Beyond Dojos and Card Charts

Last week, I was working with some second graders. We were doing a number talk and one of the students asked if he could use a hundreds chart. I said, “sure”.  All of a sudden, five kids got up and headed to the envelope of available hundreds charts. I watched the onslaught for a minute, wondering how it would work out. The first two students grabbed a chart and went back to their space on the carpet.  Then, two different students grabbed the envelope at the same time, each maintaining a fierce grip while glaring at the other. A third student stood quietly and watched.

“I had it first!” Elise said as she tugged the envelope towards her.

“No. I was waiting here. It’s my turn.” Joseph shoved his hand into the envelope and tried to wrestle one of the hundreds charts free while Elise simultaneously used her free hand to pinch the envelope closed.

I focussed my attention on Charlie, the boy who stood watching. “Charlie,” I said, “Thank you so much for waiting so patiently. It must be really hard to wait when you are anxious to get your hundreds chart and go back to your seat. I really appreciate how careful you are being.”

Elise and Joseph froze. Joseph dropped his hands from the envelope. Elise carefully took one of the hundreds charts out of the envelope and handed it to Joseph. I said, “Elise that was really thoughtful. I know how much you wanted that hundreds chart and you just gave it to someone else. Thank you for taking care of your friends.”  Joseph passed the hundreds chart to Charlie. I continued, “Wow, Joseph. You are also being really thoughtful and taking care of your friends. Thank you.”  Joseph took out a hundreds chart and handed it to Elise. Then, he took one out for himself and they all sat down.

You might be thinking, “well that only worked because Joseph and Elise don’t really have a lot of ‘behavior problems;'” Nope. Joseph and Elise have a really hard time relating to and communicating with their peers.  It is not easy for them to slow down and consider how their actions are impacting the people around them. You also might be thinking, “Well, it takes forever to establish that kind of relationship with kids. You’ve probably been working one on one with them all year.” Nope. I am only in their math class two or three days a week and I just started working with them in March.

I think my response to these students “worked” because I focussed on what the students were doing “right”. I held up a mirror for them. In the mirror, they saw who they want to be. The “reward” for their behavior wasn’t a sticker, a point, or a card. The rewards were feelings: pride, purpose, and gratitude.

Later, during the Number Talk, a different student was sharing his thinking.  Several students turned their bodies around to face that student. I took a moment to recognize their gesture. I said, “I notice you just turned around to face your friend. That shows them that you are really interested in what they are saying and you want to learn with them.” Simultaneously, three other students turned to face the speaker.

After our Number Talk, we  worked on a Same But Different Math problem that I created. As I anticipated, there was some disagreement about which numbers belonged in the blanks. We spent the remainder of class trying to prove whether or not the same number would make these two different equations true.

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Some students didn’t get to finish their work before class ended. I decided to continue the problem on Monday. For the lesson close, I asked if anyone wanted to share anything  they had learned so far. Greg was anxious to share. He couldn’t hold it in anymore. He shouted, “I learned I was wrong!”.

“Wow,” I replied, “Tell us more about that.”

He continued, “I thought the answer was 22 and Riley did too, but then Riley thought maybe he was wrong, but he couldn’t figure out why he was wrong. He asked me to prove to him how I knew it was 22. When I started to try to prove it, I realized I was wrong! Riley helped me figure out I had made a mistake!”

“That is really cool,” I said. “That is what mathematicians do. They share their thinking and make mistakes. When Riley asked you to prove it, he helped you figure out what you did wrong.”

Greg grinned, “Yeah! He really helped me.”

Riley chimed in, “Its a good thing because I thought it was 22 too.”

As math educators, we talk all the time about how important it is to get students to ask questions, justify their thinking, and critique the reasoning of others.  If we really want to make this happen, I think we need to pay attention to the mirrors we hold up for our students.  If we use extrinsic rewards and punishments are we building agency or promoting compliance?

I’m not sharing anything new, here. I’m not claiming to be doing anything new and amazing.  I just wanted to take a minute to share Friday’s math class because I’m so proud of these students. In our world at large, where adults are rarely listening to each other with the intention of understanding, I feel an urgent sense of purpose to help young people experience  a respectful community of learners.

“Math is a language, sometimes intimate, often boisterous.”

Math is a language, sometimes intimate, often boisterous, but always layered with experience and life profoundly lived.” (

Last weekend I was listening to wind chimes and wondering if there are patterns to the wind. The wind reminded me of  Jazz music. I don’t know much about Jazz, but I love listening to it. I love how it sounds simultaneously fluid and unpredictable. As I reflect on my time with students and teachers last week, I realize that there is a certain amount of necessary chaos in learning. The chaos often pushes my thinking and helps me connect ideas. 

Last week was crazy. I was all over the place. I didn’t allocate enough travel or planning time. I double booked myself. I was late to most meetings. I left personal belongings all over the school district.   I got more than one text like this one:

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This was my schedule for just one of those days:

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Truth: Last week wasn’t any different from other weeks. Scarier truth: I actually thrive in chaos. On Thursday morning, I was working with a small group of second grade students. The teacher I am collaborating with was working with a different small group of students. Picture me scrambling to get poster paper out of my backpack as I realize that I didn’t write the scenarios on paper ahead of time. Have a listen to our organized chaos:

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Admittedly, I was feeling frustrated.  I wanted to say, “just get a pencil, sit down, and listen to my story about penguins, now!” I didn’t. Instead, I tried to listen. I redirected. One of many benefits of recording myself when I teach is it forces me to be present, sort of. I spent one minute and six seconds trying to get a pencil in each student’s hand, only to tell them to put their pencils down and listen to me while I read to them about penguins.

Finally, I got them to close their eyes and listen. I said, “There are some penguins on the ice. Some are sleeping and some are awake.”  Then, I asked the students to open their eyes and tell me what they saw.

Rowan said, “Three penguins sleeping. Wait. Is it three?”

Meka added. “I see ten.”

I asked, “Rowan, are you wondering, is it three?  Meka, are you wondering, is it ten?”

Rowan said, “Is it okay if I say it’s three?”

I told him, “Sure. What did you imagine?”

“Three penguins on the ice. Some are sleeping and some are awake.”

“And what did you picture in your brain when you had your eyes closed?  What did you see?”

“More penguins awake and less penguins sleeping?”

At this point, there was a disagreement about erasers which derailed the conversation slightly.  I took it as an opportunity to build community. Have a listen.

These three students live in a small town and they go to a small school with less than 60 students. They are like a family. Even though they argue sometimes and forget to listen to each other, they really care about each other. They learn with and from each other. It is really important that I asked Rowan and Trent to listen to Meka, not me. And they did.

When I asked Trent what he pictured, he told me, “Nothing. It was just black. That’s it.”  I told him, “that happens sometimes.”

I asked the students, “Is there anything else you wonder besides is there three? Is there ten?”  They wondered about 100 penguins.

Next, I read the story again, but I changed the amount of penguins to 12.

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This time, instead of asking them to talk about what they imagined, I asked them to write it down on paper. Then, I asked them to share their thinking. Meka volunteered first. Trent wasn’t so sure her answer made sense. Listen to them as they discuss Meka’s work:

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I think this conversation is beautiful; simultaneously fluid and unpredictable. Yes, Meka convinces Trent that she also has twelve penguins and that is important, but her invitation to count with her and his respectful request to count for himself are so much more profound. The boisterous intimacy continued as Trent and Rowan described what they were seeing.

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At this point in the lesson, I might have assumed that the students understood that there could be more than one way to group 12 penguins into two subgroups of sleeping and awake. They seemed amenable to each other’s thinking. However, there is a difference between being convinced that a particular approach works and generalizing an understanding of whether the approach will always work and why. I tried to push, asking, what is the same about your solutions? What is different? Listen:

After we discussed similarities and differences, I encouraged them to see if they could find all the ways we could group 12 penguins; some sleeping and some awake. Trent asked if we could write the word “exhausted” on our poster. He said, “the penguins are awake, but really really tired. They are exhausted.” So, I did.  I offered the students the opportunity to use the base ten blocks or Digi-Blocks to help them solve and I went off to check in with some of the other students.

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I left my phone recording and captured some wonderful bits of conversation between the students. I particularly enjoy the giggling.

Trent is thinking about a “turn around fact”. He is wondering about changing the groups so that ten penguins are sleeping and two penguins are awake. At one point, he says, “now, ten are sleeping and two are awake”, but that is not what he records on his paper.  Trent is learning to take risks, but it is hard for him. I am tentative to push him too fast. I am thrilled that he is so thoughtfully engaged in learning.

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Rowan and Meka identified some other combinations of 12.

It was almost time for class to end. I asked Meka and Rowan to explain how they were able to find different ways to make 12. Rowan said, “I minused some from one and I plussed it to the other.” Meka describes her hard work.

I was tempted to record Rowan’s strategy using an equation, but I wondered if my recording would be meaningful for him.

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Instead, I asked him, “How would you record what you did on paper?”  This is what he drew:

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I wonder what to do next with Rowan, Meka, and Trent. I want to bring Rowan’s strategy to the whole class.  I am hoping if we can explore Rowan’s strategy and connect it to some of the tables that other students created, it may help all the students see, explain, and use structure when solving word problems.  
I am so grateful for Brian Bushart because he shared a wonderful routine and penguin problem, but the heart of this lesson is not the resource. The resource is just the vehicle for students to engage in meaningful discourse. The heart of the lesson is in the language; sometimes intimate, often boisterous, that we speak as we work together to seek common understanding. Language is how we communicate. It is how we create meaning. Math is a language. Name-ing things is important, but what is more important is how we come to name them.