Students are given a complex task: match two graphs to represent: 1) the flow of water out of the top of a prism, and 2) the flow of water into the bottom of the prism. We learn that there is a constraint: liquid in the top plus liquid in the bottom must equal a volume of six as measured by the height of the tick marks. The complexity comes in matching the distribution at the beginning and the rate of change over time (measured in seconds). The teacher sets the task up to include every student: students are to take turns making a match, and explaining why it’s a match. The partner then has to agree or disagree, and explain why. The requirement that they explain to each other is key; research has demonstrated that explanation is a powerful driver of learning for both students: the one giving the explanation, and the one receiving the explanation. 

The teacher tells us at the outset that they have done two examples together as a class. As the students begin to work together, they still need to orient themselves to the task, confirming for each other that there needs to be a relationship between the decreasing and increasing graphs. Often we think that if we explain or model something, students understand it. This kind of partner work reveals that students still need to put in effort just to articulate the task at hand. 

In this pair of students, one is completely on top of the task. In a whole class discussion, she would likely be the student who has her hand raised quickly—the student who intimidates other students who then choose to remain silent. By working in pairs, the quick student has the opportunity to do something that stretches her a bit: she interacts with her partner to encourage his reasoning. Her straightforward “no” when he reverses the x and y axis would not win her a gold star for teaching, but it does the trick. The student corrects his error. Both engage more fully than they would without the opportunity for partner talk. 

This third pair of students are working on the more challenging match; one of the graphs is not complete. Neither is entirely sure of themselves. But the interactions are pushing them toward greater clarity. One is working with the quantities that add to 6 (two here and four. Then there is zero…), and the other is arguing for the direction of the line. The interaction among them is spurring them toward a solution. 

As we listen in on students working together, we can hear that some students understand what makes for a match and some do not. We saw examples of students coming to an understanding as they talked together, or as one student guided the other. But sometimes neither student knows how to proceed, or both students share a misunderstanding. In these cases, it is often helpful for the teacher to provide just enough input, or to ask just the right question, to get students back on track. This requires walking a fine line: intervene too early and students miss the really important opportunity to think through why something is not working and self-correct. Intervene too late and students spend too much time spinning their wheels or, worse, reinforcing an incorrect idea or procedure. 

The teacher in this case sees two students identify a match based on both the top and the bottom graph measuring four at the outset. They are working with the idea that the “height” of the liquid needs to be the same at the start; they have not considered that the total height must sum to six. Note that when the teacher chooses to intervene, she does so with questions. When asked, the students acknowledge that the total height of the liquid is 6. Instead of using her own authority to give them the right answer, she asks them, “what are you finding that you’re reconsidering.” Having students make the statements that they do not match, and that the match to the four has to be a two may seem unimportant; after all, the teacher led the students to the answer. But it is actually a very important move. In many classrooms, when students don’t know what to do next or how to make something work, they simply ask the teacher or wait for the teacher to come by and tell them how to do it right. If students know that teacher help will come only in the form of leading questions, and it will still require them to do their own explaining of how to do it correctly and why, students will exert much more effort and, importantly, think of themselves as capable of figuring things out.