Over the last decade, much of my work has been focused on mathematical argument in the elementary classroom. Observing in our collaborating classrooms, I was struck again and again by how teachers supported students to build on each other’s incomplete ideas. Constructing a mathematical argument is difficult and challenging for elementary students and, therefore, necessarily collaborative.
When students are learning what it means to make an argument, not just about the solution to a single problem, but about something that’s generally true in mathematics, they start out by noticing a regularity across multiple examples. In this fourth-grade class, the teacher posts four pairs of equations that are examples of something that’s true about addition. She deliberately chooses addition problems that are easy for the fourth graders because the focus in this lesson is not about solving the problems but about how the operation of addition behaves, no matter what the numbers are.
In the following excerpt from their conversation, the teacher engages her students in articulating what they notice, and encourages them to do the hard work of finding words that clearly describe the common mathematical structure underlying these pairs of equations. She doesn’t expect any one student to formulate a clear and precise conjecture. Rather, she repeatedly invites students to restate and add to the evolving idea.
Teacher: We’re going to try to look at what’s happening with these numbers. We’re going to look for some patterns, and we’re going to make a guess, a conjecture, about what we think is going to be true for all numbers that have these patterns. What do you notice?
Martina [pointing to the first pair of equations]: I’m saying, in the first column [pointing to the 12 and 14 in the first pair of equations], it goes up by two. In the second column [pointing to the two 8’s], it doesn’t go up by anything because it’s the same number, so the answer has to go up by two.
Teacher: Okay, who can explain it in a different way?
Ana Sofía: These numbers [points to 12 and 14] are changing by two, but the second column, the numbers stay the same, so you know that the last digit [points to the sum] is going to have a difference by two.
Several students: Oooh!!
Teacher: Okay, Kevin, can you say it in a different way?
Kevin: Well, what I was basically saying is… I’m just saying that this basically grew. This one grew two more, so the sum from here to here would just grow two more because you added two on…
The teacher asks students to turn and talk in pairs about what they’re understanding and whether they can state a rule. Then the discussion continues.
Isabella: …the rules for the twos, I got — well like he was saying that, you’re adding the two because, um…
Isabella drifts off and stops. The teacher asks her if she wants to call on someone else to add on to what she started. She calls on Nicole.
Nicole: Well, I think Kevin’s rule with the two’s is that, you just basically—if you answer— Well, if the first or second addend— So, if you have a problem, like 12 + 8 = 20, and then you write another problem down, like 14 + 8 = 22, and then the only— I think Kevin was saying that since 2 + 2 = 4, then basically— wait, I don’t know what I’m talking about. Two more would be 4, and then the answer has to be two more.
Teacher: So, let’s take that part of yours, “the answer has to be two more.” Why does it have to be two more? Who can add on to make a short rule about what Nicole was saying?
Anthony: So, what’s happening is, if you add two to one of the addends, the answer has to be two more.
In this excerpt, Martina gets out some important ideas, but only when Ana Sofía re-states those ideas do some students begin to follow. They needed time, and more than one chance to hear the ideas, to think them through. Kevin introduces the word sum and explicitly connects the change in the sum to the change in an addend. Nicole picks up on this connection: “the answer has to be two more.” Anthony articulates a concise rule that has the essence of all these ideas. And even this is not the end of that day’s conversation—but all I have space to quote here!
Through this discussion, students’ articulation of their collective idea gradually becomes clearer and more precise. The invitation to start and stop, to be inarticulate, to get out a partial thought, even to be incorrect, results in a gradual refining and increased understanding of the mathematical idea. This invitation not to have to be right—now taken for granted in this classroom, but only after a lot of work to establish this expectation—signals that all ideas help move the collaborative effort of understanding mathematics forward. It allows all students to participate and their collective brilliance to be developed and shared.
This example is taken from But Why Does It Work: Mathematical Argument in the Elementary Classroom by Susan Jo Russell, Deborah Schifter, Reva Kasman, Virginia Bastable, and Traci Higgins. Heinemann, 2017.
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This is a great example of how kids may not be able to articulate their mathematical thinking and understanding YET, but it reminds us of the importance to continue practicing math talk. Each one of these children added something of value to the conversation. It gave the next person and the next person a stepping off point. Together, they are starting to formulate the beginnings of a conjecture. It is apparent that these kids feel safe taking risks.
Exactly, and it’s interesting to look at how the teacher supports that collaboration, for example, how she finds what’s mathematically important in Nicole’s comments, even though Nicole herself says “I don’t know what I’m talking about.”
This blog post has so many aspects of both the work of building a classroom community that thrives on building on the ideas of others, and also the work of articulating those often times, invisible ideas underlying the operations. I am so struck by Susan Jo’s phrase ” this invitation not to have to be right”!
I think back to when I was teaching my fifth grade class and was involved in the NSF Foundations of Algebra project. It wasn’t until I had a chance to listen to recordings of my students’ ideas and my own responses to those ideas that I was able to begin to develop strategies for supporting those “rough around the edges” statements that bubbled up in our discussions.
This teacher gives her/ his students the time and space to struggle, clarify and revise their ideas. There is a need to linger and let the students do the work of getting their statements to a place where they feel satisfied.
Thank you so much for sharing this glimpse of a classroom engaged in a productive discussion. Would love to hear what the others respond to Anthony!
Love the use of the word “linger.” “Productive lingering” on important mathematics has been something we’ve been talking about–maybe I’ll write a post on that in the future.
Thanks for sharing this and reminding me of how important it is to create the environment where it is not only okay to not be right, but one where students know that having an incomplete idea is important and may move the understanding forward. So hard to do when students don’t come from a classroom like that but so important.
Thanks, Karen. Every time I look at this class session, I’m struck by how Nicole keeps going, trying to get her idea out—she starts over 5 different times! Even though she says about herself, “I don’t know what I’m talking about,” she adds such an important insight to the conversation.