Updates on my problem-based experiment in algebra 2

As a means of tracking the progress I make in my newfound problem-centered classroom, I’m posting some recent developments and thoughts. These notes are incredibly informal and far from polished.

  • I’ve settled on assigning 5-6 problems for homework. When they come to class, I give the groups 20-25 minutes to peer review and make sense of the problems. They show their work on the large whiteboards around the room so everyone can see. As a class we then spend the last 10-15 minutes of the period in a whole group discussion with students presenting their solutions on the large whiteboards.
  • I’m now thinking…why can’t I use visibly-random groups as they peer review the problems??
  • I need to do a better job of establishing coherence within the problems. For the first 20 problems or so, students feel like they were doing random problems covering unconnected concepts. In some ways, they were since I was trying to establish some norms and routines through the problems.
    • Admittedly, the first 20 problems lacked coherence (and therefore meaning). It’s ok to intersperse concepts, but I should have a focus (or foci) for each problem string we go through.
    • It seems around 20 problems is a fair amount for each exam to assess.
  • Duh: class size matters! Periods 1 and 8 both downsized and it made a world of difference. I now have groups of around 5-6 discussing the problems. It’s only been a few days, but this has been so much more effective than the whole-class discussions we had at the onset. As I visit groups, small group instruction is the norm. I’m doing my best to simply ask questions and avoid direct instruction on the problems. I think I need a develop a simple protocol to follow when I approach a group.
    • One thing I should get back into is asking for “group questions” only. There are too many students doing their own thing and not all students in the groups are actively discussing the same problems each day. I need to push this more.
  • After emphasizing problem-solving and group discussion ahead of answers, I started providing correct answers on the board halfway through the period. Students were uncomfortable because there was too much ambiguity in final answers (thank you high-stakes exams), especially since sometimes I can’t get around to everyone’s work.
  • I am worried about the more introverted students in the class, those not openly engaging in group discussions. At times they seem to not be engaged.
  • How should my exit slip or “closing” to each day look? Note: I need to make time for this.
  • I haven’t been surfacing problem-solving strategies as students work through problems. Related: there hasn’t been a lot of focus on the various ways and perspectives to solve these problems.
  • I need to organize a day/lesson where students purposely make connections between problems and establish big ideas for the course.
    • Makes me think of Dan Meyer’s co-authoring the class post. I’m thinking we, as a class, can create a large concept map on the wall with paper and string making connections between key concepts and problems. In this way, instead of me saying, “all these problems belong to unit 7, exponential functions,” students can surface these sorts mathematical connections for themselves and own the content. That’s the dream, anyhow.
    • Maybe start with a table with columns for problems, big ideas, key vocabulary?
  • I’m allowing for students to create a 3×5 index card for use on the exams. I don’t do review days before exams so this is my way of getting them to prepare. It also forces me to think creatively about the problems I include on exams!
  • To break up the monotony of this structure, I need to begin planning lessons that don’t revolve the same sort of group discussions. I also want students to see that class won’t always look the same.
  • I have seen whiteboards being used very effectively. Student thinking is public. At times, students are moving freely around the room to independently seek out methods and strategies.
  • With these 12 whiteboards being actively used in every part of the room, I think I have successfully defronted the room. That’s a win.
  • Because the boardwork students are doing is so important, and since students can’t use their phones in my school, at the end of the period I want a student to take photos of the boards using an iPad. They would then email it the class. This would alleviate students’ feverishly copying correct work into their notes during the whole class discussion.
  • Another thing so far that I love is that the class has been focusing on doing and actively engaging with mathematics. Plus, there’s been lots and lots of struggle with the problems. That’s great, but now I just my students to be comfortable with being uncomfortable. Hopefully in time.
  • I managed to set up a spreadsheet aligning the problems to the standards-based grading “concepts” that I used last year. Although I don’t share this with students, I’m using it to guide the problem strings that I write.
  • I’m still far away of student buy-in — which I desperately need. This is due in part because of the rough start I had in sequencing the first series of problems.



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I threw all of my units out the window.

So I’m noticing a trend. it seems like every few years I have an epiphany that causes me to blow up my teaching and rethink what I’m doing in a major way.

Case in point, six years ago I flipped my classroom and realized what is really important when it comes to learning. Three years ago I implemented standards-based grading and learned how to be more analytical with assessment. I now find myself smack in the middle of another major shift in my practice: problem-based learning.

Attended the Exeter Mathematics Institute in August was the catalyst. Experiencing a purely problem-based classroom was new. I had known the “PBL” buzzword for a long time and thought I understood what it meant. I didn’t.

Here’s the workflow: Students explore problems for homework and we use the entire next period analyzing and discuss them. The problems are designed to enable key ideas to organically emerge during homework and class discussions. There are no units. No direct instruction. This is what they call the Harkness Method, I think.

Now I find myself thinking through and sequencing the problems I give my students like never before. This has been pretty fun. All problems need to be inherently scaffolded and since they are now a learning experience (and not just practice), they are everything. Well, I shouldn’t say everything, because the class discussions are crucial too…but without the problems, you have no meaningful discussions.

Without knowing it, I think I have been moving towards PBL for a while now. For a few years, I have been trying to think about sequencing questions/prompts to naturally guide students towards a learning objective — so many of my problems have come from handouts that I’ve developed through the years. Now I’m finding myself weaving these prompts/problems together that is problem-based and not concept-based.

And about the class discussions, that’s something that I feel I’ll be tweaking with throughout the course of this year. I’ve started out doing whole-class discussions and, with classes of 30+ students, I watched as equity quickly crashed and burned. Kids were hiding their ideas and drifting off. I’m now transitioning to smaller groups of around 6-8. I plan to move around the room to guide the group discussions. I’m still debating whether I should give solutions. Maybe towards the end of class to avoid it being a conversation killer?

If I’m honest, I’m worried. I have no idea how this will go and I’m pretty sure that I may have bitten off more than I can chew. I really believe in the process, but this is a pretty drastic change. Because I have no well-defined arrangement to the curriculum, my SBG is gone. And did I mention that I have no units?!

Patience, be with me.



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My experiences at the Exeter Mathematics Institute


For three and half days this week, I had the opportunity to participate in the Exeter Math Institute.

It took place at the Spence School, an illustrious independent school on the upper east side. I’ve visited the school on a few different occasions, and it always makes me gasp. From carpeted classrooms, busts of historic figures, marble staircases, and a grandfather clock in the welcome hall, in many ways it feels more like a museum than any school that I’m accustomed to.

Getting past my awe, I quickly learned on day 1 of the institute that this would be very different than any other professional development that I’ve experienced. The focus isn’t so much pedagogy or even math pedagogy. The facilitator, Gwenneth Coogan (who I later learned is a former Olympic athlete), was set to immerse us in a Harkness mathematics classroom for three-and-a-half days. Harkness is problem-based, so that meant that I was going to be doing a lot of math — which was actually the whole point of attending. I feel that I negatively impact my students by not mathematically challenging myself on a regular basis. Plus, I’ve heard nothing but rave reviews of the Exeter problem sets. (We worked on Mathematics 2.)

*Notes about Gwen: She had no slides. We used Desmos from time to time, but at no point did she even think about using a projector. This was refreshing as she moved us to be in the moment. Flow, anyone? Also, I found her to be incredibly personable and welcoming. Through all my struggles she provided a warm smile and wholehearted encouragement.

An unexpectedly pleasant aspect of the PD was the fact that I got to collaborate with both public and private math teachers. Rubbing shoulders with them, listening, and sharing stories was so helpful. I now wonder why more PD doesn’t cross over these public-private boundaries. Interestingly, despite Harkness being typically found in elite private schools with class sizes of 8-12 students, I learned from Gwen that Exeter’s goal is actually to develop Harkness in public schools (whose class sizes, to say the least, are not 8-12 students). With that said, there were only 8 of us at this EMI, an intimate little group. Admittedly, this helped the conversations get deep and stay deep. Call me crazy, but by the end of the institute, I thought of asking my principal if we could host an EMI at my school next summer. Why not?

Knowing very little about the Harkness method, being immersed in it taught me a lot about how it works and why it can be successful. Through independent exploration and group communication, students use problem solving to explore and learn mathematical concepts. The teacher isn’t the focus, as they’re just another person in the room who helps spur discussion. The mathematics and the interdependent nature of the class are everything. There are no prescribed notes or detailed lessons, just carefully planned problem strings that help unlock mathematical ideas for students. There is a sequence for the course (I think), but there are no units, per se. Concepts are interwoven into problems and uncovered by students little-by-little over the course of the school year. The result is unbelievably high levels of student ownership of learning. Experiencing it firsthand, it was truly liberating.

I do have a couple reservations. First, how the heck am I make work for a class of 34 students? Putting motivation aside (like, yeah), a rich class discussion is what truly makes Harkness thrive. Having high expectations is one thing, but to what extent can my 30 students have discussions at the same level of sophistication as a class of 12? I’m on board with PBL and Harkness, but that worries me. Second, selecting problem sets is critical in Harkness, and many Harkness teachers actually write their own. I may be the minority, but writing my own problems is not realistic — especially the type of problems that have a variety of solution pathways and generate real learning based on integrated mathematics. And thanks to the Common Core, I know that I can’t use the Exeter problem sets straight up. Lastly, I have a feeling that by shifting to a nonlinear problem-based approach (instead of unit-based, which is more linear), may throw my standards-based grading system for a whirl. What do I do???

Like much of anything we do as teachers do, much of my implementation of a Harkness- style of teaching and learning will rest on lots of tweaks and adjustments over time that will make it effective for students that I teach. I’ll start small and hope for the best. Geoff’s PBL curriculum might also be a big help.

A closing thought. In a Harkness classroom, there are boards all around the outside of the room. A powerful feature of the class — and one that captures the heart of what Harkness represents — is a message that Gwen relays to her students early and often: the boards are you for you, not me. In other words, the board space is used strictly for showing student thinking. It encourages students to be vulnerable, to get things wrong. I made progress in this area last year with VNPS — PBL and Harkness seem like a natural next step.



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On the intersection of being White and being a math teacher

Two of the books that I read this summer, Why are the All the Black Kids Sitting Together in the Cafeteria by Beverly Daniel Tatum and Blindspot by Mahzarin R. Banaj and Anthony G. Greenwald, utterly blew my mind. 

So while the summer winds down, I’ll leave it at the end of this month with so many concerns about my teaching and how I address racism.

A. For my entire life, just like a lot of other White people in this country, I considered myself colorblind. I claimed that I was blind to the race of the people I interacted with. And I took a lot of pride in this fact, too. If I didn’t see people’s race, I couldn’t discriminate or play favorites. I conned myself into this line of thinking. Being born and raised in the inner city, and one of the few White people in my neighborhood and school, race wasn’t a “thing” for me. This perspective continued into adulthood and pervaded my teaching. Even with students, I either claimed the colorblind stance or simply avoided conversations about race. It isn’t until now that I realize that this is, and was a huge, huge problem.

B. Most White people don’t think we live in a racist society. Most White teachers don’t either. But we do. I’m not talking about outspoken racism, like that of white nationalists. I’m referencing the systematic racism that pervades in the air we breathe here in America. In many ways, we choose to not think about it because it’s uncomfortable. White privilege is a very real thing, even if we chose to look the other way. It existent in every aspect of society. Most White people don’t see it this way because we are (myself included) inside the box — we are part of the dominant group. That inherently makes it harder to understand the advantages we have.

C. What’s especially damaging about this is that every single White teacher I know is a good person. They don’t intentionally aim to do harm to students of color. Heck, most of these teachers teach in schools with large proportions of students of color because they want to help interrupt the cycle of inequality and injustice that these kids experience. But our hidden biases, which strongly favor our culture of Whiteness, can still significantly affect our judgment in ways that we aren’t even aware of.

D. What does this mean? It means that if we teachers (and especially our school leaders) don’t develop an anti-racist stance that fosters a critical consciousness about life being more than White privilege, our schools and classrooms will be a mere reflection of the racist society in which we live. It means that if we don’t mindfully recognize the systemic racism that our students of color, and colleagues for that matter, encounter every day, how can we attempt to take a chance at interrupting it?

E. So how do we, as teachers, bring up such a sensitive topic with colleagues and administrators to help push the needle in the right direction? There’s fear, dread, and detachment in people’s eyes (not just White people, either) whenever race is brought up. I know because it used to happen to me. I have no idea how to address this, but I think open, safe conversations with one another are vitally important — like at staff and department meetings. Provocative, reflective prompts are needed (Jose and Wendy!). A simple discussion can go a long way. Norms need to be set. I would hope that administrators can be present and active. Anxiety is natural, but I like to think that if we’re sincere and honor one another, the right words will always find their way out of our mouths.

F. Self-discovery might also help. Here are various research-based tests that we can take online to help determine each of our hidden biases. They are called Implicit Association Tests. Here’s some background on them.

G. I don’t know, I don’t know. I’m at a loss here. I’m no expert on how to make this happen. Progress seems so far away, but this post is a start for me, I suppose. A grueling and uncomfortable path lay ahead.

H. One more thing that I want to add. Right now, 75% of my mathematics department at my school is White male. That bothers me. At times, I worry about the subliminal messages that this sends the 90% of students at my school who are Black or Latino — especially if we (White males) aren’t actively taking an anti-racist approach to teaching and learning mathematics.



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Happy third birthday, lazy0ch0. Go celebrate.


This blog is turning three years old in a few days. It’s still a toddler in human years and it doesn’t seem that long – until I realize that I’ve now been blogging for 27% of my teaching career.

The first thing that comes to mind is that I still write entirely for myself. Everything here, even the list of what books I’ve read, is for my own personal reference. Two years ago, I thought that I might get over writing for purely reflective purposes. I haven’t. I guess I’m selfish when it comes to writing. Once every six months, someone in real life brings up my blog to me in person. I’m always flattered, but I’m also surprised because:

  1. they actually read my blog
  2. the thrill I get from writing about my work as a teacher regularly makes me forget that everything here is public

Speaking of being public, sometimes I think that if I’m writing for me, why don’t I just do it in a journal or possibly make all my posts private? As anyone that writes publically will agree, there’s a high level of accountability that comes from clicking PUBLISH. In my case, that accountability rests on my own shoulders…and what I hope to represent as a teacher of mathematics. Because the world can read my message, by openly publishing, I’m holding myself to a pretty high standard. That’s kind of scary, but it’s also really empowering.

It’s been fun to see my writing evolve. Sometimes I look back at my early posts and realize how much I’ve changed both as a teacher and a teacher who writes. I’m a far more thoughtful and proactive teacher these days. I value my development much more and I’m certainly more socially concious. I’m also much more casual with my writing. I used to meticulously craft my posts. I used to edit heavily. Now, I’ll do a once-over, but I very much honor the informal nature of my posts. And not that I was all imagery-centric before, but I also don’t include as many photos or images as I used to.

There are truckloads of thoughts related to teaching in my head at any given time, but I usually have around 2-3 serious ideas for posts lined up. Most times they need to marinate in my head a while before I can tap them out using my keyboard. Other times, it’s more immediate. Regardless, if I haven’t published in a couple of weeks, I get an inkling to write. I get antsy. Ideas start to bottleneck. This lets me know that I need to let those ideas, no matter big or small, breathe on my blog.

Here’s to another three years of breathing.



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