My four-year-old son likes to walk over to the magnetic rekenrek (math rack) I keep on the fridge, move some of the red beads to the right, and then ask me, “How many beads you see Momma?” and “How did you get that?” My son has my routine down pat. When I ask him or his brother questions on the math rack, I keep my questions very consistent. I put up a particular number and ask, “How many beads” and then when they answer, “How did you get that? or “How did you know?” or “How did you see that?”
Teaching mathematics is a very complex act- when teaching a number string, you are listening carefully to students’ responses, carefully representing their thinking, and always at the same time looking for ways to connect to the larger mathematical goals of the string. As the mathematics educator Deborah Ball so beautifully wrote, we teach mathematics with “ears to the ground, listening to students, eyes are focused on the mathematical horizon” (Ball, 1993, p. 376). I found that as my mind does so many things simultaneously, it helps to keep my own words pretty simple. When I am teaching strings without the rekenrek, thus most number strings, I tend to use these phrases a lot:
“Give me a thumb when you are ready”
“What did you get?”
“How did you get that?”
“Talk to your partner about this one”
“Can anyone restate her/his strategy?”
“Does this match what you were thinking?”(about the representation I made of their work)
Much of my talk when representing strategies is my repeating the words of a student who shared a strategy, simply because it helps me remember the strategy if I restate it while I am drawing a representation of it on the board.
My aim is to get strategies up on the board, and then to make sure that at some point in the string, I ask a question that moves the discussion to the level of generalization. The best questions are closely connected to the mathematics of the string, and the strategies of the students, but I seem to often say,
“Will this strategy always work?”
“Can we name and define this strategy?”
“Are there helpful patterns in this number string?”
Often, a student will begin this generalization process for you, and you just need to follow their thinking. They may say, I think that this strategy works because . . . , and at that point I write their words down verbatim on the side of the board, leaving space for the group to refine the generalization.
Just like my son learned his questions from me, I learned them from watching other people teach strings. Are there other go-to questions for teaching number strings?
“Which numbers do you think I will choose next? Why?”
Teaching older students (5th/6th grade), many of the learners in my classroom are eager to spot the strategy or relationship I am highlighting and jump ahead with it. When students say “I see what you are doing!” or “That same pattern is in all the problems!” I challenge them to extend their understanding and anticipate new problems built from their generalized understanding. This works particularly well with strings designed to highlight equivalence (e.g. doubling/halving or interchanging numerators when multiplying fractions). Of course, students must justify why the new problem is equivalent or fits the same strategy. They love “taking over” from me and develop a whole new level of ownership!