problem based learning

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.”

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.

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.

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.