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.
One of the teaching problems that is particular to teaching “big ideas” like the part-whole relationship is assessing what students already understand and what they still need to learn, and doing this for many different students. Although there is no simple way of averaging their accomplishments, I need to steer the work of the class as a whole. Within that common journey through some mathematical terrain, I need to attend to who needs extra guidance and when they need it. There is no simple metric here as there is in teaching separate topics one after another, where what students have learned can be crossed off a list (238).
Lampert’s view of covering curriculum is in stark contrast with that of certain leading reform efforts today:
There is a critical difference of perspective here. Lampert is examining a series of lessons about division in different contexts, and focusing specifically on numbers that “don’t work out”. Here is some more of her language:
When I asked Sam where “ninety-six” came from, I was providing the class with an opportunity to study how to connect a problem context and what is done with numbers (235).
Moving back and forth between money and an area model of fractions, I supported talk about the common structure of ratio in both of these problem contexts (240).
I was teaching the topic of “remainders” yet again, but coming at it from quite a different direction than I had in the time-speed-distance unit in November or the cakes and bakeries unit in January (242).
Lampert is not taking a laundry-list view of student learning, where if she just checks off all the boxes students will get what they need from her class. She is intensely humble about the fact that students will not learn overnight, and also intensely focused on the broad, transferable elements of a topic. This isn’t at the expense of the details — she dives deep into several specific contexts, as indicated above, and doesn’t hesitate to probe a student’s thinking, for instance in another exchange where she examines one small piece of the long division algorithm. These details are essential to cover a curriculum and provide students with the facility to apply their thinking in a broad variety of ways. But those details come after the big ideas, in this case the part-whole relationship, and are connected to their broader place in mathematics, rather than existing as procedures that are learned for the sake of procedures.
David Wees has a great, short talk that I think is relevant here, called “From Mistake-Makers to Sense-Makers”. He talks about his shift from looking at student work as a set of mistakes to be avoided, to a set of ideas that have value in moving their thinking forward.
This is a significant shift in thinking, and reading Lampert’s account of her teaching made a connection for me. Looking at students as sense-makers clearly impacts how we respond to student thinking, and honor the ideas that they have rather than trying to pigeonhole students into prescribed modes of thinking. But it also impacts the structure of the curriculum — whether we choose to start with big ideas, and take advantage of students’ attempts to make sense of them, or teach in little pieces that likely send the implicit message that math is a set of disconnected questions that students just need to avoid making mistakes on.