Research, scholarship, and teaching are three facets of a ``gem'' professor. Each facet supports the others. Original, frontier research informs the whole process, through something intangible (but essential) that arises from direct connection to the hitherto unknown. Scholarship affects the research effort by giving a context in which the value of a particular direction may be judged. Scholarship affects the teaching (through the curriculum) with that same context.
When we do research, we teach ourselves (and the world) something new. When we work on our scholarship, we teach ourselves something new to us, and integrate it into our worldview. When we teach, we help others to learn what we have learned.
Put that way, it sounds as though teaching is the simplest of the three facets. This is not so, principally because different people learn in different ways. In fact, teaching others is an excellent tool for building your own scholarship, because it forces you to consolidate and contextualize your knowledge and to look at material from someone else's point of view. Finally, teaching others is also an excellent tool for helping your own research, not least because teaching others about the great and productive ideas brings them fresh into your mind, where they are handy for use.
Those paragraphs make it sound grand, to be a professor; and so it is. However, as I found out in my first week here, it is also extremely enjoyable. No-one had told me beforehand how much fun it is to teach! Many of the students are serious, hardworking, clever people, and there are some extraordinarily good students at Western (media perception to the contrary notwithstanding). Moreover, even ordinary students can provide extraordinary intellectual pleasure for a teacher.
Of course, no-one can deny the pleasure of scholarship---reading a beautiful book like D. F. Lawden's Elliptic Functions for no purpose except the general principle that an educated mathematician ought to know something about the subject, for example---and no-one can deny the gratification of discovering something really new.
These pleasures, I have found, are infectious. Since it's my job to encourage students to do the necessary work ( of course learning at the university level is difficult; if it was easy, we'd all be doing something else), it helps to be able to say authoritatively that one of the payoffs of all the discipline and effort is a commensurate pleasure. And when the students see for themselves that you enjoy what you do, they know that you're not lying. There is no `Royal Road' to mathematics (or indeed to any other discipline), and we have to do what is necessary to get students to go the distance the hard way. Recently I read ``Talking about Leaving'', by Seymour and Hewitt, which talks about why so many good students leave science programs; one conclusion I drew from the book was that getting the students to go the distance can't be done just with a good set of lecture notes.
One student said of my teaching style that ``I have never seen a prof so aggressive (in a nice way) about getting us to work.'' If I can make their hair stand on end in class, electrify them, get them off their behinds and moving, then I have been successful.