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} , then find the missing values:

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

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{?}{?}

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.


Meditations on a Cogen (No. 24) • Thursday, May 5, 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 24th post in the series.

Lesson, Lesson, Lesson
The focus of today’s cogen is our lesson. Last week, I realized that I failed to think through how much effort would be needed to teach the students about rational exponents, talk pedagogy with them, and then plan the lesson. As the students arrive around the table after school and one helps with the tablecloth, I gain some confidence. We can do this.

The issue that’s staring us in the face is the content. Last week, I did an abysmal job of scaffolding the examples. Instead of varying consecutive problems slightly to gradually build complexity, I found myself bouncing around from problem to problem like a mad man. There were similarities between the problems (i.e., converting between rational exponents and radicals), but the jumps from one problem to another were too big. In fact, they weren’t even jumps — they were leaps. It was the exact opposite of how I would actually teach the topic.

Anyways. We settle in and I ask everyone how they’re doing. Five students are present, one was absent from school. I hear crickets, so I poke fun at a kid by drilling into the specifics of their day. We laugh at something that I can no longer remember. The ice is broken.

We recap what we covered last week and dive into a few more examples. I give them independent think time between problems and, man, it’s quiet. The kids are into it, but I hoped they would be more collaborative. On top of this, all of the examples from this week and last are so different that I feel the math is tugging and pulling us in lots of different directions. If I feel this way, the students probably do, too. They just don’t know it yet.

With about 10 minutes left, we shift gears to discuss pedagogy. How do we want to teach this?

A couple of the kids give me blank stares and another looks down with uncertainty. At this point in the year, I figured the cogen would be firing on all cylinders, but I’m wrong. I have to remember: they’re NOT teachers. Advice on my teaching? Sure. Guidance on how to make a lesson more student-friendly? Definitely. Teaching? Not so fast.

I throw out some ideas, and we eventually land on some structures for the lesson. The kids organize the examples we did by difficulty so we can scaffold the examples for the class. (I promise to bring similar examples next week for us to peruse.) We agree to combine tables in the classroom so that each cogen student has one large table for small group instruction. The cogen students will use direct instruction and wait time to facilitate a discussion of the scaffolded examples. We end in a good place.

Left wondering
After today’s cogen, I’m left wondering: is this my last cogen cohort? It’ll be past mid-May by the time we teach this lesson and the last day of classes is June 14. I have had side conversations with a few other students about the cogen and think they would make great end-of-year additions, but what would we work on? What projects could we adopt for 2-3 weeks?

Including the students from today, I’ve had a total of 22 students from three different classes be part of my cogen this year. If nothing else, I will invite all of them to an end-of-year shindig to thank them for their service to our class. The odds of everyone being able to attend is low, but it would be fun to order pizza and get them all in one place.


Meditations on a Cogen (No. 23) • Thursday, April 28, 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 23rd post in the series.

Because of spring break, it’s been two weeks since we’ve met. Fearing that everyone would forget, a couple days ago I give my cogen students a soft reminder about our session today. All six confirmed they can make it, but I end up with only five because one of the students is absent today. I drape the tablecloth over the table, dump the snacks out, and we’re ready.

I open with three quick check-ins and updates: our recent poetry assignment, tomorrow’s pre-exam review, and Tuesday’s group exam.

Before spring break, I assigned my annual poetry task and I ask the kids for their feedback on how it went. The feelings towards it are generally warm, with one student — who made a math rap — saying it was really fun. Another student comments that he found it a little weird to be writing poetry in math class, but thought it was cool. That said, two students convince me that next year I need to use class time to model how to write the poems. Poetry in itself, for some students, is already a heavy lift. And now I’m asking students to use the medium to describe a math topic with no guidance? I can see why it would be hard to do.

For what it’s worth: Given the growing success of the poetry assignment these last two years, I actually had a goal to invest more time and energy into it this year. I wanted to pour myself into making a memorable experience for my students. But with the emergence of my farmer activity, I didn’t have the gas to go that far. Its development was placed on hold until next year.

Next, I inform the students that tomorrow I’m going to be implementing their recommendation from four weeks ago: pre-exam review handouts. I share my excitement for this and my hope that it will resonate with my classes. The cogen students nod in agreement, relishing (I hope) at their suggestion being put to action.

Lastly, I ask for tips on Tuesday’s group exam. I rarely do two of these in a single school year, but after doing one in the fall which my cogen helped me reflect on, I figured, why not? I’ve been meaning to revamp the experience using their feedback for months. This is the perfect opportunity before the year ends.

What can the current cogen offer me in terms of advice? They don’t add much, but do double down on what my fall group suggested: keep groups to 3 or 4 people — preferably 3 — and be mindful of how many problems I give. Deal.

Today’s cogen really excites me because I get to “pre-teach” the students the topic that we will eventually coteach for what will be the third cogen-inspired lesson: rational exponents. The previous two lessons were game-based and not tied to any particular math topic like this one will be, so it’s an exciting challenge.

In preparation for today, I printed some Regents problems on the topic and made copies for each student. After our opening potpourri and my mini-lesson on how to convert between rational exponents and radicals, we use the problems as practice. We solve four problems and end up staying 10 minutes after our usual time. A few things stand out:

  • I underplan. It doesn’t take long to realize that I should have put more thought into the session. I hastily choose examples in the moment and stumble over certain parts of my explanations. I could have done a waaaay better job at sequencing and demoing. With just 5 students to teach in an empty, quiet room, I was too confident. I wonder how my instruction today will shape how my cogen students’ planning. (We will start that next week.)
  • Teaching teachers is different. Today was unique in that the students I teach will go on to teach our class. I know this going in, but my planning doesn’t take it into account. Discussing potential teacher moves while introducing the content would have been helpful and not out of the realm of possibility with these students. I miss the mark.
  • This is going to take longer than I thought. I should have known better than to think we could accomplish this in two 30-minute sessions. Considering that I want the kids to write a full-blown lesson plan with me, this process is going to take at least three weeks to plan, execute, and reflect on.


Teaching’s hidden benefit

Making a difference. Job security. Lasting impact. Helping students achieve “ah-ha” moments. Summers off.

These are all benefits that come with being a teacher. The pandemic has caused many of us to question their existence these last two years, but they are significant reasons why many of us continue to teach.

All of those things resonate with me, but there’s yet another reason why I love teaching. It’s hidden beneath the surface, often overlooked, and deceptively important to my well-being. It’s not quantifiable like the two relaxing months of summer are or tangible like the warm-hearted embrace from a student at commencement.

What is this gift? What is this intangible dimension of teaching that I’ve come to view as one of its most valuable benefits?

The gift of youth.

Each day, I’m surrounded by teenagers. Eager, fearless, curious, bold, emotionally present, rebellious, unsophisticated, perceptive teenagers. By way of teaching, I walk into a building each morning and find myself immersed in their youthful worlds.

At the same time, each day I grow older. As a result, I move further away from what it means to be young. Sure, age brings its own blessings, but it also draws me away from those I serve. In this way, teaching is a paradox: I’m expected to reach today’s youth even as my body and mind naturally retreat from them.

And therein lies the gift my students offer me. They involuntarily surrender insights into youth culture, yes, but if I listen to them and observe closely, it goes much further than that: their youthfulness actually rubs off on me. My body might be aging year after year, but I’m unquestionably younger in mind and spirit because of them. They’re a big reason why I often forget how old I am.

Though many years their senior, I’ve learned a great deal about successful teaching and living a meaningful life from my students. I’m not saying that I do everything a teen would, but I have absorbed many of their finest assets. I constantly question myself, run towards risks, and embrace spontaneity. These and other qualities are transferred to me from my students in myriad ways — both in the classroom and out. It’s in their rebellious attitude and unapologetic creativity. It’s in how they audaciously seize the moment. It’s in their struggle to be understood.

I’ve never worked in any other field. If I did, having taught for 16 years, I’m confident that I would be a much older person than I am right now. I have my students to thank for that.