There Are Only Two Ways: Making Conjectures in Second Grade

Today, I led a quick “finger math” warm up in a second grade classroom. In honor of reaching “half way day” — the half way point in the school year! — I asked students to use their fingers to show half of a given number. I wrote 10 on the white board, and students quickly flashed a single hand with every finger extended. Half of 10 is 5.

16. Most students held up either 4 fingers on one hand with 4 on the other, or 5 and 3.

“I held up 4 and 4 because it’s a double, and 8 doubled in 16. So it’s all about doubles,” Gabriella explained. I think she knew what half of 16 would have been from having internalized (memorized?) 8 + 8.

“I did 5 and 3 because of how I broke up 16. It’s a 10 and a 6. Half of 10 is 5 and half of 6 is 3. 5 + 3 = 8,” Max said. He participates in a small pull out (intervention) group that I teach, so I was especially pleased with this response. I wonder if the kids can tell.

6. Students held up 3 fingers, again in a variety of ways. Madeline, who was sitting at my feet, looked around at her classmates and said, “I think there are only two ways to show 3 on our fingers.”

Bam. That’s how a “quick warm-up” gets derailed in the most beautiful way.

I asked the students to play around with their own fingers for 1 minute to decide whether or not they agreed with Madeline’s statement. As I watched them fumble, I tried to make sense of Madeline’s conjecture, too. I thought about all of the ways I could show three on a single hand, let alone adding in the second hand.

In the last few days, I read a few blog posts about making and proving conjectures with elementary students.

These posts had me thinking about ways to justify student conjectures. I had been working with my sixth grade section on claims about terminating decimals: how can we predict whether or not a decimal fraction will terminate? I introduced students to “Because/And” framework: “yes, because…” (with proof); “yes, and…” (to encourage modification) and “no, because…” (to frame disagreement with proof). Because I was still struggling through Madeline’s conjecture in my own head, I didn’t even feel ready to entertain any version of this with the second graders. I wanted to know more about Madeline’s thinking, and hoped another student might illuminate a key idea for all of us.

Dante spoke first during the group discussion.

“I think there are a lot more than two ways to show three.” He stood up to allow for everyone to see his configurations.

“You could make a three like this…” he continued.


“Or this.”


“Or you could use only one hand.”


Jackson wanted to add on. “Yes, I think there a lot of ways, too. I have some ways on just my right hand that Dante didn’t show.”

“So you think there are more than two ways to show 3,” I repeated. “How many do you think there might be?”

“So many!” Dante called out.

Jackson agreed. “It would take me a long time to figure it out. How many ways did Dante and I do?”

The ever-contentious Stella raised her hand to speak. “Dante and Jackson showed four ways.” Stella often raises her hand to speak, but rarely shares her own thinking. I started to lose myself in thinking for how to help her feel comfortable sharing her own ideas, when Madeline spoke out.

“No, no, no, they didn’t show four ways!” Madeline sounded annoyed. She crinkled her nose. “They only showed two ways. That’s what I said before.”

Across the meeting area, Amir was nodding. I called on him to share.

“There are only two ways,” Amir said.

“So you agree with Madeline,” I said. “Tell us more about the two ways.”

“Well, you can have three fingers on one hand, or two fingers on one hand and one on the other. That’s it.”

A-ha. This was a classic dilemma of combinations vs. permutations.

“So you could show two fingers on your left hand, and one on your right… what about if you showed two fingers on your right hand instead?”

Amir eagerly raised his hand, and continued: “that’s the same thing. Like a turnaround fact.” Great connection.

Jackson looked puzzled. “But those aren’t the same thing. They’re different.”

32173660500_14b443c938_oI asked students to indicate with their thumbs whether they thought they were different or the same. Most students seemed to think that they were different. Maybe they were thinking about the visual differences, or the fact that it feels different to try to hold up a pinky and a thumb rather than a thumb and pointer finger.

Ultimately, we were running short on time, and the students did not seem any closer to realizing the connection between their two ideas. They focused on whether they believe a “way” should be like Madeline’s combination conjecture, where order doesn’t matter, or a “way” is a permutation, like Dante and Jackson’s line of thinking.

“Hmm. I wonder if there’s a connection between what Madeline and Amir think counts as a way to show 3, and what Jackson and Dante think.”

All of the students looked confused, as I wondered whether my phrasing even made any sense, semantically. I wanted to let students puzzle it out, but I also didn’t want to lose them.

“I wonder if Madeline means there are two types of ways to show 3. Show me three fingers all on one hand.” In response, students showed me (thankfully!) a number of different ways to show three fingers: some used their thumb, pointer and middle fingers; others used their pinky, index and middle fingers, with their pointer finger and thumb curled into a circle. I asked three students to come up.

“What do you notice?” I asked the class, most of whom were still holding their three fingers raised up, even though I had asked them to put their hands down.

Josephine did her best to explain. “They’re all three fingers on one hand, but they’re all different. They’re all the same but different.”

“So maybe both arguments are right?”

Madeline tried to clarify, as a proud second grader does: “well, I think I’m right [that there are two types of ways]. Even though they look different.”

I started drawing a T-chart with different configurations that I had seen, organized as “3 on one hand” and “2 on one, 1 on other.” Somewhat facetiously, I reminded students about the difference between an “art drawing” and a “math drawing.” These were decidedly math drawings. Most of the thumbs I drew were smaller than the pinky fingers.


Two other students spoke in agreement of this newly revised conjecture, and we moved into the lesson. A few students did not seem fully convinced.

…but everyone agreed that my sketches of fingers were absolutely terrible.


It was difficult to decide exactly how long to give to this diversion. I did not want to derail all of the classroom teacher’s plans, especially as we reached an impasse with students not realizing that Madeline assumed that order/configuration didn’t matter. I had read all of these beautiful blog posts about students making and drawing models, and constructing their own arguments, and for the sake of time I felt like I led my students towards my own conclusion.

At the very least, this experience with the warm up had me thinking about more strategies to be “less helpful” to my sixth graders later in the day, as they worked on proving their conjectures about when a decimal fraction will terminate. 32173677310_e570d6a738_o




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