Note: This post was started in Late July, and abandoned until now in Late August
I promised three posts post-PCMI on what I’ve learned — on math talk, on lesson study, and on PCMI as a learning community. Now that it is late August, I don’t know if I can do all three, at least not immediately. So I am going to focus on the learning community formed at PCMI, via the way the program was structured.
I first found out about PCMI on twitter. I am pretty sure it went something like this:
samjshah: anyone have ideas of good summer programs for math teachers?
tweep1: PCMI
tweep2: PCMI
tweep3: PCMI & exeter
samjshah: did exeter, should check out PCMI i guess
I hadn’t heard of it. And no one really could explain to me what it was, and why it was so powerful for them. Many people on in the edublogosphere talked about going there a few summers. To me, that’s advocacy enough, because if you’re a teacher, and you attend a 3 week professional development program in Utah, and then want to do that again, well, that’s saying something to me loud and clear.
Without getting too meta or analytical or anything, I think PCMI showed us how a serious, committed learning community could look like. It mirrored what we would have wanted our school experiences to have been, and had qualities that we wished our schools had. That’s what was so powerful. We participated in this professional development around creating an engaging and effective math community, while actually participating in an engaging and effective math community.
This is how a typical day would look.
I would wake up around 6:30 and take a bath and read. Okay, that sounds like a strange way to start the day, but this was my own private heaven, because I love reading in the bathtub and I don’t get to read much during the school year. Then I would mosey on over to breakfast, where we have a daily buffet — a veritable cornucopia of muffins and eggs and cereal and soy milk and coffee and bagels and other stuffs. Over breakfast, I would generally sit with other math teachers.
You see, at this point, I’ll interrupt and say PCMI actually has a bunch of parallel programs running at the same time. Researchers have a program. Undergrads have a program. Grad students have a program. University teachers have a program. So we high school teachers are just one of a few different strands of PCMI.
Over breakfast, we’d discuss our schools, math problems, what we did the previous night, books we’re reading, TV shows we love, weekend plans, Justin Bieber, whatever! The whole three weeks definitely had a summer camp atmosphere because almost no one knew each other and so there was always lots to discover about everyone. There were about 60 teachers in the program.
Then we’d rush over to our morning problem solving session, which lasted a hair under 2 hours. There would be quick announcements, and then Morning Shorts. These are 5 minute presentations given by participants. Here’s mine. Then onto problems. We’d sit in groups of 5 or 6 (one of those people was a table leader, but they weren’t privy to any of the problems or solutions we’d be working on beforehand). We’d be given a packet of problems and then just set off to go. Nothing else. No formal lesson. Just compelling problems. Sometimes our group would work independently, sometimes we’d talk, sometimes we’d get off track (but super rarely). Connections were made, informally, independently, at our own pace.
The problems were made by Darryl Yong and Bowen Kerins, and are online here (look for the Hand-Outs section). The philosophy of the class is pretty well summed up by their “rules”:
Bowen and Daryl made each subsequent problem set after watching us, and seeing where we were at. We were going, each, at our own pace. To make sure we could keep up with the course, and not fall too far behind, there was a core idea that was put at the start of each problem set — called IMPORTANT STUFF. We had to get through that (we always did), and then we could just work on whatever. But the next day’s material would only really require us to know the IMPORTANT STUFF.
The problem sets themselves were the most well-crafted set of problems I’ve ever been given, in terms of scaffolding. I don’t think you can see really how these problems are so amazingly scaffolded until you actually work through them. Because you will start seeing cycling back to old material, little hints about connections you’ll be making (no connection was ever explicitly given to us in the problem set — we had to do all that heavy lifting on our own), and a general ramping up to some really frustratingly engaging problems.
They were also really funny. With each of our names included in at least one of the problem sets — which actually gave us a nice feeling to see in print. And lots of jokes and puns.
Examples:
Ha, this was the title of the 8th problem set. Notice the use of our names in the problem set! (And each table had numbers, hence the Table 8 in the title.)
I think I got a stomach cramp laughing at the marginalia. Um, it’s okay if your answer has some p in it? Get it? Also, notice Caro(l)’s name in the problem set? (I cut off the l, but it was there, I’m sure.)
And of course the stupid math humor I’ve come to love so much.
I would just like to point out that all these examples came from the same problem set! So imagine this, every day! Fun! Times!
Okay, so we’d work, and then in the last 15 minutes, or maybe in the middle, Bowen would talk about what people were seeing, he’d maybe throw up a Sketchpad applet he made, or a photograph of some of our work and have us explain it to the rest of the group. And then we’d move on, the next day.
There was one other thing that made this setup so well, minus the self-directed pacing, the well-crafted problem sets, the ability to collaborate. It was that we would only be with our group for 3 days. Yup, that’s it, 3 days. Then we’d be assigned a new group. The end result? After 3 weeks (15 days), we’d have been in groups with almost everyone attending.
Frequent group switching was one of the ways that I think our community was built so quickly, and so powerfully, in 3 weeks.
After some coffee and cookies (that I refrained from eating, thank you very much), we’d then go into our “Reflection on Practice: Connections to Research” groups. For these, we were put in groups of 5 or 6 again, but then we’d have a few groups in a room instead of all of them. So instead of having all 60 of us in one room, we’d only have 15 or 20 so. We had two leaders, who led us through various exercises, reflections, video-watching and transcript-reading sessions, etc. I would write more about this part of the program, but we learned a lot about a lot of things, so it would be like trying to capture a hundred butterflies fluttering about in a room. Possible, but time-consuming.
For me, the most major theme we hit upon was “math talk” — the purposes of it, how to encourage it, how to evaluate it, and the rewards of it. That’s too big to tackle for me in this post. I’ll leave it at that, until some future time.
We switched up groups and leaders in “Reflection on Practice” session every 5 days.
Then lunch. Mostly we were allowed to sit where we wanted, while we enjoyed more (really good, really filling) meals. At least for me, most conversations were based around teaching, since we had just been given lots to think about in the morning. A few times, we were assigned lunch tables. At first the thought irked me. I’m an adult! But the point was to mix all the different programs together, so we could talk to undergrads and university teachers and everyone inbetween. It was actually fun. For the most part the conversations were enjoyable and engaging, and the few times they weren’t, they were benign and innocuous.
Finally after lunch, we’d embark on the third major part of the day: our individual project groups. There were six different groups (and within some of those groups, sub-groups!):
Reasoning from Data and Chance
Exploring Discrete Math
Investigating Geometry
Implementing Lesson Study
Visualizing Functions
Learning the Math of Image Processing
My group had 9 people in it (2 of them group leaders), and focused on Japanese Lesson Study. It was here we got to create. All groups created some final product. And dang if it isn’t fun to create with other people.
My group went through a sped up cycle of Japanese Lesson Study. We put all our work on a wiki (slightly messy, since it was used a lot), focused on achieving these three goals:
- Students will develop a conceptual understanding of converse, inverse, and contrapositive statements, and will be able to use multiple models (specifically Euler diagrams and sentences) to reason about these statements.
- Given an assumed true conditional statement, students can distinguish and clearly explain the truth values of the inverse, converse and contrapositive statements — using counterexamples to show the falsity of statements.
- Students will develop an appreciation for the precision of language, and usefulness of if-then statements.
We actually got to teach this lesson twice (once to other teachers, and once to real students), and revise it once. One of the most memorable and exhilarating moments at PCMI was watching the lesson that we argued and slaved and nit-picked over come to life when taught to students. The kids were engaged, and I could see them slowly come to understanding on their own. We had come up a list of possible confusions and a list of strategies to employ if they happened. Watching that unfold was… well, you could see the impressive class that results from using collaboration, backwards design, and a focus on student understanding. And seeing kids smile, and work through frustration (productive frustration!), and get that deep a ha moment, that was powerful.
It wasn’t a perfect lesson, but it was perfect enough. A thousand times better than what I can produce in my own classroom. It was an example of the type of lessons and the type of teaching I want to do, where there is less lecturing, serious math discourse, and the teacher is merely a guide while the students are in the drivers’ seats.
I gave a little spiel about the use of Wikis in lesson study at the end of PCMI. Here is a clip:
Much like our teaching in the classroom, the nature lesson study is organic and evolving. Fundamental to the lesson study process are the dual ideas of: collaboration and continual improvement. The wikispace provides a well-suited home for this sort of work. When we meet, we never quite know what ideas will jump out, what we are going to pursue, what ideas will become central to the lesson, and what ideas will be jettisoned. But it’s important in this type of collaboration – where the creativity of multiple minds comes into play – to have a way to organize these ideas.
As I said, each working group created things which we shared with each other. And seriously, it was all amazing stuff. And then, at 3 something, we were done. That was the end of the required part of the day.
In the evenings, there were a number of fascinating lectures, informal discussion groups, formal classes — all optional, most interesting. Evenings were the social times, where we would BBQ and go into town and watch Veronica Mars and karaoke and eat at a restaurant and play RockBand and go on walks/hikes and …
And the days would repeat.
I know we had a self-selecting group, but it was a learning community at it’s finest. The program was heavily structured — as you can see, it’s a full day. And you couldn’t skip classes or arrive late. You didn’t get to choose groups (except foryour working group, kinda). But within those strict parameters, you had an informal, playful, intense, passionate atmosphere. To me the most defining features of PCMI was the group work — which had groups switch constantly (some every 3 days, some every 5 days, but our working groups stayed constant). We also had lots of different activities (we weren’t in a lecture hall all day) which broke up the time. We were given breaks and informal times to talk about what we had learned. We weren’t really lectured to at any point; it was about thinking and sharing, reflecting, and collaborating. We learned by doing, not by being talked to by Almighty Gods of Teaching. Most importantly, the program was designed for us to be engaged. Clearly. The designers made a conscious effort to be rigorous and interesting. It was differentiated, good for people at all different stages in their careers. And it worked.
If it tells you anything, I spent 3 of my 10 week summer vacation at PCMI. I am going to forego applying to the other program that I have been coveting, in hopes that I can return to Utah next summer.
Continuous Everywhere but Differentiable Nowhere 