Of equal importance to solving a problem is the communication of that
solution to others. The best way to improve your writing skills is to
practice, practice, practice.
Seize every opportunity to write mini-essays about the theoretical work
you are doing. Compose a blog for your friends, or even for yourself.
When you write programs, write literate programs.
One of the best strategies to follow while doing PhD research is to
prepare weekly reports of exactly what you are doing. What questions
did you pursue that week? What positive answers did you get? What
negative answers did you get? What are the major stumbling blocks
that seem to be present at the moment? What related work are you reading?
I discovered the great value of such weekly essays by accident in 1972,
when I spent a sabbatical year abroad. During that time I communicated
with my students by mail, instead of meeting them face to face in my
office; and I found that I was a much better advisor at a distance!
With weekly meetings in person, my students and I would mainly stare at
the blackboard and try to set goals and make conjectures. But the written
communications, when I was away, forced clarity of thinking.
After you’ve accumulated dozens of those weekly reports, you will
naturally have gained invaluable experience in expository writing.
But you’ll also have prima facie evidence of the difficulty of the
problem that you’ve been working on. Ultimately this will give you
confidence that your thesis has genuinely made a significant
contribution to the world’s knowledge.
My book Mathematical Writing is full of tips about the nuts and bolts of
expository techniques. Among the top takeaways is the necessity to
know your reader: Have a good mental model of what the reader understands
so far and of what he or she is ready to learn next. Another is the
technique of saying everything at least twice — once formally and
once informally — in order to give the reader complementary ways
to understand the concepts. Give examples of every definition that
you formulate; and exercise the reader’s brain on each new definition
by making small inferences, by which he or she learns to apply the
The best exposition can often be characterized as “telling a story.”
In fact, story-telling has for centuries been an ideal way for human
beings to share information, and to remember it afterwards.
Don’t try to impress your reader. Much better is to tell the reader
when impressed you, as you were exploring the topics.