By Michael Flynn, posted January 30, 2017 —

Note: This vignette was adapted from an assignment submitted by a student from one of our graduate courses at Mount Holyoke College. The teacher’s name has been removed and her students’ names have been changed to pseudonyms to protect their identities. Thanks to Ms. P for sharing this vignette with us. We greatly appreciate it. 

As part of my coursework, I needed to explore some regularity (SMP 8) with my second graders in hopes that they would develop a generalization about what happens to the sum when you add one to an addend. In other words, could my students develop a deep understanding that increasing an addend by ones will have the same effect on the sum?

The plan was that I would briefly explore an idea for about 10 minutes per day for a week or two as part of our morning meeting. During the previous morning meeting, students were working on solving a series of related problems. First they had to solve 6 + 6 = ___ and 6 + 7 = ___. Then they had to solve 12 + 13 = ___ and 12 + 14 = ___. I purposefully picked pairs of problems in which the second pair had an increase of one so students would begin to notice some regularity when we add one to an addend. Eventually, I wanted them to make a general claim they could work to prove using representations later in the week. After laying the initial groundwork, I began by calling the class to the floor and directed their attention to our chart paper that had both pairs of equations on it. The following conversation ensued. 

Teacher: Today we are going to explore an idea that came up yesterday when we worked on those related problems. A bunch of you noticed something happening with these pairs of problems. Can someone come up and make a story or representation to show us what’s happening in this first equation, 6 + 6? [Kim’s hand is waving. I allow wait time. Now her hand is waving frantically.] Kim, would you like to show us a representation? [She jumps up.]

Kim: Two kids had 6 marbles. Well, they each had 6 marbles. [She draws six circles, leaves a space and draws six more circles. Then she writes the number 6 above each, puts a plus sign between the two sets and writes = 12.] Altogether that is twelve marbles.

Teacher: Good, Kim. Now, how does Kim’s story change when we think of the second equation, 6 + 7? Turn to a neighbor and talk about a way to use Kim’s equation, story, and picture to explore this idea. [Students discuss ideas excitedly as they point to Kim’s picture. After a few moments, I call them back.] So, who has an idea about what happens?

Cam: The answer will be one bigger. [The rest of the class signals their agreement.]

Teacher: You all seem to agree. Why do you think that will that happen? How does the story change?

Cam: They both had six marbles. One kid goes home. [Cam draws a circle around one of the sets of six and crosses it out], and a different kid came to play. He had seven marbles. [Now Cam draws a new set of sevem circles.] Now there are thirteen marbles in all. [He writes 6 + 7 = 13.]

Luke: Can I show mine? Mine is kinda like Cam’s idea. Six plus one [making six circles, a plus sign, and one more circle in parentheses] plus six [making six more circles] equals thirteen marbles [writing 13]. See? ’Cause you made this one bigger [pointing to the 6 + 1 in parentheses]. Then, the answer is one bigger. [Heads nod in agreement.]

Teacher: You have a different way of picturing it. How does the story change with your picture? Cam had a new kids come in with more models. What’s going on here?

Luke: Well [pausing], the kids are the same [pausing], they each have six, so that’s twelve. Then [pausing], I guess the first one just gets one more marble from her mom, so now that’s seven. And that means the twelve is now thirteen.

Teacher: So, Cam has one kid leaving and another one showing up with an extra marble. And you have the same kids, but one kid gets an extra. But in both cases, the sum goes up by one? [Heads nod.] Do you think that will always happen? [Again, heads nod.] Who has a way of saying what’s happening here using an if/then statement?

Anthony: If you add one to the addend, then the answer gets bigger.

Bridgette: Well, it gets one bigger.

Teacher: I’m going to write this claim down. You’re both saying, if we add one to an

 addend, the sum will get bigger?

Bridgette: By one.

Teacher: Will that work for other equations? Does anyone else have a pair of equations that work? I’ll scribe.

Lauren: 9 + 9 = 18 and 9 + 10 = 19. Yep, that’s one bigger.

Blair: 17 + 3 = 20, 17 + 4 = 21

Rosie: 10 + 12 = 22, 10 + 13 = 23

Juan: 8 + 7 = 15, 8 + 8 = 16

Grace: 20 + 20 = 40, 20 + 21 = 41

Dae: 8 + 8 = 16, 8 + 9 = 17

Teacher: Take a moment to look at this list and think about what you notice. [I give some quiet think time so kids don’t start blurting out answers.] OK, raise your hand if you’d like to share something you noticed.

Hannah: Each time the second answer is one more than the first. Eighteen to nineteen, twenty to twenty-one. [Students signal their agreement.]

Teacher: Any other patterns?

Corinne: Oh! Everybody added one to the second addend!

Teacher: Would our claim work if we changed the first addend instead?

Kim: 8+2=10 and 9+2=11. That works.

Bella: 12+13=25, 13+13=26

Hayden: I tried it with zero and it doesn’t work! [I ask him to show us. He writes 6 + 0 = 6, then stops, looking confused.]

Teacher: Let’s think about our claim. If you add one to an addend, the sum gets bigger by one. [Hayden seems very unsure, so I proceed.) Can you add one to an addend, Hayden?

Hayden: Six plus one, [pausing] equals seven. [I believe he mentally added one to the zero from his original example of 6 + 0]. It does work! [He’s smiling.]

Teacher: I noticed Hayden was looking for an example that wouldn’t work. That is also very helpful when we explore these claims. It looks like your claim works for a lot of examples. Can I offer a suggestion that we change our claim to say that if you change either addend—since you noticed we can change either one and the sum still gets bigger by one? [The students agree.] We’ll come back to this again tomorrow, and we’ll think about how we can show why our claim works using cubes. 

I thought we had come to a good stopping point for the day and didn’t want to start having the class work on proving their claim with representations. This work is designed to develop over the course of a number of days in short spurts. I’m excited to see what they come up with during the next session. 

Reflection questions

1.   What instructional moves did the teacher employ to ensure the deep thinking about these ideas emerged from the students instead of her?

2.   What was the benefit of having the students consider contexts, connecting them to their representation, and considering how the stories changed based on how the mathematics changes?

3.   What kinds of representations might you expect students to create in their future sessions as they work to prove their general claim? Note that their teacher assigned the following criteria from Russell, Schifter, and Bastable (2011):

(a) The representation must show addition.

(b) It should show that your claim works for all whole numbers.

(c)  We should see why your claim works from your representation.

Your turn 

We encourage you to consider the reflection questions. We want to hear from you! Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk. 

References

Flynn, Michael. 2017. Beyond Answers: Exploring Mathematical Practices with Young Children. Portland, ME: Stenhouse.

Russell, Susan Jo, Deborah Schifter, and Virginia Bastable. 2011. Connecting Arithmetic to Algebra. Portsmouth, NH: Heinemann.

Michael Flynn is the Director of Mathematics Leadership Programs at Mount Holyoke College in South Hadley, Massachusetts. Previously, he taught second grade in Southampton, Massachusetts, for fourteen years. He is currently interested in how primary and elementary school students develop algebraic reasoning and how teachers can support that work. He tweets at @mikeflynn55.