I’ve had the tremendous honor of attending the IAS/Park City Mathematics Institute (PCMI) every summer since 2003. I’ve gained a “math camp family” that I treasure dearly. I have learned so much from the people that I’ve met. It’s affected my career in major ways*. I can’t say enough good things about it. If you’re a math person of any flavor, you should definitely try to attend if you’ve never been.
Since 2008, I’ve also had the honor of co-facilitating the morning math course for the Teacher Leadership Program (back then known as the Secondary School Teacher Program or High School Teacher Program). Side note: You should watch this awesome video about TLP. Our work has resulted in a series of books published through the American Mathematical Society.
Bowen Kerins has been my main co-conspirator in this course, but in 2008 I also worked with Ben Sinwell. The work that we do for this course is really intense while we’re at PCMI, but it’s one of my favorite teaching experiences.
Many people who’ve experience the course are curious about how we design and implement the course. So, this post is meant to give a bit of insight about our ridiculous process.
First, a bit about the course itself. PCMI is a three-week residential program. There are usually 14 or 15 class meetings, each about two hours long. The participants in the program are elementary, middle, and high school teachers. There is always a different research theme for PCMI each year, and this math course for the Teacher Leadership Program tries to connect to that research theme in some way. In 2016, the research theme for PCMI was big data, and the mathematics content for our course had to do with probability, games, and Markov Chains. We have the luxury that we don’t need to follow any specific math content standards–we get to teach a course on whatever we want and we don’t need to assess the learning in any formal way.
The math course is designed to help teachers deepen their mathematical content knowledge, to experience authentic mathematics in a collaborative environment, and to feel joy and wonder while doing mathematics. We want teachers to walk away with an understanding of how mathematics is socially co-constructed so as to make sense of the world around us and solve problems. We want teachers to experience the beauty and power of mathematics.
During the two-hour class itself, Bowen and I do very little talking. We start the course by passing out a problem set and we spend 95% of the time walking around, taking notes, answering questions (obliquely), encouraging people to work well with each other. Teachers work together in groups of five or six at tables; each table has a table facilitator (a past participant that has been invited back to serve in this capacity). Sometimes we end class with a short (~5 minute) wrap-up. These wrap-ups usually consists of a carefully curated and sequenced set of observations from participants in the room. We only allow participants to share about a mathematical observation about a problem to the whole class if we’re sure that everyone has already worked on that problem and had the opportunity to make that observation. The wrap-up is usually orchestrated so as to help participants make connections between different mathematical ideas or representations.
Typically, participants work on the problem set non-stop during the entire class. They are often disappointed about having to stop at the end of class. But, we never expect the participants to continue working on the problem sets outside of class and actively dissuade them from doing so.
The design of the problem sets has evolved over the years, and I cannot take credit for their genius. At first, when Ryota Matsuura, and Bowen led the course in 2001**, their design was inspired by the problem sets of the PROMYS program, which were themselves inspired by the problem sets used in the Ross Mathematics Program. Over time, the design of the course evolved to its current state. The current form of the problem sets is also somewhat similar to the Phillips Exeter Academy math problems in that they are a carefully sequenced set of tasks. Program leadership has tried to use the term “problem-based approach” to describe how the problem sets work, but I think we need to come up with less klunky.
The problem sets are divided into three sections: Important Stuff, Neat Stuff, and Tough Stuff. Because we have elementary school teachers working alongside Calculus teachers, there is a huge heterogeneity of mathematical expertise among the participants of the class. But, we view this heterogeneity as an asset instead of a challenge.
The problem sets are written in such a way that the main arc of the problem sets from the first day to the last day are woven through the Important Stuff problems. We design the problem sets so that all participants get through the Important Stuff problems, but they never get through all of the problems for the day. If we notice that any participants don’t make it through all of the Important Stuff problems on a problem set, we design the next problem set taking this into account by repeating problems. The Neat Stuff problems build on the Important Stuff problems–they’re neat, but not essential. They often extend the main ideas into other interesting cases, use alternative representations, or involve other areas of application. And the Tough Stuff problems are notoriously difficult. Sometimes they involve unsolved mathematical problems.
So how are these problem sets written? Before PCMI begins, Al Cuoco, Glenn Stevens, Bowen Kerins, and others at Education Development Center, brainstorm about ways to make the PCMI research theme accessible to teachers. They develop some of the important problems and concepts that should be part of the course and think about the mathematical prerequisites that would be required to reach those problems and concepts. In some years, they have even test out some problems with groups of teachers in Boston.
When Bowen and I arrive at PCMI, we begin working immediately. Knowing the general idea for the theme of the course, we decide on the aha moments that we want to build into the course. Since the course generally fills up three weeks, we write the problem sets so that there is a medium-sized aha moment at the end of the first and second weeks, then a big aha moment at the end of the last week. We then think about the prerequisite mathematical ideas and skills that are needed for all of the aha moments and use these learning outcomes as targets for each day’s problem set.
Each day, our routine goes something like this: Class starts at 8:30 a.m. Class ends at 10:40 a.m. Bowen and I debrief about how class went and come up with a plan for the next problem set. We each lunch at noon. We work in the afternoon. Bowen often takes a nap in the late afternoon and I keep working. We eat dinner. We keep writing. I go to bed around midnight. Bowen keeps writing into the wee hours of the morning. I wake up at around 6:30 a.m and make a final pass through the problem set for that morning. I print it and head to class. Repeat ad lassitudinem.
The three weeks are truly grueling, but also immensely fun. Bowen is a great partner in crime and there is a lot of laughter involved in the writing. He has a wicked sense of humor and inserts all manner of jokes, puns, and obscure pop culture references. We also make an effort to work in the names of every participant in the course into the problem sets somehow. References to events that take place outside of class time (like cowboy karaoke parties or escape room antics) often find their way into the problem sets. These references enhance the sense of community in the room.
Working for nearly 22 hours each day to prepare for a 2-hour class is not sustainable. I do not advocate that school teachers work in this manner. It is somewhat frightening that we approach PCMI each year with so little of the problem sets written in advance. But, one of the main reasons that the participants in the course seem to enjoy it so much is that it really is written on fly to meet them exactly where they are. Bowen and I take copious notes about what strategies participants are using, what they find interesting, what they struggle with. The problem set for the next day is written with that in mind. The design process of the course demonstrates the importance and effectiveness of formative assessment.
There are several other important principles that we try to adhere to in the construction of the problem sets.
- Every problem is deliberate and almost always has connections back to a previous problem and forward to later problems. It is not uncommon for us to write problems in the afternoon only to toss them later that evening.
- The Important Stuff problems, and especially the first problem (which we call the Opener), are constructed to have multiple entry points. We write problems that can be attacked via multiple approaches: enumeration, calculation, modeling (particularly with manipulatives), algebra, or more sophisticated approaches. This gives everyone a way into the problem and also encourages participants to compare their approaches and make connections with each other.
- Problems are often sequenced in such a way that participants can’t help but notice patterns. If you’ve done three problems in a row and the answer each time turns out to be 47, you’re going to notice that and wonder why that’s the case. We avoid being heavy handed about pointing out those patterns but instead leave lots of breadcrumbs for participants to notice. In many cases, the trail of breadcrumbs leads back to previous problem sets so there is a revisiting and reexamination of what has been learned before. We do a lot of foreshadowing of key ideas by building up a bank of examples before the key idea is revealed.
- Wording of problems is very important. Bowen is a master of efficiency and clarity. I have learned so much from him about how to say exactly what you mean and no more. We try our best to fit the problems onto 2 sheets of paper (4 pages total). Also, language is important and sometimes we need to obfuscate–If there are mathematical ideas that we think folks might recognize and be tempted to look up on Wikipedia, we sometimes purposely obscure those ideas. For example, in the 2007 problem set the Farey sequence appeared as the “godmother sequence“. In 2014, participants played with Penrose tilings for two weeks without us mentioning those words explicitly until they had discovered many of their properties through the problem sets themselves. We obfuscate in this way so as to create a more even playing field for participants and to encourage participants to make their own discoveries.
- We use computers and technology sparingly for two reasons: open computer screens often put up a barrier that impedes communication between people, and participants often have a wide range of experiences with technology. Technology should be used when it meets the learning objectives in a compelling way, not just because it’s whiz-bang neat. When technology is used, we craft problems carefully so that participants who are unfamiliar with the technology learn what they need to know in small chunks. Over time, their fluency with the technology builds. We are also big fans of using Google Sheets as a way for participants to pool their data with everyone in the class.
Over the years, we have noticed that the table facilitators play a crucial role in making sure that the participants at each table work well with each other. Sometimes there is a participant who knows a lot of mathematics and rushes through the problems to get to the Tough Stuff. In contrast, others at the table will take their time to “smell the roses” and make deep insights. These table facilitators try their best to make sure that the speed-racers are sufficiently intrigued by the deep insights made by the rose-smellers that they slow down and savor things too. It’s especially wonderful when a high school teacher with a lot of mathematical experience is humbled by the mathematical prowess of a neighboring elementary school teacher.
Part of the reason why I enjoy this course so much is that it is so gratifying to know that participants enjoy themselves and walk away with meaningful mathematical experiences. Here are some of the things they have to say.
“That’s where it’s had the biggest impact—in how I teach. Now I encourage and allow students to collaborate more in groups, to communicate mathematically with each other and collaborate to solve problems. I used to just always show my students how to solve problems, and now I encourage them to try on their own—to practice problem solving and arrive at their own conclusions, and then we come back to discuss it.”
“When I reflect on the morning sessions, the effect was positive. It was a touchstone. I think about the kinds of experiences we had and how to get more of that into my classes—sharing about math, giving kids time, allowing kids to dig in and take ownership, rather than, we have to cover this and that and this. [PCMI] helps me to stay student centered. So my experience of being in that room in those days resonates in that way.”
* In 2003 I participated in the Undergraduate Faculty Program, which is where I first met Peg Cagle, a math teacher in Los Angeles. We became fast friends and started something like a math teacher circle in Los Angeles. That math circle is how I get to meet Pam Mason, who I then asked to be the director of Math for America Los Angeles when I helped to start that in 2007. Fast forward to 2016…MfA LA has supported over 120 math teachers in the Los Angeles area and going strong.
** Some more history: The morning math course dates back to 1981 when PCMI started as a regional geometry institute. The course took many different forms. In 2001, EDC was contracted to design the course. Ryota and Bowen led the course, each running the course for half the time. In 2002, Bowen and David Offner ran the course. In 2003, Bowen and Ben Sinwell ran the course, each for half the time. In 2004, the TLP got a large NSF grant and the number of teachers doubled, so Bowen and Ben ran the program together for all three weeks. I was a table facilitator prior to helping to lead the math course. In 2006, I had the good fortune of being put in the same housing unit and Bowen and Ben, and that led to me helping with the writing.