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I am doing most of my research in a relatively new area of mathematics called finite subdivision rules. I have an undergraduate student who has begun doing research with me on some of the properties of simple subdivision rules.

He would like to get more background on this topic, but because it is such a new field, all of the references are at a research level.

In your experience, how would you deal with this situation?

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    $\begingroup$ Great question. I was in exactly the same situation (replace finite subdivision rules with magnitude of metric spaces) about two years ago. Thomas's answer basically says how I handled it. $\endgroup$
    – Mark Meckes
    Commented Mar 15, 2014 at 12:07
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    $\begingroup$ sometimes "toy problems" are very useful to teach basic principles. also "easily described open problems" (aka mysteries!) give a flavor. etc $\endgroup$
    – vzn
    Commented Mar 27, 2014 at 17:22

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If your student would like more background, then there is only one way to get that: Study a lot. There is a reason that it often takes years to get a research degree (Ph.D.).

So, if you by "bringing someone up to speed" mean that you want the student to learn the background, then I believe this would be very hard unless your student is very bright and highly motivated.

But if you by "up to speed" mean that you want the student to be able to help in carrying out the research, then one way to do that would be to do a lot of concrete computations. I mean, you forget about the background and the larger perspective and learn about a concept from doing concrete examples. Much research involves specific computations that one doesn't need much background to actually do. While doing examples you can slowly build up the background.

the challenge for you is that you have to break up the research into parts where some of these parts can be attacked by someone who doesn't have much background. This, of course, isn't easy in some areas, but maybe you could do this - in particular because your area is somewhat new.

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No way around it, have your student dig in. Perhaps discuss an overview of the paper(s) first, set aside time (and patience!) to go over dificulties afterwards. Look for accessible textbooks/lecture notes on any required material that isn't standard fare, you'll need to have it at hand (or recommend it).

If you can, get other students interested in the area. Studying new material as a group is always much more rewarding (and effective!). It has the side effect that overall it frees up some of the time mentioned above ;-)

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