I've been writing up some of my research recently, and when I go back to it after a month, the examples are the most helpful part; second is psuedo-code. That's so far, anyway.<p>I'd actually gone to a lot of trouble to define input and outputs precisely (in set notation, using LaTeX), and defining explicit constraints/relations between them, and even outlining proofs. This was a lot of work, and the process did increase my clarity of understanding, but it hasn't helped me at all in reading it - in <i>communicating</i> to myself. I guess proofs are primarily about proving, not communicating.
Another problem with proofs is that the best way to conceptualize/visualize an algorithm for the purpose of proving it may be quite different from the best way to grasp it intuitively. Proofs are not primarily operational.<p>Re: defining terms in the article: I think metaphors can act like definitions of terms, and are very helpful for communication. An apt metaphor can imply all kinds of behaviours and interactions that the reader already knows, enabling them to quickly grasp your meaning in a few words. I think it's arguable that our thinking is actually metaphorical, in that we interpret new ideas in terms of existing ideas; even to the point of tools being neurophysical extensions of our arms.
The difficulty with apt metaphors is finding them; and before you can do that, you need a <i>very</i> clear understanding of what you mean. Improving my own understanding is probably the best way to improve my write up of it... though, writing it up definitely also helps improve my understanding, as implementing it also does.<p>---<p>So, an iterative back-and-force process between notes, writing, thinking and implementation is probably the most effective route to clearer understanding and therefore clearer writing.