Watch Le Grand Content – a short, well-produced and very entertaining video. I’m not sure what it is meant to be – some kind of graphical poetry? But it strikes me as an excellent portrayal of how conscious thought unfolds when one is trying to think about something. Moments of structure mixed with free associations; occasional patches of sense but the whole thing amounts to an incoherent ramble. Conscious thought is not very good at staying on topic and organising itself into structures which “hang together” in some useful way. This is why we can generally think more effectively when conscious thought is paired with external representations which do sit relatively still and maintain their structure. It is also why Rodin’s classic Thinker
is such a misleading picture of thinking, even though it is almost always the first think people think of if you ask them to picture thinking in their minds. If you could see inside Rodin’s Thinker’s mind, you’d probably be watching something like Le Grand Content.
Here’s a much better picture of thinking:
Tim van Gelder 
Different kinds of thinking? Scribbling a proof on a board, making a case, involves the acts of (at least) remembering, explaining, interacting, and persuading. Sure, it’s “thinking”, but it’s a different flavour of thinking to that used when discovering the proof. That’s not to say that proof discovery involves sitting in a darkened room, head supporting a heavy brain. Scribbling must help, as must talking to people, and having a heap of experience of proving things that are similar to the new proof you’ve just produced.
Good point. Srikanth Iyengar, in the photo, is doing more than “just” thinking, and the diagram is playing a role not just in his thinking about the problem but in his conveying his argument to others. Still, it seems we agree than even more “solipsistic” thinking generally requires, or at least is immensely aided, by external representations. I once had a book by an Eastern European dissident. He claimed to have written the whole thing, word for word, in his head while in prison. At the time this seemed to me implausible, and it still does.