I find spoken English (or any other language) remarkably ineffective when it comes to reasoning about more abstract concepts. Many of these concepts are much easier to describe and reason around in some sort of diagram and symbol shorthand. Even when just thinking to myself I often use a lot of internal visual representations of the concepts I'm thinking about.

Also leaving a visual trail of the development of your thoughts make it easier for others (and you) to follow and gives you something to refer back, rather than trying to remember a long chain of complicated arguments.

Personally I like using a whiteboard even when just reasoning out things for myself.