When you encounter a very bright student in a first-year (college/university) class (and who is therefore bored), what do you do?
Leaving them to their own devices can be problematic. It can lead to a loss of motivation, or, at the other end of the spectrum, enthusiastic crankery (I currently have a student who is trying to define the number infinity). Therefore, some nurturing is required. However, I am busy (as I suppose most people here are) and so I cannot lend such students very much of my time and I so cannot personally "lead" their thoughts and ideas as much as I might wish. For example, I suggested that my "infinity-defining" student might want to look up ordinal numbers, but I simply don't have the time to spend the hours that it would require to sift through his workings or to find the most appropriate concept of "infinity" for what he is trying to do.
Telling them to attend second year lectures or giving them an advanced book to read is again problematic. It just defers their boredom for another year.
A final approach, and is the approach I have been favoring, is to give the students a good -but mostly irrelevant- book. That is, a book which has a small intersection with the courses they will take in the future, but one which they might still find useful. However, such books are hard to find...(my current one out on loan is Yaglom's Geometric transformations, in the hope that when (okay, if...) they encounter Fuchsian groups they will understand them a bit better...).
I would be interested to hear your thoughts and ideas on how to help nurture such students.
(Note: This question is similar, but disjoint.)