My Discovery-Based Math Elective: The Purpose of the Elective (Part 2)

This is part 2, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe the purpose of the course, as well as the fundamental curricular and pedagogical design philosophies of the course.

• This is a one year high school math course, surveying five topics selected from pure mathematics. At my school we broke the course into two unrelated semesters, where students could take either/both semesters. We had to give the semesters two different names, neither of which fit the content correctly, but we were working with a constrained district catalog). 
• Number Theory (we called it Discrete Mathematics)
• Modular Arithmetic (8 weeks)
• Alternative Number Bases (5 weeks)
• Discrete Mathematics (we called it Advanced Quantitative Reasoning)
• Combinatorics (Pascal's Triangle, primarily) (4 weeks)
• Sequences (4 weeks)
• Graph Theory (5 weeks)
• Who is this course for?
• The course is designed to be accessible in grades 9-12. This year I had grades 10-12 in the class, some of whom were 10th graders struggling in their other (primary) math class, others of whom were 12th graders crushing AP Calc at the same time. It also was mostly upperclassmen, as they tended to have more space in their schedules for electives. It was one of the options for a 4th year of math, for students two didn't want to do AP math. The course was eligible for math credit fulfilling my school's math graduation requirement.
• With some work, this course could totally be adapted to meet the readiness of middle schoolers, though it would need someone with a deeper understanding of middle schoolers than I have. Similarly, the course could be modified for post-secondary use.
• The objective of the curriculum advanced in this course is to expose students to topics in mathematics that are typically less covered in traditional K-12 coursework. Below is a pretty cool Map of Mathematics that's floating around, which you can see full-size here (and even buy a poster!) Circled in yellow is my understanding of the extent of math covered in traditional high school mathematics. Circled in blue is the math that is surveyed in this course.

• This focus on non-traditional topics has two major impacts:
• Facilitates differentiation for readiness
• By focusing on topics that are outside the traditional K-12 pathway, this course can provide a differentiated experience for an uncommonly wide range of incoming student readiness levels. The basics are Number Theory are accessible at the elementary level. For example, division with remainder is technically an elementary level standard. But by extending it through modular arithmetic, we can experience meaningful and unfamiliar work all the way up to advanced topics like Linear Diophantine Equations and the Chinese Remainder Theorem.
• Any of the five topics covered could definitely be a year-long elective course, on their own. But by including five different topics, I really only need to include the richest and most accessible 40% of each field. And then I only need to reasonably expect a student to access around half of that. But by having so much math available to be studied, it becomes incredibly easy to provide multiple options for students to find relevant math that feels interesting and possible to them.
• Facilitates re-invitation of previously discouraged students
• Traditional high school math coursework has an unnaturally narrow focus of on Algebra, with guest appearances by Statistics and Geometry. As such, we ask students to retread familiar content over and over. And yes, there is some value to that, since there is much math to be learned as we dive deeper and deeper into math we've already begun to understand. It helps us to construct an understanding of how one topic in math can have unfathomable depth, and surprising applications across many different topics.
• But this is a double-edged sword, as it also sends the message that all of math is basically fundamentals in algebra. Many students have negative experiences in some of their math classes, especially in high school. And these experiences often compound, as strict prerequisites force students to repeat classes that they have not succeeded in before. Even if a student does pass a class, they may very well feel like they're just being pushed to do more of the same math, and so have more of the same negative experiences.
• Certainly, the environment in which a student learns has a greater impact on whether or not they have a positive experience in a given math class. All else being equal, changing the topic alone likely won't radically shift a student's perspective on math. But by providing access to math that feels, looks, and is meaningfully different, we can create an opportunity for student to positively reinvent their feelings about math.
• The objective of the pedagogy advanced in this course is to prioritize authentic mathematical practices, with a focus on exploration, discovery, and proof. This focus on practices facilitates the course's ability to differentiate the course for readiness, by providing multiple means of action and expression.
• It is more natural to hold fair, but different expectations for students to engage in a given practice. If I have 9th graders and 12th graders in my class, it's difficult to ask them BOTH to engage in a problems where they have to prove trig identities, for example, because my 9th graders haven't done enough of the prerequisite work. But I also can't ask them both to solve basic linear equations with integer coefficients, because my 12th graders would (hopefully) be mostly bored.

• Instead, say I give students a class the classic Counting Trains problem, posed here by PlayWithYourMath.com. This is a classic "Low Floor, High Ceiling" problem, and a really incredible example at that. Because there is truly meaningful math to be done at levels accessible to both elementary and high school students, and beyond. The interesting math content introduced by the various parts of the problem may be different for students with different levels of readiness. But at all levels, students are asked to engage in mathematical practices (explore, discover, use repeated reasoning, etc.)

My Discovery-Based Math Elective: A Typical Week (Part 3)

This is part 3, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe what a typical week looks like in this course.

1st Day (Usually Monday)

• Alex walks in, and grabs the half-sheet by the door. They then grab their notebook from their cubby. Alex then walks to the front of the room where this week's seating chart is, to figure out which table they can sit. On their way to their table, they swing by the supply shelf to get some tape.
• Once at their table, Alex tapes the half-sheet into the next blank page of their notebook. Alex likes to tape just the edge, so they can flip it up and write underneath it. As they're taping, Alex starts to read the Big Problem printed on it. As they read over it a few times, making notes, other students shuffle in, following a similar routine. Alex says hi as some others end up at the same table, and eventually most of the people at the table of five are talking about the Big Problem.
• Sam is one of the other students sitting at Alex's table. Sam has has headphones in, and is working independently for the most part. Occasionally he might ask or answer a question of the table, but for the most part Sam keeps to himself, writing in his notebooks. Every once in a while Sam will look up and read the Big Problem off of the projector instead.
• Alex's table calls me over, cause they're a little unsure how to proceed. I ask them to summarize what they've figured out so far. I ask some further assessing questions to get a better idea of what they actually need. I ask a few advancing questions, to keep them moving in the right directions, and then continue my circulation. On the first day of any PSet, most of the questions I'm answer are clarifying questions, helping people get a feel for the Big Problem.
• As I'm checking in with another table, it's clear to me that collectively they have a pretty clear understanding of what the Big Problem's about, even if they don't have a plan for solving it yet. Any of the basic advancing questions I would ask them to keep them moving are already in the PSet, so I encourage them to go ahead and each grab a copy of the PSet from the file folder in the back. One of them volunteers to get a set for the table. I notice that a couple of the other tables have already done this on their own.
• I do a quick scan of the room to get a feel for approximately how many groups have started the PSet. I'm not particularly concerned either way, but the amount and quality of work that happens on the Big Problem before students dive into the PSet tells me something about the Big Problem, so I'm curious.
• Students start to pack up, and I realize that there are only two minutes till the bell rings. I make an announcement, reminding everyone to leave their notebooks today, because I'm grading last week's PSet after school. Some students ask if they can finish during the day, and bring it back right after school. As long as I get it by the end of the day, that's fine. I ask one of the students to file the rest of the half-sheet Big Problems in the same folder as the PSet, in the back of the room. The bell rings and I start to switch gears for my next class.

Middle Days

• The middle days of the PSet are all pretty similar to each other. Students walk in, grab their notebook and sit at the same table as the first day. Anyone who didn't start on the PSet on the first day makes sure to start it on the second day, getting a copy from the folder in the back of the room. If they forget, they're usually reminded when they look up and see it projected on the whiteboard at the head of the room.
• Most students are starting with the Obviously Related problems, though a few are skipping ahead to some of the *Probably* Not Related problems. One student is just reading through the whole PSet, before starting anything, putting a star by the problems that look more interesting or easier, and putting a question mark by the problems that they don't immediately understand.
• My first round of circulation is mostly just me saying hi to students, checking in with them and occasionally encouraging them to grab a PSet if they haven't already. Some students and tables start asking questions pretty quick, so it's it's tough to stay disciplined with my "circulation rounds." I know I don't want to just bounce from table to table, playing whack-a-mole with individual student questions. So keeping consistent loops through the room helps assure students that I will get to them soon enough.
• Some tables have a question ready when I get to them. I'll often snoop on other tables that haven't even asked a question yet. Usually, just standing next to their table will elicit some questions from them. If a table looks stuck, I'll ask them to share where they're at so far, which I can typically spin into a handful of advancing questions for them. Some tables are engrossed in their work, either independently or collaboratively, and I'd rather not interrupt their flow, so I just walk by slowly with a questioning thumbs up, which they return, signaling that they're set and I can keep moving.
• There's a rare lull, so I take the opportunity to update my circulation notes tracker. I haven't even finished when more questions have popped up, so I restart my rounds. Alex's table has a question about how to proceed with question #3, because it has some unfamiliar language. I clarify, and then they follow up with another, bigger question. As I listen to one of them explain their question and progress so far, I scan the other notebooks at the table. From what he's written in his notebook, it's clear to me that Sam already has enough of an answer to their question, though either hasn't realized it, or hasn't spoken up yet. I redirect the table's questions to Sam. He was a little unsure how to respond, so I encourage him to start by sharing with the table what he did for #3. Once he starts talking, I exit the table without saying anything, confident that they'll have at least enough to keep going until I get back to them on the next round.
• The next day, I project one of the more interesting *Probably* Not Related problems on the whiteboard. This will be enough to give some direction to some stalled-out students. Some students are still plugging away at the Big Problem and Obviously Related problems. Some have gotten to an answer for the Big Problem, and aren't sure where to go from there. Other are looking for inspiration in new problems. I've got a kid who's particularly advanced, so I recommend one of the more sophisticated Going Deeper problems.
• I've got another kid who's having a real tough time with some LCM and GCD problems. After asking some assessing questions it's clear that they could use some remediation around factoring numbers. So I show them what a factor tree is, and tell them to make a bunch of factor trees for their ten favorite numbers. We both feel good about that, and I make a mental note to add some more problems to bolster factoring skills in the next PSet.

Last Day (Usually Friday)

• It's the morning of the last day of the PSet. The morning before classes start (or sometimes the afternoon of the day before), I review my circulation notes from the week, and circle the names of two or three students who I'd like to share a piece of the work they did this week. One of the students I picked had a pretty insightful way of organizing their exploration of the Big Problem, and I want everyone to have a chance to see it. The other two students also had interesting work that'd be useful to share with the class. I also try to keep in mind who tends to speak up in class, and who doesn't, because this is a chance to disrupt that a little, in favor of equity.
• I take about twenty minutes and draft a practice copy of the notes I'm going to give in class today. I already had a pretty good idea of what they were going to be before I even gave this PSet, but I wanted to see what the class did during the week before finalizing anything. I make a note to myself when I should loop in a specific student and their work.

• This class starts like a normal day, but students know that on the last day of a PSet they'll have the first half of class to do work on their own, to try and finish their PSet. Then the second half we have a discussion and take notes. So they usually get to work pretty quick. My circulation is a little more targeted today, and I make sure to check up on kids who I know need just a little extra nudge to get to some kind of satisfying conclusion. Up on the projector I have a note that says what time we're officially going to start notes.
• As early as possible, I check in with the students who I want to share their work. I tell them I'd like them to share a specific part of their work, and I point it out. Sometimes they need a little bit of encouragement, sometimes they need a little bit of coaching, and sometimes they need time to prep. And other times they're ready to go, no questions asked.
• As discussion time approaches, I head to the front of the room for the first time all week. I get their attention, though many of them know what's coming and have already begun to switch gears. They title the next blank page of their notebooks, and turn their chairs if they need to so they can see the board. I start the notes, we intersperse it with as much discussion as necessary/possible. I take the notes in my own class notebook, which I keep for myself. I wrap up the notes with about 5 minutes left, and give students a few minutes to take care of their stuff before class ends.
• A few students come up and take a picture of the notes with their phones, a practice I encourage. I also announce that I'll post a picture of the notes on the Google Classroom, as per usual. A few students let me know that they're taking their notebooks home for the weekend to finish up some things. The bell rings, and that's a wrap for the week. During my off block later today, I'll take some more detailed notes on my circulation tracker. I won't grade the notebooks till Monday, so I want to record any important data while the week is still fresh in my mind.

My Discovery-Based Math Elective: Circulation Notes Tracker (Part 6)

This is part 6, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe how I used observation notes, taken during circulation, in order to provide further assessment data for grading and feedback.

• Assessing students is always a complicated business. It is important that students have multiple options for expression and communication. The primary method of assessment here is the work that's written in the notebook. But we can enrich our assessment of student understanding by taking notes on the things they do and say that don't make it into the notebook. This was especially important considering that there aren't any other different forms of assessment in the class, outside of the PSets and Exhibitions. By employing multiple methods for tracking/assessing students, we can layer the different data sets, and hopefully converge on a pretty clear assessment of the student's work and understanding.
• This additional layer of daily assessment was a done by "taking notes" on students, based on my observations of them, and conversations with them during my circulation. I say "taking notes" with quotation marks, because it wasn't nearly as thorough or qualitative as the term suggests. I basically had a tracker that listed every kid, and had one space, per kid, per "category," per day of the week. The categories were the four I identified in the PSet rubric: the intersections of Numericals vs. Logic & Reasoning, and Exploration vs. Depth of Understanding. You can get a full blank copy of the template here.

• As I would circulate, I would try to put a check mark when a student did a super solid job of that category that day. Then, when I sat down to grade PSets, even if I didn't have notes on exactly what a kid did that was evidence of that particular set of standards, I had some idea of how they did the previous week.
• I recognize that taking circulation notes is one of those super classic best practices that teachers are always recommended to do. And I think most teachers would agree that it's pretty unambiguously a good supplemental source of assessment data. It's also difficult to do. Class happens so fast, and there's so much to do, that it's difficult to find time to pause between students and take notes on their understanding. And it's even harder to make sure that the way you do it is systematic, equitable, and consistent, ideals we must constantly work towards in our grading and assessment of students.
• I had this elective first block of the day, and most days I wouldn't actually fill out the tracker until my prep period, after I had already taught a completely different class, which was not ideal, but mostly adequate. But I never tried to fill out the tracker for previous days--I knew my memory wasn't reliable enough to remember things that students said a day earlier. I'd rather have a blank day for everyone.
• How you make this system work for you depends on your individual practice and routine as a teacher. I know some teachers who use circulations trackers like this every day, and it's a system they use reliably and well. But in a class where there are so few different kinds of work being formally assessed for grades, and where student performance is so incredibly dependent upon the individual student, there HAS to be some system besides just assessing what is written in students' notebooks. And a circulation tracker does a decent job of that.

My Discovery-Based Math Elective: Compiling PSets (Part 7)

This is part 7, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe how students assembled and submitted the classwork in this class.

• As a math student in college, it was typical for us to do all our work on blank sheets of loose paper--personally, I prefer blank white paper, using grid paper as needed. Then at the end of the PSet, we would staple all the papers together in the proper order, and turn them in as a packet. We would then get our PSets back at a later date, graded.
• For this course, instead I opted to have each kid use a dedicated quad-ruled composition notebook. All their work for the class goes in the notebook. Only their work for the class goes in the notebook. Students could use as many pages as they needed. I just asked that they label the problem and PSet number by their work.
• I chose notebooks over loose-leaf stapled packets because the executive functioning demand is much lower. I don't want students to have to keep track of tons of loose-leaf paper. And I definitely don't want a kid to lose any of their work. I even had a dedicated space in the classroom where students could keep their notebooks easily, so they didn't have to take it out of the room if they didn't want to. I did have a student who had a particularly robust binder organization system, who convinced me to let him complete his PSets on lined paper in his notebook, and then he'd turn in the PSet as a packet, college style. He even showed me the section where he would keep his graded PSets, all organized. I didn't regret my decision, and his system worked well.
• I opted for quad-rule in the notebooks, because it can do basically everything that blank or lined paper can. And there are things I need quad-rule for, and I don't want students to waste time/energy making grids on their blank/lined paper. So if I'm going to ask them all to do their math in the notebook, they should have access to quad-rule paper if possible.
• Other things I tried, and learned from
• Submitting PSets on Google Slides
• Originally, I asked students to take pictures of their work, and then upload it into a Google Slides template. The template had a title page and a bunch of blank slides in the middle for them to fill. After the blank slides, there were a couple generic questions to summarize their learning on the PSet. Then the last slide was a blank version of the rubric. Here's an example I made using some math work I had on hand from a different class.
• I liked this system for a few reasons:
• Giving feedback on the computer allows me to use copy/paste, and the canned comments feature of Google Classroom. I can also type faster than I can write. Which improved efficiency while still offering detailed feedback and comments.
• It was a system that organically archived everyone's work, for the entire class.
• Students could upload more slides and pictures, then resubmit. Through the resubmission system on Google Classroom, and the "version history" feature, it was easy to see what had been revised.
• Students could do work with manipulatives, on the computer, or on the whiteboards, which they could upload pics or screenshots of.
• Occasionally students would use text boxes, and other slide tools to help explain what's going on in the PSet.
• I did NOT like this system for so many more reasons:
• My school kicked all the students off the WiFi, so they couldn't upload pics easily. Which was especially brutal, because my classroom was in the basement, so they couldn't even get data reception. Hassle.
• Some students would do a ton of great work during the week, but never got around to transferring it to the slides, so I didn't have much evidence to warrant as much credit as the deserved. Waste of effort.
• It's not always easy to read pictures of student work. Hassle.
• I had to have laptop access every Friday. Waste of resources.
• I had to dedicate half of Friday to students assembling their PSets, instead of doing more math. And the amount of time students needed to do this was so wildly variant, that it was hard to give them less time to do it, in order to squeeze in more math time. Waste of time.
• The generic questions at the end of the PSet rarely resulted in meaningful student responses, and often when undone. Waste of time.
• Submitting PSets on Google Classroom
• I did this during our distance learning time. I would make an assignment on Google Classroom, and I asked students to upload pictures of their work, links to any online stuff they did, or screenshots of whatever work they did on the computer. A few even uploaded videos of them explaining or demonstrating some of their work. Some made and uploaded slides like we had done earlier in the year. All good stuff. I would say that this is an improvement on the whole Google Slides system, though still far inferior to the notebook PSet system.

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

This is part 8, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe how I planned for weekly end-of-PSet discussions, and how they supported the pedagogical and curricular philosophy of the course (outlined here).

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

Link to full-sized image.

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

Link to full-size image, made on Desmos

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

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

Link to the PSet Rubric Shown

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

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

Link to full-sized image.

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

My Discovery-Based Math Elective: Exhibitions (Part 9)

This is part 9, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe how I held quarterly presentation-based projects.

• My school had a strong positive tradition around authentic alternative assessment. In particular, we had a culture of "Exhibitions." These were kind of like big projects in some classes, where there was some kind of performance or presentation component. The 9th grade ELA department did poetry readings. 10th grade history did a formal debate. 12th grade science did presentations on climate change.
• My own experience is that math classes often leave themselves out of these presentations. I think this is because live formal presentation seems like such a small part of what many math teachers believe to be authentic mathematical work. An exception here is much of statistics, where there are many rigorous applications in social science. Similarly, there are other fields of applied math that have some authentic presentation contexts. One year, the Financial Literacy teacher had students pretend to be financial advisors, and make a "portfolio" of services, like tax filing, budgeting, etc. They would then present to community members and students, to try and convince us we should hire them. Lots of fun.
• But I found examples of really great "pure math" presentations to be rarer. I experienced a decent example with the 2nd year experience of PROMYS for Teachers, which helped. The example that helped me to understand best what mathematical presentation could be, are the Numberphile videos. They're a really great example of taking classic topics and problems in (mostly) pure math, and making them accessible, while surfacing deep mathematical richness.
• All that experience in mind, in the context of this course, I have two types of exhibitions that I have asked students to do, both of which went well enough. I tried to sandwich these between units.
• Here are the slides I used when talking about exhibitions with students.
• Exhibition 1: Presenting for Depth
• Objectives
• Help students learn how to draw a Big Idea out of a Big Problem and its PSet
• Highlight the purpose of the *Probably* Not Related problems.
• During this kind of exhibition, students are asked to go back to one of the PSets they've already done, which they will return to. The first time around, students tend to focus more on the Big Problem. This time, students are expected to do a few things:
• make sure they totally understand a solution to the Big Problem
• do most/all of the *Probably* Not Related problems
• articulate the Big Idea and any relevant content threads
• show how the Big Idea is realized throughout all of the problems in the PSet, showing that they actually *Are* Related
• Logistics
• Groups of 2-3. I tried to go one-to-one, matching groups to PSets, without duplicating any PSets, because it'd be boring to see the same presentation twice in a row.
• Product: group presentation, with slides
• ~2.5 weeks total: 1 week doing the chosen PSet + 1 week making the presentation + 0.5 weeks for presentations.
• I would do this around the end of the 1st and 3rd quarter, and let them pick from any of the PSets we've done since the last exhibition.
• Exhibition 2: Presenting for Access
• Objectives: 
• Provide an opportunity for students transfer their more developed mathematical content and practices to a novel problem
• Build student capacity to share interesting math problems with others
• During this kind of exhibition, I have all the students sample a bunch of new Big Problems, for which I have made individual one-off PSets. They then pick one, get the PSet, and try to solve it. Here, they are more focused on the Big Problem, using the other problems to support/enrich their investigation. In general, their presentation will focus on a couple things:
• talk about their discovery process, what they tried, what worked and didn't work
• make sure they totally understand at least one solution to the Big Problem
• The PSets I wrote for these exhibition problems were just like the regular PSets I wrote, only they didn't have any explicit threads that connected to the other PSets. But occasionally, if beneficial, they would reference material from earlier in the class, (re)building prior knowledge as helpful.
• How to have students sample the problems, before choosing:
• I would run Stations, and at each station was a different Big Problem. The design principles were the same as any other Big Problem, which I've written about here [insert link]. Students would spend 10 minutes at each station, before rotating. We usually needed two days to get through all the stations.
• Some Big Problems are better experienced with a demonstration. For example, last year Nim was an option, so I played Nim a few times against students, had them play with each other at the front board, and then again with each other. All took about 10 minutes.
• The second time I did this, I had to be absent from school the next day. So instead I made one big packet of the Big Problems available for exhibition, and gave them two days to try them all out. This was much easier, though resulted in less real exposure to all the problems.
• This exhibition is also more generally focused on inviting the audience into the problem. Before diving into the process and solution, students are asked to have a kind of Do Now or something, where they are posing the problem (or a related one), to the audience. The idea here is that unlike the other kind of exhibition, nobody else has really spent any time doing their problem. It's also non-trivial to try to repackage a problem for maximum accessibility, and teaches you about how to consider your audience when planning a presentation.
• Logistics:
• Groups of 2-3. I tried to go one-to-one, matching groups to PSets, without duplicating any PSets, because it'd be boring to see the same presentation twice in a row.
• Product: group presentation, with slides, interactive component
• 3+ weeks total: 0.5 weeks for sampling + 1-2 weeks doing the chosen PSet + 1 week making the presentation + 0.5 weeks for presentations.
• I would do this around the end of the 2nd and 4th quarter. Since my school broke this course into two semesters, these were realized as end-of-semester projects.
• Assessing Exhibitions
• I would assess students using two rubrics. I would use the same rubric for math practices that I used for regular PSets. I would also use a second "Communicating Clearly" rubric (second page) that was pretty common throughout my school. Both rubrics would reflect both the work they presented, and the work they did in the weeks leading up to the presentation. I would keep a circulation tracker for the duration of the exhibition, to support in the final rubric completion.
• Regular PSets constituted 75% of a student's grade, so exhibitions were the other 25%. I made the first exhibition of the semester (Exhibition 1: Presenting for Depth) worth 10%, and the second one (Exhibition 2: Presenting for Access) worth 15%. It made sense to make the second exhibition a little heavier, because not only did it take longer, but it was experienced as a "final project" for each semester.
• Even before I was a math teacher, I coached high schoolers in presentation making and presentation skills. Maybe it's just because I'm teaching out of my content when I do it, but I always found it refreshing to coach students on the non-mathematical parts of presentation. (I am very passionate about title case and the Oxford Comma.) Here is a handout I gave students to help them think about making an effective presentation.
• The process of preparing a presentation is a huge skill in and of itself, and it always took longer than I originally expected. This is definitely because I've made so many presentations that I underestimate the layers and layers of necessary skills and demand. But I was also pretty clear within myself that, when possible, it's usually a good call to give students extra time when making a presentation. Presentation is a very high-leverage, high-transfer skill, and if I can support it in my math classes, I should.

My Discovery-Based Math Elective: Modifying for Distance Learning (Part 10)

This is part 10, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe how I modified the class once we had transitioned to distance learning as a result of the COVID-19 pandemic.

I can't say the "distance learning" version of this class did everything I wanted it to do. But this class made the transition to distance learning better than my Math 1 class. So after trying a few things, here's what I settled on.

• Platforms I Used
• Google Classroom
• Google Voice: for weekly phone calls/texts with students, checking in with them, and answering quick questions
• Google Hangouts, Zoom: for conferencing/tutoring
• Awwapp.com: an online collaborative whiteboard during conferencing/tutoring
• Desmos.com/geometry: a canvas upon which to record videos
• Typical "Week"
• ~Sunday Night: I posted PSets at the beginning of each week.
• In addition to posting the PSet, I found it helped to post a very quick (~2 min) video explaining the Big Problem, and doing a couple quick examples to get them started. A little bit of extra support, early on, can do a lot for facilitating student entry into a task. This was crucial, because students were doing PSets mostly on their own.
• Thursday/Friday: text/call each kid to check in about the work, school, life, etc.
• ~Sunday Morning: Post the end-of-PSet notes video
• Compiling PSets
• I kept this super wide open. Google Classroom is pretty effective, in that it allows students to basically upload anything into an assignment. Most kids uploaded the pictures directly. Some made a Google Doc and uploaded pictures in there, along with some descriptions typed in the document. Others made Google Slides, doing a similar thing. I even had one student submit a video demonstrating their solution to a problem. Another student opted to use the Awwapp online whiteboard, and they just uploaded the link to their work.
• With Google Classroom keeping everything together, it really didn't matter what the format of their submissions were. As long as I am able to make comments on the product somehow, anything goes. This is a place where I can remain flexible, hoping to make distance learning more accessible for students.
• The Architecture of a PSet
• Students were going to be completing these PSets mostly independently. During a regular PSet, the general philosophy is to provide scaffolds "just-in-time," so that students only get the minimum degree and kind of scaffold that they need. However, this is basically impossible to do with distance learning. So I erred on the side of including a bunch more scaffolding questions.
• These more developed scaffolding questions looked like smaller questions, more "easy" questions, and some questions that provide more explicit directions with how to do some of the work. They also included more examples and diagrams.
• I also included more "self-checking" questions. For example, if I wanted kids to convert numbers to binary, I could add a small question before that: "Show that 12 in binary is 1100." Thusly, the question provides an answer key with which they can double-check their process.
• With all the extra scaffolding questions and additional pictures, the PSets got longer. Back in the Normal Times, I would restrict each PSet to a single two-sided page (albeit with very narrow margins). Once we got online, with these added design principles, the average PSet was closer to four pages. I further justified this by including more, different questions, in an effort to really amp up the degree of choice and flexibility students had.
• There were some things that were actually better about the PSets, because students were accessing the PSets digitally. I was able to include more links to online math tools and videos. I was also able to include little animations directly in the PSet, by copying in GIFs. I also hoped that more students would comment on the PSets, asking clarifying questions, because I think that would have been helpful for other students as well. Next time I might advocate for this more directly?
• Grading, Standards, and Assessment
• Each PSet is graded 0, 1, or 2 points.
• 0 = no work submitted
• 1 = some work submitted, PSet off to a good start, but not "done"
• Most first submissions of PSets got a 1, so it was more like I was recognizing that the student had submitted a 1st draft of the PSet.
• I would then return the PSet to them, with comments on where they could go from there, feedback on the work they'd done, or just recommendations for other problems that build well on the work they'd done.
• I will say, this mechanism was the closest I ever got to a system for meaningful revision, in any version of this class.
• 2 = solid PSet, lots of problems done well enough, or a couple problems done really well
• In order to pass the class, students needed a certain number of points. In an effort to minimize the privilege filter that is student performance during a global pandemic and national crisis, I ran the class Pass/Incomplete.
• This is about as far from standards-based as grading gets, which is not great. But at a time when students need maximum flexibility in their work and assessment, this felt respectful. It is pretty subjective, and really requires the teacher to understand the students, both as a mathematician and person in the middle of a pandemic.
• To balance the subjectivity, it helped that I had most of the students for the time before distance learning, and I had everyone for in-person school for at least a month. So I had get an understanding of where they were going into this. I also had weekly individual text/call conversations with each kid, just so I could keep up with how they're doing. I would then take all that into account when reading PSets.
• No late penalty for PSets. I had a "due date" so that they'd get regular reminders, but I would grade a student's PSet whenever they turned it in. All-in on asynchronous learning.
• End-of-PSet Discussion Notes
• The end-of-PSet notes were much more "level to the objective." Students did different PSets at different times, so it was not possible to get a feel for which ceilings which students hit, and level to them. So I basically just leveled to the objective, and erred on the side of spoiling the Big Problem as little as possible.
• I tried to keep the videos to 10 minutes, but they often were closer to 15 minutes. 
• I originally recorded videos using my phone like a document camera. But uploading videos was a pain. So instead I made the videos on my computer, typically using desmos.com/geometry. I would record just a section of my computer screen, and then have all the objects/labels I needed just off the edge of what was being recorded, so I could bring them in quickly. The Desmos Geometry app is definitely rudimentary, and if I were to do it again, I might consider GeoGebra instead. But I didn't need that much from it, so Desmos Geometry did the trick.
• For Graph Theory: PSet 23: Bridges, the Big Problem was the classic Seven Bridges of Konigsberg problem. Instead of recording my own video, I posted Numberphile's video on the problem, which did a better job than I ever could have. I would do this as often as I could, though rarely could I find a video that did what I needed it to do.
• Exhibitions (?)
• I technically offered an option for an end-of-semester exhibition. You can see the memo I gave them, detailing all the options, here. It was worth up to 6 points (3x as much as a PSet). All but one student said they'd rather just do more PSets, and revise more PSets. Which was totally okay with me. I can't totally blame them--exhibitions are a lot of work! If I were to do this again, I would need to reconsider the weight, or potentially make it mandatory if it was that important to me.
• The biggest issue with this class, I felt, was that it didn't support students in collaboration. It was certainly possible for two students to call each other up, pull up the PSet, and do the work together. But it's pretty difficult. I need to do a lot more learning about what remote math collaboration can look like.

My Discovery-Based Math Elective: Different Things Become Difficult (Part 11)

This is part 11, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe a couple of the biggest ways teaching this class felt different from a traditional class, primarily from a planning/prep perspective.

Teaching this class was a very different experience. Some things were much easier. Some things were much more difficult. Here are a few of the biggest differences, in terms of what it was like teaching the class.

• Why does it feel like I'm doing less prep?
• I'm the kind of teacher who often feels the need to make my materials at most a day or two in advance. The making of the materials is a part of my planning process, and so in making whatever materials I need, I am able to prep myself to teach the class the next day. Sometimes I'm able to go an afternoon without having to prep anything for the next day, but not often.
• As a 3rd year teacher, I have enough background with my content, students, and routines that I can whip up a good enough lesson in not that much time. I can always sink more time to make lessons better, especially if I'm experimenting with something. But I don't always have to. I don't even always want to--I'm not trying to burn out any time soon.
• The prep load for this class was very different. This year, the first one teaching this course in its entirety, I made the first half of the PSets for this course during the summer. Since the only materials I really needed for the first four days of the week was the PSet itself, that meant that during the first few months, I did basically no material creation. It honestly felt a little weird being able to go multiple days not really having to prep anything. In fact, the time spent outside of class was so much freer than any other class I ever taught:
• Sunday: 30 minutes reviewing the PSet, tweaking and polishing it
• Monday: One hour, after school, grading notebooks (I taught one section of 28).
• Friday: 30 minutes before school prepping for the discussion.
• And that was it. Sure, around exhibitions there was much higher time demand, but that was only for a couple weeks, once per quarter. I definitely could have improved the class by reviewing notebooks in the middle of the week, providing written feedback. But overall, the class went pretty alright. It felt like I was getting away with something. Until the first Sunday morning when I realized I hadn't already made the PSet for the next week.
• Let me tell you, one whole Sunday has never felt so short. Trying to write a whole PSet, with all the layers that I want it to have, that does the job and doesn't suck, is super complicated. And some parts of it I couldn't just research and come up with an answer to. So much of the math in these PSets is there because at some point in the last five or six years I stumbled across it on Twitter, PlayWithYourMath.com, or Numberphile or something. And I couldn't really force that to happen. Needless to say, that PSet was less good than the others.
• This is all to say that this class doesn't necessarily have less prep than other classes I've taught. It just pushes the material prep work to the summer and breaks, when I actually have the time to sit down and think through a whole PSet, peruse all my different archives of interesting problems, and play around with things. In particular, trying to create content threads that wander and braid meaningfully across multiple PSets requires a huge amount of focus and time to do with any kind of intentionality. I actually have to build the PSets in one big document, because there is (ideally) a bunch of interplay between PSets.
• The heightened demand on content knowledge
• Similarly, this course, as designed, puts so much of the scaffolding into the hands of the teacher, who has to differentiate, scaffold, prompt, and redirect in real-time. Sure, we have to do that in all our classes already. But with so many options, legitimately telling students to feel free to pick whatever questions and paths that they want means that the teacher could have literally thirty students doing as many different problems, needing as many different scaffolds.
• This means that the teacher has to have some really strong content knowledge. And while I wouldn't ever recommend it in any classroom anyways, with this course it is basically impossible for the teacher to just be "one week ahead" of the students. This is especially important, since so much of the content isn't the kind of math that many of us learned in high school.
• The teacher of these PSets needs the perspective on the whole landscape of problems, their full depth and connections. I wrote these PSets, and I wouldn't even say I have 100% understanding, but every shortcoming in my content knowledge is a point where I am needlessly limiting students. This challenges was made easier for me because I was the one writing the PSets. So I was able to build PSets that reflected my own experience as a math major, #MTBoS teacher, PROMYS for Teachers alum, recreational math hobbyist, Numberphile fan, and colleague of some math teachers who shared some truly awesome backgrounds.
• My goal is that it is useful to have already packaged this course as series of pretty complete PSets. Teachers to whom this content is unfamiliar can experience this as an authentic and fun opportunity to do some interesting and developmental math. Hopefully, by presenting all of these written PSets, in one big package, that frees up teachers to take some time and be students again--do the PSets. I wouldn't ever tell a kid they needed to do all the problems, but the teacher? Go for it. Play around. Map the mathematical landscape. Try to find the narrative threads throughout the PSets. If a kid asks for help, we have to have a solid vision at the ready, because this course places a lot demand on efficiently conferencing and circulating.

My Discovery-Based Math Elective: Pedagogy of a PSet (Part 12)

This is part 12, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe the general pedagogical approach to using Problems Sets (PSets) and the primary form of classwork, and how it connects to the underlying curricular and pedagogical philosophies of the course (described here).

• The "pedagogy of the PSet," to me has felt like a reallocation of resources away from pedagogy and towards content. So many of the mechanisms and structures of the traditional math class don't have a place in this course. And I think the clearest way to describe them is by sharing the three PSet "rules" that are posted at the top of each PSet, which are borrowed almost directly from the PSets at PCMI:
• Don’t worry about answering all the questions. Don’t worry about getting to a certain problem number. Some students have been known to spend the entire class working on one problem! That’s okay!
• Many of the structures and routines I use in my traditional math classes I teach are based on principles of necessary and sufficient productivity. But "productivity" is barely necessary, and definitely not sufficient alone for great math learning to happen. I also think that if we want to build up students' attention spans, and general ability to persevere through problems that take a long time, and have many layers of difficulty, we have to give them problems that take a long time, and have many layers of difficulty...and then give them the time to do so.
• Have fun! Make sure you’re spending time working on problems that interest you. Feel free to skip problems that you’re already sure about. Relax and enjoy!
• Choice is an enfranchising experience. Every time a growing person has to make their own decision about what they do and don't want to do, they define themselves a little bit more, as mathematicians, students, and people. Moreover, they assume real control over their education. Here, I am trying to expand my students real choices beyond "do the work or don't."
• Whatever you do, do well. Flying through the problem set helps no one, especially yourself--you’ll miss the big ideas that others are grabbing onto. There is more to be found in the problems than their answers.
• This is pretty self-explanatory. And also generally good advice for life, I think.
• There is one more that I use when designing PSets, that's more for me than for the students.
• The PSet should lead to the math, not require it.
• Here's a useful example of this in action. In the Combinatorics Unit, I wanted students to discover that the coefficients of the expansion of (x+1)^n are the entries of the nth row of Pascal. But for a lot of my kiddos, I couldn't just tell them to take the powers of (x+1). So in the weeks leading up to that, I included some chill polynomial multiplication problems, using the area model. It wasn't an in depth analysis of the area model--just enough to do the job. Then over the next few PSets, the problems scaffolded them up to the powers of (x+1)^n. And a few students followed that thread across the PSets, and it was cool. But, importantly, even though they weren't engaging quite with the goal objective (coefficients of the powers of x+1), they were still doing valid, worthwhile math all the way up to it.
• This idea also helps to provide a schema for designing for re-entry. Some students will miss a PSet or two, either because of attendance, life, or just really not digging the Big Problem for some reason. We don't want that gap in engagement to perpetuate itself, by limiting future engagement.
• A goal is that students should be able to do the PSet if they randomly dropped into the class that week. This may not be 100% possible all the time. I mean, in the last paragraph I literally just described a system of scaffolds where students may have needed the work of previous PSets to build up to the more sophisticated problems. I try to mitigate this by allowing the more important questions to resurface in multiple PSets. I also give students the opportunity to go back to problems from older PSets if they want to, or if it's helpful.
• I don't want to say that these PSets are the best way to teach every high school math class--certainly not. I mean, I was still primarily a traditional Math 1 teacher this year, and there were a bunch of things I would do in this elective, that I'm not yet ready or willing to do in my Math 1 class. And that's partly because the purposes and contexts for the two courses are different. But it's also because the stakes are a little bit lower in a math elective than in a students' primary 9th grade math class. But that's not to say that I didn't learn a bunch from it. I have tried to ship over bits and pieces, when and where I feel ready.
• I also want to take the time to point out that this is only one way of doing PSets. Back when I was first trying to make my own PSets, I was doing so as part of a collaboration with Joey Kelly. It was a kind of natural application of the experience that he, Dan Henderson, and I had last year with MIST (a year-long PCMI spinoff pilot). We had similar goals, but he ended up breaking his PSets up into chunks, and rolling them out over the course of a few weeks, more as a series of mini-PSets. That definitely gave the groups working on the PSets a little more day-to-day direction, and avoided the issue of later questions spoiling earlier ones. I'd love to some day ask him more about it. Or you can ask him on Twitter yourself @joeykelly89.

My Discovery-Based Math Elective: Architecture of a PSet (Part 13)

This is part 13, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe the overall format of a Problem Set (PSet), and the different design features of the different parts. This post in particular identifies a handful of other educators who have been especially influential in their work.

• The architecture of a PSet is pretty straightforward, and is largely inspired by the PCMI PSets. Each PSet is broken into four parts, each with a different set of design principles. When we were in-person, I committed to keeping the PSets to a single two-sided sheet of paper (albeit with very narrow margins).
• Big Problem
• The Big Problem is the core of the PSet. It's the first problem students do, and the only one that everyone has to spend at least some amount of time on.
• The most comprehensive model I have of an effective Big Problem is the work done by Xi Yu and Joey Kelly with their PlayWithYourMath.com project. Joey has documented a lot of really useful reflections around the development of some of their problems. I'll summarize some of the big ideas of the work they've done, and link to where you can read more about it at Joey's blog MisterIsThisRight.com. In particular, here are all the posts where they reflect on the design of some of the PWYM problems, which has significantly informed my own understanding.
• Characteristics of effective Big Problems, from the reflection on problem 13. Thirteens
• involve some sort of “play” before choosing a specific strategy.
• have a low floor (accessibility and entry point)
• have a high ceiling (need for more complex mathematics)
• have a succinct, accessible, intuitive wording and visualization
• What is meant by "ceilings", from the reflection on problems 18 and 19
• To make success attainable. In addition to a high ceiling, I also want a low ceiling, where students can feel a sense of accomplishment. There should be lots of ceilings. 
• To make space for curiosity. Just because there is a ceiling, doesn’t mean I have to show it to them. 
• To shelter from inaccessible questions. Some ceilings are just too high for some people, and that is fine. The high ceiling is not meant to intimidate.

Link to full-size image, made on Desmos

• Here is a diagram that will come up a few times in this blog, when talking about ceilings and differentiation. The idea is that students enter the problem at whatever level they are initially ready for, and can exit the problem having made as much progress as possible. And wherever those points are, there is some satisfying and meaningful work to be done.
• I also had some interest in exposing students to some of the famous problems in mathematics. These are classics like The Bridges of Konigsberg, the Utilities Problem, Counting Trains, Fibonacci numbers, etc. Usually they're famous for good reason, and their inherent richness is usually worthwhile. Furthermore, these problems are cultural cornerstones of the mathematician community. So if our goal is to induct students into the community of mathematicians, we would do well to give them access to some of that culture.
• Obviously Related Problems
• The Obviously Related Problems are just that--obviously related to the Big Problem. They come in two categories: Scaffolding Questions and Advancing Questions
• Scaffolding Questions: Scaffolding questions are questions that are designed to break down the Big Problem, and help students work through it. Imagine yourself walking around the room, circulating and conferencing with students, helping them initially make sense of the problem.
• If you know that all of your students are going to ask some of the same introductory questions, and you're going to keep having to have the same introductory, orienting questions and recommendations, you might as well put those questions in the PSet--it's more efficient. Especially once students get the hang of the PSets, they'll start to look to this section when they get stuck, and that's something they can do independently. It's also a much quicker conversation, once you recognize where a kid is stuck, if you are able to quickly say, "Check out #3," and then walk away.
• As with any scaffolding, it is dangerously easy to add too many scaffolding questions, and in doing so, rob students of the opportunity to develop their own insight and independence, if not spoil the problem entirely. This is especially dangerous in the context of a mixed-levels class. If I was ever worried that a question was going to offer too much scaffolding, I would err on the side of not including it. Dan Meyer had a good blog post on this: You can always add. You can't subtract.
• Another helpful idea for thinking about how I managed scaffolds in this class is the different between Just-in-Time vs. Just-in-Case Scaffolding, which was introduced to me in a post by Dr. Juli K. Dixon. Highly recommend the read. In this case, the Just-in-Time scaffolds are the ones you provide through conferencing. The questions in the PSet can become Just-in-Case scaffolds if the kids don't skip them, and Just-in-Time if you are the one recommending they try them.
• Advancing Questions: These questions are less about students working through the Big Problem, and more about helping students start to find the Big Idea. They push the kid to try to refocus on the big picture, and think about the broader mathematical structure. These usually come up more often in the *Probably* Not Related Problems.
• *Probably* Not Related Problems
• The name of this section describes pretty clearly, if ironically, the purpose of these problems. They don't look overtly related to the Big Problem or the Big Idea. But with a deep enough exploration, there is a connection. The name of the section poses the challenge of trying to figure out exactly how they are related to the Big Problem.
• This section is founded on the idea that you can have a bunch of different problems that all have a common underlying structure to them. I think the best example I have of this, unsurprisingly, is in Sequences: PSet 7: Bees, which is a PSet exploring Fibonacci numbers. We can put a ton of interesting different looking problems in there. And then miraculously the Fibonacci numbers will come up again and again (as they so often do). Sometimes it'll be obvious the way they are connected, and others it will be very difficult to find a way to connect things.
• We can think of the Big Idea at the core of the PSet like a statue, set up inside of a dimly lit house. Looking through a single window at the statue, we only have a single incomplete perspective of the Big Idea. Even if it's a Big Window (i.e., Big Problem), our perspective of the statue is limited. But by looking in from other windows, we get glimpses of our Big Idea at multiple different angles, each incomplete on its own, but collectively greater than any individually. The different perspectives allow us to kind of "triangulate" the Big Idea, and understand it's construction more deeply, independent of the context of a given problem.
• We can describe these problems as "Different Surface--Same Depth." This is not to be confused with @mrbartonmath's (blog) "Same Surface--Different Depth" problems, it's actually kind of the opposite. But I believe it was the inspiration for this reconceptualization, which I first heard coined by Joey Kelly.
• Going Deeper
• Basically any kid should be able to do basically any problem in the PSet. In this section, however, I'm willing to push the boundaries a little bit, in terms of how high I'm willing to let the entry-point creep. I also tend to walk back many of the scaffolding questions I usually include in a PSet, instead usually opting to pose the problem in the simplest, "purest" form possible.
• I do this for two reasons. First, at this point students have usually built up more background knowledge by doing the earlier problems in the PSet. Moreover, the kinds of students who would jump ahead to the Going Deeper problems, usually do so as a signal of readiness. This doesn't necessarily mean that any of these questions are inherently harder or better. It just means there's less scaffolding.
• Often, these questions are also extensions of the other problems in the PSet, that are maybe a bit of a tangent away from the Big Idea, but are too rich to not include. For example, in Bases: PSet 9: Grapes, we were looking at a James Tanton classic, Grape Codes, as an precursor to diving into Exploding Dots. All I really wanted to get across was that grape codes were ways of rewriting numbers as sums powers of some base number. But a really cool related problem is actually figuring out some kind of rule to determine how many grape codes there are for any number. Great problem, but a bit of a tangent, especially considering how big it really is.

My Discovery-Based Math Elective: Content Threads (Part 14)

This is part 14, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to define a major design feature of the curriculum of the course, "content threads." I discuss in depth the content threads of the Combinatorics unit, as an example.

• One of the most challenging, and worthwhile, design features of this course was my attempt to make "content threads." The way that I've come to understand it, a content thread is a major mathematical idea that keeps coming up across different problems and PSets. What that idea looks like can very wildly in scale, and you can have some ideas come up, skip some PSets, and then resurface when they become relevant again. The visual I have in my head of this is something like the Movie Narrative Charts from xkcd, that shows how the characters gather and disperse in different ways over the course of the plot.

Link to full-size image.

• I've wanted to try to make some kind of similar diagram for the content threads of this course. But I'm not totally sure where to begin, or if this is even the best representation, even though it's the one that exists nebulously in my mind. If you come up with something cool, please let me know--I'd love to see what you come up with.
• Some of the units in this course have longer and stronger content threads than others. Here are some of the bigger ones, in each of the five units:
• Modular Arithmetic
• Looking at remainders suggests a "cyclic" mathematical structure
• If we change the setup of our number system a little, a lot of things we used to take for granted are no longer true
• Special things usually happen when numbers are relatively prime
• Bases
• Our choice of base is arbitrary, based mostly on convenience, and depending on the application, different bases become convenient
• Even if we change our base, arithmetic still works in fundamentally the same way
• Combinatorics
• Counting can quickly become difficult, but if we count cleverly, we can use structure to be systematic and efficient
• Pascal's triangle has a lot of features and patterns, and we can make sense of them.
• Lots of counting problems can be conceptualized as "choosing" problems (choosing), and so we can bring Pascal and the Binomial Coefficient to bear
• Sequences
• Sequences can be defined recursively and explicitly
• Many sequences can be characterized by their differences and ratios, but some are a little more complicated
• Graph Theory
• Representing complicated problems with an abstract graph sometimes helpfully simplifies the problem (decontextualizing makes things easier)
• With a few simple geometric rules, there are some inevitable implications for how graphs work, which extend to the contexts they model (contextualizing provides insight)
• The goal of trying to develop and articulate these content threads across multiple PSets is manifold:
• The big ideas of math are much more interesting, powerful, and transferrable than a lot of specific granular content knowledge [citation needed?]. So by building a curriculum around the articulation of these threads, we are being strategic.
• We are creating multiple opportunities for students to engage in the big idea. This system allows for spaced-retrieval, which improves long-term retention.
• By working through multiple PSets and problems, all centered around the same content thread, students gain multiple perspectives on a single idea. These multiple perspectives add layers and nuance a student's understanding of the idea. Multiple perspectives develop a more sophisticated schema for understanding. This is even more important, as these big ideas of mathematics are often quite nebulous and abstract.
• It creates multiple opportunities for representation of the idea, and engagement with it. Depending on the student, different problems will be more interesting, or make more sense. By creating multiple PSets around a similar content thread, we are providing multiple entry points for students to access the big idea.
• Some of the PSets have a much stronger, more coherent set of content threads. Some threads within a unit are more effectively developed in the PSets than others. Some threads are more general, while others are pretty specific (though I did try to keep grain sizes roughly similar). Moreover, the degree to which any of these threads "land" is largely dependent upon the teacher's execution of the day-to-day, especially the end-of-PSet discussions.
• I have done the most work to develop the content threads for the unit on Combinatorics. This was the unit I put the most time, energy, and research into. It also had one of the richest mathematical structures ever at its center--Pascal's Triangle--which certainly helped. But the content threads of this unit provide a kind of best-case scenario, where each thread is introduced with a new problem, tied back into older threads, and braided with future threads. The best representation I could find of this is a table, like the one below. These content threads are a bit more granular, than the more practice-oriented threads I listed earlier, which I'll talk about more later.

Link to full-size image.

• In this table, you can see how each content thread "runs through" each of the four Big Problems. I tried to be explicit about how each problem could be recontextualized in the new content thread, adding more layers of understanding and insight. As we accumulate more content threads, we can look at each new Big Problem through the lens of all the content threads before it.
• The yellow box indicates that that Big Problem was the one that officially launched the content thread. All of these problems technically feature all the content threads. But to introduce a content thread, I tried to choose the Big Problem that "best" captured the Big Idea. That is, I tried to find the Big Problem whose context organically encoded the thread. The goal is that this problem serves as an initial schema for future applications, like a "hook" upon which future related problems can be hung.
• It's possible to sustain a content thread that does not rise to the level of one of these big ones that cut across the whole unit. For example, in this unit there are two mini-content threads that run through the PSets, hovering around the *Probably* Not Related problems. These are the problems concerning the outcomes of coin flips and the coefficients of (x+1)^n. I opted to avoid elevating these two minor threads because:
• Didn't have enough time to dedicate to 1-2 more PSets in the Combinatorics unit
• Didn't have an good enough Big Problem that I liked for either PSet
• Didn't feel that the coefficients of (x+1)^n would be sufficiently accessible to enough students, algebraically
• Didn't feel that the number of outcomes for n coin flips was rich/interesting enough yet (as far I could realize it as a PSet that was a part of this unit)
• Including or not including the a PSet dedicated to these two content threads is the kind of planning and instructional decision you can make based on what you know about your students. If I was adapting this unit for either middle school use, or for a group of less experienced math students, it would be a good opportunity to surface the coin-flipping thread. This would do a good job of explaining why each row of Pascal sums to a power of 2.
• If I was adapting this unit for post-secondary classrooms, or a group of students with broader backgrounds in high school algebra, I might surface the (x+1)^n thread. This would be good to put at the end of the unit, because it's such a great example of how to apply combinatorics and Pascal to a seemingly totally unrelated context.
• As I mentioned before, there are two sets of content threads in this unit. The more specific content threads (identified in the table), and the more general practice threads:
• Content Threads
• Strategic Counting
• Symmetry in Pascal
• Recursive definitions in Pascal
• "Choice" problems in general
• Practice Threads
• Counting can quickly become difficult, but if we count cleverly, we can use structure to be systematic and efficient
• Pascal's triangle has a lot of features and patterns, and we can make sense of them.
• Lots of counting problems can be conceptualized as "choosing" problems (choosing), and so we can bring Pascal and the Binomial Coefficient to bear
• This combinatorics unit is definitely the unit with the more articulate content threads. This made opportunities for some really awesome mathematical connections. However, it did shift the focus of the class (and end-of-PSet discussions) a little more towards content, and further from practice, which can negatively constrain differentiation and student engagement (discussion of practice vs. content standards here).
• It also required more often that I level the discussion to the objective, which if done too much, can lead to students disconnecting from the narrative (discussion of levelling the discussions here). The narrative for this unit really "accumulates," and the more I focused on trying to sustain that accumulation of content threads, the more students disconnected, because it often failed to respond directly to their experiences.
• Reflecting on this unit, it really seemed like I had lost sight of the practice-orientation that was at the foundation of the class, which was had a negative impact. A question I still have is what it means to advance both content and practice standards, at the same time, but disconnected? Also, how to we expand content standards, so that they can be better differentiated for students with a broader range of readiness levels?
• The big idea here is that we can articulate content threads that run throughout the course, especially within each unit. These threads separate during some PSets, and braid together during others. The more precise, coherent, and authentic these threads are, the easier a job we'll have of build PSets around them, and making them visible. Also, the more completely we understand these narrative threads, the better we can facilitate the course towards their effectiveness.