certain lengths and color to measure the length of a student’s desk. Then, she divided the class into small groups and asked them to measure their desks using rods of different lengths. After she placed samples of the (magnetized) rods on the chalkboard, the following exchanges between teacher and students transpired: Example #7 Initiation Responses Feedback T: So, what I’m interested in hearing now is how you figured out how many blue, red, and white rods it took to stretch across the table. And I’m also interested in hearing how many different ways there are to think about that. So, Craig, how many blue rods does it take to stretch across the table and how did you figure that? C: Six and two thirds T: And how did you figure that out? C: Because, um there is 3 light greens that fit into the blue and then you go 3 greens “3, times 6” that would 18 and that won’t be a full rod so take off 2. So you put down 2 and take out one green. That would be 2/3, that would be 6/23 T: And Nathan, how many blue rods do you think stretch across the N: About 6 T: And explain to me how you got that answer N: We took the blues and then took the light greens and compared how many would stretch across and worked our way down and got up to 20 green. And we just stuck these on top
certain lengths and color to measure the length of a student’s desk. Then, she divided the class into small groups and asked them to measure their desks using rods of different lengths. After she placed samples of the (magnetized) rods on the chalkboard, the following exchanges between teacher and students transpired: Example #7 Initiation Responses Feedback T: So, what I’m interested in hearing now is how you figured out how many blue, red, and white rods it took to stretch across the table. And I’m also interested in hearing how many different ways there are to think about that. So, Craig, how many blue rods does it take to stretch across the table and how did you figure that? C: Six and two thirds T: And how did you figure that out? C: Because, um there is 3 light greens that fit into the blue and then you go “3, 3 greens times 6” that would 18 and that won’t be a full rod so take off 2. So you put down 2 and take out one green. That would be 2/3, that would be 6/23 T: And Nathan, how many blue rods do you think stretch across the N: About 6 T: And explain to me how you got that answer N: We took the blues and then took the light greens and compared how many would stretch across and worked our way down and got up to 20 green. And we just stuck these on top