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.