While in the library puzzling over some geodesic vector fields, I happened upon the text of a speech Oppenheimer made at Cornell in 1949 concerning teaching and research at liberal universities. He makes a deep observation which I shorten and paraphrase below, using his own words. The complete text can be found in the chapter "Science in Being" of "Oppenheimer, Metropolis, Rota, and Sharp. Uncommon sense. Boston: Birkhaeuser, 1984."
The professor who stands helpless before the unknown is much closer to the student than the teacher who knows all the answers. The experience of the student and of the researcher is to be puzzled. The ability to be humbled by a tough problem is one of the reasons why the most inspiring teachers are those who devote a good deal of time to their own researches. Respect for what other people have done before, and respect for ignorance, make them far more sympathetic as teachers than the professional pedagogues.