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.



Engaging tasks for students that I’ve never met

Anyone who teaches in New York City Public Schools knows that from time to time you get asked to cover a class when a teacher is absent. Personally, despite the hectic nature that is a school day, I typically enjoy these coverages mainly because I get to meet and interact with lots of students that I either don’t know or don’t teach.

Lots of times, there is an absentee lesson plan, but many times there’s not. What’s the result then? Me in front of a class of 30 adolescents for 45 minutes with nothing to offer them. Last year I realized that I was tired of this. Here is an attempted remedy.

I’m want to compile mathematics tasks meant engage students that I’ve never met before. Since I teach high school, I will assume nothing about prior knowledge or motivation besides the students being in grades 9-12. The tasks should be highly accessible. Also, in terms of materials, I’ll have nothing but a whiteboard and/or Smartboard at my disposal. I love mathematics, but since I’m not the best at coming up with stuff on the spot, this page will be a necessary resource for me. It’ll be updated regularly whenever I come across worthwhile ideas.

  • The Four 4’s. Express the numbers 1-20 using only four 4’s and any set of operations. Additional challenge: express the numbers 21-???)
  • Similar to The Four 4’s: Using each of the digits 1, 2, 3, and 4, once and only once, with the basic rules of arithmetic (+, –, x , ÷, and parentheses), express all of the integers from 1 to 25. Source.
  • Sprouts | A fun game that involves nothing but a pencil and paper. Get’s deep.
  • The password riddle | Connect the computer to Smartboard to show video. There are loads more like this from Ted-Ed.
  • What comes next? O, T, T, F, F, S, … | A clever little sequence.
  • Which One Doesn’t Belong? (Numbers and Shapes) | These give every student an opportunity to show off their mathematical perspective.
  • Add seven subtract one | A great problem to promote numeracy.
  • Are there any operations that make the equation 5   5   5   5  = 19 true? (Source)
  • Variable analysis game.
  • Find as many patterns as you can in Pascal’s Triangle.
  • Various problems from the Man Who Counted (book) by Malba Tahan


VNPS, VRG, and creating flow

Last summer at TMC16, I learned about vertical non-permanent surfaces (VNPS) and visible random groupings (VRG) from Alex Overwijk, which is based on the work of Peter Liljedahl. Despite the explosion of ideas that I came across at the conference, I knew that I had to implement these two.

So after the conference I visited my new school to poke around the classrooms where I was to teach to see what the whiteboard situation was like. I was discouraged. My school is a converted elementary school with gigantic windows, lots of cubbies, and lockers that take up half of the walls. Disappointed, my hopes for VNPS and VRG slowly faded away.

Fast forward to last week. We were studying advanced factoring and the pains of heavy algebraic manipulation and computation lurked. I’m not sure what triggered me to rekindle my excitement, but I reread Alex’s post and slides from TMC and decided dive in.

Each of the rooms I teach in have some whiteboard space already, but I still needed several large whiteboards. I had no time to get to Home Depot. Then I remembered seeing the physics teacher having some. He’s probably the kindest teacher in the building. He let me borrow them with open arms.

I found some guidelines from Laura Wheeler (more here) and away I went into the world of VNPS and VRG.

I randomly assigned 2-3 students to each board. I displayed the expressions that I wanted them to factor on the board and the groups immediately jumped in. The level of complexity grew slowly with each expression, some of which they had never seen before.

The clearly visible work allowed me to efficiently assess everyone in the room. I gave some hints, but I wasn’t needed much. When I felt a group was hitting a wall, I outwardly moved someone to their group who could help. With their knowledge now mobile, their insights spread throughout the room like wildfire. And despite calling out “switch” periodically to keep the marker bouncing between group members, I also moved students who looked to be disengaged in their group.

This was all to maintain optimal levels of engagement, or flow. It worked like a charm.

At the end of each period, rather than looking finished, my students looked recharged. They wanted more. I couldn’t count the number of students that declared how much they loved the structure. They were doing like never before, completely lost in the work for over 30 minutes.

I can say the same for me. I felt my senses heighten as I feverishly assessed the students. I was completely in sync with their thinking. As 30 students openly crisscrossed the room to collaborate and build on each other’s ideas, I knew that my classroom would never be the same again. It was like magic. What Alex described as flow back at TMC16 is exactly what my students and I experienced.

That was Tuesday. Thrilled, I’ve used VNPS and VRG every day since with no plans to slow down.

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