This is one in a series of posts on Teaching Problems and the Problems of Teaching, by Magdalene Lampert. In each chapter, Lampert examines a challenge of teaching in the context of her fifth grade math classroom, and I try to learn some things from her.
Lampert teaches fifth grade in this book, yet I am constantly struck by how much of what she writes applies to my high school classes. I intended to read her chapter, “Teaching to Establish a Classroom Culture”, on how she works to establish a culture conducive to learning mathematics in the first weeks of school, and find one big takeaway to write about. Instead, I have a whole bunch, but here’s my attempt at paring that down to one big idea.
Lampert spends much of her time talking about three activities that students would engage in over the course of the year, and that she teaches deliberately from the first week.
In the context of these lessons, I taught my students three new activities and named them as such for public identification:
- finding and articulating the “conditions” or assumptions in problem situations that must be taken into account in making a judgment about whether a solution strategy is appropriate;
- producing “conjectures” about elements of the problem situation including the solution, which would then be subject to reasoned argument; and
- revising conjectures based on mathematical evidence and the identification of conditions (66)
These are well illustrated by a series of problems Lampert gives her students. Here is one:
Lampert allows students to work on the problem, and challenges those who finish early to try to find all of the possible additions. She finds that, as students try to find more solutions, some use unusual approaches. Some use unit fractions, others use negative numbers, and others put multiple digits in a single box. Creating “conditions” for a problem here is making explicit a practice of mathematics to encourage precise thinking and provide opportunities to interrogate the assumptions of a problem.
In this problem, Lampert begins to make conditions explicit. She’s very thoughtful about this choice. There is clearly value to thinking creatively about different ways to answer a problem, but she points out that
These multiple interpretations pose a problem for teaching because if all these ways of making combinations are allowed, students will not be able to evaluate the assertions their classmates are making about the total number of possibilities (74).
She then takes this a step further and poses a question with conditions that seems to me to be very difficult for the first weeks of a fifth grade math class.
She makes students’ conjectures transparent to the class by putting them all on the board
Students discuss the conjectures, and in the course of doing so, Lampert provides multiple opportunities for students to revise their thinking based on the reasoning and conjectures of their classmates.
This seems to me like remarkable teaching. She introduces significant mathematical ideas — conditions, conjectures, and revisions — and uses these ideas to teach students their role in the math classroom. Students learn that everyone’s ideas have value, that mistakes are normal and are part of learning, that communicating about mathematical ideas is important, and that it is the student’s job to make sense of mathematics.
She also has an interesting way of framing student work. She refers to this as independent work, though students are working in groups and have group norms:
- You are responsible for your own behavior
- You must be willing to help anyone in your group who asks
- You may not ask the teacher for help unless everyone in your group has the same question (82)
Lampert’s choice to call this “independent work” emphasizes individual accountability, and her norms both reinforce this accountability, and provide built-in support for students without requiring the involvement of the teacher at every stumble.
I found it fascinating that Lampert launches right into norming the way her class does mathematics. She spent the first class talking about revision and the importance of revision in learning math. Even more, she makes explicit her priorities in setting routines:
The work of ‘routine setting’ is more than telling students what the rules are going to be and enforcing them. It is largely a matter of guiding student talk and action in such a way as to establish shared understandings among everyone present about what it means to teach and to study and how it is to be done, here, with this class and this teacher (93).
This obviously ignores the many challenges of maintaining these routines, and what to do when students challenge them. I have never been particularly good at classroom management, and I look forward to her insights over the coming chapters. But framing expectations in class around the mathematical norms that facilitate learning, and then using those norms as a framework for teaching students what it looks like to do school, strikes me as a more purposeful way of managing a classroom. I have looked at behavior norms and mathematical norms as separate entities in the past; I’m considering now what those norms could look like, synthesized together, in my class.
I think the other important takeaway here is that Lampert is explicit in teaching what these actions look like, and why they are important for the class. Too often, I just put students in groups, or ask them to make a prediction, or give them feedback on their work, without teaching them why and how these things are important for their learning, and what student actions will contribute to their success. My meta-curriculum project broached several of these ideas, but I think I have a ton of room to grow in thinking through a progression of how these ideas develop and how students become normed to being productive and doing math in my class.