My Discovery-Based Math Elective: Architecture of a PSet (Part 13)

This is part 13, 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 overall format of a Problem Set (PSet), and the different design features of the different parts. This post in particular identifies a handful of other educators who have been especially influential in their work.

• The architecture of a PSet is pretty straightforward, and is largely inspired by the PCMI PSets. Each PSet is broken into four parts, each with a different set of design principles. When we were in-person, I committed to keeping the PSets to a single two-sided sheet of paper (albeit with very narrow margins).
• Big Problem
• The Big Problem is the core of the PSet. It's the first problem students do, and the only one that everyone has to spend at least some amount of time on.
• The most comprehensive model I have of an effective Big Problem is the work done by Xi Yu and Joey Kelly with their PlayWithYourMath.com project. Joey has documented a lot of really useful reflections around the development of some of their problems. I'll summarize some of the big ideas of the work they've done, and link to where you can read more about it at Joey's blog MisterIsThisRight.com. In particular, here are all the posts where they reflect on the design of some of the PWYM problems, which has significantly informed my own understanding.
• Characteristics of effective Big Problems, from the reflection on problem 13. Thirteens
• involve some sort of “play” before choosing a specific strategy.
• have a low floor (accessibility and entry point)
• have a high ceiling (need for more complex mathematics)
• have a succinct, accessible, intuitive wording and visualization
• What is meant by "ceilings", from the reflection on problems 18 and 19
• To make success attainable. In addition to a high ceiling, I also want a low ceiling, where students can feel a sense of accomplishment. There should be lots of ceilings. 
• To make space for curiosity. Just because there is a ceiling, doesn’t mean I have to show it to them. 
• To shelter from inaccessible questions. Some ceilings are just too high for some people, and that is fine. The high ceiling is not meant to intimidate.

Link to full-size image, made on Desmos

• Here is a diagram that will come up a few times in this blog, when talking about ceilings and differentiation. The idea is that students enter the problem at whatever level they are initially ready for, and can exit the problem having made as much progress as possible. And wherever those points are, there is some satisfying and meaningful work to be done.
• I also had some interest in exposing students to some of the famous problems in mathematics. These are classics like The Bridges of Konigsberg, the Utilities Problem, Counting Trains, Fibonacci numbers, etc. Usually they're famous for good reason, and their inherent richness is usually worthwhile. Furthermore, these problems are cultural cornerstones of the mathematician community. So if our goal is to induct students into the community of mathematicians, we would do well to give them access to some of that culture.
• Obviously Related Problems
• The Obviously Related Problems are just that--obviously related to the Big Problem. They come in two categories: Scaffolding Questions and Advancing Questions
• Scaffolding Questions: Scaffolding questions are questions that are designed to break down the Big Problem, and help students work through it. Imagine yourself walking around the room, circulating and conferencing with students, helping them initially make sense of the problem.
• If you know that all of your students are going to ask some of the same introductory questions, and you're going to keep having to have the same introductory, orienting questions and recommendations, you might as well put those questions in the PSet--it's more efficient. Especially once students get the hang of the PSets, they'll start to look to this section when they get stuck, and that's something they can do independently. It's also a much quicker conversation, once you recognize where a kid is stuck, if you are able to quickly say, "Check out #3," and then walk away.
• As with any scaffolding, it is dangerously easy to add too many scaffolding questions, and in doing so, rob students of the opportunity to develop their own insight and independence, if not spoil the problem entirely. This is especially dangerous in the context of a mixed-levels class. If I was ever worried that a question was going to offer too much scaffolding, I would err on the side of not including it. Dan Meyer had a good blog post on this: You can always add. You can't subtract.
• Another helpful idea for thinking about how I managed scaffolds in this class is the different between Just-in-Time vs. Just-in-Case Scaffolding, which was introduced to me in a post by Dr. Juli K. Dixon. Highly recommend the read. In this case, the Just-in-Time scaffolds are the ones you provide through conferencing. The questions in the PSet can become Just-in-Case scaffolds if the kids don't skip them, and Just-in-Time if you are the one recommending they try them.
• Advancing Questions: These questions are less about students working through the Big Problem, and more about helping students start to find the Big Idea. They push the kid to try to refocus on the big picture, and think about the broader mathematical structure. These usually come up more often in the *Probably* Not Related Problems.
• *Probably* Not Related Problems
• The name of this section describes pretty clearly, if ironically, the purpose of these problems. They don't look overtly related to the Big Problem or the Big Idea. But with a deep enough exploration, there is a connection. The name of the section poses the challenge of trying to figure out exactly how they are related to the Big Problem.
• This section is founded on the idea that you can have a bunch of different problems that all have a common underlying structure to them. I think the best example I have of this, unsurprisingly, is in Sequences: PSet 7: Bees, which is a PSet exploring Fibonacci numbers. We can put a ton of interesting different looking problems in there. And then miraculously the Fibonacci numbers will come up again and again (as they so often do). Sometimes it'll be obvious the way they are connected, and others it will be very difficult to find a way to connect things.
• We can think of the Big Idea at the core of the PSet like a statue, set up inside of a dimly lit house. Looking through a single window at the statue, we only have a single incomplete perspective of the Big Idea. Even if it's a Big Window (i.e., Big Problem), our perspective of the statue is limited. But by looking in from other windows, we get glimpses of our Big Idea at multiple different angles, each incomplete on its own, but collectively greater than any individually. The different perspectives allow us to kind of "triangulate" the Big Idea, and understand it's construction more deeply, independent of the context of a given problem.
• We can describe these problems as "Different Surface--Same Depth." This is not to be confused with @mrbartonmath's (blog) "Same Surface--Different Depth" problems, it's actually kind of the opposite. But I believe it was the inspiration for this reconceptualization, which I first heard coined by Joey Kelly.
• Going Deeper
• Basically any kid should be able to do basically any problem in the PSet. In this section, however, I'm willing to push the boundaries a little bit, in terms of how high I'm willing to let the entry-point creep. I also tend to walk back many of the scaffolding questions I usually include in a PSet, instead usually opting to pose the problem in the simplest, "purest" form possible.
• I do this for two reasons. First, at this point students have usually built up more background knowledge by doing the earlier problems in the PSet. Moreover, the kinds of students who would jump ahead to the Going Deeper problems, usually do so as a signal of readiness. This doesn't necessarily mean that any of these questions are inherently harder or better. It just means there's less scaffolding.
• Often, these questions are also extensions of the other problems in the PSet, that are maybe a bit of a tangent away from the Big Idea, but are too rich to not include. For example, in Bases: PSet 9: Grapes, we were looking at a James Tanton classic, Grape Codes, as an precursor to diving into Exploding Dots. All I really wanted to get across was that grape codes were ways of rewriting numbers as sums powers of some base number. But a really cool related problem is actually figuring out some kind of rule to determine how many grape codes there are for any number. Great problem, but a bit of a tangent, especially considering how big it really is.