Have you ever been teaching (or leading professional development) and asked a really good question only to be met with silence? We all have teacher moves in our back pocket for situations like this—maybe do a turn and talk, ask the student if they’d like to call on someone to help them, or ask a different question. I recently observed a third grade lesson when I saw a teacher face this exact situation. The lesson (Unit 5, Session 3.4) focused on strategies for solving division problems. The discussion focused on the following problem:
As students worked, Mrs. N observed and interacted with students, jotting notes about how to sequence solutions for the discussion:
First, the class discussed Taylor’s solution. Taylor skip counted by 4s to 32. The class determined that Taylor skip counted 8 times, and Mrs. N recorded an 8 in the equation above Taylor’s solution—8 x 4 = 32. Next, Mrs. N. pointed to Julie’s first step, 2 x 4 = 8.
Mrs. N: So if we look at Julie’s first step, what part of the problem has she solved?
The students quickly answered that the 2 stood for two groups, and the 4 was how many cars were in each pack.
Mrs. N: What does the 8 stand for?
Students had some difficulty answering this question; they said “8 cars”, “8-packs”, etc. This wasn’t so surprising given that Mrs. N had just recorded an 8 in Taylor’s equation: 8 x 4 = 32. There it represents 8 groups. But the 8 in Julie’s equation represents something different than the 8 in Taylor’s!
Once the class established that Julie’s 8 represented the number of cars in two 4-packs, Mrs. N drew a circle around the 4 and the 8 in Taylor’s skip counting sequence. Each time Julie said she wrote another 2 x 4 = 8, Mrs. N circled the next two numbers in Taylor’s solution. After doing 2 x 4 = 8 four times, Julie said she added the 8’s and then the 16’s to get a total of the 32 cars.
Mrs. N (pointing to equation above Taylor’s solution): We’ve already established that eight times four equals 32. Where in Julie’s work do you see the 8 groups?
Take a minute to look at Julie’s equations. Where are the eight 4-packs in Julie’s solution?
A number of students immediately raised their hands. The teacher called on Jackson. He said, “It’s…” and then he stopped. The pause seemed to stretch for eternity; one I think we’ve all experienced. The teacher waited.
I found myself holding my breath! There were 2s, there were 4s, and there were 8s—which are all multiples of 2—all over the board. Mrs. N’s question was a difficult one, partly because these third graders were just beginning to understand and solve division problems, and also because Mrs. N was asking them to make sense of another student’s solution. And there were 2s, 4s, and 8s everywhere!
Finally, Jackson said, “It’s in the representation right there. The two and the two and the two and the two.” The teacher circled each of the 2’s in Julie’s solution, and the class agreed that by combining the four 2’s, you got the eight 4-packs of toy cars Gil bought.
Whew, I could start breathing again! In reality, the teacher probably waited less than 10 seconds. As I wrote at the beginning of this blog, we’ve all been there; it feels longer than just “wait time.” It can be agonizing. We want to step in. We want to “save” the child—maybe call on another student, or ask a leading question such as, “Is the 8 in these four groups of 2”? which in reality is just another form of telling.
But the teacher waited. And waited. She had confidence that her students could understand what she was asking and that they could make sense of a classmate’s solution. She understood how hard this question was. She trusted her students to figure it out. And her faith in her students paid off. It generally does. I think “And then she waited” is going to be my new mantra! And maybe a good mantra for all of us.
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I was in that room when we waited! What a feeling it is when the connection is made and the struggle was productive.
I was, too! I was definitely holding my breath and might have even pumped a fist from the back of the room when Jackson said the 8 was the four 2’s. It felt like such a hard question! What strikes me looking back – on this discussion and on my interaction with Nicole about this same problem (https://investigations.terc.edu/that-seems-way-too-big/) – is how focused the teacher and the kids were on thinking about what the numbers represent (the number of groups, the number in each group, the total number). (It also struck me that the language of 4-packs might be an additional challenge – a pack sounds like a group but here it is the number in each group!)