Update for 2018/2019: sign up for our January 2019 Math Teacher Circles in Seattle here.

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In the run up to the upcoming year of Math Teacher Circles, we decided to offer a kickoff workshop for 4th – 8th grade teachers to drum up some excitement for the upcoming year. (The “we” in this case is myself, from Math for Love, and Jayadev Athreya of the Washington Experimental Math Lab, run through the UW.) This partnership (M4L + WXML) led a series of eight Math Teacher Circles last year, for K – 8th grade teachers, and they were great. The model is one of my favorites: teachers opt in to meet once a month to study topics in math and pedagogy, then try things out in their classroom between sessions. The circles are a fantastic way to meet and collaborate with colleagues from other schools, focus on fascinating and critical issues in math teaching, and energize your practice. They’re also free! Sign up now for elementary or middle school math teacher circles.

This year, we’re hoping to put together resources that will allow others to take this model and adapt it around the state/country/world. To that end, I wanted to summarize the last week’s kickoff workshop, especially because it was a remarkably fun workshop to be a part of.

**Part 1: Introduction and Motivating Examples**

We had about 35 teachers in attendance, teaching math for 4th – 8th graders, plus a few mathematicians. The workshop began with conversations among groups about people’s math autobiographies. What were the most memorable experiences in people’s mathematical histories, positive and negative?

Some of the big ideas that came from these discussions:

- Math is emotionally charged. We feel a lot – pride, shame, satisfaction, etc. – in the course of doing math.
- One teacher can have a huge impact on how someone views themselves mathematically.
- Those moments where the math is meaningful creates a sense of empowerment; alternatively, those moments when math is empty and we’re working without understanding undermines that sense of personal power.

After the discussion we showed Robert Kaplinsky’s mind-blowing *How Old Is the Shepherd *video. This video always baffles me, especially because I suspect that younger students wouldn’t respond the way these 8th graders (they’re eighth graders!) do.

There’s an idea that certain institutions create a mirror image of the very outcomes they’re trying to prevent: there is a certain kind of sickness that exists only in hospitals, and a certain kind of ignorance created only in schools. We need to take this seriously. If we’re not careful, we estrange students from their own sense of knowledge, and they end up not being able to make the obvious observation: that something doesn’t make sense.

**Part 2: Conjectures and Counterexamples**

I’ve been developing a structure to help students to help invite students into a genuine mathematical process, starting with their own understanding. We call it **Making and Breaking Conjectures**. In the next section, we talked about this structure.

First, what are *conjectures?* What are *counterexamples?*

*Conjecture. A mathematical hypothesis. A guess of the underlying structure or pattern based on what we know so far.*

*Counterexample. An example that proves a conjecture false*.

Mathematics as a field progresses by way to conjectures and counterexamples. The good news is, we can use them even with very young kids.

We played the game Counterexamples to get a sense of how this works. (Lesson plan PDF here. Online here.) The game is super-simple: the teacher makes a false conjecture, and the students prove it false with a counterexample.

Teacher: All pets have four legs.

Students: No! Because birds have two legs!

Teacher: Okay – refined conjecture: all pets have two or four legs.

Students: What about a snake?

Teacher: That’s a pet with no legs. So I’ll refine my conjecture again. All pets have *at most* four legs.

Students: What about a spider?

And so on. So compelling is the game that we almost got off track when we considered conjectures about area and perimeter. But there was work on conjectures to be done first.

To warm up our observing/noticing/wondering/conjecturing muscles, we started with a paperclip.

This is a classic exercise, and Kindergartners tend to beat the pants off of adults. After spending ten minutes or so, groups had come up with everywhere from 20 to 65 uses for a paperclip. That was just the warmup, though, so we didn’t go too deep into what the uses actually were; the main event was still on its way.

Considering this question was trickier, and required a little more discussion to draw out observations and questions. By the time the conversation was done, however, we had a tidy collection of questions to consider.

- How many rectangles can be formed by connecting dots with straight lines?

(Refinement: diagonal lines vs only horizontal and vertical) - How can you connect all the dots with connected straight lines drawn in a continuous path? How many lines does it take?
- How many sides could a polygon have on the grid? Could it be all 16 dots?
- Can you make a square with any number of dots on its perimeter?
- What grid came before? What grid comes after?
- Maze: how many different paths are there from the bottom left to top right? Refinement: direct route or can you loop back?
- If you can define “1” differently (length or area), what it the area of the whole grid?
- How many different angles can you form if you connect three points?
- How many different areas can you get if you form a triangle with three points on the grid?
- How many lines of symmetry can you make? What if the lines don’t have to be on the dots?
- How many symmetric shapes can you make with corners on the dots?
- How many lines would there be if you connected every dot to every other dot?
- Can you find two triangles with the same area that are not similar?
- How many different lengths could you find by connecting two dots?
- Can you make a triangle with area 1/3? (If the whole is 1)
- How many fractions could you represent on the grid?

Take that list in for a minute. There are weeks of beautiful, high level problems to explore here. For each, you can start by casting around. Soon, you’ll find you have conjectures. Once you have a conjecture, you can try to break it right away by looking for a counterexample. Refine and repeat until you end up with something that seems to be true, and then you can put together an argument, with luck and a little insight, into why it actually might be true. We’re really doing mathematics!

With all these questions to consider, we wanted to provide some guidelines. So we posed a choice: teachers could choose the problem that inspired them most and work on that one. Or they could work on the question of how many different areas a triangle could have if its corners are on the grid. (Having a default question is something we’ve found can help prevent groups from becoming aimless. With an actual classroom, you might want to skip some of these steps, and start from a more tightly focused, teacher-chosen task in the first place. Still, opening up the entire thinking process can definitely be worth it, if you’re ready to take the step.)

And with that, we sent the groups out to work on their own. However, we had one more idea ready at hand to ensure things went as well as they could go.

**Aside: Thinking Classrooms**

Peter Liljedahl has been developing a series of concrete steps to change the classroom to support genuine thinking in mathematics. The first, central step is to use tasks, that is, make real, meaningful mathematical experiences the heart of what you ask students to do. But you can help students actually transition, and the steps Liljedahl has determined to encourage this transition are bizarrely simple to employ. Here’s how you begin, according to Liljedahl:

- Use Tasks
- Use vertical, erasable surfaces (i.e., whiteboards)
- Assign groups in a visibly random fashion

It seems almost too easy, but the room was electric with thinking energy. Here’s what it looked like.

It’s hard to overstate how getting everyone standing and working together impacts the quality or thinking and engagement in the room. One of Liljedahl’s recommendations is to use just one marker per group. You can see the value of that suggestion in the video below, as the carrier of the marker is drawn back up to the board to write in details of someone else’s computation.

Don’t have enough whiteboards? You can write on windows too!

With so many teachers considering such a range of problems, we decided to let groups pair up to share what they had discovered with each other.

All in all, we were totally energized by the workshop, and I think the teachers were too. They’ve got a whole bunch of great questions to pose about grids, and the motivation to use task, whiteboards, and visibly random groups in their own classrooms.

And we’ve got a goal for this year’s math teacher circles, especially for the middle school level: create a series of PD sessions that will help everyone turn their classroom into a Thinking Classroom, where students use Conjectures and Counterexamples to power genuine mathematical experiences.

You can sign up for our 2017 – 2018 Math Teacher Circles now:

**Part 3: Epilogue – the Global Mathematics Project**

There was one more takehome point for the teachers: October will be the launch of the Global Mathematics Project! We shared another grid-related resource from the GMP.

Not only is this kind of problem a super-fun way to launch an exploration with students, it’s also grid-related! And it’s also just a taste of the kind of math that’s going to be highlighted during the Global Math Project in October, 2017. We’re looking to give 1,000,000 a genuine math experience the week of October 10. Interested? You can learn more and sign up here.

So, a long post, but hopefully just a beginning of what will be a framework to help others lead their own Math Teacher Circles, and help making authentic and beautiful mathematical experiences the center of math class.

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