Short note with a couple of important points about my hierarchy of hexagons post.
- An essential piece (perhaps the essential piece) of this whole teaching sequence was that it was based on what my students saw in those hexagons. I designed a diverse set that I thought had interesting properties, but this wasn’t about them guessing my properties. Instead…
- This was about my helping them to better articulate exactly what they saw in these hexagons. I helped them to develop good language for describing properties (this is not easy to teach, nor to learn!). And I helped them to understand differences between properties and definitions (again, not easy for either party).
- Josh wrote in the comments, “I would have worried about leaving so many types of hexagons uncategorized or about not recognizing some of the important types.” Which reminds me to state that this activity is all about process, and not at all about content.
- Perhaps that last claim is overstated. I had a few items of content to sneak in there. Vocabulary such as concave. The difference between equilateral and regular. That sort of thing. But there was no one thing about hexagons that I needed to get on the table. Instead, hexagons were our territory for exploring mathematical practices.
- This was my second time through the activity. Last semester, my students noticed somewhat different things in the hexagons, and we ended up with candlesticks, chevrons, shields, starfish and rectilinear hexagons. Again, the exact properties and resulting categories aren’t what matters here.
- I did a much better job of integrating the hexagon sorting into the proof work. The Venn diagrams and the hierarchy were important to that.
- If you read carefully, you’ll see the van Hiele model play out pretty tightly as the underlying model for the sequence of activities. We did not proceed to Level 4.