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 any of this.
- 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.
- 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.
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| 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.
- 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!
HUGE UPDATE (6/9/2020): In the year since first drafting this post, I've taught the elective again, and more, and have done a BUNCH of reflection on it, and have made a bunch of posts about it. Check them out here.
HUGE UPDATE (6/9/2020): In the year since first drafting this post, I've taught the elective again, and more, and have done a BUNCH of reflection on it, and have made a bunch of posts about it. Check them out here.
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