Category Archives: PBL

Math journals and editorial boards, round 1

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Back in October, I wrote about the ambitions I had for journal writing in my class this year. Well, this week the students submitted their first journal. And learning from last year, instead of trying to read 120 multiple-page journals myself, I formed editorial boards in each class where students would peer-review and critique the journals of another class.

How’d it go? Well, the journals themselves were awesome. I’ll have to upload a few at some point. It was obvious that some kids did a rush job, but the majority of them took it seriously and put thought into their reflections.

And everything else? Meh. I seriously underestimated the organizational nightmare of setting up a blind review of the journals. Because everybody knows everybody, I didn’t want the boards to have any impartiality when it comes to seeing a name at the top of the journal (they were all hand-written) and having to assign a grade. Consequently, I scanned the journals, blocked out the name of the author, and reprinted them for the boards to read and assess.

At the same time, I didn’t want the authors to know who reviewed their journal either. So that meant I had to scan the individual review/feedback sheets used by the boards (there was one peer journal), block out the names of the board members, and reprint them for the authors when I returned their journal.

Given the headache that was brought on from organizing this, I didn’t get a clear sense of whether or not the kids valued the process of writing the journals. I’ll have to survey them at some point after they’ve written a few. I’d like to think they appreciate them, but who knows. Plus, I think it’ll be easier for me next time around.

What was cool was that each board had to select 1-2 of the journals they reviewed for “publication.” I’m going to compile a bunch of journals throughout the course of the year into a little book and print it off in the spring.

When I handed the journals back, I publicly celebrated each of the students whose journals were being published in front of the class. They also got this swanky handout detailing next steps. It felt professional although it wasn’t. And it wasn’t the highest performing students whose journals got selected, either. Students who may not get to shine as much as others were spotlighted for their mathematical thinking and writing abilities. This was really, really nice. Best of all, this recognition wasn’t a result of their teacher cheezily trying to boost their morale. Their peers genuinely saw greatness in them and let them know by choosing their journal for publication.

 

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One graph. Ten minutes. An important conversation.

At the beginning of class I showed this to my students:

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They came up with lots of interesting things.

  • There are three variables
  • They are functions
  • They are different colors
  • The units are millions and years
  • The scale for the millions is by 500,000’s and for years is decades
  • The domain of all 3 functions is 1920 to 2010
  • The range is 0 to 2.5 million
  • All of the functions are positive over their domain
  • The average rate of change for the red graph from 1920 to 2010 is positive
  • The average rate of change for the light purple graph from 1920 to 2010 is close to zero
  • The greatest average rate of change for all functions appears to occur from 1980 to 2000.

Then I asked them to predict what the graph was about. Most felt it detailed some sort of economic situation. Or population. Then came the reveal:

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They were shocked. We talked about possible causes for this situation, like the school-to-prison pipeline and the privatization of the prisons. More eyebrows raised. I brought up the question of what the racial breakdown of the prison system might look like. It was an important conversation.

And then we moved on to the regularly scheduled program: the lesson.

That was this week. While this class opener wasn’t directly tied to work that we’ve been doing in algebra 2 and was relatively brief, I felt compelled to have this conversation with my kids. This summer I began thinking about how to deepen the connections between social issues and math. Since I suck at projects, I thought about making these connections in smaller, bite-sized ways — comparable to problems found on a typical NYS Regents exam. In an ideal world, I would find (and write some) problems around social issues that are directly tied to the algebra 2 curriculum and discuss them with students. But this is really, really hard. Factoring by grouping doesn’t exactly lend itself to talking about racial inequities.

I was upfront with them. I said that its hard for me to relate some of the mathematics we learn to their daily lives, but we can do it in other ways. I told them that it was my responsibility to help you see how math can you uncover your world. Graphs are one way.

Through this graph of incarcerated Americans, I’ve myself learned that periodically presenting an interesting graph or data can be another way to build in time for important discussions around social justice and empowering students through math — even if the discussion isn’t wrapped up in a “problem” or directly tied to what we’re studying. This is not unlike What’s Going On in this Graph from the NY Times.

 

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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]

 

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