This is one of a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines the one challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.
This chapter of Lampert’s book has been the most interesting to me so far, but also the most complex. There is no three-step approach to connecting content across lessons. Instead, it’s a great deal of thinking about reoccurring representations, connecting contexts to content, using language purposefully, and formative assessment. I don’t know how I could summarize the chapter as a whole, but Lampert does investigate one domain that I had not thought much about before.
Lampert poses a problem to students:
She spends the class discussing the first problem, which students struggled with, and uses this diagram, which she calls a “journey line”, to relate the context and the mathematics.
Looking at student work after the class, she notes that only a few students make assertions about the second problem, and only one student, Charlotte, tries to figure out precisely how long it would take the Mimi to go 60 nautical miles at 8 knots. This problem is significantly more complex because the numbers don’t divide evenly. Lampert’s students are still struggling with the their knowledge of division, but Lampert knows she wants to move into teaching about remainders in a future lesson, and uses this as an opportunity to think about how to structure that teaching.
Here is Charlotte’s explanation for part two of the problem:
Charlotte’s interpretation of the remainder is wrong, but that’s not Lamperts focus. She writes:
Using the problem context to give meaning to the remainder is something I would want everyone in the class to be disposed and able to do (200).
This isn’t something I saw when I first read Charlotte’s work. I think it’s an important thing to notice because it is a focus not on the how, the mechanics of a computation or procedure, but instead a why, a habit of making sense of mathematics and relating mathematics to the world. It’s a change in perspective from looking at Charlotte’s reasoning as lacking because her answer is wrong, to looking at her reasoning as useful because it is attempting to make sense.
But more than using this piece of work as an evaluation of Charlotte’s thinking, it provides a “way in” to future mathematics.
Examining the mathematics involved in Charlotte’s assertion led me to think that I should pose problems that would engage students with these ideas. Although Charlotte may have been ahead of everyone else in confronting the problem of giving meaning to the remainder, I could use her work to clue me in to the kinds of things other students might do and what they should be able to do. The challenge for me would be to structure the learning environment so that neither Charlotte nor anyone else would be satisfied with an assertion about the remainder like the one Charlotte ventured without returning to the constraints of the problem situation to see if it fits.
Here, I expected Lampert to launch into a problem the next day to explore this mathematical territory. Instead, the class spends two days working on problems where a remainder is not necessary, so that students can continue to study division and its relationship to a context — dividing 135 by 5, and 180 by 6. Then, on the third day, she poses a problem about a rate. She uses the same distance, 180 nautical miles, as the day before, but changes the speed to 8 knots so that the division will not “come out even”. This problem was not entirely successful. Several students offered interpretations of the remainder, but the class ended before anyone could offer a coherent explanation for why the remainder of 4 represents half an hour.
The next day, Lampert chooses a similar problem, but instead chooses 20 nautical miles and 6 knots, reducing the algorithmic load on students and providing an opportunity to focus more mental effort on the meaning of the remainder. Lampert prompts students to figure out how long it will take for the ship to go one mile, and to use a diagram. This allows her to launch into teaching about what that remainder means, and to relate the hour to parts of the hour, and to do the unit conversion necessary to move between hours and minutes.
This is a bit of an anticlimactic end to a fascinating series of instructional decisions, but I learned several important things from seeing Lampert’s thinking. First, she is enormously patient, and content for students to leave class, perhaps for several days in a row, unable to solve a problem using a certain piece of mathematics. She does this because she has a laser-focus on sense-making, creating situations where students can believe that math makes sense, and they are capable of figuring out problems by making connections with what they already know. Finally, she launches this entire series of problems with an observation from student work, and uses that work to guide her planning for future lessons and frame her thinking about how other students are likely to interpret a new idea.
I’ve never done this level of thinking in my teaching, and it’s a bit intimidating, but also seems like an enormously valuable way to both examine student thinking, and use that thinking to inform future planning. It also highlights the complexity of connecting content across multiple lessons. Fun stuff to get better at.