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.”

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.

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:

Screen Shot 2017-10-14 at 9.52.11 AM

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:

Screen Shot 2017-10-14 at 10.01.43 AM

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:

Screen Shot 2017-10-14 at 10.17.29 AM.png

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

Screen Shot 2017-10-15 at 6.31.25 AM

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:

Screen Shot 2017-10-14 at 10.17.17 AM

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?”

Screen Shot 2017-10-15 at 5.08.30 PM

(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.

Screen Shot 2017-10-15 at 3.51.51 PM

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.

Screen Shot 2017-10-15 at 3.53.51 PM.png

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.

Screen Shot 2017-10-15 at 3.58.26 PM

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:

Screen Shot 2017-10-15 at 4.07.55 PM

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:

Screen Shot 2017-10-15 at 4.10.18 PM.png

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

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

Screen Shot 2017-10-15 at 4.18.39 PM

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.

 

 

 

 

Limits

II am a K-12 math coach and I never took Calculus. I hated math in High School. I hated it even more in college. It is really challenging for me to memorize formulas, vocabulary, math facts…. anything really. I am a slow processor, but a fast talker.  I won’t remember anything unless I truly understand it which takes a looooonng time.

When I was a kid, I never knew my multiplication facts. I just couldn’t memorize them. So I taught myself some tricks. Trick #1: count really fast. If I knew 6 x 5 was 30, then I could just count up really fast to figure out 6×7.

You know what I am talking about: 6×5 is 30 so six times 7 must be 31,32,32,34,35,36, 37,38,39,40,41,42. In my head it sounded more like this

“thirtyone thirtytwo th-three,

th-four, five, th-six!” (head bobbing every so slightly- almost imperceptively.)

“th-eight,th-nine, fty, ftyone, forty two!”

Super fast. Right?

Speed matters. Right?

At some point, someone tried to teach me a trick to remember the nines, but it didn’t make sense to me so I never remembered it. Eventually, I figured out that I could use 9×9=81 to figure out 9×8 by subtracting 10 and adding one because that was the same as subtracting 9.  That was as close as I got to using the properties of operations and, at the time, I had no idea that what I was doing had anything to do with the properties of operations. I just thought it was handy that 9 was so close to ten.

Anyway, I didn’t really learn a whole lot of math as a K-12 student. Most of the understanding that I have now has been developed since I started teaching. Now, I love math. I love learning. Now, I know my math facts. I don’t need to use tricks. I don’t even need to use the properties. Using and understanding the associative and distributive properties over time eventually led me to recall multiplication facts. I own them. I can even recall them pretty quickly.  More importantly, now, I have a deep understanding of the properties of operations and the major role they play in our number system.

Every year, I stretch myself to expand my understanding of the properties of operations. I read about them, I ask questions about them, and I play with them A LOT.

This year, I decided it was time I played with Calculus. I thought about taking a Calculus class, but I don’t really want to take a class. I just want to learn; on my own, in a messy, not linear, no-time-crunch, kind of way. I just want to play with Calculus.

I am a little nervous. Sometimes, Calculus seems like a big, burly, sand throwing, dodge-ball-pelting, kind of playmate. Usually this happens when I let myself get into a space where I think I should know something. I get distracted by the little voice in my head that says, “you should know that word.  You should know that symbol.  You should. You should. You should.” Then, it isn’t fun anymore. So, I stop. I walk away. I wait for the curiosity to bubble up again.

The first day of school, my colleague/friend/mentor/math-mate Robyn told me she was trying something a little different with her AP Calculus class this year. She said she was going to give her AP Calc students a problem from Paul Forrester that she adapted. She was hoping her students would struggle a bit and, hopefully, develop an appetite for some important information. She seemed excited about starting the year with a problem. She hoped her students would ask a lot of questions as they solved this problem.  I asked if I could try it. She scribbled the problem on a piece of paper for me.

Screen Shot 2017-09-09 at 9.19.54 AM

Suppose a door that is pushed open at time t=0 and slams shut again at time t=7s. While the door is in motion, the number of degrees, d, from the closed position, is modeled by this equation. How fast is the door moving at the instant when t=1s?

One of the first things that I said was, “If  the door was slammed, why did it take seven seconds to close?” She laughed. “Okay,” she said, “maybe it didn’t slam.”

Over the next few days, whenever I had a moment- sitting in the car while my kids were at soccer practice, while I was running, driving around the school district – I would think about the door.  Finally, when I had a minute, I opened Desmos and started a graph.  You can take a look at it here, if you want. Please remember, I am sharing my messy learning with you. I am not looking for feedback, right now. I am happy to answer questions.

Seeing the graph really helped me wrap my head around the problem. I wondered a couple of things:

  • Does a negative exponent always signify exponential decay?
  • Can I call the curved part of the graph a “parabola” even though it is attached to a decaying exponential curve?  What do we call a function that is part quadratic and part exponential?
  • Does the “peak” of the “parabola” represent the moment in time when the door started closing? Is that why 100 degrees shows up twice? Once, when the door reached 100 degrees on the way out and, again, when the door reached 100 degrees on it’s way back to the door jam?
  • Can I use the formula for speed to solve this problem?  If speed is measured by distance traveled divided by time spent traveling, can I measure the difference in degrees between two moments in time and divide it by the time it took the door to travel that distance?

I decided to try this. I think I found the difference in degrees between the door’s location at 0 seconds and 7 seconds and divided it by 7.  I can’t find the scrap of paper that I worked on. I texted Robyn.

Screen Shot 2017-09-09 at 10.07.03 AM

She responded:

Screen Shot 2017-09-09 at 10.07.11 AM

I decided to call her because I couldn’t explain it in a text.  She asked me to explain what I had found. As I was explaining, I realized that my time span was too long. She reminded me that I was trying to find the speed at 1 second. I was averaging 7 seconds. She re-iterated the word, “instantaneous.”

Instantaneous.

I put the problem aside for the night. The next day, after school, I went back to Desmos.

Screen Shot 2017-09-09 at 10.07.46 AM

Screen Shot 2017-09-09 at 10.07.55 AM

I added some more points to my table. What if I found the degrees at .99 of a second? That is pretty close to instantaneous, right? This is when the problem solving started to get really fun.

Screen Shot 2017-09-09 at 10.08.16 AM

Screen Shot 2017-09-09 at 10.08.34 AM

I went back to the drawing board.  I started plugging in points with ridiculous amounts of nines in them. At one point, I went too far. I was using the calculator on my computer. It got tired of my nines. It started rounding.

Screen Shot 2017-09-09 at 10.42.43 AM.png

Screen Shot 2017-09-09 at 10.09.07 AM

Screen Shot 2017-09-09 at 10.09.21 AMI walked away again. I went back to work, soccer practice, dinner making, life, but I kept thinking about the door. I couldn’t wait to find those few minutes in my day where I could open my Desmos graph and tinker around some more.

Screen Shot 2017-09-09 at 10.09.35 AM

Screen Shot 2017-09-09 at 10.09.48 AM

At this point, I found a serindipitous mention on my Twitterfeed. My friend Chase Orton had shared a blog post about the importance of collaboration between HS teachers and elementary teachers.  I shared my thinking with Chase. He asked me to see if I could use a visual approach to the problem.  I thought about that for awhile.

Yesterday, I stopped by Robyn’s office and we chatted about what I had learned so far. We captured our conversation on her whiteboard:

Screen Shot 2017-09-09 at 11.14.56 AM

As we were talking, I noticed that self doubt creeped in every time Robyn used a math term to capture what I had described.

I described the change in distance divided by the change in time.

She wrote a “triangle d” symbol on the board.

That little voice started whispering, “You don’t know what that means.”

I described how I was  finding the slopes of really short lines on the graph.

She asked me what we called the line that went through a circle.

I almost panicked, but I didn’t. I was with Robyn. I don’t need to panic with Robyn. Instead, I told her about the voice. I told her I struggled with recall. I told her I wanted to understand but I couldn’t remember what it was called.

She said, “it starts with an “S”.”

I remembered a problem that I was solving with Chase last spring. “Secant?”

“Yup. What about the line that touches a point on the outside of the circle?”

I asked, “What does it start with?

“T”

“Tangent?”

“Yup.” Then, she talked for a little while about how the slopes of the little lines were related to secants and tangents.

I told her what I had noticed about place value patterns when I was solving the door problem. I noticed that I was trying to find the difference between a number super close to 1 and 1.  The delta change (am I using this correctly?) is always going to be 1 whatever the furthest place value to the right is called: one hundredth, one thousandth…..one millionth……one infiniti-th?

Robyn helped me articulate that I was trying to get as close to zero as possible. She introduced a new word.

Limit.

I wondered if I needed a symbol to represent this distance that I can’t really pin down.

Robyn challenged me to try  this:

Screen Shot 2017-09-09 at 11.12.19 AM.png

She also gave me the second problem that her AP Calculus class tried. I am excited to work on both. I will let you know how it goes.

For what it is worth:

I wrote this in April and never hit publish. I have been letting it simmer. It seems like it might be worth the conversation.

I am on my way home from the #NCSM17 and #NCTM17 conference. I had the privilege of being part of a team of educators who presented at both of these conferences.  We shared a story about how we collaborated K-12 to improve our ability to teach math.

During our story, we shared some powerful #MTBoS experiences that transformed our teaching.  These experiences were so transformative because they simultaneously lived inside and outside of the boundaries of our zipcode.  We worked together in our district, but we also worked together in the Math Twitter Blogoshpere.  We reflected with each other and we reflected with educators all over the world.

I have been sitting in the Charlotte airport for 5 hours. My connecting flight doesn’t leave for two more hours. I have spent the majority of my time in the airport reflecting.

This morning, on Twitter, I shared links to our NSCM and NCTM presentation with several #MTBoS folks who were instrumental in the evolution of my (and my colleague’s) learning this year.  I engaged in a thoughtful, and somewhat provocative conversation with some twitter friends, none of whom I have ever met in person.

Take a look:

Screen Shot 2017-04-08 at 7.12.11 PMScreen Shot 2017-04-08 at 7.12.21 PMScreen Shot 2017-04-08 at 7.12.33 PMScreen Shot 2017-04-08 at 7.12.46 PM

As we talked, I wondered:

  • When and how do we establish relationships on Twitter?
  • Does humility and vulnerability impact who we engage with on Twitter?
  • Does Twitter have a culture?  Is it persavise or incidental?

Shortly after I wrapped up a two-hour reflection session with @Simon_Gregg@nomad_penguin@KentHaines@TAnnalet, and @m_pettyjohn, I read Dylan Kane’s blog post: On NCTM and MTBoS

I found Dylan’s post fascinating. It seems like Dylan is wondering if people who are new to MTBoS know the purpose of MTBos.  I have only been engaged with MTBoS since last May so I would consider myself on the “new” side of the experience. I am not sure what the collective purpose of #MTBoS is, but I have a pretty clear vision of my own purpose.  I actually tweeted it during one of the conversations that I mentioned above.

Screen Shot 2017-04-08 at 7.37.51 PM

Sometimes I do this by just engaging in a problem solving experience.  Other times, I ask for help with a teaching struggle.  I don’t actively remind myself of my purpose, but I think it is usually a motivating factor of my engagement.

I guess I wonder if it is possible for #MTBoS to have a collective purpose.  It is a community right? But it doesn’t have a structure or rules.  There are no bylaws or sign up sheets.  You can’t point to it or place it on a map. It doesn’t really live anywhere.

Yet, it most certainly feels alive.

It evolves.  Doesn’t it?

Shouldn’t it?