I had the pleasure of promoting one of my favorite books on Twitter last night, which got me thinking about my own use of the Five Practices for Orchestrating Mathematics Discussions.
This week, we were working on problem solving in the first of three courses for future elementary and special ed teachers. We were discussing solutions to this problem:
Roberto is shopping in a large department store with many floors. He enters the store on the middle floor from a skyway, and he immediately goes to the credit department. After making sure that his credit is good, he goes up three floors to the housewares department. Then he goes down five floors to the children’s department. Then he goes up six floors to the TV department. Finally, he goes down ten floors to the main entrance of the store, which is on the first floor, and leaves to go to another store down the street. How many floors does the department store have?
It’s not a deep, rich problem. But everyone can get a start on it, no matter their math background; it doesn’t feel to students like there is a particular piece of mathematics content that they are supposed to apply, etc. For these reasons and more, it’s a good first task for these students in this course. I did not create it.
Here is my artifact of selecting and sequencing.
I wanted students to see a variety of diagrams (going from most concrete in representing the problem to most abstract so that we could notice this as a difference).
I wanted to identify two key issues (both come from experience with this problem and this student population): (1) That we need to account for the precise meaning of the phrase the middle floor, (2) That there are assumptions about the problem that remain unstated (e.g. that there are no subterranean floors), and (2a) That identifying these assumptions, stating them and dealing with their consequences is a mathematical task.
As for connecting? Classroom Discussions by Chapin, O’Connor and Anderson is the ticket for making that happen.