Sometimes at the graduate level, students are intimidated to ask simple questions because they are worried that they should know the answers. Yet some of the simplest questions are the most elegant. These are the questions people who are deep into their field have forgotten to ask. This enthusiasm and excitement about new ideas is infectious, and keeps you young. The direct interaction, the ability to have students experience what the research endeavor is all about, allows you to harness that energy, and it's wonderful for them.
Undergraduates can see what's beyond their classes, and that can be very productive.
Tom Banchoff is known for making math come alive in the classroom. His Web-based classes feature problems that spark collaborative thinking and provide opportunities for students to generate ideas based on those of their peers. Sometimes a problem or idea discussed in class leads to a research collaboration between Banchoff and one or more of his students. Endeavors turn into all sorts of study projects, including research papers published in mathematical journals. The last thing I want to do is consider students as fodder for my conjectures.