Mathematics of Democracy: A Story of Three Weeks (part 1)

Imagine that you had three weeks with a group of ~20 students. You can do WHATEVER you want with them during those three weeks, because you have them all day long. There has to be some academic rigor to your choice, but that is the only restriction. What would you do?

This is what my school asked us to imagine a little over two years ago. We had voted (unanimously, which is unheard of) to move to a semester schedule with two “Immersive” courses, in January and June, during which students would take one class,  equivalent to a one-semester credit. We were encouraged to work together, to dream big and to think about doing things in this course that we couldn’t otherwise do during “business as usual” semester courses.

For a variety of reasons our school decided to wait for two years before implementing the change to a semester schedule (prior to this, we were on trimesters). During those two years we were first asked, in year one, to plan a “practice” immersive course and then, in year two, to write a proposal for a course we wanted to teach the following year a.k.a. the 2018-2019 school year a.k.a. THIS YEAR!

Well, friends, I knew immediately what I wanted to do: Mathematics of Democracy.

The name hadn’t yet been found, but what I wanted to do was a Voting Theory course that explored the math behind elections. A course in which students learned about civics and math. I needed a partner, someone who taught social studies, and luckily I found one in my colleague, Craig Butz.

In year one, we mapped out our course: the essential questions, the arc of the course, how the different strands of representation, voting theory and civic engagement would play off of one another. In year two, we planned in more detail, what a typical day looked like, who would teach what when, what kinds of trips we wanted to go on. We gave ourselves large To Do lists before parting ways for the summer.


And then I had a baby…

So that was cool.

Craig did whatever he did to make sure all of our field trips and guest speakers were arranged for. A parent helped a TON with this too, doing things like putting us in contact with staffs for various city and state elected officials so we could coordinate meetings and finding people who’d worked on the SF city redistricting project in various capacities to create a panel of experts for our students. She even found me a lactation room I could use on our trip to the State Capital! Truly a valuable asset to our course planning process.

Before we knew it, the first day of the course had arrived. Which was also my first day back at work following maternity leave. Teaching an all-day, three-week long course is certainly an…interesting…way to come back into full-time teaching. In some ways it was really perfect, though. Because we had the students all day, we started school later than usual and ended a smidge earlier. Because we were co-teaching, I could leave to go pump whenever I needed to.

Screenshot 2019-04-05 10.23.23

However, I’m guessing the Venn diagram of teachers who would love to add Mathematics of Democracy content to their curricula and the teachers who have tiny babies at home is fairly sparsely populated, so I think I’ll move on.

When we initially planned out our course, this is the map we made outlining the big ideas we’d bring in/out/through the course:


As we all know, nothing goes exactly to plan…

But that is the story for a second post.

#TMWYK – Overheard on MUNI Edition

This is from a while ago, but I never #pushedsend

Dad: What’s 152 divided by 8?

Girl: …

Dad: …

Girl: 19!

Dad: Great!

*pause during which I bite my tongue*

Dad: How did you get that?

Girl: I did it 20 times and then I realized it was 8 less.

Girl: But first I tried it times 10 and tried to add up.

Dad: That was kind of a slog though, wasn’t it?

Girl: Yeah–

And then I had to get off of the bus.

Week Eleven–More Important Than Math

I want to honor my commitment to posting weekly, but this week I can’t simply report on what happened in my classes. This week was too much. This week was too hard.

I live in the great state of California, in the amazing city of San Francisco. We are by no means universal in our beliefs, political or otherwise, but there is a definite liberal leaning around these parts. Many people in my life are reeling right now from the outcome of the election. I am as well.

Our students were hurting this week. They were confused, they were angry, they were disappointed, some of them were scared. We teach our students to be honest, to be kind, to conduct themselves with integrity in all that they do. They make plenty of mistakes, but they also come to expect this of themselves and of one another. They certainly expected the same from the leaders of our country.

On Wednesdays our school schedule begins with advisory. We meet in small groups that stay together across all four years a student is at our school. This year I have a new (and awesome) group of freshmen. There was a plan–an agenda–for the week, but that all went out the window Tuesday night as the election results rolled in. When I walked into my advisory room on the 9th, one of my advisees was in tears. She’d stayed up late talking and texting with her friends, one of whom is terrified that his parents will be deported.

I can’t just tell these kids that everything will be okay.

I wish that I could. I wish that I believed that this was true. But I just don’t know if it is.

Instead, I asked everyone to check in with how they were feeling. I went first. I shared how I was scared and sad and frustrated and angry and confused. My voice shook and my eyes welled up with tears, but still I spoke. I told my students that I would be looking for ways to stand up and make a difference. To do good in the world in whatever way I could.

Each student shared their thoughts, which ranged from deep sadness to outright confusion and disbelief. Afterwards, I asked everyone to go around and share something they were grateful for. Despite the strong emotions that we had shared with one another, we ended our time together with gratitude. For family and friends; for a future where this group of young people will have more of a voice; for the outcomes in this election that we do celebrate; and for chocolate chip cookies, which were very much needed at that moment in time.


The Student Life Council recently put up a blackboard in our dining hall for students and staff to write in response to a variety of prompts. Wednesday’s prompt was “What are you scared of?”

I wrote: Apathy.

Week Ten–A Day Late…


Howdy All,

Did you think I’d forgotten to post about last week? Well, sorry to burst your bubble–or rather, happy to give you the goods!

Last week was a little odd. First we had Halloween. I got observed on Monday too. That was fun–no, really; it was! I mean, it’s not every day you get video-taped wearing a chef costume with a banana sitting in the shot. 🙂 And I did great. My students are awesome. ❤

Then, on November 1, our school had a PD day (genius move, btw). Then, I took a personal day on Friday to fly to Palm Springs for the CMC-South conference. The added stress of packing for a trip (with a baby) on top of planning for a sub day (even though I was just giving a test in most classes) was crazy-ville. Also, not a whole lot of school days in there, in case you didn’t catch on.

It was Mara’s first trip to go to a conference–though it wasn’t her first conference. She attended part of the Desmos pre-conference and ShadowCon last year at NCTM. She was a bit more alert this time for her first Ignite session.

Poor child. This is her life for the next 18 years.

Okay, okay…at some point we may let her stay home. We’re not monsters.

Though according to Patrick Callahan, we are going to hell.

Week Nine–Step By Step

So, this week my students continued their work with congruent triangles. The thought-process behind this part of the unit has been intentionally moving up from physical/concrete reasoning to more and more abstract and rigorous reasoning. Going step by step along the way.

We started last week by creating and measuring physical triangles in order to establish sufficient criteria for showing two triangles to be congruent. Then, this week we used diagrams of triangles with sides and angles marked to determine whether or not the two were congruent. After we got good at that we moved on to writing proofs that two triangles were congruent. Finally, we added on to that the next day by using CPCTC to prove various things were congruent. By the time we got to CPCTC students were like:

That’s it?

And I was like:

Yep. That’s it.

And they continued to be awesome.

I don’t know what it is that I’m doing right–whether it’s my trust in students’ writing and critiquing skills, or my amazingly coherent lesson planning which allows students to build skills incrementally, or if it’s the insightful feedback I give on student work, or if I just got lucky (again)–but I feel like when I teach proofs, students do well (a feeling that some of my colleagues do not share).

I feel particularly successful this year. And I do think it has to do with the first item on my (somewhat tongue-in-cheek) list up there. The students read and commented on A LOT of proofs over the course of this unit. They saw good stuff; they saw mediocre stuff; they saw some pretty crappy stuff too. And seeing work that is effective, alongside work that isn’t, is one of the best ways that I know to become a better writer.

Turns out it works with mathematical writing too.

Who’da thunk?


Also, this week I got a great email from the mom of one of my students, letting me know that her child was really excited about what we’re doing in class and was eager to tell Mom all about it–something that this child apparently doesn’t do very often.

Talk about making my day week!

Week Eight–Trust, But Verify

Our wifi has been on the fritz for several weeks (it seems like the issue may be resolved for the time being, but it was pretty hairy for a while there) which was a pain in the you-know-what. But like all storm clouds, it had its silver lining.

From my first year at my school, seven years ago, whenever I taught triangle congruence I did it using an interactive Geometer’s Sketchpad file that had students manipulate virtual triangle configurations to try to make non-congruent triangles. This year I knew I needed a back-up plan in case the wifi was down and they couldn’t download the document onto their laptops. So it was back to good ol’ rulers and protractors!

Talking the idea over with my office-mate, I decided that I didn’t care whether the wifi was out or not, I was going to do the paper and protractor version anyways.

Kinesthetic learning! Using appropriate tools with precision! Decorating triangles!

The decorating was very important, btw.

I had read this blog post, which for some reason I didn’t clip to Evernote (why?!?), that talked about giving students the following scenario:

Let’s say I ask Suzy to create a triangle with sides measuring 3 inches, 4 inches and 5 inches for homework. Now, I know Suzy is really amazing with a ruler and scissors, so she is going to come back tomorrow with a really great triangle whose sides are 3, 4 and 5 inches long. Let’s say I also ask Stewart to do the same task. Stewart’s skills with ruler and scissors are equally amazing, so his triangle will also be great.

I want to know, will Suzy’s and Stewart’s triangles be exactly the same or could they be different?

The class discussed this, as well as the second scenario, which is the same except that now the triangle being made is 45 degrees, 45 degrees and 90 degrees. Then the teacher threw out another scenario (don’t remember what it was) and kept doing this until a student suggested “Let’s try it!” Then, out came the rulers, protractors and scissors and away they went.

I didn’t wait for students to suggest the cutting–I gave them time to discuss the first and second scenarios and then I had them construct the two triangles, decorate them and cut them out. In my first class I had them tape their triangles to pieces of paper all at the end of class (it was kind of a last-minute thought) but for the second class I had them do this for each triangle as they went along. I taped up papers labeled “1”, “2”, etc. around the room and wrote out the triangle’s criteria next to the paper on the whiteboard. This worked really well; students went up and taped their triangle to the paper as they finished and then they moved on to the next triangle. I was able to take down papers that everyone had finished and put up the new ones as we went along, which helped keep everyone moving forward without having stress to rush through because they hadtofinishinthreeminutesohmygahhhh!

There were multiple (!) exclamations of “this is fun” from (unexpected!) students as they were cutting and measuring and taping and decorating. When class was over we had multiple pages worth of triangles for many of the criteria and a full page worth of all the rest.

That afternoon/next morning I selected triangles to put on the “final sheet”–really I just took off triangles so that there was one page per set of criteria–and photocopied each page. This is where the decorations came in handy. That was my selection criteria–not accuracy, but aesthetics! [also what will show up on a photocopy best]

Day two was our day to come up with some conclusions: Was SAS enough information to determine whether two triangles would be congruent? What did SAS mean, really?

I had students Trust, But Verify one another’s work. Which means I passed out the photocopies of the triangle sheets and check to see whether each triangle on the page satisfied the given criteria. This means students needed to measure the side lengths and angle measures as well as figure out if they were in the correct order. So, not only were students getting additional practice with protractors–something some of them really needed–but they were forced to confront the idea that AAS meant something specific about where the 30 degree angle went and realize that which angle the 5 inch side was touching made a difference.

I didn’t anticipate that this activity would address this misconception, but it did. Big time! Students were figuring out on their own that if the 5″ side was touching the 40 degree angle rather than the 30 degree angle this was not the same triangle as the one that was doing the reverse.

By the end of the class we had a set of criteria for determining triangle congruence.

And I could positively kiss the kid who, when we were talking about SSA, asked:

Isn’t this like that problem we did in our last unit?


Yes, little darling.


Yes, it absolutely is.