Student Coteachers Part 3: Introducing Rational Exponents

Over the last three months, my cogen has grown from a place where we generate insights into our class to one where we co-plan lessons together. So far, we’ve co-planned and co-taught two lessons. The first was a Bingo review and the second involved an original board game we created called Infinite Levels. I shouldn’t have been surprised with how much planning was needed to pull them off, but somehow I was.

For the third and final cogen-inspired lesson of the year, today my students and I used it to introduce rational exponents. In contrast to the the previous two lessons, this one required us to delve into the pedagogy behind a particular mathematical concept. For Bingo and Infinite Levels, we got away with glossing over content because the class wasn’t learning anything new. This time was different. I had to teach the cogen students about rational exponents and then, based on how they saw it playing out in class, sit with them and figure out how we were going to teach it. In all, we spent the good part of five cogens to accomplish this, with a several of the sessions running over our allotted time of 30 minutes. I reflected on the planning process in my Meditations on a Cogen series (those posts can be found here, here, here, here, and here).

As for the lesson itself, we set up the room in two large groups, each led by one cogen student. Groups faced 2-3 large whiteboards. At the start of class, I introduced the cogen students as my coteachers for the day and gave them the reigns. They designed an opening slide to acquaint their peers with the conversion between rational exponents and radicals. They used it as the warm up.

The cogen-designed opening slide

After they tag-teamed the opening, which included a short mini-lesson on the missing index for #2, my cogen students varied greatly in their instructional approaches. In 1st period, both students took a more traditional teacher role and used the large whiteboards to walk their groups through various examples we planned beforehand. In 3rd, one coteacher took a similar stance, but mixed in getting kids out of their seat to explain work to each other using the large whiteboards. The other coteacher in 3rd floated around to various clusters of students and engaged in small group instruction. In 7th period, both coteachers mainly used small group instruction.

Other than having to step into a demonstration when a coteacher made a mistake in her example during 1st period and two instances where I had to encourage my coteachers to check-in with other members of their group, my coteachers held it down wonderfully. I floated between groups to provide scaffolding to individual students, but my responsibilities were reduced significantly in support of the cogen students. Exit Ticket data was solid.

Something I wish I would have paid more attention to during the lesson was the explanations my coteachers offered the class. The pedagogy we decided upon wasn’t anything out of the ordinary: modeling, direct instruction, small group instruction, etc. Where the magic happened, I think, is how my coteachers chose to relay key ideas to their peers in the moment. It was in their less technical, more student-friendly language. It was in their informal phrases and idioms. This is one of the most intriguing and useful aspects of learning in a student-led lesson such as this one.

Adding to this, other than a 2-minute conversation after the lesson, there won’t be any time for my coteachers and I to reflect on how they think it went. Considering that 70% of students reported on a post-lesson survey that they “definitely want to do this again,” not finding time to debrief with the cogen students is a missed opportunity.

One of the less visible outcomes of the lesson was how it placed some quieter, low-performing, and math-resistant students in positions to teach their peers. This is a direct consequence of the make-up of the cogen since, by its very nature, it includes a diverse cross-section of students. What a beautiful thing it was to witness students of all ability levels and dispositions moving the class forward. A teacher’s dream!

The lesson took a lot to pull off, but was totally worth the time investment. It’s these types of lessons that help me reimagine what my classroom can be, empower my students, and maximize the potential they walk in with. It’s not the only way to accomplish these goals, but through my cogen, it’s the one way I’ve discovered works for me.


Meditations on a Cogen (No. 26) • Thursday, May 19, 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 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.

Our lesson’s opening slide, designed by my cogen students

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.

Cogen reunion flyer
Invite to be given to all of my cogen students for an end-of-year cogen reunion


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.


Cogens for Social Justice • Part 4

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.