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I teach at a small liberal-arts college and advise our Math Club. In past years, I have talked a few students into taking the Putnam Exam. I've found that I pretty much have to "coerce" them into taking it at all, let alone prepare for it adequately. In the month or so leading up to the exam, I hold evening sessions after classes where I go over past "easy" problems and remind them to really focus on problems 1 and 2 in each session since those are more doable. This proves helpful for some, but other students just don't bother to attend.

I'd also like to encourage our students to work on problems posed in journals, like the Pi Mu Epsilon journal, as well as the MAA's Mathematics Magazine and College Math Journal. Working in groups on these problems can really foster the students' "ownership" of their mathematical expertise. This can even lead to a mention in a journal, which can boost their confidence and make a nice addition to their CV. Even Project Euler problems would be great, as well.

In short, I want to encourage our math majors to work on problems outside of their course work, for the reasons mentioned above as well as just to strengthen their education. What are some effective methods for fostering students to work on challenging problems, like those mentioned above, in groups outside of their standard course work?

I'm interested in hearing about your personal experiences with this, and would love to hear about methods that have/haven't worked. I'm also interested in any research about intrinsic student motivation and how to encourage that, because I imagine that's germane here. (If students want to solve problems outside of their courses, then they'll make time for it.)

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    $\begingroup$ FYI: I'm surprised that there was not Putnam-related tag, so I added the putnam-exam tag. $\endgroup$ – Brendan W. Sullivan Jul 24 '17 at 16:29
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    $\begingroup$ Come in the evening to do more math? When I could go to a party instead? When I could go to a movie? When I could take a walk in the park? You must be crazy! How many of your students go on to graduate school in math or computer science after they graduate? Those students (if any) are the few you may be able to fire up about this. $\endgroup$ – Gerald Edgar Jul 24 '17 at 16:45
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    $\begingroup$ Previous comment is true but ... it's also true that many smaller colleges not in the "top 25" lists have active such groups. So a worthy question. $\endgroup$ – kcrisman Jul 24 '17 at 16:54
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    $\begingroup$ In my personal experience, only the kids who have previous experience with Math Olympiads in other countries (well, really just Vietnam) get any traction at all with contest math. Each year we do the VTRMC (local contest for my school) and even my best students rarely earn a point or two. These students deemed unworthy by the contest are far more intelligent than I and will go on to do deep research in graduate school. Honestly, I'm not convinced the whole contest thing is worth it. That said, for the kids with special training, it does seem to attract attention of... $\endgroup$ – James S. Cook Jul 25 '17 at 4:14
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    $\begingroup$ government agencies... soliciting them for future employment. But, again, without special focus and training on how to do such contests, it is a rare student who can hack such problems. So, to be successful, it probably requires an honest to goodness course at the institution. Indeed, you find such courses at some of the bigger schools. Also, a prof. who gets actual pay to work on the contest math. But, to be clear, if I was king of the world, I'd probably spend the money elsewhere. For example, getting external speakers about cutting edge math seems way better. $\endgroup$ – James S. Cook Jul 25 '17 at 4:18
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Our MAA section (North Central) has an annual team math competition for undergraduates. It has proven more attractive to the students than the Putnams (which we haven't tried for many years). The reasons for this are: (1) It's just a Saturday morning (9:00-12:00), rather than the 6-hour Putnam format. (2) The problems are more accessible, although some are quite challenging. (3) The students seem to like working in groups.

Here is a link to the competition website, including links to past tests.

http://sections.maa.org/northcen/teamcomp.html

P.S. It also doesn't hurt to offer milk, juice, and doughnuts to 'bribe' the students.

P.P.S. Some professors offer extra credit in their classes for participation; although this sometimes brings in some not so serious students.

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    $\begingroup$ Totally agree about bonus points bringing in "not so serious" students. I have experienced this first hand. $\endgroup$ – James S. Cook Jul 26 '17 at 15:35
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Not a high-quality answer, but based on some decades of observation: among the kids' "peers" (and you have to figure out what that means), participation in the program needs to be positioned to enhance status (as they say). Otherwise, you will only (potentially) get "loners", etc. Of course, the "loner" demographic is not the worst population to look into for high-performance Putnam Exam takers, but I think this is sorta the opposite of your question.

In my own direct experience in trying to generate student (undergrad or grad) enthusiasm, free food (pizza...) will get you great mileage. After that, you can do your spiel.

(I've seen the perhaps-silly success of pizza-as-bait in the grad program at my own R1 uni, where more-senior grad students lured-in beginning grad students to prelim-prep activities simply by pizza... and game-show-styled formats! People like "winning", apparently.)

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One reason students might not be motivated is that the Putnam is too hard and the other options come with no tangible rewards.

You might want to consider setting up a competition within your school with even token prizes. Perhaps a set of weekly problems which would be graded, with a regularly updated high scores table and prizes awarded to the top n people at the end of the semester. Once you have a small group of people going at this, you could set up your group and hopefully these students will now be more receptive to doing maths in their free time.

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    $\begingroup$ Honestly, I did exactly what you're suggesting. But I ran into the same lack-of-participation problem. I even made the problems an easier "logic puzzle" style, as opposed to math competition style, hoping to encourage participants. $\endgroup$ – Brendan W. Sullivan Jul 25 '17 at 5:47
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I would recommend adding some structure since many students respond well to an activity if it makes it onto a transcript or resume, due to being trained this way in high school.

  • The way they got me into the competition scene as an undergraduate was to offer a 1-credit course called a "problem seminar" that was explicitly a fun time to work on hard competition-type problems. Then all math majors were encouraged to enroll in it. Once we registered, we had external motivation to show up, try, and accomplish hard things and then it was natural to try the Putnam afterwards.

  • At my current institution we proctor a national math contest but have the prep for that contest tied up in student clubs (the Math & Engineering Club) and we offer a prize for the highest scorer at our school even if they do not place nationally. I had some luck getting the Software & Tech Club to complete some Project Euler problems as well by making it a weekly meeting; see also this and this for more on Project Euler.

The common thread here is that convincing students to do things on their own time due to intrinsic motivation doesn't work so well. Instead, setting up structures so that students put themselves into a situation with extrinsic motivators seems effective in my experience. Good luck!

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  • $\begingroup$ The problem session course was Math 289, Problem Seminar, by Harm Derksen at University of Michigan. Grading policy was "If on average you correctly solve 3 problems each problem set, you'll get at least a B. If you regularly solve some of the harder problems, you may get an A." $\endgroup$ – Chris Cunningham Jul 27 '17 at 14:33
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I would like to suggest there is a significant psychological difference between solving Putnam-like competition problems, and working on open problems. For example, the Canadian Conference on Computational Geometry has an annual open-problem session that poses problems accessible to undergraduates:

PDF download of full proceedings, Open Problems p.73ff.

To get a flavor without downloading that (large file), see the MO posting, Sofa in a snaky 3D corridor.

Working on these problems requires faculty guidance, but it has a different feel than competition problems. My experience is that some students get totally hooked on this frontier of the unknown in a way that differs from their reaction to a competition.

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    $\begingroup$ Oh-so-true: the "tame-but-nasty" problem-solving thing has a vastly different psychological feel from "open problems". A hugely significant difference, on many levels. $\endgroup$ – paul garrett Jul 28 '17 at 0:08

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