I welcome suggestions from experienced student coaches as well--feel free to post yours in the comments.
Dear new student coaches and math circle mentors:
I'm delighted to hear that you have decided to work with some younger students in a MATHCOUNTS group or small middle school math circle. And I can completely understand that it can be tough to keep the energy going at first.
I only have a bit of time right now, but here are a few ideas off the top of my head:
1) Remember that encouragement and cheerleading is really important. MATHCOUNTS or AMC8 problems can be so much tougher than what the students are used to in their regular math class. It's easy for them to get discouraged.
Empathize with that--tell them that you also found MATHCOUNTS problems when you began yourself. When they make silly mistakes, tell them that even the best of students do that sometimes. If it's true for you, tell them you make silly mistakes sometimes too, and share your strategies for avoiding them.
It's certainly true for me. I make lots of silly mistakes, and I'm not shy about sharing that! Mistakes--and half-baked ideas--and learning from them--is part of the Aha! problem-solving process.
2) Ask the parents of your students if they'll take turns sending in refreshments! That helps a lot, especially if you try to hold practices right after school.
3) Get your students to savor and enjoy and celebrate their Aha! experiences. The picture above comes from the legendary Harvard physics professor, Howard Georgi, who has a wonderful approach of going around to different groups of students during physics problem-solving sessions, watching students develop cool insights, and he'll periodically exclaim "Eureka!" That might sound corny to you, and maybe it's not exactly your particular personal style, but you can figure out something else that works for you.
4) If some of the students are veterans and some are rookies, get the veterans to help the rookies.
5) Tell the rookies not to be shy about asking for help. Everyone was a rookie once. Tell the rookies that they are actually helping the vets when they ask questions, because the vets deepen their own understanding by explaining things.
6) Build in games as breaks in your practices. I posted some ideas on the Math Circle blog and I'll post more when I have time. Hangmath is a fun and easy game to run. The directions are here.
7) Another good game is the Factors game. The version of the Factors game at this link is a computerized version, but once you yourself have played it against the computer a few times, you can see that it's easy to figure how to run the game with pairs of students playing it on a chalk board (with two different colors of chalk) or just on a piece of paper (with two different colors of writing instruments.)
There are lots of good games (that just require pencil and paper) in Marilyn Burns' classic book, About Teaching Mathematics K-8. It's a relatively inexpensive paperback and it's chock full of great problem solving games and activities.
8) Tell stories. One of my favorite stories (possibly apocryphal, but still fun!) is to talk about the famous mathematician Karl Friedrich Gauss. Here's a great opening to the story from Jim Loy:
There is a story about Carl Friedrich Gauss. Supposedly, when he was a little boy, his teacher asked the class to add up the numbers one through a hundred (1+2+3 etc., all the way up to 100). The teacher wanted to get some work done, or get some sleep, or whatever. Anyway, to the teacher's annoyance, little Gauss [Here the lecturer holds his hand out to show that little Gauss was about 2 feet tall, to the amusement of the audience]... To the teacher's annoyance, little Gauss came up to the teacher with the answer, right away. The teacher probably had to spend the rest of the class time verifying little Gauss's [2 feet tall] result.
Some people find that story hard to believe, even impossible. I think that the story has the ring of truth to it. I believe that the story is true, or close to it. There are versions of the story, in which the numbers are one to a thousand [murmur in the audience].
I think that you people can duplicate little Gauss's [2 feet tall] trick [doubt in the audience]. I'm going to give you two very small hints. But, that's all you will need, to be just like little Gauss [2 feet tall].
Then you can show them the trick. I really like the way Jim Loy does it, because he first tries a few really inefficient ways first before getting to the snappy efficient way. That's a great way to motivate problem-solving.
9) Another story I like to tell students who are getting discouraged because they keep trying things that don't work is about Albert Einstein. Supposedly when he was first coming to work at Princeton, they asked him what furniture he needed for his office. His answer: all he needed was a desk and a "very large trash can," for all his false starts and mistakes!
10) You can get more great stories from reading Richard Feynman's books--he's a great storyteller, and I think part of being a great teacher is being able to talk about the process of problem-solving, the stories of the mistakes you made and the insights you got from making them. Popular math biographies can be great sources of inspiration and stories too--see what you can find at your local library.
11) You can find lots of little gems to share in Martin Gardner's recreational math books. The Aha! Insight and Aha! Gotcha! books are great places to start.
I've got a bazillion more ideas to share, and no time to write more at the moment. But I welcome other ideas in the comments. Experienced coaches can contribute ideas that have worked for them. Even if you're not an experienced coach, think back to when you were a new student, what ideas worked for you?