Paul R. Halmos (1916-2006) had strong views on the communication of mathematics, whether in written form, in the classroom, or in lectures. See, for example, “Paul Halmos on Writing Mathematics.”
Here’s his reminder to lecturers:
“Some lecturers defend complications and technicalities by saying that that’s what their subject is like, and there is nothing they can do about it. I am skeptical, and I am willing to go so far as to say that such statements indicate incomplete understanding of the subject and of its place in mathematics. Every subject, and even every small part of a subject, if it is identifiable, if it is big enough to give an hour talk on, has its simple aspects, and they, the simple aspects, the roots of the subject, the connections with more widely known and older parts of mathematics, are what a non-specialized audience needs to be told.”—Paul R. Halmos, “How to Talk Mathematics,” Notices of the American Mathematical Society, Vol. 21 (1974), pp. 155-158.
Did Halmos follow his own advice? You can sample some of his classroom lectures, recorded in Australia in 1975, in a collection of videos at the Archives of American Mathematics.
Additional material:
“Paul Halmos: In His Own Words” by John Ewing, Notices of the American Mathematical Society, Vol. 54 (2007), pp. 1136-1144.
I Want to Be a Mathematician: A Conversation with Paul Halmos, Zala Films (George Csicsery), 2009. Trailer. Review.