thinking algebraically or geometrically

I was recently talking with an historian who mentioned to me that when Stephen Hawking lost the use of his hands (so could no longer write on a whiteboard), he had to switch from thinking algebraically to thinking geometrically. The historian asked me about this distinction, and I attempted to explain these alternative modes of mathematical thinking. We discovered an analogy that is so cute that I have to write it up: the historian, who specialises in shipping in the Mediterranean, often visualises the scenery, and life on board the ships, but does not end up illustrating his books with his imagined scenes. Visualising the past environment presumably helps to understand the narrative of what was going on at the time, and of course, that corresponds to the geometrical thinking.