I Just Took What I Needed

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

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

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

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

Here is what Deb and I noticed:

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

Here is what we wonder:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


I am Those Kids.

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

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

“What happened?”

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

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

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

How do you feel about math class?

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

She said, ‘numbers don’t inspire me.’

I asked, ‘what inspires you?’

She said, ‘poems’.

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

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

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



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

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

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

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

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

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

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

Screen Shot 2017-11-24 at 8.21.12 AM

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

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

And here is what they wondered:

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

Listen to what happened next:

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

I will call him Colin. Remember him.

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

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

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

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

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

Screen Shot 2017-11-24 at 8.43.50 AM

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

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

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

They said, “Yes! He does.”

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

He counted sixteen because he forgot the number thirteen.

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

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

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

Evan beamed. He didn’t mind at all.

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

Screen Shot 2017-11-24 at 8.44.01 AM

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

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

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

He said, “We have the same amount!”

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

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

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

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

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

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

I wondered, “How many shapes is that?”

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

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

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

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

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

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


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

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

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

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

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



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

Yesterday, I got this text:

Screen Shot 2017-11-18 at 8.36.01 AM

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

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


Decimals, Backwards Slashes, and Giggling in Math Class

Recently, I read a blog post by Andrew Gael, Our kids Are Not Swiss Cheese.  Some quotes that stuck with me from Andrew’s blog:

  • “Maybe it is not the learners; maybe it is the way that we conceptualize learning…”
  • “Learning is complex, multi-leveled, and no one is all the way “filled in.””
  • From Megan Franke, “How do we notice and use what students DO know to support them to make progress in their thinking?”

Last week, during our 5th grade collaborative planning session, we discussed how to introduce decimals to our students. We decided we wanted to start by unearthing what the students already understood about decimals.  I was really excited to approach “decimals” as a concept that connects to prior knowledge, instead of a series of disjointed procedures.

So, today, we started our journey. I co-taught with Mrs. G. and Abby, our school based math specialist, co-taught with Mrs. C.  We wrote this question on the board, “What are decimals?”  We told the students we would ask them for their thoughts about this question at the end of class. Then, we counted.

We started counting by ones and tens. Then, we asked them to count by tenths.  That is all we said, “let’s count by tenths. Who wants to start?”  Matt said he wanted to start.

“one tenth.”

I asked, “how would you like Mrs. G to record that?”

“Just write one tenth.”

“Can you tell us what that will look like?”

He went over to the white board and wrote this:

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“Okay,” I said. Mrs. G recorded ‘one tenth’ on our chart.  Then, I asked Gary to continue the count.  We continued the count, each time asking the student how they would like us to record what they said.  This is what they told us:


If someone was unsure, we told them we could put a question mark above their suggestion and we could come back to it.  It was so interesting to me how quickly the students began referring to each other as authors of ideas. When asked, “how would you like us to record that?”  They said, “like _______ did.”  When we got to nine tenths, one student told us he would like us to record it ‘the same as six tenths, but with the slash the other way.’  I actually have a voice clip.  Listen.


As I listen to this clip now, I am smiling. I love it. I hear confidence and creativity. The first time I heard it, I was nervous. I wondered, should we put that on the chart?  What if the students think it is an acceptable way to write a decimal?  What if I ruin them forever by supporting this backwards slash business?  I almost panicked. There were so many times during this routine that I almost caved. I almost said, “actually, that is not how we write decimals.” But, I didn’t. I am so glad that I didn’t.  Look at this chart! I mean really look at it.  What do you notice?  What do you wonder?  What do these students know about decimals?  What do these students know about our number system?

Yes. There are definitely some partially formed ideas here. There is no doubt that we need to continue our study of decimals.  Of course we do. It is only day one.

After we finished our count, we told the students that we will continue to look at this anchor chart and we will continue to count by tenths. We also asked them to write down something they noticed and wondered about our chart.

We asked a few kids to share their thinking: B shared his thoughts about 2.5 = 2.50, S shared his question about whether we can write decimals in word form and we confirmed that we can, K asked about writing decimals in exponential form and we told him it is possible, but we would talk about that in more detail later. At the time of the lesson, Mrs. G and I were so bummed that no one noticed the ten in a row pattern. As I write this blog, I realize Molly DID notice it. Arghh!  We will have to ask her to explain her thinking tomorrow.  Maybe we can compare and connect Molly’s, Patrick’s,  and Gabe’s responses.

Next, we split the kids into two small groups.  Mrs. G took half and I took half. When we made our heterogeneous groups, we considered processing time, distractibility, schema, perseverance, expressive and receptive language, etc. We spent less then 5 minutes, but we considered all of these criteria as we tried to form groups that amplified the learning experience.

Mrs. Gordon and I each facilitated a Number Talk using the following images:

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My group had some really interesting conversations.  I was worried that, if we used money, it might limit our conversation to one context, but I don’t think it did. One of the most interesting questions they asked was when they wondered whether 1.50 and $1.50 were the same or different. What a deceptively simple thing to wonder about. This next clip is really interesting.  I like it beacause I think it is an example of what Andrew discussed in his blog. Can you hear the non-linear complexity of ideas being formed?


To close out the lesson, we asked the students to do two things. First, we asked them to answer the question, “what are decimals?”

We also gave them a question to think about.  We asked them to tell us whether they thought the statement was true or false and explain why. We didn’t expect them to get the answer *correct*.  In fact, we were less interested in correctness and more interested in how they explained what they understood so far. Here is what they came up with:

After the lesson, Abby and Mrs. C check in while Mrs. G brought the kids down to lunch. Then, Abby told me how it went  in the other 5th grade class. Then Mrs. G and I checked in while Abby went to lunch. We were all wondering what to do next. We decided we would continue with the plan we had sketched out last Thursday. Tomorrow, we will introduce the kids to the Zoom in on the Number Line routine. We will try to connect the magnitude of tenths and hundredths as we compare decimals and place them on number lines with varying intervals. We also decided we would re-use our artifacts from the lesson close.  We are going to give the exit ticket and sticky note answers back to the students throughout the week and ask them what they would add and/or change.

Finally, on my way to Wayne Elementary School, I stopped at varying spots to collect artifacts that reminded me about decimals.  Here is what I came up with.

I texted my artifacts to Mrs. G and Mrs. C and asked them to ask the kids to go on their own scavenger hunt for decimal related pictures or conversations that they wonder about.

I got my first response:

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So, yeah, it is scary to invite kids to put a bunch of partially formed ideas on the table. It is messy and it will take us awhile to sift through them and make connections, but I think it will be time well spent.

Last week, I was reading Tracy Zager’s book, Becoming the Math Teacher You Wish  You’d Had.  I tweeted her to let her know that it was going to take me years to finish her book because it so rich with provocative ideas.  Here is one that I have been mulling over for days now:

“Above all else, maintain your focus on developing young mathematicians who listen to and refine their internal truth detectors. Encourage them to be skeptical and allow them to remain in doubt until they are genuinely convinced. Do not apply pressure to concede, even, if you’d like to move on.”

I just love that. I would be pretty psyched if, some day, some thirty-something mathematicians tracked me down to thank me for helping them refine their internal truth detectors.  Thanks again for the push Tracy.

Crossroad: The Point at Which a Vital Decision Must be Made

Last weekend, I wrote about my experience doing Number Talks in two high school classrooms. I got some really helpful feedback.

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After I read the blog post Pam recommended, I read the pages from her book that were referenced in the blog post. Then, I started to plan the string I would use in the Transitions to Algebra class. I thought about, why would I use the string Pam recommended?  How does that string fit with what we are trying to do with our students?

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Then, I anticipated what our students would do and say when I presented the first problem in the string:

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So, that is how I planned to introduce the string. Here is how it actually went down:

When I walked into the classroom, Ms. S wasn’t there. The bell hadn’t rung yet. All nine students were scattered around the room. Some were sitting in pairs and others sat by themselves. I told the students how glad I was to be back.  As I connected the Smartboard, to my computer, I chatted with the students. I checked in with each of them to see if I remembered their names correctly.  A group of girls tested me.

I asked their friend, “I can’t remember, is your name Megan or Meegan?”

They laughed and said, “Megan”.

Someone said, “no. Her name is Meegan.”

More laughter.

At this point, Ms. S came back in the room. As soon as she sat down across from Riley,  he looked at her and said, “I hate math. I just hate it.” Ms. S reminded Riley how hard he had been working and how much he is learning this year.

I asked her, “Is it Megan or Meegan?”

She said ,”Meegan.”

“Thanks.”  Then, I officially started my lesson.  “So, last time I was here we were doing some multiplication. Today, I planned a Number Talk that is a little different. Today’s Number Talk problems are going to be about a situation. We are going to talk about bags of m&ms.  Not the big bags. The little ones.”

Ms. S chimed in, “The fun size bags.”

“I have one of those!” Hayley said, as she dug into her sweatshirt pocket. She held up a little red bag. “Forget it. These are Skittles.”

Several students erupted, “Let’s do Skittles. I hate m&ms. Skittles are so much better than m&ms.”

“Not today,” I said, “maybe next time we will talk about Skittles. I am so glad Hayley had a bag of Skittles in her pocket because the m&m bags look a lot like the Skittles bag. Now we all have a clear idea of the size of the bags. Thanks for adding to our context, Hayley.”

“Okay. So , today, I want you to consider bags of m&ms. Each bag of m&ms has 17 m&ms in it.” I wrote in the table as I spoke:

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“Think about how many m&ms would be in 2 bags. Please don’t say your answer out loud. You don’t have to raise your hand. Remember, you can show me a thumb when you have a possible solution.”

Some of the kids held thumbs in front of their chest. A few kids raised their hands.  Meegan blurted out, “like 30.”

I said, “So Megan…”

Laughter. I looked at Ms. S and she said, with a smile, “it’s Meegan.”

I smiled, “See what happens when you mess with me? I have trouble remembering names when I hear them correctly the first time.”  I continued, “So Meegan, can you use your thumb next time. I totally appreciate that you are participating, but I want to make sure everyone in the room gets enough think time.” I wrote her solution on the smart board.

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“Are there any other solutions?”

“30 something.”

I was about to ask Riley why he said 30 something, but Max started talking, “34 because 10 plus 10 is twenty and 7 plus 7 is 14 and 20 plus 14 is 34.” As Max was talking, I started recording what he was saying:

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In the middle of Max’s explanation, Riley started a monologue of verbal processing, “No. I did 20 plus 7 is twenty-seven, but now I have to add seven more and I lost count. You need to add the 7 to the twenty….”

I need you to picture those conversations happening simultaneously, while Meegan is having a side conversation with Katie and drawing on her hand. I wish I had an audio of it.

The two boys were talking at the same time. They weren’t trying to be difficult. They weren’t intentionally ignoring each other. They were just being impulsive. My greatest struggle with this class is that they are incredibly impulsive. If you know me at all, you can feel free to chuckle right now. I get it. The impulsive leading the impulsed.

“Okay,” I said, “hold on. I really want to hear all of you and I want you to hear each other.” At this point, Ms. S went over to Meegan and asked her to put the marker away.  I continued, “Max was telling me that he thinks the answer is 34 because he added 17 plus 17 by decomposing the 17s. Max, did I record your thinking correctly?


“Okay. Riley, it sounds like you did not solve the problem that way. Am I correct?”

“Hold on,” Riley said, “Ten plus ten is twenty and then plus seven is twenty-seven. Now, I have to add 27 plus 7. Ugh. That is the worst.  27, 28, 29, 30, 31, 32, 33, 34. Okay, yeah, 34.”

I tried to record his thinking:

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I asked, “Does my recording represent what you were thinking?”


“Okay. So, we just spent a lot of time talking about addition. Can anyone see a multiplication problem in this situation that we are talking about?”

Matt said, “17 x 2”

Riley asked, “Is that the same as 2 x 17?”

“Yes!” Said Ms. S. Later, Ms. S told me that Riley has been thinking about the commutative property a lot lately.

“Okay,” I said,  “so we can say that 17 + 17 = 17 x 2?”

“yes,” they agreed.

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Next, I asked, “What if there are 4 bags of m&ms?”

Olympia blurted out, “41”.

I recorded it and reminded her to please show me a thumb.

I think she said, “sorry.”

I saw a few thumbs and a few raised hands.  Several students, to include Samantha, had yet to participate. I asked Porter, “Do you have a solution? You can pass if you want, but I would love to hear what you are thinking.”


“Hayley, do want to share a solution?”


Riley said, “I got 68.”

Hayley said, “me too.”

Olympia said, “I got 41. It’s wrong.”

Riley added, “I think you only added one 17.”

Max and Matt were having a side conversation about why the answer wasn’t 41. Meegan and Katie were giggling about something that I am pretty sure had nothing to do with math. I was trying desperately to NOT lose the exchange that just happened between Olypia and Riley.

“Riley, are you saying that you think Olympia got 41 because she only added one bag of m&ms, instead of two?”

“Yeah. She only added one 17. She needs to add another one.”

“Okay! So we can add that to our table. Where can I put that?”

“you can put a 3 in between the 2 and 4.”

I did.

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So… I did NOT anticipate this conversation happening. That might be why I didn’t actually notice that 34 plus 17 is NOT 41. It is 51.  I was so excited that Riley was thinking about Olympia’s mistake in the context of the problem that I didn’t even catch the arithmetic mistake.

Keep in mind, Meegan is still having a side conversation. Samantha is drawing on her whiteboard (where did she get a marker?).  Max and Matt are explaining their solutions to each other, completely ignoring the rest of us. Oh…. and Bill, Hayley, and Porter haven’t said a word in a long time.

“Can everybody listen for a second?” I asked. “I absolutely love coming into this class to do Number Talks with you. It is the highlight of my week. I learn so much from you, but I get really frustrated when you are all talking at once. I want you to be able to hear each other. Can we please try to take turns when we talk?”

It got quieter.  It was not totally silent, but everyone was making eye contact with me and attempting to pay more attention then they were before I started speaking. I’ll take it.

At this point, I asked, “where is the multiplication in the work we are doing?”

Matt mentioned doubling again. I asked him to show me where the multiplication was, in regards to doubling.

He explained, “17 x 2 is 34 and 34 x2 is 68.”

We continued to discuss the amount of m&ms in eight packs.

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Then, I asked about ten packs. Several students said, “170.”

Riley said, “one hundred something.”

I asked, “how do you know it is one hundred something?”

He explained, “well ten times ten is 100, but then I have to add ten sevens and I can’t do that in my head.”

Matt said “dude, you don’t have to do that. There is a much easier way. Whenever you have something times ten you just add a zero to the number. Seventeen times ten is seventeen plus a zero. It is 170. You are making it harder than it needs to be.”

I said, “okay, can we slow down a second? What Riley is saying and what Matt is saying actually go together. Matt is talking about the procedure and Riley is talking about why the procedure works.  Riley, can you repeat what you started to say?”

As he spoke, I drew an area model on the board.

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Halfway through my drawing, Matt spoke up again, “you don’t have to do all that. You just add a zero. You are making it harder than it needs to be.” Matt is not trying to be disrespectful.  Matt is trying to help his classmate.

I said, “Matt, you are trying to show Riley a trick that helps you. Riley is trying to understand where the answer is coming from. I think Riley’s strategy is connected to what you are saying.”

Riley said, “I just don’t know 7 times 10.”

Meegan said, “it is 70. You just add a zero.”

Riley agreed, “Oh yeah! Okay. Yeah. The answer is 170.”

At this point, I REALLY wanted to revisit the commutative property with Riley, but there was so much else to consider. Matt was getting frustrated that we were spending so much time discussing a strategy that, in his mind, was needlessly cumbersome. He wasn’t frustrated with his classmates. He was trying to help them. He was frustrated with me.  He didn’t understand why I was “wasting” all this class time talking about something that had no relevance to him.  Why didn’t I just tell Riley to “add a zero” and move on with the Number Talk?  So, for better or worse, I moved on.

“Okay, let’s talk about 12 packs. Ms. S, can you give everyone a marker? I would like all of you to record a solution on your whiteboard. Try not to use the whiteboard to solve the problem, but use it to record your solution. I want everyone to at least try, please. You can write your answer really small, if you want. I just want to see something so I know that you tried.”

Right away, Meegan wrote 204 and then covered it.  She looked up at me and whispered, “Do I have to keep it uncovered?”

I said, “I saw it. You can keep it covered it, if you want. Just don’t erase it.”

Hayley wrote twelve seventeens on her whiteboard and started to add them. Olympia had three hundred something written on her whiteboard. I can’t remember the exact number. I waited about 2 minutes.  Hayley was still adding twelves. Bill hadn’t written anything but had that “mental math” look on his face (eyes looking at the ceiling, lips moving, head nodding in sync with a count,).  Matt was describing to Max how he added 34 to 170. Meegan, Olympia, Samantha, and Kate were doodling.

“Okay,” I said, “you might not have a solution yet and that is okay. I want us to start a conversation about what we think so far. Meegan, can you tell us where you got 204?”

“I added 34 and 170.”

“Can you tell us why you did that?”

“Because that is 12.”

“What is 12?”


“Twelve what?”

Meegan responded, starting to get frustrated, “you asked how many are going to be in twelve!”

“Right. I did. Okay. So you added the amount of m&ms in ten bags to the amount of m&ms in two bags?”


At this point, there was a lot of agreement about Meegan’s answer. Everyone thought it made sense.

“Okay,” I said,  “Does anyone see a multiplication problem in the problem we just solved?”

For the next five minutes, we engaged in a round of Guess What the Teacher is Thinking. I hate this game and I try so hard not to end up playing it, but sometimes, I get caught off guard. The kids were not really sure what I was asking.  I should have just said, “You told me that 2 bags of m&ms was 2×17 and ten bags of m&ms was 10×17. So…..”  But I didn’t say that. I actually don’t remember what I said. I just remember the distint feeling that the kids were trying to guess what I wanted them to say.

I am not sure where it came from, but, eventually, Riley said,  “so, do you mean twelve times what is 204?” I wrote it on the board.

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“Let’s go with that. So, what did we multiply 12 by to get 204?”

This is where the lesson got really interesting. No one saw the 17. Meegan mentioned that she added 34 and 170. Matt and Max were discussing how they added 170 and 34.

I asked Hayley, “Do you mind sharing what was on your whiteboard before you erased it?”

She smiled and said, “It was so stupid. I am so stupid.”

I said, “I don’t think it was stupid. I don’t think you are stupid. In fact, I think it is going to really help us answer this question. Right now, we have an answer that we know makes sense, but we are struggling to figure out how to write the problem using multiplication. I think what you had written on your white board will help us. You can pass, but I would love it if you shared your work with us.”

She said, “pass.”

“Do you mind if I talk about what was on your white board?”

“That’s fine,” she agreed.

I said, “Hayley had the number 17 written on her white board 12 times,”

Riley interupted me, “It is 17! Of course it is 17. It is 12 x 17!”

“Yes,” I said. “It is 12 x 17. Hayley, your work was really important because you were the only one who thought about the problem as 12 groups of 17.  We needed to hear about your work to remember that we are multiplying 12 x 17.”

Then, I asked, “Where is 12 x 17 in Meegan’s solution? Meegan said that she added 34 and 170. Where is 12 x 17 in her work?”

Matt said, “Well 10 plus 2 is 12 so there is the 12.”

I asked, “where is the seventeen?”

Somebody responded, “There is a 17 under the one, but that isn’t the seventeen that Meegan used.”

We had been Number Talk-ing for awhile.  I decided to wrap it up. I drew another area model of 17×12. I explained that Meegan’s strategy works because of  the distributive property. Our table showed 10 bags of 17 (or 10×17) and it also showed 2 bags of 17 (or 2×17).  Meegan added the partial products to find out how many m&ms would be in 12 bags of m&ms.

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I added that Riley was using the distributive property earlier when he decomposed 17 into 10+7.

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Riley erupted, “can we call that ‘Riley’s Law’?”

“Sure,” I said. “It is also called the distributive property, but Riley’s Law works for me.” I wrote Riley’s Law next to the area models.

After class, I chatted with Robyn, the high school Math Specialist who has been collaborating with Ms. S. Robyn was in the math office and I asked if she had a minute to reflect with me.  I described how our class went. I asked her to help me think about what we might do next time.  Here are the notes I took:

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After I chatted with Robyn, Ms. S came into the office. We started chatting.

She said, “I am sorry they were rude to you. I talked to them after you left. I have been wanting to revisit our classroom expectations. I told them if they acted like that next time, there would be consequences.”

I said, “We should also revisit our Number Talk expectations. How about next time we create an anchor chart that defines the purpose of the Number Talks and the expectations. We did that in September, but we didn’t write it down anywhere. Also, now that we have done several Nubmer Talks together, the conversation about norms and expectations will have more meaning.”

“I like that idea,” said Ms. S. “Can you lead that conversation?  I would really like to see what it looks like. I will support, but I would love to watch you do it.”

“Sure,”  I said.  I took some notes as we processed the different behaviors we saw today and how they impacted our learning.

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Then, I shared the conversation that Robyn and I had about the content of our next Number Talk. Ms. S agreed that the plan Robyn and I came up with would be a good next step. We both had to run to a meeting, but I am going to ask her if we can set up a time next week to anticipate student responses and create a monitoring sheet. Maybe Robyn can join us.

My favorite moments are when frenetic, seemingly unrelated experiences lead to serendipitous learning.  This week, I engaged in several different reflective conversations on Twitter. One was about my high school Number Talks experience. Another was about teacher collaboration, and the third was about a little boy’s response during a Which One Doesn’t Belong routine.  In my non-virtual life, I was reading chapter 11 of Tracy Zager’s book Becoming the Math Teacher You Wish You’d Had and facilitating several really hard conversations about why collaborative interventions are best for kids.

I wonder, how do I stay intentional when real life veers from my plan?  I think, after writing this really long blog, the answer lies in honing my intuition. There are a million things I could have done differently during the Number Talk that I just described. Okay, maybe not a million, but I have thought of at least five just while I was writing this blog.

My best attempts at reflection usually involve finding a crossroad in the journey, revisiting the path I took, and exploring the path I might have taken. If the reflective process works, it leads me to a truer understanding of my intentions.

The most important teaching decisions are made in those micro seconds, when things don’t go as planned and we have to use our intuition to decide what to do next. My intention is to shine the light on those moments so we can all think about them together.  I try to write about the messy stuff: the moments of uncertainty, confusion, and frustration.  I try to write about the times when things didn’t go exactly as planned because, in my teaching and coaching experience, they never do.


Making A Hodgepodge of High School Number Talks Matter

As a K-12 math coach, I am all over the place.  We have about 50 teachers in our district who teach math in some form.  I haven’t found a way to divide my time equally among those 50 teachers and make a difference. My superintendent tells me to work with teachers who want to work with me.  So I do.

In the beginning of the year, I had two high school math teachers ask me if I would do Number Talks with their classes. I was thrilled. First, I gave them both a copy of the book, Making Number Talks Matter by Cathy Humphreys and Ruth Parker. (If you don’t have this book, go get it.) Then, I added both classes to my calendar and told them I would be there, once a week,  for the rest of the year.

I didn’t have a pre-planning meeting with them. I didn’t write down formal goals.  I probably could have. Should have? The reality is I don’t always make the time to follow elaborate coaching protocols. I am not saying that I shouldn’t. I probably should, but I don’t.  My number one priority is getting into a classroom, as quickly and regularly as I can.  My number two priority is getting invited back. All the formal protocols in the world aren’t (necessarily) going to help me build relationships, BUT building relationships might help me use coaching protocols more meaningfully.

Ms. S and Ms. K, were really excited about me doing Number Talks in Algebra II and Transitions to Algebra.  They talked about wanting to build their student’s number sense and get them to have more meaningful math conversations.  Those sounded like great goals to me. My unofficial goal for the first few Number Talks was to cultivate a space where the guiding principles for Number Talks could bubble up.  In chapter three of Making Number Talks Matter, Cathy and Ruth introduce ten guiding principles:

  • All students have mathematical ideas worth listening to and our job as teachers is to help students learn to develop and express these ideas clearly.
  • Through our questions, we seek to understand student’s thinking.
  • We encourage students to explain their thinking conceptually rather than procedurally.
  • Mistakes provide opportunities to look at ideas that might not otherwise be considered.
  • While efficiency is a goal, we recognize that whether or not a strategy is efficient lies in the thinking and understanding of each individual learner.
  • We seek to create a learning environment where all students feel safe sharing their mathematical ideas.
  • One of our most important goals is to help students develop social and mathematical agency.
  • Mathematical understandings develop over time.
  • Confusion and struggle are natural, necessary, and even desirable parts of learning mathematics.

The first couple of Number Talks I did in both classes were kind of a hodgepodge. I was trying a bunch of things, looking for the “sweet spot” of just enough disequilibrium to prompt some spontaneous questions, revisions, and “wait… what?” moments.  I was less interested in the content of the Number Talk. I was cultivating the process. I started with dots and subtraction.


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I saw all kinds of strategies from counting dots one at a time to subitizing.  Most kids said they used the algorithm for the subtraction problems. A few subtracted too many and adjusted. I thought one student used constant difference, but it turned it I was just projecting my thinking onto her strategy.  Many of the kids seemed to feel comfortable sharing their ideas.  Some were really open about changing their thinking.

The next time I came in, I used only dots and a Number Talk Image.  I wondered if the images would “nudge students beyond the algorithms.”


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The highlight of this class was when I heard one student say, “I used ________’s method. I added dots in the corners, multiplied, and then subtracted the dots that I added.”  I hadn’t officially named the strategies. I hadn’t even “officially” encouraged them to try each other’s strategies.  They just thought “R’s strategy was cool.”

Cathy and Ruth talk about the importance of “helping students develop social and mathematical agency”.  They say,  “Students with a sense of agency recognize that they are an important part of an intellectual community in the classroom; that they have worthwhile ideas to contribute, and that they learn from considering, and building on, the ideas of others.”

For the next Number Talk, I decided to try some multiplication.  I know, I am all over the place.  Remember, I am just poking around right now.  I am trying to see what these students are willing to share, what they know, and what they are not sure of.

A lot of the strategies used during this Number Talk were based in addition and many students struggled to figure out why their partially correct approaches  were not working.  There was a whole lot of talk happening, which is why it went way beyond 15 minutes. It might have even lasted 30 minutes. Don’t call the Number Talk police, yet!  These kids – all of them – were so present and invested.


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When it came time to discuss 35 x 24, they were talking over me and each other.

They were arguing.

There was one girl, who always sits outside of the group, by the wall. She hadn’t said anything in a while – maybe not ever.  I heard her whisper, “I don’t think you can do that.”

I barely heard her because two other students were going back and forth about whether you should multiply 35 x 2 or 35 x 20.

I had to raise my voice a little, ” Hold on! You need to be respectful of each other’s ideas.  I want to hear all your thoughts, but you need to be respectful and I can’t hear what _____ is saying.” They listened! I swear to you that they actually listened to me, and there is no doubt in my mind that the only reason they listened to me was because I said I couldn’t hear their classmate.

It was silent. I was super nervous, but I asked anyway, “Do you mind saying more? You don’t have to, but I would love to hear your thoughts.”

She said, “I am just not sure if you can do that. I don’t know the answer, but I am not sure you can just add the four like that.”

“Do you think we should multiply 35 by 20 or 2?”

“I think 20 because it is twenty four, but I don’t think we can add the 4 after that. It doesn’t seem right.”

“Yeah,” the boy said – the one who originally suggested an answer of 704 – “I don’t think it makes sense either, but I am not sure what else to do with it.”

It was silent. I thought about drawing an area model, but I just couldn’t bring myself to do it. I didn’t think it would mean anything to them. I am not saying I will never show them an area model, but I just couldn’t commit to it at that moment. Instead, I waited.

A different girl said, “I got 840, but I think it is wrong.”

I asked, “how did you get it?”

“I did it on my whiteboard. 4 times 5 is 20, carry the 2, 4 times 3 is twelve, plus 2 is 14, 2 times 5 is ten, carry the 1, 2 times 3 is 6 plus 1 is 7. 140 plus 70 is 840.”

I was kind of bummed. Where did she get a whiteboard? It must have been in her desk.

At this point, I remember feeling completely overwhelmed and exhilarated at the same time. How can I get them to see the groups of 35? I decided to show them what they already knew.  We went back over the other problems in the string.

“We know that 35 x 20 is 700. We talked about how multiplication can mean ‘groups of’. We can see the twenty groups of 35.  How many more groups of 35 do we need?”

Someone said, “I think we need to do 35 times 4, not just plus 4.”

I asked, “what do other people think?” I really wanted to scream, “YES! You are so right! We definitely need to multiply the 35 by 4!” I didn’t.

“I don’t know,” somebody said. “It seems like we could add the 4, but then it doesn’t.”

I was exhausted.  They were exhausted. I didn’t know what to do. Should I leave them hanging? Should I draw an area model?  Should I add 35 twenty-four times?

I think I wrote the partial products on the board and asked, “Why does this makes sense?”

I think at least one or two kids were able to articulate that we needed to multiply 35 times 4. I am pretty sure one of them referenced 35 x 20 being 700 so 35 x 24 had to more than 704.  There were definitely more than a few kids who were still unsure, but they looked like they would take our word for it.

One boy said, “that was so hard. My brain hurts.”


“Can we do one more?”

I laughed. I said, “No. I want you to want me to come back. Everybody looks pretty fried right now.”

On my way out, I said, “I think multiplication is a good place for us to spend some time. Do you mind if I bring another multiplication string next time?”

They nodded. I think one of them even thanked me for coming. Their teacher, Ms. S, said, “that was so cool. I am learning so much from these Number Talks.  I didn’t learn how to do math like this when I was in school.”

I have read several books about math coaching and I have found them helpful.  However, my favorite way to think about math coaching is through the lens of a teacher.  Often, when I read books about teaching math, I replace the word “students” with “teachers” and I find provocative advice for myself:

Teachers with a sense of agency recognize that they are an important part of an intellectual community in the classroom; that they have worthwhile ideas to contribute, and that they learn from considering, and building on, the ideas of others.”

So, at this point, I have done about four Number Talks in each high school math class. Tonight, I emailed Ms. S and Ms. K. I asked them,

  • What have you noticed?
  • What do you wonder?

I hope they mention the guiding principles, in their own words, of course.



The Unorthodox Guide to Math Coaching

What do you notice? What do you wonder?

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I noticed that I scheduled two events for the exact same time. I wonder if I am ever going to stop over-extending myself and learn how to just say, “no. I am sorry. I cannot do that.” My inability to set boundaries comes from a good place. I want, so badly, to be in as many classrooms, working with as many teachers and students, as I possibly can.  That is great. However, when I overcommit myself, I inevitably end up letting someone down. That is not great. It is not good. It is something I want to change.

First thing Tuesday morning, I canceled my co-planning time with a 6th grade teacher so I could meet with Deb, the Math Interventionist at one of our elementary schools. The 6th grade math teacher didn’t mind. We meet often.  Our next scheduled time was on Wednesday.

Next, I drove out to Mount Vernon Elementary School to tell Deb that I couldn’t make our pre-planning session later that morning because I was scheduled to teach a 7th grade math lesson at the same time. Deb and I just started working with Katie, a kindergarten teacher.  Deb and Katie have taught and planned together a few times.  I have worked extensively with Deb over the years.  Katie and I have a good relationship. Tuesday morning was the first of many monthly planning meetings that we had set up to plan and teach together.

As soon as I told Deb I couldn’t make it to the planning meeting, I noticed she looked totally overwhelmed.  I wondered,  what the hell is wrong with me? Why do I keep over scheduling myself. It is not helpful at all.  I said what I usually say, “It is going to be okay. I have a really good plan.”

Deb, being the amazingly patient, understanding, and trusting colleague that she is, actually listened to the “big plan” I created for her and Katie.


  • Watch the the Counting Collections video.  15 minutes
  • What do you notice?  What do you wonder?  10 minutes
  • Next steps: 10 minutes
    • What parts of the routine you saw in the video would you want to try to incorporate into your classroom?
    • How do we do that?
  • Read plan for today
  • Look at and discuss the counting and cardinality progress monitoring sheet.
  • If there is time, you can try to use it with some of the kids from the video.

Deb looked a little less uneasy. I asked her, “what do you think?” She said, “I love this. I think I can do this. I just needed you to talk me through it. I panicked when you said you weren’t go to be here.”

Of course she panicked. Why wouldn’t she?  I told her I was going to be here.  Believe it or not, I have actually read books about math coaching. I have gone to conferences and taken classes about how to be a good math coach.  All of these experiences taught me that keeping commitments is essential to being a good math coach. I almost convinced myself that I made up for having to cancel the preplanning meeting by meeting with Deb to go over the preplanning meeting. Then, I reminded myself that Deb could have been doing something else instead of meeting with me from 7:30 – 8:30.  She could have been meeting with kids or teachers.

I left Deb and headed to the middle school to teach a 7th grade math lesson.  I had met with this team of teachers last week and they were struggling to find good lessons for their students. They have a small group of multi-aged students who have some “holes” from prior years. They told me they are struggling to use our district curriculum because the 6th, 7th, and 8th grade standards are “beyond what their students can do right now.”  I was thrilled when they asked to meet with me. They were reflective and asked for help.  I asked if I could come in and teach a lesson so they could observe their students. Then, we could talk about what they noticed.

I chose to do lesson 1 from unit 1 of Open Up Resource’s Illustrative Mathematics  Middle School Math Curriculum because I knew the students were going to start their unit on scale soon and it is one of my favorite lessons to teach.  I also knew they would love the interactive apps that are part of the lesson.

I thought the lesson went great. Some kids got frustrated and shut down, but they came back.  Kids were talking over me most of the time, but it was always about math. Some kids were totally playing with the GeoGebra app instead of drawing scaled versions of the letter F, but that was my bad. I didn’t take 3 minutes to just let them play with the app before I started the activity.  Oh, and they were “playing with math” so who cares? Every kid matched up the pairs of scaled figures correctly and most made mistakes while they were doing it or had to justify their reasoning because their partner made a mistake.

After the lesson, the teachers and I talked. They said,

  • “I can’t believe _______ and ______ volunteered to share their thinking.”
  • “How about ______? He really struggles with math and usually doesn’t talk. “
  • “How cool was it when________ shared________?”

I was so psyched and proud that these comments were all about what the kids knew and could do.  My favorite part was when one of the kids who they said really struggled in math handed in his exit slip:

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After his teachers commented that he didn’t know his multiplication facts, I said, “Yeah, but he reasons multiplicatively! It is so much “easier” to teach a kid like this to learn his facts, then it is to teach a kid who memorized a bunch of facts how to reason multiplicatively.”  At this point in the conversation, I panicked.

“Oh my god,” I said.  “What time is it?”


“Phew. I have to go teach Kindergarten in Mount Vernon. Can I come back?”

“Anytime. That was awesome. We would love to get our kids to talk more.  I loved how you taught that lesson, but if I tried to teach like you, they would just tell me I was trying to act like Mrs. Caban.”

“What do you mean?”

“I love how you kept saying, ‘say more’ and ‘how do you know?'”

At this point, one of her colleagues spoke up.  She said, “You say some of those things. I hear you ask kids to explain their thinking. You just might say it differently.”

They thanked me again and asked me to come back anytime. I asked them if I could email them to set up a time where we could meet to plan. They said, ” You are welcome anytime. It was really cool to see our kids learn today.”

I got in my car and it wasn’t until I was halfway to Mt. Vernon that I realized I forgot my bags of shapes for counting collections routine. I wish I could say that doesn’t happen to me all the time, but I can’t. I forget things a lot. So, I drove back to the middle school, picked up my shapes and headed to Mount Vernon.

When I arrived at Mount Vernon, Deb shared how the pre-planning session went.  She said it went great. She and Katie spent most of the time reflecting. Check out some of their responses:

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Now, it was time to teach Kindergarten.  I am absolutely terrified of teaching Kindergarten.  My wheelhouse is grades 3-8.  Deb and Katie knew that I was scared and they agreed to carry on the lesson if I ended up in the fetal position under a table.

I started the Kindergarten lesson by sharing a few questions that I have been wondering about lately:

  • What makes counting hard?
  • How can we make it easier?

I wish I had a picture of the posters we created. They had some really thoughtful answers to my questions. I will ask Deb to take a picture and I will add it to the post later.

We started the lesson by noticing and wondering about berries (Thanks Number Talk Images):

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Then, I modeled how I would count my collection of tiling turtles (thanks Christopher Danielson)

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I thought out loud while I counted my turtles. I counted really fast and I did not touch a turtle every time I said a number. I think I counted 39 turtles.  Before I could finish counting, at least three indignant 5 year olds interrupted me:

  • “You’re not doing it right!”
  • “You said too many.”
  • “You’re going to fast!”

I asked them for some advice.  They told me to move the turtles and make sure I touch each one. So, I did. I wish I had a voice recording of the next part because it sounded beautiful.

I started counting more rhythmically, intuitively aligning the cadence of my voice with the slide of my finger:

“one” (sliiiide)

“two” (sliiide)

Right about here, I think, is where a few students started counting with me. I didn’t ask them to. They just did. By the time I got to 5, we were counting together. Then, I started to play.

I slid the sixth turtle reallllllllly slowly and I didn’t say anything. I heard a cacophony of this:


“six, sev- (pause) six”

“six, seven,”

I kept going. I alternated between speeding up my turtle slide and slowing it back down. I even paused a couple of times. Most of the kids kept the cadence of the count. A few didn’t. Deb noticed it and she intentionally observed those students during the collection count.

After we established that I had 19 tiling turtles, I said, “hmmmm. I would like to record my thinking so Mrs. Reed can see it later.” I wrote 19 on my recording sheet.

Then I continued, “It says that I should show how I counted.”  I started drawing a turtle. “This is going to take me awhile. I don’t really want to draw all 19 turtles. I wonder if there is another way I can show how I counted.”

One girl spoke up right away. She said, “You just have to draw 2 turtles. Draw a 1 in that turtle and a 9 in the other turtle. Then, you have 19.”

I did what she said. Inside, I started to squirm.  “Okay. Does anyone have any other ideas?”

“You could draw a person with a long arm and then draw a turtle in the hand.”

I said, “I could do that.” I thought, where is the nearest table to crawl under?  I asked, “Is there anything else I could do?”

“You could draw a speech bubble that says ’19′”

“You could draw unicorns!”

“I could draw unicorns. That seems like it might be harder than turtles.”  Mrs. Hatt and Mrs Reed were smiling at me. I was sending subliminal cries for help.

“You could draw horses!”

“Unicorns can turn into turtles.”

I said, “that is true.”  Seriously. That is how I responded to the comment about unicorns turning into turtles. I said, “that is true.” I didn’t even realize I said it. I was looking for the nearest table. Deb told me after the lesson. Apparently, she wrote it in her notes.

I was stuck. None of these children were telling me anything that I had hoped I would hear. So, I asked again, probably louder and slower this time,  “I wonder how I could show my thinking without drawing all of the turtles.”

The five-year olds wiggled and squirmed. We had been sitting for what felt like at least three days. There was no more criss-cross-apple-saucing. Nobody’s hands were in their cookie jar anymore.

Out of nowhere, I heard “We could use tally marks.”

“YES! YES WE CAN,” I said in my not-so-neutral voice. “We can definitely use tally marks!” I drew 19 tally marks and decided to move on.  Deb, Katie, and I had talked previously about how we don’t want to force tally marks, circles, or ten frames on kids as a recording strategy.  It is October. We have plenty of time to let recording strategies evolve. We agreed that if it came up, we would highlight it, but we didn’t want to force it. I highlighted the bejeezus out of it.

I asked, “Who wants to help me count my shape collection?”

A resounding, “me!”

Phew.  I handed each child a bag of shapes and a recording sheet.  They scattered.

Deb, Katie and I circulated and conferenced. I intentionally gave the students more than twenty objects. I was hoping for the counting to be hard. I was hoping they would have to count their collection multiple times.  I was hoping to challenge them. The truth is I don’t know how to “teach” 5 year olds how to record their thinking.  All I know is that Deb, Katie, and I are all really interested in figuring it out together. Here is some of the work we collected from the lesson.

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We plan to go over the work and talk about next steps at our next meeting, which is November 7th.  In the meantime, I had an awesome email exchange with Deb and Katie.  It went like this:

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Yup. It’s true. I told Katie I love her. In fact, I tell teachers I love them all the time. Sometimes, I hug people, without asking first. Today, I spontaneously told a student I love her. Yikes. Not the kind of behavior you read about in math coaching books. I am not your typical math coach. I really struggle with:

  • being on time
  • finishing what I start
  • filtering my emotions
  • listening without interrupting
  • overcommiting myself
  • drying my hair before I go to work
  • wearing my name badge
  • keeping track of my belongings
  • blurting out ideas


Here is what I do really well:

  • build relationships
  • advocate for kids
  • take risks
  • share my mistakes
  • ask questions
  • pay attention to the positive
  • think big
  • think mathematically
  • love, love, love my job
  • learn
  • try to be a better math coach








I just started taking the last of my classes to obtain a Certificate in Math Leadership. The name of the class is The Art of Math Coaching and Supervision. I was offered the opportunity to “design” the course so I had some ownership over the work I did. I asked if I could use my blog as a platform for reflection, instead of writing papers.  The six blog posts are supposed to “Set the Stage – Describe relationships I have as a K-12 math leader to advance student performance in mathematics.” 

The first blog post I wrote, Limits, was about me being vulnerable.  I set the stage. I peeled back layer after layer of the shame and frustration that has accumulated over the 20 years of my formal math instruction and exposed the mathematician in me. The mathematician who didn’t know the answer, but wanted to. I thought the message was obvious; being a math coach means admitting what you don’t know.

My professors thought the blog post was written by one of the teachers that I work with. They said she showed a lot of reflection.  They said my blog posts fulfilled some of the “setting the stage” assignment, but they asked me to write a 1-2 page paper describing my professional relationships with other people in my district.  They said the 1-2 page paper would show “how math coaches/specialists become a part of the school culture.”

I can’t help but wonder, why didn’t they think I wrote that blog post?

I hope I have supported the teachers I work with to admit what they don’t know and  learn a tremendous amount with me and that is the culture that I hope to cultivate in math class.


Rife With Conflict

For the past five years, every February, I have presented data to the school board. The purpose of these presentations was to use “data” to convince the school board that my job, and the job of the math interventionists in our district, mattered. I would show them all kinds of colorful charts in the hopes that they would “see” us making a difference.

“Look!” I would say. “All these kids went from red to green!  Isn’t that amazing?! We matter!  We are doing a good job!  Right?”

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Most of the time, the school board was thrilled with my colored charts. “Yay, Sarah! Look at all that progress.”  Last February, one school board member, I will call him Mr. G,  had a different reaction. He wanted to talk about the kid in the bottom row:

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He said, “I am looking at these three words, ‘did not improve’ and I am thinking that tells me that you didn’t do your job.”

How would you have responded to this comment? What would you have said?

Honestly, I don’t even remember what I said. I probably said something like, “there are so many factors that impact student learning,” or maybe even “well, we shoot for 80 percent of students to meet grade level standards and look at all the students that did improve, look at all that green, blah, blah, blah, blah…..”

I left that board meeting fuming mad.  I heard those words every second of every day for weeks.

You are not doing your job.

So many people told me to forget about that comment. They said, “he doesn’t know what he is talking about” ,”he is grumpy,” “he never has anything positive to say”, “you are doing a great job” “the interventionists are doing a great job”.

Those words rattled around in my brain for months. At some point, I reframed them as  a question,

Am I doing my job?

My reflections became more transparent with every day that passed.  I began to realize that I had created a monster, a data monster.  I had conditioned the school board to expect colorful charts. Every year, my main focus was getting one more math interventionist position into the budget.  What was the quickest way to show the board we needed one?  Data.

The tricky thing about data is that I can pretty much make it say whatever I want it to.  A few years back, we had a math interventionist who was only able to visit one of our elementary schools once a week. Come February, I made a bunch of colorful charts to show the board that the kids she saw on that one day weren’t making progress, but the kids she saw 4 days a week at her other school were making progress.  The data I showed the board was real. It was based on screeners, common assessments, NWEAs, etc.  I didn’t manipulate the data. I manipulated the story.

What was the story I was telling the board? It was a story about how the math interventionists job was to “fix kids”.  I was showing the board a bunch of numbers and colors. I wasn’t showing them kids, teachers, and math classrooms. It makes total sense that Mr. G didn’t think any of us were doing our jobs.  I had been spending years “showing” the board that our job was to “fix” kids.  The worst part about the story I was telling is that it wasn’t true. If you follow me on twitter or have read even one of my other blog posts, you will know that I am not in the business of “fixing” kids.  I hope you will also know that I am not in the business of “fixing” teachers. So what happened?  How did the story I was telling become so far removed from the story I was living?

I think the gap was born out of simplicity, efficiency, and trust. I thought I needed to “sell” the board a quick and easy need for more math support positions.  I didn’t trust that they would understand the uncomfortable truth of working with kids and teachers. The truth? We have been trying to collaborate, but collaboration is messy, uncomfortable, and rife with conflict.

This year, I decided to change the story I was telling. I asked if my first presentation could be early in the year. Last Wednesday, I started to tell a different story. A story that matched the truth about the work the math interventionists and I have been trying to accomplish.

We, as a math support team, have been trying  to collaborate; with each other, teachers, parents, administrators, and studentsLast spring, we created a vision statement:

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Then, we started to draft a document that defines what math class should look like. Many of us had read Minds on Mathematics by Wendy Ward Hoffer. We used this book as an anchor when we defined the essential elements of a math class: Challenging Tasks, Collaborative Community, Intentional Discourse, Conferencing, and Reflection.  Below is a screenshot from this document. Keep in mind, it is a DRAFT.  We didn’t create this work in a vacuum. As a district, certain buildings and groups had done some important work in the past that is reflected in the chart we created. Some elementary schools had done learning rounds on the Common Core Math Student Practices and the NCTM Teaching Practices.  The elementary and middle school teachers had spent time trying to answer the question, “what is a workshop model?” I have facilitated learning labs at the K-8 level during grade level meetings for two years.

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As a team, the interventionists and I started the year with an agreement that all math interventions would start with, and be anchored to, classroom instruction.  We re-defined the “data” we wanted to collect.  Yes, we are still going to look at data from universal screeners and common assessments, but what else are we going to look at?  How are we going to know if collaboration is making  a difference?  How are we going to know if we are doing our jobs?

We each created an excel spreadsheet that we will all use to collect “data”.  The first page looks like this:

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Yes, it also has a bunch of test scores on it. There are many columns to the right of the ones you see above.  The columns you see above are the most important. The second page of the document looks like this:

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Recently, we have noticed a glaring problem:

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We never established goals and norms with some of the teachers we are collaborating with.  We didn’t think we “had” to.  Some of us have worked together in the past.  We knew each other well.  We had established relationships with these teachers.  We never sat down and formally articulated norms and goals.  We made assumptions.  This was a bad idea.

Right now, some of us are feeling uncomfortable and frustrated.

  • “I feel judged.”
  • “I just want you to pick up my kids and work with them outside of my classroom.”
  • “I don’t want you to see my kids. I will just work with them myself.”
  • “There are too many adults in my room.”
  • “I am doing all the planning. How do I get the teachers to work with me?”
  • “How are we supposed to schedule this? I don’t have time to wait for a mini-lesson to end.”
  • “I just need a break. I have 26 kids and I feel like I don’t even know them, yet. I have to share them with too many people. I just want two weeks with my students, just me and my students.”

All of these statements are true.  All of them are valid. We are at one of many pivotal points in our journey.   We are at a point where we can give up or reflect, revise, and move forward.

Even though we feel frustrated and uncomfortable, there are some wonderful things happening in our math classrooms.  Did you see that collaborative planning doc above?  It is awesome!  Teachers and interventionists are meeting regularly to plan and teach children together.  We are all trying to improve our craft. We are trying to do this work together.  At my board presentation, I shared an example of this collaborative work.  You can see it here.  I also shared the struggles we are having.

We are all feeling incredibly vulnerable.  This is good!  Vulnerability is at the heart of true collaboration.  However, feeling vulnerable is scary.  We might need to back up; maybe slow down, ask for help, and be courageously honest. Collaboration is messy, uncomfortable, and rife with conflict, but it is essential for equitable and effective math support.

Save the Least Efficient Strategy for Last

Recently, I have been thinking a lot about the 5 Practice for Orchestrating Productive Mathematics Discussions by Margaret S. Smith and  Mary Kay Stein.  In particular, I have been thinking about sequencing.  Often, I hear people say that we should sequence student work from the least efficient strategy to the most efficient strategy.  I agree that there are times that we might want to sequence work from least efficient to most efficient, but I think we should be intentional about when and why we chose those times. I also wonder about the times that it might be more useful to save the least efficient strategy for last.

Look at the image below and write down one thing you notice and one thing you wonder:

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When I asked a group of 6-12 teachers what they noticed, they said:

  • “I notice missing dots in all the corners.”
  • “I notice each image has a middle part and then dots surrounding the middle part.
  • “The middle part is a square in three of the images.”
  • “I notice the number of the dots on the outside sections increase by one each time.”
  • “I notice that the number of the dots on the top and sides tells you how many rows and columns there will be.”
  • “I notice the number of rows and columns increases by one each time. So the step number tells you how many rows of three dots you need
  • “I notice a pattern. It goes (1×1)+(1×4), (2×2)+(2×4), (3×3)+(3×4), (4×4)+(4×4).”
  • “I notice little white squares in between the block dots.”

Then, I asked them what they wonder.  They said,:

  • “I wonder what the hundredth one would look like?”
  • “I wonder what the one before the first one would look like?”
  • “I wonder what the next one would look like?”

Perfect timing. I showed them this slide:

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They all went to work.  As they worked, I monitored.  I had anticipated that, in a room full of 6-12 educators, several people would see the following expressions in the dots:

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The original version of this Illustrative Mathematics task actually gives these two expressions:

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I decided to remove the text from this task because I was hoping  to lower the floor and raise the ceiling. My goals were to get us  thinking more deeply about equivalency and the language of math. How do we use symbols to convey relationships that we see in images?  How does the structure of an expression help us represent what patterns and relationships we see? How do we know that we truly understand what another person sees and thinks? Often, I wonder if we assume we know what our students are thinking when, in fact, we are merely projecting our own thinking onto their words.

So, with those goals in mind, I set out trying to find:

Someone who quickly pulled the following expression out of the images:

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I was thinking that this is one of the more efficient expressions for finding the total number of dots in any step.  One person had already eluded to it during our notice and wonder phase. I planned on sharing it first because I wanted to use it as an anchor for equivelance.  I also wanted us to look beyond efficiency.  I wanted us to wonder if there were times when a long clunky expression with many terms might serve a purpose. Maybe? Maybe not? I didn’t know the answer, but I wanted to explore the question.

So, our first friend, Lily shared where she saw Screen Shot 2017-10-14 at 10.17.17 AM.

When Lily finished, I asked Rita, “What do you notice about the pattern you noticed earlier and the expression that Lily just shared?”

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(1×1)+(1×4)        (2×2)+(2×4)          (3×3)+(3×4)           (4×4)+(4×4)

She said “Oh. Well they are equivalent. You can see the Screen Shot 2017-10-15 at 6.54.26 AM in the first term of my expression and you can see the Screen Shot 2017-10-15 at 6.54.37 AM in the second term in my expression.”

“What does n represent in the image?”

“It is the number of dots in the bottom row.”

I asked, Lily, “Is that what n represents in your expression?”

“No. n is the step number, but it doesn’t matter because the step number is equal to the number of dots in the first row.”

Next, I asked Chris to share. When I first checked in with Chris he told me, “I am thinking about the way that Jared saw the sequence when we noticed and wondered. It sounded like he saw the three horizontal dots in the first image as the constant. I am trying to create an expression that matches his description of the pattern.”

Chris explained to the group that the expression he created was a little clunky, but he was trying to capture what Jared saw staying the same and changing.

3n + n (n+1)

Chris explained that he heard Jared referring to the three horizontal dots as the constant.

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Then, he noticed that the stage number told us how many rows of the constant we needed. The term 3n represented the array formed by n number of 3 rows.

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Then, he had some extraneous rows and columns that he needed account for.  He realized that there were always n+1 groups of n dots.

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So, he came up with the expression: 3n + n (n+1).  He asked Jared, “Is this how you were seeing it?” Jared said, “yes.”

Then I asked, is 3n + n (n+1) equivalent to Screen Shot 2017-10-14 at 10.17.17 AM?

Everyone agreed.  Someone shared that they could “see” it. “If you distribute the n, you have n squared plus n. Then, you just add the n to 3n and you have n squared plus 4n.”

Next, I asked two people to come up to the document camera, Rachael and Max. Earlier, when I was monitoring, I found Rachael describing how she saw the pattern changing and staying the same.  She said, “I can see a relationship, but I can’t find an expression to represent my thinking.”  Max asked her to describe the relationship she saw. She said, “I can move one of the dots from the top row down to the lower left corner to make a square and then just add the dots that are on the border.” Max suggested the expression  Screen Shot 2017-10-15 at 4.04.22 PM and described how he saw it in the dots:

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Rachael said, “I see where your expression is in the dots, but your expression doesn’t represent what I was seeing.”

Max and I were determined to find the expression that matched Rachael’s thinking.  We tried to rephrase what we thought she was saying.  We marked up the picture as we spoke:

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She said, “Yes! That is how I see it.”

After some false starts, questions, and dead ends, we figured it out!

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We decided n would represent the number of dots in the top row, which is also the step number. I asked Max and Caitlin to share our experience and speak to how they knew their expression was equivalent to the others. They said they were able to prove equivalence algebraically.  They shared their work.

As I sit here now, I am realizing that I can “see” the structure of Caitlin’s expression in the image. I don’t think I need to verify equivalence algebraically.  I think all the dots are accounted for.  Each term represents a group of dots.  n-1 represents the top row of dots. n+1 squared represents the lower left corner of dots (arranged as a square). Is the “action” of the dot moving represented by the subtraction symbol or is the subtraction symbol really a negative sign? I still have so many questions.  The final n represents the column of dots all the way to the right.

I wish I could say that I had made this connection during my session with the 6-12 teachers, but I didn’t. At the time, I was primarily focussed on creating a situation where a group of educators would share their thinking, listen for understanding, and change their perspectives, which is why I saved the least efficient strategy for last.