My Discovery-Based Math Elective: Content Threads (Part 14)

This is part 14, 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 define a major design feature of the curriculum of the course, "content threads." I discuss in depth the content threads of the Combinatorics unit, as an example.

• One of the most challenging, and worthwhile, design features of this course was my attempt to make "content threads." The way that I've come to understand it, a content thread is a major mathematical idea that keeps coming up across different problems and PSets. What that idea looks like can very wildly in scale, and you can have some ideas come up, skip some PSets, and then resurface when they become relevant again. The visual I have in my head of this is something like the Movie Narrative Charts from xkcd, that shows how the characters gather and disperse in different ways over the course of the plot.

Link to full-size image.

• I've wanted to try to make some kind of similar diagram for the content threads of this course. But I'm not totally sure where to begin, or if this is even the best representation, even though it's the one that exists nebulously in my mind. If you come up with something cool, please let me know--I'd love to see what you come up with.
• Some of the units in this course have longer and stronger content threads than others. Here are some of the bigger ones, in each of the five units:
• Modular Arithmetic
• Looking at remainders suggests a "cyclic" mathematical structure
• If we change the setup of our number system a little, a lot of things we used to take for granted are no longer true
• Special things usually happen when numbers are relatively prime
• Bases
• Our choice of base is arbitrary, based mostly on convenience, and depending on the application, different bases become convenient
• Even if we change our base, arithmetic still works in fundamentally the same way
• Combinatorics
• Counting can quickly become difficult, but if we count cleverly, we can use structure to be systematic and efficient
• Pascal's triangle has a lot of features and patterns, and we can make sense of them.
• Lots of counting problems can be conceptualized as "choosing" problems (choosing), and so we can bring Pascal and the Binomial Coefficient to bear
• Sequences
• Sequences can be defined recursively and explicitly
• Many sequences can be characterized by their differences and ratios, but some are a little more complicated
• Graph Theory
• Representing complicated problems with an abstract graph sometimes helpfully simplifies the problem (decontextualizing makes things easier)
• With a few simple geometric rules, there are some inevitable implications for how graphs work, which extend to the contexts they model (contextualizing provides insight)
• The goal of trying to develop and articulate these content threads across multiple PSets is manifold:
• The big ideas of math are much more interesting, powerful, and transferrable than a lot of specific granular content knowledge [citation needed?]. So by building a curriculum around the articulation of these threads, we are being strategic.
• We are creating multiple opportunities for students to engage in the big idea. This system allows for spaced-retrieval, which improves long-term retention.
• By working through multiple PSets and problems, all centered around the same content thread, students gain multiple perspectives on a single idea. These multiple perspectives add layers and nuance a student's understanding of the idea. Multiple perspectives develop a more sophisticated schema for understanding. This is even more important, as these big ideas of mathematics are often quite nebulous and abstract.
• It creates multiple opportunities for representation of the idea, and engagement with it. Depending on the student, different problems will be more interesting, or make more sense. By creating multiple PSets around a similar content thread, we are providing multiple entry points for students to access the big idea.
• Some of the PSets have a much stronger, more coherent set of content threads. Some threads within a unit are more effectively developed in the PSets than others. Some threads are more general, while others are pretty specific (though I did try to keep grain sizes roughly similar). Moreover, the degree to which any of these threads "land" is largely dependent upon the teacher's execution of the day-to-day, especially the end-of-PSet discussions.
• I have done the most work to develop the content threads for the unit on Combinatorics. This was the unit I put the most time, energy, and research into. It also had one of the richest mathematical structures ever at its center--Pascal's Triangle--which certainly helped. But the content threads of this unit provide a kind of best-case scenario, where each thread is introduced with a new problem, tied back into older threads, and braided with future threads. The best representation I could find of this is a table, like the one below. These content threads are a bit more granular, than the more practice-oriented threads I listed earlier, which I'll talk about more later.

Link to full-size image.

• In this table, you can see how each content thread "runs through" each of the four Big Problems. I tried to be explicit about how each problem could be recontextualized in the new content thread, adding more layers of understanding and insight. As we accumulate more content threads, we can look at each new Big Problem through the lens of all the content threads before it.
• The yellow box indicates that that Big Problem was the one that officially launched the content thread. All of these problems technically feature all the content threads. But to introduce a content thread, I tried to choose the Big Problem that "best" captured the Big Idea. That is, I tried to find the Big Problem whose context organically encoded the thread. The goal is that this problem serves as an initial schema for future applications, like a "hook" upon which future related problems can be hung.
• It's possible to sustain a content thread that does not rise to the level of one of these big ones that cut across the whole unit. For example, in this unit there are two mini-content threads that run through the PSets, hovering around the *Probably* Not Related problems. These are the problems concerning the outcomes of coin flips and the coefficients of (x+1)^n. I opted to avoid elevating these two minor threads because:
• Didn't have enough time to dedicate to 1-2 more PSets in the Combinatorics unit
• Didn't have an good enough Big Problem that I liked for either PSet
• Didn't feel that the coefficients of (x+1)^n would be sufficiently accessible to enough students, algebraically
• Didn't feel that the number of outcomes for n coin flips was rich/interesting enough yet (as far I could realize it as a PSet that was a part of this unit)
• Including or not including the a PSet dedicated to these two content threads is the kind of planning and instructional decision you can make based on what you know about your students. If I was adapting this unit for either middle school use, or for a group of less experienced math students, it would be a good opportunity to surface the coin-flipping thread. This would do a good job of explaining why each row of Pascal sums to a power of 2.
• If I was adapting this unit for post-secondary classrooms, or a group of students with broader backgrounds in high school algebra, I might surface the (x+1)^n thread. This would be good to put at the end of the unit, because it's such a great example of how to apply combinatorics and Pascal to a seemingly totally unrelated context.
• As I mentioned before, there are two sets of content threads in this unit. The more specific content threads (identified in the table), and the more general practice threads:
• Content Threads
• Strategic Counting
• Symmetry in Pascal
• Recursive definitions in Pascal
• "Choice" problems in general
• Practice Threads
• Counting can quickly become difficult, but if we count cleverly, we can use structure to be systematic and efficient
• Pascal's triangle has a lot of features and patterns, and we can make sense of them.
• Lots of counting problems can be conceptualized as "choosing" problems (choosing), and so we can bring Pascal and the Binomial Coefficient to bear
• This combinatorics unit is definitely the unit with the more articulate content threads. This made opportunities for some really awesome mathematical connections. However, it did shift the focus of the class (and end-of-PSet discussions) a little more towards content, and further from practice, which can negatively constrain differentiation and student engagement (discussion of practice vs. content standards here).
• It also required more often that I level the discussion to the objective, which if done too much, can lead to students disconnecting from the narrative (discussion of levelling the discussions here). The narrative for this unit really "accumulates," and the more I focused on trying to sustain that accumulation of content threads, the more students disconnected, because it often failed to respond directly to their experiences.
• Reflecting on this unit, it really seemed like I had lost sight of the practice-orientation that was at the foundation of the class, which was had a negative impact. A question I still have is what it means to advance both content and practice standards, at the same time, but disconnected? Also, how to we expand content standards, so that they can be better differentiated for students with a broader range of readiness levels?
• The big idea here is that we can articulate content threads that run throughout the course, especially within each unit. These threads separate during some PSets, and braid together during others. The more precise, coherent, and authentic these threads are, the easier a job we'll have of build PSets around them, and making them visible. Also, the more completely we understand these narrative threads, the better we can facilitate the course towards their effectiveness.