Category: pedagogy

My experiences at the Exeter Mathematics Institute

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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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I’ve contracted a SEVERE throat infection. I cannot speak at all.

I recently wore this sign on my neck during class. I placed it around my neck and went about the business of being the teacher in the classroom.

At first there was high levels of concern. There were comments like “Are you ok?” or  “Stay away from me, I don’t want it.”

And then one student remembered to having seen my speaking just fine earlier in the day. There were snickers, laughs, and smiles all around. They were picking up what I was putting down.

I silently gestured them through a slide that explained that although I couldn’t verbally communicate, this didn’t mean that they couldn’t learn. There was a handout. They have brains. They had each other. No excuses. The focus was recursive sequences.

I don’t think this idea would work well with any class, but it was an awesome experience with those I tried it with.

There were natural leaders who volunteered to demo examples on the board for the class. Some kickstarted conversations about what they noticed and wondered. There were others who made their way across the room to seek help or give it to other groups. And yet others did their thing on their own or with a partner.

Throughout all of this, I said nothing and did very little. For the entire period I peered over shoulders, pointed out (literally) interesting steps in student work, and wrote on the board the time remaining before the exit slip was to be administered. I reminded them (with a slide) that they needed to ensure everyone in the class could be successful on the exit slip, as it was the measure of their success as a team. (They ended up being 90% proficient.)

I closed by reiterating with them the fact they need me far less than they think they do. I’m no gatekeeper of knowledge.

 

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Developing rational exponents through geometric sequences

Soon I’ll be teaching rational exponents in algebra 2. I’ve never found an intuitive way of teaching it…until now. Thanks Bowen.

The approach leverages geometric sequences. I’d love to regurgitate it, but this tweet from Bowen is my source and sums it up:

The unit prior focused on sequences with a heavy emphasis on geometric sequences, so this is the perfect bridge to developing this idea that most students find confusing. It all comes back to repeated multiplication, as it should.

In the past, I’ve usually had students enter various expressions (e.g. 100^(1/2)) into their calculators to stumble upon the relationship between rational exponents and radicals:

But this painfully ignores the mathematics behind exponentiation and instead lures them into believing that these two concepts are magically connected through a few keystrokes on their calculator. It treats rational exponents as an isolated concept and unrelated to repeated multiplication.

After discovering my new strategy for teaching rational exponents, I found this video from Vi Hart on logarithms. The similarities run deep.

 

Now I can’t wait to teach logarithms.

 

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Checkpoints and homework, circa 2016

Here’s my current structure for exams checkpoints and homework. Everything is a work in progress.

Checkpoints

  • First off, terminology. Formally known as exams, I now call these summative assessments ‘checkpoints’ to further establish a low-stakes classroom culture. It feels much less formal, but I still reference them as ‘exams’ when in a rush. Plus, my frustration with the Regents exams is at an all-time high, so distancing myself and my students from any term that references them is a good thing.
  • I really liked how I lagged things last year, so I’m going to continue with this routine. This means that each checkpoint will only assess learning from a previous unit. In most instances this will be the previous unit, but once a month there will be a checkpoint that only assesses learning from material learned at least two units back. With my standards-based grading, students can lose proficiency on a standard at any time during the course of the year. The hope is to interweave what has been learned with what is currently being learned to help improve retention.
  • Speaking of SBG, I’m reinstituting mastery level achievement in 2016-17. I have yet to work out the kinks regarding how this will impact report card grades.
  • I will not review before any checkpoint, which is what I started last year. Instead, that time will be spent afterwards to reflect and relearn.
  • I make these assessments relatively short, they take students roughly 25-30 minutes to complete…but my class period is 45 minutes. I’m still trying to figure out how to best use that first 15 minutes. Last year I didn’t have this problem because my checkpoints always fell on a shortened, 35-minute period. Right now I’m debating over some sort of reflection or peer review time.
  • I have begun requiring advanced reservation for every after school tutoring or retake session. I learned very quickly at my new school that if I don’t limit the attendance, it is far too hectic to give thoughtful attention to attendees. Right now, I’m capping attendance at 15 students per day with priority given to those who need the most help.

Homework

  • Disclaimer: developing a respectable system for homework is a goal of mine this year.
  • Homework assignments are two-fold. First, students will have daily assignments from our unit packet that are checked for completion the next day. Second, they will have a DeltaMath assignment that is due at the end of the unit, again, checked for completion.
  • Homework is never accepted late.
  • Homework is not collected.
  • To check the daily homework, I walk around with my clipboard during the bell ringer. While checking, I attempt to address individual questions students may have. This serves as a formative assessment for me gauge where they are on the homework. After the bell ringer, but before any new material, I hope to have student-led discussion around representative problems, depending on the homework that day (I haven’t gotten here yet). The goal is to have students write on the board the numbers of the problems that gave them a headache…so we know which ones to discuss.
  • I’m going to do everything I can check it this year. It sounds simple, but over time things can slip away from any teacher.
  • I’m posting worked out homework solutions on our class website. I used to include the solutions in the back of the unit packet. This is an improvement on that, but also requires students take an extra step. Students must check their thinking, assess themselves against the solutions, and indicate next to each problem whether or not they arrived at the solution.

 

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