PCMI Part 1: Morning Math

This summer, I spent three weeks in Park City, Utah as part of the Park City Math Institute (PCMI) Teacher Leader Program. PCMI is a teacher math camp of sorts, and is sort of like if PROMYS for Teachers and Twitter Math Camp had a baby. PCMI captures the intense but thrilling mathematical experience of PROMYS with the pedagogical reflectiveness and social bonding of TMC. This post focuses on the mathematical experience.  

What is Morning Math?

Every morning at PCMI, we did two hours of math. We sat in assigned groups of six, and were given a new problem set each day. The problem sets were masterfully written and facilitated by Bowen and Darryl. These problems sets were so interesting and engaging that we often booed when told to stop or take a break. Bowen and Darryl wisely warned us that this experience was not meant to be a literal model for own classrooms, but then we were left to wonder, as a teacher, what should I learn from this awesome experience?

What kind of math were we doing? 

I guess we were studying number theory, but it feels wrong to say there was one topic. Instead, it felt like we were bouncing between the topics below and slowly seeing connections between them.

  • Approximating irrational numbers
  • A laser bouncing around a rectangular room
  • The greatest common divisor of two numbers
  • Cutting squares out of rectangles
  • Continuous fractions
  • The Area of Parallelograms in the coordinate plane
  • Linear and quadratic Diophantine equations
  • Numbers with friends and “bad adding”
  • Lattice points on hyperbolas

But the content really wasn’t the point – the experience was worthwhile because of how we were thinking, not what we were thinking about.

How was morning math facilitated?

Bowen and Darryl’s style was most distinctive for how little talking they did. On the first day, they introduced us to the norms, but after that, they never read us directions – they passed out the problem sets and just told us to “do math.” There were several reasons they stayed quiet.

  1. They wanted to maximize the time we were doing math.
  2. They wanted to maximize the time they were doing formative assessment.
  3. And most importantly, they did not want to rob us of the “joy of discovery.” Bowen once said that he “will be supportive but not answer your questions.”

Only a few times did Bowen and Darryl pull the whole group together to solidify what we had learned. When they did, they focused on content everyone had already experienced (what Peter Liljedahl calls “leveling to the bottom”).

How were the problem sets structured?

The problem sets were split into four sections: opener, important stuff, neat stuff, and tough stuff. The opener and important stuff set the stage with a low floor and the neat stuff and tough stuff raised the ceiling. Nobody ever finished all of the problems, and that was liberating. We were free to spend the whole session playing with one problem, taking our time, trying multiple methods, and exploring our own extensions until we got bored or stuck. Sometimes, dwelling led us in an interesting and unplanned direction. Other times, we felt an odd joy at seeing a problem we had already asked ourselves on the next day’s problem set.

In a weird way, trying another problem was the main source of scaffolding. There was no direct instruction –  when we needed to know something, Bowen and Darryl created a problem in which we needed to discover something new. Problems were not a burden but a path toward a new discovery. We could always depend on the opener and important stuff to help us discover something clear and connected.

Did everything wrap up nicely in the end?

The last couple problem sets asked us to list connections, recall problems from early on, and create our own review questions. Those prompts gave us a chance to notice and appreciate all the connections we had made along the way.

Only on the last couple days were we introduced to the formal vocabulary and historical context for what we had been doing for three weeks. Bowen explained why they save vocab for the end with an analogy about meeting new people, and it really resonated with me. Often, when I meet someone, we introduce ourselves, talk for a while, and then I feel awkward because I forgot their name. It is only when I have talked to them for a while that I will remember their name, and it is only after I have played with some math for a while that I am ready to learn its vocabulary.

How did morning math compare to PROMYS?

In style, morning math felt like a bite-sized PROMYS. In content, this year’s PCMI overlapped significantly with PROMYS, which had its pros and cons. I enjoyed reflecting on what I remembered and actually understood from PROMYS (like continued fractions of square roots) and what I had forgotten (e.g. the magic box) because it was a procedure that I had used without understanding. Sometimes the familiarity of the content was frustrating for me. It was hard for me to collaborate authentically when other people were seeing this content for the first time, and I felt like I needed to remind people that I was picking up on things quickly and going ahead because I had previous experience and not because I was “smart” or something.

What was missing?

Morning math was thrilling, and we needed time to process what it meant to us as learners and educators. We made space for reflection by talking about morning math during meals, on the long walk back to our rooms, and in the hot tub, but we would have been better served if part of our structured day was devoted to reflecting on the experience of morning math and working to integrate its many strengths into our own classrooms.

What am I still wondering?

Were we meaningfully collaborating? While we were sitting together and checked in with each other often, we all worked in our own notebooks. Sometimes, collaborating felt like we were robbing our peers of the chance to make their own discoveries. I wonder what the experience would have felt like if we had been standing and working on whiteboards together.

IMG_1558
One afternoon, we decided to continue working after class and wrote on the glass sliding door with whiteboard markers. It was fun to work together on a VNPS, but we didn’t really figure anything out …

Which comes first – procedural fluency or conceptual understanding? In another part of PCMI, we read the common core’s description of “rigor” and wondered about the relationship between procedural fluency and conceptual understanding. NCTM says, “the development of students’ conceptual understanding of procedures should precede and coincide with instruction on procedures,” and we debated whether these problems sets supported or contradicted that view. Some problems helped us discover how, why, and when the procedures we had already been using worked.

What about application? On Day 5, I was surprised to see that the opener was a word problem because real world context had been all but absent thus far. What made this problem valuable, and why were there so few applied problems? Was application rare because our content was number theory? Because the audience was adult math teachers?

Should content goals be explicit? Each lesson had it’s important stuff, and in those problems, there was something we were supposed to learn. For instance on day 3, it felt like the goal was to connect numerical and geometric versions of the Euclidean Algorithm. I say “it felt like” that was the goal because Bowen and Darryl never made their content goals explicit. If they had presented them up front, that would have been a huge spoiler, but would we have benefited from seeing them later on?

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