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.
Five Twelve Thirteen 
It would honestly never occur to me to contrast Lampert’s teaching with Sal’s Khan game, which is the antithesis of what Lampert is about. Consider this piece I co-wrote with Christopher Danielson 3 1/2 years ago for the WaPo’s online education blog: https://www.washingtonpost.com/blogs/answer-sheet/post/how-well-does-khan-academy-teach/2012/07/27/gJQA9bWEAX_blog.html
The answer, to kill the suspense, is that Sal Khan doesn’t teach at all well. No surprise: he’s not a teacher, no matter what Bill Gates bleats about how ab-fab Sal’s mathematics teaching is. He throws lessons together thoughtlessly, with no regard for potential students: he just mouths the steps for solving a few poorly-chosen, ill-ordered examples. How can anything be less like what Lampert, Deborah Ball, and other deeply reflective practitioners do?
Just sayin’.
Agreed that the approaches are totally different — but I’m specifically contrasting an approach to student understanding, and even folks who strongly disagree with Sal Khan could look at student understanding as a set of red, yellow and green boxes. That’s a more subtle distinction — about our perspective on what we think of our students understanding, and what we do about it.
I’m really curious how this interplays with current trends in grading towards SBG. While the idea of teaching from big, connected ideas really appeals to me, I feel like it’s harder to wrap my mind around how to give useful feedback to students (and many of them) in this mode. An SBG model often looks no different than the Khan Academy spreadsheet of colored boxes in this post. When you’re teaching 150 students, how do you effectively communicate progress and what each one needs to do to improve outside of checkboxes or red/yellow/green squares?
I agree that this has a challenge. I think that, from a teaching perspective, this affects how I approach a topic, even within an SBG framework. I begin with big ideas, and don’t teach skills one at a time, but instead spiral through them while focusing on big ideas. Then, later in a unit, I begin assessing. While I don’t love the message that SBG sends about what it means to learn math, I do think that SBG-style feedback is much more actionable for students.
I think the other corollary here is that skills-based SBG alone is insufficient as a grading system, in terms of the messages it sends to students. I’ve been trying to come up with a better system that integrates some broader topics and different types of thinking, but I haven’t been very happy with what I’ve come up with so far.
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