Agenda

1. Agenda • 9:00-9:15 Announcements • 9:15-9:50 Sums Group Papers • 9:50-10:00 MTR • 10:00-10:30 Staircase Group Papers • 10:30-10:45 Break • 10:45-10:55 MTR • 10:55-11:25 Working with Classroom Tasks • 12:00-1:00 Lunch

2. Agenda • 1:00-1:20 Listening Interviews • 1:20-1:50 Listening for a Thought Process • 1:50-2:20 Carnival Bears Problem • 2:20-2:35 Break • 2:35-3:05 Carnival Bears Discussion • 3:05-3:15 MTR • 3:15-3:45 Listening Interviews

3. Looking at Student Written Work Goals • Use student written work as data during the process of exploring student thinking and examining the potential to further that thinking. • Explore how the structure of mathematical tasks—what they ask students to do—is related to the algebraic thinking that can be seen in students’ work done on those tasks.

4. Directions of Student Thinking • Use the Student Work Analysis Sheet to analyze examples of student work for algebraic thinking, noting what you see in the work and what inferences you have. • Each person should present a brief explanation of why he/she chose to bring that piece. • Group members can comment on what they wrote about the piece on the Student Work Analysis Sheet.

5. What evidence do you see in these papers of Building Rules to Represent Functions?How do students organize information?How do they describe any rules they are building? • What evidence do you see of Abstracting from Computation?What computational shortcuts do students take in order to generalize?What justification do they give for these shortcuts? • What can you say about the direction each of these students is heading with their thinking?

6. MTR—Sums of Consecutive Numbers • What would you like to recall about the different strategies and/or solutions used by your colleagues? Record the approaches and strategies you would like to remember. • What would you like to recall about the algebraic thinking? Record the specific features of habits of mind that you have seen in the different solutions. • What would you like to recall about the different strategies and/or solutions used by your students? Record the mathematical approaches or strategies you would like to remember.

7. Staircase Problem Group Papers • Use the Student Work Analysis Sheet to analyze examples of student work for algebraic thinking, noting what you see in the work and what inferences you have. • Each person should present a brief explanation of why he/she chose to bring that piece. • Group members can comment on what they wrote about the piece on the Student Work Analysis Sheet.

8. What evidence do you see in these papers of Building Rules to Represent Functions—in particular,How do the students organize information?In what ways do they describe any rules they are building? • Some students like to rely on a recursive rule. What could you do to help those students get beyond a recursive representation to a closed representation? • How might you change the wording of the problem to encourage to go beyond a recursive representation to a closed representation?

9. MTR—Staircase Problem • What would you like to recall about the different strategies and/or solutions used by your colleagues? Record the approaches and strategies you would like to remember. • What would you like to recall about the algebraic thinking? Record the specific features of habits of mind that you have seen in the different solutions. • What would you like to recall about the different strategies and/or solutions used by your students? Record the mathematical approaches or strategies you would like to remember.

10. Working with Classroom Tasks • Solve this typical algebra question:G(X)=x2 + 2xWhat is G(5)? • What is the problem asking students to do? • What kind of algebraic thinking does it require?

11. Working with Classroom Tasks • Create an “Altered Problem 5” and an explanation to accompany it for the original problem “G(x)=x2 + 5; what is G(5)?” • Be prepared to discuss these questions:1. How did you choose to alter Problem 5, and why?2. How will you choose tasks and questions to ask your students in the future? • Post your group’s work.

12. Implications for Instruction • During this module you have examined written student work for evidence of algebraic thinking, reflecting on the directions of students’ thinking. Discuss what you have learned about examining the directions of students’ thinking, such as the limitations or confusions you notice students having, or the ways you think you can help students improve their algebraic thinking, based on what you see in their written work.

13. Implications for Instruction • How will you take what you have learned during the course of the Analyzing Written Student Work back to the classroom? • Has your perspective on the range of algebraic ideas that can be seen in your students’ thinking changed? • How will you look for evidence of algebraic thinking in your students’ written work?

14. Listening to StudentsGoals • Investigate the kinds of information about student thinking you can gain from listening to students as they work on math problems. • Contrast the information you get from looking at students’ written work with the information you get by listening to students working on math problems. • Continue to use the A-HOMs to describe and understand algebraic thinking.

15. Listening Interview Video Simulation • As you watch the video, listen to what the students say, both to one another and to themselves. • Try not to worry about whether students are getting the right answer. • Try to notice:(1)Thought processes—how the students’ solution evolves(2)How students’ written answers compare to the ideas you find in their discussions leading up to the solutions

16. Listening for a Thought Process • What is the students’ first strategy? • What generalizations do they make early on? • What makes them decide that it might be possible to make all amounts above 23¢? • What do they do to try to confirm this hypothesis? • How do they build on one another’s ideas? • When do the different students pursue different ideas? • Does the group have a common understanding of the solution, or do different students have different ideas at the end?

17. Listening for a Thought Process • If you had tried to answer these questions by just looking at the written work students turned in, without having heard what they said, which questions would you have been able to answer? • How would your analysis of the students’ thinking have been different? • What kinds of questions might be easier to answer by looking at students’ written work rather than by listening to their conversation?

18. Carnival Bears • Work on the Carnival Bears activity alone or with one or two others. • Keep track of your thought process and the strategies you try as you work.

19. Small Group Discussion • What was the first thing you noticed or started exploring? • What questions did people ask themselves about the problem? • What did people pay attention to as they proceeded? • In what different ways did people represent things in the problem? • When and how did people generalize to help them answer the question?

20. Carnival Bears Discussion • How did group members use some of the features of Building Rules to Represent Functions? • Did group members display any Doing/Undoing? • Did anyone use features of Abstracting from Computation? • What happens if there are an unequal number of bears on each side?

21. MTR—Carnival Bears • What would you like to recall about the different strategies and/or solutions used by your colleagues? Record the approaches and strategies you would like to remember. • What would you like to recall about the algebraic thinking? Record the specific features of habits of mind that you have seen in the different solutions. • What would you like to recall about the different strategies and/or solutions used by your students? Record the mathematical approaches or strategies you would like to remember.

22. Listening Interviews • Read the Pre-Module Homework. • Read Ruth’s and Andrew’s Listening Interviews.

23. Listening Interviews Discussion • Ruth’s approach to the interview was more like a classroom approach than Andrew’s. What evidence from the transcripts supports this statement? • Why do you think Andrew might have chosen this particular segment to transcribe for his colleagues? • What questions do you have about what you are expected to do when conducting a listening interview and why are you being asked to do this particular activity with your students?

24. For tomorrow… • In Asking Questions of Students, read Pre-Module Homework, pages i-iv. • Read “Question Types”