Let’s stop condemning kids to a life without math.

Recently, during a Number Talk, I wrote this problem on the board and asked students to show me a quite thumb if the problem made sense. Then, I encouraged them to try to find a solution using some of the strategies we have been working on.

26 + 10 = ________

Calvin mumbled something under his breath. Students started raising their fingers to indicate how many strategies they had used. Calvin crossed his arms and started to kick the air as he slunk down into his chair. I asked the students to whisper the answer on the count of three. I heard a chorus of “36”. I wrote the next problem on the board.

26 + 12 = ________

Calvin glared at me. He growled. He turned his back to me. Mrs. X went over to him and he whispered to her as I continued the number talk.  When Mrs. X came back over to me, she whispered, “he’s mad at you because you won’t stack the numbers.” I nodded to her and continued on with the next problem.

26 + 22 = ________

Calvin glanced at me with angry eyes. Then, he looked at Mrs. X and snarled, “She’s not stacking it!” My quiet thumb dropped to my side. I took a breath, looked Calvin in the eyes, and responded, “Calvin, if you would like me to write the problem a different way, I can do that, but you need to use your words and ask me. I am happy to help you if you ask me to.” Then, I went back to thinking about 26+22. Calvin kicked the air and turned away from me. After some quiet think time, I collected solutions and asked for volunteers to defend them. Jason told us that he decomposed and added. I recorded his thinking as he spoke. Screen Shot 2018-02-08 at 4.52.25 PM

I said, “This strategy reminds me a little bit of the one Calvin likes to use.  We don’t have it listed on our strategies menu, but I think maybe we should. Calvin, did you want to talk about stacking?”

Calvin turned around. His shoulders settled. He asked, “can you write the numbers so they are stacked? That is the only way I can do it.”

“Of course I can. Thank you so much for asking. I will write the numbers so they are stacked. Can you tell us how you would solve it?”

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Calvin explained, “Six plus two is eight and two plus two is four. The first time I got 82, but then when Jason said 48, I figured out I was wrong.”

I rephrased, “So you added the six ones and the two ones and got eight ones. Then, you added the two tens and the two tens and got 4 tens.”

“No. It’s just four. Two plus two is 4. The answer is 48.”  He beamed.  “Can you write my name next to it, like you did for the other kids?”

“Of course.”

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I asked Jason if he noticed anything that was the same about his strategy and the strategy that Calvin used. Jason said, “we both decomposed the numbers. We both added the tens and the ones.” I asked Calvin if he understood what Jason meant. I motioned towards the similarities as Jason explained them.

 

 

 

Calvin smiled, “We both got 48.”

“Yes,” I agreed. “Calvin I am going to continue to write the problems horizontally, but I can also write them stacked, if you would like.”

“Yes,” Calvin replied.

This number talk took place about three weeks ago. I think about it a lot. I think about Calvin a lot. Usually, I get really angry when I think about Calvin. I’m not angry at Calvin. I’m angry for Calvin. Calvin wasn’t always in our district. He transferred here from somewhere else. Counting is challenging for him. His only experience with place value seem to be to “stack” numbers and use his fingers to tally up the digits in each column.

Calvin clings to stacking like a life boat. I picture him, tethered about 100 feet off shore, clinging to the lifeboat because no one taught him how to swim. I imagine, and maybe I’m wrong, that no one taught Calvin how to swim because they firmly believed that he couldn’t learn how to swim. They probably thought it was safer for Calvin to just cling to his lifeboat.  I can imagine the types of conversations that happened in regards to Calvin’s potential for learning:

“Calvin is just so low.”

“Calvin just needs to be taught a procedure. He can’t think ‘like that’.”

“Poor Calvin. His life is so hard. It’s not his fault, but it makes sense that he is so far behind.”

“Calvin can’t be in class with his peers. He is just so far behind.”

“Math just isn’t Calvin’s thing.”

“Calvin should be tested.”

Bullshit.

Who the hell are we to decide what Calvin can’t do? Maybe it isn’t Calvin’s problem. Maybe it is our problem. Maybe we need to ask ourselves, what can we do to help Calvin think deeply about mathematics?

After this number talk, I added stacking to our anchor chart of strategies. The week following this number talk, I wrote the Number Talk problems both vertically and horizontally. If I forgot, Calvin grinned and politely reminded me.  Gradually, Calvin stopped asking for the numbers to be stacked. He still clings to his strategy, but he seems to be thinking about trying to count up by tens. Yesterday, we checked in on his counting skills. Look at his work and ask yourself, what CAN Calvin do?

Screen Shot 2018-02-09 at 9.53.46 AMCalvin can count forwards and backwards by tens, off the decade! He might not be able to do it all the time, but he can certainly do it. Now, we have to help him develop his place value understanding and connect it to what he knows about counting.  I have been carefully observing Calvin during class lately. Here are some other things he can do:

  • Calvin notices patterns when we do choral counts.
  • Calvin always raises his hand when I ask if anyone wants to defend a solution.
  • Calvin takes risks. Yesterday, one of Calvin’s peers used a compensation strategy to solve 19 + 19 = ______. Joey said, “I took one from the 9 and gave it to the nine in 19 and then I had 10+10+10+8 so I got 38.” I asked if anyone else could explain what Joey meant. Calvin’s hand shot up.  He had a big smile on his face. I called on him. He thought for a while, smile never disappearing. He said, “I am not really sure.” I offered, “help or time?”. He asked for help. He loves being in charge of choosing who gets to help him.

Let’s start looking at all of our students in regards to what they can do. Let’s stop finding excuses for why we can’t teach students. Calvin is far behind his peers.  He didn’t have the exposure to the math practices that his peers had, but Calvin has potential. Calvin has a voice. Calvin is capable of greatness. I will admit that Calvin terrifies me because it is going to require a lot of work and reflection for me to figure out how to help him think deeply about mathematics. I have a ton of questions:

  • Should I give him place value blocks during the Number Talks?
  • Should I give him Digi-Blocks?
  • What if the other kids want to use them? Will it lower the cognitive demand of the Number Talks?
  • How do I help Calvin connect counting to place value?

I don’t know exactly how to how help Calvin. I have a lot to learn about supporting K-2 students. So, I read. I try things out. I reflect. I ask for help and feedback.

Today, I tried to help Calvin, and the rest of the class, understand Joey’s compensation strategy. I put nineteen place value blocks in each of my hands. I asked Calvin to count them to make sure I was correct. Thank goodness he did because I was one short.

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Then, I asked the class if anyone could use the blocks to show us Joey’s strategy from yesterday.  Ben volunteered. Ben picked up one of the cubes from my left hand and said, “Joey took one away from the nine,” he placed the cube in my right hand, “and he gave it to the nineteen.” Calvin watched carefully.  He looked up at me and said, “That’s a ten! We can trade it for a stick!” I smiled and asked him if he would like get the ten stick for us. He did. When he came back, he said, “that makes three tens. So it is 38!”

I said, “your darn right it is,” and I gave him the highest of  high-fives.

All of our students deserve respectful, engaging, math instruction that requires them to think deeply.  How can you help make this happen?

Counting Collections: When does a “nudge” become a “shove”?

Kindergartners simultaneously terrify and inspire me. They terrify me because they are so candid and unencumbered by humility. They won’t hesitate to look you right in the eyes, in the middle of a conversation, and say, “this is boring.” They inspire me because their sense of wonder is raw, and, also unencumbered. Five year olds wonder as naturally as they breathe. Being curious isn’t something they have to practice or strive towards. It is just what they do; breathe, sleep, eat, be curious.

Once a month, I meet with Deb Hatt, one of our building based math interventionists, and Katie Reed, a Kindergarten teacher. We co-plan and co-teach a Counting Collections routine. We are trying to learn more about how Kindergartners record their thinking and justify their reasoning. We have so many questions:

  • What is the difference between “how did you count?” and “how do you know your answer is correct?”
  • What do Kindergartners understand about the word, “prove”?
  • How do we honor student thinking while also nudging it forward?
  • What is the difference between a “nudge” and a “shove”?

During our last planning session, Mrs. Hatt and Mrs.Reed discussed a recent blog post by Heidi Fessenden.  They were so appreciative of Heidi’s honest reflections about how she let go of some control in order to make space for the opportunity to learn about her students. Earlier in the year, I had shared some resources from Kassia Wedekind. We have found them incredibly helpful. Mrs.Reed encouraged us to try Kassia’s guide to conferring during our routine today. We discussed places that we might try to nudge student’s thinking.

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Katie wondered, “What kind of questions should we ask that might nudge the students?” We decided we could start by just saying, “I noticed (something about how they counted). I wonder how you are going to record that?”

We knew it was still a challenge for many of the students to accurately record their count with a picture. Most of them had no problem recording the number they counted, but they were still struggling to show how they know their answer is correct. Some of them have started to record how they counted, but aren’t necessarily depicting an accurate number of objects. The last time we did this routine, I asked one of the students if she had drawn a circle for each one of the shapes she had counted. She looked at me, exasperated, and said, “No. You told me to show you how I counted, not how many I counted. This is how I counted, see?” She touched one of the shapes and then touched one of the circles she had drawn. “This (shape) is this (drawn circle).” A few students are pretty content to count the objects, write a number, and move on. Mrs. Hatt, Mrs. Reed and I have spent a lot of time thinking and talking about how we can help Kindergartners find a purpose for proving to themselves that they counted correctly.

Katie started the lesson by reminding the students about the Counting Collections routine, “We are going to do our counting collections routine today. Do you remember? We have some collections that we count and then we talk about how we counted. Today we are going to spend some time exploring how we can tell we have the right answer.”

She and Mrs. Hatt made a space for themselves on the floor, in front of the students.  Katie continued, “We have some shapes that we are going to count. Mrs. Hatt is going to count them first. Then, I am going to try. We are going to see if we get the same answer because, sometimes, when we are counting, we get different answers. Has it ever happened to you, when you are counting with somebody else and somebody else gets a different answer?”

A sympathetic chorus of “yes”.

Deb shook the shapes onto the carpet and counted. She waved her finger over the pile as she quickly and haphazardly accounted for each of the shapes. “I got 21,” she told Katie, “Now, you count them.”

Katie pulled the pile of shapes into her space and began to count. She got a different answer than Mrs. Hatt did. The children were riveted. Listen.

 

Mrs. Reed still wasn’t sure what the answer was so she decided to check again.

 

At this point, Mrs. Hatt and Mrs. Reed agreed that they had 19 shapes. Deb held up the recording sheet and asked the students how she might fill it out.  “It says, ‘how do you know?'” She wondered, “How do I know?”

Several students responded, “Because you counted them.”

Deb described how Katie’s arrangement really helped her see that there were 19 shapes. She drew a sketch of the group of ten shapes and the group of 9 shapes. Then, she asked the kids about the representation she drew on her paper. “Does this drawing look kind of like Mrs. Reed’s arrangement?”

“Yeah. It needs one more to make 20.”

“That’s right,” Mrs. Hatt agreed, “My drawing also needs one more to make twenty.” She asked the students if they could tell, by looking at her paper, that there were 19 shapes in her collection.

 

Katie introduced the visual anchor charts that we got from Heidi’s post. Deb introduced the tools; cups, ten frames, plates, and hundred charts. She reminded them,
“You are welcome to take anything that is up here. You have lots of choices.”

The kids went off to count.  Katie, Deb, and I circulated and watched. After a little while, I decided to check in with Alissa.  I reminded myself to take the time to notice what Alissa had done before I attempted to nudge.

I said, “I noticed that you were drawing lines on your paper, instead of drawing the shapes, is that right? Am I correct about that?”

She nodded yes. I continued, “It looks like you wrote that you have 17 shapes. I am wondering how do you know that the lines on the paper match how many shapes are in the bag?”

Alissa responded, “because I counted. I looked for the number 17 on the number chart.” She pointed to the 17 card sitting on the table. She had taken it out of the hundreds chart and used it to record the number on her recording sheet.

I decided to nudge a little bit. I asked, “What if someone came over here and said, ‘I don’t think there are seventeen shapes in that bag. Can you show me how you know there are seventeen in there?’ What if I said that? What would you show me?”

Alissa dumped the shapes out of the bag and placed them in the cup, one at a time, as she counted out loud. This time she counted 29. Then, she counted a third time, and got 19.  She chuckled and said, “I counted wrong.” I wondered how we could figure out which count was the correct one. I think at this point, I transitioned from “nudging” to “shoving”.  I was grasping at straws; “how can we organize?”, “can we make groups?” Alissa would humor me with a glance in my direction, then go back to counting. Listen to me fumble. Can you hear me sweating?

 

Did you hear what she said at the end? She said, “I need another one.” She filled up one of her ten frames. Then, she wandered across the room to the table with all the tools on it.  When she got back, she picked up right where she left off. I decided maybe I should go back to noticing, instead of shoving, so I attempted to be a mirror for her. I was pretty enthusiastic. Maybe too enthusiastic?

 

I don’t know if I did the right thing. I wanted so badly for Alissa to convince herself she was right. I wanted her to trust her intuition. She knew it was 19. I could tell. How could I get her to show it on her paper?  I knew I had to wrap it up. This conference with Alissa had lasted at least three days (or ten minutes).  Ethan was waiting patiently to show me what he found out. After Alissa changed the number of items on her recording sheet, I asked her how she was going to show that she knew the answer was 19?

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I asked, “What if someone came over and said ‘I had bag L and I got 17. I think it is 17.'”

She responded, “Then I would say, ‘Okay. Let’s just do another one.'” She was just going to consent to 17. Oh no! After all this work, she was just going to consent to 17! I started nudging (shoving?) again. I said, “How could you convince them? What if you said ‘no. I am pretty sure it is 19.’ and they said, ‘well show me your paper. Show me how you know it is 19.’ What would you write on your paper. How did you figure out that it was 19?”

She said, “A ten frame.”

“Okay,” I gave one last push,  “so what could we put on your paper to show that? Think about that, okay Alissa. Think about how you are going to show here that you know it is 19. I am going to check in with Ethan and then come back and see what you came up with.”

I didn’t get a chance to check back in with Alissa before the closing circle. When I look at her recording sheet, I can tell she changed her thinking. She tried to erase some of her earlier lines. She still has more than 19 lines on her paper, but do those top two sets of hash marks resemble groups of five? Might that be a group of ten?

A K CC 2:4:18

I don’t know. I have so much to learn about conferencing, but I am so grateful for my time with Alissa. She taught me a lot:

  • Noticing is really important. Maybe more important than nudging?
  • I can always go back to noticing, even after I try a nudge.
  • A nudge can linger.

I really enjoyed using Kassia’s recording sheet. I would like to try noticing and nudging with all my students, K-12.

Notice Nudge Next screenshot

 

Kindergartners remind me that deep learning takes time: intuition isn’t built in one day.

 

 

 

Mathematicians Think About What Their Partner Means

I have been thinking about trying to use Desmos with elementary students for a while now, but I was struggling to commit. Honestly, I wasn’t sure if I trusted it. Would it amplify the learning or just be another fingerprinted screen? Mostly, I have used Desmos as a learner. I play with the graphing tool to help me see patterns, build my intuition, make connections, etc. Once, when I was trying to deepen my understanding of repeating decimals, I asked for help on Twitter.  Christopher Danielson gave me something to play with.

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I explored for a while.

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Here is a screenshot of something else I was exploring on Desmos once. I don’t remember what it was. It is rough draft thinking. I was just playing around.

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I have used a few Desmos activities with Middle School students and teachers. I have played around with the teacher platform a lot, but, like I said, I had trouble committing.

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This week, I finally tried it. During a 3-5 grade span meeting, I shared, planned and taught a polygraph that I created. A group of us went into a third grade classroom together to investigate this questions:

How do we know if technology has amplified the learning?

I spent a chunk of time preparing for this experience. The first thing I did was connected with the third grade teacher with whom I would be working, Ms. M, and I shared my thinking. We agreed to have the students try a different polygraph the next day so they would be familiar with the routine.  We settled on a polygraph called Kittens because it wasn’t explicitly “math-y”. We wanted them to focus  on the structure of the activity and “try on” the important characteristics of the Polygraph game: asking and answering questions and using deductive reasoning.

We also talked about which polygraph to use during the grade span meeting. I told her I wasn’t really sure that any of the polygraphs I had found were the right ones. I was hoping to find one that elicited big ideas about multiplication and division because that is what the students are investigating in class right now. The only one I found was a multiplication practice task.  It didn’t seem like a good fit. I wasn’t as interested in exploring the potential of Desmos to help students practice a skill. I wanted to see whether or not Desmos amplified deep understanding of a concept. Ms. M and I agreed that, over the weekend, I would create a polygraph that was made up of arrays. (As I am writing this, I re-discovered Michael Wiernicki’s Arrays Polygraph. It looks similar to the one I created, but with area models instead of dots. Maybe a next step for us?)

The next day, I stopped by math class while the students were playing Kittens. Each pair of students had an iPad. They were having fun; trying to figure out who they were playing against, talking about how cute the kittens were, trying to think of questions to ask that would eliminate kittens. There were some arguments starting to bubble up.

“Hey! You said it wasn’t a fluffy cat!”

“It isn’t. That cat isn’t fluffy.

“Yes it is!”

“It isn’t as fluffy as the other cats.”

We talked a little bit about why “fluffiness” was subjective and collected a few questions that were not as subjective.  Ms. M and her students had no problem accessing Desmos and getting into the game.

Last Sunday, I sat down to create a an array Polygraph. First, I asked myself, what big mathematical ideas do I want this activity to illuminate? Here is where I landed. (You can click here for a link to my whole planning doc.):

Screen Shot 2018-01-17 at 10.24.18 AMI opened Powerpoint and started messing around with arrays. The first few images I created only had 6 dots in them.

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I wondered what kids would do with these images. What kinds of questions would they ask to distinguish between the images?

  • Does your card have 2 rows?
  • Does your card have 3 columns?

What if they don’t know the terms rows and columns or maybe they have heard them but aren’t using them – they don’t “own” them, yet? Maybe they would say “groups”.

  • Does your card have 3 groups of 2?

At this point I wondered if I would have to use numbers other than 6. Polygraphs need 16 cards.  Could I make 16 different representations of 6 dots? I might be able to, but would the polygraph still illuminate the ideas I mentioned above? I wasn’t sure, but my gut told me to stick with one number. I decided to try a bigger number. I ended up using 48 dots to create 16 different representations. Actually, one of them has 36. I lost two groups of sixes somewhere amidst the copying and pasting. I noticed the lone group of 36 while I was playing my first round of Polygraph: Arrays with my friend Jocelyn Dagenais.  Here is what is great about #mtbos: on Monday morning, I tweeted this:

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Within minutes, Jocelyn and I were playing a game of Polygraph: Arrays.  Later that night, Laura Wageman and her daughter played a round. Laura also shared a great resource for finding other Elementary Desmos tasks.  Many other folks helped me out. Dianna Hazelton shared a slide show that she presented to teachers earlier in the day.  Jenna Laib and Lana Pavlova gave me some feedback. Jessica Breur, a Desmos fellow, checked in with me to let me know that she was interested in hearing how the lesson went. Wow. That is a ton of support. Great stuff.

Later that night, I played the Polygraph with my 8-year-old son. Playing with Max was really helpful because I got to see what kinds of questions he asked. Below is a screenshot from the teacher dashboard.  His sister was helping him type.

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At one point, he asked me if my card equaled 48. I said yes.

He said, “Oh! I know which one it is.”  Then, he paused. He looked at the cards that were left and started thinking out loud, “8 and 8 is 16. 16 and 16 is 32 plus another 16 is………oh no. That is 48 too.?”  This was a cool moment. Is it evidence of amplification? I don’t know. Would Max have had a similar moment if he and I were playing the same game with paper cards?  Maybe.

After playing with Max, I spent some time anticipating what questions other third grade students might ask:

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Then, I thought about how I might connect student learning.

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Finally, I felt like I was ready to try this Polygraph with a group of students and teachers.  I made a Google Slide presentation for our meeting the next day.  Looks great on paper. Then, reality set in.  We only had two and a half hours for the whole meeting. The meeting started at 7:50. The third graders had to go to Art at 9:00.  I wanted the teachers to try the activity first. I wanted them to anticipate student questions and brainstorm ideas for connecting student thinking. The third graders needed at least thirty minutes to try the task. I am not know for my punctuality. How the hell was I going to squeeze any meaning out of this meeting/lesson?

As soon as the teachers came in, I told them to sit with a partner and one computer, go to the website on the board, and type in the code. Each team was able to finish one round.

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We discussed what big Math ideas this activity might elicit. The teachers shared the types of questions they thought students would ask.  We briefly discussed some ideas for connecting student work. Then, we headed down to third grade. This is when things got interesting.

The kids got right into the game. They were familiar with the platform. They had no trouble signing in. Immediately, they noticed I had anonymiz-ed them.

“Who is Katherine Johnson?” somebody called out as they looked around the room.

I explained that I had used Desmos to randomly assign them fake names. They seemed fine with that and went back to their game.

The first round was really interesting. Many of the pairs picked the wrong card.  A few guessed correctly. Some of them didn’t complete the game. Take a look at some screenshots from the class. What do you notice? What do you wonder?

 

I notice that the questions Katherine Johnson asked were very different from the questions Pythagoras asked. Katherine was thinking about ten frames. Pythagoras was thinking about groups. I wonder how this difference in perspective impacted the game? I wonder what Pythagoras means by “ten frame”. I wonder a lot about what these students know and understand.  I am definitely grateful to have their questions captured on screen.

Throughout the class, we paused the game to check in about a few things. Every time I used it, the class erupted into a chorus of ,”Hey! What happened?”  During pauses, we had some brief conversations about which questions were helpful and why. As I  monitored, I conferenced with students about what they were thinking.  The students who were guessing took some time to think of a question. I used it as an opportunity to model what I might think about if I was a “picker” waiting for my partner to ask a question. I huddled with one group and whispered, “I am trying to think about what questions they might ask. What questions do you think they might ask us?”  Later, during the debriefing, the teachers mentioned the “pickers” had a lot of “down time” while they waited for questions to be asked. I wonder if down time is an opportunity to engage in the standards for math practice.

Towards the end of class, I paused everyone and asked them what they noticed and wondered so far:

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The consensus was that Polygraphs can be challenging. Here is why:

  • Sometimes, you write the wrong answer by mistake.
  • Sometimes, you  mis-understand what your partner is asking.

I asked, what can we do about these challenges?  One person said, “We can read the question twice.” I asked, “Is there anything else we can do?”

One girl spoke up, “You can think about what your partner meant.” I love this. Admittedly, I got pretty excited when she said it. I hadn’t anticipated it. I hadn’t thought about it before and she was spot on. Think about what your partner meant. Wow.  What if we all did more of that?  What if we really tried on each other’s thinking, not to judge it, but to get a different perspective.

It was time to wrap up. We were running late. I have been trying hard to not forget the lesson close so I tried to pull some of the pieces together. I asked everyone to turn and talk to their neighbor about what they think Sally meant when she said, “Some cards had the same number of groups, but a different number of dots in each group.”

Silence. No movement.  The kids looked at me with blank faces. I repeated my prompt.

Nothing. I realized that these kids were not sure how to think about what their classmate meant. I told them what I thought she meant.  Then, I asked Sally if I was rephrasing her thoughts correctly. She said I was.  I asked the students to talk to each other about what they thought she meant. A few of them seemed to be making some connections.

Finally, before we left, I asked them to see if they could find an example of what their classmate meant. Could they find two cards that had the same number of groups, but a different number of dots in each group? Here is what they came up with:

After we got back from the lesson, we discussed what we noticed and wondered. One of my colleagues, Abby, said she wondered if Sally meant something different from what I had interpreted. She wondered if Sally was referring to different arrangements of the same number of dots. When we were in the classroom, Sally seemed to go along with my rephrasing. She nodded and smiled at me, as if to say, “Sure. What you say sounds great.” Was I really listening to Sally?  Was I really trying to understand what she meant? Or did I hear something that sounded like the “big idea” I was looking for?

I don’t know.

During our debriefing, we discussed what we might do next with these students. We decided it would be meaningful to continue to use this Polygraph.

  • We could use individual cards as Number Talk Image prompts.
  • We could ask the students to sort the questions from the lesson into groups and discuss the characteristics.
  • We could play another game – the teacher vs. the students.
  • We could show the students the questions they asked and ask them what they would change.
  • We could continue to have the students play this game – maybe once a week and monitor what strategies they were using to find the total number of dots per card.
  • During the next round, we could look for opportunities to structure a Same, but Different Math conversation.

I am not sure if using technology amplified those third grade students understanding of multiplication, but it definitely amplified my own understanding of how to teach third grade students.

Trust Your Gut and Grow Into Your Heart

My twelve-year-old daughter couldn’t fall asleep. She was agitated and restless. I laid next to her, hoping my proximity would be calming. After a few minutes of tossing and turning, she said, “I don’t want to go to school tomorrow. I am afraid there is going to be a shooting.”

I waited.

Her words tumbled out in one long strand of jumbled terrors.  “My teachers aren’t going to be there tomorrow.  I will have two subs. A lot of our subs have grey hair. I had a dream once that there was a shooting and right before the shooting, I saw a woman with gray hair.  What if there is a shooting tomorrow?”

I wanted to say, “don’t go! Don’t ever go! Stay here with me and dad and your brother.  We will homestead and I will teach both of you. We never have to leave again. Don’t go. Whatever you do, don’t go.”

Instead, I said, “you don’t have to go to school tomorrow. You can stay home if you want.”

Tomorrow was going to be the last day before Thanksgiving vacation. She was immediately relieved by the notion that there was a choice. She wondered whether she would miss anything. I think she said something about a movie being shown in one of her classes. I told her she could decide in the morning. She fell asleep.

Later my husband confirmed that he would have said the same thing I did.  For the rest of her life, we are going to be telling our daughter to “leave if you ever feel uncomfortable”, “call us if something doesn’t seem right and we will come get you”, and “always trust your gut“.  Neither of us thought there was going to be a shooting at school the next day. We didn’t think our daughter was having premonitions. We didn’t think anything. We felt. We looked into our guts. Our daughter is feeling scared, vulnerable, unsure, and alone. What should we do? We decided we had to give her the choice.

The next morning, my husband and I whispered to each other as we peanut buttered English Muffins and reheated our coffee.

“Should we just keep her home?”

“If she stays home, he (my son) will want to stay home.”

“So let them both stay home.”

“What if she wants to go?”

In the end, we left it up to our daughter. She said she wanted to go.  She said she thought it felt okay.  I confirmed for her that I would be at her school in the afternoon. She could come say “hi” to me in my office. Dropping her off at school that day was one of the hardest things I’ve ever done.

On the way home from school that day, she brought up her dream. She said she was worried. She wondered, what if she could see the future? I told her I thought she was experiencing intuition. I tried to explain it, but I am not sure I did it justice. I told her intuition is made of feelings and experiences. It lives in a space where our heart and our brain intersect. It gets more finely tuned over time.  I tried to describe how intuition can be wrong or partially wrong. It can miss the mark sometimes. We can miss the mark. She didn’t say anything. I followed her lead.

I wonder whether our mathematical intuition lives separately from our other intuitions.  Over the past six years, I have gradually wandered deeper and deeper into the center of my math self. I came to terms with some pretty deeply rooted shame and insecurity.  I also discovered all kinds of knots and snarls in my intuition.  My whole life I was told that I was too sensitive, too emotional, too naive.  It was often suggested that I think too much.  I got these messages early and often in my life.

Eventually I started to own them. After all, they are true. I am sensitive, pensive, emotional, naive. Unfortunately, I came to think of those traits as something that needed to be changed.  It wasn’t until pretty recently that I have re-framed these traits as gifts.  I get it now.  I am a thinker. I question everything. I am always wondering. I trust easily. I am intense. I feel things deeply. It turns out, this makes me a natural mathematician.

My daughter and I have a lot in common.  She is intense, sensitive, innocent, driven.  However, she doesn’t see herself as a mathematician. Last year, she told me she hated math. This year has been hard for her too, but in a different way.  Her teachers have tried hard to establish a math culture where students feel safe to take risks. Her struggles have been with her peers and her self. She says she is tired of them finishing before her, asking her if she wants a hint or help. She is tired of not being “good at math”.  She cried one day and told me she was so sick of everyone else getting things faster than her. Through her angry tears, she said, “but I love my teacher. I told her about it. She is helping me.”

Sometimes, at night, the whole family sits in our big king sized bed.  My husband is reading or sketching (he is a carpenter). I have my big sketch book out and I am usually anticipating student thinking.  Lately, my 8-year-old son has joined me. One night, I was working on designing a Three Act Task about a ladybug. Max peered over my shoulder as I sketched number lines and tables.  He started to sound out the words in the problem at the top of the page:

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He thought about it for a while. He mumbled some numbers to himself. He manipulated his fingers. After awhile, he said, “Is it 84? Did she walk 84 cm in 1 minute?” Then he paused and said, “I don’t agree with my answer so far.” I asked him why? He wanted to borrow my notebook. I gave it to him and he played with numbers for awhile:

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Eventually, he figured out the answer was 96.  My son plays with math a lot. When we play Yahtzee, he likes to calculate his own score. It takes him awhile.  Usually, I go do other things. Then, after about 20 minutes, he will come find me and say, “I won.”

My daughter plays with math differently and much less frequently.  Sometimes, if she is in the right space, she will join me as I play with shapes. Recently, she made this:

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She notices details in nature.  She takes brilliant photos. Once when we were walking on the frozen pond, she found a clear patch of ice.  She and my son examined the intricacies of that frozen water patch for a long time.

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I can live with it if my daughter just doesn’t like math. I hope someday she will find a way to enjoy it, but maybe she won’t. Maybe it just isn’t her thing.  But, what if she doesn’t like math because she thinks she can’t do it?

Recently, I took a rare weekend nap. It was amazing. I dozed off reading a book and my husband took the kids sledding so I could stay asleep. When I woke up, I saw this:

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My daughter had found my new compass and pens. She must have curled up in front of the fire and played with them while I slept. My daughter is an amazing kid. She is in a really tough space right now. She isn’t a little kid. She isn’t a teenager. She knows she is growing up and that terrifies her. She is navigating all of the awkwardness and insecurity that goes with being a middle schooler, but I think she is also becoming aware of some pretty deep and scary truths about life: we can never truly know what another person feels and, eventually, we all die.

The other day, I was talking to a friend about how hard it is to watch my daughter feel so anxious and afraid. My friend asked me, “what is the best thing about Lily?”  I said, “She has an enormous heart.  She feels everything. It is her greatest gift and her cross to bear.  I  can only imagine what she will be like when she grows into her heart.”

 

 

 

 

I Just Took What I Needed

Recently, I have been thinking about how coaching isn’t something you do “to someone”. It is something you do with someone. I used to think coaching was about structures and protocols, but now I think the heart of coaching is in the conversations I have with my colleagues, the learning we do together. My colleagues push me. They ask me questions that I don’t know the answers to. I used to think I had to know the answers, but I realize now that the most important learning happens when we co-construct the answers together.

This blog post is co-written by Deb Hatt and I. This year, I am working with the math specialists in our district to design and implement a collaborative intervention model.

I am a math specialist at one of our elementary schools. Sarah and I have worked together for a long time. During Sarah’s last year in the classroom, we co-taught fourth grade. I work closely with the third grade teacher in my building. We meet weekly to plan together, and we co-teach daily.  I also work with some of the third grade students during the school wide intervention block for grades 3-5. This morning I showed Sarah this representation of a 3rd grade student’s work.

Screen Shot 2017-12-12 at 10.58.39 AMTake a minute to notice and wonder. What do you think this student is doing? 

Here is what Deb and I noticed:

  • The student is flexibly regrouping based on what he needs.
  • He knows how to keep track of his regroupings using expanded form.
  • He might prefer subtracting multiples of tens.
  • His regroupings are based on the properties of operations, as opposed to the place based structure of our number system.

Here is what we wonder:

  • How would this student use the same strategy without expanded form?
  • Will this strategy become cumbersome with larger numbers?
  • If he tries to use this strategy without expanded form, will he forget about the true value of all of the digits he is subtracting?
  • How do we honor student agency while simultaneously introducing other perspectives and strategies that highlight the useful structures of our number system?

Deb and I talked for awhile about what to do next. Neither of us knew the answer. I asked Deb what she was thinking about doing.

I talked to Carolyn about this strategy at length.  I really wanted her opinion as an experienced 3rd grade teacher.  We both thought it was fascinating, but had some qualms about what comes next.  We see a lot of place value understanding in Sam’s work in general, but we worried that might be lost if he tried to make his strategy look more “compact”, like the algorithm.  Would he remember that the 6 was worth 6 tens when he was presented with 38-6?  We thought it would be interesting to explore further with him.  

Now I’m thinking that I want to encourage him to continue using this strategy, but also use a strategy based on place value.  

I said, “Tell me more about why you want him to use the strategy based on place value.”

“We emphasize these strategies based on place value and I am wondering why,” I said to Sarah.  “I feel this strong urge to teach him place value strategies, but I’m asking myself why?”  I felt like using place value strategies helped me so much as a mathematician, but in that moment,  I was struggling to see why Sam would want or need to use them if this strategy is working so well for him.  

“I am wondering the same thing,” I said, “ it seems like we should teach him to use the place value structure, but I don’t know if I can say why. Let’s play it out. Why does he need to know place value strategies?”

“Well, I really want him to use numbers flexibly,” I started. “I don’t want him to only think about adding and subtracting what he needs.  We have this wonderful system of tens. The ten structure is so helpful when we think about multiplication, division, exponents. I want him to have flexibility with numbers, not just be confined to adding and subtracting small bits.”

I paused and then continued, “I think the part that gets to me is that I almost shut it down. We were talking about using the specific subtraction split strategy.  I had asked the group to try it, allowing that they could use another more familiar strategy to check their work once they had attempted this one.  When he was showing me his work, I saw that he had the correct answer. But then I saw how he had solved it and said, ‘Hold on. What did you do here?’  

He said, ‘I just took what I needed.’”

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“At this point, I almost said, ‘Well, we’re trying this new strategy today.  Can you show me how you would solve it using subtraction split?’ I stopped myself. Instead, I said, ‘Tell me what you did here.”  I thought I understood pieces of it, but I didn’t know exactly what he had done. After he had explained his thinking, I took it and said, “This is really cool!  I am going to have to think more about it.”  I really needed time, without other kids in the room, to really think about how he had solved the problem.

I think you showed a lot of respect for Sam and his work.

“That is my big thing this year,” I responded, “ I teach these kids and they are so worried about math. I have fourth and fifth grade students who have gotten a message, somewhere, that they are not good at math and their ideas aren’t as good as other people’s ideas. I don’t want to perpetuate that at all.”  I paused. “I was so close to screwing it up.”

“But you didn’t screw it up. I so appreciate your honesty, Deb. I have felt the same way so many times. I see myself in you. Don’t be so hard on yourself. It is what you told me the last time we reflected together.” 

Deb and I continued to reflect on how we have changed as teachers and learners.  We used to think teaching and learning math was all about decision making.  We were always thinking about which step to do next, following a prescription.  We never used our intuition.  Now, we are re-training ourselves to build and use our intuition.  We look towards ourselves, our students, and each other to figure out what to do next.  The decision making process is much more complex.

As we were wrapping up our meeting, I said, “I’m very glad I caught myself and listened to Sam. I am thinking a lot about how I introduce strategies. There are some strategies that lend themselves to students discovering them on their own. I have found that subtraction split isn’t necessarily that type of strategy. But is a strategy that I have found helps many students when they learn it. I think I would like to reframe the way I introduce strategies. I think I need to present new strategies as more of an invitation, instead of a prescription.”  I told Sarah that I was so grateful that I was able to take the time to think about Sam’s strategy on my own and reflect with both her and Carolyn about our next steps.  I’m learning so much about being a more reflective, responsive teacher from both of them.  

Later this afternoon, I was talking with Abby, one of our other school based math specialists. I was relaying the conversation that Deb and I had. Abby agreed that Sam’s strategy was unique. I asked her what she would do next. She wondered if we could compare and contrast Sam’s work with a student who used the place value structure. We could ask him what is the same?  What is different? Both strategies are decomposing and finding equivalent ways to represent the subtrahend.  The only difference is that Sam is only taking what he needs. He is taking smaller amounts.

She also wondered whether Sam was actually using the place value structure, but in a different way. I hadn’t thought about that before. She said her gut says this kid is thinking flexibly about numbers. It might not be a far leap for him to find the commonalities between his strategies and the place value strategy.  

I am so grateful for Deb and Abby. They care deeply about the students they work with. They show so much respect for student thinking. They push me to question what I think I know about teaching and learning. I just love this strategy that Deb shared with me. It is like a little nugget of truth. What if we all just took what we needed?

 

I am Those Kids.

Yesterday, after I taught math in Kindergarten, I went into my colleague’s office and cried. I didn’t see it coming. I was not prepared for it. Once I started, I had a hard time stopping. My friend Deb came in and asked, “what is wrong?”

“I still carry shame about learning, school and Math.”

“What happened?”

I told Deb. “It just sneaks up on me. I think I am over all that shame baggage, and then it comes back. I have just been thinking a lot about “those kids” lately. This past week I have worked with so many kids who hate math or think they are stupid.  Kids who don’t fit into “the mold” for one of a million reasons; because they don’t think like we want them to, they don’t learn fast enough, they don’t learn in a straight line, and maybe they can’t remember things or they mix things up.

“I am those kids!” I started crying again. “I was an English Major. I hated math. I was miserable at it. It was my worst nightmare. I got the message that I was not smart.  But, here I am! I am a district math coach! That’s great, right? I moved past it. I am in love with math. I am okay with confusion. I am drawn to things I don’t understand. I can be a role model, right? But still, deep down, sometimes, there is that little voice that wonders, ‘what if I’m not good at math? I know. There is no ‘good at math’. I know that. My rational brain tells me that ‘good at math’ is a farce, but sometimes, I just let that self-doubt creep in.

Last week I was in a high school math class. I asked the kids to tell me some words that described how they felt about math. They told me, ‘it is hell’ and ‘it sucks’. Some kids actually said things that were positive which was awesome.  One girl told me math was fun if she was baking. But another girl told me it was loathsome.  When I asked her to think of a word that described how she wanted math to be, she said, ‘tolerable’. That is it. That is all she wants; for math to be tolerable.

How do you feel about math class?

I told her, ‘I want to show you that Math can be inspiring.’

She said, ‘numbers don’t inspire me.’

I asked, ‘what inspires you?’

She said, ‘poems’.

I told her I was going to challenge myself this year to help her find math inspiring.

Then, last week, when I was in 5th grade, you know the blog post I wrote? Those boys have confessed to hating math in their lives.  They have told me more than once that they can’t do it. Those boys are amazing thinkers! Seth’s mom texted me after that math class and told me he talked about math the entire ride home!

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Max, the other boy, asked Mrs. G and I , ‘why can’t math class be like this all the time?’ We said, ‘it can! You just need to keep asking questions.’ That is awesome, right! I should be happy about that. I am happy. I just had a hard time this weekend, while I was doing the math. Those boys inspired me. They inspired my friends on Twitter. We spent all weekend trying to solve problems about repeating decimals. I kept thinking about Seth and Max and their questions. At one point, I couldn’t tell where my thinking stopped and Max’s began. But, I made some mistakes. The people I was working with seemed to ‘get it’ a lot faster than me. I asked questions, that afterwards, seemed obvious.

(WARNING: UNFINISHED THINKING BELOW)

 

I am not sure what happened, but I started to doubt myself, somewhere in the middle of all those decimals.

After we taught Kindergarten, I came in here, composed a tweet, and started to cry.”

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I think at this point, Deb said really nice things about me.

I know those nice things are true.  I know I am good at my job. I am pretty sure I inspire people to think differently about how they teach and learn math. Maybe it is not a bad thing that I am so sensitive to feelings of inadequacy.  Sometimes, I think it would be better if I could block them out, but maybe it is okay to let them in. I just need to remember:

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-Becoming the Math Teacher You Wish You’d Had by Tracy Johnston Zager

So, I need to share one more story with you. Remember the Kindergarten class that I mentioned above?  Deb, Katie, and I have been meeting monthly to plan and teach a lesson together.

We are trying to learn more about how kindergarteners learn to record their thinking. Today, we decided to introduce a body sized ten frame. ( I forgot to take a picture of it before I left it with the kindergarteners so here is a picture from last year:)

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These kids haven’t done a ton of work with ten frames, yet, so it was really interesting to hear what they noticed:

  • It is pink and blue
  • I notice squares.
  • I see lines.
  • I see rectangles.
  • It looks like a window.
  • The whole thing is a square cuz the long lines and short lines and long line and short lines.
  • No, it is a rectangle because this side is 1,2,3,4,5,6,7 seconds (starts crawling around the perimeter of the ten frame) and this side is 1,2,3,4,5,6 seconds, then this side is 1,2,3,4,5,6,7 seconds again and this side is 1,2,3,4,5,6 seconds. Because one way is shorter and one way is longer.
  • He noticed it is a square and there is one long second and one short second.
  • You could put Xs inside the boxes.
  • Yeah, you could put X, X, X, X, X, X, X, X, X, X
  • You could do jumping jacks on them.
  • I notice it is the yellow brick road because I was the Wizard of Oz for Halloween.

And here is what they wondered:

  • I wonder if you could put them together to make a circle.
  • I wonder if you could make the pink or blue a different color.
  • I just noticed that there is ten squares of pink.

Listen to what happened next:

When I listened to the whole recording, I  heard a boy in the background saying, “1,2,3,4,5… 1,2,3,4,5”

I will call him Colin. Remember him.

After we introduced the body sized ten frame, we handed out collections of shapes. All of the collections were less than twenty.  We invited the students to use paper plates, cups, or ten frames to help them organize their count, if they wanted to.  We also gave each of them a recording sheet. Then, they went to work.

As I was circulating and chatting with students, I came upon Evan.  Evan’s shapes were scattered across the table. His recording sheet was on the floor and he didn’t have any organizational tools. I asked him, how many shapes do you have?

He said, “I don’t know. I can’t do this. I don’t know how to count.”

I said, “I think you can do this. I wonder if one of the organizational tools would help you.” I invited him to visit the table and see if any of the tools interested him.

He came back with a paper that had three ten frames printed on it. He sat down and began to put one shape in each of the squares. When he finished, he looked up at me and smiled:

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I asked him, “how many shapes do you have?”

He gestured with his hand, sweeping it down the entirety of the full ten frame and said, “one full rug”. Then, he quickly waved his hand over the rest of the shapes and said, “and one half a rug.”

I smiled. I couldn’t help it. I am miserable at maintaining a poker face. I said, “You do, don’t you! You have one full rug and one half of a rug.”  At this point, I got the attention of the other friends at the table. I asked Evan if he would share what he discovered. Then, I asked Evan’s friends, “what do you think? Does Evan have one full rug and one half of a rug?”

They said, “Yes! He does.”

Then, I asked Evan, “how many shapes is that?”

He counted sixteen because he forgot the number thirteen.

I said, “I think you might have forgotten the number 13. Let’s count together and see what we get.”

We counted together, Evan smiling the whole time, and got 15.

“So,” I said, “When someone asks you how many shapes you have, you can say 15 or you can say one full rug and one half of a rug. That is pretty cool. Do you mind sharing that with the whole class when we come back to the circle?”

Evan beamed. He didn’t mind at all.

I walked away and was about to call everyone back together. Then, I spotted Colin’s paper:

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I asked Colin, “how many shapes do you have?”

He tried to count them, but he got 24.  He counted to 11, but then continued with 20, 21, 22, 23, 24.

I asked Colin to look at Ethan’s shapes and tell me what he noticed.

He said, “We have the same amount!”

I asked Ethan, “How many shapes do you have?”

He said, “one full rug and one half a rug.”

Colin said, “Yeah! I have that too!” He gestured with his hands, just like Ethan did and said, “one full rug and one half a rug”.

I asked Colin if he would mind sharing what he noticed during our closing circle.

Originally, I thought I was going to facilitate the closing circle to highlight the ten structure. Ethan and Colin threw me a curve ball. They were thinking differently. They were thinking about one whole. I changed my plan. I drew models of Ethan and Colin’s ten frames on the board. I asked Ethan to tell us what he found out. He said, “I have one whole rug and one half of a rug.” Then, I asked him, “how many shapes is that?” He counted 15. I wrote it under his ten frame.

Next, I asked Colin to share what he learned. He said, “I had one rug and a half a rug.”

I wondered, “How many shapes is that?”

He counted, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 22, 23, 24”

I said, “you told me that you and Ethan had the same amount of shapes. It looks like you got different amounts. What do you think about that?”

“We do,” he said, “one whole rug and a half of rug.”

“Yes! I see that. You both filled in one rug and one half of a rug. Ethan says that is the same as fifteen shapes. You counted 24 shapes. Can we count your shapes again?  We can count together.”

I wrote the numbers 13, 14, and 15 next to three of the dots. I am not sure if this was the correct thing to do because I might have further confused the matter by assigning a dot a number name, but I was trying to give him a visual reminder.

I said, “Let’s count together.  We choral counted to 15.”

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Then, he said, “I only need five more! I only need five more until I get to….” He started counting again from the beginning. “I only need 5 more until I have 24!”

At this point, Katie asked the students, “how many shapes do I have if I fill one of the rugs?”

Some of the students started counting. Some of the students were spinning around. One student was very busy trying to convince Colin that he counted wrong. It was 1:30 on the Wednesday before Thanksgiving vacation. They were done. However, right before we pulled them back together to close up, one little friend said, “ten”. “There are ten shapes in a whole rug.”

I thanked Ethan and Colin for sharing what they discovered. I said “we learned that there are ten boxes in each rug and that the rug can help us organize our counting.  We also learned that there are different ways to count. Many of us were counting each shape, one at a time, but Ethan and Colin showed us that we can also count by the number of rugs we fill. We learned that 15 shapes fill up one rug and one half of a rug.”

On my way out, Katie approached me. She said, “Thank you. Thank you for highlighting Colin and Ethan. They really struggle with counting. They felt so good about themselves today. I am so glad that I got to see what you saw. They counted!”

 

 

Smaller, Bigger, or More Precise: Refining Our Internal Truth Detectors

Yesterday, I got this text:

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It was from Mrs. G, the teacher I worked with on Tuesday.  I really wanted to check in with these kiddos and talk with them about these questions.  Today, I stopped by to ask them to tell me more.

First, I checked in with Ms. G. She told me that these questions came about when the class was doing a choral count.  She said the class was counting by hundredths. When they got to nine hundredths, someone suggested the next number might be one whole. Then, the students had a conversation about how it wouldn’t be one whole. It would be ten hundredths or one tenth.  They continued counting until sixteen hundredths. At this point, Max said, “wait! When we are doing this, are the numbers getting smaller or larger?” This question prompted a different question from Seth, “If whole number places can go on and on forever, can decimal places too?”

At this point, Ms. G wrote a long decimal up on the board and asked, “can I do that?”  Charles said, “I think that number would still be between 2 tenths and 3 tenths.”

Max responded, “wait! Can you just keep putting places because once you get ten of them, it is going to go into the next place and once you get ten more it will go into the next place and on and on.”

After Mrs. G caught me up, I turned my attention to the kids. I asked them if we could talk about this number again.

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Max immediately jumped in to the conversations. He said, “yesterday, Seth asked if decimals go on forever and I was the only one who said, ‘no.’ I don’t think they would go on forever because eventually they would all add up to one.”

I wasn’t sure if I understood what Max was saying so I asked him to explain it. Listen.

 

“That is why I don’t think that you could count forever in decimals because I think eventually this will add up to one like I did here. I think that wouldn’t work. Eventually, no matter how big the number is, if you’re still adding, eventually, even if it takes years, it will eventually make one whole.”

At this point, the students went back to their Social Studies lesson. Yes, it is true. I totally interrupted Social Studies to revive a math lesson. I love Social Studies, but sometimes, I think it is okay to Drop Everything and Do Math.

Mrs. G and I went over to the kidney shaped table to reflect a little. Mrs. G took some time to share the back story of these questions. She explained what happened the day before. Listen.

I wondered, what is Max’s claim?  Is he claiming that decimal numbers DON’T go on forever or is he claiming that all unit decimals (is this a thing?) will eventually add up to one whole? Mrs. G and I wondered how language was impacting our conversations with Max.

Mrs. G said, “I think I kept saying “adding” a place value.  Can we keep “adding decimal place values”? Max is hearing the word ‘adding’. Maybe he is thinking about counting as adding.  We decided to ask Max a few more questions about his claim, but try to use more precise language this time.

When Max sat down, I said, “I want to try to understand the question you are asking.”

He said, “Well. I only half understand it myself.”  Have I mentioned, yet, how much I absolutely love this kid?

I tried to rephrase Max’s claim without using the word ‘adding’. Listen to the conversations:

I have listened to this clip several times and I wish I had done something differently. When Max says, “so you are just adding place values. You are not adding the numbers one by one.”, I wish I had not said anything. I wish I would have waited and let the magnitude of his statement settle into the silence. 

Max goes on to rephrase his claim. He says, “no matter how small the number is, you are eventually going to get to one whole, no matter how long the number is, even if you give up, if you didn’t give up, eventually it will go back to one whole.”  

Now that I think I understand Max’s claim, I am wondering how it fits in with Seth’s original question about whether or not decimals can go on forever.  Listen as Max invites Seth into our conversation:

So, at this point, I am still wondering about what Max is disagreeing with. When he is talking to Seth, he says, “And I asked, without it going into wholes? Did you mean adding?” These words make me wonder if Max is still talking about the cumulative addition of unit decimals, as opposed to the literal writing or naming of a decimal number.

I told Max that I was still unsure. As we talked, I wrote the number below. Listen.

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Next, Max catches me off guard. He is thinking so fast, I have a hard time keeping up with him.  I was trying to see if we all agreed that I could keep writing digits forever. However, I got lazy and just started writing zeros. Well, that added a whole new layer to the conversation.  Max didn’t miss a beat. Listen.

This kid is thinking so fast and so deep that I can’t keep up.  I started using the word “adding” again which didn’t help with clarity.  Fortunately, Max persevered and straightened me out, at least as far as the whole “zeros question” goes.

I was still unsure about whether he thought decimals could go on forever. He keeps bringing in these other nuanced constraints: “without it going into a whole”, and “you have to count with decimals by one.”

I asked him, “what if I didn’t write zeros. Couldn’t I just keep writing digits forever?”

He said, “That is not correct. You’ve got to do one and then another plus one to make a zero. You can’t just add ten numbers at a time or seven numbers at a time.”

At this point, I tried to sift through what I thought were two different claims- one about writing/naming decimals and one about counting/adding decimals. Listen:

I am not totally sure we all ended up on the same page about understanding our claims, but this conversation with Seth and Max was one of the highlights of my career.  I could probably spend the rest of my day just reflecting on this conversation.  These boys pushed me to think differently and to try to truly understand them. What if we all did more of this?  What if we dropped everything and did math? What if we dropped everything and listened to understand each other’s thinking?  I am so grateful for these boys and their amazing thoughts. I tried to conclude our conversation by letting them know how much I appreciate them. Listen