I am so freakin’ excited!

I am part of a group of inspiring K-12 teachers in my district who just embarked on a crazy yearlong professional development journey together.  We designed a collaborative professional development certificate that will earn us the equivalent of three credits for re-certification.  Here is our plan:

We are going to meet monthly to choose a task that has a low floor and a high ceiling.  We are going to do the task together and anticipate what our students will do and how we will push their thinking without stepping on it.  Then, after we do the activity with our students, we are going to blog about it.  Before we meet next, we will all read each other’s blogs.  Next month, we meet again and start the cycle again.  Who are we?

  • A District Math Coach
  • A Life Skills Teacher
  • 2 Middle School Math Teachers
  • A Middle School Special Ed. Teacher
  • 3 Fourth Grade Teachers who teach in different Elementary Schools
  • A High School Math Interventionist
  • An Elementary School Teacher who teaches a combined 2/3 classroom
  • An Elementary School Math Interventionist
  • A High School Math Teacher

We had our first meeting this week.  We  all set up our blogs and Twitter accounts. We chose our first task from the fantastic math site, Math Pickle.

We chose to play A Little Bit of Aggression with our students.

After we played the game ourselves, we wondered :

  • Is it an advantage to spread your armies out so they are not bordering each other?
  • Is it good to put a high number of armies in the middle territory?
  • If you put a big number of armies in the mix,  should you get rid of your armies quickly so you can go first?
  • Does the first or second person always win on a certain map?
  • Could you play with multiplication instead of addition?
  • Could you use fractions of armies?
  • Is there a rule you could use to determine how to win, given a certain board and a certain number of armies?
  • Is it ever an advantage to put one army in a territory?

This is what we wonder about our students:

  • Will they stay with it long enough to develop a strategy? 
  • How will they decide where to put their armies?
  • Should I introduce it as a whole class or in small groups?
  • How can I scaffold it so my kids can access the reasoning part of the task?
  • What happens to the student who doesn’t persevere when their classmate steps up and takes over?

My favorite part of our meetings, besides getting everyone into the blogoshpere of course, was when we all started to organically help each other plan.

“Could you use blocks for the armies so the kids could manipulate them easier?”

“You could introduce the game over a week – a little bit at a time to build anticipation.”

“We could ask our students, ‘what would you do differently next time'”

“Yes! Or maybe ask them to write down questions they are thinking about as they are playing.”

“Or what if you paused throughout the game and asked them what they notice and wonder.”

“How did you decide where to put your armies?”

“Maybe you could make a bigger game board – like poster size”

Finally, I asked them, how does this activity fit in with math?

Somebody said, “It build perseverance.” Yup. “It also builds community because they will be talking about their reasoning.”  Hmmmm. Building community by talking through our reasoning. What a fantastic idea.



I think it is about the conversations.

For the last two weeks, I have been mulling over my first blog post of the year.  Every time I sat down to write, it felt clunky and half true.  I didn’t blog much this summer. I tried to write about the fantastic experience I had working with K-12 educators in my district, but I fell short on the follow through.

At the end of the first day of school, I read Tracey Zagar’s provocative post about what NOT to do on the first day of school.  As always, she made me think about whether my beliefs and practices are aligned.  Part of my job is to coordinate curriculum for our district.  Several years ago, I pushed hard to get us to collaboratively design curriculum, instead of purchasing a textbook or program. Prior to the adoption of the Common Core Standards, our four elementary schools were each using a different outdated textbook as their math “curriculum”.  I saw the transition to new standards as an opportunity for us to work collaboratively to understand the standards and build a focussed, coherent, and rigorous curriculum around our understanding. It is really challenging for teachers to NOT use a math program, especially elementary school teachers.  There are several reasons for this, including limited planning time, shallow pedagogical content knowledge, and a disconnect between their own math identity and the identity cultivation that needs to happen in math classrooms today.  I am not blaming or judging teachers, at all.  This is just the current reality of education.

In the last few weeks, I have had the pleasure of teaching, planning,  assessing, and (most importantly) reflecting with a variety of K-12 teachers.  Throughout these experiences, I fluctuated between feeling sure and unsure.   In our district, for each of our units, we give a common unit assessment that was collaboratively designed.  For some units, if the standards are closely aligned with standards from the prior grade level, we give a pre assessment.  It is the same assessment that we give for the post assessment.  The purpose behind giving the pre assessment is to see what prior knowledge the students are bringing to the unit. What do they know? What are they still confused about?  How can we build on what they know?

In the second week of school, I met with the second grade teachers.  We discussed the common Unit 1 pre assessment that they had given the week before. Several teachers were overwhelmed.  They conveyed their worries about all the assessing we are doing – not just in math, but all subjects. They talked about how some of their students were overwhelmed by the Unit 1 pre assessment.  As I listened, I felt unsure.  I questioned all the work we had been doing in an attempt to create consistency in our math instruction. We created these unit assessments together in an attempt to understand the standards and guide our conversations about what it looks like and sounds like to meet the standards. We have also been doing learning rounds to focus on the standards of math practice.  Last year we started doing learning labs at some of our grade level meetings so we could teach and learn together in actual classrooms with live students. Were these  assessments hindering the work we had done on cultivating a math culture that honors inquiry and risk taking?

Then, I thought about the teachers with whom I am currently collaborating. In one of our buildings, the math specialist and 2/3 teacher are co-planning and co-teaching the 2nd graders. They have been using Jo Boaler’s week of inspirational math videos and tasks. They invited me in to teach one of the tasks. When I entered the room the day of the lesson, the students were scattered around the room. Many of them had headphones on to dampen the noise. Some were sitting on t stools or exercise balls.  All of them seemed to be engaged in what they were doing.  One of them looked up at me and said, “Mrs. Caban. I just made three mistakes.  My brain is growing!”  The teacher informed me that they were finishing up their pre assessments.  I commented that they didn’t seem worried or upset.  She said she told them that they may not know the answers to the questions, but she wanted them to give it a shot because if they tried, even if they got it wrong, it would help her understand their thinking.

Back at the grade level meeting, the same teacher asked her peers about one of her student’s work.  She was not sure if the student is demonstrating an understanding of this standard:

    Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
    100 can be thought of as a bundle of ten tens — called a “hundred.”
    The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

Here is my drawing of what the student did. I have been meaning to get back to the teacher and take a picture of the actual student work, but I just haven’t had time.


So, this wonderful group of second grade teachers proceeded to engage in a conversation about what this student knows about place value.  One teacher didn’t think he demonstrated understanding of the standard because he included the values of all the digits and the question specifically asked for the value of the 2 in the hundreds place.  Another teacher disagreed.  She said he is showing the value of the 2 in the 200.  I think at this point I may have asked them what this standard had to do with the other standards that they teach.  They started discussing decomposition and how it relates to work with addition and subtraction. The teachers asked, “What did the student do for the other numbers?”  (there were 4 different three digit numbers).  He did this for one of them:


Looking at actual student work is incredibly important.  It is essential if  teacher’s are going to calibrate their understanding of student understanding. The irony is:  I don’t have it.  BUT, you are looking at my understanding of the student work that I analyzed with my peers and that is what this blog post is really all about.

The above story is one of many similar stories that I could tell about my first three weeks of school.  I have met with every grade level in our district. At every meeting, we took out pre assessments, analyzed them, argued about them, adapted them, and used them to plan.  There is a lot about our common assessments that bugs me:  some are too long, some of the questions stink, it takes a long time to score them because we use standards based rubrics, and my least favorite thing: they are stressing some people out.

What do I like about them? We created them together and they evolve with us.  Here are some examples of how they have evolved this month:

  • The first and second grade teachers decided to only give the interview section of the unit one pre assessment next year.  IF a students gets a “3” on the standard that is assessed in the interview AND seems comfortable with persevering, they will continue with the rest of the assessment. If not, they will use the information they get from the interview to plan instruction.  They said this will be less stressful and still informative.
  • The 6th grade teachers seemed to enjoy watching a teaching channel video on one of the standards before we commonly scored.  They want to do more common scoring because it really helps them reflect on the implications for instruction and what changes we need to make to the assessment and rubric.
  • The K-12 district interventionists want to devote some of our meeting time to making an assessment bank for the standards so teachers have a variety of vetted assessments to use to inform their instruction and measure progress.
  • Next year, we are going to try to postpone any pre assessments long enough so more people can devote the beginning of the school year to using Jo Boaler’s week of inspirational math without feeling rushed.

I am not claiming that our assessment system is awesome. I feel pretty crappy that there are kids in our district who felt stressed out in math class, but I think the changes that we are making will be an improvement.  What is most important is that WE are making the changes.

Grateful. Humbled.

My name is Sarah Caban and I am a district math coach. Phew. You have no idea how difficult it is for me to say that out loud.  It has always seemed like such a loaded statement.  I rarely say it.  Usually, I default to “I teach math.” or “I teach math with other teachers.” I always worry, if I say “district math coach”, what do people hear?

Do they hear DISTRICT math coach and think “oh, she must know all the math – high school – we should probably ask her to solve some complicated Calculus problem. Yikes. Don’t say District.

What if I just say MATH coach?

“She must be a “math” person.  She probably doesn’t make mistakes.  She knows her facts, she does them fast.  She is THE math person. We should probably blame her for all this Common Core nonsense.”  Ahh!  Don’t say Math.

Okay. Let’s go with COACH.  Hmmm. What might people say?

“Oh!  Lacrosse? Soccer? Track?  That’s great.  Good for you.”

Then, I try to explain, “well, I am an instructional coach.  I coach teachers.”

“Oh. Coach teachers to do what?  Play lacrosse?  Soccer?”

Me, hesitantly: “Coach teachers to teach math?”


I have had this elusive title for 5 years.  One would think I had come to terms with it by now.  Identity is complicated, particularly when it comes to teaching math.  For more on this, read Hannah Mesick’s post about Jumping Into the Ball Pit.  Thanks for being a mirror, Hannah.

As part of my job, I attend weekly administrative team meetings.  Last spring, as a team, we read the book School Culture Rewired. I was simultaneously reading NCTMs’ Principles to Action. Our superintendent gave us a homework assignment:  Identify a part of our culture that we want to change and explain how we would cultivate  that change.  I think she just wanted us to sketch out some thoughts, but I saw an opportunity.  I decided to write a proposal.

I focussed on the professionalism section from PtA. This was my purpose:

“It must become the norm for teachers to believe that they have a professional responsibility to collaborate with their colleagues and open their practice to collective observation, study, and improvement. In short, we must move from “pockets of excellence” to “systemic excellence” by providing math education that supports the learning of all students (and teachers) at the highest possible level.” – NCTM Principals to Action

How would I work towards achieving this purpose?

  • Invite K-12 teachers do math together this summer.

What would we do? 

  • Read Principals to action
  • Read Progressions documents
  • Watch Shadowcon talks and Ignite talks– choose a couple – reflect on calls to action
  • Understand progressions of learning
  • Plan (DO MATH) and practice  implementing tasks (ANTICIPATE HOW OTHERS MIGHT DO MATH).

I submitted this proposal to my superintendent, along with a budget, which included teacher stipends and copies of Principles to Action for each participant.  My superintendent approved it by the end of the day. She was thrilled and supportive.  She asked me to please send her the dates ASAP so she could make sure she attended at least one of the days.  Also, would I consider inviting Special Education staff to join us? Absolutely.

There are about 60 classroom teachers in our district, K-12.  This includes math interventionists, special educators, and G/T teachers. It doesn’t include ed. techs or specialists. Twenty teachers signed up for three days of summer math.  Note to self: next time include ed.techs and specialists (gym, art, music).

Wow.  Twenty teachers, representing K-12, including interventionists and special ed. Awesome.

Now what do I do?  I felt pretty overwhelmed.  How do I makes sure that every minute of these three days moves us towards “collective observation, study, and improvement”?

It has been over two weeks since the completion of the these three days. I have been trying to figure out how to describe the experience.  I am not know for my ability to think linearly.  I am more known for swan diving into rabit holes. In the next several blog posts, I am going to attempt to describe the incredible journey that I experienced with 20 fearless, curious, and creative educators.

Stay tuned.

What’s my purpose?

Recently, a middle school math interventionist came to me with a problem.  She has a small group of middle school students who she teaches every day.  She worries a lot about whether or not they like math. She says they don’t persevere.  She wants them to enjoy problem solving but they give up so easily. I asked if I could join her in class one day so I could experience, first hand, the dispositions of her students.  She welcomed me with open arms.

At this point, I wondered, what is my purpose?  Clearly my purpose was not to miraculously, in sixty minutes or less,  get these students to love math and see themselves as vital contributors in a math community.  Right?

I was pretty sure these students knew that they struggled with Math.  I don’t think they needed me to point that out.  I wondered if it would be worthwhile to ask them about their struggles. Maybe we could identify some times that struggle could be helpful.  I decided to keep an open mind and try to do more listening than talking. The cool thing about my job is that I work K-12 and I have worked K-12 for the last 5 years. So, when I walked in to Miss. R’s  math class, I was greeted with ,”Hey Mrs. Caban.”  I have known these kids since they were in third grade.

I wrote on the board:

Screen Shot 2016-06-06 at 10.17.50 AM

I asked them to talk to me about anything in their lives that they struggle with.

Right away a boy said, “Football.”

I asked, “Can you tell me some more?”

“I love football. When I make a mistake on an important play, it is hard for me.  I screwed up the play.”

“So making a mistake in football is a struggle.”


“Anyone else have something you struggle with?”

Three different students started talking at the same time.  I asked if we could take turns because I really wanted to hear everyone. First I called on Ashley.

“I struggle when I am reading a really good book and the character dies. I know it is essential to the development of the plot and usually important to the theme, but it is still so hard. Usually, I cry.” She chuckles to herself, “It is just so hard.”

“Thanks for sharing that, Ashley. I totally agree. Sometimes when a character dies, it is like I lost a friend. What else do you struggle with?”

“Friends.”  Charlotte said, as she glanced at Joanne across the room.

“How so?”

“Well, sometimes, friends (another sideways glance at Joanne) assume things about you that aren’t really true and they don’t ask you so they don’t really know.  People shouldn’t make assumptions about other people.”

“Wow. I agree that friendships can most definitely involve struggle.  What do other people think about this?”

Lots of agreement.

“Joanne, did you want to add anything?”

“Yeah. I struggle with not being able to see my family and friends a lot because they live far away. I miss them.”

“That is hard.”

Finally, Eve said, “So are we not doing math today? Are we just going to do counseling?”


“That is such a good question, Eve. What does struggle have to do with math? Before we talk about that, though. I have a different question for you. What does struggle look like and sound like?  Tell me what struggle looks like and sounds like.” Here is what they said:

Screen Shot 2016-06-06 at 11.04.55 AM


Then, I put a continuum on the board and asked them to place their struggle on it.


No struggle at all                                                                                              Struggle all the time

They discussed which of their struggles were more challenging and why. Then, I asked them, would it be better if we never had to deal with struggles?   They unanimously agreed that it would not be better. I asked why?

They said struggle can be good because it is necessary, it helps us learn from mistakes, we get through it and build confidence, and we become better communicators.

“So… do you ever struggle in math?”


“I do too.  I was wondering if I could share a struggle with you and you might help me with it. I have been thinking about a problem, but I don’t know the answer. I want you to help me solve it, but I can’t tell you if your answers are correct or not because I haven’t solved the problem yet.”

“What do you mean you don’t know the answer?  How do you not know the answer?”

“I don’t know the answer because I just invented the problem and I haven’t solved it yet. I thought I would present the problem to you and see what you came up with.  Do you want to try it?”


“Okay. Thanks. So here it is.  Yesterday, I was in a second grade classroom. The teacher was giving her students pieces of fabric and asking them how many peace signs were in each array.  I don’t have the fabric with me because she wasn’t able to let me borrow it. She is still using it in her classroom.  I took some pictures of some of the pieces. They looked like this:

Right away, one of the students said, “So, you want us to do a second grade problem?”

“No. Not at all. The problem is one that I made up.  As I was looking at all these pieces of fabric, I wondered whether it was possible to sew them all together into one big rectangle, without changing the pieces. I wondered if I would be able to match up all the arrays to make a rectangle. I don’t know if it is possible. I thought you could help me figure it out. I don’t know if there is a solution or not. That is going to be the tricky part.”

“You really don’t know?”


When I was planning this lesson, I thought a lot about Tracy Zager’s NCTM Ignite talk: Going Beyond Group Work. My favorite quote: “They (Julia Robinson and Yuri Matiyasevich) managed to do all this fantastic collaboration without a teacher saying, ‘okay Hilbert’s tenth is going to be a group project.'”  I reminded myself not to squeeze the life out of this organic problem that I had stumbled across. In her presentation, Tracy challenged us to teach our students to “collaborate in ways that are fluid and powerful”.  Yikes. In order to do that, I might have to give up some power.  That sounds scary.

I explained to the students that I wasn’t able to bring the pieces of fabric, but I was able to use graph paper to cut out pieces that were the same size as the ones in the photo.  I only had time to make 4 sets of graph paper arrays.  There wasn’t enough for everyone to have a set.

“Can me and Liz work together?”

“I want to work by myself.”

“Me too.”

“Me and Jon can be partners.”

And there you have it.  Within 30 seconds, these students had organized themselves: some into thinking partnerships, some not.

They all  jumped right into the problem.  Some were moving the graph paper pieces around. Some were looking at the picture of the fabric pieces. All were engaged in problem solving.

They were acting like mathematicians.

  • They were asking questions:  “Does it matter how long the sides are?  Is there more than one way to do it? Does it have to be a long skinny rectangle?”
  • They were sharing observations: “Hey! A lot of these rectangles have a side length of six. Let’s try to match up the sixes.”
  • They were critiquing each other’s thinking: “That won’t work because you will end up with a tail.  You have to combine the rectangles so you can line up their sides.

As they worked, the excitement built. The kids were frantically moving graph paper around.  Every few minutes, somebody would ask me, “are you sure you don’t know the answer?  What if it isn’t possible?” They never disengaged from the problem solving.  They didn’t even look up.  They just asked the questions. I can’t say for sure why they kept asking, but I can speculate. I think they were asking because they couldn’t believe that they were actually being asked to solve a problem that might not have an answer, a problem the teacher had not attempted to solve. This was a new problem, a raw problem. Could they really have ownership of this problem?  Was I genuinely offering it to them with no strings attached?

“We got it!  We did it! It is possible”  I looked over and two of the girls had this in front of them:

Screen Shot 2016-06-06 at 12.37.56 PM

I asked, “are you sure?”

“Yes.  Look at it.”

At this point, I expected to hear sighs of frustration.  I thought the other students would be angry that these two girls had found the solution and now the problem solving was over. I was wrong.  I looked around the room and noticed that everyone else was still working on the problem.   Some had looked over, interested in what the girls had found, and then went back to their work.

From the other side of the room, Jon said, “I think there is another way.”  In front of him, were the beginnings of a rectangle with different dimensions than the one the girls had just completed.

“I just can’t figure out how to get rid of the tail, but I am almost positive it will work.”

The girls came over to watch them and said, “That won’t work because you have to use sixes and one of your sides is 14 squares long. 14 isn’t a multiple of six.”

Someone else came over. “I don’t think it has to be a multiple of 6 because one of your (the girls) sides is 46 and that isn’t a multiple of six.”

At this point, I thought about calling everyone together to discuss what we knew so far, but I decided against it.  There were only a few minutes left in the class and these kids were locked in – they were totally engaged in a beautifully raw math moment.  I couldn’t bring myself to take that away from them.

In a truly synchronistic way, the bell rang just as Jon declared, “I did it!”  He stepped back and let us all admire his work:


Nobody packed up.  Nobody ran out the door.  We all just stood there.

“There is an answer,” somebody said.

“Actually there are two,” someone else added.

Finally I asked, “I wonder if there are any other ways to do it?”

“Mrs. Caban, will you leave the graph paper pieces with us so we can see if there are any other ways?”

“Absolutely.  Keep me posted.”

As I re-read this blog post, I realize that I started the blog with a dangerous assumption.  I said, “Clearly my purpose was not to miraculously, in sixty minutes or less,  get these students to love math and see themselves as vital contributors in a math community.”

Wow.  If that isn’t my purpose than what is? I realize now that my purpose is most definitely to spend each second of those sixty minutes getting students to love math and to see themselves as vital contributors in a math community.   Who am I to think that anything less than that is possible?


Thank you Tracy Zager for asking about the Math!  It prompted me to double check the arrays in the picture.  I think they are askew!  We left this lesson in such a rush that we never double checked the size and number of the smaller arrays inside the big array. Fortunately, I took pictures. Look what I found:

Screen Shot 2016-06-07 at 5.27.06 AM

See comment below for reflections about whether or not to beat myself up about missing the opportunity to discuss this major difference with my middle school math buddies.

Simmer Down.


Recently, my colleague Mrs. Shink,  showed me this picture. She was baking macrons with her family. She told me she was going to use it with some of her second graders.  I asked her if I could use it too.

I brought the picture into a second grade class. I gave each student a color copy of it. I asked, what do you notice? What do you wonder? Here is what they said:

I wonder why there is a spot with no circle.

I wonder if someone ate one.

I notice there are 7 on the top row and 8 on the other rows.

I notice they look like cinnamon buns.

I notice some have holes and cracks.

I notice they are different sizes.

I notice that if you go down there are 5 and the one with no spot, there is 4.

I notice there is a metal thing underneath the cookies. I wonder if they just came out of the oven.

I notice there is one small one and the rest are big.

I notice that under the top, there is one, two, three, four, (runs her finger across each row under the top row).  They can be lined up into an 8 row and a 5, I mean 4 (runs her finger down each column). There are 8 going across and 4 going down.  If you take the top (row) away.

At this point, I noticed that some students were thinking about the context of the picture, some were organizing the objects into groups, some were noticing characteristics of the objects. I was particularly interested in the last student who spoke. I wondered if this student was thinking about decomposing the array to make it “easier” to see.

I gave the students more information about the picture.  I told them where it came from.  I also told them that one was missing because Mrs. Shink had eaten one.  Then, I asked them if they could figure out how many macrons were left after Mrs. Shink ate one.

They got right to work.  Many students started counting by ones.  Some students organized the cookies into two groups – ones that were darker and ones that were lighter – and then counted each group and added them together. A few students skip counted by groups or added groups. I noticed some students orally skip counting by fives.  A different student was adding 8s on the side of his paper. I asked the class if they could somehow show me their thinking so that when I took their pictures with me, I would be able to understand how they counted the cookies.  Many students started labeling each cookie with a number: 1, 2, 3, 4, etc.  The students who were orally skip counting wrote an addition equation:  5 + 5 +5 +5 + 5 +5 +5 +4 = 39.   I wondered, “what is the difference between skip counting and using repeated addition?”

When I originally planned this lesson, I was anticipating that the routine would inspire discussion about multiplication.  In my head, I remembered a second grade standard about introducing arrays as representations of multiplication.  I was wrong.

Here is the standard that I was thinking of:

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

What do I notice?

The word multiplication is never mentioned.  The size of the array is limited to 5×5.

As I was reflecting about this lesson, I realized that I was remembering the story wrong. Recently, I watched Graham Fletcher’s shadowcon talk about becoming a better storyteller.  In it, he challenged us to find one standard in the grade level you teach and discover something new about it. Is there something you currently teach that isn’t in the standards? He suggested:

  • Find that standard and explore it more deeply.
  • Draw it or use tools to support understanding.
  • Be dumb. Surround yourself with brilliance.
  • Be vulnerable.

This was my chance.  I decided to start by re-reading the Common Core Progression for Operations and Algebraic Thinking.  I have read this document many times and I will read it many more.  I always learn something new. Immediately, this sentence jumped out at me.  “In Equal Groups, the role of the factors differ.  One factor is the number of objects in a group (like any quantity in addition and subtraction situations), and the other is a multiplier that indicates the number of groups.”  We have spent a lot of time in our 3rd grade meetings discussing how three groups of 5 items is different than 5 groups of 3 items, but 3×5=5×3. This is one of the topics we always leave simmering.  Our understanding has gotten richer and more dense over time.

However, I haven’t taken the opportunity to explore this topic with second grade teachers, especially as it relates to addition, as it relates to cookies cooling on a rack.  As I re-read the progressions, I wondered how many students were being given the opportunity to simmer in repeated addition for awhile?  How many second grade teachers had been given the opportunity to deeply explore the relationship between addition and multiplication, and what about subtraction and division?

Recently, I have been watching a few fourth  grade teachers bang their heads against the wall as they try to convince their students that adding 13 forty seven times is not efficient. I have felt helpless. I ask,  “Have you tried showing them arrays with smaller numbers?  Have you tried using the Number Talks Images site?  Have you tried having them compare strategies and articulate someone else’s understanding?”   Yes. Yes. and Yes. Commence banging.

As I write, I am wondering, maybe what we need to do is put these students back on the stove?  We keep trying to pull them towards multiplication, but maybe what they really need is to dive a little deeper into their understanding of repeated addition.  Last week, one student was trying to use an open array to show 26 x 8.  He didn’t want to break the 26 into 20 + 6.  He wanted to use 26+26.  What should I do with this? Graham Fletcher might say, “Draw it. Use tools to understand it.”

Here is my example of what I see these students doing when they are confronted with a two digit by one digit multiplication problem.

Screen Shot 2016-05-14 at 7.55.08 AM

We want them to do this:

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When we try to convince them to decompose by place value, they push back.  They don’t want to.  They want to use repeated addition.  What if we honored that?  I mean, what if we honored that while simultaneously trying to moving them forward.  What if we let them simmer?  What if we intentionally tried to connect multiplication to repeated addition?

When I sat down to draw in an effort to further my own understanding of how repeated  addition connets to multiplication, here is what I came up with:

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Wow!  Graham Fletcher was right. I learned a ton.  I realized that decomposing 26 into 20 +6 might not be helpful to students if they don’t understand the 20 as being equivalent to 2 x 10. I also learned that there are a lot of steps in between using repeated addition and using an open array and the distributive property to solve multiplication problems.

Let me be clear. I am NOT suggesting that we teach students all the steps that I just did. The drawings above represent my journey towards understanding the story better.  My next step is to go back to the fourth grade teachers.  I will retell my story of repeated addition and multiplication.  Then, I will listen carefully and unassumingly to their stories. Together, we will map out the next chapter in the book.




Learning Rounds: Watch and Learn

I work as a K-5 math interventionist in two rural Maine schools. Last week I helped to facilitate a staff meeting at one of the two schools I work in. Both schools have implemented Learning Rounds or Instructional Rounds as part of their professional development. This staff meeting was meant to be a time of reflection for this staff as they concluded their first year of learning rounds in their school.

This was quite an accomplishment for both staff and administrators in this building. Learning rounds have not been met with open arms in all of the buildings in our district. In fact this is only the third of four elementary schools to adopt them for math. In order to facilitate the discussion and reflection at this staff meeting, our district math coach and I decided to revisit concerns voiced by this staff at the beginning of the year. We rephrased statements they had placed on a pro/con chart at a September staff meeting into belief statements and gave them back.

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Teachers found that many of their concerns were outweighed by the benefits reaped from the learning rounds. Teachers had the opportunity to observe and to be observed at least twice over the course of the year. One of the things that surprised many teachers in this building was that they learned more from observing their peers (even if they were a 5th grade teacher observing a Kindergarten teacher), than they did from being observed.

The focus of our learning rounds are the 8 Mathematical Practices. We had chosen two practices on which to focus at the beginning of the year. This staff had chosen to look at MP#1 Make sense of problems and persevere in solving them, and MP#3 Construct viable arguments and critique the reasoning of others. For each round in which they were observed, teachers worked hard to consider these practices and how the tasks they chose  and planned would provide an opportunity for their students to engage in them. They also participated as observers and had a chance to look for evidence of one of the practices in their colleague’s classrooms using a “Look For” sheet similar to the one below.

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This “scavenger hunt” has resulted in a much deeper understanding of the practices for the teachers that have taken part.

Learning rounds have proven to be a powerful tool used to  improve mathematics instruction in our district. As facilitators, we have learned important lessons about how to keep them positive. For instance, setting and reviewing norms such as confidentiality and active listening before each round, providing teachers with PD on the practices we have chosen, and ensuring there is plenty of time to debrief in small observation groups and as a whole staff when possible, are all extremely important.

Personally, my favorite part about them is seeing lightbulbs come on for teachers as they take ownership of their own teaching and learning.  The process requires teachers to take risks in their teaching, and in their learning of new mathematical ideas. Through these risks, we are growing as teachers and learners.

Power or Influence?

Recently, I watched an NCTM shadowcon talk by Robert Kaplinsky. I can’t stop thinking about it.  His words echo in my thoughts.  I am a district math coach. I have no administrative power.  I only have the opportunity to influence.  Sometimes, my influence is positive. Sometimes my influence is negative. Often, the difference between the two is how honest I am with myself about my intentions and how intentionally I reflect.  

Last week, I led a learning round in an elementary school. I watched an exceptional lesson in a third grade classroom. What made the lesson exceptional was how little the teacher said.  For ten minutes, 7 different students participated in a student generated investigation about how many lines you need to draw to show fourths on a number line.

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When we walked in, we heard:

Student #1: “You have to draw more than three lines because if you only drew three lines, you would have thirds.”

Student 2: “I agree. If you didn’t have the end line, you would have thirds.”

Student 3: “You don’t count all the lines.  You don’t count the 1.”

Student 4:  “I see both sides of the story.  What does the zero stand for?”

Pause. A long, silent pause.

Student 5:  “If you count the 1 line, you have to count the zero line.”

Student 6: “I disagree with you… about not counting the 1 line.  If you didn’t count the 1 line, you would only have three fourths. You would only have three pieces.  It really wouldn’t make sense without the end line.”

Student 7: “Yeah!  If you don’t have the zero line and the 1 line, the numbers would go on forever.  The zero and the one are like the start and the stop.”

Several Students:  “I agree with (student 7).  A number line goes on forever.  When you make fractions on a number line, it is kind of like you are showing the pieces. You need the zero line and the 1 line to show  the “piece ” of the whole number line.”

Another long, silent, pause.

Mrs.Watkins:  “I think I heard you say a couple of things.  When showing fractions on a number line, it is really like showing a piece of the whole number line – a line segment.  We need to draw the zero line because it tells us where to start. We need to draw the 4/4 or 1 whole line because it tells us where the whole ends. You taught me something today. I could be more specific when I am using number lines to show fractions.  I could call the fraction pieces line segments.”

These students presented, questioned, and defended their own and each other’s reasoning.  The teacher listened.  

I was participating in this learning rounds with an interventionist from another building. She didn’t know these students at all.  Afterwards, I asked her, “Would you be able to guess which students received “gifted and talented” services?”  She said she would have no idea. Then, I asked her, “Would you be able to guess which students receive interventions?”  She said she would have no idea.  I pushed.  I asked her, based on the thinking she just observed,  choose a student who you think sounded like a typical “gifted and talented” student.  She chose a student who receives math interventions. 

After I left this classroom, I couldn’t stop thinking about these students and their teacher. Their words were echoing in my thoughts.

During the debriefing session of our learning rounds, I asked my peers who observed after we did, “What happened next?”  I was hooked. I only saw ten minutes of this math lesson. What else did these students do and say after I left?

They shared with us that, after we left, the teacher presented the students with a problem about cupcakes.  “There are 3 cupcakes and  4 kids.  How should they share?  I wonder if they should each get half?” She sent the students off to work in small groups to come up with a solution.

As the observing teachers circulated, they noticed the level of engagement in math talk. One student said to her partner, “no offense to Mrs. Watkins, but I think her estimate is off. I am pretty sure each kid could get more than half a cupcake.”

Later that day, I checked in with Mrs. Watkins.  I asked her what the kids came up with for solutions.  She showed me the white board below and said, “this was really interesting.”

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“Is carot a type of cupcake?”, I asked.

“No. It is the name of the kid. They chose their own names.”

Of course. Carot, Tomato, Charlie, and Joe.  Just your typical group of nine year old names.

She told me that these students presented to the class that the solution could either be 3/4 or 1/4 or 3/12.  She endured yet another silent pause as I processed.

“1/4?”, I asked. She was being really patient with me.

“1/4 or 3/12 of all three cupcakes. These students explained to us that the answer depended on what you considered to be the whole.”

Again, she waited for me to catch up.  I felt humbled.  I wouldn’t have thought to consider either of those answers. She might not have thought of those answers herself, but she was open and quiet enough to allow her students to consider them.

Yesterday, I went back into third grade.  I can’t stay away.  Mrs. Watkins has inspired me. She’s got me thinking more deeply about fractions.  I shared with Mrs. Watkins that, since I left her, I saw this message on twitter:

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She and I had a great conversation about how and why we label fractions on a number line. She told me that one of the first lessons she does with her students is create fraction strips.  She has the students label each interval as a unit fraction.  She doesn’t introduce labeling the hash marks on a number line until later.  The lesson that I described above was her first introduction of labeling the hash marks. Again, I learned from her – the progression matters.

Mrs. Watkins has been teaching third grade for 30 years.  She often talks to me about how much her math instruction has changed since she first started teaching.  When I first started teaching, I worked across the hall from her.  She is quiet.  I think she would consider herself an introvert.  This year was the first year we completed learning rounds in her building.  Learning rounds are opportunities for teachers in a building to observe short segments of each other teaching with the intention of learning together to improve math instruction. Some of the teachers that Mrs. Watkins works with said, “We have worked together for 16 years and today was the first time I saw her teach.”  I am so grateful that we are all able to see Mrs. Watkins teach.  Thank you, Mrs. Watkins.

The Essence of Mathematics, in One Beatles Song

This is amazing. I didn’t write it, but I want everyone to read it and I want to re-read it, a lot.  It speaks to me and for me.  Please share.

It was posted by Ben Orlin from Mathwithbaddrawings.com

Math with Bad Drawings

Okay, here’s a life regret: No one has ever stopped me on the street, grabbed me by the collar, and demanded that I explain to them the essence of mathematics.

I’ve envisioned it many times, though.

What math teacher hasn’t?

20160425071213_00003Me: So, you want to get math?

Assailant: Obviously! Why else would one human being violently accost another, if not for the acquisition of knowledge?

Me: Easy, then! All you need to do is listen to Sgt. Pepper’s Lonely Hearts Club Band.

Assailant: [arches eyebrow] You can’t be serious. The Beatles album?

Me: [easing out of their grip, brushing my collar] Naturally! The whole album is trippy and spectacular, of course. But I’m talking about the final moments of the final track, a song that Rolling Stone has hailed as the Beatles’ greatest: “A Day in the Life.”

Assailant: [listening on an iPhone] This better be good…

View original post 715 more words

modeling with fractions, revisited

In my last post, I described three boys attempting to model a problem about servings of chicken and potatoes.  My colleague and I were watching the boys wrestle with how to represent the potato servings with fraction circles.  At the end of the post, I “saved” the boys from crossing out their representation and starting all over.

I tend to consider myself a pretty decent math teacher.  I try not to say anything that a kid can say. I ask a lot of questions.  I use rich tasks that promote deep thinking.  I encourage my students to persevere and engage in productive struggle. And, yet, here I was rescuing these boys from their learning.  What, exactly, was I saving them from?  Maybe I thought I was saving them from starting over.

A mathy friend read my last post and she asked me, “why didn’t you let them cross it out?”

“because I….. Well, I thought I…..hmmmm.  Oh, man. Why DIDN’T I let them cross it out?”  I sighed as I realized that  I may have hijaked their learning. I saw those boys on the edge and I yanked them right back away from it.  Why did I do that?  Maybe I felt rushed.   I was super excited to watch them in the thick of making some really cool connections. Maybe old habits are hard to break. Did I ruin their learning?  Of course not.  Would it have been interesting to see what the boys would have done if I hadn’t said, “wait!”?  I think so.

It is really challenging to maintain the balance of providing and removing scaffolding for students.  How do we keep the struggle productive?  What makes a struggle productive? I have been thinking about this for awhile now.  Here is what I came up with:

A struggle is productive when learning is happening, connections are being made,  questions are being asked, and  doubt and uncertainty are being examined.  

My colleague and I were on a learning round. We were “looking for” evidence of students checking to see if an answer makes sense within the context of a situation and changing a model when necessary.  In my last post,  I commented that I thought the boys were abandoning their model if they crossed it out, but I have changed my thinking. I don’t think they were abandoning their model. I think they wanted to change their model so it more accurately reflected the context of the situation.  They were sense making. They were struggling productively.

I love looking at student work. It is amazing.  I often walk away with more questions than answers, but that is okay.  I also walk away with a deeper appreciation for how my students are making sense of math. Below is the poster that the boys shared with their teacher and their classmates after they had finished solving the problem. The top portion is their work on the potatoes part of the problem.  The bottom portion is their work on the chicken part of the problem. They completed the bottom part after I left. I have been looking at their work for awhile.  I think it is beautiful.

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My favorite parts are the “cross outs”.

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modeling with fractions (twice)

Yesterday, I participated in a learning round.  I visited several k-5 classrooms with two other teachers.  We looked for evidence of modeling with mathematics.  In third grade, we found an opportunity to observe what it looks like when a model pushes students to the edge of their understanding. Incidentally, we found ourselves on the edge too.

When we walked into third grade, there was a quite hum of math talk and it took us a minute to find the teacher.  She was on the floor with a small group.   Several other students were working independently at their desks.  A group of boys caught my attention across the room. They were gathered around a large piece of poster paper having an intense conversation about something.

Tina, a Kindergarten teacher, and I went over to the boys to get a closer look at the poster.

“Do you guys mind if we watch you do math?”

“No.  We are working on this problem.”

(Thank you k-5mathteachingresources for sharing this problem with us!)

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As Tina and I pulled up chairs and read the problem, the boys went back to their Math. Joe was writing on the number line and narrating his thinking as he counted the groups of partial pounds (2/3) of potatoes.

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Bobby was coloring in groups of 2/3 on the circles that were drawn on another section of the poster.

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I asked, “How do you know how many people there are?”

“We read the story. There are 6 people at the dinner. We are figuring out how many pounds of potatoes she needs.”

“What do you have so far?”

“Each person gets 2/3 of a pound so we think she need 4 pounds of potatoes.”

(He pointed to the groups of two thirds on his number line as he counted.”

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“Wait a minute,” his partner said.  “I think we need to change something.”

“What do you mean?” he asked.

“Something doesn’t seem right about our circles.”

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I asked Bobby to tell us some more about what he was thinking.

“Well. We already have 4 pounds of potatoes, but we aren’t done showing all the people. We still have to show potatoes for 2 more people. We need to draw some more circles.”

Mike thought about it. “We can’t draw any more circles or we will have more than 4 pounds. Each circle is a pound. Our number line says we need 4 pounds.”

I asked, “How do you know you have 4 pounds?”

They showed me again on the number line. “Why do you need two models?”

“Mrs. T said we needed to create two different models.”

I decided to push more, “Where are the pounds in the circles?

“See,” (they pointed to each circle) “1, 2, 3, 4”

“Oh!”  Bobby said.  “We need to move that (points to 1/3 in second circle) over there. (points to the blank space in the first circle.”

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One of the boys reached for a marker and motioned as if he was going to scratch out all the circles and start again.

“Wait!” I said.

My colleague and I came into the classroom looking for students who might be “checking to see if the model makes sense within the context of the situation and changing the model when necessary.”  These students realized that their model didn’t make sense.  Their solution was to abandon the model.

Welcome to the edge.

What are these boys trying to understand? What are they confused about? What would you do next?  Post your thoughts.