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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 26th post in the series.

Final Preparations
With our lesson on rational exponents going down tomorrow, today is important. When all the students arrive, we greet each other and dig into snacks. I seemed to have planned things well because my snack reservoir is running low at just the right time of the year.

The kids still have clarifying questions on the math, so we dig into the problems I gave them last week. I pepper the students with questions and they return the favor. I work out a couple of problems for them on a nearby board.

We go over the opening slide, which I helped a student create and share during period 8.

It’s pretty good. Front-loading the conversion formula was a smart move. This way, the class can jump right into applying it and gaining fluency. In past years, I’ve spent time developing it through a geometric sequence, but this year that ain’t happening. The cogen students from periods 3 and 7 decided to create student groups, so I give them a printed class roster and ask them to type group member names directly on the slide before class tomorrow.

All that’s left is to move furniture. It’s awkward moving our two-person tables into two large groups. We change the layout at least three times in order to make it work. After about 10 minutes, the room looks like a different place. Weeks ago, during the early stages of our planning, we envisioned precisely this type of physical transformation of the room. I’m glad to see it come to fruition.

As the students head out, I dish out high-fives and reiterate how proud I am of them. The previous two cogen-inspired ventures had similar pre-lesson vibes, but this one feels different. Maybe it’s because of the lesson itself or because it’s because this is the last one, I’m not sure. Either way, I couldn’t be any more satisfied with their commitment, ingenuity, and willingness to make this lesson a reality.

Last Cogen
With Regents prep tightening its grip on the remaining days of the school year, I’ve decided that today will be the final cogen. At some point during today’s session, I mention this to the students. I also highlight the 25 previous cogens that brought so much life to my school year and how, unlike their predecessors, this group will not be afforded the opportunity to choose their replacements.

As I share this with the group, I am disheartened. It hits me that our table will be empty next week, that there will be nothing left to discuss, and that we will have exhausted our pedagogical problem-solving. I’m thankful for how my cogens have sharpened my instruction, but low-spirited that it has to end. My cogen has traveled with me and helped me navigate the highs and lows of this unforgettable school year, so when the students depart today, I feel a companion has left my side. I am left to face the last several weeks of school on my own.

Refusing to accept an anticlimactic exit to the cogen, I immediately plan for one last gathering: a cogen reunion. I’m sure not all of my students will be able to attend, but it will still help bring closure to this memorable experience.

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

1. [katex] \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. [katex] \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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This is the fourth and final post of a four-part series where I explore planning and implementing a social justice-themed activity in Algebra 2. In addition to traditional collaboration with colleagues, my use of three cogenerative dialogues to develop and reflect on the activity were critical to its design and execution.

Part 4: Reflections on Ghouldy Muhammad’s Five Pursuits

Last month I taught a multi-day lesson that honored farmers of color and interrogated the injustice they have faced in this country. The lesson applied compound interest, average rate of change, and exponential regressions to the lives of seven fictional farmers. The lesson continued the work that my students’ did last year in Geometry when they used triangle congruency to explore the same issue.

After combing through survey data, grading post-lesson assessments, and speaking with students, I’ve humbly concluded that the lesson was a success. Looking back, a key to its success was undoubtedly Gholdy Muhammad’s Culturally and Historically Responsive framework. I discovered it after reading her book Cultivating Genius: An Equity Framework a few years ago. The guidelines she lays out in the book were a guiding light that steered my planning and something I found myself constantly referring back to as I met with my cogen students.

Her framework has five pursuits: identity, skills, intellect, criticality, and joy. Each is rooted in the research Muhammad has done on Black Literary Societies and how they served Black folks in the 1800s. The ideology behind the pursuits challenges the notion that learning standards should focus primarily on skills and knowledge.

While I love the theoretical foundations of Cultivating Genius, I needed more specific examples of how the pursuits look in practice. I found many of the examples in the book too broad to be meaningful when I was planning my lesson. Luckily, I supplemented my reading by attending one of her trainings and one of her talks which clarified a lot. These experiences gave me a better sense of how her framework looks on the ground and to strive to make each pursuit a goal of my lesson. My efforts were modest, but I tried.

When it comes to Identity, Muhammad’s guiding question in Cultivating Genius is, “How will my instruction help students to learn something about themselves and/or about others?” (p. 70) While my lesson does an admittedly poor job at helping students explore their own identities, it does a respectable job at helping them understand the experiences of farmers of color who were at the mercy of the USDA and its discriminatory practices. It achieves this through a compelling juxtaposition: my students, all urban youth, play the role of several fictional farmers. Assigned the role of a farmer and given personal details about their life provides historical context and, I hope, helps my students “become” the farmers. This naturally lends itself to perspective taking.

The Skills component of the lesson develops my students’ abilities to understand compound interest, average rate of change, and exponential regression. As a skill-driven teacher and a foot soldier for the New York State Regents, Muhammad speaks directly to me when she claims, “Teaching skills is important, but teaching them alone is problematic and also should not be privileged over other goals or pursuits.” (p. 96) With her inspiration, the lesson was my first real attempt to map key ideas from Algebra 2 to an issue of social justice. Armed with their loan and land information, students use mathematics to run all kinds of numbers on the farmers. The regressions were the only thing that was new — everything else was review, but somehow it didn’t feel like it because of the unique context.

According to Muhammad, Intellect “is the understanding, enhancement, and exercising of mental powers and capacities that allow one to better understand and critique the world.” She goes on to say that it “creates space for students to apply their learning in authentic ways connected to the world.” (p. 104) For this lesson, my students draw connections between exponential functions, the financial realities of the farmers, and the inner workings of the farming industry. They do this by completing a USDA debt-relief application designed to mimic the restitution granted to socially-disadvantaged farmers in the American Rescue Act. The application, which has four sections and is the crux of the lesson, requires students to use their knowledge of Algebra 2 to analyze farm loans, property values, and crop yields.

Muhammad defines Criticality as “the capacity to read, write, and think in ways of understanding power, privilege, social justice, and oppression, particularly for populations who have been historically marginalized in the world.” (p. 120) The historic mistreatment of farmers of color at the hands of the USDA is the centerpiece of lesson. The carefully staged role-playing and mathematics of the debt-relief application bring this to light and nurture criticality. Both of these pedagogical strategies surface historical context for students and reveal how our government has wronged farmers of color, many of whom are still alive today. In this way, mathematics serves as a tool for justice.

Joy was one of the tricker dimensions of the framework to address in the lesson. At one of the workshops I attended with Muhammad, I distinctly remember her emphasis that the work we do with students should be rooted not in oppression and loss, but instead with joy. Oppressive currents direct much of how social justice is studied in schools, and this can be easy to overlook. For me, it wasn’t until after my second cogen that I realized how role-playing and the debt-relief application could not only honor farmers of color, but also simulate justice and spark joy amongst my students. The farmer bios highlight the farmers’ passions while the immersive environment — including everyone wearing name tags and me donning a suit and tie — made it fun and engaging from the moment students walked in the door. At the end of the lesson, every student (a.k.a. farmer) received an actual check in the amount of the farm-related debts, courtesy of the USDA.

Looking forward, not every lesson I teach will be connected to a social issue. All things considered, that’s impractical and even undesirable. But for those lessons that are, including this one, Muhammad’s framework has proven to be an outstanding resource to guide my planning. It’s not enough for me to have an idea and the willpower to see it through. I’m not skilled enough to pull that off. I need a systematic and informed blueprint to turn to as I look to try this again in the coming years. Fortunately, coupled with my cogenerative dialogues, Muhammad’s framework provided exactly that.

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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.

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