Teaching Math with Examples

I’ve appreciated Michael Pershan’s writing for as long as I can remember. It always gives me something deep to think about, blows me away with its clarity of thought, and is terribly persuasive. I usually walk away from it feeling like I’m missing something, that I’m not seeing the bigger picture. To top it off, Michael’s wit never fails to make me smile. He’s a great guy and friend.

I can’t help but teacher-crush on Michael right now because of how thrilled I am that he wrote a book! It came out earlier this year and is called Teaching Math with Examples.

Despite the fact that I maintain a blog and use Twitter, I don’t enjoy reading online. I much prefer newspapers, physical books, and anything printed. This is why reading Michael’s thoughts in book format is a dream come true for me. It allows me to indulge in his genius without having to stare at a screen in the process.

I must say, the book doesn’t disappoint. Highly readable, practical, and smart, it’s exactly what I expected from Michael. It starts with him making a case for worked examples and detailing a classroom routine that he’s developed that uses them. He goes on to discuss how worked examples can foster problem-solving. Michael then explores how worked examples can be used as a form of feedback after assessments like quizzes and exams. I was lucky enough to learn about this him firsthand at a MƒA workshop he ran a couple years ago. It’s dynamite. He closes with tips on designing worked examples and exploring geometric proof to show that worked examples can be effective in the most abstract mathematical settings.

A big part of what the book does right is how actionable it is. No matter where I was in my reading, I felt I could put the book down and try out what I just read. In fact, I did this several times! This was empowering. The morning after I finished reading about Michael’s classroom routine in chapter 3, for instance, I paired two of his self-explanation prompts with a worked example on trig ratios. It worked well with my students. This happened again a few days later in chapter 5 after Michael put me on to the interesting distinction between “part-task” practice and “whole-task” practice. There he offers up five adaptations of worked examples that are painfully simple to use. With a delicate sprinkling of examples from most levels of K-12 mathematics, I see Teaching Math with Examples appealing to a wide array of educators.

A theme I found throughout the book that I appreciate, and one that Micahel highlights, is how worked examples promote access. By showing a worked example, we invite more students into a discussion about the complex mathematical ideas we teach. I see this each time I use one; no matter where they are in their understanding of a concept, somehow kids get really into discussing math work. Access is a particularly important point this year since a lot of my students have been struggling to bounce back from remote learning. Worked examples have been a pathway into learning for many of them.

In the end, I came to think of the book as a field guide to worked examples. As such, the only thing missing for me is an index. This would make referencing it in the future that much easier. Heck, my cover is already tattered from flipping through it so much. It helps that it’s small enough to fit into my back pocket. As a fierce hoarder of student work, this little book is going to travel with me and come in handy for years to come.

Other than its practicality, one aspect of the book I really appreciate is how Michael marries research to classroom practice. He judiciously cites articles that have impacted how he uses worked examples, but doesn’t do it in a way that feels heavy or too academic. Though he has read a bunch of research and is considerably more informed than many, including me, he never talks above the reader. He presents research swiftly and ties it directly to his experiences teaching math. His approach is lean, focused, and encouraging. Truth is, his seemingly casual references to this article and that article make reading research feel like a thing that all teachers should be doing more of. By the time I got to the last few pages of his wonderfully annotated bibliography, I wanted to print out some articles for myself and dive in.

At one point in the book, Michael gives advice on how to design worked examples. After showcasing one that is perfectly balanced, he says, “Reading this problem feels, to me, like having my hand held by a comforting teacher, leading me right into the middle of a terrific and complex set-up for a fascinating problem.”

He is referring to a specific example, but it resonates because this is how Michael’s entire book felt to me. Chapter by chapter, I feel like he is holding my hand, leading me through the art and science of worked examples. I never feel alone or that he doesn’t see me, the reader. His writing is personal, but tactically reveals a world of worked examples that is much bigger than himself. He welcomes me into that world warmly and instructively. I didn’t find an exhaustive list of worked examples in his book, but Micheal — like any good teacher — knows this wouldn’t be helpful. Instead, he masterfully provides just the right amount of variety — and then lets my hand go. He shows me what’s possible, why it’s possible, and backs it with evidence. In the process, he gives me the confidence and ability to do it on my own. It accomplishes a lot in 126 pages.


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Meditations on a Cogen (No. 6) • Thursday, November 18, 2021

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 sixth post in the series.

New members
This was an important week for the cogen: it was the last scheduled meeting for the initial members. After a six-week commitment, it was transition time. Having mentioned this to them a few weeks ago, earlier this week I reminded them individually to identify their replacement. Who would it be? Who might be a good fit? I meant to discuss this with them at last week’s session, but got too caught up with the journals. I noticed some hesitancy for two of the students when it came to picking their replacement. I want to offer them suggestions on who to choose, but also want them to own the decision. It’s their seat at the table and they need to ensure it gets filled.

I’ll be honest, all this talk of transition and finding replacements has me a little anxious. Will we have to coordinate a new meeting time? Will there be the same amount of buy-in from the new members? Just when I was getting comfortable, there’s suddenly so many variables that could derail us.

My hope for this week was that some of the new members would be able to join us for today’s cogen. I shared this with the cogen, but knew that scheduling conflicts might prevent it from happening.

Fast forward to 2:45p today and — bam! — there was two new members sitting around the table. I was pleasantly shocked. The initial members actually recruited! They care! One of the new members even had a friend with them and asked if they could stay too. The friend was a student I taught last year and I immediately said yes. How fascinating would it be to get his perspective on things after having been in my class for an entire year? I learned that the other newbie would be sacrificing 15 minutes of basketball practice every Thursday so that she could take part in the cogen. It was only her first meeting, but it already seemed worth it. I did my best to contain my excitement as we got started.

My giddiness in check, I asked everyone how they were feeling. There were casual nods and grunts of fatigue. I empathized. I was exhausted too. Poor sleep has ravaged me this week and my eyes have been burning for the last three days. But going back to last year, my cogen has been a highlight of my week for a while now. The dialogue heals and leaves me energized. So despite my tiredness, I was elated to spend 30 minutes building with them again.

As we went around sharing about our days, I sensed awkwardness in the two new members. This was to be expected. Cogens are unusual and untraditional spaces — especially if you’re new to them. To help welcome the new members, I asked the initial members to describe the cogen and share its purpose. They talked about some of the changes that were made as a result of the cogen, like the Deltamath review assignment and the after-school tutoring scaffolds. They also mentioned the time commitment and perks. Towards the end of the shareout, one student said confidently, “He actually makes changes based on our opinion — it’s not just us talking about it.”

I must say, hearing the students talk about the cogen in this way made me proud. They believed in the space we created together and in their authority as change-makers. Our dialogues activated their ideas for improving the classroom and helped their solutions manifest in concrete ways. I don’t know where things will go from here, but for now, it seems like the cogen is making a difference.

Quizzes
I treat quizzes as bite-sized check ins. They’re based on an individual students’ understanding of a single concept or, at most, two concepts. They’re always one problem and take around five minutes to solve. I call them “quizzes,” but other teachers would probably call them exit tickets.

Anyway, this week, somewhat spontaneously, I tried two new structures for quizzes. My students have struggled this year with collaborating, so I decided to use our quizzes to help them improve. The first thing I did, on Tuesday, was to have a “Class Quiz.” I stole this idea from another teacher at my school and it asks each group to solve a problem on a large whiteboard. The problems were from the homework and there were four of them (I had four large groups, too). After students finish and have an opportunity to revise their work, they listen in as I grade the problems in front of everyone. I choose a random problem to count as the quiz grade. The grade that problem earns is what everyone in the class receives for the quiz.

The second thing I did, today, was what I called a “Group Quiz.” For this one, I gave everyone in a given group the same quiz. The quiz was one problem, just like the regular quizzes. In a twist, at the end of the class, I asked for one random student’s paper from each group. The grade this student earned on the quiz would be the grade everyone in the group received.

I’ve never used either of these quiz formats with students before, so I was curious how my students experienced them. Were the quizzes fair? Did they serve their purpose? Which format did students’ prefer? The bulk of our time in the cogen today was spent with me picking the students’ brains with these types of questions.

Interestingly, the first comment came from one of the new members. She said that, during the class quiz, another teacher who does it asks the class to vote on whether the problems should be revised (this is the teacher I stole the idea from). For mine, I made revision mandatory. Another student commented that the revision process didn’t enable her to adequately take notes on the problems. This made sense because, after putting up their solution, each group was assigned one problem to revise. After that, I graded them. If I stay with four problems, one idea that was mentioned by the crew was to have students rotate so that every group gets a chance to visit each problem before the grading. (Ironically, after this comment, I managed to do just this in one of my co-taught classes the next day.)

The kids also gave me some quality feedback on the group quiz. What stood out most to them was how hard they worked on it. They were hovering over one another feverishly checking each others’ papers to ensure that everyone’s work was representative of the group. They felt pressure to work together and praised this aspect of the quiz. While the students appreciated being forced into collaboration of that degree, they said I should have ensured more time for them to work (many had to stay a few minutes after the bell to finish). Because I collected one random students’ paper, one cogen member pointed out that he was uncomfortable with his grade being dependent on another student. This was a fair critique.

Overall, the students appreciated the spiced-up versions of the quizzes. They recognized that the quizzes gave them a better understanding of the mathematics as compared to a traditional quiz: they got to communicate about it and compare ideas in the moment. They wanted me to use both again. We acknowledged that the students’ grades on these quizzes are not always going to be indicative of what each individual knows. This can be unfair and inequitable. From a teacher’s point of view, these quizzes also make it impossible for me to assess individual students’ understanding of the mathematics we learn. That said, I commented that my quizzes are a relatively low-stakes assessment; you have to take five of them for it to be equivalent to an exam. I reminded them how hard they worked on the quizzes; sometimes it’s worth sacrificing individual accountability if what results can promote cohesion and teamwork. In a system focused on individuality and students working in silos, our talk today evoked the “Are you responsible for the others’ learning?” conversation the cogen had a while back.

Near the end, I asked the students to choose which quiz format they preferred. The group quiz won 5-2. One students’ comment after we did this made me think: higher-performing students probably prefer the class quiz over the group quiz because they have more influence over what gets graded; they can write the correct work on the whiteboards. I left today’s cogen searching for small tweaks to make both quizzes better. The students’ criticality today definitely opened the door for that. The quizzes will surely be a topic of discussion at a future cogen.

Revisiting the math journals
At last week’s cogen, the students had a hearty discussion about the upcoming math journal assignment. They showered me with suggestions about it and I told them I would use their feedback to create a draft of the assignment. In the last 10 minutes of today’s session, I presented them with the assignment and asked for their feedback before I finalize it and assign it to my classes.

The students liked what they saw. They felt their ideas were reflected in the options students had for writing the journal. I reiterated that in previous years there was only one option for the journal and thanked them for their input in opening it up.

Saying goodbye
It was bittersweet that I had to say goodbye to some of the initial members at today’s session. I told the cogen today that these students have officially graduated and are now “cogen alumni.” The departing students asked if they could come back in the future and I replied with an enthusiastic yes. They will always be welcome; they are the founding members, where it call began. They filled me with the conviction that in-person cogens can and should happen. Truth is, these students have set the tone for an entire year of critical conversations and student-empowered decisions about our class. I couldn’t be more thankful for them. The day after today’s cogen (Friday), one of the graduating cogen students wrote me a Friday Letter that warmed my heart:


Next week is Thanksgiving break and I’ve decided that instead of holding a cogen, I will try and touch with the students individually as best I can. I’m doing something funky with next week’s exam and I’ll need their feedback.

bp




Meditations on a Cogen (No. 5) • Friday, November 12, 2021

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 fifth post in the series.

Lunch and a debate
We usually meet on Thursdays, but because of Veteran’s Day, that couldn’t happen this week. I remembered that we all have lunch 4th period and asked my cogen students if they could do Friday during lunch. They said ok. I don’t like sandwiching the cogen between classes like this, but I had to take it over not meeting at all. I spiritedly promised to bring the students Slim Jims for their flexibility, but ended up forgetting them. I now owe them double next week.

As I began dumping all the snacks onto the table, the kids trickled in. Two forgot that we were meeting today, but someone ran down to the cafeteria to get them. After a few minutes, we were all sitting around the table again. This was our 5th session together. I was unexpectedly comforted by this.

We checked in on how everyone was feeling. As a means of breaking the ice, I like picking up on whatever energy, vibe, or idea reveals itself in a given moment. I tend to be pretty observant, so it’s not that hard. I do this in class, too. It’s my way of staying present, acknowledging one another, and warming up to dialogue. Today, one of the students brought a cup of water they got from the cafeteria. For whatever reason, something spoke to me about that cup and we spent the next few minutes airly debating water dispensers vs. water fountains. It was fun.

Collective brainpower
We opened the formal dialogue by recapping what we discussed last week. I mentioned the feedback quiz that I tried last Friday (the day after our last cogen), and how that led to Monday’s lesson on trig ratios in the coordinate plane. I used VNPS and VRG for that lesson — which is the structure the students told me they learned a lot from. We ended up spending two days this week using it as well, which the students confirmed were effective. Last Friday the cogen also persuaded me to bring out the dry erase sleeves for worked examples and practice problems in small groups. I botched the timing so it wasn’t the best lesson, but it was cool. I don’t think the cogen students even realize it, but I often use worked examples at the start of class now. I didn’t mention this last point at the cogen today, but it’s interesting how our talks are impacting class in unforeseeable ways. Our ideas are helping me rethink areas of the class that we don’t explicitly touch on in our dialogues.

At one point during our recap, one student pointed out how another student was responsible for a couple of the ideas. “That was from her…” the student said casually a couple of times. I was proud that the student vividly remembered our last talk, but commented that those ideas, though one person may have said them, were the direct result of our dialogue. Our collective brainpower, problem solving abilities, and creativity birthed them; through our dialogue, we co-created the context that enabled the solutions to emerge. The students nodded.

Making journals relevant
I left last week’s cogen desperately wanting to discuss metacognitive math journals with the group, so that’s what we did for the remainder of today’s talk. I’ve done metacognitive journaling with my students for the last several years and have even used editorial boards to peer-review them, but the journals themselves tend to be fairly dull. They need some sprucing up.

I overviewed the journal process to the cogen and asked for their suggestions. I told them I was looking for ways to make it more meaningful, but also for ways to make it fun and engaging. After some clarifying questions, they let loose. One recommendation was that students turn the journal into a written dialogue between two people. If this is the case, maybe students should be permitted to work in pairs to do the journal? This is a fascinating idea! Other ideas were allowing the journal to be a letter to self or even a short story. The journal requires a student to reflect on their solution pathway to a problem, but could they integrate drawings or other art to do this? Instead of students choosing any problem (what I have always done), one student commented that they should have to choose between a handful of challenging problems that I curate for them. In years past, I have required that the journal entries come from our standard problem set, but why can’t it be one from an exam that the entire class struggled with? Or an entirely new problem they haven’t seen yet? Our recent usage of worked examples even made me think about students using one for the journal. Why not? Given the right problem and work, it could make for juicy reflections.

An assumption throughout our dialogue and their recommendations was choice. In my three years of doing journaling, crazily enough, I never considered giving students a choice in the matter. But why not? Reflecting on one’s mathematical thinking should not be uniform. Putting their ideas in a box is probably why the journals have failed to come alive for my students these last few years. I need to throw out the box. As we wrapped up, I thanked the students and them that I would use the ideas we generated today to create an updated version of the journal assignment. I would present it to the cogen next week to get their feedback before I assigned it to all my classes.

During our discussion, I noticed one student in the group was kind of quiet and I encouraged him to share what he was thinking. He remarked that he didn’t really like writing. I sensed that he wasn’t looking forward to this assignment and withdrew from giving me feedback on it. I respected this, but told him this is precisely why I wanted him to give his opinion. What could make the journal worthwhile for him? At a minimum, what could make it less annoying for him to do? Of all the students sitting around the table, his opinion mattered a lot because of his overall disdain for writing. I even followed up with him after the cogen to reiterate this message and hear his thoughts.

I got the feeling that, for the cogen students, they felt they didn’t do much today. They were just offering up their opinion on some writing assignment. This was true, they were. But I shared that I never considered any of these types of ideas in past years. There have been hundreds of students who have written journals in the past — and all of them were confined to one way of expressing their metacognitive reflections. Our conversation today changed that. As a result of our talk today, I hope more students will find meaning in the assignment.

Post-cogen
After the students left, I noticed a colleague in the room. I believe she was there the entire time. I explained to her what it was. She remarked that she really liked the idea and wished she was “at the point” where she could do something like it in the future. We chatted briefly about the difference between giving kids a survey to gather their feedback and being in dialogue with them. One provides real-time opportunities for follow-up questions, divergent thinking, and adjacent ideas, and the other one doesn’t. Our exchange reminded of me last year when I asked a few colleagues to join the cogen to provide me with an outsider’s point of view.

Last two things. Our meeting today felt a tad rushed and it caused me to forget to inform the students that they need to choose their replacement next week. I emailed them afterward about this. All good things must come to an end. Secondly, I realized that we had an exam this week and didn’t revisit the DeltaMath review/exam score correlation idea that was brought up a couple weeks back. Today would have been the perfect time to discuss it, but I had to prioritize the journals. So much to discuss, so little time.

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Deprioritizing relationship building

In listening to colleagues tell it this year, I’m getting vibes that people are overwhelmed. Students have skill gaps because of remote learning, we’re all rediscovering school again, and the copy machine is always breaking down. Many teachers are feeling the strain.

What’s really interesting to me is how this plays out in the classroom. What I’m finding is colleagues who are struggling to do the one thing they said was their biggest takeaway from remote learning: foster relationships with students. They’re unable to nurture strong relationships with students because there’s no time. They’re too busy reteaching lessons that went poorly or encouraging students to show up for tutoring. We left last year believing that bonding with students, listening to them, and earning their trust was of chief importance, but now we’re up to our neck in content. We want to do it all, but the kids are behind. Something has to give.

What I find deeply troubling here is the idea that relationships are an aspect of teaching that can be skimmed off the top when things get hard. That without them teaching and learning can still occur in meaningful ways. When this happens — when we deprioritize relationship building — I think we lose a key ingredient of what makes for quality teaching. Though they are often unobservable and hard to measure, student-teacher relationships are not fat that should be trimmed in order to make our teaching lean and more fit. For me, they are what makes teaching teaching. They are essential.

Students may be struggling to meet content standards right now, but that doesn’t mean I can sacrifice getting to know them and opening non-academic lines of communication. I can’t look past who they are in favor of seeing my content in greater clarity. This only hurts my cause of helping my students use mathematics as a vehicle for personal growth. During these challenging times, it’s critical that I fight the urge to hustle my students along to a higher test score. My relationship-centered pedagogy and routines must remain firmly intact. In the end, by moving them to the fore, this will not only fortify my teaching, but also nourish me and serve as an antidote for the stress I’m under this year. Exercising compassion is cathartic. It’s a win-win.


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