"There is no one way"

Monday, December 29, 2014

Geometric Construction, continued

Readers of this blog probably realize that I'm very much into geometric construction. This is in part related to my general interest in puzzles, in and out of the classroom. (In my other life, I construct cryptic crosswords.) My first math education publications were books of geometric puzzles for grades K-10. My pentomino puzzle books stayed in print for almost 30 years, and my most recent publication is an update of one of those. In fact, my geometric puzzle page remains one of the most visited on my Web site.

Scott Kim says that puzzles are to math as literature is to reading. That's probably not the whole story, but it's definitely part of it. This is confirmed by the fact that given the right tools and the right problems, students are tremendously motivated to stick even with very tough construction challenges. Some of the most thrilling moments in my teaching involved remaining silent as students helped each other through the  activities in this construction unit.

One paradox is that I have found the very same unit to be too difficult for teachers when I've presented it in my summer workshops. I can't really explain this. Part of it may be that in many teachers' mind, geometric construction is about memorizing a few techniques, and perhaps understanding why they work. My approach is rather different, as I see geometric construction as a puzzle environment. (Alas, the Common Core may reinforce the "how to" mindset, as geometric construction is mentioned only as a list of basic construction techniques.)

What is prompting today's post is that I have renewed hope that construction may be able to make a comeback in US classrooms, in spite of the overall decrease in geometry content. This is in part because of the availability of excellent geometric construction software (Cabri, Sketchpad, GeoGebra, and others.) And especially because there are now some very good construction games available.

- Euclid the Game has been recommended to me by many. It is simultaneously an introduction to construction, and an introduction to GeoGebra. I cannot vouch for the whole program, but I like the concept and the way it's set up. Basically, as you figure things out, you are given more tools.

- A similar philosophy informs GeoCon, a game for iOS. Again, I have not gotten very far into it, but I love that very early on it proposes a number of non-standard challenges. (I am currently stumped by challenge 2.11. Let me know if you have a hint for me!) However note that after a certain point, to get solutions to problems that stump you, you have to pay for "keys". Like the online construction game from Science vs. Magic, this app may be better suited for teachers and math clubs than for general classroom use.

Check out challenge 4.6, which I've just started to think about: