My Discovery-Based Math Elective: End-of-PSet Discussions (Part 8)

This is part 8, 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 how I planned for weekly end-of-PSet discussions, and how they supported the pedagogical and curricular philosophy of the course (outlined here).

• If there are 20 problems in a PSet, even if all students do the Big Problem, there are still over half a million different combinations of problems that a student could do. And since we are prioritizing freedom in when and how students do their independent exploration in the PSet, you can be sure that students are going to be all over the place, trying all different kinds of things, and talking about them in all kinds of different ways. I think a useful analogy for how to handle this can be found in bunting...as in the decoration.

Link to full-sized image.

• Each week is a cycle where everyone starts together, diverges over the course of the week, and then ends the PSet, with the teacher gathering everyone to a common understanding. Then the next week, we start again.
• Students will all start the PSet with a common shared experience--the Big Problem. That's why I give everyone just the Big Problem on the first day, and encourage them to really dive into it before trying the other problems. If students are talking to each other at the same table, then they might continue in the same direction, but not always. Over the course of the week, the scaffolds I provide via circulating and conferencing may also lead to different students pursuing similar paths. But for the most part, due to broad capacity for choice for which problems they do and how, students generally diverge over the course of the week. And that's not only okay--it's ideal!
• The PSets are designed such that even if you're doing problems that look different on the surface, the underlying mathematical structures being engaged (the "depth") is the same. So the discussion at the end of the week is the key moment in which to connect the different threads that students followed. Then by connecting them, and holding them all together at the end of the week, we surface that broader general structure--the Big Idea.
• More advanced students may begin to detect this underlying structure, and will probably be able to recognize it within the subset of problems they did. It's definitely difficult to see the Big Idea when engrossed in specific problems. Moreover, what exactly a Big Idea looks, sounds, and feels like isn't always obvious. So it's helpful for the teacher to do some of that heavy lifting, take the big complicated mixed-up thoughts and experiences of the students, and package them into a clear, relevant Big Idea.
• What needs to happen in the end-of-PSet-discussion?
• To be clear, it is not the goal of the discussion to spoil the answer to the Big Problem. But it IS the goal to provide some kind of resolution, in order to end on a satisfying note, and validate all the great work that students did. Best case scenario, a Big Problem even has multiple "endings" or "ceilings," and you can feel alright spoiling the most advanced ended that everyone got to. Or maybe you just spoil a part of the answer--just enough to get the point across. As much as possible, we want this discussion to provide just enough resolution to satisfy, while also inviting students to recognize the further vastness of the math they are doing.
• Consolidate and focus student discussion on the Big Idea.
• I once sat in on a science pedagogy class for preservice teachers, and the instructor emphasized the importance of what they called the "ABC: Activity Before Concept." The idea is that if students already have some experience with the content when they go into the discussion, not only are they more likely to have something to say, but they'll also already have some kind of schema for understanding what's being discussed. So instead of spending cognitive capacity on figuring out what you're talking about, the they can think about how to engage in the discussion and help make higher-level connections. So the goal is that most students have already had a pretty rich experience with the Big Idea. The teacher's role is then to help make some connections, polish some ideas, and stoke some further curiosity.
• With that in mind, there are two ways I think about how I want to discuss the Big Idea. It's important to have both ways, because students don't always end a PSet where I thought they would. Depending on where the class goes with the PSet, I may stick to my original objective, and sometimes I may flex a little bit. It depends on how I am trying to level the discussion.

Link to full-size image, made on Desmos

• "Level to the Objective"
• Going into the PSet, I have an idea of what I want students to get out of the PSet--I made it after all. Especially if I'm trying to build a narrative that builds across multiple weeks, it's important for me to make sure everyone gets to the checkpoint--the place "Where everyone 'needs' to get." But it's totally possible that not everyone has gotten their on their own. So sometimes the discussion needs to be a bit of a "lift," to get everyone to that point. The pros of doing this are obvious: everyone in the class has at least seen and heard you demonstrate the intended Big Idea.
• The cons are also usually pretty easy to guess: not everyone is going to understand what the heck you're talking about, and students may feel that their weeks-worth of work is invalid. Which sucks. What you don't want is a room full of students watching you talk them through some really great math...with them as spectators. There's a time, place, and way to do this a little bit that may be more help than harm--but it's definitely not "always."
• "Level to the Bottom"
• Alternatively, maybe it's the morning before class on the last day of the PSet, and your read of the room is that not enough students have engaged deeply enough with the Big Idea for it to be feasible to go ahead with the discussion as planned. So you look along the Curve of Meaningful Work, and you find the highest point that enough people got to, and reorient your discussion there. Sure, it's not where you planned initially, but depending on your goal, that may not be an issue.
• I remember this past year, for the Modular Arithmetic: PSet 5: Tables. They were making multiplication tables, then taking mod of the whole thing, and color-coding it by hand. Lots of fun, beauty, and really great math. I don't have a student copy, so here's a digital copy of one:

• I planned for it to be an introduction to the idea of zero divisors, where two non-zero numbers have a product of zero, something that doesn't happen in the regular number system, but does when you look at remainders. I was also hoping they'd figure out which numbers could be zero divisors (numbers that aren't relatively prime to the divisor).
• But by the end of the week, most students hadn't quite gotten deep enough for that to be something that very many of them had thought deeply about. What they had thought deeply about though was the idea that every multiple of the modulus essentially behaves like a zero, under multiplication. So I was able to pivot our discussion to that slightly less complicated idea, and we still had a good discussion.
• It's not always better to choose one over the other. Depending on the week, PSet, Big Idea, the students, and you, your decision may change from week to week. Listen to the students, listen to your gut, read the room, and make the call.
• Obviously, the best case scenario is when "the Bottom" is at least "the Objective." There are some things we can do to bring these two endpoints together:
• Higher quality PSets, in terms of overall setup, coherence of underlying Big Idea, quality of individual problems, and effectiveness of the Big Problem in particular.
• More effective conferencing during circulation. In conferences with students during the week, the teacher can "put their thumb on the scale," so to speak, and nudge students in certain directions. You can also do this with the questions in the PSet. And not only does this help students get to the Big Idea, but it also can help shield students from going too far down a path doomed to be unproductive. How much you do that will depend on the PSet and the students. But we also want to keep in mind that independent discovery and inquiry are the supremely valuable here, and in this class we're willing to kind of trade a lot for it.
• Share great examples of the practices in action
• In a course as practice-oriented as this one, it's also important to spend some time explicitly talking about the practices. Depending on the content advanced in the PSet, and the practices that came up during the week, you may want to gear the discussion more one way than the other. These are the practices I've used, which I've talked about before.

Link to the PSet Rubric Shown

• For the most part, in order to decide which practice to focus the discussion around, I would look at the Big Idea, Big Problem, and the work that the students did that week--and then try to figure out which practice "fit" all of that the best. I tried to prioritize talking about some of the higher leverage practices from each section. I especially tried to talk about these at the beginning of the course, and again later when appropriate. Some of the practices I prioritized were:
• "Uses concrete computations to strengthen understanding."
• "Seeks and uses counterexamples."
• "Finds patterns and makes generalizations."
• "Understands and uses implication (If...then...)
• I also tried to point out the practice out on the rubric when I mentioned the practice, as a way of increasing the utility and meaning of the rubric.
• Surface and universalize important conventions and language
• Yes, this is a discovery-based math class, and I'd rather not tell students something if it's possible or likely that they discover it on their own. However, it's unreasonable to expect the average person to guess or innovate many language and notation conventions. This is why it's important for to surface them in the end-of-PSet discussion. And it doesn't have to be any big complicated thing--often, simpler is better with these things.
• Here's an example: in the context of modular arithmetic, when calculating remainders, as far as I know, it is simply a matter of convention to refer to the divisor as the "modulus." As the mathematician representing the broader mathematician community, it's useful and important for me to tell my students that this is an arbitrary thing we do, so they can talk like all the mathematicians before them.
• Maybe they come up with their own word for it, which is likely if the concept is super important (like "modulus"). They might remember that the word "divisor" is pretty relevant, and often does the trick. Or they'll probably default to using some clunky, but totally legitimate circumlocution, like "the number you divide by." But mathematicians name things because it's useful--both because it's more efficient to say, and because it helps to have some common language. So we might as well embrace the formal language that mathematicians use--after all, there is often a good reason it's called what it is.
• A similar argument can be made for explicitly providing clarification or interpretation of formal notation, which is often also pretty arbitrary. For example, it's helpful to share that the notation (a,b) represents the greatest common divisor of a and b.
• Sometimes this notation or language is presented in the PSet, interpreted as a problem. For example, consider this example of me introducing floor/ceiling notation. Here, I have made decoding the notation the problem itself. This works well when the notation has at least some layer of common sense, like the floor/ceiling notation does.

• Logistics around the discussion
• I tried to keep the discussions around 15 minutes, but they almost always ended up closer to 25. More than 25 minutes is super tough. 
• I try to take notes under the document camera during the discussion. I do this because illustrating the discussion feels useful in the moment, to track the discussion. The process of "dual-coding," integrating verbal and drawn/written representations can help enrich student reception.
• Notes are is also helpful for posterity. Students can refer to them if they need to (often with my prompting). I can also post a picture of them in Google Classroom, so that if students missed class that day, they can still get some benefit from the notes. And if I was really slick, I might even record the notes/discussion, so that I could post a video to really archive the discussion (not that I ever did that, but remote teaching has developed my video skills).
• [Update 6/20]: The kinds of notes I did could best be described as "Sketch Notes." Something I learned from Deanna Rice at the Inclusive STEM & CS Summit is the value in explicitly identifying the kinds of notes that we're using. The goal is that by modeling and talking about the styles of notes we use, we are helping students develop and expand their own toolbox of note-taking strategies. So maybe it'd be interesting to try a few different note-taking strategies, to show students some of the possible ways they can take notes. Other kinds of note-taking options are Harvard Notes, Cornell Notes, and Concept Maps. 
• I use a document camera because I personally find it easier than using a whiteboard. I also try to restrict the notes to whatever I can fit reasonable on one piece of paper. Here's an example of what a page of notes might end up looking like.

Link to full-sized image.

• I tended to encourage students to more or less copy down everything I wrote. This dramatically consumed their cognitive capacity during the discussion. Honestly, they definitely started out more as notes than discussion. Over the course of the year, the notes wandered further in the direction of discussion. Especially since I spent the year getting PD on the 5 Practices for Orchestrating Productive Mathematical Discussion. Honestly, employing the 5 Practices in this class was a great context, because it actually stretched out the enactment over a whole week, which slowed it down for my awkward novice attempts.
• It was common for students to not take the notes during the discussion. Instead, many would just pay attention to the discussion, and then take a picture of the notes afterwards, to copy it into their notebooks afterwards. That might be the best of both worlds.