Tuesday, July 30, 2019

A Tool for Developing a "Curriculum Portfolio"

HOT TAKE: There is No “Magic” Math Curriculum
  • I don’t think that there is ever going to be one-perfect curriculum that works well for all teachers, students, and contexts. This is because teachers, students, and context are so diverse across the world, and math is so rich, that trying to say that there is one “best way” to articulate a math curriculum is folly. Maybe you get lucky, and find one curriculum that works for you, your kids, and your context. But even that can be hard! Teaching is hard!
  • I think one way to manage this is to instead develop a “curriculum portfolio” that you use. You’ll probably start with one curriculum (formal or otherwise) that you are OK with. Then you branch out and deliberately investigate and analyze other curricula, one by one, adding them to your portfolio. You take the things from each curriculum that you like, leave the things you don’t, and try to incorporate these desirable elements into your classroom and practice. There are countless curriculum you have to add to you portfolio. Naming just the national high school math curricula I know off the top of my head, you have CPM, IMP, CME, Exeter, and Illustrative. Then you have the countless thoughtful educators on the #MTBoS and beyond sharing their lessons, projects, units, materials, and reflections.
  • I recognize that some teachers are forced to adhere to a given curriculum with fidelity, which hampers their ability to make use of even the best/smallest parts of other curricula. While curricular coherence across years, schools, and districts is valuable, I think that restricting teacher curricular autonomy to this extent does more harm than good. But I guess that’s between you and your administrators? Good luck?
  • Given how many math curricula there are, and how complicated curricula can be, it is helpful to have an efficient and reflective process for analyzing and cataloging curricula, and organizing your curriculum portfolio. I’d like to share with you the process that I have developed this summer. This is just a snapshot of my current process, and with every portfolio addition, the process is refined further.
My Process for Analyzing Curriculum
  • It helps to have a general structure that you are connecting your analyses to. This is the “skeleton” of your current portfolio. This skeleton can be as extensive or sparse as you want. The more extensive it is, the less arduous your curriculum analysis is going to be. This is because you have a better idea of what you are looking for in the curriculum--what is relevant. That doesn’t mean more is better, though. You may not want or need to constrain your curriculum analysis so much in this way. You may be open to investing time and energy into questioning/challenging your existing portfolio. It depends on what you have time for and what you’re looking for.
    • For example, the curriculum skeleton I had when analyzing IMP (and that I generally have) was this: I follow the units as outlined in CPM Int 1, skipping over a couple that I have already identified for the sake of time. For these units I have a set of unit plans where I keep track of what I have done. I am pretty clear on what Big Ideas I want to cover in each unit, but am super flexible in what day to day lessons I use.
    • With this skeleton in place, I’m generally looking for:
      • Strong tasks and problems for a given topic
      • How they design for interleaving of topics, and mixed-spaced practice?
      • Understanding certain (challenging) topics of interest
        • These are a small handful of topics I’m particularly interested in, either because they’re going to be new for me, they continue to be challenging, or I’ve heard the curriculum does it well. The grain size depends on what’s interesting, but is generally unit level. For example, this year I analyzed IMP with an eye for Similarity, because I’m teaching it next year, didn’t love the one way I did it in the past, and heard that IMP did it well. Solving linear equations is always on the list. And I teach mostly 9th graders, so linear functions is usually up there, too.
        • Some things I will try to analyze in the curriculum is how the curriculum builds schema for those challenging topics. For example, how does a curriculum introduce solving equations? Do-undo tables? Algebra tiles? Make equivalent equations until you know what the variable is? Graphing?
  • At the same time, I also try to analyze curricula on a more general level. The goal is to advance what I know about curriculum, pedagogy, and math in general. These days, I’m usually looking for: 
    • The relative valuation of conceptual understanding, procedural fluency, and application, and how that’s realized.
    • How do they develop a course narrative (as communicated/summarized partially in a ToC)? How explicitly is it stated?
    • The unit narratives--how do they make a unit feel like it builds cohesively? Again, how explicit is it?
    • How do they differentiate through universal design? In particular, I’m interested in how they design for daily accessibility of the content.
    • Any great problems in general, whether they’re relevant to any math I’m teaching now or later
  • In order to help structure my curriculum analysis, I will have a document that essentially has space for all of these components:
    • Course Narrative (ToC and Problems)
      • Here is where I identify the sequence of big topics (basically unit topics, and maybe sub-units if they’re meaningfully distinct)
      • Here is also where I keep track of good curricular problems/tasks
    • Tracking Big Ideas
      • Here is where I have my pre-identified topics of interest beforehand, and populate it with notes and reflections as they come up. I try to organize and connect ideas within each problem as they go along
    • Other Cool Problems
      • Here is where I note any problems that seem generally cool in general, along with a link to the original, and enough of a description/picture to remind me what they are at a glance
    • Reflections on the curriculum
      • Here’s where I track and organize my thoughts on the bigger ideas of the curriculum. Again, I may just write down any relevant notes, but I try to connect and cohere it to the other relevant notes and ideas as they come up.
    • Reflections on teaching
      • Anyone who has taught alongside me knows that my reflections tend to get really “big” really fast, and so I’ll often be drawn down a reflective rabbit hole at a moment’s notice (whether or not it’s really the best time for that!) This frequent big picture reflection has felt super valuable here in my early stages of development as an educator. So I try to create space for this in my document as well, so that I can keep track of my reflections, and provide them with space to grow. Or if right then isn’t really the best time, at least leave myself a note that I may want to go back to it at some point.
      • This is actually how this very blog post came to be! I had just finished perusing IMP Years 1-3, during which I had organically developed the curriculum analysis document. I was then going to transition to analyzing some of Illustrative Math’s curriculum, and wanted to evaluate/reflect on how the structure of the document aided my process. Then this reflection kind of started, and the genre of “blog post” felt like a useful way to force myself to be thoughtful and clear. A lot of the topics that tend to come up in this section are fodder for blog posts/drafts.
You can see a blank template for my curriculum analysis here. You can see an example of me using this template to analyze IMP Years 1-3 here. This analysis of IMP represents ~10 hours of work. Feel free to leave comments on the documents if you want!

As always, I hope that this blog post was helpful for you--I know it really helped me to reflect and organize my thoughts around curriculum analysis and my own “Curriculum Portfolio.”

Tuesday, July 23, 2019

My Concerns About "Problem of the Week" and Related Systems

It is a common practice in different math classes and curricula to occasionally feature "random" big interesting math problems. IMP realizes this as a "Problem of the Week." Other curricula have "Thinking Problems" where students explore great problems in the middle of an unrelated unit. (I'll refer to these as POWs for the rest of this post.)

  • PROS: Why I think educators do this:
    • Educators want to honor the vastness of math. They know that math is full of wonderful rich problems that students deserve exposure to.
    • Educators want to foster "general problem solving skills." A lot of educators (myself included) believe that these skills can only be developed by tackling a diverse collection of big messy math problems. Educators realize that by introducing these non-curricular problems, they can diversify the kind of math that students do. They also tend to focus more on mathematical reflection and "big picture thinking" in these problems than in a regular math class.
    • Educators want students to have a chance to do new and worthwhile math, even if they are feeling disconnected from the general thread of math they're covering in class. Some days, you read the room as a teacher and KNOW that if you don't do something big and different for at least one day, the students will revolt. That's a super real feeling, and sometimes taking a break from things is a necessary decision.
  • CONS: Why POWs as a system can be problematic:
    • If a student's experience with all the biggest, best, most fun problems in math is only in the context of a POW, it will create a false dichotomy between "fun and interesting POW math" and "not-fun and not-interesting regular math."
    • Many educators are pressured to observe a fairly constrained set of content standards, and with not enough time to do so. As such, POWs are often the first to get cut. Or they are pushed into the realm of homework, where they are given much less value, structure, and support.
  • What we could do instead, (ideally):
    • Best case scenario, we DO center our math courses around big, rich, messy problems. This way we can teach to those "general problem solving skills." But we ALSO make sure that these problems transmit the curriculum that is at the center of our courses. This way the problems are connected to what students are doing every day in class, adding value and depth.
    • This is WAY harder for teachers to do. There is a pretty narrow strand of algebra/geometry content that is commonly considered mainstream secondary math standards. Forcing our POWs to be within this limited context does constrain the kind of problems we can give them. It is then incumbent upon the teacher to push the boundaries of their math class by finding problems that feel different and diverse, but ALSO rigorously include mainstream secondary math standards.
  • What I'm NOT saying:
    • I'm NOT saying that we shouldn't do POWs, or have random days when we do math that is disconnected from the unit/course we're teaching that week. As with every other decision we make as professionals, educators have to weight the pros and cons of each decision in the context of who we and our students are, what our goals are, and what the context is.

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.

Thursday, July 18, 2019

Planning Math Electives

I've been privileged in my short teaching career to have had a lot of exposure to math electives in high school. In particular, I've had a rare experience with "pure math" electives. During my student teaching, I got to watch Joey Kelly teach two different versions of a semester Discrete Math elective (modular arithmetic, bases, and graph theory). Then last year I got to teach that class for myself. It was great for a bunch of reasons:

Especially as an early career teacher, observing, planning for, and teaching an elective math class was super beneficial for my development. I was looking for a "different" experience teaching the course, and students were looking for a "different" experience as a student. It's the perfect space to be radical and experimental in both curriculum and pedagogy. I will say, my administration and math team were also clear in their confidence and comfort giving me the opportunity to teach Discrete Math, and do with it what I wanted (provided kids learned worthwhile math).

I think students appreciated the chance to do some super different math, in a pretty different way. Most of them said as much, anyways. As a teacher, I know I had a lot of fun. By removing the (often harmful) pressures of a "regular" math class, and my admin empowering me to make bold teaching decisions, I learned a lot professionally.

Maybe later I'll try to consolidate my reflections on what I learned. Right now though, I'd like to look at some ideas of how I can advance the course, in both curriculum and pedagogy. Because my school doubled-down on it, and I'll actually have the chance to teach two different semester-long electives this year.

Course Descriptions
These are the topics I plan on covering. The topics I've highlighted are the once I haven't taught and/or know the least about, and am looking for resources and ideas!
  • Semester 1: Discrete Math
    • Modular Arithmetic
      • Calculating remainders
      • Applications in time and cryptography
      • Operations in the integers (mod m)
      • Solving equations in (mod n)
      • Inverses, quadratic residues, and zero divisors
      • Systems of congruences
      • Representing functions on circular graphs
    • Group Theory
      • Rubik's cubes
      • Intro to group theory (closure, associativity, inverses, identity, commutativity, order)
      • Symmetry groups, mod groups
      • Weird "multiplication" tables
    • Number Bases
      • Counting in alternate bases
      • Converting bases
      • Operations in alternative bases
      • Applications of binary and hexadecimal in computers
  • Semester 2: Advanced Quantitative Reasoning
    • Combinatorics
      • Fundamental Counting Principal
      • n Choose r
      • Pascal's Triangle
    • Sequences
      • Fibonacci (and related sequences)
      • Josephus Problem
      • Polygonal numbers (and painted cube?)
      • Other cool/weird sequences?
    • Graph Theory
      • Representing real world applications
      • Paths and circuits
      • Graph coloring
      • Geometric solids and planarity
    • Game Theory
      • Impartial combinatorial games
        • Nim and its Variants
        • Nimbers and Nim addition
      • Decision theory
        • Prisoner's Dilemma, Battle of the Sexes
        • Equilibria and Pareto efficiency
Just some notes:

  • Honestly, I think the first course should be called "Number Theory." The general theme of the course is that it's focused on "pure math" and topics in number theory. But that's just how it's rostered--so it goes!
  • The second course should really be called "Discrete Math." If I just want it to be an honest "discrete math" course, the topic "Decision theory" really doesn't belong as much. I could also elect to observe the actual course title "Advanced Quantitative Reasoning," and stay more within the realms of applied math, at which point the discussion of "Sequences" fits in less. I can't say I'm super concerned about either of the
  • If you've got some great ideas for math that you think could fit well into either of these courses, it might be helpful to know that these are two of six total math electives available to students. The other four are Statistics, Financial Literacy, Mathematical Biology, and Economics. Preferably, my course would intersect with the other courses as little as possible.

Math Practices Over Procedural Fluency
  • Perhaps even more than "big picture" mathematical content understanding, I'd like to prioritize my students developing their ability to engage in mathematical thinking, work, and play. It's pretty tough to articulate exactly what I mean by these things. But the Standards for Mathematical Practice are a decent partial-articulation.
I got this from a NYSED presentation, but they are a national thing.
  • In an effort to prioritize these, the bulk of my assessment, grading, and feedback will be centered around them. The biggest way I will use these is in the assessment of BPs/PSets. For most BPs/PSets, I will identify two target MPs that seem particularly relevant to the problem--one of which is always MP1 Perseverance. I'll assess their products and take observation notes, with respect to those two MPs, mostly using this super-simplified rubric I made.
    • The more I do it, the more I have come to value observation notes as evidence for assessment/grading. I'll walk around with a clipboard that has a list of all the students, with space for me to track evidence of the student showing elements of proficiency in the indicated MPs.
  • I don't do tests. I do very few quizzes, and even that felt too much. I prioritize them much less than in my regular Math 1 class:
    • I did them less and less as the year went on, instead prioritizing BPs (and eventually presentation). I think that this was a result of a waning utility of data/feedback on procedural fluency, which my quizzes tend to target, because I got a clearer vision of how to prioritize BPs and MPs.
    • This year, my goal with quizzes will be as follows:
      • Each unit will have 2-3 quiz standards. Only the most central and specific content standards will be realized in quizzes. Examples of these are as follows:
        • Modular Arithmetic
          • Calculating Remainders
          • Finding GCD and LCM
          • Solving Systems of Congruences
        • Alternative Bases
          • Counting in Alternative Bases
          • Converting to and from Decimal
          • Addition and Subtraction in Alternative Bases
          • Converting to and from Hexadecimal
      • Each standard will be assessed at least twice before the end of the unit and at least once after the unit ends. This will probably shake out to 1-2 times a month.
      • Quizzes will try to focus on literacy and conceptual understanding, as opposed to procedural fluency. Here is are some examples and non-examples of my "Calculating Remainders" from my Modular Arithmetic unit:
      • Some standards are easier than others to realize as quizzes that don't measure only procedural fluency. Generally, it is much more difficult to write and grade quizzes that assess literacy and conceptual understanding of standards.
  • Grade Breakdown: I'm thinking my grade breakdown will be 60% BPs, 15% Quizzes, 25% Exhibition (broken into 10% for the first one, and 15% for the second one). I think this breakdown represents my real relative valuation of the different components of the class.
Questions I Have
  • Managing Tardiness
    • Last year, 20% of your grade was "Participation," which was largely based on attendance. This was an effort to use the (often harmful) economy of grades to incentivize attendance and timeliness to this elective, which was an issue as it was the first class of the day. I think none of us will be surprised when this structure failed to make anybody show up on time who wasn't already going to show up on time. It was basically just another privilege filter. It WAS a conversation starter sometimes. But I think I'd like to get rid of it. Because at best it wasn't useful, and at worst it was an active pedagogy of oppression. I want to bail on this "Participation Grade." But I DO feel like I need to do something extra to help manage what is often pretty high tardiness/absence rates. I know I'm not the first teacher to have this issue, but yeah.
    • Question: How can I design a course/culture so that it both incentivizes timeliness/attendance, AND is respectful of students when they aren't on time or present?
  • Mathematical Presentation
    • I'd like there to be at least two "exhibitions"--one at the end of the semester and one at the middle. I think an exhibition consists of students attacking a PSet, and then presenting their work. Ideally, all of the PSets resulting in presentations are new to 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.
    • Purpose of the Presentations
      • I like the idea of consolidating a semester or PSet with presentations. But my biggest frustration with presentation in math class (and class in general) is that I don't want to see ten different groups all present how they did the same problem in slightly different ways--or even worse, the exact same way! It is important to me that when doing presentations, that the content of each presentation is largely unfamiliar to most of the students in the room.
      • Sharing of (and invitation into) an exploratory problem
        • Each group presents a totally different problems
          • Explain and invite everyone into your problem
          • Share what you discovered (or didn't discover!)
          • Share how it connects to other math we've done (if it does)
          • Say where you could go next
        • Two different “ending paths” for the same problem.
          • Example: We could both start with PlayWithYourMath.com's X-Factor as a BP, but one of us could end with a presentation proving the non-planarity of the complete graph on five vertices, and the other could present on the graph isomorphism of products of distinct primes.

          • Even though the underlying problem is common to everyone, the math that any one group ended up spending most of their time on, and presenting, is pretty different from what groups students did.
        • Question: What else can the purpose of a mathematical presentation be?
There's definitely more questions, comments, and concerns that I have in the planning of these courses. But these are the biggest questions in my head right now, and I appreciate having you as an audience to help push me to be reflective and thoughtful here!