Sunday, October 13, 2019

Student Perception Surveys

Have you ever watched video of yourself teaching? It's painful. Even on days where everything's goes well, it's a pretty humbling experience. You watch yourself miss opportunities, spend too much/little time on things, and as clear as you think you're being when you give directions, it's incredible how easy it is to be super confusing. Even hearing a recording of your own voice is the worst thing. And this is the person in front of your students every day!

In the name of reflection, growth, and professional development, understanding how your students perceive you as their teacher and leader is important. Trying to see yourself the way that your students see you, is a humbling experience, and it forces you to reflect and be constructively critical of yourself and your practice. One way that I try to do this in my practice is by giving Student Perception Surveys.

What Are Student Perception Surveys
As a graduate student, we were asked to give a student perception survey. The survey was designed by Tripod Education Partners, and was intended collect data on how students perceived their teacher, and how the teacher/class made them feel. Some example questions are:
  • My classmates behave the way my teacher wants them to.
  • This class does not keep my attention--I get bored.
  • My teacher takes the time to summarize what we learn each day.
  • In this class, we learn a lot almost every day.
  • My teacher seems to know if something is bothering me.
Students then respond on a scale from "totally untrue" to "totally true." The survey has some great design features, and some mediocre ones as well, which is what you'd expect from a private education business thing. But the spirit behind the survey is what matters. Just like you would collect data on which of your students can identify the roots of a polynomial, and adjust your practice to suit, you can collect data on how you make your students feel--because it matters.

I'm not making a pitch for this particular Tripod Survey they made us give. Tripod doesn't know I'm writing this post about their survey (I hope they don't mind), and I won't post my survey or results because it's technically a private product that someone has to pay for. IDK, maybe you can convince your admin/district to sponsor your school or district. Or you can make your own, or maybe find a free one online. If you do, let me know because I'd love to see what kind of items you put on your survey.

Example of How It Can Help--A Personal Case Study
Last year, I taught two sections of Integrated Math 1 for 9th graders, and two sections of a Discrete Math elective for juniors. As I was looking at the data after the first survey, I was particularly surprised by student responses to the statement: "My teacher seems to know if something is bothering me." Fewer than 50% of students agreed with this statement. I was discouraged by this response, because I try to put a lot of work and thought into making sure that my students feel emotionally attended to, and it was something I thought I did well. After seeing this data, however, I tried to incorporate more check-in questions on my Do Nows asking students how they're doing.

I also modified one of my quick check-in routines that I use. At some point during a class, when I'm scanning the room, or students are taking a quiz, or working independently, any time I see a student who has their head down (in a "sad" way), or who looks worried about something, or if they're even just "spacing out," I try to check in with them non-verbally. I do this by making eye contact with them, and then giving them a thumbs up, sideways, and down, non-verbally asking them how they're doing. This gives them a chance to simply respond with the appropriate thumbs-up/sideways/down.

Before, I considered the check-in itself to be the primary means of attending to the student. Letting them know that I see them, and want to see if they're okay, especially if it looks like they're struggling. In an effort to be more attentive to students who might be struggling, now if a student gives me anything but a thumbs-up in return, I'll go kneel next to them and check in with them privately, asking what's up and if there's anything I can do to help.

After these two modifications, I saw a moderate improvement in teacher-favorable responses, up from 49% to 55%. This is still much lower than I was hoping for, and this area continued to be a weakness for me after the third administration of the survey. Nevertheless, in general, the survey directed me towards a blind-spot I had, which I appreciated, even as it continued to challenge me.

Things I Have Done In The Past
  • Give the survey multiple times over the course of the year. I typically give the survey three times, once each in November, February, and May. This give students some time at the beginning of the year to cultivate some feelings about me, and gives me some time after each survey to reflect and try to improve things.
  • Divorce the survey from big tests/grades. It's super tempting to give the survey after a big test, or at the end of each quarter. But this might bias the data by drawing a connection between how some test just went, or what letter you sent home on a report card. And conversely, it also increases the temptation for teachers to tweak grades or routines in an effort to inflate their own survey results.
  • Make it a normal day. Like I mentioned above, if you want the survey to be an accurate and honest assessment, you don't want to make a big speech beforehand, or do it the day after a field trip, or some other random day.
  • Make sure students know the responses are anonymous. Data could be biased in either direction if students thought you could see who responded what. Also, especially if you have smaller groups of students, don't play the game where you look at the data and try to guess whose responses were whose.
  • Offer the survey in native languages. My first year teaching, I had a section where almost the entire class had moved to the country in the past year. I wanted to assess their perceptions of me, not their language. So I made translations of the survey. Yes, this might have introduced new biases due to imperfect translations, but I figured it was worth it.
Things I Haven't Done (But Want To This Year)
  • Give the survey with a team of teachers. Find a team of like-minded colleagues in your building, and form a working group. You can all give the survey to your students around the same time. Then you can reflect on the data and experience together. You don't have to share all your data with each other--it is pretty sensitive and personal data. But you can if you feel comfortable--it's an intense exercise in professional vulnerability. Especially if the results hit a sensitive nerve, like mine did in the personal case study I gave above. You can use the data as a starting point for discussing problems of practice, or brainstorming ways to respond to the data. You can also help each other stay accountable to the efforts you make to respond to the data.
  • Share some of the results with the students. It would probably be overwhelming to share all of the data with your students, but maybe you can share an interesting or challenging cluster of data, and ask students what they think. If you're feeling really stuck on how to respond, this can help you to brainstorm ideas. It also messages to the students that you are actually attending to the data, and care about what they have to say. If you do this, make sure it comes from a position of you accepting responsibility for this, and seeking their support and input. It must not be about you blaming them or trying to justify why their responses were not favorable to you.
Using This Survey in Teacher Evaluations
I am one of the advisors for my school's student government. Every year, students share that they are frustrated that they have little to no input on teacher evaluations. They are certainly the people most proximal and most affected by the quality of the teacher. And they often cite the fact that student evaluations are common at the college level. I always see their point, and every year respond with how complicated and litigious the teacher evaluation process is. So from an administrative perspective, making student perception surveys a mandatory part of the teacher evaluation process is basically impossible, and definitely problematic.

Regardless, I voluntarily share my survey data, and my reflections on it, with my administrator as a part of my annual professional evaluation. It makes for a great artifact, I gotta say. But, I also trust my administrator to look at the data holistically, in context, and with an eye for me trying to be reflective. I trust that my administrator will not use the data to penalize me for the areas in which I need to grow. I know that not all teachers are privileged to have this kind of evaluator. Share the data with those who you feel safe doing so.

It would be inappropriate for an administrator to evaluate a teacher, or for teachers to evaluate each other, based solely on the raw data from this kind of survey. All this data exists in a context from which it can/should not be extricated. As such, it would also be inappropriate to use this data to compare teachers with each other, even those with the same students. The degree to which an administrator attempts to do this, is the degree to which they incentivize a teacher to game the survey, and is the same degree to which they erase the value of the survey as a professional exercise.

Indeed, it is very common for teachers to avoid these kinds of surveys for evaluative purposes, because they are concerned that the surveys end up as more of a "popularity" or "likability" assessment, which are not necessarily meaningful assessments of the quality of a teacher. It is impossible to avoid this bias, true. Nevertheless, I claim that students are much better at divorcing their student experience from their personal experience than we give them credit for. So we can feel comfortable giving this kind of data some real professional credence.

Regardless, and this is more of a personal opinion that not all teachers have to hold, I DO think that it is important for teachers to have a modicum of accessibility, likability, and collegiality. Despite the popular belief that students like "easy" teachers, I do think that students also find it easy to "like" the teacher that teaches them respectfully and well. Kids WANT to learn. Kids WANT to work hard. And kinds appreciate teachers who help them to do that.

Long story short, yes, a student perception survey will be biased in some way by whether or not students "like" you. No, that does not invalidate the professional utility of the data. Yes, it does complicate it. And if you want to invite this data into your professional evaluation, that's your professional prerogative to do so, recognizing that the evaluative focus should be on your reflection on the data, and your response to it, instead of raw performance.

If you end up putting together a working group at your school to develop or use some kind of student perception survey, let me know! I'd love to hear how it goes, what your assessment looks like, and how you or your team reflect on, and respond to the data.

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'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!

Wednesday, June 26, 2019

Discussion in Math Class: Part 1 -- Why Bother?

I just finished my second year teaching. Pretty much the whole time I've been thinking hard about class discussion in math class. My teacher prep program (Boston Teacher Residency) was BIG on class discussions, so from the beginning of my career it's been a big thing I've thought about. That's not to say it's been a big part of my practice. I've got a lot of questions I'm trying to work through, a few answers, some resolution, and a bunch of unresolved issues. I'll try to present not only my current conclusions, but some of the narrative as well. I'll try to break this reflection into chunks. This is Part 1.

To be clear, when I say "classroom discussion," I mean a class full of students, all talking about a topic, or series of topics, for a sustained time (like...more than a couple minutes?). This looks mostly like one student talking at a time, and everyone else listening. The teacher may or may not be involved, and there may or may not be any other participation structures.

Experiencing Classroom Discussion as a Student

In math class, the discourse was almost entirely centered around the teacher. I never found discussions particularly educational. I'm not sure who did. For the most part, the content seemed pretty cut-and-dry. Image of a traditional classroom, the teacher's modeling and lecture provided all the information. Any small details, confusion, or minor discovery came out in the problems sets completed alone at home. I never feel the desire or need to wrestle over the math in discussion.

In my smaller math seminars later in college, where the professor explicitly invited discussion, we were all so intimidated and self-conscious that we didn't want to participate for fear of revealing how little we understood. By "we," I mean my friends and I who bonded over how lost in the sauce we felt so often. I presume there was at least a handful of people who felt, as I did in high school, that discussion was unnecessary or uninteresting.

These experience as a student led me to form the following schema about classroom discussion:
It felt like there was a pretty narrow target of students who benefitted from discussion. I would say these students are in the "Zone of Utility," where classroom discussion provides some kind of utility or benefit. They are relatively small "Goldilocks" group where the content is neither so hard it's intimidating, or so easy it's uninteresting to discuss. One assumption of this model is that understanding is one-dimensional spectrum, which is an oversimplification for sure.

What They Taught Me In Grad School

What was told to me about classroom discussion in grad school:
  • It should be something that happens in almost every class, almost every day.
  • It is an effective way to make sure everyone engages with the lesson objective or Big Idea.
  • It is useful in an of itself, because it requires students to make sense of other people's thinking.
  • It is a form of communication that can honor the discourse culture of the individuals and whole (classroom) community.
This teacher-side experience of class discussion led me to develop another model for thinking about classroom discussion, which (noticeably) isn't incompatible with the model I had developed as a student:
  • Without discussion, many students can engage in a common math experience, but fail to arrive at the teacher's learning objective, these students are, as I described myself earlier, "Lost in the Sauce." Only a small subset of students managed to meet the teacher's learning objective.
  • With discussion, many students can engage in a common math experience. As shown in the "Without Discussion" model, probably not all students will arrive at the teacher's learning objective independently, so the discussion after the math experience serves as a way to "funnel" most students to the learning objective.
What I learned from experience about classroom discussion in grad school:
  • I didn't have a super clear idea about what a classroom discussion CAN do, and what ONLY classroom discussion can do.
  • Discussion for its own sake is a palpable waste of time.
  • Class discussions aren't "free learning." Just because students are talking about what they did or what someone else did doesn't mean the speaker or listener are learning from what's being said.
This discouraging experience led me to form the additional model about how my classroom discussions went (and how I thought they went down in many other classrooms):
  • With ineffective discussion, many students can engage in a common math experience, but fail to arrive at the teacher's learning objective independently. These students are, as I described myself earlier, "Lost in the Sauce." Only a small subset of students managed to meet the teacher's learning objective. I then manage to lead an ineffective classroom discussion, which at best serves to validate and confirm those few students who had pretty much met the learning objective already, and at worst further alienates students who felt they had not met the learning objective.
Why bother? During my residency year and the vast majority of my first year teaching, this is what it felt like I was doing. It felt like there was no new learning happening during the discussion. Or, that is, it felt like the thing that my students were learning was that discussions are not useful for learning math. So I would bail on them. Pretty much every time. I'd rather let students keep working right up until the end of class, because I felt it was more likely that they learn something during the independent-/group-work time than during a discussion.

My end-of-class discussions weren't going particularly well. Where any of my classroom discussions going well?

To be continued!

Sunday, June 16, 2019

Proposal: A Less-Sequential High School Math Experience

How can we design a high school math department (and graduation requirements) so that we are accurately reflecting the vastness, diversity, and non-linearity of mathematics? Consider this representation of a hypothetical high school math department (Image here):

Understanding the Map
  • Each course is a semester-long math course
  • If there are two separate arrows from A to C, and B to C, then that means either A or B are required for C, not necessarily both. If the arrows meet before C, then that means that A and B are both required for C. Consider this helpful diagram (Image here):

Graduation Requirements
To go with this more open curriculum offering, is a graduation requirement system that tries to honor both the openness of the program, and the necessity of certain math topics in high school.
  • You must pass at least eight (8) distinct courses, including:
    • Proportionality, Linearity, Financial Literacy, Probability, Correlation & Causation
  • There are four Fields of mathematics, and students must take the appropriate number of courses in each field. Some courses satisfy multiple requirements (Image here.).
    • (A) Algebra (take at least 4)
    • (G) Geometry (take at least 2)
    • (P) Applied Math (take at least 1)
    • (U) Pure Math (take at least 1)

  • You must pass any and all prerequisite courses before taking a course.
  • Students are permitted to "test out" of prerequisites provided they demonstrate sufficient proficiency, as determined by the math department.
    • If a student tests out of a course, they are not awarded the credit for that course, nor does that satisfy any Field requirements.
Logistical Comments I Have
  • The standard pushback here is that this kind of system would require teachers to teach lots of different courses in a year. More preps makes things more difficult for teachers.
  • Lots of students might be forced to take certain courses simply based on the fact that it was the only course available at a certain time in a certain semester. The bigger the school, the better the school's capacity to support all these different courses.
  • It might make more sense to default all 9th graders to starting in "Linearity," and only walk them back to "Proportionality" if necessary.
  • It would take a TON more resources to make this work if you have special populations of students who might need to be grouped (students with learning disabilities, students learning English as a second language, etc.)
  • This would work best if the other departments (ELA, History, Science, etc.) used a similar system.
Questions I Have for Y'all
  • What courses would YOU make required?
    • I thought Proportionality, Linearity, Financial Literacy, Probability, Correlation & Causation were the most important, recognizing that most high schoolers would already have proportionality from middle school, and some might already have linearity from middle school.
      • Those five courses I identified that everyone had to take are based solely upon my own value judgements of what math content I think every person should know before leaving high school. They definitely need to do more math than just those courses, but they definitely have to do at least those courses.
  • What other courses would you add?
    • These were all the courses/topics I could think of that I've either taught, seen, or felt spanned the traditional high school math curriculum.
      • Some courses I would need to figure out how to add are all the AP's and Computer Science.
      • I'd also like to add some kind of geometry-centered course where students are getting lots of geometry measurement and whatnot. Like, where in this course offering are students going to think hard about the volume of geometric solids (except calculus)?
      • I think Economics might better be split into Micro and Macro? Is one a prereq for the other?
      • If other departments get in on this, then that opens the door for requirements to be met across entire departments. For example, lots of schools consider Econ to be a social studies class. Next year we are teaching Mathematical Biology, which could count as a science credit.
    • Best case scenario, we think of the biggest reasonable course map possible, with all the appropriate relationships thought out, and then schools would pick some subset of that map.
  • Do these course descriptions make sense?
    • Some of them I don't even know how to describe (e.g., Proportionality)
    • Some of them I'm not sure are a whole semester of high school level work (e.g., Linear Algebra, Periodicity)
  • What do you think about the Fields?
    • I made the Field requirements so that they were roughly proportional to the course offerings. That is, there are way more Algebra classes in the map, so I required way more Algebra classes. But this is a reflection of my own limited ability to create course offerings. If I could think of more geometry courses, then I could make one.
    • I'm lumping statistics in with Applied Math, which would include Econ and Financial Literacy. The implication of this is that kids could take Projectile Motion or Financial Literacy and never HAVE to take any other stat classes. Except that I've already said that half of the Applied Math Field is a requirement on its own. I guess that says something about my own feelings about Applied Math in high school?
  • How does this gel with our understanding of intellectual development of kids?
    • By this model, you would have 9th graders taking Probability. You could also have 10th graders taking Linear Algebra. Is there a reason this couldn't be the case? It certainly feels a little weird, expecting 9th graders to do content that is typically assigned to 11th and 12th graders.
      • Is that because they are literally not OLD enough to do that kind of math? At which point, we should place another pre-requisite there, to slow them down? Or have an age requirement? But that feels wrong.
      • Or is that because we typically haven't taught them "enough math" until they are older? At which point, we should place another pre-requisite there, to make sure they get all the required math.
    • This would put kids of all grades/ages in the same classes. Lots of classes might be mostly grade-grouped--Linearity and Proportionality would have mostly 9th graders. Polynomials and Solving Triangles would have mostly 11th and 12th graders.
  • Where does solving equations go?
    • I think that for the most part, most of the big first "this is how you solve equations" would come out in "linearity" maybe? And then as appropriate, as you run into different contexts for solving equations, you learn new methods for solving equations?
  • How can you make sure that students are making informed decisions?
    • They would have a good idea about what each course is about, so that they can pick the classes that are the most interesting to them.
    • They would have to have a good understanding of the requirements system overall, so that they could "see" the paths and scheduling required to get to certain outcomes. Which is complicated, for sure. I mean, it's like trying to plan a major for college--only you have a major for math, science, history, ELA, and whatever else.
  • What do you think?