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:

Screen Shot 2017-02-18 at 7.48.41 AM.png

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.












It wasn’t long after I asked students to share their thinking about the problem above that one of the students started commenting on her classmate’s solutions.

“That is wrong.  You are wrong.”

I was recording possible solutions on the board while she was holding court behind me.

I wrote :

10 1/2

11 1/2

10 1/3

“Those are wrong.  The right answer is 10 1/2. Those other answers don’t even make sense.”

This voice had been the most prominent voice since I walked into the 6th grade classroom. It was loud and disruptive.  It overpowered the other voices and it demanded attention. It was accompanied by evasive eyes, sneering, and whispers to a friend.  I had been trying not to pay attention to it.  I was waiting for her to contribute something authentic so I could recognize her as a contributing member of our community, but it wasn’t happening.

I asked the class, “does anyone want to defend any of these answers?”

One girl came up and drew a number line  on the board. She said she thought the answer was 10 and one half. She circled each of the ten groups of two-thirds. She circled the one-third cup at the end of the number line and told us she thought it represented one half of a meal.


One boy, Max,  raised his hand.  He said, “I don’t think it matters if you say 10 1/2 or 10 1/3. You can call it one half or you can call it one-third. Maybe they are the same thing?”

“What?!” She interrupted, “That doesn’t even make sense. That is wrong.” Her voice was getting louder and less respectful.

At this point, I had to say something.  This one voice was starting to supersede all of the other voices in the classroom.  It was eroding the fabric of trust faster than I could establish it.

I raised my voice. I said, “We are sharing our math thinking.  You are being disrespectful.  I need you to listen to your classmate’s ideas without putting them down.”


I made eye contact.

More silence.

She looked away.

I felt bad.  I didn’t want to call negative attention to her. I think she felt embarrassed because I called her out in front of her peers. Did I lose her? I wanted to build her trust, but  the time and space that I was offering to her was being used to damage the relationships I was trying to build with the other students. I don’t think it was intentional.  I think it was coming from a place of mistrust.  What did she mistrust?


The math problem I proposed.

The model her classmate was drawing.

This girl does not have an easy life.  I don’t need to go into the details. We all know this girl.  She has every reason not to trust me, her classmates, or the math we are doing.

What does she trust?

I think, in this particular instance, the one thing she might trust is the algorithm. That is all that was on her paper. In the beginning of class, I asked the students to draw a picture of the situation described above. She didn’t. Maybe she couldn’t.

Before I arrived in this math class, these students were taught the algorithm for dividing fractions. Their math teacher asked me to help him teach these students how to model fractional division. He said they all know the algorithm, but they are struggling with the modeling part.  My goal was to teach them how to model a fractional division situation  so they could explain and interpret the units they were working with.

As I sit here, I wonder if my goal should have been to get them to trust themselves as mathematicians.

At this point, we had three different answers on the board.  It was up to the students to decide which answer made the most sense.

I asked Max if he would come up to the board and show us his thinking.  He did. He drew seven circles. He partitioned each circle into three parts.  He colored in groups of two-thirds.  He counted 10 groups of two thirds and then said,

“See.  There  are ten.  And, then, there is one-third of a cup left so I think the answer might be 10 and one-third, not 10 and one half.”

“I got 10 and one half,”  she said. “I don’t get it. This math class is making me feel retarded.” Her voice was softer than it was before.  She was talking to her teacher now- my colleague.

I felt uncomfortable again. I don’t like it when people use the “r” word. I don’t think she meant to be disrespectful when she said it. I think she meant to be self deprecating. I don’t like self deprecation either.  I was trying to think of what I should say to her – should I call her out on using the “r” word?  Would that make things worse? Before I could say anything, my colleague spoke:

“Don’t use that word.  Can you think of another word?”

“Fine. This math class is making me feel stupid.”

Ouch.  That is just about the worst thing I could hear in a math class.  She sat down next to her teacher.

I wondered, what should I do?

I looked right at her. “Did I hear you say that you are not sure why your classmate is getting 10 and one third for an answer because you got 10 and one half?”


“THAT is a really interesting question. Let’s talk more about that. Can anyone answer that question?”

A different boy volunteered to come up to the board and share his thinking. His writing is in the dark blue at the bottom of the picture. He drew rectangles.


As he presented, we discussed our thinking.  I asked him, “What does the one half represent in the story and the picture?”

“One half of a meal.”

“Some people are saying one half is the same as one-third because 1/3 is half of 2/3. What do you think about that?”

Other students chimed in, “Yes. But the question asked about meals and you are going to have one half of a meal, not one-third of a meal.”

“Oh.” I said, “So one-third is one half of two-thirds but the answer to this question is 10 and one half because that is how many meals we will be able to feed the dog.”

Some nods of agreement.

The girl was quiet for the rest of class.  When I left class that day, I worried that I should have done something different.  I decided that, no matter what, when I went back to class the next day, I was going to try to help her believe in herself as a mathematician.

The next day, we were using virtual Pattern Blocks to model how many sixths were in one-third.  I introduced the Illuminations website to the students and encouraged them to explore. Then, I asked them to use the Pattern Blocks to show how many sixths were in one third.

I immediately went over to my friend.  This is what was on her screen:


My first instinct was to say, “You can’t use the squares.” or “Don’t you want to use the triangles?”  My second instinct – my growing intuition – reminded me to listen.

I asked, “What are you thinking?”

She said, “There are six of them.”

“Do you think you could use the squares to show one sixth.”

She repeated, “There are six.”

I was thinking, “don’t shut down. Please don’t shut down.”  What can I say to get her to trust herself?  I wondered, “So… what is one sixth?”

She pointed to one of the squares.

“Okay. So there are 6 squares and you are saying one of them would be one sixth. What was the other part of the question that I asked?”

She looked at the board.  “How many sixths are in one-third?”

“So where is one-third in your picture?”

“I don’t know.” She gestured towards three of the blocks. “Forget it. I don’t know how to do it.”

“Yes you do.  Can I try something?  Do you mind if I move your blocks around a little bit?”


“Show me one sixth again.”

She pointed to one of the squares.

I asked, “How do you know that is one sixth?”

She said, “Because it is one out of 6 parts.”

“What if we wanted to show one out of three equal parts of the same rectangle.”

She thought about it for a while.  Then, she slowly traced the line that marked off one of the thirds of the rectangle.

I wanted to jump up and down and make a really big deal out of her success, but something inside of me told me not too.

Then, I asked her, “Can you see how many sixths are in one-third?”

“One?  No. Two!  There are two.”

“Are you sure?”


“I am going to go check in with some other kids. Why don’t you share your thinking with Jamie.  How do you feel about sharing what you found with the class?”


“I understand. I only ask because you taught me something. I hadn’t thought to use the squares to answer this question. I thought you could only use the triangles and the rhombus.  I am wondering if other people thought the same thing. They might learn something from your strategy.”

“I don’t think I want to share.”

“Okay. If you decide to change your mind, let me know.  You can bring a friend if you want.”

The next pair of students that I checked in with had this on their screen:


Wow. I was starting to wonder if anyone was going to use the ol’ triangle and rhombus combination.

It was time to share our thinking. I decided to check in one more time with my new math friend.

“Do you want to share your thinking?”

“Okay. But can I bring Jamie?”


She and Jamie projected her computer screen up on the white board.  She started to explain her thinking. Two students kept whispering. She tried to talk over them.  Another two students started a side conversation.  She looked down at the ground and said to her shoes, “no one is listening to me.”

“Hey!” I raised my voice again.  “She didn’t want to come up here. I asked her to share her thinking and she originally said, ‘no’.  Then, she changed her mind. She is taking a huge risk right now and you are being disrespectful.  Stop talking and listen.”


“Well. I thought I needed six squares so that I could show one sixth.”

“And how many sixths are in one-third?”

“One. No.. two.”

“One or two?”

“Two. It is two. See them?”  She pointed. “One. Two. Two sixths.”

“Does that make sense to you?”

“Yes.”  Was that a smile I saw?  I can’t remember for sure, but what I do remember is that she went to her seat in the back of the room, picked up her belongings, and moved to an open seat in the second row.

She moved closer to me.






Upside Down and Super Fast

Recently, my colleague, Dan, asked me to help him help his students understand how to model division of fraction situations.

When I walked into Dan’s classroom, he was asking his students a question about decimals.  He wrote this number on the board:


Dan asked his students to divide this number by 100.  He said this was a short routine they had been doing during transition time for the last couple of days. Dan said they enjoyed being able to quickly do a problem that looked really hard at first glance.

I asked, “Is the answer going to be bigger or smaller?”

Many of the students said “smaller because you are dividing.”

I asked, “can you think of a time when you would use a number like this?”

One student said, “When you are counting sand.”

Me: “Why?”

“Because grains of sand are sooooooo small.”

Another student said, “I think I know what the answer would be, but I don’t know how to say it.  How would you say that number?”

Me, thinking out loud: “Hmmm. Let’s see, tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, ten millionths??  I am not really sure. Maybe fifteen thousand eight hundred ninety-two ten millionths??? I need to ask my math friends for some help with this.”

I told the kids about the Math Twitter Blogosphere while I typed my question on twitter. I am so glad I asked for help. I hadn’t even considered scientific notation until I started reading some of the twitter feedback.  I wondered “if scientific notation is the aspirin, is 0.0015892 the headache?” I haven’t had time to dive into the twitter feed. I am looking forward to revisiting it. I would also like to revisit the comment above about sand.  I want to ask, “tell me more about how 0.0015892 could represent sand.”

At first, I thought, this student was wrong.  He is thinking that really small numbers represent really small objects.  0.0015892 can’t represent sand. Then I realized maybe I don’t know enough about what, exactly, this student was thinking. Can 0.0015892 represent sand?  What if .0015892 represented the size of a grain of sand in relation to the rock that it was once a part of?  Could a grain of sand be .0015892/10,000,000 of a rock? How do I get the students to consider the  relationship of a unit this size?

My colleague Dan and I are pretty similar.  We tend to live “outside the box.”  In fact, at times, we humbly admit that we need help getting back in the box.  The boundaries of the box are often elusive to us.  We don’t even know how we wandered outside of them.  We were probably following something shiny. For example, when I arrived to teach in Dan’s classroom yesterday, I asked him to video tape the lesson for me.  This morning, I anxiously sat down to watch myself teach and reflect on the lesson.  This is what I found:

I love this video.  At first, I didn’t love it. I thought, “Dan!  What the heck did you do?” Then, I laughed.  Next, I tried to edit the video. Can I slow it down to “normal pace”? Then, I laughed some more.  There is no sound.  I can’t read lips. What is the point of slowing it down? Finally, I settled into leaving it the way it was.  I wondered, what can I learn from this super fast, upside down video?

At first, I thought, “bummer.  All I see is me in front of the class the whole time.  Was I being the ‘sage on the stage’?”

Then, I looked a little closer and noticed something interesting.  There are a lot of students bobbing their heads and turning around.  There are a lot of students moving their mouths. This part of the lesson is when I was doing a Number Talk from the book Making Number Talks Matter. I am trying to record student thinking on the board.

You can’t see what I am recording because my Google Slide presentation for the next part of the lesson is taking up most of the space on the white board.  Here is the perfect example of me not even being slightly aware of the boundaries of the box.  As I watch myself now, I am thinking (kind of yelling), “TURN OFF YOUR SLIDESHOW!”  “Hello??? You are monopolizing most of your board space with a presentation that you aren’t even using right now!”

See. I can learn stuff from watching a super fast upside down lesson.

I remember this part of the lesson.  In fact, If I blow up a screenshot from the video and rotate it I can see what I did.


See. Doesn’t that help you?

What do you mean, “no”.  It helps me.

Down on the bottom, I see the number 99 and I am remembering the great conversation we had about whether 50/99 was more than one half or less than one half.   Some students said less than one half.  Some said more than one half.  I highly suggest you take a minute to figure it out for yourself before you move on.  I asked the students if anyone wanted to defend either of the solutions.  One boy said he wanted to defend less than one half.

He said “half of 100 is 50 and 99 is less than 100 so 50/99 would be less than one half.”

I remember thinking really carefully about how I should record what he said.  I chose to use words.  Immediately, one of the other students said, “but half of 99 is 48.5”. At this point, I chose to just stand there and wait.  Then, I said, “can you tell us more?”

“Well, half of 99 is 48.5 and 50 is more than 48.5 so 50/99 has to be more than 1/2”

Then, the first boy chimed back in.

“I was wrong. It is more than one half, but it is 49.5, not 48.5.”

Me: “What do you mean?”

“half of 99 is 49.5, not 48.5.”

other boy:  “oh, yeah.”

Me: “So, why did you change your mind?”

“because 49.5  would be half and 50 is more than 49.5.”

At this point, I am noticing that no one has actually referenced the units. They aren’t talking about the fraction in it’s entirety. The students are referring to the relationship of 1/2 to the numerator or to the denominator, but no one has actually said, “49.5 ninety ninths or 50 one hundredths”  You can’t see this because the video is upside down and super fast. I remember it because I was trying really hard to listen “to” and not “for” answers. I was also trying really hard to cultivate my students’ math intuition.

I remember thinking, “how do I get them to think about the unit without telling them to think about the unit.”

Finally, I said, “how many hundredths would be equal to one half?”






Wow. This is most definitely NOT what I anticipated for answers.  I thought I would hear a loud chorus of “fifty.”

So. I recorded those answers on the board, somewhere up in the top corner, smooshed in, next to the magnetic marble tracks. (Get back in the box Sarah!!).

I asked, “which of these answers is correct?”

Several students said “fifty.”

Me:  “Why?”

“because fifty is half of one hundred.”

“fifty cents if half of one dollar.”

A lot of “oh yeahs….”

“Does anyone still think it is something other than 50?”

Silence and heads nodding “no”.

I asked the students, “Does anyone want to share why they changed their minds?  Some of you originally thought the answer would be 1/100 or 2/100 or 10/100.  Can you tell us why you changed your mind?”


Okay. What if I try this, “Does anyone want to share why you think someone might have gotten one of these other answers the first time (1/100, 2/100, 10/100)?”

One boy said, “I could see why someone might say 2/100 because 2 x 50 is 100 and 50 is half of 100.”

Several students nodded in agreement.

I silently did a giant happy dance.  I think I am doing it!! I think I am helping my students develop their math intuition.  Wow. I need to ask those kinds of questions a lot more.

At this point, we had to move on.  This was a nice segue to the division of fractions exploration that I had planned for today. (I could write a whole other blog post about the second part of the lesson.)

We definitely need to offer these students more opportunities to explore and discuss the relative magnitude of numbers. We also need to cultivate more opportunities for them to explore and communicate the units they are working with.  At first glance, this post may seem like a series of upside down and super fast examples that are unrelated, but I actually think they are the beginning of a journey towards deeper understanding of when and why the unit matters.