Monday, September 21, 2009

Geometry brain warmup for high school students

The standard American high school math curriculum seems to regard geometry as something to be gulped down as quickly as possible, rather than savored and thought about in order to develop geometric intuition. Many bright high school students take geometry in 8th or 9th grade and their teachers then believe they are "done" with geometry.

Award-winning math teacher Pat Ballew, who taught in schools in Japan and England for 16 years, has a different perspective.

Mr. Ballew posted the following:

In the January, 1929, issue of the American Mathematical Monthly, there appears a problem submitted by J. Rosenbaum of Milford Connecticut. The problem begins, "It is well known that the radius of the inscribed circle of a right triangle is equal to half the difference between the sums of the legs and the hypotenuse." I ... suggest that the theorem suggested may be less well known now than it might have been in the past.

If it's been a while since you've taken geometry, and you're feeling uncomfortably rusty, see if you can prove this so-called "well-known theorem" by playing around with it, then look at Mr. Ballew's very nice presentation and discussion of several approaches to proving this theorem. Even if you came up with your own proof, you may find you get new insights by looking at his approaches.

If you liked working on this problem, there are more where that one came from.

Tuesday, September 8, 2009

Advice for new student coaches

Several new student high school student coaches have written to me for coaching advice, so I've decided to post a few of my thoughts on this subject. I'll be posting more thoughts this month, as I'm preparing to give a talk to Bard College students who are planning to start up math circles near Kingston.

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?

Monday, September 7, 2009

Congratulations--and Happy Birthday! to Mr. Babbitt!

Our high school students make OUTSTANDING coaches and mentors for middle schoolers!

Last February, the top six teams at Chapter MATHCOUNTS were all mentored by high school student coaches from the Albany Area Math Circle!

And all 12 of the top individual students in the Chapter CountDown Round were members of Albany Area Math Circle's middle math circle and/or had AAMC student coaches! Nine of the 12 had worked with our student coaches before chapter, and the others joined our middle school math circle to prepare for the state contest in March.

Many of our student coaches and middle school math circle mentors from last year are pictured in our banner photo at the top of this blog and a complete list, including those not pictured can be found here.

Mentoring younger students is a great way for high school students to strengthen their own problem-solving skills and develop valuable leadership skills at the same time. It's also fun and rewarding--a win-win opportunity all around.

High school student coaches are also helping to build future AAMC members. We are excited about the prospect that many of the 8th grade students coached by our high school student coaches will join AAMC's high school math circle this fall! And some of our terrific veteran student coaches who are now high school seniors were themselves coached by former AAMC student coaches when they were in middle schoool!

Our AAMC high school student members who would like to help with coaching teams or mentoring math circle students should send an email to with the header "student coach/mentor available."

If your local middle school wants a student coach for MATHCOUNTS, please put them in touch with us, and we'll try to find a nearby high school student member to help out with coaching. Send an email to with a subject line that says "MATHCOUNTS student coach needed."

Middle school students can also join our AAMC middle school math circle, which will begin meeting on Friday afternoons from 4:00 to 5:30 in Niskayuna starting Nov 6.

We will also try to start up some smaller satellite middle school circles in other locations. Parents of interested middle schoolers can email us to look for opportunities near them. Send an email to with the subject header saying "middle school math circle info request."

All emails should describe your geographic and schedule constraints and desiderata.