pedagogy – lazy 0ch0

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]

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]

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

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]

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]

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.

#blackbrillance + social justice + problem-based learning

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):

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:

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.

PBL v2

So my yearlong experiment with problem-based learning has concluded.

After attending Exeter Math Institute last summer, I decided to overthrow my units and use problems as the foundation of how my kids learned each day. Throughout the course of the year, week by week I wrote a bunch of original problems, edited others that I already had, and stole the rest. In the end, there were 349 problems which I would classify as mediocre at best. These problems (and other practice, including DeltaMath) were the vehicle that my students used to learn algebra 2…and be adequately prepared for the Regents exam on June 14. The 12 whiteboards wrapped around the walls of my classroom provided the platform for my students to dig into these problems each and every day.

It was idealistic, but this change was inspired to help my kids be more independent and interdependent problem solvers. I took a huge risk because I didn’t know what heck I was doing. Despite some early struggles, I stuck it out because I believed in the process and knew that real change would take time. I constantly adjusted to support my kids as they pushed themselves out of their comfort zones. There were tears. There were instances where I felt like I bit off more than I could chew. Despite support from my admin, I still felt alone because I was doing something so different, so radical, from the rest of my colleagues. It wasn’t their fault. I was hard to relate to. Mine was a messy, nonlinear pedagogical stance to teaching mathematics and, as such, others stayed away. In the end, although folks wished me well, I had no one to talk to about the day-to-day, nitty-gritty roadblocks that I ran into. Other than an independent trip to Exeter and an awesome visit from one of their teachers, I worked in isolation. This only intensified my struggles.

Anyhow, the result of all this was an uplifting, rejuvenating, and stressful school year. I have some major takeaways that will inspire next year’s work, PBL v2. I’ll let them breathe here.

Don’t think that students will value my perspective on learning simply because I say its valuable and worthwhile. There was going to be a natural struggle involved with learning through problems, but I did a poor job of setting them up for dealing with it. Next year, the first 1-2 weeks will be all about helping them meet my expectations. This may include modeling how they should approach the problems using prior knowledge and independent research, encouraging uncertainty, showing them how to document their thinking, and how to use classmates as resources. I also want to present the research behind how and why I’m structuring their learning experiences.
A more diverse set of instructional routines to discuss problems. This year I used student-led Harkness discussions, rotating stations (group speed dating), Desmos Activity Builder, structures unique to the specific problems, and traditional, teacher-directed lessons that focused on anchor problems. Before the year began, I was worried about having the right problems as they are so pivotal in this setting. As the year progressed, I realized that I overlooked the pedagogy behind implementing the problems. Even with a focus on small groups, uniform Harkness discussions simply won’t cut it for a class of 30 every day. While it is and will continue to be a foundation of what I do, students quickly tire of the routine. I’m also thinking that exploring the use of protocols may be worthwhile.
Better engagement during group work. On most days, I gave students lots of freedom when discussing the problems of the day. For much of the period, they were on their own to construct their own (with guidance from me) understanding of the problems and the related concepts. Trust was baked into each day’s discussions; their thinking inspired the success we had each day. Some days were great, but on plenty of occasions, they did what teenagers do: be lazy. I’m wondering what else I can do to foster more consistent engagement during these small group discussions.
More metacognitive journaling. I did one in the spring and I liked it. They chose a recent problem and analyzed their own thinking around it. They told the “story” of how they arrived and understood the solution. They were a lot to grade though. Maybe one per marking period?
Be better with parents. I need to have a much more transparent and stronger relationship with my parents. I almost got around to inviting one into my classroom. Nonetheless, I need to clearly communicate how students are learning, why it’s important, and how I will support them along the way. Some parents had reservations about my approach and they definitely didn’t hold back from sharing their thoughts.
Use standards-based grading. Because I didn’t have explicitly defined units for students, when they encountered the problems, they didn’t have the crutch of knowing they were working on “section 2-4,” for example. They needed to use the context of the problem (and work done on previous problems) to discern what to do. I really like this because it made more challenging for students, but it handcuffed me when because I couldn’t find a way to accurately identify and document their understandings on exams, other than a vague, overarching percentage like “74%.” I thought deeply about this a lot and decided I will need to sacrifice a little PBL to assess meaningfully and authentically. Next year, I still don’t see having units, but I do think I will attach concepts to problems, at least to start. At the start of the year, when I give them their problems, I will also give them an exhaustive list of concepts that the problems elicit over the course of the year. I will number the concepts (eg 1-52) and each problem will have an indicator showing which of the concepts the problem connects to. Maybe over time, I can move away from this and students can make the problem-concept connection on their own. Either way, with well-defined, itemized concepts, I should be able to assign qualitative measures to each student’s understandings (needs improvement, developing, proficient, mastery). Whew.
The above would allow for more meaningful retakes of exams. With “corrections,” this process was a joke this year. There was no meaningful learning and we were all simply going through the process of applying an informal curve to their exam grades. With SBG back in the fore, this means that my post-exam procedures will look more like last year.
A nonlinear approach to learning mathematics. A huge plus of the PBL as I implemented it was that it gave me the opportunity to interleave concepts like never before. Not only did I marry concepts together in natural ways that are harder to achieve with discrete units, but I was able to space out concepts over the course of several months when it would traditionally be crammed into a three-week unit and subsequently forgotten. The most obvious example of this is trigonometry. We did many problems over the course of four months, each being a small step that got us closer to learning all the concepts from the unit. All the while, students were learning about other concepts as well. I can definitely improve my sequencing of problems but, again, since concepts learned are nonlinear, this makes recall more challenging for students and harder to forget.
One formal group assessment per marking period. These are just too valuable to not include on a regular basis. The kids love them. Plus, real learning happens during an assessment! They include two-stage quizzes, group quizzes, and VNPS quizzes.
Assign problems that will be formally collected and graded. In addition to the daily problem sets that are worked on for homework and usually discussed the following day, I want to give one meaty problem that’s due every two weeks. I’ll expect integrity and independent solutions, but students are free to research how to solve them using whatever resources they want. This will hopefully promote deep thought and a formal write up of math on a complex problem. I would love to have students type up their responses. I foresee using the Art of Problem Solving texts to find these problems, at least to start.
Using DeltaMath as a learning resource, not just practice. I was surprised by how big of a role DeltaMath played in my students’ learning. Given the lingering Regents exam, my kids relied heavily on the ‘show example‘ feature of the site to explore and solidify key ideas brought out by problems that we discussed during class.
Check homework randomly, I think. Because I didn’t check homework at all, the majority of students didn’t do it. Since the homework consisted of problems that were the centerpiece of following day’s discussion, it was a necessary component of the class. I wanted students to internalize that if they didn’t do it, they would be lost the next day. It’s ok if they didn’t understand, but they had to try. Well, that didn’t happen. Most kids just tried the problems in class the next day and set us all back. A colleague gave me feedback that students will give priority to things that have incentives, like points. I get it, but refuse to accept giving a carrot for homework. To compromise, I may check the homework of a random set of 5-7 students each day. Any student is fair game and, by the end of the marking period, every student will have roughly the same number of homework checks. I had tested this out in May and I think it triggered some initiative amongst students to do homework. I also like the idea of possibly administering a homework quiz that’s based on the previous day’s homework. If they didn’t do the homework, they’ll struggle…and I’ll offer tutoring for them to make it up.
Deliberately teach problem-solving skills. I had a flawed expectation that students would somehow become better problem solvers by simply solving a bunch of problems and have discussions about them. While that happened for some, at the end of the year most of my students grew minimally when it comes to their actual problem-solving abilities. I’m still trying to figure out exactly how to get better with this, but I know purposeful reflection will play a big role. I will also need to help surface specific PBL skills for kids. I want to bring in the question formulation technique and problem posing. This is still up the in air…and I’m reading a lot about this right now.
Be uncomfortable. It’s a great thing. In past years, I unequivocally strived to have students that were comfortable and at ease with everything we did in the classroom. I hoped they would find what and how they learned as easy and unproblematic. If I’m frank, I did a pretty good job of that. This year, I landed on the cold realization that, in many ways, my students should be uncomfortable. How else will they grow? As this post showcases, I led by example.

That’s all I have for now.

A lingering thought. Years from now, I’ll probably look back at all this and realize that I was fighting a losing battle, that I was too idealistic, that my time with students could have been used more effectively. I’ll look back and see how foolish I was. Yes, foolish to think that I could somehow establish a subculture within my classroom of independent and interdependent problem solvers that relied more on themselves than on the teacher. A subculture that places little value of remembering a formula or procedure for a quick fix, but instead focused on the mathematical relationships, collaboration, productive struggle, and prior knowledge to own what and how they learned. I’ll laugh at myself and shrug it off as me being ignorant. I’ll recognize that my goals were too lofty and practically impossible in a day and age of teacher-driven learning, high-stakes exams, and point-hungry motivations.

With this in mind, I can’t help but quote Maya Angelou: “I did then what I knew how to do. Now that I know better, I do better.”