Thursday, March 25, 2010

Automata Antics: Ball Throwing

I recently played this game with my students, once again found in Solve This by J.S. Tanton. They had a good time exploring a problem to which the answer was unknown and taking a break from the everyday math to play a game. Play went as follows:

A number of students stand in a circle, each with a card showing either “left” or “right”. John begins a ball game by tossing a ball across the circle. If Beatrice catches the ball she throws it back across the circle one place to the left (in her perspective) of John if she is holding left, or one place to the right if holding right. Then Beatrice flips her card over (“right” becomes “left” and vice versa) and waits for another turn. Each person receiving the ball operates in this way.

There is one complication: If Maxine is holding “right” and receives the ball from Christian directly to her right, the rules do not make sense. We make the convention that in this predicament Maxine holds the ball, turns in place to the right more than 180⁰, and throws the ball to the first person she sees (and still changes the word she holds). This is of course, equivalent to Maxine passing the ball to the person directly on her left.

Similarly, if Keone is holding “left” and receives the ball from Shannon directly to his left, he turns left and throws the ball to the person on his right. This is a little confusing at first, but it doesn’t take long to get the hang of it.

Here is the question: In this game is everyone guaranteed a turn? Will the ball eventually reach everyone, no matter the choice of word initially on everyone’s card?

(A lightweight ball is recommended.)

Note: The book with which this game was found often presents similar problems in the same section. There were two related problems presented before this one which involved a grid-like playing board and proposed the situation of an ant walking along that grid alternating vertical and horizontal steps or a grid with random “L” and “R” which dictate which direction the ant is to turn 90⁰. In both the these the question is similar: will the ant walk through every box in the grid, and will the ant ever return to a box from the vertical direction if it originally left that box from the horizontal? Having done these activities first, we might have had an idea of what would happen in the circle, however I chose the circle because I wanted a whole class activity. Have fun exploring any or all of these and report your findings!

My class has yet to discover a "solution," and I'm not sure what we did find out, but many predictions were made, variables changed, and a modeling tool like Excel has yet to be utilized to try multiple cases quickly and efficiently.

Friday, March 19, 2010

Spring 2010 has been a challenge for our Circle to meet, but we have met a couple of times and a new member from Laramie has joined, Julia. We also decided to continue meeting on the 2nd and 4th Wednesdays, but not as a credit-earning class. We'll offer credit again for our July face-to-face workshop at Ranch A near Devil's Tower. Watch for a directions to sign up and recruit for this SOON. For now, mark your calendars for Thurs July 15 through Saturday July 17, 2010.

I promised (myself) to post some of what I have gathered from attending two national meetings, so let me start with the most recent: Circle on the Road at ASU in Tempe this past week. Pam joined me on Monday morning to see a little, and perhaps she'll add to this note. It was inspiring to meet so many people who have had the courage to start circles, and see how they have grown in time. One of my favorite new friends also runs teacher circles: Ginny Bohme from Arizona. I asked Ginny to join us for an Elluminate session, and she's planning on it.

There were quite a few who wanted others to know how rural and small their communities, but none had us beat, and everyone agreed that the beginning is hard. Now so many are ready to share resources, and this workshop has probably encouraged many more to share! With the new found support of the national mathematics groups like MSRI and AIM, the web resources should get better real fast. Of particular interest is perhaps the elementary grade materials. It seems that many parents with Eastern European roots have translated Russian texts for older students, but lots of creative folks, especially parents running circles that were created for their own kids,

There are a few especially interesting ideas I want to share now:
1. When we post problems for use in classes, include a description of the Standards met so that other teachers more readily see how to justify using the problem in their classrooms.

2. A measure of "success" derived from problem solving work include risk-taking, readiness for tests such as the ACT, and increases in proficiency within high-needs areas. No one seems to have much data (yet) but the ASU group seems intent on gathering this from us and anyone else who might join them. The circles in Charlotte NC offered that they have some evidence of the risk-taking and another promised some data on the second. I got the impression that he was one who told of problem solving work breaking through the boredom that under-achieving students find in too any of our classes.

3. Models for student math circles in more urban and established places include summer "camps" and math Saturdays. Many of the latter stay in contact with the students directly via email. Elementary age circles are shorter, and often held in a person's home near the schools. Those running these young-age circles believe in having a few visitors, but mostly agreed that having one main person is important for the younger children. The Kaplans run an amazing number of circles in one week, and they describe the importance of a non-competitive "congenial setting". One of the Bay Area parents arranges with teachers from neighboring schools to transport students to her home.

4. Some practical advice is available for parents who might be anxious during the initial weeks, and slides from various parent presenters should become available on the MSRI or TC sites soon.

5. Others urge "recreational math" as the content so that every problem is new and no one needs to catch up. Others have quarterly or yearly themes, but agree that adjacent sessions are best when they stand alone. Those running circles for older children break many of these rules.

6. Lots of people are creative in a humorous way with the problems they pose to children. A favorite was from a Mom who asked "If 1 kid can annoy 1 Mom in 1 hour, how many hours can 10 kids annoy 10 Moms?" Another person introduced us all to the resources from NACLO, the North American Computational Linguistics Olympiad. If you haven't seem their sort of challenging, go look soon! Many agreed that these challenges met everyone's criteria for engaging and entry-level.

7. In some places, Circles partner with other groups such as the Boys & Girls Club and government agencies that have an outreach mission.