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.

Similar Triangles

So now I'm going to blog about something that I'm just starting to think about.

For two days, I've had a tab open with a neat Futility Closet post. (So many clever bits of mathematics and reasoning there.) It has this image:

A pleasing fact from David Wells’ Archimedes Mathematics Education Newsletter

I immediately made it up in GeoGebra, but being the start of a new semester, hadn't really thought about it yet. I didn't see what parallel lines had to do with it, nor being equilateral. About to close the tab finally, I shared it on Twitter. Boom!

Matt and John jumped in. And then Henrí... 

I love the cycle of generalization in math! Get rid of this restriction, and that restriction.

Get rid of the 2nd line.

 

Get rid of the shared vertex.

 

And then the what ifs. What if we restricted a vertex in the preimage?

 

Surprise!

Lines and circles to lines and circles... must be complex. But John had already gotten there!

 

Then Simon found circles another way!

Now I'll go find someone to talk with locally about the complex transformations here. I could do it alone, but I prefer math in dialogue! I think I want to see someone else get excited the math, too.  I also enjoyed this coming up so soon after the post on Willingham's 4C's of story. Great illustration of the causality and complications inherent in an interesting mathematics situation.

It's available in GeoGebra if you want to play, too.

Story Teaching

Quick post thinking about Dan Willingham's post on The Privileged Status of Story, which I got to via Dan Meyer's post Study: Implicit Instruction Rated More Interesting Than Explicit Instruction

What constitutes a story?

"The first C is Causality. Events in stories are related because one event causes or initiates another. For example, "The King died and then the Queen died" presents two events chronologically, but "The King died and the Queen died of grief" links the events with causal information. The second C is Conflict. In every story, a central character has a goal and obstacles that prevent the goal from being met. "Scarlett O'Hara loved Ashley Wilkes, so she married him" has causality, but it's not much of story (and would make a five-minute movie). A story moves forward as the character takes action to remove the obstacle. In Gone With the Wind, the first obstacle Scarlett faces is that Ashley doesn't love her. The third C is Complications. If a story were just a series of episodes in which the character hammers away at her goal, it would be dull. Rather, the character's efforts to remove the obstacle typically create complications—new problems that she must try to solve. When Scarlett learns that Ashley doesn't love her, she tries to make him jealous by agreeing to marry Charles Hamilton, an action that, indeed, poses new complications for her. The fourth C is Character."

At the end, Willingham challenges us to incorporate these C's into lessons. In particular, the most important C, Conflict. "Teachers might consider using 10 or 15 minutes of class time to generate interest in a problem (i.e., conflict), the solution of which is the material to be learned."

I think this is compatible with several MTBoS approaches, in particular & obviouly, 3-Act lessons.

Character - my biggest question after my first read was who are the characters? Not in a heavy handed Life of Fred way, but in the story. I think it must be teacher and students for us. We resolve the conflict, after all.  Probably one of the inherent advantages of inquiry teaching is making the students the central characters. Not that we teachers can't be involved - I think we have to be ready to jump in, too. But we can't be Deus Ex Machina everytime, and let the students know there's always an out.

Math lessons are well set up for storytelling otherwise, I think.

Causality - why does this work is a great basis for an investigation. Add up the digits - if that's divisible by three the original number was, too. What? How could that work? Look - these three centers of a triangle are always on the same line. Why on earth...? Of course, if we make it out that knowing the fact is more important, we're killing the story. This is historically a great spark for mathematical developments as well. While I was writing this Sam Shah posted this image which got my mind wandering, making me go off and do some GeoGebra.

Conflict - I have no idea if this is unusual, but I try to get good math arguments going every chance I get. I usually refuse to be the authority. ("Is this right?" What do you think? "I think so..." Well, let's ask the class!) Plus anytime I ask for an answer, I always ask if there are any other answers. And when the students propose answers, there's a chance for a math argument. It also makes me think of Chris Luzniak and his Math Debates.  Even whether a particular topic is math can be a great argument. There's a course I start with Sudoku, and the last question is, were we doing mathematics? I have never had a class agree on this answer.

Complications - is there anything more mathematical than this? Oh, that worked. What if we added this? Could we do it still if we didn't know that? Messing around with conditions is prime mathematical behavior. And if the problem is problematic enough, this happens by itself. I could solve it if I knew that, now how do I find that out? Or you're trying all the cases and get to one where the freak out lives. Or you're practicing the very mathematical habit of mind of trying to find counter-examples to your own idea.

Where I think these C's might be helpful to me is in being more intentional about the type of math the students are working on, and using this structure to help design how I'm going to try to get my lead characters to find the problem.

For my first math for elementary teachers tomorrow, I want to create the conflict for my students between what they know about elementary mathematics and what they need to know. I loved Graham Fletcher's progression of multiplication, so I'm going to try to use that in contrast with their native ideas about teaching multiplication. (Also such a nice synthesis of understanding to model for them.) In the past I've mirrored mathematics development in children and schools, starting with number concept and building up. This will essentially be going in reverse, but will hopefully be a more obvious need to know that will motivate the deconstruction on supposedly simpler topics to follow. Wish me luck!