The National Council of Supervisors of Mathematics

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

2. 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. Common Core State Standards • Mathematics • Standards for Content • Standards for Practice

4. 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 Today’s Goals

5. 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. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Standards for Mathematical Practice

7. Structuring the Practices

8. Standards for Mathematical Practice • Individually review the Standards for Mathematical Practice. • Choose a partner at your table and discuss a new insight you had into the Standards for Mathematical Practice. • Then discuss the following question. What implications might the Standards for Mathematical Practice have on your classroom?

9. Sidewalk Patterns 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. 1. Draw the next pattern in this series.

10. Sidewalk Patterns • Complete the table below. • What do you notice about the number of white blocks and the number of gray blocks?

11. Sidewalk Patterns • The total number of blocks can be found by squaring the number of blocks along one side of the pattern. • Fill in the blank spaces in this list. 25 = 52 81 = ___ 169 =___ 289 = 172 • How many blocks will pattern number 5 need? • 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. • 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. Sidewalk Patterns • Individually complete parts 1-5. • Compare your work with a partner’s work. • Consider each of the following questions and be prepared to share your thinking with the group: • What mathematics contentis needed to complete the task? • How are the mathematical practices related to seeing structure and generalizing relevant to the task?

13. Tasks as they appear in curricular materials Student learning The Nature of Tasks Used in the Classroom … Will Impact Student Learning!

14. Stein, Grover & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen & Silver (2000) Tasks as enacted by teachers and students Tasks as they appear in curricular materials Tasks as set up by teachers Tasks as they appear in curricular materials Student learning Student learning But,WHAT TEACHERS DO with the tasks matters too! The Mathematical Tasks Framework

15. Oral Language Verbal - Written and Oral Real-World Situations Pictures Geometric/ Graphical Contextual Written Symbols Manipulative Models Symbolic Tabular Representation Stars Elementary Secondary 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.

16. www.InsideMathematics.org

17. Student A

18. Student C

19. Student F

20. Student D

21. 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. 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. 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 3 by 3 and a 7 by 7 BLUE =4•3•3+7•7 4 sets of 2 by 2 and a 5 by 5 BLUE =4•2•2+5•5 4 sets of 1 by 1 and a 3 by 3 BLUE =4•1•1+3•3

24. 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. Student A

26. Student B

27. Student C

28. Student E

29. 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. 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. 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. End of Day Reflections • 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. Join us in thanking theNoyce Foundationfor their generous grant to NCSM that made this series possible! http://www.noycefdn.org/

34. 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. 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