Going back to the video clip in the first part of this example, reflect on students' knowledge, understanding, and use of mathematics content using the following questions:

• What does David appear to be thinking? What does he possibly understand? What is your evidence?
• What can you not tell about what he is thinking? What is your evidence?
• What does the teacher do with David's explanation? What could be a mathematical reason for this move? A pedagogical reason? What is your evidence?
• What are some alternative possible moves that the teacher could have made and why?

Using the video clip in the second part of this example, consider the following questions:

• What new evidence do you have about David's possible understanding based on this clip?
• What does David appear to be doing to find the volume of the large cube? What appears to be his way of solving the problem?
• When the teacher says: "See how you're, what you're thinking. Now you have heard all those different explanations. What do you think?" What could be her reasons for asking this question? How might that relate to her mathematical learning goal? What could be the teacher's reason for getting many different explanations out for discussion? What could be a reason for not?
• What appears to be the mathematical logic in Eddie's explanation and thinking? What does she appear to understand? What does she appear to be struggling with? What does the teacher do with Eddie's explanation? What could be possible reasons for this move both mathematical and pedagogical? What are some possible alternative moves and why?
• What does Kevin appear to be thinking? What is his strategy for solving the problem? What is your evidence of this?
• What possible misconceptions did these students bring to the problem? What is your evidence? What other possible misconceptions might students bring to this problem? What possible teaching strategies might you employ with these misconceptions and why?
• Are there ways of categorizing possible ways of thinking about this problem? How might the teacher utilize this knowledge in using a variety of ways of thinking about the problem? Are there more efficient ways? Less efficient ways? Problematic ways? What would be a reason to use this in relation to the mathematical learning goals?
• What homework might the teacher give? What would the next day's lesson look like in building on this lesson?
• What might be the larger mathematical point over time? Why is it important?

References and Credits