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

Yesterday, I got this text:

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

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

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

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

At this point, 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 your 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:

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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?

“yes.”

“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?”

“yes.”

“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.”

“pass.”

“Hayley, do want to share a solution?”

“pass.”

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?”

“204.”

“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?”

“yes.”

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 ten groups of 35 and the 4 groups of 49.  How many 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.”

Silence.

“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.

10/24/17 PRE PLANNING MEETING

  • 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?”

“11:00”

“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:

“Siiiiiiiiiiiiiiiiix”

“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

BUT

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

 

 

 

 

 

 

Culture

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.