1. Be explicit at the start that getting more voices, more ideas into our conversation is our goal, because we believe it helps us of all learn and make sense of new ideas. “We’re going to learn a lot more today if we work together as a class….let’s try to build on ideas from many of us, not just a few.”
2. Remind kids that a number string isn’t a “race” or a “test” and there is no “prize” for having the quickest response. Slow down the pace — “Let’s take our time to really think about this idea….” This move potentially makes space for others to enter the conversation.
3. Use turn and talks in a variety of ways*, but particularly to listen carefully to what kids say (not to interrupt or to “manage” their conversations). When you hear the glimmer of an important idea from a mathematician who hasn’t spoken much in the whole group, elevate this idea. “Are you saying….? What an interesting idea! Would you be willing to share that with the class?” If they are not, ask if the partner could share the idea (giving authorship to the mathematician or co-authoring the idea) and if neither is willing to share, ask if you might share the idea (also giving authorship where it belongs).
4. Celebrate good listeners and give them a role. One way to disrupt the existing talk patterns during a string is to ask, “Who can share their partner’s idea?” or “Who had a partner who just said something really interesting?” When I see the same kids wanting to participate, I often ask this question instead. This move has the potential to bring in either new ideas or new voices, or both.
5. Move out of the paradigm of “being confused.” Instead reframe these moments as “not being sure about an idea” or “disagreeing with an idea” or “not yet being convinced it is true.” Because we process ideas at different speeds and in different ways, we shouldn’t assume that those who don’t “get” an idea immediately after it is spoken or represented visually are confused. They may need more time, a different voice to paraphrase it, another example, a different model, or simply to let go of it and come back to it later.
6. Encourage skepticism and doubt. They may be skeptics (a beautiful role to play in math class), who simply are not as easily convinced that ideas are true and/or generalizable. It’s not our role, when facilitating a number string, to explain ideas to these kids, but rather to name that “some of us are not yet convinced” and give the community the chance to do the convincing. Defending another person’s idea, re-explaining it, extending or critiquing it, crafting of an argument is all mathematical work — let the kids do it!
7. Use kids’ names strategically to elevate their status when others may perceive them as having low status. “As we are solving this problem, I want you to think about how Sandi would solve it?” or “I wonder if Jinoos’s strategy would work on this problem. Hmmm….” or “Remember how Joaquim and I were not convinced by the last example? So think about how you might convince us on this new problem…”
8. Care less about the right answer (it’s never the singular goal/focus of a number string) and shift into all the other rich mathematics that is at play. Especially as the problems get trickier, I will say, “Okay, let’s not worry about the answer right now. Who can get us started?” And then I notice that focusing on strategy, not final answer, typically brings more/different kids into the conversation.
9. Consider your pronouns. If, in fact, you want to work as a collective and to include kids in mathematical thinking and decision-making, then it’s helpful to shift your pronouns from the singular (teacher) to the plural (the class). “What do we think of this idea? Who could convince us? Tell us why you think that might work? What do you all think of Ramon’s array?”
10. Teach kids how to speak to each other. Particularly when kids disagree, it’s important to frame this as “disagreeing with an idea, not a person.” Also, we don’t need to triangulate the conversation by having kids tell us who they disagree with and why — “If you disagree with Iyanna’s idea, tell her. She’s right there.” Support kids to look at each other as they speak about each other’s ideas: engendering more kid-to-kid talk instead of kid-to-teacher talk.
* This assumes that you have assigned everyone a math partner in advance, which is critical. Letting kids randomly choose someone near them to talk isn’t as purposeful and often leaves kids out of important sense-making and learning. In most classrooms where I work, kids know that they are assigned a just-right math partner by their teacher, and when they come to the meeting area, they sit right next to this person.
I’m forwarding Kara Imm’s latest post in NCTM on 10 Ways to Develop a Community During Number Strings. Her thoughts apply to not only number strings, but to ways we as teachers can foster rich math discussions in our classroom. I especially like her idea of changing our pronouns to what do “we” think about, tell “us” about… to work as a collective group and include all students’ thinking in the classroom.
Math Specialist / Coach Colorado Academy