I have trouble talking about my teaching

I have trouble talking about my teaching.

Part of the reason for this is that teaching is so damn complicated. This makes it hard for me to have conversations about how my students learn — especially to teachers that I don’t know. Sometimes I just avoid talking about myself because I fear the incoherent answers that I’ll provide to the questions about what I do every day with my students. This got much worse last year when I started teaching through problems.

To get better at talking about myself, my classroom, and my students, I’m going to simulate such a conversation. I’m speaking to a fellow algebra 2 teacher. The scene opens when I make a comment about how they’ve set up their units.



Me: I like how you’ve structured your units. After your review unit, you start the year with exponential functions? That’s interesting.

Teacher: Yeah, it’s only the third year we’ve taught the course with the new standards, so our math team likes starting with exponentials and then diving into logarithmic functions. We then go into polynomial and rational functions and end with stats, probability, and trig.

Me: I like that. The standards place a huge emphasis on exponential functions now…it’s good to get that off the ground early. This is also my third year teaching with the new standards. There’s so much content in algebra 2 that I’ve run out of time each of the last two years. I couldn’t teach it all.

Teacher: There’s so much! How do you do your units?

Me: [Feels uneasy] Umm…I don’t have units.

Teacher: [Confused look on face] What do you mean?

Me: I suck at doing it, but I teach through problems.

Teacher: 

Me: It’s confusing, even to me. All the topics that were in my traditional units are now all mixed up…but it a way that helps bring them closer together. Instead of having discrete units where topics are isolated from one another, the problems allow for the concepts to be easily interleaved, spiraled, and married in ways that I found hard to do when I had units. I’ve realized that a lot of what and how students learn in math class can be studied nonlinearly…and that’s what my classroom reflects.

Teacher: So, wait, are the kids just solving random problems? How do they learn?

Me: Sort of, but I think a lot about how I sequence the problems. I’m very intentional about which problems kids do and when they do them. So while on the surface the problems may look random, underlying themes and concepts from algebra 2 emerge for students through the problems over time.

Teacher: … [still confused]

Me: Here, let me show you what I mean. In a typical math class, the units are sequenced and taught linearly. [Gets paper and begins drawing] For example, take four units from the school year. Traditionally, when we finish with one unit, we move on the next. [shows drawing below]

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Instead of using that model, I interleave the topics, skills, and vocabulary from each unit to span the entire school year. My old units are now parsed. Think of the first unit in purple as broken up into smaller pieces and spread out over the course of the school year. [Shows drawing below]

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Then the same for the 2nd unit in gold. [Updates drawing]

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And so on with the remaining two units. Notice that some pieces are bigger than others. In the end, it might look something like this: [Updates drawing again]

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One of my goals in using this model is that, since students are frequently revisiting key ideas from the units, it helps with retention. With all the units mixed up, it makes it harder for the students to remember what they’ve learned…but that’s the point. It’s messy by design. That said, I build coherence by thoughtfully sequencing problems.

Teacher: Hmm…I’m wondering how your lessons look?

Me: Well, I typically assign them 3-5 problems for homework. The problems aren’t “practice,” as homework is traditionally viewed. They are more like puzzles or explorations that I ask the kids to do before class. It’s not expected that they understand and speak to all of the problems when they walk into class…I fully expect them to have questions. I also expect them to do individual research to help them figure out the problems. And because of how concepts are interleaved, the problems are usually all on different concepts — and have roots in different units. We don’t typically study one idea per day as is customary in math class. Instead, we study several ideas — and sometimes they are not directly related.

Anyway, we’ll spend the entire period discussing the homework problems in small groups and as a whole class. I have large whiteboards all over the walls that help with these discussions. Students are fully responsible for putting up problems and trying to gain a better understanding of them together. If they cannot (or do not) put up meaningful work to drive our thinking for the day, then they don’t learn. Also, I put few constraints on how the discussions look and feel. The kids typically move about the room freely.

Teacher: So where do you come in?

Me: Most days I help students make sense of the problems while in small groups. I also sequence student presentations of solutions for the whole class discussion. Equity of voice is important here — I keep track of who presents and how often. I also step in with direct instruction on the problems when it’s needed.

On other days, usually 1-2 times per week, things will look more like a traditional lesson where the problems focus only on one key concept. I consider these my anchor experiences that usually focus on high-leverage concepts (like sequence notation or logarithms). I also bring in Desmos Activities all the time.

Teacher: I wonder, where do you get the problems that you use?

Me: All over the place. I steal most of them from other teachers online, but I do write some myself. Those suck. I use Regents problems, too.

Teacher: How did you learn about all this?

Me: Two summers ago I attended the Exeter Math Institute. It blew my mind. As an immersive PD experience that pushed me beyond my comfort zone, it helped me completely reimagine what math teaching and learning can look like. It was different and challenging. It was led by a teacher from Phillips Exeter Academy who used one of their problem sets with us for a week. Exeter has pioneered the problem-based model that I’ve adopted…and they are well known for their problems — they’re tough, but they’re rich. I have included a couple of them in the problems that I give my students.

Teacher: This sounds interesting…I would love to see it in action.

Me: You are welcome any time. I must say, though, there are tradeoffs to using this model. Lots of them. First, students generally don’t like it…at least initially. Giving them so much control and disrupting what they know to be “math class” causes plenty of frustration and discomfort. And they are regularly confused and don’t always leave each day with a “clean” answer or understanding of a problem or concept. This can be hard for everyone — them, me, their parents. Last year, I wasn’t prepared for the amount of dislike and pushback I got. Second, since students learn content nonlinearly, it’s a mess for me to plan and sequence. Also, each day can be somewhat unpredictable because what we do each day is largely dependent on students’ independent work before class and the motivation to drive learning during class. Our discussions can suffer as a result of kids not doing their part…which happens A LOT. What makes this worse is the fact that I’ve never met another public school teacher using this approach…so I haven’t been able to critically bounce ideas off anyone. This makes it very hard to improve. I miss co-planning. There’s more, but, yeah…[awkwardly changes the subject]

 

bp

The case for disorder in the classroom

I’m going to go ahead and say it.

I think there needs to be more disorder in our classrooms.

By disorder, I don’t mean kids throwing chairs and running amok. Instead, I’m thinking about those instances when teachers give students largely unstructured time and space to land on their own ways of thinking the content. Those instances when the teacher, by design, fails to impose a rigid learning structure on student learning.

This is not a popular idea. It goes against pretty much everything teachers are told must happen in their classrooms every minute of every day. We must have structures, routines, and systems. We need tidiness. Students need to learn concepts linearly, there must be an obvious beginning, middle, and end to everything. It is our job to provide managed, predictable spaces for our students to work together and exchange ideas. For if we don’t do these things, our students’ will be distracted. They won’t learn. Unless its art, a mess is not welcome in the classroom.

Now I’m not saying that there’s no value in structured pedagogy. There is. I have lots of structure in what I do with kids. This includes approaches that range from “traditional” teacher-directed lessons to instructional routines to Desmos Activities to debate-oriented strategies like Talking Points. These are great and serve a purpose. They work to establish outlets for students to explore concepts in safe and dependable ways.

Yet with all the value of structured time, I would argue that messy, unpredictable time is equally important to our lessons and student learning.

By consuming ourselves with algorithmic structures, we teachers sometimes take away opportunities for our students to face problems openly. At times neutralize their brilliance and rob them of their natural inclinations — both intellectually and socially. By giving my students pedagogically less and expecting more individually and collectively, I’ve realized the importance of allowing my students to own their learning — to own our classroom.

For example, it’s now a regular thing for me to give my students a set of carefully constructed problems, whiteboards, and random groups as a means to learn new concepts. They are free to do whatever to understand the problems, including each other and the internet. If it was up to me, I’d even let them leave the classroom. Nonetheless, they are out of their seats for the entire period. I will aid with the math, but I indirectly encourage struggle. I’m there to help, but mainly around to support them to summarize and reflect on their work. It is their energy will make or break the room. It’s on them.

The result is often an untidy and confusing classroom. The uncoordinated, ambiguous, and disoriented learning environment it creates relies heavily on the cognitive diversity in the room. It’s an intentionally unpredictable and flawed approach, but something I’m learning to be good with. For me, it’s worth the tradeoffs.

Rarely does it end in rainbows and butterflies. But that’s kind of the point, though. Often times the kids walk out more confused than when they walked in. We might not get to an answer, let alone a correct one. This usually means that they don’t like me for a while (sometimes all year), that I won’t be on their list of favorite teachers. But in long run, it’s my belief that their discomfort will not only help my them understand the responsibility they have to themselves and their classmates when it comes to learning, but also the responsibility they have to make our classroom go.

Formal schooling sucks the instincts out of our kids. I teach high school and by the time my students get to me, they’ve internalized the classroom as a place where the teacher is supposed to direct their every action. They lose their ability to sense-make because they’re only concerned with “doing school.” They would probably stop breathing if I told them to (and then run to the principal’s office).

So while we thoughtfully select safe and comfortable approaches to student learning this school year, let’s make sure we don’t deprive our students of something they desperately need, which to experience disorder and be pushed out of their comfort zone. This means that they’ll be more tension, messy interactions, and awkward moments in our classrooms. And this will most likely require us to be pushed out of our comfort zones. And that’s a good thing.

 

bp

#blackbrillance + social justice + problem-based learning

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One of my summer reads has been The Brilliance of Black Children in Mathematics by Jacqueline Leonard and Danny B. Martin (inspired by Annie Perkins). I’m almost three-quarters of the way through it. It is rather dense because it’s packed with research, but I’ve been enjoying it.

Chapter 6 has stood out. It focused on the development of culturally relevant, cognitively demanding (CRCD) mathematical tasks. The authors of the gave this definition of CRCD:

Culturally relevant, cognitively demanding tasks should be mathematically demanding tasks and embedded in activities that provide opportunities for students to experience personal and social change. The context of the task may be drawn from students’ cultural knowledge and their local communities. But, the use of context goes beyond content modification and explicitly requires students to inquire (at times problematically) about themselves, their communities, and the world around them. In doing so, the task features an empowerment (versus deficit or color-blind orientation) toward students’ culture, drawing on connections to other subjects and issues. CRCD tasks ask students to engage in and overcome the discontinuity and divide between school, their own lives, community and society, explicitly through mathematical activity. The tasks are real-world focused, requiring students to make sense of the world, and explicitly critique society — that is, make empowered decisions about themselves, communities, and world. (p. 132)

The authors go on to caution the reader that finding/creating a CRCD task isn’t enough:

It should be reiterated here that task creation is by far only the beginning. Culturally relevant pedagogy necessitates that teachers learn about students, their culture, and their backgrounds. Ladson-Billings (1994) indicates that the teacher must be the driving force to creating a culturally relevant classroom. The contexts of the tasks alone will not necessarily make for the culturally relevant environment. It is the thinking behind the tasks and the actions during the implementation that make them culturally relevant. Without the appropriate set up of the task and the accompanying discussion and connection to the students and/or their communities, the task although created as culturally relevant, will lose its relevance. (p. 134-135)

This all got me thinking about all of the problem-based learning that I did last year with my kiddos. Our focus all year was thinking about, discussing, and solving problems that built on each other. As such, the big ideas of the algebra 2 curriculum were slowly uncovered through the problems. I used a range of pedagogical approaches but mainly leaned on whiteboarding (VRG and VNPS) to foster small and whole group discussions. On top of all this, back in June, I learned of Brian Lawler, who has done work around how teaching mathematics equitably requires problem-based learning. It’s an interesting take and learning from him provided even more incentive for me to improve my PBL approaches. Here are the slides to a presentation that he gave at the PBL Summitt in 2016.

So reading through chapter 6, it hit me that the PBL setting that I’m constantly improving affords my kids frequent, bite-sized opportunities to have meaningful discussions about relevant, empowering mathematics — exactly what I didn’t do last year. I centered all of the problems in contexts typically found on the Regents exams, which surely has its place, but when considering that 90% of my students are either Black or Latinx, it is an issue. The bottom line was that there was a strong disconnect between the problems I curated and my students’ lived realities. Here’s an example from last year’s problems (I could have chosen many more):


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While fairly procedural, it’s a pretty standard Regents problem. Most algebra 2 teachers in New York wouldn’t complain too much about it.

Other than the unrealistic nature of the problem, what I’m coming to grips with is that the discussion we have a problem like this involves just mathematics, not the implications of the mathematics and how it directly affects how my students view themselves and/or society. The challenge I’m setting forth to myself now is to find ways to change the narratives that my problems present to my students that will help us have more meaningful, transformative conversations.

For instance, after combing through the website Radical Math, I found myself thinking about all those payday loan joints that are everywhere in the city, especially in Black and Latinx communities like where my school is located (and where I myself live). With interest rates as high as 400 percent, they help create a wicked cycle of debt that cripples many folks who are struggling to make ends meet — some of whom are quite possibly parents of my students. In addition, they target people of color. I’m thinking that instead of focusing on Bella, Ella, and their mythical interest rates, I could help my students explore about the damaging impact these lenders have our communities through introducing data from the above sources and through a series of problems that they grapple with. It’s not perfect, but here’s an example:


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I’m pretty bad at using math to generate discussions about broader social issues like race. But then again, apart from beyond the white dudes, I’ve never had math problems to catalyze such discussions. I hope I’m better with facilitating discussions about problems like this, to help students see how they can better identify with math. If so, the result could be something important, relevant, and empowering.

This is a long post.

Last thing. The authors shared some examples of these sorts of problems that were created by graduate students who were also teachers. What was interesting was that, after studying the problems, the authors found that “very few of the teachers used race as a basis for their culturally relevant tasks.” Instead, the primary culture the teachers relied on was age. For me, it’s easy to get excited about some other aspect of problem set and get swept away in White culture, so this is a reminder to deliberately seek to address race in the problems and activities I use.

Through all of this, I feel like I’m getting closer to where I need to be, but I’m still left thinking about the many ideas in algebra 2 and how I might address them in the midst of the looming Regents exam.

 

bp