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Some Days

Some days are good days. Some days are okay days. Some days are rough. And then, some days are really, really good days.

Today was a really, really good day.

A really, really good day is a day spent in classrooms, with kids and their teachers, who are engaged in thoughtful mathematical conversations. Classrooms that are part of a system of classrooms in a school (from Necessary Conditions, written by Geoff Krall) that is working really, really hard to offer their students opportunities, each and every day, to talk about their ideas, to think about how their ideas are similar and different to their classmates’ ideas, to notice connections between ideas, and to build on each other’s ideas.

Details of the day

  • Each grade level hosted a number talk (K – 5)
  • Each grade level team had planned a number talk during the district work sessions held the previous week
  • One educator facilitated the number talk
  • Colleagues (from the grade level, the site instructional leadership team, the district, and outside the community) collected data and noted the teacher moves used during the conversation
  • A reflective conversation was held at the conclusion of each number talk, usually just outside the door on the sidewalk


A glimpse inside a team’s number talk

When planning for their number talk, the grade 5 team decided they wanted the students to explore division.  They selected the following number string from Number Talks, by Sherry Parrish (note: the district has provided each teacher participating in the project work with a copy of the book. So awesome.)

30 ÷ 3

18 ÷ 3

48 ÷ 3

It would be the children’s first experience thinking about division during a number talk. The team was very interested in the ideas and strategies students would use. They anticipated students might use their knowledge of multiplication and its relationship to division, multiplication and/or division facts they just know, skip counting and keeping track of the number of groups of 3, and partial quotients.

The kiddos did all of that and more. Because, when kids know you really want to know what they know and how they think, they show you. This means that they are total and complete MATH ROCK STARS.


We asked them to think about 30 ÷ 3.

They said:

“I know that 30 is three tens. So, each group has 10.”

“I know that 3 groups of 10 makes 30. So, it’s 10.”

“I know that 3 x 10 = 30, so 30 ÷ 3 = 10 and 3- ÷ 10 = 3.”


We asked them to think about 18 ÷ 3.

They said:

“I just know that 18 ÷ 3 = 6.”

“I know that 3 x 6 = 18. So, the answer is 6.”

“I skip counted: 3, 6, 9, 12, 15, 18. I used 6 fingers to count, so it’s 6.”


We asked them to think about 48 ÷ 3.

They said:

“I counted by 3s. I think it takes 16.”

“Well, I thought about 48. I noticed that 30 and 18 is 48. And, I already know that 30 ÷ 3 = 10 and         18 ÷ 3 = 6. So, I added the 10 and the 6 and got 16.”

And this final idea was shared by a student who had listened and nodded and agreed with his classmates’ ideas throughout the entire conversation.

He said, “I split 48 into two 24s. And, 24 ÷ 3 is 8 and another 24 ÷ 3 is 8. So, 8 + 8 is 16. So, I think        48 ÷ 3 = 16.”

There was silence.

And then gasps—from both kids and teachers.

The kids were like, “Oh, wow! That works!”

The teachers were like, “Oh, wow! We didn’t anticipate that idea!”

Both kids and teachers said, “Oh, wow! I never thought about it like that before!”

The number talk was wrapped up.


Then, the teachers stepped outside to chat. All eight of us were talking at once. CELEBRATION time!!

We celebrated:

The hard work the classroom teacher has done each and every day over the past 3 months to create a classroom culture where kids know that there is a space for each of their voices.

The children’s confidence in the safety and security the number talk routine offers them. They know how the routine works, so they can just focus on the mathematics and their ideas and not have to worry about what comes next. They know what comes next. They know they will have sufficient think time, all answers are accepted and respected, those who would like to share an idea can do so without ridicule—no scoffing or eye rolling, and their ideas are celebrated, discussed, and tried out by others.

The hard work the grade level has done so that this is true in all of their classes

The welcome to our learning community that is the campus vibe

A visiting teacher facilitated the number talk and the kids didn’t miss a beat—they did what they do best—think deeply about the math and confidently share ideas

Seven educators listening in on the conversation; they were there as learners


Then, we talked about:

  • what the next number talk might be for this class
  • what we each learned
  • what we each are going to take back to our classes
  • that this really, really good day is the result of the intentional and thoughtful work done by the teachers, the school’s Curriculum Coordinator, and their instructional coach to create a schoolwide culture that extends into each classroom: a belief, a conviction, one that highly values students, their teachers, and their work

These amazing teachers went on to the rest of their day to do what they do best, each and every day; teach their kids, care for them, and encourage them.


Lucky Kids

Yesterday a math team spent the day creating the road map for a group of kids they have yet to meet. The students won’t be on campus until August 7th. Their teachers don’t know anything about them, but they do know what they want for them.

During the year that the kids spend with them, the math team wants to build a course and a learning space in their classrooms that supports kids

  • to be confident mathematicians
  • to take the tools and knowledge they have and use them in new situations
  • to analyze their work
  • to persevere, and
  • to defend their ideas

The team also decided that they can’t and won’t start class working on whatever the textbook says. They are starting from the premise that they need to create opportunities for kids to investigate interesting ideas that cause conversations about relational thinking. So, the kids and their teachers will start the year exploring visual patterns.

Lucky kids.

Thank you, @fawnpnguyen.

#noticeandwonder and #ChiLSconf

Some noticings and wonderings from two days of learning at The Chicago Lesson Study Conference.


  1. Looking at learning over the shoulders of children provides us with a brutally honest reminder of what we ask of them each day. We demand they grapple with and make sense of the many layers of complex ideas, often with partially developed sets of understanding, and to do so on a pretty tight timeline.
  2. Kids whose teachers’ regular practice is built upon the idea that problems worthy of investigating may take more than one work session develop a confidence to leave an idea in mid-sentence and a willingness to pick up the conversation again the next day. They can do this because their teachers always close the day’s learning, and in doing so, position them to be ready for the next step in their learning.
  3. Learning is beautiful. It is really, really beautiful when it takes place inside a classroom community built on care, compassion, and kindness.


  1. What drives some educators to demand that reflective practice be a strong component of their professional work, and to do everything possible to ensure this happens?
  2. How do some administrators keep their passion for teaching as their raison d’être?

For the record, the whole reason this wonder even exists is because the principal at the Helen C. Peirce School of International Studies in Chicago gave 120 educators the most amazing gift on Thursday, May 11th. She taught the public lesson at the Chicago Lesson Study Conference that provided the context for small group conversations around focused questions, the wonderings that were explored during the panel discussions, and the final comments provided by Dr. Tad Watanabe. Talk about learning being beautiful. WOW. Just WOW.

  1. How can we begin to ask more wondering questions of each other about our work with the same commitment to grace and honest curiosity that was demonstrated during the two days of the conference?

The Chicago Lesson Study Conference

For the past 15 years, The Chicago Lesson Study Alliance has hosted a Lesson Study Conference in early May. The conference offers educators the opportunity to observe two public lessons, participate in panel discussions, evaluate the data collected, and listen to and reflect on the final comments provided by Dr. Tad Watanabe of Kennesaw State University and Dr. Akihiko Takahashi, of DePaul University.

On Thursday, 120 educators spent part of their morning at the Dr. Jorge Preito Math and Science Academy watching the learning of a class of fifth graders evolve. The planning team from the Helen C. Peirce School of International Studies created a research lesson entitled Measuring and Expressing Capactiy with Liters and Milliliters. The lesson focused on students’ understanding of how to choose an appropriate measurement tool when working in different measurement contexts, determining when larger/smaller measurement units are appropriate to use when expressing values for measurement, and understanding the proportional relationships between units and working within conversion systems using proportional reasoning. The lesson was taught by Lori Zaimi, the principal of the Helen C. Peirce School of International Studies.

On Friday morning, a group of second graders from O’Keefe Elementary School joined the conference participants at the Dr. Jorge Preito Math and Science AcademyThe planning team designed a research lesson entitled How can we organize this information? The lesson focused on exploring how students might visually represent data they collected as a class. It afforded us the opportunity to investigate how students organize information with a bar graph, understand how a bar graph represents data, understand that bar graphs help find information quickly, and answer questions about the data. The lesson was taught by Meghan Smith, the classroom teacher.

Thank you to the students, staff, and faculty of the Helen C. Peirce School of International Studies, the O’Keefe Elementary School, and the Dr. Jorge Preito Math and Science Academy for so generously sharing your work so 120 educators could listen, observe, discuss, ponder, and reflect on what it means to learn.

Much Ado about Watermelons

Day 2 of #summermathcamp

To feed our math brains at 8 am on a Tuesday morning in the summer, we showed this photo of some watermelons in the hopes of generating some conversation.

How many watermelons are there? How do you know?

Screen Shot 2016-06-14 at 7.21.07 PM


This was just going to be 10 to 15 minutes of a notice and wonder conversation. Yea. We were wrong. Fifty minutes later we were still chatting about watermelons. Who knew a pile of cut up watermelons could keep 45 educators engrossed. Really, what is there to talk about—it’s just a bunch of watermelons, people.

Car full of watermelons

Image from:


So, here’s what we chatted about.

We predicted that the most common answer would be to cut up two of the one-half sized pieces to make the missing one-fourth pieces, slide those around to make 4 whole watermelons. Then, add the other two halves to make another whole watermelon. So, the sum of 4 whole watermelons and the other whole make 5 watermelons. We thought that some version of that idea would be a good start to the day.

And that’s exactly what happened. One of the campers shared her version and just about every person in the room said, “Yup, I thought about it that way, too.”

answer #1


And then Shannon said, “I saw it another way. I saw 4 groups of 3/4 of a watermelon and then I added the 4 one-half sized pieces. So, 3 whole watermelons and two more means that there are 5 watermelons in the picture.” The conversation then moved to connecting Shannon’s use of the algorithm to the photo and then adding in the notation.

answer #2

Since many of the campers were happy to think about the quantity of watermelons using the algorithm in Shannon’s explanation (4 x 3/4), we thought we would pause here and push on the idea of equivalent representations. [We much preferred thinking of the watermelons as 3 groups of 1—written as 3 x 4/4.]

This provoked lots of conversation on how come we can do this.

  • How does the image show (4 x 3/4) = (3 x 4/4)?
  • How does the picture show Shannon’s equation: (4 x 3/4) + (4 x 1/2) = 5?


The third way that surfaced was to simply add all the pieces of the watermelons visible in the photo. We recorded it like this: ¾ + ½ + ¾ + ½ +¾ + ½ + ¾ + ½

answer #3


Here are some of our takeaways.

If it’s important to the kids it MUST be important to us

  • We need to listen to what our kids are saying and not be on the lookout for the answer listed in the TE or for the student response that matches our preferred strategy.

Questions and their power

  • Your questions need to offer ALL kids a place in the conversation. So, in this situation we could have asked a couple of questions.
    • How many watermelons are there?
    • How many whole watermelons are there?

           Which question lets the kid who says there are 8 watermelons be in the conversation?

Dots on ten frames and Photos of watermelons, almonds, and tangram puzzles

  • Math is an active subject—it’s interesting, irritating, perplexing, confusing and invigorating. It makes your head hurt when you are in the midst of the struggle and then you get to embrace the high fives when that last piece falls into place and the connection appears as a result of your hard work.


Reflections on Summer Fun

The last few days spent in classes have been lovely. I knew they were going to be. Spending time in classes with kiddos and their teachers trying out some ideas that were new to my colleagues, and then chatting about what we saw and heard, and what we learned was just wonderful.

1.  Kids (and their teachers) are super interested in containers filled with stuff. It is nearly impossible to resist joining in the conversation about “how much stuff” is in the container, especially when the stuff is the lovely orange cheese balls that you can buy in the mega-container at Target.


It offers kiddos who really, really, really struggle with math anxiety a way to join in the conversation. Everyone made guesses and built number lines and laughed and shared ideas.IMG_2161


These are the sights and sounds of what learning should be for our students.

2.  Games provide an awesome arena for kids to explore numbers and their structure, especially the game we shared on Monday morning, What Fits Between?  We taught them how to play the game. They had to figure out how the three numbers created are related (largest number, smallest number, and the number that fits between) and organize the results.


Then, we put the students into groups, distributed the record sheets and the sets of UNO cards, and they were off and running. We listened in on their conversations and watched to see what decisions they made about their numbers. Occasionally, we answered questions about situations that arose in their games that hadn’t in the few rounds we modeled. There were lots of conversations: kiddos asking each other for help, doing the if I had made 48 with my cards instead of 84, my number would have been in the middle and I would have won, and deciding what to do if two of the three players in a group made the same number. (They gave each player one point for the round because no one had created a number that fit between the other two.)

  • We learned that if we leave them alone to figure things out, they do. The adage, be less helpful, comes to mind.
  • Kids are able to explore ideas more deeply when they engage in conversation with classmates than when they engage, individually, with worksheets.
  • It’s great fun to have a group of teachers in a classroom when you want to try out new things especially when you are able do some planning together before hand.
  • It’s really nice to have a couple of days to work on new ideas and have time for people to talk together on each of the days.
  • Kids are amazing. If we choose the right tasks, they aren’t even nervous and anxious about the math part of their day.

3.  Using games as assessment is an informative practice. Notes and photos and conversation bring the next instructional steps to light.

  • Of the 12 kiddos playing, nine were ready to move onto playing the next level of the game where each player must build a three-digit number and determine who had created the number this fits between the other two. Three students needed additional opportunities to play the original version of the game.
  • The “top” math student got to play and interact with the ideas just like everyone else, as did the kiddos who are often not interested in math. All 12 students were invested in the game. The students who usually need to get a drink, sharpen a pencil, or use any number of distractions to help them get through math did not need to employ any of those strategies.
  • During the reflection time at the end of the class period the kids decided that they wanted to change the criteria that determined who won the game. Now, in their class, the winner is determined after each player counts up the number of rounds s/he won. The player who has the number of wins that is between the other two, wins.

How do we use the learning we saw today to support tomorrow’s instruction?

The collaborative work started when we noticed that kids were having a difficult time estimating height in an estimation warm-up. We decided to address the manner in which they worked with the concept of estimation. We selected a graphic organizer for them to use, determined that the quantity of objects for them to estimate would be less than 100, provided some boundaries to the quantity, and put the cheese balls in clear, plastic containers so the students could see them. With these adjustments the students were more comfortable in making estimates and discussing their ideas. After the team talked about the work the students did that day, it was decided our next move would be to use a series of photographs from We selected the group that asks kids to estimate the number of cheese balls on plates and trays. The students will play What Fits Between? and do some other math work.

I am looking forward to hearing how the day went, and what the team is going to tackle next.

Summer Fun

One of the highlights of the summer is being part of a project with a math team at an independent school. We are focusing on collaboration and open dialogue in instruction to support student learning. The project is structured so that there are daily opportunities to experiment with lesson design and pedagogy.

The first day I was on campus, the head teacher and I spent time in each of the math classrooms. We were interested in learning what instructional practices and teacher moves are used during mathematics instruction, and the connection of these elements to the work students were doing and the mathematical ideas they were investigating.

In one class we visited that morning, the day’s learning opened with a question from The students were asked to determine Mr. Kraft’s height. As we listened in on the conversation we noticed the students’ guesses about Mr. Kraft’s height were very random. The explanations contained little comparative thinking or relational thinking. This notice was a priority in our team planning conversation.

During our planning conversation, we determined the students were more confident with estimating quantities of objects rather than the measurement situation they were asked to investigate using a photograph from the website. They are working on how to use given information to create reasonable estimates and are building deeper understanding of magnitude.

So, here’s our plan for Monday.

PART 1: Estimation–how many plastic ninjas are in the middle container?

The three containers shown in the photo will be on the front table. Students will estimate how many plastic ninjas are in the middle container.


They will record the information about the containers and their estimate in a specific way. The number of plastic ninjas in container 1 and container 3 are written in the boxes. The estimate of the number of plastic ninjas goes on the number line.

Screen Shot 2016-06-26 at 10.08.50 PM(The number line is adapted from Graham Fletcher’s 3-Act Task recording sheet.)

The students will then discuss the information they used to create their estimates.

PART 2: Estimation--how many cheese balls are in the middle container?


The second round of estimation uses cheese balls from the mega-size container of cheese balls from Target. There will be 10 cheese balls in container #1 and 75 cheese balls in container #3. Students will estimate how many cheese balls are in container #2.

They will record the information about the containers and their estimates in the same way they did for the plastic ninjas. The number of cheese balls in container 1 and container 3 are written in the boxes. The estimate of the number of cheese balls in container 2 goes on the number line. A class discussion on the information used to create the estimate will wrap up this part of the day’s work. 

Part 3:  We’ll play an adaptation of a game from Box Cars and One-eyed Jacks called What fits between? 

In this game, students work in groups of three. Each player takes two cards from the stack of UNO cards (only contains cards 1 -9), placed face down, and creates a two-digit number. When all three players in the group have created their two-digit numbers, they each lay down their cards and read their number. As a group, they decide who has created the largest number and who has the smallest number. The player who has created the number that fits between the largest and the smallest numbers earns the point for that round. The player who has the most points at the end of the work time, wins the game.

We will end this part of class with a conversation about how they decided to create the numbers in each round, and what they noticed about the game.

Final Thoughts: We are exploring how the use of a specific structure in the lesson supports students as they build a more comprehensive sense of number. Offering the “too low” and “too high” in an estimation task is intended to provide some reference points for the students to use as they create an estimate. The plan is to use this structure and its variations, and then gradually remove the two estimates. Our goal is to ensure that students can construct a sense of the quantity that is represented by the written form of a number and to use this sense of quantity in other situations. We liken it to what happens when we read a word and we create an image. We want that to happen for our students when they are working with number.

I can’t wait for school to start this morning.



The marvelousness that is the #MTBoS

Because of #MTBoS, I met @AlexOverwijk at Twitter Math Camp in Jenks, OK in July 2014. One afternoon, Alex, @gwaddellnvhs, and I spent time talking about Lesson Study.

Because of #MTBoS, I am on my way to Ottawa, Ontario to spend the next two days participating in a Lesson Study at Glebe Collegiate Institute; otherwise known as Alex’s school.

Lesson Study offers educators an incredible opportunity to learn from and with other educators. The collaboration, the conversation about the important work we do, the opportunity to explore ideas and try stuff out and see what happens, and to examine how all of the layers, pieces, and components of instruction move students’ learning during a lesson is something more of us should have the chance to do.

It’s going to be AWESOME.




containers and estimation

Quite awhile ago, I facilitated a day-long workshop for grade 2 teachers. During the morning break, I had the opportunity to chat with one of the participants about the work she was doing with her students. She talked about a routine she was using to open up conversations in her math class.

Fast forward about 10 months.

While looking through my photos, I came across the set of pictures I had taken with the intention of writing up this post. The following is my version of the really nice task she shared.

Step 1:  Share this stack of containers.


Ask the students what they notice. Record their ideas. Next, ask them to share their wonderings and record their ideas.

Step 2:  Tell them that the bottom container holds 5 crayons and the top container has 32 crayons. Ask them to think about how many crayons might be in the middle container, and prepare a statement that would convince the group that their idea could be true.

Step 3:  Have them share their idea with one other person; a test run to practice their statements before sharing with the whole class.

Step 4:  Ask the class, “Who’s partner had something interesting to say?” Step back and listen. (Totally stole that line from Cathy Fosnot–she uses it so strategically and magically in her workshops to create great conversations.)

Step 5:  Ask for some more ideas. Record the quantities the students share.

I am taking a teacher time out here.

A teacher time out is nice strategy we’ve used in Lesson Study. It is a coaching move that Elham Kazemi has talked about. The purpose of the time out is to literally take time, right in the middle of class, to reflect and decide on the next set of teacher moves you are going to use. It is incredibly powerful to have the option to talk in the moment rather than saying after the lesson is over, “I wish I had…” 

The intent of the teacher time out I inserted here (much, much later, long after the lesson was over) is to consider the idea that I am currently wrestling with: what to pursue. This idea is courtesy of Tracy Zager Johnson’s blog post Which Mistake to Pursue? Looking back on this conversation with the second graders, I want to think about how to pursue the ideas of quantity and magnitude.

How formal should this be?

Should we just open up the middle container and count them because this is the first time we have used this structure to pursue the idea of how quantities relate to each other? (The thinking behind selecting how many crayons were in each container was to be sure and use quantities of crayons kids could easily imagine.)

Do we have a sequence of teacher moves ready, like those listed below, and see what the kids think and say and do, and trust them to lead the conversation?

Have all kids record, on post-it notes, the quantity they each believe might be in the middle container and stick them up on the board. Talk about the estimates.

Talk about the number of crayons that couldn’t be in the middle container and why that could be true.

Have a few more students share what his/her partner said, and chat about the possibilities.

Have your kiddos do a short writing piece that captures the ideas explored so far:  If you had to choose one of the amounts written on the board (and it can’t be yours), which one you would choose? Why do you think that could be the amount in the middle container?

Maybe I am over-thinking this whole deal and should just enjoy the conversation. (Because when you start to explore all of the possibilities as to how and where the conversation might go, it can get crazy and it starts to take on a life of its own.)

Step 6:  What I actually did was collect a few more estimates, open up the middle container, and count the number of crayons. Lots of ohhing! and ahhing! and groaning! and I was so close!  A very nice way to move into the next part of the day’s math work.


For those of you who want to check your estimate, the number of crayons in the middle container was:

A task for a Wednesday morning: nine and three-fourths

Nine and three-fourths is another task on my list of favorites. Students are always intrigued with it, as are colleagues who tackle it in workshop settings. It has been one of my go-to tasks for quite awhile. I do have to be honest and say that I have no idea where I found it. However, I want to extend my thanks to the author, as well as, my sincere apologies for losing track of whose work it is. It is an awesome task.

It is an awesome task because everyone can play. Everyone can join in the conversation and the learning. Everyone has ideas to investigate, think about, and share with others.

This past June, more than forty K – 5 teachers spent some time with this task during a five-day summer math camp. We used nine and three-fourths to kick off our two-day conversation centered around the big ideas in fractions.


Choose one or more pattern blocks to represent one unit. Based on the unit you selected, create a picture that is worth 9 and three-fourths units.

Screen Shot 2015-10-05 at 6.12.35 AM

  • Identify the unit you selected
  • Trace and color your picture
  • Justify that your picture does have a value of 9 and three-fourths units

The morning session began with an exploration of the patterns blocks and the relationships that exist among the various blocks in the set. The set of pattern blocks we used included all of the blocks shown, as well as, the brown trapezoids from the add-on set. We talked about the task itself; which including a discussion of an example of a picture worth nine and three-fourths, when the yellow hexagon is worth 1. The picture was comprised of nine yellow hexagons and 3 brown trapezoids (not shown in the picture above). We agreed that we would not use the yellow hexagon, or an equivalent, as the unit. And they were off; working, building, talking, re-starting, recording, and considering how to share this with students, and what aspect of the task should be used to begin the class discussion. At the conclusion of the morning’s work time, a few of the workshop participants presented their pictures.

The following are representative of the pieces that were shared.

Screen Shot 2015-10-05 at 5.30.48 PM



Using the 5 Practices for Orchestrating Productive Mathematical Discussion by Margaret Smith and Mary Kay Stein to build our conversation, we talked about the unit each picture was based on, how three-fourths was represented in each drawing, how and why each picture met the criteria of the task, and how each unit was selected. We discussed the teacher moves used throughout the morning session, the impact they had on the learning, and what adjustments might need to be made when sharing the task with students. The morning session concluded with a conversation focused on how the task provides opportunities to discuss and tackle some of the misconceptions students have about the big ideas of fractions.

It was a great way to spend a Wednesday morning.