1. Mathematical Reasoning

2. Mathematical Proof: Review & Sharing What is Mathematical Reasoning? Inductive and Deductive Reasoning How to Assess Reasoning Skills Problem Solving: Exploring to Generalization Today’s Agenda

3. Proof Review It is “proofs rather than theorems that are the bearers of mathematical knowledge” (Rav, 1999)

4. Proof Review Pass around and look at student baseline assessments within your group. Assess and sort according to students’ understanding of proof and language ◦ 0: no understanding ◦ 1: some understanding (use of examples) ◦ 2: understands the need for proof and attempts abstract arguments, but struggles ◦ 3: high level understanding Describe some of your students’ views on justification and proof

5. Proof Review What did you learn most from the Proof baseline and summative assessments? Describe at least two ways that you helped your students understand what mathematical proof is. Describe one classroom situation where you saw a student exhibit growth in thinking about proof.

6. What is Mathematical Reasoning?

7. Initial Discussion With your neighbors at your table, try to write a definition of of mathematical reasoning. How does it differ from mathematical proof?

8. Definition Mathematical reasoning involves gathering evidence, building arguments, and drawing logical conclusions about these various ideas and their relationships.

9. Types of Reasoning There are many types of mathematical reasoning, including: Inductive (e.g. pattern generalization) Deductive (logic) Geometric Algebraic Can your group think of any others?

10. Inductive and Deductive Reasoning

11. PISA Apple Orchard Problem

12. Logic Puzzle Jerell has three children, Jordan, Nastasia and Logan. They attend Central, Humboldt and Arlington and play Soccer, Tennis and Basketball. a.Jordan does not attend Central. b.Nastasia is not at Humboldt. c.The child at Central does not play soccer. d.The student at Humboldt plays tennis. e.Nastasia does not play basketball. Which child attends which school, and what sport does each one play?

13. Mathematical Methods Some versions of the NAEP assessment have marked certain problems as measuring “mathematical methods.” ( = “mathematical reasoning” ) (Silver & Carpenter, 1989) analyzed the results from the 1986 NAEP exam.

14. Based on this limited data, what conjectures might you make about students’ reasoning abilities in different contexts and with different types of problems? NAEP Discussion

15. How to Assess Reasoning Skills

16. Analyzing Reasoning Skills

17. ADW: Analysis of Paul

18. ADW: Analysis of Ben

19. RND: Analysis of Paul

20. RND: Analysis of Ben

21. SAS: Analysis of Paul

22. SAS: Analysis of Ben

23. Synthesis Work with your table to create a summary of Paul’s and Ben’s abilities in the following areas: Foundational Knowledge Mathematical Reasoning Communication Then create a condensed summary of their overall reasoning abilities with help from your colleagues and the presenter.

24. Problem Solving: Exploring to Generalization

25. Problem Which natural numbers can be written as the sum of two or more consecutive natural numbers? Experiment on the table in your handout!

26. Baseline Assessment

29. Special Assignment The baseline assessment contains one of the problems you analyzed as part of the Peressini & Webb framework. The Interview Protocol includes the other two. Choose a variety of students for your interviews. Combined with their baseline assessments, you have all three of the problems we used to analyze Paul and Ben’s reasoning abilities. Perform the same analysis for the students you interview, using all three problems.