing at the denominator over and up). Even if I'm working on paper, this helps me see one or two steps further ahead than I could do so using internal mental imagery alone. I don't remember explicitly learning this.

I move my fingers (and hands or a prop wand if I'm carrying one) to "write" stuff in the air when I'm doing serious thinking. The way that helps me is that I can keep more thoughts in my head. This doesn't (just) apply to math problems (since I hardly know any math and can't do much calculations in my head). My current hypothesis for why this works is that it couples certain actions to certain ideas and repeating the action makes it easier to recall the idea. If I'm right about that it might be learnable and useful, to a similar extent as mind palaces. By coincidence, I've been thinking about trying to formalize this technique in some way since Saturday.

I have the belief that I solve math, design, and logic problems more rapidly when standing/pacing in front of a whiteboard with a marker in my hand, far out of proportion to any marks I actually make (often no marks), possibly because the physical motions put me in the state of mind I developed during university.

(I don't know if it actually helps; I have not tested it)