Defining the Problem:
In a recent workshop, a third grade teacher shared the following: “I think I’m doing it right, I ask a lot of questions like, ‘How did you solve?’ and ‘Can you explain your strategy?’ or ‘What do you think about _______’s approach?’ But the kids often just stare at me blankly. Only a handful will jump into the conversation. What am I doing wrong?”
This teacher’s story is not, in fact, an isolated experience. Any one who has ever designed and planned a beautiful string, only to have it fall flat, understands this lament. I have designed and led strings that seemed mathematically perfect — they solidly built on a big idea while developing one or two key strategies. In my planning, I had anticipated the model(s) that students would use to solve, and I thought about how I might help the kids refine and clarify their thinking. I had made sure that the problems in the string were scaffolded in such a way so that every student had an entry point somewhere. I anticipated arousing conversation and big mathematical epiphanies — I was so excited!
But then — with a roomful of kids in front of me — everything I anticipated would vanish. Very few students shared strategies, or even answers. Kids struggled to add to, comment on, or challenge each other’s thinking. In the midst of this, I found myself talking WAY TOO MUCH. And it was frustrating! And yet, despite little concrete evidence, I felt pretty certain that the kids were solving, thinking, and using some sound strategies. What was the problem?
Our approach:
My fellow coach, Monica Mendoza, a team of McKinley Elementary School teachers and I diagnosed the problem at a workshop here in Santa Barbara, CA a few years ago — many children were simply unaccustomed to talking in math class. They had simply never done it. Certainly not more than the sharing of a single answer (e.g., “What did you get for problem number one?”). On top of that, many of our students were English Language Learners. Not only had they rarely discussed mathematics before, they were now expected to have these brand new kinds of conversations in a second (or third or fourth….) language. We recognized that this was a tall order.
In order to coax children into talking more during strings, we needed to create some structured “jumping off points” in our conversations. We realized that sentence starters, like those used to guide conversations in Language Arts, might be helpful to get things rolling. We began by introducing very simple prompts for kids:
My answer is: _________.
My strategy was ____________________.
We posted them on the board, practiced using them together and tried them out during a string. Lo and behold, new voices joined the conversation. We (the adults) were talking less, and the children were talking more. Students started to describe and even name their strategies. For example, when posed with the problem 23 + 100 one student responded that, “I just moved the numbers around to make it easier to add. I did a ‘flip.’” Amazingly, with these two short initial prompts, students began to talk a bit more.
But we wanted to develop really good mathematical conversations, where kids not only shared their own thinking, but learned to listen for and respond to others’ thinking, too. So, we developed some more sentence starters:
I thought about it this way: _____________________________________________________.
I agree with _______________ because _________________________.
I disagree with _______________ because ________________________.
I want to revise my answer because ____________________________.
Now we were cooking! In all honesty, there were — and still are — plenty of long silences, moments when tumbleweeds rolled by and crickets chirped, but a key development was occurring: the students had developed facility with some key language used to discuss mathematics, comfort with the culture of sharing and debate during a number string (and eventually, in the bulk of their math experiences).
More recently, I visited a group of fifth graders who have done strings regularly for the past two years. Nearly all of the students in this particular classroom are not native English speakers, and all started working through strings using the aforementioned (and other) sentence starters. And now they no longer need them. Discussing mathematical ideas, critiquing the reasoning of others, reflecting on their own thinking all comes quite naturally to them now. I heard the following ‘sentence starters’ that were entirely their own yesterday: “I used a different strategy than Jorge, I ….” and “that’s actually not the way I thought about it; I thought about it like this….” and “I thought of another way you could do it! You could…” There were no longer sentence starters on the classroom wall, these students were on a roll all of their own!
With the Common Core State Standards for Mathematics upon us, teachers are increasingly mindful of both the value and the joy of engaging mathematical discussions. The Standards for Mathematical Practice insist that students critique one another’s reasoning, make use of mathematical models and support statements with evidence. Daunting? Sure. But starting students with something as simple as “my strategy was_______” can be a surprisingly eye-opening, gentle first step.
Number Strings 
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