In my view, the primary duties of the college professor are to be a conduit as well as a generator of knowledge. In practice, this general means teaching and engaging in research. I have a passion for each of these activities, and this is what drew me to my profession.
I find that often these activities reinforce each other: I am a better researcher when I teach, since teaching encourages me to consider questions I may not otherwise have asked. I am a better teacher when I am active as a researcher, since the research mentality encourages me to look at old material in new ways.
I view the mentoring of undergraduate research as lying at the intersection of teaching and research. I have seen that undergraduates who thoroughly engage in interesting research problems in mathematics understand the material at a much deeper level than one generally achieves in a standard classroom environment. Moreover, such students develop an autonomy for learning, gain an appreciation for knowledge-generation, and refine critical thinking skills to a degree rarely seen at the undergraduate level. As such, I hold the mentoring of undergraduate research in high regard.
I believe a good research problem for early researchers in mathematics should be (i) tractable and (ii) interesting (to someone other than the researcher). These are subjective and often competing properties that are difficult to simultaneously achieve, particularly at the undergraduate level. I think this goes a long way to explain why there are relatively few opportunities for undergraduates to engage in research in mathematics, as compared to many other disciplines. I personally believe that we in the mathematics community can do far better at generating meaningful undergraduate research experiences for a much wider range of students, and I think much of this starts by identifying as many good research problems for early researchers as we can.
The above poses a challenge: Find as many tractable and interesting undergraduate research problems as I can.
I am always on the lookout for such problems, and I have discovered a little trick that has helped me identify a number of them. Perhaps it will help others too, so I want to share it. The trick is that I look for pieces of my own research projects that an undergraduate can understand with a few weeks of background training. Such a problem tends to automatically be interesting (by construction it is related to broader or deeper questions), so the only constraint to be satisfied is that it needs to be tractable. If an undergraduate can understand the statement of the problem, then its likelihood of being tractable is often much higher (of course, as we know from Fermat, this is no guarantee; it is a challenge, after all).
To wrap this up, let me take a step outside of my department and briefly discuss the school I call home: James Madison University (JMU). One of the main reasons I chose to work at JMU is precisely its focus on the undergraduate experience, while simultaneously having resources to support advanced research. Indeed, JMU appreciates and respects excellence in teaching and excellence in research at the same level, and it also provides many opportunities and resources for undergraduates to engage in research.