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!
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:
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.
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?
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?
