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 K-5 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 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 Building Walls Maya is using blocks to make a wall grow.

10. 10 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Building Walls 1.Draw the 4 th wall below. 2.How many blocks are needed to make the 4 th wall? _____ blocks 3.Tom wants to build the 5 th wall. Tell Tom how to build the 5 th wall. 4.How many blocks would it take to build the 7 th wall? _____blocks Show how you know your answer is correct. 5. Tom and Maya’s teacher gave them 21 blocks. Can they build a wall with exactly 21 blocks? Yes or No _______ Why or Why not?

11. 11 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Building Walls 1.Individually complete parts 1-5. 2.Compare your work with a partner’s work. Look for as many plausible explanations for part 5 as possible. 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) Which mathematical practices are needed to complete the task?

12. 12 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!

13. 13 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

14. 14 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

15. 15 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing www.InsideMathematics.org

16. 16 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student F

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

18. 18 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Comparing Students F and G Try to translate each student’s explanation into a mathematical expression. What would these expressions be if the 5 was replaced by n ? How do these students differ in the way they see the structure of the walls?

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

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

21. 21 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Comparing Students Z and L Do you think Student Z is seeing structure or regularity in repeated reasoning? Justify your position. Do you think Student L is seeing structure or regularity in repeated reasoning? Justify your position.

22. 22 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student K

23. 23 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Student E

24. 24 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing Comparing Students K and E Do you think Student K is seeing structure or regularity in repeated reasoning? Justify your position. Do you think Student E is seeing structure or regularity in repeated reasoning? Justify your position.

25. 25 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.

26. 26 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 plan next steps In particular, participants will Examine opportunities to develop skill in seeing structure and generalizing

27. 27 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.

28. 28 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/

29. 29 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

30. 30 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