Yesterday I wrote about trying out Notice and Wonder Graphs with 4th grade teachers preparing for their unit on data. In yesterday’s post I wrote about how these teachers noticed, wondered, and drew conclusions about a bar graph entitled “More then half of Brits like pineapple on pizza”–an interesting, if not earth-shattering topic. But we didn’t want to only analyze graphs about pineapple and pizza.

I’ve noticed that in elementary school, at least, the data we collect and analyze often falls into the broad category of “interesting survey questions.

“Did you buy milk today?

“Who is your favorite Pokemon character?”

“Do you prefer winter or summer?”

And, of course, “do you like pineapple pizza?”

I don’t think these kinds of survey questions are bad! However, I think that if this is the ONLY kind of data that students are exposed to, they’re getting an incomplete picture of what data is and why it is useful.

Hoping to have a conversation around these ideas, I showed 4th grade teachers this series of images. Again, I asked them:

“What do you notice?”

“What do you wonder?”

“How does this new information change your thinking?”

First we looked at this:

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Then we looked at this:

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And finally, after thinking about the question, “Given what we know, what might be the title of this graph?” we looked at the unmasked graph:

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We had lots of great conversations after looking at each image, but the most interesting came when we were trying to make sense of the full graph image.

The subtitle of the graph says “Percent of drivers pulled over by police in 2011, by race. We wondered: does that mean that this graph represents all people pulled over by police in 2011 and each bar is the percentage of those total people people pulled over? But then someone pointed out that the bars only add to about 70%. We also noticed that the bar representing Native Americans was the highest. That seemed surprising considering that Native Americans represent a pretty small percent of our overall population. (My quick googling said 2% in 2014.) Could more Native American people be being pulled over than black people? Something seemed off in our interpretation.

We realized we needed to go back to the source of this graph, a report from the Justice Department. Reading into the report, this was our new understanding of the data. The graph is based on a survey with people who had “involuntary contact” with the police in the last year (2011, in this case). Of these people (grouped by race in this graph), the percents on the graph represent the people whose “involuntary contact” was a traffic stop. The report summarized that Relatively more black drivers (13%) than white (10%) and Hispanic (10%) drivers were pulled over in a traffic stop during their most recent contact with police. There were no statistical differences in the race or Hispanic origin of persons involved in street stops” (1). (Does this jive with your interpretation of the data? I’m open to the idea I misinterpreted it or didn’t understand something!)

Really making sense of this graph required a good amount of work for us, as teachers, and this routine support the slowing down of our thinking. The topic of this graph helped us discuss the idea that data is a way of investigating our world. We wondered about the kinds of data and graphs we could have students interact with around social justice and civil rights issues. We talked about giving kids data about important issues with which to grapple. Racism. Sexism. Climate change. Kids are thinking about these issues. Some of the amazing children’s literature out there helps kids consider these ideas in age appropriate ways. How can we do this with math too?

A few weeks ago I was lucky enough to hear Brian Bushart (@bstockus) and Regina Payne (@reginarocks) talk about Numberless Word Problems at NCTM in San Antonio. Brian and Regina left me with lots of ideas to think about, and one I’ve started working on already is numberless graphs. By the way, Brian blogged about this idea  and I found that post to be a really helpful starting place when trying to figure out how to do this.

I happened to have a planning meeting with 4th grade teachers coming up right after NCTM in which we were planning a unit on data, so I thought I’d try this routine out with them. (By the way, I called it Notice and Wonder Graphs because it seems to be about more than being “numberless” to me.) This was the first image I showed teachers. I asked them:

What do you notice? What do you wonder?

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Here’s what they said:


“I think there are two groups being compared.”

“It looks like the difference between the dark and light green are shrinking over time.”

“The third pair of bars looks different that the first two.”


“What do the numbers on the bars represent?”

“Do the dark green and light green go together?”

“Are the same two things being compared each time with each pair of bars?”

Then I revealed a little more information with this image.

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I asked “What do you notice and wonder now?” “How does the new information change your thinking?”

Teachers wondered:

“Are three different things/categories being compared? Or is the same thing being compared over three different times?” 

Are the numbers percents? Why don’t the three pairs of bars add up to the same number? Why don’t they add to 100?”

Is this graph based on a survey question?”

Next I revealed the title of the graph.

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Again, I asked “What do you notice and wonder? How does the new information change your thinking?” (What a beautiful routine for practicing revising your thinking, right?) and “Considering the title of the graph, what do you think the x-axis labels might be?”

The teachers thought the title was referring to the third pair of bars, considering 53% is close to half. However we all agreed that “More than half” could apply to the 84% or the 82% as well.

Finally, it was time for the big reveal!

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“What surprises you? What are you still wondering? What questions might you ask about this graph?” I asked.

Teachers still wondered why the first two sets of bars adds to 96% but the third adds to 94%. We decided that some people were so disgusted by the idea of pineapple on pizza they refused to even answer the question.

I love this routine for many of the reasons that I love Brian’s numberless word problems–it slows the thinking down and focuses on sense-making rather than answer-getting.

But I also love it because it brings out the storytelling aspect of data. So often in school (especially elementary school!) we analyze fake data. Or, perhaps worse, we create the same “What is your favorite ice cream flavor?” graph year after year after year for no apparent purpose.

I’ve decided to make it a goal to think more about data as storytelling, data as a way to investigate the world, and data as a tool for action. In my next two posts (YES, people! I’m firing the ole blog back up again!) I’m going to delve into the idea that we can use data to discuss social justice ideas and critical literacy at the elementary level. I’m just dipping my toe into this waters, but I’m really excited about it!

Have you tried this routine with students or teachers? I’d love to hear more about your experiences.


How do play and sense making intermingle in math? What does this look like for the youngest and oldest students at our school? How and when does school math get in the way? And how can we get out of the way a bit and use the power of play to guide the sense-making process? These are questions I’ve had bouncing around my head for a while that I want to investigate a little bit this year.

This week I visited a preschool class at our school with a big tub of tiny pumpkins and gourds. I have to admit that I didn’t have any great plan for what I wanted to get out of pumpkin and gourd play, I just wanted to play alongside the kids and see what they noticed and wondered. Also, tiny pumpkins and gourds are just the best. It’s impossible to not have fun with them. I asked to come to the preschool class during their open play and explore time of the day, so pumpkin and gourd play was a center kids could visit and engage in if they chose and many were intrigued by the tub o’ gourds.


As the kids played, I tried out some questions to see what kinds of pumpkin and gourd attributes they were noticing:

What do you see/notice?

What’s special about this one?

Which ones go together?

Which one doesn’t belong? 

(What other questions would you have asked?)

Most of our students are English Language Learners, so part of the joy for me was listening to them working to create ways of talking about the pumpkins and gourds in a new language.

Some of the more poetic groupings they created were “teeny tiny green” (a pumpkin or gourd with only a teeny tiny bit green), broken stems, and long bumpys.

After a little bit, I asked if anyone wanted to play a game with me. I asked one child to choose a pumpkin or gourd to put in the center of table. “Tell us about it,” I asked. The child would usually say or point to some sort of attribute (bumpy, stem, long, orange) and then I’d ask the other kids, “Do you have one that goes with this one? Why?” Sometimes they’d echo back the same attribute of the person who placed the first pumpkin or gourd, but sometimes they had their own ideas. “Which ones belong? Which ones don’t belong?” I asked as they played.

My favorite part of the pumpkin and gourd play happened when I was watching some kids decide to sort into pumpkin and gourd groups. There was one small white pumpkin. One kid said, “Yes, this is a pumpkin. Like these,” he said pointing to the shape of the others. Absolutely incredulous, another child responded, “No! This is WHITE! Pumpkins are ORANGE! This is NOT a pumpkin!” What makes a pumpkin a pumpkin? We debated back and forth. The white pumpkin was added to the pumpkin group and pushed away. No consensus was reached. But it was authentic and lovely. And mathy in its own way.


I tried to guide some talk, ask some questions, but I also tried to just let them play and ask their own questions through play. There was some counting, some stroking of pumpkins against cheeks to determine the degree of bumpiness, some trading of pumpkins and naming which ones were our favorites.

I’ve read Christopher Danielson’s (@trianglemancsd) article, “The power of having more than one right answer: Ambiguity in math class,” several times this week. He writes, “but ambiguity—a messy place—can be where important mathematics begins.” One of the beauties of play is that no one is ahead or behind in play. There is no wrong way to make sense of a tub of pumpkins and gourds.

I plan on keeping up my visits to the preschool class. I want to know more about “where math begins,” how to muck around in important math ideas with small people and let their play lead the way.

I want to keep wondering about this place for ambiguity and questions and play as kids get older and school math seems to take over. What would the 5th graders say about the white pumpkin?





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Last week I got to visit three wonderful schools in Michigan to think and learn with teachers about math talk, reasoning and number sense. We got to plan, teach and debrief together, which is one of my favorite ways of learning with a group of teachers!

With one group of 3rd grade teachers, we decided to think more about choral counting routines and how we could use them to get kids talking, reasoning, and looking for patterns and structure.

The team chose to count by 3s starting at 21. Before teaching, we thought about what mathematical ideas and patterns this count might bring out and how we could record the count to support those ideas. This group of teachers also wanted to think more about students journaling around math, so we thought about taking time at a certain point in the count to have kids write about their thinking.

In the above picture are the numbers we said and recorded in our count. Students immediately noticed how the digit in the ones place repeated at a certain point in the count, as well as patterns related to how many times numbers in a particular decade would appear. Students noticed that while we were skip counting by 3s, there was a jump of +30 between the numbers in the same column. When we asked students to predict the number that would go in the orange box below 60, some counted by 3s from 78 (the last number we said together), but many used the +30 noticing.

But things got really interesting when one student spoke up and said “Hey, and 30 + 60 is 90! You can add the first two numbers in the column together to get the one below.”

“That does work here, doesn’t it?” I said, writing it down. “I wonder if that works everywhere or just here? I wonder how we could figure that out.”

In many ways I wish we had had more conversation here, especially since the students were new to this routine and to the idea of pursuing and investigating claims and conjectures, but for the sake of time and getting to the journaling we decided to move ahead with our plan to have them write.

We asked students to write the number they thought would go in the second orange box (the blank box in the picture) and write about the patterns they noticed that helped them figure this out.

As a group of teachers we read through the students’ journal entries. Some kids wrote about skip counting from 90. Some kids wrote about how they knew the ones place would be a 2 and the number would be 100 and that 2. But a good number of kids wrote that the number would be 114 because 42+72=114. This was fascinating because they ignored the previous patterns they had noticed in favor of following this claim set forth by a student. This claim had power! (I think the idea that you could notice something that would lead to a claim was powerful for this group.) The students wanted to make a generalization, find a pattern, find a rule so much that they ignored evidence against their claim. It was fascinating and a beautiful error, in an inaccurate sort of way. 🙂

This count and journaling also provided amazing material for teacher discussion. The thoughtful teachers I worked with talked about how they might push students towards ways of investigating conjectures and claims, how you might prove or disprove this claim, and the importance of this kind of reasoning for wider mathematical understanding.

All this great thinking, teaching, and debriefing in a brief little time with this team of teachers. I was only sad that I didn’t get to stay for next class meeting to see where students would take this next!

But it has pushed my thinking as well. First, it made me reach for Russell, Schifter and Bastable’s Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades. It also made me reach out to Twitter-thinkers Kristin Gray (@mathminds), Mike Flynn (@MikeFlynn55), Simon Gregg (@Simon_Gregg) and Elham Kazemi (@ekazemi) to clarify my thinking on claims and conjectures (which I’m still working on!). I can’t wait to think and teach more.


I just love counting routines. I’ve learned a tremendous amount about counting routines from Jessica Shumway’s Number Sense Routines and the teachers featured in Elham Kazemi and Allison Hintz’s Intentional Talk(Several of these teachers are featured in videos you can view online for free. This one is my favorite!)

Today I got to do a Count Around the Circle with a twist in a 3rd grade class launching its fraction unit. While 3rd graders have a good amount of experience with fractions, this group’s teacher was wondering about their understanding of the part/whole relationship and relationship among unit fractions. We decided to do this count to investigate these ideas!

After we gathered students in the counting circle, we told them that today we’d be counting around the circle in a different way. We’d be counting brownies today and each person would have one whole brownie. We explained that we’d be laying out the brownies (post-its!) on a big imaginary plate and we wondered how many brownies we’d have after we had each contributed one brownie to the plate. The students quickly sized up that we would have twenty brownies–one for each person in the circle. As students counted and placed their brownies in the middle of the circle, I helped them organize the brownies onto the plate in rows of five. I knew this would help them see some important relationships once we had counted by halves.

Next we got ready to count by half brownies. These students already had experience looking at different ways that two people could share one brownie equally, so they quickly suggested a way for cutting brownies into halves. Once each student had their half brownie in their hand we said, “We’re going to put these half brownies on another imaginary plate. Do you think we’ll have more or fewer brownies on the plate after we all count around the circle?” We first asked for a quick thumbs up or thumbs down to signify more or fewer brownies. The class was split! These were some of the ideas they shared after a quick turn and talk.

“It’s the same. There’s still twenty people and we still each have one piece to put in the circle.”

“More. This time we’ll have 40 because that’s double the twenty brownies.” (This one fascinated me! Playing around with doubling and halving, but still developing that fraction number sense!)

“Fewer. It will take two people to make a whole brownie.” (Some people had an idea of what number we’d land on. Others just knew fewer.)

As we counted, I again helped students place their brownies in rows of five. After the brownies were laid out we simply asked, “What do you notice? What do you wonder?”

There was a lot of clarifying talk about halving and doubling, two halves equally one whole, and twenty halves being equivalent to ten.

We moved on to our next task by saying, “Let’s save our plate of whole brownies and half brownies. I wonder what would happen if we counted by quarter/one-fourth of a brownie?”  (Which we will do tomorrow! Can’t wait for the discussion of the relationship between the denominator and “size” of the pieces in unit fractions.)

Working with halves and wholes might seems overly simply for 3rd graders. But I’m reminded again that sometimes “simple” numbers offer an opportunity to explore big ideas and important relationships.

[Special thanks to 3rd grade teacher, Ellen Rogers, for planning, teaching, and reflecting on fraction counting circles with me.]

Math in Life Photo Series

The grocery store is one of my favorite places to think about math. Aron, my husband, has gotten used to me saying, “Oh, I’ve just got to take a picture of this!” on our weekly grocery shopping trips. Today I’m sharing a picture I took at the grocery store this week.

This is the first photo I’m sharing as a part of a Math in Life photo series I’m starting. In my book, Math Exchanges, I encourage teachers to find their own rich mathematical life, share that life with their students, and make space in their classroom communities (and beyond!) for students to live their own rich mathematical lives. Sharing photos has always been a part of this practice for me. Include a math in life picture you take (or borrow mine!) in your morning message and ask students a couple of questions that make sense for the photo:

  • What do you notice?
  • What do you wonder?
  • What math does this make you think of?
  • What does this picture make you want to figure out?

I took the photo I’m sharing today because I thought the array were particularly interesting. But it might speak to you and your students differently.

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When I saw these chicks, I wondered how the 3rd graders I work with would mathematize this picture using multiplication.

Would they think of the rows–8 groups of 6 (8×6)?

Would they group the chicks by color and column?–6 groups of 8 (6×8)?

Would they see each individual box first–2 groups of 6 (2×6) and then four of those boxes 4(2×6)? (What a nice what to start playing with the distributive property!)

Tune in soon for more Math in Life photos!

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The first ten minutes of math workshop are often my favorite part of the day. This is a time when we use number sense routines to build math understanding and also to build communities in which students are talking about and playing with math. We try to make this time a time when every single student feels a sense of belonging, efficacy and participation in the mathematical community.

As our 5th graders have begun thinking about addition of fractions, we want students to learn more than procedures for fraction computation. We want them to reason about fractional equivalencies and relationships as they solve problems.

In this version of Over or Under, students are asked to consider whether the sum will be Over or Under 1. Students are presented problems one at a time (we usually did three problems each day). Students have time to think about each problem on their own. No one raises hands yet. We just think in silence (this takes practice!). Then we either have partners do a turn and talk or share ideas as a whole group. Finally, for one of the problems, we  have students share their thinking through drawing and writing in their notebooks.

Here are the three problems we presented on the first day of this routine and some of the students’ thinking. (I admit we didn’t capture a lot of the thinking on our chart paper because we were just really into discussion! We want to get better at this though.)

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Question 1:

2 1/2 + 1/5

We decided to start each day with a problem that would seem fairly easy to most students, but would open conversation up about an important idea. In this case we wanted students to think about mixed numbers and how the whole numbers can help us make reasonable estimates.

Question 2:

1/12 + 1/6

We’ve been doing a lot of work helping students think about which fractions are close to 0, close to 1/2 and close to 1. We want them to create mental images that include area and linear models. Some of the student thinking that came out of this was that 1/12 + 1/6 were both “kind of near zero” and “wouldn’t even be 1/2.”

Question 3:

8/9 + 1/4

This was my favorite question for eliciting fascinating student thinking. Many students talked about what an area model of 8/9 would look like and what the “missing piece” (1/9) would look like. Many said that 1/4 was too big to fit in this “missing piece” and thus the sum would be more than 1.

Here are some of the other problems we used throughout the week. We recorded some student ideas as they talked, but we got so into the talk at times, we didn’t record everything.

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We did notice that students still need more experience thinking about linear models. This week we started playing the awesome Investigations game “Fraction Track” which encourages just this kind of thinking. I’ll be interested to see how experience with this game shifts some of their thinking and reasoning about fraction computation.

Today students took a district-wide multiple choice assessment. One of the questions asked students to solve 4/5+1/3. Only one the answer choices was over one! Many of the students wrote that they didn’t need to solve the whole problem or worry about finding common denominators because they knew that 4/5 + 1/5=1 and since 1/3 >1/5, the sum would be more that 1. Boo ya! Way to outsmart a test question with reasoning and number sense, kids!

Besides building number sense, this routine really built community. The game aspect of the routine (Is it over or under?!) really appealed to the kids. They loved arguing for their ideas. Also, not having to arrive at an exact answer was very inclusive of all students. Even students just developing strategies for addition of fractions were able to share their hunches. “I think it might be over because…” The risk of sharing this idea was low and students refined important ideas about fractions even around the “easy” problems.

I’m looking forward to extending this routine into subtraction with fractions!

How else would you use this routine?