Satisfied

This post is a contribution to the amazing, wonderful Virtual Conference of Mathematical Flavors hosted by the amazing, wonderful Sam Shah. I spent much of yesterday reading everyone’s sessions and it was the best PD I’d been to in a while. Highly recommended.

Sam asked us, “What flavor of mathematics are you serving up in your classroom?” First off, I don’t have my own classroom anymore. This coming Tuesday is the first day of school this year, and I always cry on the first day of school because I miss it so much. I miss the energy, the nerves, the long pencils that still have erasers, the excitement about the new year, the relationship-building. I miss it all and feel no small amount of grief. But, gratefully, I get to spend most of my school days in my colleagues’ classrooms or working with teachers as a coach, and I continue to work on my teaching practice in that different setting.

As this school year gets going, I find myself reflecting on an experience I had last week at Math on a Stick at the Minnesota State Fair, and how it relates to a practice I see in lots of classrooms. I spent a fair bit of time watching a four-year-old-boy play at the eggs station.

I love the eggs table. It’s a phenomenal place to watch young mathematicians while they play. This particular child chose two colors of eggs and put them in rows, alternating colors in each row. When he was done, he studied his crate for a while. He made this tiny, almost imperceptible nod with his chin after a bit. The word that flew right into my mind was satisfied. He was satisfied. He then turned his attention to another crate and made new rows with two new colors, but this time on the diagonal. When he was done, his two crates looked like this:

 

He stepped back again and studied what he made. A long time went by and I wondered what he was thinking. He finally put a hand on each crate and began rotating both crates, first in the same direction, then in opposite directions, then adjusting both until his rows went perfectly straight across both crates, pink-to-orange, and blue-to-yellow. He studied them again. This time, there was a tiny smile in the corners of his mouth as he nodded his chin once, decisively. Now he was really satisfied. He turned the crates back before I could take a picture and moved over to the pattern machines. I didn’t ask why he turned them back so I don’t know, but my instinct at the time was that his solution was his. His discovery was a secret, one he wanted to treasure and keep, just for himself.

In math education circles, I increasingly hear words like agency and ownership. I can’t think of a better example of these words than the boy with his eggs. The ideas were entirely his. The discoveries were entirely his. The decisions to (1) do math publicly, but then (2) keep a bit just for himself (he was so engrossed he never noticed me watching) were entirely his. The locus of control for deciding what was interesting, valuable, worth doing, or “solved” resided within him. It was beautiful, and a lump grew in my throat as I watched him.

Know what I didn’t do? I didn’t walk over and talk about his eggs, or other people’s eggs, and express a value judgment about which crates of eggs were better, more efficient, more abstract, more impressive, or more sophisticated. It wasn’t tempting at Math on a Stick.

In classrooms, I see those value judgments become tempting. I absolutely love the 5 Practices for Orchestrating Mathematical Discussions by Smith and Stein, but I worry about the “selecting and sequencing” steps of the share-out devolving into just this sort of judgment and ranking. The authors talk about crafting a storyline when selecting student work to share, and I think they genuinely mean it. I can think of so many possible storylines to craft when looking at and discussing a range of solutions (Intentional Talk by Kazemi and Hintz blew my mind on this front). But in practice, somehow “stories” are reduced to “ranking” more often than not. And ranking is treacherous.

Don’t get me wrong. I sequence sometimes. For example, when I work with teachers, we often work on the Noah’s (or Nora’s) Ark problem, which I learned about from Fawn Nguyen and love with all my heart (I love her too).

(By all means, if you haven’t worked on it yet, stop right now and play. I mean it!)

When I facilitate this problem, I strip all the text off and ask folks what they notice and wonder for a long time. They talk in groups, we share in the large group, ideas get flowing. Then I finally share the question and everyone’s pumped to work. I give them five full minutes of silence to work on their own, and I’m strict about that. If somebody starts talking, I’ll even use proximity to quiet them down. Everyone needs a chance to get oriented. While they’re working, I walk around looking for some possible first steps. Not solutions; first steps. For example, usually somebody tries putting everything in terms of one animal–usually polar bears. I snap a picture of their first swaps. Somebody else starts using symbolic notation, writing 1E = 2B. I snap a picture there too. Somebody else might start assigning values to animals, such as, “If a polar bear is worth 3, then a zebra is worth 1,” and they go from there. I grab a picture. And somebody usually starts throwing animals overboard, crossing them off if they balance each other on the ship.

Again, I grab a picture before they get too far. After five minutes, I pull everyone together and I ask these folks to share their first steps so that, if people haven’t found a toehold in the problem yet, they might hear an idea that appeals to start. I also talk about how most people find they need to switch strategies partway through this problem, so listening to some of these other approaches might help them get unstuck. When I share these first steps, I do sequence them in one important way, which is that I share the symbolic language last. In my experience, folks who haven’t found a way to start yet would shut down if I started with pages with equations. So I hold that one for a sec. My goal here is to increase access to the problem, not to intimidate. That, to me, is a reason to sequence carefully.

But then I cut everyone loose to work, talking or not talking as they organically feel the need. We’ve already talked about norms a lot, and I hope folks state their needs, (“I’d love to hear your strategy but I’m not ready to listen yet. I’ll be with you in a few minutes, once I’ve got my ideas sorted out.” “If anyone is ready to talk, it would help me to think this through together.” And so on.). And they work. I don’t stop them until everyone is satisfied.

At the end, I try to share some strategies we haven’t seen yet. And the lovely twist to this problem is that people who use a lot of algebraic expressions are often stunned by the beauty of a purely visual strategy. It’s a great surprise, especially for folks who assume that the solutions that look the most like 8th grade math will be deemed the most sophisticated.

My point is that, when I sequence, I am doing so for mathematical and cultural goals. I’m trying to share solutions that will reveal the mathematics, will surprise and delight, will enable students to make connections, will spark new questions and conversations that will keep going after we pause our discussion. I’m trying to build a conversation and understanding, not build toward some culminating, “best” solution.

The first time I played with this sort of conversation was with my colleagues and the cheese problem:

Take a sec and anticipate how students just beginning to study division might solve it.

We saw a range of solutions, as we expected. Lots of circles and stars and skip counting by fours. We saw one student use multiplication to solve it, keeping track of his partial products. If we were ranking, we would have called this the “most sophisticated” strategy. But we weren’t ranking. We were exploring division. We ended up deciding to put his strategy next to a grouping strategy, hoping students would see some connections:

It took a while, but then they saw it. “I see the ten fours! It’s like a ten-frame! And the four fours!”

Suddenly, students who had no idea where those equations came from could literally see them. The student who had written the equation might have been the most surprised.

What was really important here was these solutions weren’t ranked from least to most sophisticated. These solutions were side-by-side because looking at them together was mathematically productive. The student who had drawn the blue dots helped her peers learn, which is something we value. Her solution enabled the class to see new mathematics, to learn something, to connect ideas. Her solution revealed the mathematics. During and after the share-out, both students felt respected, and both students were satisfied.

Toward the end of my time at Math on a Stick, I hung out by the Stepping Stones a lot.

I saw such beauty. I saw a young child walking carefully, one leg forward only. He’d stand on one and take a step with his left foot. Say “two” out loud. Take another step with his left foot. “Three.” He went up and back and up and back, over and over. Eventually, he started striding normally, alternating left and right feet, saying, “One, two, three…” A little while later, I saw him walking like that with his hands in his pockets, (“Look, Ma! No hands!”), thinking hard but with confidence, no longer vocalizing the numbers. He was satisfied.

A few minutes later, a grandmother cutting through the booth hopped along from 23 down to 1. She turned to me and said, “I just can’t help myself.” We shared a grin.

Later in my shift, I saw two young adults jumping from step to step, counting on their fingers, loudly. The young man was on 12. “1100!” he said, fingers out. She was one step behind him: “1011.” They were laughing, delighted. Each time they stepped, they’d look toward the sky or the ground, concentrating hard. Partway through, she said something about how she was adding one each time as she counted in binary. He said, “I can’t do that yet. I have to derive every single number.”

I was struck by how similar this pair of friends looked to the young child from earlier in the day. Each person was working on a new-to-them number system. Everybody went to the sure ground of using fingers, talking out loud, and taking as much time as needed. They were all working on mathematics that felt novel and interesting and enticing, to them.

Was one more sophisticated? Efficient? Abstract?

Honestly, who cares?!

Answers to those questions would necessarily be subjective, and would be relative to my mathematics, not theirs. Asking those questions would have stopped their play, ended their ownership, taken their agency.

So I didn’t. I was just the lucky person standing by, bearing witness to their joy. And that, to me, is the greatest privilege of teaching. I will forever be addicted to being in math classrooms with children and teachers because some of the time, I’m the lucky one who gets to bear witness to someone else’s joy and discovery. Some of the time, students’ and teachers’ discoveries will lead to new mathematical thinking in me. But all of the time, their discoveries belong to them.

I go into this year more wary than ever before of the moves we make–with the best of intentions–that take over those discoveries, that diminish that satisfaction, that co-opt that joy for our own instructional purposes. I continue to love strategy share-outs. They’re some of the most complex teaching to facilitate, and “orchestrating” productive conversations for mathematical and social goals takes a lot of finesse and skill.

The moment I’m attuned to is when sequencing a storyline turns into a temptation to rank student work. That’s the moment to pause and think about the folks on the stepping stones, giggling as they tried to figure out 19 in binary. Or the young child with his egg crates, who explored his own ideas because they were his, and thinking about them was interesting and joyful.

That little chin nod he made, that moment when he stepped back and looked and was pleased with what he’d done? That’s the thing. That’s the flavor I want to serve up. I want mathematicians of any age to experience that deep satisfaction for themselves.

 

Postscript: I found myself thinking, this morning, about a reference that’s implicit in almost everything I wrote here, which I wanted to make explicit. When I ask teachers what they need from one another in their groups, I often find myself talking about my friend and colleague, Christopher Danielson. He’s my favorite person to do math with, and I talk about why. What’s interesting is that Christopher knows much more advanced mathematics than I do, but I never, ever feel dumb when I work with him. Over the years, I have learned how to articulate what feels different about doing math with Christopher: he takes delight in the discoveries of others. This is the biggest lesson he has taught me, by example.

When I work with teachers on setting norms before we engage in something like the Nora’s Ark problem, for example, I ask what we should do if one person in the group finishes sooner or knows more or goes faster or something like that. Teachers often suggest that person should bite their tongue, sit on their hands, walk away so they won’t be tempted to blurt out a way to deal with those pesky kangaroos. And yes, this is a start. We need to not just tell people how to solve it or “what I got” or “how I did it.” When someone does that with me, I feel like they are unwrapping my birthday presents for me. They are depriving me of the satisfaction I described above–satisfaction that often follows a period of being stuck.

But what’s interesting about Christopher is he is never tempted to tell his solution. He is, instead, interested in my solution, and on the experience I’m having while working toward it, (whether I get there or not). I have observed that my solution paths are often very different than his precisely because I don’t have the advanced mathematics background he does, and he finds the questions I ask or the approaches I try refreshing and surprising. Sometimes we are then able to connect what he knows and what I know and we each learn something new, like the students did with the cheese problem above. Other times, my discovery is fully settled mathematics for Christopher, and he just takes delight in watching me wrestle with something that is not yet settled for me. Bearing witness, as I wrote about above.

I was recently reminded of all of this–of Christopher’s spirit and purpose and genuine curiosity about other people’s thinking and discoveries–at Math on a Stick, which is where Christopher is most at home. He’s made something marvelous, and he shared it with my children and me. We are filled with gratitude.