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. [S*hows 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*]

bp

You and I see things and feel things the same way!

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Well, I owe you a lot for helping me grow Joseph!

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Some years ago, I was in that place, and ended up developing a unit-based approach that still allows a central role for problem solving and for the needed spiraling. As you know, I wrote about it here: https://blog.mathed.page/2018/10/spiraling-out-of-control.html

and here: https://www.mathed.page/teaching/full-range.html

Perhaps those articles will be useful to others in your situation.

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Thanks Henri!! Until reading your post I hadn’t really thought about what I do as spiraling. (Previously, I had only lagged homework for my lessons.) This is only year 2, so I certainly hope to iterate and improve upon it over time. Along with reading your blog, thinking out loud on my blog helps with this. Lately, I’ve been thinking a lot of instructional related instructional strategies to teaching through problems. We’ll see where it goes!

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