What a great day. After housekeeping was handled, and everyone knew how to communicate within our Elluminate class, we started sharing. Kirk shared a GREAT source of inexpensive equipment: Radio Shack. Gigaware: 2 headsets and 2 web cams for $14.95?!
1. Bertine is organizing math events to mesh with the Science Olympics meetings that already happen in Gillette.
2. Cindy has started a program she's calling "Math in Action" and she told of her students roaming the Sheridan area, capturing pictures of things related to Cartesian coordinates.
3. Dave, Kendall, and Ariane told of problems they've started posing their students, and they also shared a question about "what makes a good question". In part their interest is in guiding their own search for effective problems. Part too is the need to appeal to colleagues and have a good cogent statement about these problems we've come to appreciate. Note that on the blog mentioned later, a man named Larry Zimmerman proposes that these (good ) problems have the following elements:
1. a goal (construct, prove, maximize, minimize, classify, compare, compute)
2. given information
3. special rules (sometimes)
Dave told us about his upcoming geometry meetings using Zome tools that he's purchased.
4. We shared which of the problems from our Jackson days have served us since. These are ones we recommend as starters: the human tangle, Conway's tangle of ropes, Mark's numerical problems, the pennies problem, the pills. When Ariane & Kirk discussed have a simply stated purpose with problems like the ropes, Kirk volunteered that for him it's the inverse aspect. Lynne said it was about feeling that a solution is possible without knowing what it will be. Kendall suggested persistence with math challenges. Kirk suggested problems using chess or Go moves. Steve shared a rubric for ranking problems. The 8 categories were whether the task:
- is relevant to experiences of target population;
- can unfold over time and is not sequential and closed;
- is cooperative, drawing on current knowledge;
- has outcome accessible to self-verification;
- has accessible extensions;
- has a problem solving trajectory that can be shared;
- has a solution involving fundamental math;
- involves learning that's detectable on standard ed measures.
5. We looked at a problem based on one from Marilyn Burns: Number Bracelets. There is a great discussion at http://www.geom.uiuc.edu/~addingto/number_bracelets/number_bracelets.html. Note that under the link What's going on here? you'll find a lot of the math extensions!!! There is also a great blog called "Math Be Brave" that mentions this game: http://mathbebrave.blogspot.com/2009/07/larry-zimmerman-plus.html. Any one can join their blog, and most interesting!!!! They discuss the nature of and how to establish good problems: check out the old post on Larry Zimmerman II. It's a very nice discussion of how to invent good problems.
6. Kendall shared his first attempts using Excel to search for number bracelets. The command that he used reveals that this is a "mod problem". Kendall placed the first 2 numbers in A1 and A2, then A3=mod( sum(A1,A2),10) was the formula he put in the column of derived values. Mod(???,10) gives the last digit (base 10) of the value ???.
By the end of class we had resolutions and ideas of where to go. Most importantly, coming together seems a positive and helpful effort. We want to continue helping ourselves and some of the ways will be to
- try some problems with kids and share -- for now we all try the bracelets problem
- work some problems in class
- post problems on blog --- rate them with Steve's rubric or one that we evolve
- find a couple of new problems on our own to share & invite others to share before next mtg
- look for problems with a "theme" like Kendall & Lynne looked for mod problems
- get feedback on how problems "roll"
- bring problems we find and get suggestions for implementing & extending