1. Developing Teachers’ Understanding of Proof Developing Teachers’ Understanding of Proof Peg Smith University of Pittsburgh Teachers Development Group Leadership Seminar February 17, 2011

2. Overview of Session Provide a rationale for focusing on reasoning and proving and describe the CORP project Solve and discuss the “Odd + Odd = Even” task Engage in an analysis of student “proofs” and discuss the opportunities for learning afforded by such work Discuss a framework for thinking about reasoning and proving activities Consider the potential of the activities to foster teacher learning and discuss situations in which the materials might be used

3. 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 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).

4. 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

5. 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

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. 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

8. 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

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. 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?

11. Construct a proof for the following conjecture: The sum of two odd numbers will always be an even number. Private Think Time – spend five minutes thinking about the task individually before beginning work with a partner or trio. Small Group – discuss different approaches with your partner(s) and jointly create a proof. Once you have proven it one way, see if you can come up with an alternative approach.

12. Analyzing Student Work (Part 1)

13. Criteria for Judging the Validity of Proof The argument must show that the conjecture or claim is (or is not) true for all cases. The statements and definitions that are used in the argument must be ones that are true and accepted by the community because they have been previously justified. The conclusion that is reached from the set of statements must follow logically from the argument made.

14. Analyzing Student Work (Part 2) For each response C, E, H and I consider: What is the limitation in the current argument that is being made by the student? (Refer to the Criteria for Judging the Validity of Proof list to help pinpoint what might be missing or incorrect.) What would it take for the current argument to be classified as a proof? What question(s) could you ask the student that would help improve her argument so that it would qualify as a proof? (How could you bridge between where the student currently is and where you want them to end up?)

15. 15 The work in which mathematicians engage that culminates in a formal proof involves searching a mathematical phenomena for patterns, making conjectures about those patterns, and providing informal arguments demonstrating the viability of the conjectures. Lakatos, 1976 Reasoning and Proving: Connecting to Literature

16. 16 Reasoning and Proving: An Analytic Framework Making Mathematical GeneralizationsProviding Support to Mathematical Claims Mathematical Component Identifying a pattern Making a conjecture Providing a proof Providing a non-proof argument Plausible Pattern Definite Pattern Conjecture Generic Example Demonstration Empirical Argument Rationale Stylianides, 2008, p. 10

17. 17 Reasoning and Proving: An Analytic Framework Making Mathematical Generalizations Providing Support to Mathematical Claims Mathematical Component Identifying a pattern Making a conjecture Providing a proof Providing a non-proof argument Plausible Pattern Definite Pattern Conjecture Generic Example B, D Demonstration A Empirical Argument C, G Rationale H Stylianides, 2008, p. 10

18. 18 By focusing primarily on the final product - that is, the proof - students are not afforded the same level of scaffolding used by professional users of mathematics to establish mathematical truth. Therefore, reasoning and proving should be defined to encompass the breadth of activity associated with: identifying patterns, making conjectures, providing proofs, and providing non-proof arguments. Stylianides, 2005; Stylianides & Silver, 2004 Reasoning and Proving: Connecting to Literature

19. 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

20. 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

21. Plausible Pattern InputOutput 11 24 3 4 5

22. Definite Pattern For the pattern shown below, compute the perimeter for the first four trains, determine the perimeter for the tenth train without constructing it, and then write a description that could be used to compute the perimeter of any train in the pattern. [The edge of the hexagon has a length of one.]