My Discovery-Based Math Elective: Pedagogy of a PSet (Part 12)

This is part 12, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe the general pedagogical approach to using Problems Sets (PSets) and the primary form of classwork, and how it connects to the underlying curricular and pedagogical philosophies of the course (described here).

• The "pedagogy of the PSet," to me has felt like a reallocation of resources away from pedagogy and towards content. So many of the mechanisms and structures of the traditional math class don't have a place in this course. And I think the clearest way to describe them is by sharing the three PSet "rules" that are posted at the top of each PSet, which are borrowed almost directly from the PSets at PCMI:
• Don’t worry about answering all the questions. Don’t worry about getting to a certain problem number. Some students have been known to spend the entire class working on one problem! That’s okay!
• Many of the structures and routines I use in my traditional math classes I teach are based on principles of necessary and sufficient productivity. But "productivity" is barely necessary, and definitely not sufficient alone for great math learning to happen. I also think that if we want to build up students' attention spans, and general ability to persevere through problems that take a long time, and have many layers of difficulty, we have to give them problems that take a long time, and have many layers of difficulty...and then give them the time to do so.
• Have fun! Make sure you’re spending time working on problems that interest you. Feel free to skip problems that you’re already sure about. Relax and enjoy!
• Choice is an enfranchising experience. Every time a growing person has to make their own decision about what they do and don't want to do, they define themselves a little bit more, as mathematicians, students, and people. Moreover, they assume real control over their education. Here, I am trying to expand my students real choices beyond "do the work or don't."
• Whatever you do, do well. Flying through the problem set helps no one, especially yourself--you’ll miss the big ideas that others are grabbing onto. There is more to be found in the problems than their answers.
• This is pretty self-explanatory. And also generally good advice for life, I think.
• There is one more that I use when designing PSets, that's more for me than for the students.
• The PSet should lead to the math, not require it.
• Here's a useful example of this in action. In the Combinatorics Unit, I wanted students to discover that the coefficients of the expansion of (x+1)^n are the entries of the nth row of Pascal. But for a lot of my kiddos, I couldn't just tell them to take the powers of (x+1). So in the weeks leading up to that, I included some chill polynomial multiplication problems, using the area model. It wasn't an in depth analysis of the area model--just enough to do the job. Then over the next few PSets, the problems scaffolded them up to the powers of (x+1)^n. And a few students followed that thread across the PSets, and it was cool. But, importantly, even though they weren't engaging quite with the goal objective (coefficients of the powers of x+1), they were still doing valid, worthwhile math all the way up to it.
• This idea also helps to provide a schema for designing for re-entry. Some students will miss a PSet or two, either because of attendance, life, or just really not digging the Big Problem for some reason. We don't want that gap in engagement to perpetuate itself, by limiting future engagement.
• A goal is that students should be able to do the PSet if they randomly dropped into the class that week. This may not be 100% possible all the time. I mean, in the last paragraph I literally just described a system of scaffolds where students may have needed the work of previous PSets to build up to the more sophisticated problems. I try to mitigate this by allowing the more important questions to resurface in multiple PSets. I also give students the opportunity to go back to problems from older PSets if they want to, or if it's helpful.
• I don't want to say that these PSets are the best way to teach every high school math class--certainly not. I mean, I was still primarily a traditional Math 1 teacher this year, and there were a bunch of things I would do in this elective, that I'm not yet ready or willing to do in my Math 1 class. And that's partly because the purposes and contexts for the two courses are different. But it's also because the stakes are a little bit lower in a math elective than in a students' primary 9th grade math class. But that's not to say that I didn't learn a bunch from it. I have tried to ship over bits and pieces, when and where I feel ready.
• I also want to take the time to point out that this is only one way of doing PSets. Back when I was first trying to make my own PSets, I was doing so as part of a collaboration with Joey Kelly. It was a kind of natural application of the experience that he, Dan Henderson, and I had last year with MIST (a year-long PCMI spinoff pilot). We had similar goals, but he ended up breaking his PSets up into chunks, and rolling them out over the course of a few weeks, more as a series of mini-PSets. That definitely gave the groups working on the PSets a little more day-to-day direction, and avoided the issue of later questions spoiling earlier ones. I'd love to some day ask him more about it. Or you can ask him on Twitter yourself @joeykelly89.