How to nurture a good student?

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.)

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"It just defers their boredom for another year" (sorry, my first response along these lines wrongly thought you were at high school level). I don't think a student will mind too much reaching the start of their final year already having covered everything they need to graduate. They can take extra subjects, or by that point they can start reading graduate-level material. A problem occurs however if they see their "official" classes as easy, tiresome work, and their private reading as the only real learning they do. –  Steve Jessop Mar 25 at 17:11
@SteveJessop I had a specific example in mind when I wrote that, although it is a rather extreme example. The example is of Simon Norton, whose rather entertaining biography I once read. Apparently (according to this biography), his grades were "good" as opposed to "exceptional" in his undergrad at Cambridge because of this. The issue is that although they have covered everything they still need to pass the exam. Continued... –  user1729 Mar 25 at 17:14
Cont. However, this is an extreme example because he already had a first-class degree from London University when he started his studies, and because, well, he is Simon P. Norton, who wrote the ATLAS and came up with Monstrous Moonshine...However, I think the issue holds true for even the least of the "good" students: If you want them to do well they still need good grades. Therefore, they need to be running at exam time and learning all this interesting stuff as fresh material rather than jogging along with stuff they vaguely remember from last year. –  user1729 Mar 25 at 17:24
... for another example of someone who doesn't respond well to typical undergraduate study, Stephen Hawking scraped a first in Physics from Oxford. Not because of advanced study as such, because he was calculatedly lazy at that time. Ultimately, "good but not exceptional" undergraduate performance didn't harm either of them. There are worse things to do at university than cruise your degree and develop other interests :-) Of course the people who were harmed by it, with talent that was unrecognised and never developed, aren't so famous. –  Steve Jessop Mar 25 at 17:44
suggest giving them a test, letting them test out of stuff they already know or given them resources to learn/test out, and let them use that time independently to explore advanced stuff while rest of class sticks to the curriculum, & feed them advanced stuff suited to their personal interests/wonders, possibly work on a co-designed project, & get class credit for those projects. –  vzn Mar 27 at 17:16

I don't see my thoughts expressed by anyone here so perhaps I can chip in! I'm currently a graduate student but I was once that kind of bright bored student you are talking about. I had my fair share of boredom, lost motivation, exciting learning, disappointment and enthusiastic crankery. I can tell you that taking more interesting courses and finding good books to read will happen naturally - you don't have to help too much with that. Of course you can keep it at that but there's so much more you can do. Perhaps it won't take more than a couple of hours a week.

If the students don't need more courses or books what do they need? They need a mentor and a friend. They need someone like themselves but who has achieved something in life (If you are a professor that's probably enough success right there) despite difficulties.

They need a mentor to help them become a better student. They may be bright but how is their work ethic? How is their stamina and long-term motivation? Does the student know how to keep working on a problem day after day AFTER they've lost the motivation? Does the student know how to openly complain about their difficulties? Those are very necessary skills that they are not likely to develop naturally. An hour a week can make a huge difference for the student over the course of their undergraduate years and it probably beats answering another few e-mails. Not to mention that you get to help shape a promising student into something more than they would become on their own during the natural course of their studies.

On of the most important things a mentor can do is to take initiative. Schedule to meet each other but not in your office. The office, the desk and the chairs are barriers to communication. With these things you are an authority figure and they'll have a hard time relating to you well. Meet elsewhere and eat lunch together. Talk about stuff - ideally it's not going to be just about math the first few times. The more real and honest you are about things, the better it'll be.

Bright students need to hear your stories of how you had difficult times or how you tried to prove insert-big-open-problem-here at some point before you even had the tools to work with. They need to hear about procrastination and what they can do about it. Ideally, you'll share some memories of failures past and what you did to cope with them. Hopefully the student will be inclined to share their personal story as well. You might even have trouble shutting them up. If the student complains to you about their difficulties you are on the right track. You'll be someone they can talk to and confide in when the difficult times come. This is especially true if they end up doing some sort of undergrad research and even more true if they go to graduate school later on.

If they have a good time with you at your meetings they will want to come back for more and they'll want to learn from you and perhaps become more like you. From there on it won't be difficult to inspire them and teach them how to become better mathematicians.

Unfortunately my experiences with professors have rarely been this good and I only know about the things I'm sharing with you because of my own struggles as a student and because I've had amazing mentors outside of academia.

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The book that I found as a young student that propelled my mathematical career forward was On Numbers and Games by Conway, and then later Winning Ways for your Mathematical Plays by Conway, Berlekamp, Guy.

The fact that games can be solved by mathematics, and that mathematics can be built out of games, is the perfect kind of ridiculous abstract concept for an accelerated student to grapple with. If you want the same topic but in a more entertaining way, Surreal Numbers by Knuth is excellent, and has exercises!

I'd actually argue that my undergraduate research projects were mostly useless, as far as "nurturing" goes, compared to the extreme mathematical entertainment I got out of those books.

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This answer might work for your specific student more than intended -- one of the best things successfully accomplished in ONAG is a definition of ordinal numbers, and a description of a game in which someone is omega moves ahead. Then a game where someone in omega - 1 moves ahead. Yes, I said it. Infinity minus 1 moves ahead. And then why not infinity divided by 2? :) –  Chris Cunningham Mar 25 at 21:26

The undergraduate-research tag suggests you know the answer already. Get this student doing something engaging and interesting. It doesn't have to be you that mentors this person. Can you help match him or her with someone who can spare a little time?

In the US we have a set of Research Experience for Undergraduates Programs. That might be a way to go. If you are not in the US, perhaps you have something similar?

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I put the tag on suspecting someone would answer as you did :-) However, I an unconvinced that this is the answer. I am hesitant for two reasons. Firstly, the vast majority of papers from undergraduate research projects look like they are from undergraduate research projects. Research for the sake of research is pointless: it needs motivation, and especially first year students (usually) do not have the mathematical maturity to work out what the motivation should be. Continued. –  user1729 Mar 26 at 11:24
Cont. Secondly, I am unconvinced that this is the most beneficial thing for such students. As I said, if it lacks motivation...Perhaps you could argue the case more? –  user1729 Mar 26 at 11:26
Every student is different, and I have to leave it to you to judge this one. Finding a "good fit" research question for a student early on in studies is challenging. It might take a lot of time to figure out an appropriate question. You have already said you don't have the tome to do that. But maybe another person will. –  TJ Hitchman Mar 27 at 3:13

In defense of enthusiastic crankery, I feel it's very valuable to try to define "infinity" as a number. You can ask the student critical questions, like how to deal with the "$\infty=\infty+1\Rightarrow 0=1$" dilemma. Let him resolve the apparent contradictions on his own; he'll either conclude that infinity isn't a number, or make up some other algebraic rule preventing you from making that equation. You could indirectly motivate, for example, Rudin's number system $\mathbb{R}\cup \{±\infty\}$, where $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$, and help him develop the necessary rules of algebra to make it work. It's extremely instructive to a student to invent his own mathematical language, and especially with gifted students it's important to allow creative exploration.

You say you don't have time to sift through his stuff: I think that's good. Make him figure out the parts of his theory that do and do not work. It's more valuable for him to do so than you, and if he's already motivated enough to pursue this independently, he'll only go after it harder once you poke a few holes in his theory. If his ideas really are crankish, it should only take a minute or so to find an inconsistency, something that I'm sure could be managed during office hours.

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