tzager

The Steep Part of the Learning Curve

Posted on November 22, 2015January 6, 2017 by tzager

I’m still coming down from the high of the awesome NCTM Regional conference in Nashville. So many good things were happening, especially in the coming together of NCTM and the MTBoS (Math Twitter Blog-o-Sphere). I’ve been thinking about my thinking and learning, and I find myself working over a few meta-observations. Here were the sessions I chose, plus the awesome #MTBoS mash-up keynote:

“Motivating Our Students with Real-World Problem-Based Lessons.” Robert Kaplinsky, 6-8.
“Get Your Model On: Mathematical Modeling in the Elementary Classroom.” Graham Fletcher and Mike Wiernicki. 3-5.
I tried to go to “Model with Mathematics Using Problem-Solving Tasks” from Andrew Stadel, but I couldn’t get past the bouncer. 6-8. Instead, I hung out with Dane Ehlert, Michael Fenton, Michael Pershan, and Malke Rosenfeld.
“Plan a Killer Lesson Today.” Kate Nowak. 9-12.
“Using a Three-Lens Approach in Mathematics Professional Development.” Mike Flynn. 3-5.
“Empowering Students with Rich Online Algebra Activities.” Christopher Danielson. 9-12.
“Desmos and Modeling: A Mathematical Match Made in Heaven.” Michael Fenton. 9-12.
And it killed me to miss “Fumbling toward Inquiry: Starting Strong in Problem-Based Learning” because of my flight. Super high on my wishlist. Geoff Krall. 9-12.

What do you notice and wonder? Here are mine:

I went with all twitter folks this time. Normally I have more of a mix, but I follow all these people closely online and there were several I’d never seen present in person. I had to address that.

Lotta guys. Hmm.

I checked out a lot of modeling, eh? Interesting. Unintentional.

I went to two desmos sessions because, as a mostly K-6 girl, I don’t play with it nearly as much as I’d like. I knew I’d have a ton to learn.

I picked eight sessions. Two of them were 3-5, which is my home base grade band. Two of them were 6-8. And four of them were 9-12.

What is up with that?

In the car back to the airport, Michael Fenton asked me what my conference highlights were, and the first thing that came to mind was playing with Christopher Danielson’s pentagons with Geoff Krall, Max Ray-Riek, Michael, and Christopher in the hotel bar Thursday night. I remember feeling incredibly happy exploring math with these people.

The other one that comes to mind, which I didn’t have time to tell Michael, was sitting in the back row of Christopher’s desmos session between John Mahlstedt and Julie Reulbach. Cathy Yenca and Mike Wiernicki were also there and lovely. I’d gotten to know both Cathy and Julie better the night before during amazing math education conversations over hot chicken, and it was great to connect with Mike in person as well. Christopher had us play Central Park, which I’d seen several times and played the first few screens of, but had never explored all the way through. Julie and John have used it lots, of course, but they were happy to be there and supportive as they watched me figure it out for myself. Julie played too, trying out new things and possible student solutions to see what would happen. And John was a great thinking partner for me as a newbie.

I’m not sharing these stories to drop names. I’m sharing these stories because there’s a common thread that interests me. I find myself hanging out with a lot of secondary people these days, and I’m happy to be there for a few reasons. Here’s why:

My Own Mathematical Learning

Over the past 15 years, since I first walked into Elham Kazemi’s math methods class, I have been re-learning elementary mathematics. My first time through, I was taught elementary math procedurally, like everyone else. It’s been a delight to figure out why the algorithms work, what the operations really mean, how ideas connect, and what strategies make sense in which situations. I still have tons to learn, which is why I am super bummed I missed Max Ray-Riek’s fraction division session. There’s way more ore to mine for me there.

That said, I have yet to re-learn most of the rest of my math, especially algebra. I learned a lot of formulas. Given the powerful experiences I’ve had re-learning elementary math, I know the delight of the a-ha moments ahead if I plunge into that stuff I memorized and figure out why it all works. I did learn calculus in an experimental, conceptual college class where we worked in groups and designed things and it was fantastic. It was also in 1991. I used a lot of calculus afterwards as a science major, but left that world in 1996. There’s 19 years of dust sitting on that thing.

So my first reason for attending all these 6-12 sessions is that I know I’ll learn and think about mathematics that I don’t get to learn and think about very often. I have long since stopped worrying about whether that kind of choice has practical application to my work in elementary schools. I know it does. The more math I learn, the better math teacher I am. I keep growing as a learner; I know more about where my kids are headed; and I understand more about what building is going on top of the foundation we construct in elementary school.

Solutions to Problems I Didn’t Know I Had

Dan Meyer was up in my neck of the woods a couple of weeks ago for the ATMNE conference, and I went to all three of his sessions. In Nashville, I went to Kate Nowak‘s session, where she posted this slide:

Dan and Kate both talk this way regularly. Motivating disaffected students to care about boring bits of math teachers have to cover is a pressing issue for them.

It’s not a pressing issue for me, in elementary school. It’s an issue, but not in my top five because we haven’t totally killed kids’ curiosity about mathematics yet. We’ve gotten better at covering less and spending our time on big ideas. The math we do actually does exist in students’ daily lives (kids have lots of experience joining, sharing, comparing, scaling, etc.). And, as elementary school teachers, we’re getting better at building off the mathematical ideas and questions students already have. All those things help with motivation.

So why do I go to talks about a problem I don’t have? Because the solutions to the problems are pedagogically powerful whether I’m facing apathetic teenagers or squirrelly first graders. For example, Dan’s done a lot of work (e.g., this series) around how we can create intellectual need so students actually have a problem they’re motivated to solve. (The ref on intellectual need is Harel 2013 and it’s awesome.)

Over the last couple of years, I’ve been mulling over how this idea of need applies in elementary school. Kids are more willing to do the work, but does that let me off the hook on need? I don’t think so. I have been toying with creating need – creating headaches, as Dan likes to say – in my teaching and it’s incredibly powerful. I’m thinking about conversations like this twitter thread, starting with these tweets:

By the end of the conversation with all sorts of K-12 tweeps, we’d decided that, if tens frames were the aspirin, a disorganized heap of stuff made a great headache. We decided to introduce tens frames by making them available – alongside cups, bowls, and plates – during counting collections, hoping some kids would do what Joe Schwartz‘s kids did here:

Anyway, my point is that I wasn’t feeling this problem of the unmotivated student the way my high school colleagues were, but I am really feeling the power of their solutions to this problem. Their work is impacting my teaching in all kinds of good ways.

Here’s my wondering about that. What’s the corollary for my secondary friends? What pedagogical solutions do 6-12 teachers learn from K-5 colleagues about problems they didn’t know they had? I have lots of ideas about that, but I’m hoping you might share yours in the comments first.

The Question on My Mind

That wondering is all tied up with what I’m thinking about these days, from a professional point of view. This year, I began a new chapter in my working life by joining Stenhouse Publishers. As of January, I’ll be working half time as a Math Editor. I’m all kinds of excited and have lots of dreams about how I can use this position to give teachers platforms for sharing their ideas. I do a lot of thinking now about whose voices are missing from the professional development conversation, and whose voices are needed to impact education in positive ways that will help lots of kids and their teachers.

One of the great mysteries, to me, is what kind of professional development 6-12 teachers need. Elementary and secondary share so many instructional challenges, but others are so different and I have lots to learn. For example, I am so curious how teachers teach multiple sections. What’s it like to get to revise your lesson right away? I have to wait a year before I get to take another crack at most of my lessons. Julie Reulbach said, “Oh my God, you’re whole life is A period!” On the flip side, if I need 10 more minutes because of what’s happening in math, I can take it. I have flexibility, and I thrive on the variety of teaching 5 subjects. Another example: when I think about discourse with 30 kids I know really well, that’s totally different from discourse with 120 kids. Isn’t it? What happens with relationships with so many kids? And then there’s content. If I were to teach the same lesson 4-5 times this week, I expect I’d dig into that content and plan that lesson really deeply, in a way elementary teachers juggling 5 subjects rarely get to do. But then would I become committed to performing a lesson repeatedly, and get less responsive to where the kids actually are?

You could say the main question on my mind is how is instructional decision-making the same K-5 and 6-12, and how is it different? It’s not a long boil to get from that question down to this one: What do we have to learn from each other?

If the #MTBoS is any indication, a lot. My favorite picture from NCTM Nashville is this one:

Among these four people (and two pairs of cutiepie twins), I’m pretty sure you have deep experience teaching math at every grade level. In my professional life, I’ve only seen that kind of vertical collaboration in two places: conferences and the #MTBoS. The combination of the two is out of this world.

So, while I totally love my elementary peeps and tweeps and treasure the time we have to talk about both the content and pedagogical challenges we share, I also really love stretching myself by going to 6-12 sessions and learning about other content and pedagogical challenges. I’ve always been happiest on the steep part of the learning curve. Don’t get me wrong. I have tons more to learn about teaching elementary math, but I get a lot of value out of stepping out of my comfort zone.

One last word. This all only works if we support each other. When I was playing around on desmos among a group of people who know the task and the math inside out, I felt perfectly comfortable because I knew they were getting a kick out of me being there and stretching myself. Mistakes and misperceptions are part of the territory and everybody is down with that. In Michael Fenton’s session, there was a whole lot of math I haven’t touched in ages and is far from automatic for me right now. No sweat. That’s a fun prospect, and I look forward to taking the time to play with it when I can.

What’s the corollary? When middle and high school people want to learn about questioning strategies, academically productive talk, inquiry, the use of materials, small group work, listening to kids’ thinking, and looking at student work, elementary people are eager and willing to share. At least the ones I want to hang out with are.

Because, see, competitions over who knows more, who works harder, or who has a more challenging job don’t get us anywhere. What does? When we go to each other’s sessions, sit next to each other at the bar, do math together when we can, and interact with each other (not just follow) on twitter and in blog comments.

We each get to create the faculty lounge of our dreams here on the interwebs (h/t Christopher). Mine is populated with people who teach from kindergarten through college, who like to learn together and from each other, who make others feel welcome, and who listen with respect. That’s my dream.

It was awesome to connect in person with so many colleagues in Nashville. Thank you to the committee for all your hard work. I’m already pumped for Oakland/SF. See you there. You’ll find me in sessions both in and out of my comfort zone, meeting and talking with colleagues who span math education, picking up new ideas wherever I can.

Harel, Guershon. 2013. “Intellectual Need.” In K.R. Leatham (Ed.), Vital Directions for Mathematics Education Research (119–151). New York, NY: Springer.

ATMNE15 Handouts

Posted on October 29, 2015January 6, 2017 by tzager

In case we run out, here are the handouts.

#atmne15risk

Thanks for coming!

Which Mistake to Pursue?

Posted on October 16, 2015January 6, 2017 by tzager

Yesterday, I was the lucky duck working with a team of three teachers who each teach multi-age 3rd/4th grade. We were all in one class together in the morning, and then got to meet during their common planning time later that day.

The kids have been working on perimeter, mostly of rectangles but also of other polygons. The first problem yesterday emphasized decomposing a 14-sided polygon into smaller rectangles in order to find the lengths of all the sides and then calculating the perimeter. Rectangles were in the air.

The second problem was this one from Investigations:

“The perimeter of Pilar’s yard is 100 feet. Draw a picture of what her yard might look like, and label each side.”

Unsurprisingly, we mostly got rectangles from kids, and mostly these two variations:

There was a little discussion at tables about whether this 25 x 25 square is actually a rectangle. Lots of kids have the classic misconceptions there. Oh how I wish we called squares “square rectangles” or “equilateral rectangles.” Sigh.

One student made a pentagon instead, which I totally loved.

And then one student, pseudonym Jesse, made this shape:

Look closely. I noticed the dimension on the left-hand side is 0 yards. So are some of the ones on the right, but not all. I also noticed the long line across the top is 30 yards, and the series of short line segments on the right hand side that are all about the same length start with labels of 30 yards, 6 yards, 1 yard.

After the gallery walk, I headed over to Jesse. He made me laugh because he said, “I could see you squinting at my board from across the room!” I told him how cool it was that he took a risk and tried out a different shape, and I asked him how he did it.

“I made a cool shape, and then I started putting numbers down. But the problem is I used up my 100 and I still had a lot to go, so I made some of the lengths zero.”

“Interesting. I’m wondering what would happen if you tried to build this yard?”

Jesse said, “You couldn’t. It wouldn’t work. The zeros would mess you up.”

I asked, “Do you think you could make a yard this shape, that does have a perimeter of 100 yards, that you could actually build?”

Jesse said, “Yeah, but I’d need to erase all the numbers and start again. I’d try smaller numbers first so I wouldn’t run out of fence.”

We had to stop talking then, but I grabbed his board to discuss with the team later at common planning time. The kids had moved on to talking about the third problem they’d worked on, which was to draw three different rectangles that all had a perimeter of 20 units. The teacher had them post different rectangles on the whiteboard, and we saw all the whole number solutions: 1 x 9, 2 x 8, 3 x 7, 4 x 6, 5 x 5. The teacher asked a money question:

“Do we have all of them?”

Love that.

One student, pseudonym Zeke, said “No. There are lots more. Like, I could make one that’s 15, 2, 2, and 1. They add up to 20 also.” He held up a piece of paper that had a mass of zigging and zagging and curving lines and pointed to it as evidence.

This was the most interesting moment of the math class. Both the classroom teacher and the paraprofessional physically moved toward Zeke, lowered their voices and started talking at the same time:

“No, you can’t because…”

“Remember, you have to make a rectangle and rectangles have…”

In keeping with the norms we teachers have as a team of learners, I paused my colleagues for a second and asked if kids could discuss Zeke’s claim instead? Thumbs up. I asked the students whether they agreed or disagreed with Zeke’s statement that a rectangle could have side lengths of 15, 2, 2, and 1?

Students discussed the claim at their table groups. As we listened in, we heard a few useful arguments pop up. Some tables talked about both pairs of sides of rectangles needing to have equal length. Other tables talked about whether you could make a closed shape with those dimensions. Other tables were full of kids trying to draw a solution to his challenge.

We also noticed, though, that several students said you could make a rectangle with those dimensions. They had different, confusing arguments, and they weren’t leaning on the properties of rectangles at all.

And we noticed that no group had put together all the pieces. There might be one student talking about opposite sides having equal length, but they weren’t putting that property together with other properties of rectangles in a coherent way.

Time was up and we left it there.

At the end of the day, we were so glad to meet at common planning time. The classroom teacher thanked me for stopping her from correcting Zeke, and thought his claim made for a worthy discussion. We talked a bit about this—about how to view mistakes as opportunities in the classroom, and to free ourselves of the pressure to correct kids right away. We talked about not worrying that posting or sharing wrong answers will confuse kids. We talked about this lovely feature of mathematics, which is that we don’t have to jump in as the authority figures of what students “can” or “can’t” do; rather, we can trust that, in almost all cases, students can determine the truth for themselves by delving into the mathematics. This quote from Making Sense captures this idea beautifully:

“In traditional systems of instruction, teachers are asked to provide feedback on students’ responses, to tell them whether or not they are right…this is almost always unnecessary and usually inappropriate. Mathematics is a unique subject because…correctness is not a matter of opinion; it is build into the logic and structure of the subject…There is no need for the teacher to have the final word on correctness. The final word is provided by the logic of the subject and the students’ explanations and justifications that are built on this logic” (Hiebert et al. 1997, 40).

We also talked about the importance of non-examples, which made me think of this perfectly put tweet from Kate.

I told the teachers that, as a 4th grade teacher, I always had to teach kids what rectangles are. Kids pick up the crummy board-book definition of rectangles as shapes with 2 long sides and 2 short sides and square corners. (All the more reason why I can’t wait to publish Christopher Danielson’s better shapes book next year!) As teachers, we need to create conditions for kids to add depth and sophistication to their understanding about shapes. I thought Zeke’s claim was a fantastic opportunity for students to think about what a rectangle is and isn’t by focusing on the properties of rectangles, not just visual arguments.

The four of us kicked ideas around for where to go next. We had this open question: could you make a rectangle with side lengths of 15, 2, 2, 1? We all felt like the kids should take that question up by trying to build the rectangle. We settled on this plan for today:

The teacher would give each student 20 toothpicks and define each toothpick as one unit long. She’d ask students to make all the rectangles they could using all 20 toothpicks, without breaking toothpicks. They’d synthesize their list of rectangles to see if they had all the whole number solutions. Then they would take up Zeke’s challenge. Could they use their toothpicks to construct a rectangle with side lengths 15, 2, 2, 1? The teacher would use this exercise to draw out the definition of a rectangle, focusing on properties.

I wish I were going to be there today so I could see how it goes!

The four of us moved on in our planning time, and I showed them Jesse’s diagram for the 100-foot perimeter problem. This one.

Nobody had noticed it during the lesson, and we had a great conversation about it. And then the classroom teacher laid a whopper of a question on me:

“Do you think I should take up this work with the class?”

Holy smokes. So much ran through my head right then. Mostly, I was thrilled, overjoyed, elated by her question. Remember, earlier that day she had started to correct Zeke instead of consider his claim. I had just spent all this time trying to help this group understand the value of a piece of student work that reveals mathematical misunderstandings, or provides an opening to dig into a rich piece of mathematics. She was obviously listening and learning!

Once we decide to teach this way, though, the questions get harder, not easier. If we’re going to be responsive to what our students do and say in mathematics, how do we decide what to respond to? What makes for a worthy mistake to take up as a whole class? How do you know? By what criteria? We obviously can’t launch deep investigations into every single piece of student work, so how do we decide which ones to pursue?

This is one of my favorite things to think about. There’s a penciled-in plan for me to write a second book after I finish Becoming the Math Teacher You Wish You’d Had. The working title in my mind has always been Teaching from Mistake to Mistake. I want to get into these thorny questions of how we respond to students’ mistakes, what mathematics lies underneath different mistakes, what mathematical goals we have for our kids, how we decide which mistakes to take up in a big way, and how we decide which mistakes not to take up with the whole class. These kinds of instructional decisions are the ones that keep me learning and growing as a teacher. They’re also the kinds of complex instructional decisions politicians with simple answers and writers of scripted curriculum DO. NOT. GET.

So, as a group, we talked about what mathematical issues would emerge if we took up Zeke’s claim or if we took up Jesse’s work. My counsel ended up being that more students would benefit from the exploration of Zeke’s claim because the kids really need to investigate the properties and attributes of a rectangle. In my conversation with Jesse, it became clear to me that he wanted to head to a place of guessing and checking different perimeter lengths to see if he could sum to 100. There’s lots of fun math in that, but I think we’d end up investigating addition more than geometry, and the teacher’s goal is a deep investigation of geometry right now. I also think Zeke’s claim is accessible to every student in the class, and all students would benefit from exploring it. We talked about giving Jesse time to try to solve the problem he’d set out for himself because there’s powerful learning he can do. Of course, if his work spreads so other kids are trying to solve such complex problems during their next rainy-day recess, awesome. But Jesse’s diagram is not nearly as accessible to all kids, and I’m not convinced the time it would take for all students to make sense of it would have enough of a mathematical payoff to justify that use of class time.

It’s not a clear-cut call, though. I have mathematical reasons to say class time would be better spent testing Zeke’s claim rather than trying to make sense of Jesse’s work. But am I considering the right factors? Is it the same call you would make? How do we answer this question: which mistakes are worth pursuing as a whole class? I hope you’ll give me some pushback and ideas to think about in the comments.

*Update*

The teacher wrote me and told me how the toothpicks exploration went:

“The toothpicks worked great! I encouraged them to move them around and see if they could make Zeke’s suggested rectangle measurements work. It didn’t take them long to realize there was no way to make it work and most kids said for one reason or another. Yet I pushed them further to list ALL the reasons that it could not be a rectangle, based on what they have previously learned about rectangles. It was a really good math discussion to not only clarify all the characteristics that make a shape a rectangle, but to demonstrate how you need to prove it with facts – plural! That the argument one way or another gets stronger with more facts.”

References:

Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray, Alwyn Oliver, and Piet Human. 1997. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann.

Comparisons: A Little Bit More Older

Posted on June 6, 2015January 6, 2017 by tzager

Ask any elementary school teacher, and she or he will tell you that comparison problems are much harder for most kids than operations with other actions. For example, fourth-grade-teacher Jennifer Clerkin Muhammad asked her students to draw a picture of this problem from Investigations:

Darlene picked 7 apples. Juan picked 4 times as many apples. How many apples did Juan pick?

Her students are pros at representations and skillful multipliers, but we saw a lot of this:

Kids who were used to thinking of multiplication as either groups, arrays, or repeated addition had some productive, furrowed-brow time ahead of them. How can you represent comparisons? How is this multiplication? How do you draw the 4?

Then they started to get somewhere.

With each representation they discussed, Jen asked the excellent question:

“Where do we see the 4 times as many in this representation?”

Seriously. Jot that one down. It’s a keeper.

Jen and her class played with a series of problems involving multiplicative comparison, (height, quantity, distance, etc.). For each problem, she asked student to create a representation. And then she’d choose a few to discuss, asking, “Where is the ______ times as ______ in this representation?” Lightbulbs were going off everywhere as students broadened and deepened their understanding of multiplication.

Teachers often see the same struggle with comparison situations that can be thought of as subtraction or missing addend problems, like:

Marta has 4 stuffed animals. Kenny has 9 stuffed animals. How many more stuffed animals does Kenny have than Marta?

If kids have been taught “subtract means take away” or “minus means count backwards,” then this problem doesn’t fit the mold. And if kids have been taught keyword strategies, they’re likely to hear “more,” pluck those numbers, and add them together.

Knowing this as I do, I jump on opportunities to explore comparison problems with my own kids. I want them to have a fuller sense of the operations from the beginning, rather than developing too-narrow definitions that have to be broadened later. So, subtract doesn’t mean take away in our house. They’re not synonyms. I actually don’t worry about defining the operations at all: we just play around with lots of different types of problems, and I trust that my kids can make sense of them.

Yesterday, I was talking with Daphne about a neighbor visiting tomorrow. Our neighbor has a little girl, whom I’ll call Katrina. She’s about 2. We also have more awesome girls who live across the street. Let’s call them Lily, 15, and Gloria, 13. Gloria helps out once a week, and my girls idolize her. Here’s how the conversation kicked off:

Me: “If Katrina’s mother comes over to help me on Sunday, could you play with Katrina for a bit so her mom and I can work?”

Daphne: “Yes!!! We love playing with her. And she loves us. She cries when she has to leave us. We’re her best friends.”

Me: “I think Katrina feels the way about you and Maya that you and Maya feel about Lily and Gloria. It’s fun to hang out with older kids. You learn a little about what’s coming for you.”

Daphne: “Yeah. But for Katrina, Lily and Gloria would be TOO old. Like, she’d think they were grown-ups because they’re SO much older than she is. We’re older than Katrina, but we’re less older than Lily and Gloria are.”

So there’s my opportunity. She was comparing already. With a little nudge, I could encourage her to quantify and mathematize this situation. I thought I’d start with the simplest numbers so she could think about meaning first.

Me: “How much older are you than Katrina?”

Daphne: “I think she’s 2. So…”

There was a pretty good pause.

Daphne: “Are we counting on here?”

That’s paydirt.

Me: “What do you think?”

Daphne: “I think so. Because I want to know how many MORE. So, 2. 3, 4, 5, 6. I’m 4 years older than she is. Yeah. That makes sense.”

Me: “OK. So how many years older is Maya than Katrina?”

Daphne: “6.”

Me: “You didn’t count on that time. What did you do?”

Daphne: “Well, I knew that I was 4 years older than Katrina, and Maya is 2 years older than me, so 4, 5, 6. Because Maya is OLDER than me, I knew they had to be farther.”

Me: “What do you mean farther?”

Daphne: “Like, it had to be more. I was starting with 4, and then it had to be more farther apart.”

Me: “Oh! So you were trying to figure out whether to go 2 more or 2 fewer?”

Daphne: “Yeah. And it had to be 2 more, because Maya’s more older than Katrina than I am. I mean, I’m older than Katrina, and Maya’s older than me, so she’s more older than Katrina than I am. It had to be 2 MORE.”

Hot damn!

Daphne’s life might have been easier right here if she had some tools she doesn’t have yet. I’m thinking about number lines, symbols, and a shared mathematical vocabulary including words like distance or difference. Those tools might help her keep track and facilitate communicating her thinking to somebody else. But I love that we were able to have this conversation without those tools, while driving in the car where I couldn’t even see her gestures. Her thinking is there, and clear. The tools will come as a relief later on, when the problems are more complicated.

We kept going.

Me: “OK. So you’re 4 years older than Katrina and Maya’s 6 years older than Katrina. I wonder how that compares to Gloria? Is she about the same amount older than you two as you two are older than Katrina?”

Daphne: “Well, she’s 13. This is gonna be hard.”

Long pause.

Daphne: “She’s 7 years older than me.”

Me: “How’d you figure that out?”

Daphne: “I went 6. 7, 8, 9, 10, 11, 12, 13.” She held up her fingers to show me.

Me: “So how much older is Gloria than Maya?”

Daphne: “Umm…so Maya’s 8…umm…5. She’s 5 years older than Maya.”

Me: “How do you know?”

Daphne: “Because Maya’s 2 years older than me, so it’s 2 closer.”

This is the kind of problem where kids lose track of their thinking easily. Do we add or subtract that 2? In the last problem, she’d added. This time she subtracted. And she knew why!

When I glanced at Daphne in the rearview mirror, she had this awesome look on her face. She was looking out the window, but not seeing what was outside because she was so deep in thought. I wish I could have peeked inside her mind, and seen what she was seeing. Because, to use Jen’s question, where do we see the 2 years older? How did she see it? There was no time to ask, because she was onto pulling it all together.

Daphne: “So, Gloria’s a little bit more older than us than we are older than Katrina, because Gloria is 5 and 7 years older, and we’re 4 and 6 years older. But it’s pretty close.”

We pulled into the school parking lot on that one.

In our family, we count and manipulate objects all the time. Counting more abstract units like years, though, gives me a chance to open the kids’ thinking a little bit. In this conversation, Daphne was both counting and comparing years. Tough stuff, but oh so good.

Shadow Con

Posted on June 4, 2015January 6, 2017 by tzager

Back in April, I had the honor of speaking at Shadow Con, a teacher-led mini conference held after hours at NCTM in Boston. The three organizers, Zak Champagne, Dan Meyer, and Mike Flynn, spent quite a bit of time re-imagining conferences, thinking about how they could have bigger, more lasting impact on our teaching. They also thought about how we could use technology to open conversations between and among speakers and session attendees. Speakers need feedback to make their ideas better. Attendees would benefit by having a specific charge–something to go home and do that would “extend our learning” beyond the conference, as Kristin Gray put it in her Teaching Channel blog about Shadow Con.

Dan, Mike, and Zak chose six speakers and asked each of us to give a short, provocative talk about a topic of our own choosing. The only requirement was that we each end with a Call to Action–a specific assignment for people to try. They gave each of us a webpage that would serve as a forum for discussion, feedback, sharing of ideas, criticism, and collaboration. They assigned a live-tweeter to each of us. Lucky me, I was tweeted by my hero, Fawn Nguyen. And then they cut us loose.

@TracyZager Tracy said ASS in her presentation. I love this woman. #shadowcon15 #NCTMBoston

— Fawn Nguyen (@fawnpnguyen) April 16, 2015

It was great fun. This storify from Taylor Williams will give you a sense of the energy in the house.

Beyond that night and our presentations, though, I hope Shadow Con was a provocative call to action for conference organizers, presenters, and attendees. We need to talk, and think, and learn, and grow more than once a year. That’s the big idea.

So, here’s my talk:

If you choose to take up my Call to Action, please join in the conversation about it at my Shadow Con page. And while you’re there, check out the other great talks from Elham Kazemi, Laila Nur, Kristin Gray, Christopher Danielson, and Michael Pershan.

I was in truly great company.

Number Lines, Addendum

Posted on May 18, 2015January 6, 2017 by tzager

Given the ridiculous length of my last blog post, I can’t believe I forgot something, but I did!

(To begin at the beginning of my wonderings about number lines, start here.)

When the K-2 teachers were discussing contexts for equally spaced intervals, Becky Wright, a truly amazing kindergarten teacher, thought of a brilliant one: seed spacing!

http://www.portlandediblegardens.com/blog/2014/5/2/the-pleasures-of-growing-from-seed-part-2-methods-for-planting-seeds

She and her class were already planning to build gardens this spring. She was already planning to bring in members of the community to teach kids about gardening. But now, when planning the garden, Becky has many opportunities to think about intervals and scale with kids. For example:

Broccoli seeds are spaced 12″ apart.

Radish seeds are spaced 3″ apart.

Why?

http://urbanfoodorchard.com/uploads/3/2/6/5/3265114/6052476_orig.jpg?385

There is great math to explore whether you choose to plant by arrays or rows. For kids working on number lines, I love rows. Think about questions like, how many tomato plants will fit in a 10′ bed? How many carrots?

Home and professional gardeners know, the spacing isn’t arbitrary. What happens if the plants are too close together? Too far apart? This isn’t pseudocontext: look how seriously gardeners take spacing.

Once Becky and her kids figure out what they want to plant, they’ll need to do the math for each vegetable, and then get outdoors and measure the spacing when they’re ready to plant. It will be great, dirty, wholesome work, and the kind of work that helps kids develop the Equal Interval Principle.

http://www.plantanswers.com/Images/5.--Plant-seed-at-least-two-inches-apart.jpg — http://www.plantanswers.com/Images/5.–Plant-seed-at-least-two-inches-apart.jpg

Number Lines, part 2

Posted on May 13, 2015January 6, 2017 by tzager

Between NCSM/NCTM, spring break in New York, and then a big surgery, I’ve been off this space for a while. The teachers with whom I collaborate–the amazing staff at Rollinsford Grade School in New Hampshire–have been going full steam ahead, though, and they’ve taught me so much more about young kids and number lines! I’ll do the best I can to recreate what went down after my last post, Building Number Lines in Kindergarten, where we played with number lines in Becky Wright’s class.

In both the exceptional comments on that blog and through our experience, I learned that the big ideas would take a lot more time. If I’d mistakenly thought we were done, though, Daphne was going to school me. She’s my 5-year-old daughter, and she had joined me in Rollinsford for the number lines day. The next morning, she approached her regular kindergarten teacher and asked if she could set up a station to teach number lines to her class, like she’d learned in New Hampshire. I can’t begin to describe how happy these pictures made me, both about Daphne and about her kindergarten teacher’s openness.

At home, Daphne asked to make more number lines. I put tape up the stairs to see what she would do. Perhaps the rhythm of the stairs would come into play? Nope. She measured by marker lengths instead.

Then she decided she wanted to fit 100 numbers, so she started squishing:

Super fun.

Back in Rollinsford, Deb Nichols had loved the lesson, and decided to try it out with her 1st and 2nd multiage class. She put tape lines down for students to work on, and found her kids did just what the kindergarteners had done: they created number lines where all the numbers were squeezed together. When she asked the kids to spread the numbers out, the students weren’t any more concerned about equal intervals than the kindergarteners had been. Deb decided to put together a slideshow of things that come in regular intervals to see what the kids would notice. (We’ve been gathering more pictures at #intervalchat over on the twitters):

As students were noticing and talking, they developed a need for a word to describe these spaces, so Deb introduced “interval.” The kids headed off to lunch, and we met as a team to brainstorm. We were curious if exposure to all these intervals would infiltrate their thinking about number lines. After lunch, we all went in to co-teach, and Deb asked the essential question, “How do we know how to make the spaces on a number line?” Kids had some suggestions, like using rulers, or sticks if we were in the woods. One student talked about turning his line into a measuring tape. This student turns out to know quite a bit about measuring tapes because he lives on a farm, and they frequently use “cow weight tapes,” which are my new favorite measuring tool.

By measuring the length around the barrel of the cow, the farmer can get a good measurement of the cow’s weight. Beyond awesome.

Anyway, we asked the kids if the intervals had to be even, and the students were mixed. We decided to leave that question in the air, let them at their number lines and see what happened. Before they went, we reminded them that they could use tools around the room or their bodies to decide where the numbers should go. To open the possibilities a bit, we asked students how they might use their bodies. Several students stood up and demonstrated ideas like tip-toeing, crawling, jumping, or using their feet or forearms to measure. And then they got to work.

Interestingly, all the groups started working on evenly spaced intervals. I wonder if the rhythmic movement of their bodies on the number line got them thinking that way? At any rate, Deb had made the number lines around 8 feet long and asked for 11 numbers, so nothing obvious worked. Every pair got partway down their line and realized they would either run out of room or not have enough room to finish. And here’s where things got really interesting. In most groups, students went back to the beginning and tried again, with a different sized interval. For example, this group realized putting one cube between numbers wasn’t enough.

They tried over and over, adding cubes to increase the interval lengths each time.

This group was using a ruler (to create a descending number line–interesting).

They realized a full ruler was too much, and decided to switch to parts of rulers. They worked for an incredibly long time, trying different intervals. 9 was too much. 7 was too small. 8 was too much. They fine tuned their intervals into fractions of inches, which worked out great.

This pair tried using magic markers.

One marker was too small, so they tried to measure using lengths of 1 1/2 markers. Really challenging, especially on a wall.

Some groups changed strategies partway through. This pair used rulers. When whole ruler lengths didn’t work, they switched to their own feet.

To my eyes, these groups were all trying a similar trial-and-error approach, with varying degrees of success but exceptional perseverance, communication, and thinking. Great stuff. Two pairs did something very different, though. When they realized the original interval length they picked wasn’t going to work out perfectly, they didn’t go back to the beginning and try a new length. Instead, they changed intervals partway down the line.

This pair started with one student’s feet, without sneakers. At about this point, they realized they were going to have too much tape sticking out on the end. So, they got thinking.

They measured, and realized the other boy had longer feet. He took over for a few intervals. As they got to the end, they realized they still were coming up short. Their last interval was the two students’ feet, added together.

Kids are brilliant.

In another pair, they got about 2/3 of the way up the wall using 12″ rulers when they got to their “uh-oh” moment.

I asked what was happening, and this student explained that the tape would have to go all the way to the ceiling, or higher, for him to fit all the numbers. The teachers and I thought he really understood the problem, and was going to go back and create smaller intervals. Instead, he switched to using a bolt instead of a ruler, and finished his number line with a series of smaller, equal intervals.

Wow. Ever since this lesson, I’ve been noticing how often I see interval lengths adjusted like this. For example, check out the rivets on these posts in a New York City subway station. Look at all similar to his number line?

I have come to notice that we change interval lengths all the time in construction, when spacing telephone poles, when putting stripes on shirts. But, these examples are not number lines. There’s an important difference. On a number line, lengths are arbitrary, but they mean something. Ha! How’s that for a philosophical statement? But, I mean it.

When we make a graph or a number line, we set the scale. We decide what the unit is. How big is 1? Or, how big is 1,000, or 0.2, or 1/8? We get to decide our scale, but then we need to stay consistent with it. If the distance between 1 and 2 is 1/4 inch, then the distance between 17 and 18 is 1/4 inch, because a length of one is a length of one, wherever it occurs on the number line. I learned from reading Andrea Petitto’s “Development of Numberline and Measurement Concepts” that Piaget termed this idea the “equal interval principle,” and it’s key to children’s developing understanding of measurement. As far as the number line goes, kids need to learn the equal interval principle and conservation of length in order to move beyond thinking about position and sequence on the number line, and start to think in terms of proportion, distance, and space.

Also, I’m always excited when we have an opportunity to introduce the idea of authorship in mathematics. So often, math is presented as if it’s separate from us, and immutable, and impersonal. This little primary number line lesson, though, shows that each of us has choices to make when doing math. Whoever authors that number line decides what one is. That’s a pretty big thing to decide.

So, all this was rattling around in my head for a bit, and then my daughter Maya’s teacher asked me to come teach a lesson in 2nd grade. Last year, I helped her out once a week in math, but had to stop when I was diagnosed with cancer this September. This would be my first time in the math class all year. Maya had told her about the number lines, and she wanted to try it out. I knew from homework that Maya’s class was working on money. I thought it might be a kick to see what happened if we put money on the number line.

I started out asking the kids about the number line. They’re given number lines on worksheets all the time, but they’ve never made their own. After some chatting, I asked the kids what questions they had about the number line. Two standouts:

What do those little lines mean?

If we don’t write the numbers on the number line, are they still there?

Holy crow do I love that last question. These kids have been adding and subtracting on the number line for a while, so they can kind of use the number line, but they clearly want to think about the number line. I was psyched.

I put tape lines down and told them they were “one dollar long.” (I had made the lines long enough that kids could have put 100 pennies down. That’s important.) I asked pairs to choose a denomination (pennies, nickels, dimes, or quarters), and then grab a bag of play money and place the coins where they belong on the number line.

This was hard stuff, especially for kids who did not yet have the equal interval principle:

But, after some monkeying around, all the pairs placed their first set of coins, and then taped where the coins had been. Now, this was my favorite part. We asked them to choose a second set of coins to place. All kinds of good stuff happened.

Some groups completely ignored their first set and placed the second set without concern. But then, they started to think about the numbers related to each other. Where should 25 go in relation to 20 and 30? Before long, kids were using one set of numbers to orient themselves on the line and place the other set.

Seriously, how beautiful is that? I heard kids really connecting money and number lines and skip counting by 5s, 10s, and 25s:

“If you count by nickels first, then you’ve already got all the numbers there for dimes and quarters.”

“In between every set of dimes is a nickel.”

“Quarters always land on nickels, but only sometimes land on dimes. That’s because of the 5.”

I didn’t expect Maya’s teacher to have much time to continue to play with the lines, but she left them down and said she’d try to find time. The next time I stopped by, she told me the kids all wanted to add the other coins, and had:

Looks to me like they developed the need for those little lines after all. They even developed ways to demarcate different units with different tickmarks, eh?

I wish I’d been back in the class. Of course, I would have revisited their questions:

What do those little lines mean?

If we don’t write the numbers on the number line, are they still there?

And I think I would have asked them whether the intervals need to be equal, and why. I wonder what they’d say?

—

Updated here: https://tjzager.wordpress.com/2015/05/18/number-lines-addendum/

—

FYI, @bookgirlkpr shared this lesson study around number lines and kindergarten on twitter:

@MatthewOldridge @TracyZager @drcathybruce Here is an interesting lesson study for number lines in kindergarten. http://t.co/l2WWsSMNSX

— Sandy (@bookgirlkpr) May 5, 2015

http://www.tmerc.ca/m4yc/peterborough.html

Petitto, Andrea L. 1990. “Development of Numberline and Measurement Concepts” Cognition and Instruction. 7(1):55-78.

NCTM Boston

Posted on April 16, 2015December 30, 2016 by tzager

Hi folks,

If you’re here in Boston, check me out at:

Developing Risk Takers: Teaching Students to Give it a Go, #nctmgiveitago

12:30 – 1:30, Room 253B

Here are the handouts

Here are the slides

And…

At Shadow Conference, along with Dan Meyer, Zak Champagne, Mike Flynn, Elham Kazemi, Kristin Gray, Laila Nur, Michael Pershan, and Christopher Danielson. #shadowcon15

5:00 – 6:30, Room 258A

Hope to see you there!

Tracy

Further thoughts on the New York Times piece

Posted on April 8, 2015December 30, 2016 by tzager

It’s always fun to talk with Jessica Lahey, and I was honored she included me in the company of Steven Strogatz and Christopher Danielson for her last piece, “The Problem with Math Problems: We’re Solving Them Wrong .” I have been reading some of the comments and twitter chatter, and I’m so glad people are talking. Here was the original question from a parent:

My husband and I talked to our daughter’s pre-calculus teacher about her poor grades. He said that many students hit a wall at this point in math, moving from memorization — apply this theorem to this problem — to more abstract how-can-I-solve-this-problem thinking. I accepted that because that’s what is happening for her. What I thought later was that why can’t we find a way to help these many students get over that wall, instead of using it as a tool to weed out less developed brains? I really feel I have no way to have an impact on this teacher’s blind spot since it is shared by all math teachers and so many other teachers: If you don’t understand, it’s your fault.

Jess has to write under a strict word limit. Happily, I have no such limits here! I thought it might help the conversation if I posted my full comments to her, edited just a tad for this different format:

My first quick reactions to her email, which I typed from the Maine woods on my phone on a freezing cold day:

The entire idea behind reform mathematics is to eliminate this problem. Kids should not start with memorization they don’t understand. Kids should understand from the beginning. I just got to the woods where I’m going to walk the dogs in the snow. If you want to chat about it in more detail, I would be happy to talk with you later on.

A few minutes later:

P.S. Too cold to have fingers out of mittens for long, but I wanted to send one quick clarification. I am NOT saying kids should understand elementary math because they might need it for calculus. I’m saying if a student is going to learn something, she should understand it, whatever grade or subject. That is the first goal. And then, an important result is we would end up with students that have foundations upon which other powerful ideas can be built.

It was never a sensible idea to try to have students memorize first and understand later. Its just structurally flawed! So, I totally feel for these parents and this kid. This is what happens. Inevitable, if you try to build on a house of cards.

When I got back to the car, I wrote again:

I know there are many directions you could go with this question. It’s a fantastic question. For me, at the end of the day, this question exposes what is wrong. So many parents are saying, “Just give them the basics when they are young. They will learn how to do things with them later.” The math education world has to teach families why this approach doesn’t make sense. For example, most families do not understand that the ideas underneath algebra start very early in elementary school.

The next morning, I was able to write a little more in depth:

I woke up this morning thinking more about this question, and I’m finally off my phone and at a computer. I wanted to share a connection, not because it’s cute or quotable, but because I find it helpful for framing the situation.

When Sam started medical school, the dean gave a speech in which she told a version of this story (parable?):

A man and a woman were walking and talking along a river bank. Suddenly, they noticed a body in the river. The man took off his coat, jumped in the water, fished the person out, and administered CPR. Just as he was finishing up, he noticed another body floating down the river. Again, he jumped in the water to rescue the drowning victim. As he was dragging this second person to shore, he saw a third body in the river. At this point, the woman turned and ran away. He yelled, “What are you doing? I need your help! Where are you going?”

She yelled back, “I’m going upriver to see why people keep falling in!”

The dean’s point was that there’s a place in medicine for everyone. We need doctors to care for people who arrive in emergency departments, or who have contracted Ebola, or who have heart attacks. And we also need doctors who practice preventive medicine, research causes of disease, and engage in public health campaigns.

This letter from these parents, for me, is like the moment by the river. One response is to think about supporting students who have “hit a wall” so we don’t lose them–and they don’t lose STEM either. This response is especially important for underrepresented populations. I think this is what the parents were asking you to do.

The second response, though, is to think about why kids are hitting a wall, and whether that wall can be removed altogether?

We need good people working on both responses.

As someone who is inclined to run up the river and see why people are falling in, and as an elementary person, I tend toward the second response, which is what you got yesterday. When students leave my classroom, I want no weed-out walls in their futures. That’s why I’m on the team looking for systemic solutions to this problem.

Not sure if that’s helpful, but that’s how I think about their question. And I’m sorry. I hate that the odds are good this girl will end up thinking she’s “not a math person.” If she doesn’t, surely some of her classmates will. That’s why we have so much work to do.

As for what to do for this particular family, now, these were my quick thoughts:

1. For the parent. I am truly sorry that she and her daughter are in this situation. There are lots of us working to improve the system so it doesn’t happen. All is not lost yet, though! With support and advocacy, her daughter can learn the foundations and continue on in STEM! First step is working with her classroom teacher. What advice does he/she have about what would help her?

2. For you. This isn’t my age group, so I’m not as familiar with what teachers do for students who arrive without the foundations. BUT, I could point you toward some great HS math teachers who think about this kind of thing a lot, and they may have more specific advice. Would that help?

Middle and high school teachers are the ones fishing kids out of the river. I haven’t experienced the problem from that side as much. But, from either side, it’s not this student’s fault. It’s our fault.

I am so glad Jess wrote a piece looking at the larger system. It’s only by stepping back and looking at it as a whole that we can think in terms of what’s good for kids in the long run.

Building Number Lines in Kindergarten

Posted on March 21, 2015December 30, 2016 by tzager

I’m so stinking lucky. I’m working with an amazing staff at a lovely school over weeks, months, and years, and I learn so much every day we have together. Yesterday, I spent my afternoon with the K-2 teachers, and we continued our ongoing discussion about number lines.

The kindergarten teacher at this school thinks so powerfully about how her kids think! The week prior, she’d told the rest of us that she’s been modeling using number lines as representations and tools, but she wasn’t sure how to help kindergarteners really own the number line. I loved that thought.

I turned to some of my favorite sources to see what I could find about introducing number lines. Kassia Omohundro Wedekind, thinker, writer, and kindergarten teacher extraordinaire, helped me tremendously. She emphasized the importance of teacher modeling, a la, “This is one way you could show what you just did on a number line.” She suggested using lots of board games where kids move game pieces and count jumps. She brought up developmental issues around drawing the number line with paper and pencil, and how it would be hard to tell if kindergarteners were struggling because of the math or fine motor skills. She talked about building a giant number line.

The teachers and I had also thought about constructing a big number line, so I also wrote Malke Rosenfeld, who is always thinking creatively about the intersection of math, dance, and painters’ tape! She was generous with her time and gave me a lot to think about. My biggest takeaway from our conversation was to avoid turning a giant number line into a giant worksheet. I can see how easily we could have moved right to having the kids “acting out” what we did with markers and pencils on paper. That would have been unfortunate. As Malke wrote me, “Working at moving-scale is really about opening up possibility for new insights about math ideas that is not accessible at hand-scale or in symbolic form.” Hmm. That’s interesting.

I also turned to my bookshelf, and spent a fair amount of time looking for good thinking about introducing the number line as a model. If anyone has good references, I’m all ears. I found most books just jump into using the number line, without thinking about how students would make sense of it. I did find a really interesting chapter in Cathy Fosnot and Maarten Dolk’s Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction called “Developing Mathematical Models.” They focused on contexts for learning the number line, and included examples using bus stops and city blocks. Those examples were both from schools in New York City, where students have a tremendous amount of experience with buses, city blocks, and subways—all rich contexts for discussing the number line. The school where I’m working is in a small town in rural New Hampshire. The kids don’t ride buses or subways, and the numbered streets in the town center start at 2nd street and end at 4th street. These contexts weren’t going to work at all!

None of us could think of a familiar context for number lines in these students’ home lives. (If anyone has ideas for contexts, I am all ears. Please leave them in the comments.) I still felt like it was important to connect to something the kids were already doing or thinking about, so I went into the kindergarten room to look for opportunities. That’s when I saw this chart:

The teacher told me they keep track of how many days they’ve been in school using this chart. Just that morning, she’d added the second piece because it was day 121, and there’d been a lot of conversation about it. They also use the chart to look for patterns in number, practice skip counting, etc.

When we met as a team, we talked about using this chart as context. We thought about asking students what would happen if we stretched the numbers out so they were all in a line?

A quick thought about “real world” here. This chart is very different from bus stops and city blocks in that it’s already an abstract representation of numbers, rather than experience from students’ daily lives. What mattered to me wasn’t the “real-worldness” or not of the context, though. What I wanted to do was give students a chance to connect one idea to another; to make a bridge from something they knew to something new. Seeing math in the world around us is one huge type of connection (to borrow literacy terminology, it’s a math-to-world connection). Making connections from one mathematical idea or representation to another is also a huge type of connection (math-to-math, right?). Some of the best math lessons I’ve ever taught have involved comparing two different representations. I was happy to start with the chart.

As a team, we made some decisions:

We’d have the kids work in small groups on long number lines, and see what happened.
We decided that each group should pick a starting number off the chart. It could be any number they liked, but they had to come to consensus on that number.
We decided having several different sections of the number line would work better than trying to have the groups each take a segment of one long line, and then attach them together somehow. An added bonus was we want students to know they can zoom in on any section of the number line and use it. They can start and stop anywhere, rather than starting at zero or one. We wondered if this approach might help them with counting on?
We’d ask students to put eleven numbers in order on their number line because we wanted them to be able to make ten jumps.
We’d give students sticky notes to use to work out their ideas, and then tape when they were ready.
We decided to line the tape up with the floor tiles and see if any students noticed their regularity, and used them to place their numbers.

We expected students would struggle working together. That’s OK. Teaching how to work together is part of teaching, and teaching math. We expected sequencing might be hard for kids, especially because the chart is organized vertically. Otherwise, we weren’t sure what to expect!

All groups started the same way:

They chose their starting numbers first (1, 2, 6, 13, 14, 15, 121—this lesson was inherently differentiated in that regard).
They wrote their numbers on sticky notes (sometimes after arguing, sometimes after taking turns).
They sequenced their numbers correctly (some groups used the chart to figure out the sequence—awesome).
They placed them side-by-side on their number lines, all scrunched up on one end:

Fascinating! We started asking kids, “Can you use more of the line?” That’s when the really interesting discussions started. One group immediately understood they could put more space in between their numbers. They started by inserting a “finger space” like they put between words. When they’d finished, they still had lots of line left. They tried again, this time inserting a “hand space” between sticky notes. Still too much room. At that point, they noticed the tiles, and decided to use them.

Their line was one tile longer than they needed, however, which brought up some interesting questions.

When we pointed out that the space between 15 and 16 was wider than the other spaces, the kids discussed what to do. They ended up asking if they could have a 17 so they could go all the way to the end but make them even. Great solution.

At this point, we decided to have the kids take a “walkaround.” We wanted them to learn from each other’s ideas. After kids had looked, we heard students say, “We could spread them out more!” Great! What we noticed, though, was most groups spaced out their numbers, but didn’t distribute them evenly.

Teachers ended up pointing out the uneven spaces to kids. Some of the kids were fine with the spaces being uneven, while others wanted to revise their thinking. In one group, a student used his sneakers to prove the spaces were uneven.

“See? On this one, my foot touches the two numbers. But on this other one, there’s some extra sticking out.” I was thrilled with this development. I hadn’t really thought through the measurement aspect of this lesson, but here it was. A groupmate of his took out some cubes, and snapped together a stick of them. I thought she would use her cube-stick as a consistent unit, and space all the numbers that distance apart. Instead, she created lengths to match each interval.

Not what I expected, but totally cool too.

We paused the kids here for just a minute, and had students share how they were using tools they found in the room—floor tiles, feet, cubes—to measure. We gave students another chance to work on their number lines. One group loved the feet idea, and successfully worked together to make an evenly spaced number line using their bodies.

I was so happy, because now there were at least two number lines with evenly distributed intervals of different lengths: the one marked off by tiles, and this one marked off by one student’s foot. What a great opportunity for kids to compare and connect! And, of course, I love me a good use of non-standard units.

Other groups taped their lines down with irregular intervals.

I’m realizing how powerful it is to use removable painters’ tape to make these lines. I wonder, as the kids explore with their lines in the coming days, if these groups will want to revise their thinking. The teacher works hard to create a culture where revising thinking is a big part of doing math, so it’s possible.

When we discussed the lesson afterward, we were happy with a lot of it. We had all watched the kids sliding up and down the tape, moving from one side to the other of the line, and jumping from number to number. Everyone was happy to keep the lines down on the floor for a while, and see what happens. We talked about different ideas to create more opportunities for the kids to play with body-scale number lines, and decided to add a permanent one all the way down the K-2 hallway. The principal is going to have some number lines and hash marks painted on the blacktop outside, and teachers and students will be able to use sidewalk chalk to write different numbers on them. They’re going to paint some board-game paths outside and get some giant dice. And they’re going to add numbers in each stairwell, going up and down (yay, negative numbers!) from the 0 level.

Students asked great questions during the lesson. For example, one student talked about their first number line, when the numbers were squished together. She asked, “If we kept the numbers that close together and just kept adding sticky notes, I wonder how high the numbers would go.” Me too!

Our one discomfort was we felt like we imposed the idea of regularly spaced intervals on the kids, without really developing a need. We had planned the lesson quickly, and were much more focused on ideas of sequencing and magnitude than we were on measurement. We didn’t anticipate that the spacing of the intervals would be the big issue. Now that we’ve seen what happened, we’re all left wondering how we could have the kids realize the numbers need to be evenly spaced? What would help them see a reason?

Ideas?

—

Update: Part 2 is posted.

Author: tzager

My Own Mathematical Learning

Solutions to Problems I Didn’t Know I Had

The Question on My Mind