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.

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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

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…

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