One Group of Itself

Have you ever thought about how difficult it is to truly understand another person’s perspective?  I think about it all the time, especially when I am working with students. As a facilitator of learning, it is my job to interpret what students are thinking. It seems so attainable when I type the words, but my experience tells me that the quest  is actually quite elusive because interpreting is different from judging and thinking is often different than saying.

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Last week, I showed this picture to a group of fourth graders and asked them, “what do you see and how do you see it?”  I chose this image because I was hoping it would prompt students to use strategies and numbers that would encourage an exploration of the associative property.

I went into the lesson trying to capture two strategies that we could compare and connect during our next math class. I stumbled upon something else.  As always, my students got me thinking more deeply.  This time it was about units.

J was the second student to share his thinking. He doesn’t usually volunteer to talk. In fact, when he first came to this class, he hardly talked at all. It is often difficult to truly understand what J is thinking and I have to tread carefully when he shares because I don’t want him to shut down.  As soon as I saw his hand up,  I knew it was important honor his initiative.  When he started to describe that he saw 9 by 12, I almost assumed that I knew what he meant. Fortunately, the little voice in my head told me to shut up and listen because my assumptions would have been wrong. Check it out:

Screen Shot 2017-03-19 at 9.33.54 AM

Did you hear what he said?  He said he saw 9 by 12 and his classmates and I assumed he saw the larger rectangle made up by the holes in the pretzels, but when he described what he saw in more detail, what he really saw was 9 pretzels, each with 12 holes in them.

After J shared his strategy, several other students shared how they saw the pretzels. Here’s what they said:

  • “I see 3 by 3.”
  • “I see 9 three by fours.”
  • “I see twelve holes in each pretzel and 9 pretzels.”
  • “In each pretzel, I see three groups of four.”

One student said she saw nine 1 by 12 pretzels. She struggled to articulate it. First she said 12 times 1. Then, she said 1 times 12. Finally she settled on nine 1 by 12 pretzels.  Then, a different student explained, “I see one group of itself. I see 1 times 12 in E’s pretzel.” Again, they are mucking around with this word “by” that they have heard and are now trying to make sense of. Is she wondering about the relationship between the individual pretzel and the larger picture of pretzels? I wish I had asked her to tell me more.  After listening to her, I see something new: I could see one group of 12 nested inside a larger group of 12 if there were 4 pretzels in each of the three rows. I could see one group of itself.

I asked each of the students to show us how they saw what they saw. I had intentionally chosen this image because I was hoping students would use three factors to describe it. I asked one more time if anyone saw it a different way.  One student said he saw three, three by twelvesScreen Shot 2017-03-20 at 1.47.31 AM

Another student said he saw three, four by nines:

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The last student said that he saw 88 holes. He went on to describe how he calculated 88. He said “I knew that five 12s is 60 and ten 12s is 120 so if I take away one of the 12s… oh wait. I think there are 108 holes.”  I thought is was interesting that he was the only student to calculate the total number of holes.

We ended our number talk here.  I did use two of the strategies to plan a number talk for a 4th grade learning lab the next day, but that is another blog post. Right now, one weeks later, as I sit and listen to the number talk, I still wonder so much. I wonder about the word “by”.  I have always associated the word “by” with measuring area.  If I am shopping for  a rug that is 5 by 8, I am looking for the rug to be 5 feet long and 8 feet wide. J and E were thinking about the word “by” as a general way to describe a multiplication situation. We had some trouble interpreting what they were saying.  We had to navigate from the math to the pretzels and then back to the math again.

I used to think Number Talks involved me bringing the numbers and my students operating with them, but now I realize that there is so much value in letting my students bring the numbers to the Number Talk.  By starting with an image, I put the students in charge of using math words and symbols to describe what they see. Will they describe it as a relationship or a quantity? Or maybe a relationship between quantities? Starting with an image also lowers the floor and raising the ceiling of the task.  I am not assigning an operation to be used. The students choose which operation will help them make sense of and describe what they see and, as they describe, I can listen and try to interpret what they understand.

I am not saying I will never use numbers in a number talk.  I am just enjoying the opportunity to change my perspective, push my thinking, and be slightly uncomfortable with the math I think I know.

I used to think the properties of operations were all symbols, numbers, and rules, but they are so much more than that. I think, maybe, they are more like filters with which we interpret and explain our world. I have often heard people say that math isn’t “breakable”.  At first, I thought they meant that Math was impenetrable, predictable, solid. I never really felt comfortable with that definition. Now, I wonder if they meant something different. I am realizing that math is so much more bendable than I ever knew.


Thank you Christine Newell and Number Talk Images for the pretzels.

The Sound of Not Settling

Last night, at 10:00 p.m., I left the Phoenix airport.  Eleven hours and three airports later, I arrived at my home in Maine.  I made myself a sandwich and a cup of coffee, changed my clothes, brushed my teeth, and headed to school.  I was exhausted. I had a headache.  My brain was pretty foggy.  I was scheduled to meet the 6th grade teachers at one of our elementary schools at noon.  They were going to observe me do two number talks in two different 5th grade classrooms in two different buildings.  All I could think was, “What the hell was I thinking? These are going to be the worst number talks ever.” I tried to plan them on the plane ride, but I kept falling asleep.

I was so tired on the drive to Wayne that I had to turn the music up really loud to keep me awake.

I was late. I was kind of on the verge of tears – not because I was sad- but because I was just so exhausted – physically and emotionally.  I was tapped out. How was I going to model a successful  number talk when I could barely keep my eyes dry and open?  On the way into the building, I started to berate myself:

  • Why do I always schedule too much?
  • Why don’t I leave more time to plan?
  • Why I am always late?

Fortunately, it was a short walk to the building.  When I arrived at Sue’s room, she looked up and smiled. I said, “I am so sorry I am late.” She told me not to worry about it and I could tell she meant it.  The sixth grade teachers spilled in behind me. I introduced everyone.  I asked the kids if they would like to do a number talk with me?  They were all smiles.  Sue walked over to the white board and grabbed a marker. She asked, “do you want me to record?”


I showed Sue the number string I was thinking of using and told her we could go as far as she thought we should. She said she thought it was a good fit and wrote on the board:

Screen Shot 2017-03-21 at 9.01.47 PM.pngQuiet thumbs.  Thoughtful eyes.  Squished up faces and moving lips. I felt myself breathe.

“What do you think?” I asked.

Somebody said, “4.”  Lots of agreement.

“Did anyone get a different answer?”


“Who wants to tell us how they solved it?”

P nodded and smiled.

Sue said, “do you want to come up and show us or would you like me to record for you?”

“I would like to record it. I used the number line,” he smiled and looked at Sue “Mrs. Hogan has been working on this one. It isn’t her go-to strategy.”

I asked him if he could tell us what he meant.  He looked at Sue and she nodded. She said, “it is okay. You can tell them.”

He continued, “the number line is a challenging model for Mrs. Hogan because she didn’t learn to divide fractions using models.  She likes to use the bar model, but lately she has been stretching herself.”

I asked him, “do you like using the number line model to divide fractions?”

“Yes. It is kind of my go-to strategy. It makes the most sense to me, but I can use other strategies too.”

“Cool,” I said. I sat down on the floor beside M so P had room to share. He drew a number line and told us, “You see.  I divided the two wholes into halves.  So I have four.”

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“Four what?” I asked.

“Four groups of one half.”

I asked if anybody else wanted to share a different way that they came up with the answer.

W drew a bar model.  M drew two rectangles and divided them into halves.

We moved on. Next we tried two divided by one third.  All the kids got six for an answer. A couple of different kids shared their thinking.  I asked them if they wanted to try something a little more challenging. I said, “I kind of want to find a problem where we get some different answers. So we can argue about them a little.”

Someone chimed in, “politely argue.”

I checked in with Sue about the next problem. She thought it looked great.  She said, “This one will be interesting.”

She wrote:

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It was quite for awhile. The faces were more squishy. Somebody said, “hmm.” I said, “I am having a hard time doing this without a white board.  Does anybody else need a white board?”

I don’t think I was finished asking the question before everybody was grabbing white boards and markers out of the container.  I got to work.  After a few minutes, I looked up and noticed several kids had started to check in with each other.  M was sitting next to me. She had drawn three different models on her whiteboard.  Each one had shown the answer as 4. Her board kind of looked like this:

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She said to me, “I am trying to use an area model, but it is really hard to use an area model to solve fraction division problems.” I looked at her white board. I think she was referring to the diagram in the lower right. I sidled up next to her.  I told her, “I struggle with using the area model to show division problems.  I have a hard time remembering which part of the model is missing.” She looked at me with the biggest smile and said, “you just made me feel really good. I don’t know how to do something and it felt good to hear that you didn’t know how to do it either.  You know how that happens sometimes?”

“I do.”

At that point, she looked at my white board. I had drawn a number line. She said, “you got 6 for an answer and I got 4.”

“Yeah,” I said. “I noticed that too.”

She stared at my work for a little while. She said, “Your answer makes sense to me, but so does mine.”

I asked her to tell me about hers.  She said, “Well, I drew four wholes and then I shaded in sections of two thirds.  There are 4 sections of two thirds.”

At that point, Sue called everybody back to the rug.  She said, “I noticed that some people got different answers. I heard a lot of you talking about your answers and trying to help each other see whether or not they made sense.”

W said, “At first I thought the answer was 4 twelfths, but then E helped me figure out what I was doing wrong.” He walked up to the board and started to draw a long skinny rectangle.

“Oh!” K added, “You forgot about the whole.  You thought it was 1, not 4 wholes.”

“Yup.” W divided his four wholes into thirds and colored in four two-third sized pieces.

Screen Shot 2017-03-21 at 9.56.11 PM.pngM said, “I got 4 too, but not twelfths.  I got 4 groups of two thirds.”

Now, somebody else chimed in.  She said, “you can make more groups of two thirds.  You can put those thirds together.”

M said, “Ooohhhhhhhh! It is six!”  She started coloring in her extra thirds to make two more sections of two thirds.”

“Yes!”  W agreed.  “That is exactly what I did.” He proceeded to show us how to combine the extra thirds in his diagram.

At this point, Mr. Getty, one of the visiting sixth teachers asked the kids, “How do you know which strategy to use?”

Somebody said, “it depends. Sometimes you use a strategy because you are really comfortable with it so you are pretty sure it will work, but other times, you use a strategy because you are trying it out. Maybe it is hard for you but you want to try it to understand it better. Kind of like Mrs. Hogan with the number line.”

M looked at me and said, “or like us with the area model.”

I smiled.

The discussion went on to include how some kids like to use multiplication to check their answer or how it is helpful to make the groups of two thirds next to each other in the picture.  We muddled through some stuff.  We noticed a pattern that if you are dividing a whole number by a unit fraction you can multiply the whole number by the denominator and get the answer. Sue told us the kids have noticed this before.

“But!” somebody countered, “That won’t work for 4 divided by two thirds because 4 times 3 is 12.”

“Hmmmm,” I said. “Maybe we should try this out for awhile and see what happens.”

On the way home, I was debriefing the lesson in my head.    I wished I would have asked the kids if they ever chose a strategy based on the number. Nobody saw a relationship between 2 divided by 1/3 and 4 divided by 2/3 – including me because I didn’t take enough time to plan.  But how about M telling me I made her feel good!  That just made my week. I wondered if I let the number talk go on too long and should I have suggested whiteboards? But W got up on his own to tell his whole class that he got the wrong answer!  That kid has come so far in 5 years. I wondered if it was okay to use the language “groups of” when we describe dividing fractions. But Sue!  How about Sue?  Three years ago, when I walked into her classroom, her kids were sitting at desks lined up in rows and listening to her explain how to do the algorithm while they waited patiently for there practice worksheets.

We laughed about this at our debriefing meeting with the sixth grade teachers.  They asked her, “what made you change?”  She said, “I stopped being scared. I found a place where I felt safe to take a risk and just be honest with my kids. I tell them, I am learning too and sometimes I have to go back to the algorithms because it is all I know, but I never stop at the algorithms. I always keep trying other strategies and models because that is how I learn to understand the math better and I love it!”

And that made me feel really good.


What is Algebra? (continued)

Last Thursday evening, I met with the K-12 collaborative group to discuss the most recent task we had taught. Our K-12 group meets once a month. We choose a low-floor/high ceiling task, do the math together, plan how we will adapt the task for our students, teach the task, and then reflect on the lesson together. The task we were debriefing is from the Georgia Math Curriculum. The task is based on a visual pattern that looks like this:


The first thing we did was look at the content standards for our grade level and reflect on how we may or may not have addressed them during our lesson. I chose to try this pattern with first grade students. It was not my finest teaching. I am not upset about it. My comfort zone is grades 3-8. It was a real stretch for me to try this task with first graders. That is why I did it. I learned a ton. I haven’t had a chance to write a separate blog post about it, but you can read Jamie’s, Abby’s, and Sue’s if you want more background about the task.

Here are my thoughts about my lesson:


You can take a look at everyone else’s reflections here. We were able to introduce the task at every grade level, K-12, including a Special Education classroom and a Life Skills classroom.

Next, we shared our thoughts.  Our conversation reminded me of the conversation I was having with third graders in Cassie’s classroom on Wednesday. Jaime started the conversation by sharing a question that one of her third grade students asked, “Can you ever have a problem with two unknowns?” Listen to where the conversations leads:


In my last post, I described how several third grade students were wondering about Algebra.  What is it?  Do we do it or is it something that lives at the middle and high school?  Is it just a bunch of expressions with letters or boxes or smiley faces in them? Talking with my colleagues about our lessons left me wondering about how I would define Algebra.  Maybe define is the wrong word.  I don’t want to look it up. I want to own it.  I want it to be alive for me, like it was when I was talking with my colleagues.  I want it to be alive for my students.



What is Algebra?

Last Wednesday morning, I texted Cassie and asked her if I could do a number talk in her classroom. I wanted to see if I could use number talk images to create spontaneous true/false statements out of student thinking. Cassie welcomed me into her classroom.

When I got there, Cassie  asked me about two of the questions on our benchmark assessment.  She had given it to her students and they hadn’t done well on these two questions. She asked me if we could talk about how she might revisit some of the concepts that the questions were supposed to assess. These are the questions:



I have seen this assessment many times. I helped create it. I think these particular questions are actually  Smarter Balanced released items. They are supposed to assess 3.OA.5 and  3.OA.6.

While I stood there, in Cassie’s room, waiting for her students to come back from music,  I listened to Cassie explain her concerns. She wondered:

  • I haven’t presented letters as unknowns. Do third grade students need to use letters to represent unknowns?
  • Are my students struggling with the language “the product of 7 and 9”?

I wondered:

  • What are these questions actually assessing?
  • Are these good assessment questions?
  • Does the language have to be so cumbersome?
  • Do these question offer an opportunity to engage in the math practices?
  • Should we be assessing these standards in isolation?
  • How do our benchmark assessments help and/or hinder our instruction?

She asked me if we could adapt my Number Talk to probe her student’s understanding of unknowns, products, expressions, and equivalence? I thought that was a fantastic idea.

Quickly, we brainstormed how we could adapt the lesson. First, we looked at the standards and the assessment and formed a guiding question. We asked ourselves, what do we really want to know about student thinking?  This is what we came up with.


Then, we came up with some more specific questions that we hoped would get the students talking.

  • What are some math symbols that we use?
  • What do the symbols mean?
  • What are some words that we are using in math?
  • What do the words mean?

We started class by handing out sticky notes and asking students to share their thinking about the following question:screen-shot-2017-02-15-at-9-29-27-am

In my quick sweep of the poster, I noticed that no one mentioned any symbols for “unknowns”. I decided I was going to ask about unknowns.  I started the conversation by listening for these answers.

I was NOT listening for this answer:


I was going to dismiss the smiley face because it wasn’t the answer I was looking for.  Boy was I wrong.

Watch what happened:

When I watch this video, I think I can see myself listening for understanding.  It happens when the talking shifts from me to them.  They start wondering, debating, conjecturing:

  • “You can use anything for an unknown, except for a number.”
  • “Actually, you shouldn’t do letters for an unknown.”
  • “You can do letters.”

I am listening for understanding, but, in the moment, I don’t know what to do with their thinking.  I don’t even know which poster to write it on.  All I know is that I should try to follow it and write it down somewhere.  I have watched this six minute video clip at least five times.  Every time I watch it, I notice something else. Then, I wonder something else.  Here are just a few examples:screen-shot-2017-02-15-at-2-44-39-pm

I decided to tell the students that we would come back to their ideas and questions. I wondered if the students would make connections between their questions about Algebra and our number talk.  I wasn’t sure how to answer their questions. When Cassie and I planned the number talk,  we tried to anticipate how the students would approach equivalence.  We chose to use images of eggs. You can see our plans here. We started with a dozen eggs. When I projected the image, kids just started sharing what they noticed and wondered:screen-shot-2017-02-15-at-3-19-01-pm

“Why are there shiny ones?”

“Why is there a bigger one?”

“Why is there green and red ones?”

“Those aren’t green. They are brown.”

Then, someone noticed that it was an array. This prompted other students to bring the numbers to the table.


One student said, “Yeah. One plus one equals two. Two by six. You are just using the two since the two is broken up, you are just adding the one and the one for the two.”

When I heard him, I thought he might be trying to explain the distributive property. Was he thinking about two groups of six? I was going to use the opportunity to connect what he was saying to the expression (1 x 6) + (1 x 6), but another student joined in the conversation and it went in a slightly different direction.

When I watch the video, I wonder what would have happened if I had asked the original student to record his own thinking.  It is so challenging to record someone else’s thinking. Was I really capturing what he was thinking when I represented the decomposition of 2?

As I listened to the students share their thinking, I was trying to find someone who did not have to compute both sides of the equation. Was there anyone who was seeing that both sides were equal – using the properties – without having to do all the arithmetic?  After I listened in on a few conversations, I noticed that Cassie was having a very interesting conversation with two boys.  Listen in:


Two by six.

We showed the students another image and discussed their thinking:

Screen Shot 2017-02-21 at 7.08.52 AM.png

Then, we showed them this:

Screen Shot 2017-02-21 at 7.09.32 AM.png

We asked them what they noticed and wondered. Without missing a beat, many of them started talking about how there were probably 3 cartons of eggs under the grey rectangle. We asked them to use numbers, symbols, or words to convince us there were 3 cartons. They did not have a problem with this:

Most students used multiplication to justify their thinking. One student used addition.  As a whole, the students seemed to have a pretty good understanding of all the ways you could decompose 48. Most of them were more comfortable with adding partial products (distributive property) than multiplying partial products (associative property). One student tried to use the associative property but I think she lost track of how many times she had actually multiplied 12. She is thinking about “groups of groups”, but her understanding is incomplete. It would be helpful to ask her to find the expressions in her picture.  She will probably realize that she has too many groups of 2 x 12.screen-shot-2017-02-21-at-7-15-22-am

Finally, we revisited our original question about whether the vocabulary was hindering the students on the assessment. We asked them:

Is 2+2+2+2+2+2 equivalent to the product of 6 and 2?

Many of the students said yes and justified their thinking:

One student changed his thinking:

Several students asked some great questions:

We talked about the definition of product and how it was similar to, but different from the definition of sum. We learned that we needed to revisit what a “difference” is. One of the questions that came up in our discussion of this vocabulary was  “Can two statements be equivalent if they have different operations in them?” We explored this question, but I am not convinced that the students have convinced themselves of the answer.  How can we explore this more deeply?  Why is it important for students to understand that expressions with different symbols can be equivalent? Several students also wondered why this isn’t true:


One student tried to explain why she thought it was true by using smaller numbers and an image. She said, “see.  If I have 6 and I divide it into groups of 3, I have two groups.  This picture shows both expressions.”


This was a pretty tricky claim to navigate.  Cassie and I weren’t sure where to go with it and we would love some advice if you have any. I wish I had done a better job of closing the lesson. I wish I had revisited our guiding question and asked the students to write their thinking in a journal.  I think we still have a lot of wonders about Algebra. I think we can dig deeper here.  Grab a shovel. Help us out.







God, Love, and Counting Birds

Do yourself a favor.  Close your eyes.  Picture walking down a street.  When you look up, you see some birds on a wire.  Pause for a minute and get a clear image of what you see. Try to describe it in words – maybe jot down some notes – but DON’T try to draw a picture. Just use words.  It would be wonderful if you could share your thoughts,  in the comments, when you are done reading the blog.

I spent a good portion of Valentine’s Day engaged in a fascinating conversation on Twitter. It was one of those conversations that had a lot of twists and turns.   The conversation started like this:


Then it went here:

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and here:


Simon has blogged about a lot of fascinating math explorations.  When I mentioned that I wanted to explore geometric representations in other base systems, Lana told me that Simon had written a blog post about that.  “Of course he did”, I said.  Then I wondered, “Is there a question that Simon hasn’t written about yet?”

screen-shot-2017-02-18-at-7-58-01-amHere is my response to Simon:


Here is Telanna’s (Lana’s) response:

Screen Shot 2017-02-18 at 8.12.16 AM.pngSimon told us that this question was inspired by a quote from Borges:screen-shot-2017-02-18-at-8-04-15-am

He said he wasn’t necessarily thinking about this quote metaphysically.  I couldn’t help but think about it metaphysically.


If, in my minds eye, I see more birds than I can subitize, can I ever truly count them in their original form? Can I capture them? Or will they always be a “clump” somewhere between 10 and 15?  When I try to count them, do I change them?  By assigning them a number, do I bring them into existence?

I decided to ask my husband what he saw.


What does our mind do with indefinite numbers? Can we ever truly communicate what we see in our mind’s eye?

Lana and I decided we wanted to explore this with children.  She shared some beautiful pictures that her students had drawn when she asked them about the birds. These were my favorites:


Lana got a lot of definitive answers from her students. I wondered if the pictures were what the students originally saw or were they manifestations of what they orginally saw. Did the question, ‘how many?’ change what they saw?

I decided to ask my own children to answer the question.  Based on what Lana found, I changed my approach.  I interviewed my kids separately and I devised  a follow up question. We don’t often see groups of small birds sitting on telephone wires in rural Maine.  When I think of birds on a wire, I think of a lone hawk or an owl.  I wondered what my kids would say if I asked them about turkey vultures. We often see large groups of turkey vultures circling in the sky.

First, I asked my 11-year-old daughter. Here is what she said:

Then, I asked my 7-year-old son. Keep in mind, this is unedited.  Therefore, you get to experience my home at breakfast time.  Yes, my son is taking this opportunity to talk about bird poop “on the record”. Yes, my husband is yelling at the dog in the background. This is my life and I love it. Here is what my son said:

He continued talking about the turkey vultures and he said something that I found fascinating. I asked him if he saw ten the first time or if he “made” ten so that he could count them more easily.  He told me he made ten so there could be an even number for his circle. I asked him if he could draw a picture for me.  He tried, but he got really frustrated:


He was about to give up because he couldn’t make the picture represent what he was seeing in his mind. I gave him some pennies. Watch what he did:

He miscounts the pennies in the circle.  I don’t think that is important right now.  He is grappling with bigger questions. He is trying to communicate bigger ideas.  He is trying to translate what he sees so that I can understand it. It is time for me to listen. I love this video.  I have watched it many times.  I will watch it many more.

Some nights, after dinner, My husband and I walk around our driveway. We live in the woods. Our driveway is long.  We look up at the stars.  Sometimes we walk together and sometimes one of us stops to look up at something and the other one keeps walking. Later, we meet back up again.  Last night, he was showing me how the big dipper and the little dipper are related. It is hard to have a conversation about stars because they are so far away.  It is challenging to decipher each other’s perspectives.  I was trying to find the little dipper.

“Is that it or is that the seven sisters?”

“That is Orion’s belt.”

“Where are the seven sisters?”

“Over there. They are also called the Pleiades.”

“They look like a little dipper.”

“They are smaller than the little dipper.”

“Like a tiny dipper? Is the little dipper the one that looks like it is pouring water on the trees over there?”

“I guess it could look like that.  It depends on how you look at it. The handle on the little dipper points to the North Star.”

“I think I see it.”  We kept walking.

I was watching Orion’s belt.  Each time I walked a loop, I would look up at Orion.  After the third time, I saw what I thought was his bow.  I asked my husband, “Am I supposed to be able to see Orion’s bow? I think I can see it in that line of curved stars.  Can that be his bow if his belt is so much smaller?” My husband just listened.  “I’m not sure what you mean,” he said, and we kept walking.

“Did you see that?!,”  He asked

“No.  What?”

“A shooting star.”

I looked up, but it was too late. I missed it.

He stopped and stared for a while.  He said, “I am going to make a wish.”

Over the last year, I have thought a lot about the overlaps between my relationships with students, my relationships with teachers, and my relationship with the people I love. I have been trying be a better listener.  Recently, I noticed that I have gotten better at listening to students.  I still need to work on being a better listener to teachers and loved ones.  I wonder if love lives in the space where we try to understand each other?  What would happen if every single  person in the world decided to simultaneously listen to every other person?  Would there be a profound silence?  What would follow the silence? Who would speak first?  What would that person say?  Would we listen?

“… Like a tree extending its roots”



Yesterday morning, Dan, Bill, and I were discussing this pattern as we planned the lesson we were about to collaboratively teach to Bill’s 6th grade math class.

Bill: “What I wonder is what happens if I go below zero? Can you go in both directions.”

Dan: “As a teacher, I wonder about the word wonder. If you have kids wondering at all, you are in a good place and it is a hopeful statement. There are kids who won’t wonder and the goal is to get them to that place.What about the kids who says, “why are we doing this?”

Bill:  “And I would say that since I have been using this language this year, because I didn’t use it at all last year, they are really into it because it is so accessible to all of them.”

Me: “And there is no wrong answer. Actually, if somebody said, ‘I wonder why I have to do this’ I would probably say that is a great question. I hope you can answer it by the end of the class.”

Dan: “Bill, you have me curious right away.  I want to know the body of the language you are talking about.”

Bill: “Those two questions. What do you notice? What do you wonder?  I love them because, for kids who might struggle with getting an answer, I’ve got them hooked! At least for the beginning.”

Sarah: “So the point of starting with this is that the answers are already there. I am not asking them for answers. I am asking them what they wonder about those answers.”

At this point, I showed them the next slide:



Bill started the conversation, “To arrive at the same number for two different expressions, you need to increase by the same number….”

Dan: “……The increase and decrease must be equal.”

Bill: “Will it work with a decrease? I wonder.”

Sarah: “so, to arrive at the same number in what…”

At this point we realized that we didn’t have the vocabulary we needed to state our claim. We all remembered the answer being called the “difference”, but we wondered, “is the first number in a subtraction problem the subtrahend or the minuend?” We looked it up online.  We decided we would use these words throughout the rest of our discussion and with the students. We wondered if we might remember them in the future because they came out of the problem solving. We needed them. We weren’t just asked to look them up along with 50 others.

I restated our claim,”okay so what you are saying is to arrive at the same number for the difference, you need to increase the minuend and subtrahend by the same amount.”

We all agreed.

The next step was to find evidence to support our claim. We all tried different problems. We explored going below zero.  We felt pretty confident that our claim worked.

Then, I showed them the next slide.


Dan described how he was “seeing” it: “Eleven is a higher elevation so base camp is seven and the farther you climb up, the greater the distance to base camp. So if it is 11,000 foot peak, it is 4,000 feet down to the 7,000 foot base camp.”

I drew what he was describing on the whiteboard.


This is where the conversation got really interesting. I was picturing a different representation in my head so now I had to figure out how to understand Dan’s thinking.

I asked, “Where are our numbers in your representation?”

“Well,” Dan said, “It is not the same because the difference is ever increasing.”

I wondered, “So where is that in our numbers?”

“It is the difference between the minuend and the subtrahend… it is increasing all the time… No. No it is not by definition because it is 7…. because they are increasing by the same amount….”

“You originally said base camp represents 7. Where would 8 and 1 be?”

“8 would be here… and the  distance from here to here is only 1.” Dan recorded his thinking as he spoke.  Below is my rendering of what was on the whiteboard.


I still couldn’t see it. I asked, “Where is our claim?”

Dan said, “I am not sure the picture translates to our claim.”

I asked, “Could we change it to represent our claim?  Could we use your context and make it represent our claim?”

Dan thought aloud, “Difference and distance really do mean different things. So I look at that map and it makes sense to me and I look at (our claim) and I can make it work, but difference and distance mean different things.”

He paused.

Then continued, “The range or the difference say would be three or 4 and the camp for the night is at ten or eleven. So base camp is 7. The range is, the distance traveled, is 3 and the camp for the night is at ten. Now, how do we get 7 to stay the same?”

He added, “It stays the same because it is not…. if you looked at it as what is it going to take to get back to base camp…it will be a different amount, but base camp is always the same so if you go up farther, you have to come back farther. If you go out 27, you have to come back 27, but 27 is going to be a measure of, in this case, elevation from sea level which is the distance you’ve gone plus the 7,000 feet of base camp.”

I was still struggling to see it.”So, I need your help now because what your saying makes sense to me.”

“But I am not translating it well to the claim.”

I decided to share my thinking. Maybe I could adjust my representation to accomodate for Dan’s context. I shared, “The representation that I was picturing is a slide.  When you originally started talking about base camp, I was thinking of that (slide) happening on my representation, but now I can’t see it. I pictured the slide being the same distance traveled by people starting and ending at different locations on the mountain.”

I drew this on the board:


We all thought for a while.

“Well…base camp is not going to move.  So you can go up as far and come back as far as you want, but you are always going to have the same difference from sea level.”

I still wondered, “So what would sea level be over here.” I pointed to the original equations.


“It is the difference between 7 and 0, but I am doing that knowing the visual – zero is not represented in that chain of problems.”

Bill added, “You gotta start at sea level. You gotta start at the ocean. I think you’ve gotta start here (sea level) and go to 8 and come down to here (seven). That is 8 minus 1. ”

“Yes,” Dan said, “If you want to go to the base camp, why climb 9,000 feet?”

Bill  agreed, “So 7 minus zero is equivalent to 9 minus 2. So why go 9 and then down 2 when you can just go to seven.”

Wow. I was really struggling to wrap my head around how Dan’s visual represented our claim.  I kept asking myself, “where do I see this in that?”, but I was struggling.

At this point, we had to move on. We were supossed to be teaching this lesson in 20 minutes and we hadn’t even gotten to the visual pattern part of the lesson – the main part of the lesson. I decided I was going to let this idea simmer and come back to it when I had some time to think.

Fortunately, we have a snow day today. I woke up early and  reread our conversation from yesterday. I thought about what Dan was trying to say. I tried to adapt my representation to show  what Dan was describing. This is what I came up with:



I still wonder, does the context of sea level and base camp represent our claim?  Are  constant distance and constant difference related?  If yes, how? How are they similar? How are they different?

I don’t know the answers to these questions, but I am so grateful that Dan and Bill and I explored them before we went into the classroom. We didn’t get to the representation part of the lesson with the sixth graders, but they had some interesting questions of their own.

Many of them noticed the constant difference pattern and were able to articulate it as a claim.  They also found evidence to support the claim. They, like us, were prompted to think deeply and wonder about relationships.

  • Can you go below zero?
  • Does the same thing work with addition?

And my personal favorite:

  • Why doesn’t it work if you multiply the subtrahend and minuend by the same number. If multiplication is repeated addition, shouldn’t it work?

One student actually provided and supported an answer to this question. What do you think it is?  Go ahead.  Extend your roots.


Modeling problems

This week, I attended the High School Math team’s common planning time.  We were trying to select some common assessment tasks for this standard:


We started by looking at the tasks from Illustrative Math.  The first task we chose was:


As soon as this problem came on the screen,  I started whispering excitedly to my colleague Robyn.

“Robyn! It is a visual pattern!  This is great!”

Robyn smiled. Robyn and I are in a K-12 learning group. Recently, we have been discussing the value of using visual representations in math class.

Robyn and I both went to work solving the Illustrated Math problem.  I could “see” the first expression right away.

Here is a rendering of what was drawn on my paper:

I thought, “So n must be the number of dots on the bottom row which also corresponds with the step number.”

Then I started thinking of the second expression:



I couldn’t “see” this expression in the first image, but I could see it in the subsequent steps.


I checked in with Robyn. “Is the square in the middle of the image?”

She said, “I think so.”

I went back to thinking of the first step. I realized the first step was a little trippy because the square I was seeing in the subsequent steps was actually made out of circles.  That is why I can’t see it in the first step. It isn’t there. In the first step, n is the number 1 so 1 dot squared is still one dot. Weird. I was about to check in with Robyn again, but I missed my chance because it was time to discuss the problem as a whole group.

Somebody said, “Time’s up. What do we think?”

One teacher said, “I don’t see it. I haven’t done any dot problems.”

Another one said, “I am voting down this problem.”

And finally, “This is over the top. I would have to spend a lot of time to teach this and it would take away from what we have to do.”

I couldn’t say what I wanted to so I wrote it on my paper:


Not enough time?  Over the top?

In my head, I was thinking “this is what we have to teach.  This is where we have to spend our time.”

As a learner, I was feeling really frustrated inside.  When I took Algebra I in High School, it was all procedures.  I never understood one bit of it because procedural recall isn’t my strength.   If I don’t understand something, I won’t remember it.  I hated Algebra, but I loved Geometry.  Geometry made sense to me. I could see it.  I remember thinking “Why can’t Algebra be more like Geometry?”  Back then, I thought Algebra and Geometry were two completely different subjects that had nothing to do with each other.  Now, I realize that Geometry made more sense to me because I could “see” it. I wonder if Geometry and Algebra are more intertwined than I ever realized. I would love to take these classes again, but with an integrated approach.

So…. I chose not to say anything.

One of the teachers said, “I think I can see it, but I don’t know how a student would explain this. How would you answer this question?”

Robyn spoke up.  She asked, “How would you explain it?”

He started  to explain where he saw n squared in the image.

Robyn kept asking questions to draw out his thinking.

She asked, “Why?”, “Can you explain where +2 is?”,  “What does the (n+2) squared represent?”

Robyn’s questions helped me understand my colleagues thinking.

Finally, someone said, “I don’t know how we would expect students to write all that.”

I said, “I didn’t write it. I drew it.”  I held up my rough sketch.


I think Robyn said, “If you do these types of problems with kids on a regular basis, they get good at seeing the expressions and explaining their thinking.”


“I would not use this or teach this because we have so much to do. I am not going to waste time teaching this when I barely have time to teach everything else. This is “over the top”. It is nice to have over the top, but I don’t have time for it.”

I didn’t say anything.

“Well,”  someone else said, “It is time to vote. We have other problems to look at.”

We lost.  3 to 2.

We analyzed the other Illustrative Math problems and we chose to use  The Physics Professor and Mixing Fertilizer.  These are both great tasks. Why do I still feel like we are jipping our kids because we left out the Dots tasks?  It is one task.  Leaving out one task can’t have that much of an impact on our Algebra curriculum?

Or can it?

Maybe it isn’t about the task.  Maybe it is about what the task represents.  The reason we are choosing common assessment tasks is to deepen and calibrate our understanding of the standards.  Unfortunately, I think calibrate has become somewhat of a loaded word in our district. When I say “calibrate our instruction”, I think some teachers hear “stifle, homogenize, anesthetize” our instruction.  I think, sometimes, my definition of calibrate gets lost in translation.

To me, calibration is an ongoing process. You can read more about my definition of calibration here, but I think the dual purpose of calibration is professional growth and equitable math experiences for students. At the heart of calibration is transparent and collaborative reflection.   Calibration means continuously and collaboratively asking, what do we want our students to learn?  Why do we want them to learn it?  How do we want them to learn it?

I guess the reason I can’t seem to “let go” of the Dots task is because it represents a crucial answer to the questions I just asked.  It represents the integration of visually representing Algebra.  I feel pretty strongly that all students should be able to “see” Algebra.  Earlier, I refered to myself as a “visual learner”.  I often wonder about the value of this term. Should there be a certain kind of learner who sees things visually or should visual learning be an expectation for all students?  Is there value in being able to move from the abstract to the visual and back to the abstract?  As a learner,  I realize now that it is really important for me to able to move fluidly from the visual image  to the expression and back to the visual image. Isn’t this the essence of modeling with mathematics?

Dan Meyer describes modeling  as “the process of turning the world into math and then turning math back into the world.”

I think I can see how the other tasks we chose would offer opportunities for modeling with mathematics, but I still want the Dots task.  I can’t seem to let go of it. Maybe it is personal.