1. Warm-Up Begin reading The Case of Nancy Edwards on the green handout. As you read, consider the following questions: Did the teacher's decision to have students actually work on creating a proof -- rather than just fill in the blank with the word "even" -- appear to be a good use of class time? Why or why not? What did the teacher do to support her students as they worked on this task? Be sure to cite evidence from the case (i.e., line numbers) to support your claims.

2. The Task, Tools, and Talk Framework Peg Smith University of Pittsburgh Supporting the Development of Students’ Capacity to Reason-and-Prove Teachers Development Group Leadership Seminar February 18, 2011

3. Overview of Session Provide a rationale for focusing on reasoning-and- proving and describe the CORP project Engage in two project-developed activities Compare and discuss three sets of tasks Analyze a narrative case Consider the potential of the activities to foster teacher learning and discuss situations in which the materials might be used

4. Why Reasoning-and-Proving? Core practice in mathematics that transcends content areas Often conceptualized as a particular type of exercise exemplified by the two-column form used in high school geometry rather than as a key mathematical practice Difficult for students (and teachers) Growing consensus in the community that it should be “a natural, ongoing part of classroom discussions, no matter what topic is being studied” (NCTM, 2000, p.342)

5. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Common Core State Standards for Mathematics, 2010, pp.6-7

6. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Common Core State Standards for Mathematics, 2010, pp.6-7

7. CORP: Cases of Reasoning-and-Proving Focuses on reasoning-and-proving across content areas Supports the development of mathematical knowledge needed for teaching (see Ball, Thames, & Phelps, 2008) Features different types of practice-based activities Solving, analyzing, and adapting mathematical tasks Analyzing narrative cases Making sense of student work samples Provides opportunities for teachers to apply what they are learning to their own practice

8. Three Guiding Questions What is reasoning-and-proving? How do high school students benefit from engaging in reasoning-and-proving? How can teachers support the development of students’ capacity to reason-and-prove?

9. Three Guiding Questions What is reasoning-and-proving? How do high school students benefit from engaging in reasoning-and-proving? How can teachers support the development of students’ capacity to reason-and-prove?

10. select high-level (Stein & Smith, 1998) tasks that require students to reason about and make sense of the mathematics, and have the potential to leave behind important residue about the structure of mathematics (Hiebert, et al, 1997); encourage the use of tools (i.e., language, materials, and symbols) to provide external support for learning (Hiebert, et al, 1997); and engage students mathematical discourse or talk in order to make students’ thinking and reasoning public so that it can be refined and/or extended. This includes student-to-student exchanges as well as teacher-to-student exchanges.

11. Connecting to Literature: Mathematical Reasoning …it’s important for students to gain experience using the process of deduction and induction. These forms of reasoning play a role in many content areas. Deduction involves reasoning logically from general statements or premises to conclusions about particular cases. Induction involves examining specific cases, identifying relationship among cases, and generalizing the relationship. Productive classroom talk can enhance or improve a person’s ability to reason both deductively and inductively. Chapin, O’Connor, & Anderson, 2003, p. 78

12. Connecting to Literature: Mathematical Reasoning …both plausible and flawed arguments that are offered by students create an opportunity for discussion. As students move through the grades, they should compare their ideas with others’ ideas, which may cause them to modify, consolidate, or strengthen their arguments or reasoning. Classrooms in which students are encouraged to present their thinking, and in which everyone contributes by evaluating one another’s thinking, provide rich environments for learning mathematical reasoning. NCTM, 2000, p. 58

13. Activity 1: Compare and Discuss Three Sets of Tasks Compare each task to its adapted version (A to A’, B to B’, C to C’) Determine how each original task is the same and how it is different to its adapted version Look across the three sets and consider: Do the differences between a task and its modified version matter? What were the modifications in the tasks trying to accomplish?

14. Task A and A’ Same  Both ask students to complete a conjecture about odd numbers based on a set of given examples Different  Task A’ asks students to develop an argument  Task A can be completed with limited effort; Task A’ requires considerable effort to explain WHY this conjecture holds up

15. Task B and B’ Same  Same geometry content  Based on the same construction  Relate to a specific theorem about the diagonals of a parallelogram Different  Task B asks students to perform the construction once and then explain why the figure is a parallelogram; Task B’ asks students to perform the construction several times and make a conjecture about the type of figure produced.  The way Task B’ is phrased makes it more likely for students to engage in trying to develop a proof (“make an argument that explains why the same figure is produced each time” vs. “state a theorem”).

16. Task C and C’ SameDifferent  Both ask students to find the perimeter of the figure containing 12 trapezoids.  Both require students to identify the pattern of growth of the perimeter.  Task C’ provides more scaffolding by first asking students to find the perimeter of the first four figures.  Finding the perimeter of the 12th figure comes in different places: after finding the general rule in Task C vs. before the rule in Task C’.  Task C’ asks students to explain why the generalization always works.

17. Activity 1: Compare and Discuss Three Sets of Tasks Compare each task to its adapted version (A to A’, B to B’, C to C’) Determine how each original task is the same and how it is different to its adapted version Look across the three sets and consider: Do the differences between a task and its modified version matter? What were the modifications in the tasks trying to accomplish?

18. Do the differences matter? What were the modifications trying to accomplish? 1. Press students to do more than in the original versions of the task. 2. Engage students more in the development of arguments, including proofs (without actually saying “prove”). 3. Give students the opportunity to do more investigation. 4. Give students more access to the task. Underlying idea that runs across the modifications: To engage students in a broader range of activities as they investigate whether and why “things work” in mathematics.

19. 19 Making generalizations proofs Developing arguments investigation of whether and why “things work” in mathematics (both at the school level and at the discipline) reasoning-and-proving (Stylianides, 2008) patterns, conjectures

20. 20 Making generalizations proofs Developing arguments investigation of whether and why “things work” in mathematics (both at the school level and at the discipline) reasoning-and-proving (Stylianides, 2008) patterns, conjectures Reasoning-and-proving is defined to encompass the breadth of the activity associated with: identifying patterns making conjectures providing non-proof arguments, and providing proofs.

21. Reasoning-and-Proving: An Analytic Framework Making Mathematical Generalizations Providing Support to Mathematical Claims Identifying a pattern Making a conjecture Providing a non-proof argument Providing a proof Stylianides, 2008, p. 10

22. Reasoning-and-Proving: An Analytic Framework Making Mathematical Generalizations Providing Support to Mathematical Claims Identifying a pattern Making a conjecture Providing a non-proof argument Providing a proof A, A’ C, C’ A, A’ B’ C, C’ A’ B, B’ C’ Stylianides, 2008, p. 10

23. Activity 2: Analyze a Narrative Case Read The Case of Nancy Edwards on the green handout. As you read, consider the following questions: Did the teacher's decision to have students actually work on creating a proof -- rather than just fill in the blank with the word "even" -- appear to be a good use of class time? Why or why not? What did the teacher do to support her students as they worked on this task? (Pay particular attention to how the task, tools and talk supported students’ learning.) Be sure to cite evidence from the case (i.e., line numbers) to support your claims.

24. Was this time well spent? Sharpened students’ skills related to proof in a familiar domain. This might support later use of these skills in a less familiar domain. Made it clear that examples are helpful but not sufficient to prove. This idea will come up repeatedly and this task can be referred to. Introduced a variety of approaches that can be used to solve this task. This might help focus future discussions of proof less on form and more on substance. Surfaced many issues that could be used to develop criteria for proof that could be used throughout the year.

25. What did the teacher do….? Selected a “Good” Task Built on prior knowledge Encouraged multiple approaches Laid groundwork for discussing characteristics of proof Let Students Select Tools Group 1 drew pictures Group 6 used tiles to build a model Groups 3 and 4 used symbolic notations Let Students Do Most of the Talking and Most of the Thinking Asked questions to challenge students and to get them to talk to each other Encourage students to respond to questions posed by other students Instructed each new group to relate their approach to the other approaches that had been discussed

26. Take a few minutes to consider… The learning opportunities afforded by activities such as those we discussed today The situations in which the activities might be used The ways in which the Tasks, Tools, and Talk framework might support teachers’ development as well as students’ learning

27. CORP Project Team PIs: Peg Smith and Fran Arbaugh Senior Personnel:Gabriel Stylianides, Mike Steele, Amy Hilllen Jim Greeno, Gaea Leinhardt Graduate Students:Justin Boyle, Michelle Switala, Adam Vrabel, Nursen Konuk Advisory Board:Hyman Bass, Gershon Harel, Eric Knuth, Bill McCallum, Sharon Senk, Ed Silver