At the JMM this year, I had the pleasure of attending a minicourse on “Designing and Implementing a Problem-Based Mathematics Course” taught by Bowen Kerins and Darryl Yong, the masterminds behind the legendary PCMI teachers’ program Developing Mathematics course, with a significant assist from Mary Pilgrim of Colorado State University.

I’ve been wanting to get a live taste of Bowen and Darryl’s work since at least 2010, when Jesse Johnson, Sam Shah, and Kate Nowak all came back from PCMI saying things like “that was the best math learning experience I’ve ever had,” and I started to have a look at those gorgeous problem sets. It was clear to me that they had done a lot of deep thinking about many of the central concerns of my own teaching. How to empower learners to get somewhere powerful and prespecified without cognitive theft. How to construct a learning experience that encourages learners to savor, to delectate. That simultaneously attends lovingly to the most and least empowered students in the room. &c.

I want to record here some new ideas I learned from Bowen and Darryl’s workshop. This is not exhaustive but I wanted to record them both for my own benefit and in the hopes that they’ll be useful to others. In the interest of keeping it short, I won’t talk about things I already knew about (such as their Important Stuff / Interesting Stuff / Tough Stuff distinction) even though they are awesome, and I’ll keep my own thoughts to a minimum. Here’s what I’ve got for you today:

1) The biggest takeaway for me was how exceedingly careful they are with people talking to the whole room. First of all, in classes that are 2 hours a day, full group discussions are always 10 minutes or less. Secondly, when students are talking to the room it is always students that Bowen and Darryl have preselected to present a specific idea they have already thought about. They never ask for hands, and they never cold-call. This means they already know more or less what the students are going to say. Thirdly, they have a distinction between students who try to burn through the work (“speed demons”) and students who work slowly enough to receive the gifts each question has to offer (“katamari,” because they pick things up as they roll along) – and the students who are asked to present an idea to the class are only katamari! Fourthly, a group discussion is only ever about a problem that everybody has already had a chance to think about – and even then, never about a problem for which everybody has come to the same conclusion the same way. Fifthly, in terms of selecting which ideas to have students present to the class, they concentrate on ideas that are nonstandard, or particularly visual, or both (rather than standard and/or algebraic).

This is for a number of reasons. First of all, the PCMI Developing Mathematics course has something like 70 participants. So part of it is the logistics of teaching such a large course. You lose control of the direction of ideas in the class very quickly if you let people start talking and don’t already know what they’re going to say. (Bowen: “you let them start just saying what’s on their mind, you die.”) But there are several other reasons as well, stemming (as I understood it anyway) from two fundamental questions: (a) for the people in the room who are listening, what purpose is being served / how well are their time and attention being used? and (b) what will the effect of listening to [whoever is addressing the room] be on participants’ sense of inclusion vs. exclusion, empowerment vs. disempowerment? Bowen and Darryl want somebody listening to a presentation to be able to engage it fluently (so it has to be about something they’ve already thought about) and to get something worthwhile out of it (so it can’t be about a problem everybody did the same way). And they want everybody listening to feel part of it, invited in, not excluded – which means that you can’t give anybody an opportunity to be too high-powered in front of everybody. (Bowen: “The students who want to share their super-powerful ideas need a place in the course to do that. We’ve found it’s best to have them do that individually, to you, when no one else can hear.”)

2) Closely related. Bowen talked at great length about the danger of people hearing somebody else say something they don’t understand or haven’t heard of and thinking, “I guess I can’t fully participate because I don’t know that idea or can’t follow that person.” It was clear that every aspect of the class was designed with this in mind. The control they exercise over what gets said to the whole room is one aspect of this. Another is the norm-setting they do. (Have a look at page 1 of this problem set for a sense of these norms.) Another is the way they structure the groups. (Never have a group that’s predominantly speed-demons with one or two katamari. If you have more speed-demons than katamari, you need some groups to be 100% speed demon.)

While this concern resonates with me (and I’m sure everybody who’s ever taught, esp. a highly heterogeneous group), I had not named it before, and I think I want to follow Bowen and Darryl’s lead in incorporating it more essentially into planning. In the past, I think my inclination has been to intervene after the fact when somebody says something that I think will make other people feel shut out of the knowledge. (“So-and-so is talking about such-and-such but you don’t need to know what they’re talking about in order to think about this.”) But then I’m only addressing the most obvious / loud instances of this dynamic, and even then, only once much of the damage has already been done. The point is that the damage is usually exceedingly quiet – only in the mind of somebody disempowering him or herself. You can’t count on yourself to spot this, you have to plan prophylactically.

3) Designing the problem sets specifically with groupwork in mind, Bowen and Darryl look for problems that encourage productive collaboration. For example, problems that are arduous to do by yourself but interesting to collaborate on. Or, problems that literally require collaboration in order to complete (such as the classic one of having students attempt to create fake coin-flip data, then generate real data, trade, and try to guess other students’ real vs. fake data).

4) And maybe my single favorite idea from the presentation was this: “If a student has a cool idea that you would like to have them present, consider instead incorporating that idea into the next day’s problem set.” I asked for an example, and Bowen mentioned the classic about summing the numbers from 1 to n. Many students solved the problem using the Gauss trick, but some students solved the problem with a more visual approach. Bowen and Darryl wanted everybody to see this and to have an opportunity to connect it to their own solution, but rather than have anybody present, they put a problem on the next day’s problem set asking for the area of a staircase diagram, using some of the same numbers that had been asked about the day before in the more traditional 1 + … + n form.

I hope some of these ideas are useful to you. I’d love to muse on how I might make use of them but I’m making myself stop. Discussion more than welcome in the comments though.