“It Has All the Numbers on the Clock”: Context and Connections

I’m fascinated by how student ideas are shaped by context: what are the big ideas in our current unit? How did we launch the activity? We make connections — constantly! — and so the warm up we use, and the ideas that are shared, can have serious impact on the trajectory of the lesson.

Our sixth grade teacher wrote a unit on number theory that she teaches between Unit 1 (Area and Surface Area) and Unit 2 (Introducing Ratios), and, because of it, her students leverage multiplicative opportunities in their work with ratios that I think they might have otherwise missed. In eighth grade, we launched a lesson introducing scatterplots with a slow reveal graph, and discovered that we could skip many of the more introductory questions. The concept of thin slicing hinges on this.

Today, in grade 1, an unexpected conversation about time influenced how students interacted with the game board for an addition game.

“We don’t have a lot of time.”

“Because we came in a little late from recess, we’re going to have to adjust plans for math,” I said, removing some images from the board. “It’s not a big deal. We still have 30 minutes!”

“That’s half an hour!” Violet spot called out, nearly leaping out of her red rug space.

“Yes!” I smiled. “That’s absolutely right!”

“Why is it called half an hour?” Declan asked.

Ooh! It had been a while since I’ve led a first grade lesson, so I jumped at the chance to take the small detour into fractional thinking. There are also some first grade standards about time that we sometimes miss. I was about to draw a clock, when I noticed Grace raising her hand. I yielded the floor to her.

“That’s because there’s 60 minutes in an hour,” Grace stated crisply.

“Yes, there are 60 minutes in an hour,” I repeated. “That’s a very helpful start to understanding why 30 is half. And if you look at the clock, there are all these little tic marks.”

“Tally marks?” Ramsey asked.

“Oh, tic marks. They’re little marks that break things up evenly.”

“They’re making the clock look furry!” Anton called.

I chuckled. “Oh, yes! My clock does look very furry because of all the wavy tic marks. I’m drawing fast.” My clock looked so full of cilia that it might as well have been a muppet. I narrated as I continued my terrible drawing: “So here’s where the hours are marked: 1, 2, 3, 4, 5, 6… count with me.

Finally, we were onto the relevant stuff: “And there are 60 tic marks to show each of the minutes.” Right now, it’s 11:30, so the minute hand is here. We are going to finish math in 30 minutes, at 12:00, when the minute hand is here.”

I swept my hand from 6 to the 12, and shaded in the area. “So here’s those 30 minutes.”

“It looks like you colored in half!” Tamar said.

“Precisely! Those 30 minutes are half of the hour, which is 60 minutes.”

“And 30 + 30 = 60,” Grace continued.

Meena raised her hand. “And 3 + 3 = 6!”

I recorded the thinking as best I could, and then called student’s attention to the real focus of our math block: addition within 12, using the game “Five in a Row.” (from Investigations)

Notice and Wonder

I projected game board using the doc camera, and asked students to think about what they notice and what they wonder.

Just as I had anticipated, they had a lot to say. However, I never would have come up with what was shared first. I called on Sara.

“It has all the numbers on the clock.”

As I was writing this down, Sara continued: “it counts up to 12, and doesn’t go any higher, just like a clock.”

“Just like a clock, it has odd numbers,” Anton added. I asked students to help me identify the odd numbers, to which another student immediately said that “there are even numbers, too.” I knew we didn’t have time to. do an investigation on the properties of odd or even, so this would have to serve as a small preview for some kids.

There is something particular to the numbers used, and it has nothing to do with a clock. However, our quick conversation about time before the lesson started seemed to have influenced them. They were thinking about chronology. Sequencing. Numbers. (This is also an important reminder that how we begin a lesson communicates what might be most important, and so detours should be taken with care!)

Continuing Ideas

Students offered more ideas:

“The numbers go in order. Like a clock.”

“…except it’s missing the number one!”

“The numbers go up, like 2, 3, 4, 5,” and then down.”

“They start going down at 12.”

“There’s only one 12, but there’s more of every other number.”

I paused the students there. “This is all so interesting! I will tell you that, using these specific dice, there is a reason that our board does not have numbers greater than 12. And there is a reason that there isn’t a number one. As you’re playing, maybe you can figure out why.”

Then I demonstrated how to play the game. The game uses two dice (two dot dice, per the teacher’s guide, but we differentiated by having some students use one cube with numerals and one with pips, or even two cubes with numerals). Students roll the dice, and then find the sum. They cover the sum, and attempt to cover five numbers in a row. I told the students that our big focus was on how we figure out the sum. We would talk about strategies after the lesson.

As I circulated around the class, I took note of who was counting all, and who was counting on, and who was using derived or known facts. (I would have students share. the first two during the class discussion.) But, truthfully, I was also probing to see if anyone had figured out why we didn’t need the number one. Many of them assumed that adding the number 1 would mess with the perfect array. “There would be six in the first row, and this game isn’t called six-in-a-row!” Catalina stated confidently.

It turned out it was easier for students to identify we wouldn’t need to go past the number 12. Three different kids figured out that possible sum was 12: 6 from one cube, and 6 from the other. One student added that maybe we would need a one if our dice had negative numbers. “2 minus 1 is 1.”

We had moved from clocks to addition to probability, and it had felt surprisingly smooth.

Anticipating Student Connections

Before launching with the Notice & Wonder routine (Annie Fetter), I anticipate what students will say. With this game board, I thought they’d notice that the largest number was 12, and the smallest was 2. I thought that they’d notice that some numbers appeared more than others. (There are four 7s, and only one 12.) I thought they’d notice that the numbers increase and then decrease. I did not anticipate any of the clock connections. While they weren’t particularly salient to the task at hand, they did not distract from the work.

Using the notice & wonder launch also allowed me to pose a question to extend thinking — why aren’t 1 and 12 included? — to all learners, and not just the ones who I observed fluently adding as I walked around the class. These little decisions that we make about content and pedagogy can and will shape how students interact with the material.