“… what is a teacher? You see, mathematicians don’t think of themselves as teachers. That should be clear. They lecture. Sometimes they like to and sometimes they don’t. But that’s not what you [as a teacher] think of. You think of yourself as doing mathematics [while students observe].”
One of Gustin’s participants believed that mathematicians don’t really teach: rather they imitate and demonstrate and encourage others to do the same.
Gustin, W. C. (1985), The development of exceptional research mathematicians, in B. S. Bloom, ed., Developing Talent in Young People, New York, pp. 270–331.