## Sunday, October 4, 2015

### Six Sides to Every Story

I finally got a chance to teach the Hexagons.

Christopher Danielson has a distaste for boring old quadrilaterals, so he came up with teaching hexagons. (Blog posts: one, two and three.) Dylan Kane wrote briefly about his experiences with them at NCTM:NOLA, and Bredeen Pickford-Murray has a three post series on them.

For me, the idea of type in geometry is pretty complicated. You need to have a large variety of the thing that will have types, you need experience with that variety, you need to be doing something with them - often describing them. (Boy, you're needy.) Types are the way we sort, which means attention to properties and attributes. Definitions come after types, as the more we work with something the more we need precision. Making a good definition is a great task for forcing you to realize some of your assumptions about the objects.

With a group of preservice elementary teachers, quadrilaterals offer a lot of these opportunities.   Mostly they are at the visual geometry level, and debating whether a square is a rectangle is a good discussion. (Most have been told that, but don't really believe it.) This class I'm teaching is all math majors who have had a proof writing course and many of whom have had a college geometry course. Their quadrilateral training is not complete, but they are strong in the fours.

Sorry.

For me this made the conditions right for trying hexagons. I was up front with the idea that part of this was to put them in their learner's shoes, about trying to make sense of this material seeing it for the first time. So we started with variety and description.

The first day we played guess my shape on GeoBoards. One person describes and another or a group tries to make what they're describing. For more challenge, the describer can't see the guesser's geoboard either.  I limited us to hexagons. I also tried to bar instructions; you weren't trying to tell them how to make it, but rather describe it so well they could make it.

After playing in their groups, we tried one person describing while all of us tried to make it. When she was done, several people presented their guess. That gave us a sense of the holes in her description. But on the reveal, there was appreciation for what she was getting at, and several good suggestions about more description that would have helped. Then we looked for what made descriptions helpful.

 Brainstorming
With that experience, we brainstormed some possible types. The homework was to draw two examples of each possible type, and come up with reasons that is should or shouldn't be a type. Also they read Rubinstein and Crain's old MT article (93!) about teaching the quadrilateral hierarchy.

The next day I gave them some time to make classes of hexagons within their table, with the idea that we'd come together to decide on class types. We went totally democratic: each group would pitch a type to the class, and then we voted yea or nay. The one type we started with was a regular hexagon. Mostly I was quiet in the discussions, and I abstained from the votes unless I felt strongly about it. Once we had a pretty good list of types, we did class suggestions for names. I shared how historically names are often either for properties or what they look like. The table that proposed the shape had veto authority over the name. But Channing Tatum almost became a type. Several people reflected that this was a pretty powerful and engaging experience.

Homework was to make a set of 7 hexagons. At least one is exactly three types, three are at least two types, two are exactly one type and one of your choice.

The third day we opened with one of my favorite activities: Circle the Polygons.

We do rounds of finding out how many polygons people or groups have circled, and then they can ask about one of them. Then they recallibrate. If students have no experience with polygons (2nd or 3rd grade) I will start with some examples and non-examples. Once we're agreed with what a mathematician would say, I asked them to define 'polygon.' Closed and straight edges are pretty quick, and someone usually thinks about 2-dimensional. (Although I usually wonder if they have a counter example for that in mind.) The last quality is a struggle. After we have a couple student descriptions, I share that mathematicians also had trouble expressing non-intersecting, no overlap, no shared sides and just made a new word: simple. Then I asked a spur of the moment question about what did they see as the purpose of definitions and, wow, did they have great ideas.

So we worked on definitions of the hexagon types, sorted them, and looked at each others' sets. Super nice variety. People brought up a few that were tough to type, and they caused nice discussions on which properties were possible together and some good informal arguments on connections. The note to the side was a discussion of Dan Meyer's "If this is the aspirin, what's the headache?" question for definitions. We agreed that most lessons give the aspirin well in advance of any headache they might prevent.

To sum up the experience, I asked them what they noticed about what they had done, especially in connection with the Common Core standards for shape, and what advice they would give to teachers who were asked to teach these things.

To finish up, we did one drawing Venn problem together. Come up with labels, and draw a shape in each region that fits or give a reason why you can't.

So thanks to Christopher and this great group of students for a great three days of mathematical work.

P.S. Student reaction: several of these preservice teachers chose to write about the hexagon experience for their second blogpost. If you want to hear about it from their perspective...
• Jenny thinks "It is time for teachers to get away from the cute activities that are fun for the students and get into the real meat of teaching these shapes."
• Chelsea is thinking about adapting it to quadrilaterals.
• Heather wrote about the inquiry aspect.