My Favorite (Teaching): Improvisation

Sign in data. Most of the variety is size.

This is one of those things that's both a strength and a weakness of my teaching. I have a lot of ideas about things to try, but that is not especially professional. When people talk about the profession of teaching, we always seem to compare to doctors. You don't want your doctor making up treatments on the fly. Research! That's on what doctors should base treatment. Maybe it's okay because the stakes are less high - one topic in a math class vs. your health and well being, or because in teaching we are the researchers, too. Or maybe medical doctors are not the right comparison.

Regardless, I like to improvise! This is the story of two of those moments, in the same class period.

The characters, preservice elementary math teachers; the content, learning quadrilaterals with a focus on reasoning with properties; the setting, they've gone from describing quadrilaterals to thinking about their properties. Day 1 was spent describing quadrilaterals on geoboards to make, and thinking about a variety of different possibilities that still fit a type. They took home geoboard dot paper to make their own quadrilaterals, one of 11 types. (For us: square, rectangle, rhombus, parallelogram, isosceles trapezoid, right trapezoid, trapezoid, kite, chevron, convex, concave.) In general my teaching here is guided by the Van Hiele levels, in particular activities that give students reasons to transition from visual to analysis, from analysis to informal reasoning and then informal to formal, depending on the level. This is K-5 focused, so we don't push at formal reasoning too hardly.

I have an old set of quadrilateral cards that has a lot of visual confusion. Looks like a rhombus but is a parallelogram, etc. They've been good for me in the past, but the longer I teach, the more I want the students involved in the manufacture of math materials. So this time, they made the cards for homework. I made a couple of extra sets in case someone hadn't had a chance.

The first activity was the same as I've done in the past: Quadrilateral Concentration. Players shuffled up their cards (I had them put initials or a symbol on their own so they could get them back, but style would have been enough for most), and dealt them out into a grid. Turn over two cards and if there's a match - two quadrilaterals of the same type - you can pick them up. Most people knew the game already. The conversations were amazing. First question, "I turned over a square and a rectangle, can I pick them up?" The table ruled no, and I supported. The best arguments are over type, though. Either while playing, "hold on, I don't think that is a rhombus," or at the end... "Why don't these have matches?"

My fair quadrilateral

To summarize, I brought up to the document camera the ones that provoked the most discussion. We also used them to talk about variety again - as there were MANY congruent examples. And we discussed the most ambiguous case, which is probably now my favorite geoboard quadrilateral. People thought it was a kite, people thought it was a right trapezoid, people argued about the length of sides, people argued about the size of angles, people compared it to a square... loverly. It was especially nice because people kept cycling back to earlier claims, which seems to prove what I was suggesting about the power of our visual processing.

Another game I've played with the old cards is quadrilateral Go Fish.  We played with the same rules as concentration, using the most specific names possible. Suddenly it occurred to me that we could play concentration as we had, but switch the Go Fish rules to allow for more mathematical subtlety and strategy. Not everyone had played Go Fish, but enough had to make the rules explaining go smoothly in each group. To play a match, they had to be the most specific type. But when your opponent asked you for a type, you could give any that fit the characteristics. BAM! This was great, and almost instantly a better game than the original Go Fish. There was the start of some strategizing, where some people weren't asking for exactly what they wanted, and the conversations were spot on. This is a trapezoid, are you sure you don't have anything that fits, etc.

That's an improvisation that paid off. Better than what I had before.

The other idea on the spot was to go farther into combining properties. I wanted to make it natural to think about what if a quadrilateral was a this and a that. So spur of the moment, sidebar into a weird movie and TV discussion. I asked, what was an adjective that described a show or movie that you liked to watch. Then I shared how my wife's favorite genre was funny + scary. "Like Krampus?" (Side discussion on Krampus, which we recommend. But only one person had seen it, so...) I wrote down the 'equation' funny+scary=Ghostbusters. (Best example is probably Buffy, though.) Then they discussed at their table until each person had one to put on the board. I was worried about = abuse, so I did mention that what we're really doing is finding examples in the intersection.

And one table really got into trying to do adjective arithmetic. We talked about the examples & shows for a bit and then I transitioned to the purpose: what if we combine the quadrilateral types this way? Each table I wanted to come up with one quadrilateral equation. Got some good ones, and I shared about the role of conjecture in mathematics. To their list of four conjectures, I added some questions.

I connected this to the homework, which is to try the very challenging problem of a Venn diagram for all the quadrilateral types. We'll discuss those and the conjectures next class.

Passed it around again, and got much more
variety of property and orientation.

This improvisation was okay. Don't think I did much harm, it was a moment of high engagement, but not necessarily in mathematics. Well, it was mathematics, but not our quadrilateral content.  The disappointing thing is that the conversation about the shows - reasonably analytical - didn't carry over to the conversation about the quadrilaterals.


I'm okay with this, however, because even a bad result is going to happen sometimes. The same activity that is a gas burner with every class that has ever tried it can crash and burn. So the improvisation increases my store of supplies, keeps my interest, and gives me things to think about for student thinking.