Faculty Handbook: Mentoring Undergraduates in Research and Scholarship A few days ago I was asked by the director of the Center for Undergraduate Research and Scholarship at Georgia Regents University (formerly known as MCG and Augusta State) to contribute an article for the new Faculty Handbook: Mentoring Undergraduates in Research and Scholarship which is to be written along the lines of the University of Alaska Anchorage Handook. In my short time at Georgia Regents University I was indeed one of the first mathematicians who engaged in "Undergraduate Research" in part due to the nature of my subject Dynamical Systems and easiness with which one can do some very law quality numerical experiments. However, I think that I am far a way from the point that I can give any meaningful advise to anybody else. I hastily authored this short document more or less written for myself while applying to UN Lincoln IMMERSE program which fully exposes my lack of competence to contribute above article. To make matters worse despite 30 contributing articles to the University of Alaska Anchorage Faculty Handbook not a single one was written by a mathematician. What bit of advice or particular detail would you include if you had to write it? I don't know if I can acknowledge you properly in the handbook, but I'll refer to this question somehow. 
 A: I'd recommend asking for input from professors who have a track record of supervising successful undergraduate research, such as Joseph Gallian  or the people who run the Ross Program.
Frank Morgan also has some good advice for planning the nitty-gritty details of an REU program.
One general comment that I have, which is somewhat specific to mathematics, is that it is important for the advisor not to underestimate the potential for original undergraduate research.  In your article, you wrote, "Only in a very rare circumstances exceptionally talented undergraduate students guided by some of the world's best research mathematicians can produce a genuine results."  However, Gallian would be the first to admit that he is not a top research mathematician, yet his program has produced some of the world's best undergraduate mathematics research.  And even Gallian admits that he has sometimes underestimated the students in his program.  In my experience, professors are far more likely to create failure by expecting failure than they are to give students an exaggerated view of their own abilities.  Mathematics research is difficult, but there's no reason to make it even more difficult by creating an atmosphere where failure is the expected norm.
See also this MO question.
A: I can offer some general common sense that might apply; it is derived from life
experience and not from the specific setting of my mentoring undergraduates
or being mentored as an undergraduate.
I find that practice is one way to develop ability in a particular skill set.  I note
that many of the more respected answers on this forum are not just those that
are clear examples of communication: they have specific references and show
quality of research and scholarship.  Precision and clarity are important, but
providing the links to the existing and relevant literature so that others can
follow, repeat, and confirm or correct the argument presented is a hallmark
of decent research; high school is not too early to start practicing such skills,
even for those not destined to a profession in the sciences, engineering, or
education.  Even documenting and keeping journals on small projects is
good practice for those aiming to produce good research.  Mentors should
do what they can to encourage such practice.
Gerhard "Aiming To Produce Good Research" Paseman, 2013.03.03
