1. 1 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing The National Council of Supervisors of Mathematics The Common Core State Standards Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing In Grades 9-12 www.mathedleadership.org

2. 2 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Module Evaluation Facilitator: At the end of this Powerpoint, you will find a link to an anonymous brief e- survey that will help us understand how the module is being used and how well it worked in your setting. We hope you will help us grow and improve our NCSM resources!

3. 3 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Common Core State Standards Mathematics Standards for Content Standards for Practice

4. 4 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Today’s Goals To explore the mathematical standards for Content and Practice To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and to plan next steps In particular, participants will Examine opportunities to develop skill in seeing structure and generalizing

5. 5 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Standards for Mathematical Practice “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education.” (CCSS, 2010)

6. 6 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing 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.

7. 7 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Structuring the Practices

8. 8 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Standards for Mathematical Practice What implications might the Standards for Mathematical Practice have on your classroom? 1. Individually review the Standards for Mathematical Practice. 2.Choose a partner at your table and discuss a new insight you had into the Standards for Mathematical Practice. 3.Then discuss the following question.

9. 9 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Sidewalk Patterns 1. Draw the next pattern in this series. In Prague some sidewalks are made of small square blocks of stone. The blocks are in different shades to make patterns that are in various sizes.

10. 10 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Sidewalk Patterns 3.What do you notice about the number of white blocks and the number of gray blocks? 2.Complete the table below. Pattern number, n 1234 Number of white blocks1240 Number of gray blocks13 Total number of blocks25

11. 11 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Sidewalk Patterns 4.The total number of blocks can be found by squaring the number of blocks along one side of the pattern. a.Fill in the blank spaces in this list. 25 = 5 2 81 = ___ 169 =___ 289 = 17 2 b.How many blocks will pattern number 5 need? c.How many blocks will pattern n need? 5. a. If you know the total number of blocks in a pattern you can work out the number of white blocks in it. Explain how you can do this. b.Pattern number 6 has a total of 625 blocks. How many white blocks are needed for pattern number 6? Show how you figured this out.

12. 12 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Sidewalk Patterns 1.Individually complete parts 1-5. 2.Compare your work with a partner’s work. 3.Consider each of the following questions and be prepared to share your thinking with the group: a)What mathematics content is needed to complete the task? b)How are the mathematical practices related to seeing structure and generalizing relevant to the task?

13. 13 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing The Nature of Tasks Used in the Classroom … Tasks as they appear in curricular materials Student learning Will Impact Student Learning!

14. 14 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing But, WHAT TEACHERS DO with the tasks matters too! Stein, Grover & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen & Silver (2000) The Mathematical Tasks Framework Tasks as they appear in curricular materials Student learning Tasks as they appear in curricular materials Student learning Tasks as set up by teachers Tasks as enacted by teachers and students

15. 15 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Adapted from Lesh, R., Post, T., & Behr, M. (1987). Representations and Translations among Representations in Mathematics Learning and Problem Solving. In C. Janvier, (Ed.), Problems of Representations in the Teaching and Learning of Mathematics (pp. 33-40). Hillsdale, NJ: Lawrence Erlbaum. Geometric/ Graphical Verbal - Written and Oral Tabular Contextual Symbolic Pictures Oral Language Manipulative Models Real-World Situations Written Symbols Representation Stars Elementary Secondary

16. 16 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing www.InsideMathematics.org

17. 17 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student A

18. 18 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student C

19. 19 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student F

20. 20 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student D

21. 21 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Comparing Students A, C, F and D What evidence supports the conclusion that these students are using structure? What evidence supports the conclusion that these students are using regularity in repeated reasoning?

22. 22 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Watch the animation on the following slide and consider: What aspects of seeing structure and generalizing are evident in this work? What are some possible next steps for this student?

23. 23 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing 4 sets of 1 by 3 rectangles RED =4 1 3 4 sets of 2 by 5 rectangles RED =4 2 5 4 sets of 3 by 7 rectangles RED =4 3 7 4 sets of 1 by 1 and a 3 by 3 BLUE =4 1 1+3 3 4 sets of 2 by 2 and a 5 by 5 BLUE =4 2 2+5 5 4 sets of 3 by 3 and a 7 by 7 BLUE =4 3 3+7 7

24. 24 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Sidewalk Patterns A student showed the calculations at the right for finding the number of gray squares. What structure was he attending to? What questions might help him use his thinking to generalize the number of gray tiles?

25. 25 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student A

26. 26 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student B

27. 27 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student C

28. 28 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student E

29. 29 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Comparing Students A, B, C and E Compare the explanations given by these four students. How is each using regularity in repeated reasoning? What evidence might indicate that some or all of these students are making a generalization?

30. 30 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Next Steps and Resources Review the implications you listed earlier and discuss with your table group one or two next steps you might take as a district, school, and classroom teacher.

31. 31 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Today’s Goals To explore the mathematical standards for Content and Practice To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and to plan next steps In particular, participants will Examine opportunities to develop skill in seeing structure and generalizing

32. 32 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing End of Day Reflections 1.Are there any aspects of your own thinking and/or practice that our work today has caused you to consider or reconsider? Explain. 2.Are there any aspects of your students’ mathematical learning that our work today has caused you to consider or reconsider? Explain.

33. 33 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Join us in thanking the Noyce Foundation for their generous grant to NCSM that made this series possible! http://www.noycefdn.org/

34. 34 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Project Contributors Geraldine Devine, Oakland Schools, Waterford, MI Aimee L. Evans, Arch Ford ESC, Plumerville, AR David Foster, Silicon Valley Mathematics Initiative, San José State University, San José, California Dana L. Gosen, Ph.D., Oakland Schools, Waterford, MI Linda K. Griffith, Ph.D., University of Central Arkansas Cynthia A. Miller, Ph.D., Arkansas State University Valerie L. Mills, Oakland Schools, Waterford, MI Susan Jo Russell, Ed.D., TERC, Cambridge, MA Deborah Schifter, Ph.D., Education Development Center, Waltham, MA Nanette Seago, WestEd, San Francisco, California Hope Bjerke, Editing Consultant, Redding, CA

35. 35 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Help Us Grow! The link below will connect you to a anonymous brief e- survey that will help us understand how the module is being used and how well it worked in your setting. Please help us improve the module by completing a short ten question survey at: http://tinyurl.com/samplesurvey1