NCTM is the forward-thinking younger sibling of NCSM and was, therefore, much more progressive about wireless Internet access.
How to Develop Computational Skills without Drill, During Problem Solving. Jerry Becker.
So Here Are A Few Interesting Problems That Permit Constructive Solutions While Still Assessing Basic Skills
Becker brought out two problems that a) assessed both computation and reasoning and b) scaled all the way from basic counting through limits, which isn’t a small trick.
First, arithmogons. Add adjacent circles to get the middle rectangles. Then do it in reverse. Develop a rule for solving them quickly. They scale from easy to algebra at whatever speed your students want.
Second, the Christmas problem, which goes from counting all the way up to limits as you add more and more rows to the pyramid.
The Really Curious Part
The session was remarkable mostly for the one hundred copies Becker made of a 55-page handout he spread across ten chairs. “Take one from each pile,” he said as I walked in.
He only used (conservative estimate here) seven of those pages. When he ran out of packets, he promised the remaining attendees he would mail them a copy (as in postal mail) if they left him their addresses. I swear I am not making this up.
Here’s an excerpt of an e-mail Becker sent out to the group several days after the conference:
I apologize for running short of the handouts at our session. But I will have the handouts duplicated in the next couple days and then put them in the U.S. Postal Services mail to you – snail mail, so it might take a few days. But I am working on it already. The address labels will be typed up tomorrow.
So I don’t know.
It struck me several times throughout both conferences that we need to counter-program a session across from the “Newcomer’s Orientation.” I’m not talking about “Rolling Your Own Backchannel with Twitter.” Scale that back. Way back. Something more like, “How to Make National Presentations a Lot Less of a Chore for Presenters,” featuring URL shorteners, Delicious, PDFs, basic FTP. maybe drop.io. You name it.
(BTW: here’s a PDF of Becker’s handouts.)
David RadcliffeMay 5, 2010 - 12:32 pm -
Thanks, this is great stuff! I have saved Becker’s handout for future reference. By the way, the arithmogon problem has an interesting geometric interpretation via inscribed circles. I created a GeoGebra worksheet to demonstrate this at http://cmath.org/incircle.html.
vlkMay 14, 2010 - 6:38 am -
Sorry, I don’t get the illustrated arithmogon. If I understand the rules correctly, there is no solution to it. If the vertices are A,B, and C, it results in 3 equations,
hence B-C=B-A-8=9 ==> B-A=17, but B-A is already stated to be -7.
DavidMay 14, 2010 - 11:59 am -
vlk: It’s a typo. The number in the middle rectangle is the sum of the numbers in the adjacent circles, not the difference.
Dan MeyerMay 14, 2010 - 7:32 pm -
Got that fixed now. Thanks for the edit.
EileenMay 17, 2010 - 10:06 am -
I love these! I have a bunch of this type of stuff I have collected over the years, some of it ancient, for ways to include practice without sacrificing higher-level thinking. The kids love it too and they start to see math as the fun game that it is. I think half my job as a middle school teacher is to help kids have good feelings about math so that they actually desire to continue on with it. And that doesn’t mean I have to sacrifice rigor or learning to do so.