Friday, July 19, 2019

Big Problems vs. Problem Sets

There are two big structures for "Doing Math" that I really value in math class, and want to understand better. This post is basically a reflection and narration of me working through the relationship between those two structures. They are "Big Problems" and "Problem Sets." The kind of math class I'm trying to learn how to lead is centered around these two big structures, and I need to do some reflection to figure out what they even are.
  • Big Problems
    • I consider a "Big Problem" (BP) to be one task that is so deep and rich that it could take 2-4 days to complete. Consider my BP-interpretation of this classic problem, given as is:
    • The best BPs are problems where the "Big Question" kind of asks itself as you start to play around with it. They tend to start out with an image or description of a context, and then I may or may not follow up with some scaffolding questions. Here are a few examples like the one above. My biggest inspiration for BP Design is Play With Your Math. One of the leaders, Joey Kelly (@joeykelly89) reflects on the design process for some of the problems in his blog, if you wanna learn more.
    • I might give kids just this for a day or two, and then slowly give them some follow up problems to guide them towards whichever of the million content objectives I could have with this problem. The students' product is some recording of their work in their notebooks, whatever whiteboard work they did. I track their work with some observation notes and asses their products.
  • Problem Sets
    • I consider a "Problem Set" (PSet) to be a collection of related problems that all get. Here are a few examples I've used in the past. The header on each PSet give the big idea:
      • "These problems are meant to help you if you get stuck, or would like some help organizing and thinking about your work. Do some of them, all of them, or none of them. The order isn’t SUPER important. The problems do tend to build on each other as you go."
    • My PSet design is based on what I've experienced from a limited exposure to the problem sets I've experienced at PCMI and PROMYS. The PSet is broken up into three different sections:
      • "Definitely Related to the Problem" (A Big Problem At Heart)
        • These PSets were designed as follow-ups, after students had already spent a day with a related Big Problem-looking task. The questions in this section are directly designed to help them along the way with the main part of the task.
      • "PROBABLY Not Related to the Problem" (Different Surface, Same Depth)
        • These questions are where I will engage students in problems that have a "Different Surface, Same Depth." That is, problems that aren't obviously related to the main problem, but share some deep structure at the deeper level. This design feature is related to, but converse to Craig Barton's (@mrbartonmaths) work on "Same Surface, Different Depth" problems, which you can learn more about here.
      • "Finding More Spice in the Problem" (Building Rigor & Adding Advanced Math Tools)
        • These questions are designed partially as extensions and partially as differentiation. These questions often require more specific background knowledge, like alternative bases, multiplying and polynomials, or modular arithmetic. This is in contrast to the previous two sections, where I design so that pretty much any high schooler can start. They are also designed to help increase the mathematical rigor with which students discuss the main problems.
  • Big Problems vs. Problem Sets
    • Question: What is the practical difference between a BP and a PSet?
    • I'm not totally sure! I know there is a meaningful difference of scope. That is:
      • I think that BPs are more focused on specific nuggets of knowledge. For example, I did a BP this year where students discovered and articulated the Handshake Theorem (which upon research just now, is only a lemma--oof). That's pretty targeted content.
      • As designed in the PSets I shared above, the first part of a PSet is the breadth of the problems, all of which point towards a pretty big representation or idea (Pascal's Triangle, Fibonacci, the solvability and equivalence of all impartial combinatorial games, GCD and LCM, etc.)
    • I think it's possible to experience the same narrative of a PSet, without actually doing a PSet. We all start and finish a BP together, then in each of the following days we tackle a a different "Probably Not Related" problem, or "Finding the Spice" problem. We're basically all just walking through a PSet together.
    • Why not just give them a PSet? The PSet is a structure that really gives students a chance to experience independence and ownership of the process, which can be an enormous feature of differentiation.
    • The PSet is an "idealization" of mathematical exploration--the mathematical narrative. You got some good math problems, you attack it, connect it to other problems, and use your mathematical machinery to rigorously analyze and articulate the problem and its underlying mathematical structure. These courses then are basically a sequence of scaffolded PSet--scaffolded by our moving together as a class. As such, the structure of the class can be represented below:

    • The course is built around a sequence of BPs, with smaller supplemental lessons between them. Then a big PSet towards the middle and at the end. Sometimes the green "regular lessons" between BPs can follow the same narrative of a PSet, just without the explicit structure. Then over the course of the semester, I just giving them the PSet (which we do before each exhibition, at the middle and end of the semester). Having experienced that same "PSet narrative" in class many times, they have some practice and experience with it when they are given a PSet, and they can rely on that to help them navigate the much more open design of a PSet.
    • I haven't really thought about how to incorporate PSets into my regular Math 1 class. It feels super reasonable to at least bring in bits and pieces of the philosophy of PSets into class. By my own observance of the pressures and expectations of a standard 9th grade, Math 1 class make me less comfortable introducing such a radically different structure.
  • BPs, PSets, and Mathematical Presentation
    • I talked about this a little in my previous post describing how I want to do Exhibitions in my math electives. But here I talk about how PSets and BPs relate to Exhibitions in that course specifically. I think an exhibition consists of students attacking a PSet, and then presenting their work. Ideally, all of the PSets resulting in presentations are math new for everyone. 
      • I imagine that I might be okay with some students deciding to do one of the BPs we already did as an exhibition. At which point they'd basically be starting the PSet at the "Probably Not Related" stage. Which is okay? But I might permit this only for the subset of BPs that I think we can meaningfully extend to PSet-level scope.
    • Question Concern: By design, I'm trying to integrate all of my biggest best problems to serve as the anchoring tasks throughout these courses.
      • Side note: Say a student took both of my courses. They would expect to have 20-25 BPs. Then they would experience four different exhibitions, each of which features probably 8-10 different problems. So I'm looking at 52-65 big, juicy, high-quality problems. That's a lot. I got some work to do!
    • Question: Where/how am I going to find enough problems so that I can have a complete set of novel exhibition-style problems?
      • Please share whatever you have! I'm looking for big, rich, deep math problems that a group of kids could conceivable spend 1-2 weeks studying it and its extensions. Comment here or tweet at me @bearstmichael
    • Question: How can you consolidate a PSet, or sequence of PSets, besides student presentation?
      • PROMYS really seemed like they used exams and lectures to bring about some feeling of resolution to the sequence of PSets. I've only half-done a full sequence of PCMI PSets, so can't say I really know how they do it.
I realize that this is a public reflection about some pretty non-universal ideas, so I'm not sure how helpful this post and reflection will be to other people. I hope it is! Let me know what you think on Twitter @bearstmichael.

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