Meditations on a Cogen (No. 25) • Thursday, May 12, 2022

During the 2021-22 school year I’m having weekly co-generative dialogues (or cogens) with my students. In an effort to help me process these talks and document progress, I summarize and write reflections after each cogen. This is the 25th post in the series.

The Math
With no reminders, all but one student shows up. The one student who is absent left school early. This is incredible.

We do a quick check-in and get right to work. Based on what I’ve taught them about rational exponents over the last two weeks, I put together a worksheet that has 8 problems on it. They’re sequenced by difficulty. I hand it to the students upon their arrival and ask them to begin working on them. These are the problems they are going to use to teach the class. The cogen crew has to be able to handle them.

The 15 minutes I give them to work on the problems turns into a lot of time answering their individual questions around the table. It was unplanned and highly productive. Thinking back to my poorly planned mini-lessons with the cogen in each of the last two weeks, I’m glad this one was much better. My confidence in them swells.

The Pedagogy
After the math is squared away, at 3 pm I help pivot us towards the pedagogy. Last week, we tentatively agreed that we would break up the class into three large groups: one for each co-teacher (me and two cogen students). After I gave it some thought this week, I ask the students if we had two groups instead, with them leading both and me floating between the two. This will give them more control and facetime and affords me the flexibility to check in with any student in the room. The kids like it.

We start talking about how the lesson will open. The assumption is that the cogen students will use facilitate discussion in small groups around the problems, but will students begin in their groups or will we start in whole class? Thinking through these types of details is crucial, I tell them. We quickly draw consensus to start class as we normally do: with a whole class warm-up that gets everyone involved. One student asks if we can put up the conversion equation on the SmartBoard at the start of class and pair it with a simple problem that applies it. The others nod. I find it surprising that they want the conversion up to start class — it’s not something I would’ve thought to do. I hop out of my seat and make a sketch on a nearby whiteboard of what this might look like.

If \sqrt[b]{x^a}=x^\frac{a}{b} [\katex], then find the missing values:

1. \sqrt[5]{x^3}=x^\frac{?}{?} [\katex]

After seeing it written out, I buy-in. It’s straightforward and quickly defines the rational exponent-radical relationship. We feel great about it, but with only the conversion and a simple fill-in-the-blank, one student suggests we add another question. What we have is not enough to effectively warm the class up. One of the quieter students recommends that we add another fill-in-the-blank question, but this time use the square root. I sketch a quick example on the board.

2. \sqrt{x^7}=x^\frac{?}{?} [\katex]

It’s a dynamite idea because it will manifest a teachable moment for the cogen students: the class will inevitably get the exponent wrong. We discuss teaching moves when this happens and how the cogen students might introduce the index of two to the class.

At this point, our collective juices are flowing. The kids are asking questions and building on each other’s ideas. The lesson is really coming to life. I use our momentum to launch us back into a discussion around the small-group instruction part of the lesson, and it doesn’t take long for us to map it out. The kids want students to have individual whiteboards so they can assess student understanding. They also want the groups at opposite ends of the room and for me to add a few more problems to the worksheet in case they fly through them. Individual preferences start to surface for the cogen students, like whether they should stand or sit while leading their groups. I steer us away from getting lost in weedy discussions like this. I assure them that facilitation will vary slightly from person to person — and that’s ok. In fact, it’s more than ok: it’s one of the beautiful aspects of teaching.

A few times today, the students mention adding a competitive aspect to the lesson. They want to learn, but they also understand that healthy competition can boost morale and elevate everyone’s learning. They even reference Infinite Levels as an example. The idea lingers throughout our talk, and near the end, we figure it out. We’re going to administer both groups a small quiz at the close of the lesson. Whichever group performs better will earn a prize. It’s perfect. The cogen student-teachers will act like coaches helping their team succeed.

We run 15 minutes over, but leave bolstered by our productively. We decide that the lesson will go down next week, on Friday, May 20. This gives us one more cogen next Thursday to iron out final details and for me to answer any last-minute questions they have about the mathematics.


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