The National Council of Supervisors of Mathematics

1. The National Council of Supervisors of Mathematics The Common Core State Standards Illustrating the Standards for Mathematical Practice: Getting Started with the Practices www.mathedleadership.org

2. 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)

3. 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

4. The Standards for Mathematical Practice What are the verbs that illustrate the student actions for your assigned mathematical practice? Circle, highlight or underline them for your assigned practice… Discuss with a partner: How does this practice compare to what your students currently practice?

5. The Standards for Mathematical Practice #1: Explain and make conjectures… #2: Make sense of… #3: Understand and use… #4: Apply and interpret… #5: Consider and detect… #6: Communicate precisely to others… #7: Discern and recognize… #8: Notice and pay attention to…

6. The Standards for Mathematical Practice On a scale of 1 (low) to 6 (high), to what extent are you or your school/district promoting allstudents’proficiency in the practice you discussed? What evidence might you site for your rating?

7. One one hand, the Standards for Mathematical Practice describe mathematical content students need to learn. SP1. Make sense of problems “… students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.” Standards for Mathematical Practice

8. On the other hand, they describe the nature of the learning experiences, thinking processes, habits of mind, and dispositions that students need to develop a deep, flexible, and enduring understanding of mathematics; in this sense they are also a means to an end. SP1. Make sense of problems “….they [students] analyze givens, constraints, relationships and goals. ….they monitor and evaluate their progress and change course if necessary. …. and they continually ask themselves “Does this make sense?” Standards for Mathematical Practice

9. Standards for Mathematical Practice • Individually consider the question below and make a few notes about your current thinking. • Choose a partner at your table and discuss your thinking about the implications these Standards for Mathematical Practice might have in your classroom. What implications might the Standards for Mathematical Practice have on your classroom? (Phil Daro video)

10. Structuring the Practices

11. Maria and Wayne Task Maria saved $24. She saved 3 times as much as Wayne. How much money did Wayne save? Work the task. What representations and strategies did you use to solve the task?

12. Teacher Set UpTask as Set Up Maria saved $24. She saved 3 times as much as Wayne. 1. In what way(s) did the set-up of the task differ from the “traditional” way we experienced the task? 2. Why do you think Ms. Sherman might have set up the task the way she did? 3. What Standards for Mathematical Practice might benefit from her delivery?

13. 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

14. www.InsideMathematics.org Four boys discuss their understanding of the task • What representational tools do the boys use to engage in the task? • What mathematics do they use to model the real-world problem? • How does their conversation demonstrate any of the Standards for Mathematical Practice?

15. Closer Look at Content 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. • What about the content of this problem made it so difficult and led students into thinking the problem might be 3 x 24? • What about student approaches reassures you? • What about their thinking on this standard needs refinement?

16. Closer Look at Practices Model with mathematics. Use appropriate tools strategically. How does the task, as it was posed, support the development of students related to these two Standards for Mathematical Practice?

17. 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.

18. Using Formative Assessment to Plan Instruction Carlo thinks he will go to the gym about 20 times a month. Calculate how much each of these options would cost Carlo for one month. Which of these options is the least expensive for Carlo? Pay as you go Pay only $6 each time you work out Regular deal Pay $50 a month and $2 each time you work out All-in-one price! Pay just $100 per month for unlimited use of our great facilities

19. Using Formative Assessment to Plan Instruction Consider the summary of 6th grade results of the Gym task. What are some possible next steps to deal with misconceptions or misunderstandings documented in the table?

20. Opportunities for Content and Practice in Tasks Create verbal and tabular representations of the 3 DVD rental plans below. Do the three plans ever cost the same? Mail Flix $18 per month regardless of the number of movies rented Online Flix $12 per month plus $1 per movie rented Movie Buster $3 per movie rented

21. Individually, complete the task. Then work with a partner to compare your work and discuss the following questions: What mathematics content is needed to compete the task? When using tables to model and compare the dvd plans, what information and processes will students need to use strategically? What aspects of the explaining and modeling Standards of Practice might students need to complete the task? Opportunities for Content and Practice in Tasks

22. Standards for Mathematical Practice: Opportunities in Task Enactment In what ways did the teacher’s launchincrease students’ opportunities to begin explainingandmodeling? What evidence do you see that students are building these standards of practice?

23. Using Student Work to Develop Standards for Mathematical Practice Use the questions posed by Mr. Dimas to analyze the tables generated by Student H on the following slide. Do the tables make mathematical sense, and do they match the plans? Would these tables help us tell if the three plans will ever cost the same?

24. Using Student Work to Develop Standards for Mathematical Practice Student H Movie Buster Online Flix Mail Flix

25. Using Student Work to Develop Standards for Mathematical Practice As you watch the video consider: What evidence do you see that suggests students are developing competency with explaining and modeling? In what ways did interactions between students support their ability to develop competency with explaining and modeling?

26. Using Student Work to Develop Standards for Mathematical Practice Use the questions posed by Mr. Dimas to analyze the table generated by Student A. Does the table make mathematical sense, and does it match the plans? Would this table help us tell if the three plans will ever cost the same?

27. Using Student Work to Develop Standards for Mathematical Practice As you watch the video consider: What evidence do you see that suggests students are developing competency with explaining and modeling? In what ways did interactions between students support their ability to develop competency explaining and modeling?

28. Considering Next Steps Create verbal, graphical, and tabular representations of the carnival ticket plans below. Will any of the three plans ever cost the same? Bracelet Unlimited tickets with a $12.00 bracelet Dollar Deal No Entrance Fee $1.00 per ticket Discounted Plan $4.00 Entrance Fee, with discounted tickets ($0.50/ticket)

29. Standards for Mathematical Practice The eight Standards for Mathematical Practice place an emphasis on students doing mathematics and demonstrating learning. Equitable achievement will begin with an understanding of how the selection of tasks, the assessment of tasks, and the student learning environment can support or undermine equity in our schools.

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

31. 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 teacher do with the tasks matters too! The Mathematical Tasks Framework

32. Practices & Assessment Tasks • What might we expect mathematics assessment tasks in the new national assessments to look like given the CCSS Standards for Mathematical Practice?

33. Partnership for the Assessment of Readiness for College and Careers (PARCC)http://www.fldoe.org/parcc/ SMARTER Balanced Assessment Consortium http://www.smarterbalanced.org/ Assessment Consortia

34. Practices & Assessment Tasks As you read through the sample assessment tasks from SBAC and PARCC consider the following questions with a partner and be prepared to share your thinking with the whole group. • Which of the Standards for Mathematical Practice do you see being utilized in these tasks? • How are these tasks similar to and different from the mathematics tasks on your current classroom and state assessments?

35. Sample Grade 8 Mathematics Task

36. Join us in thanking theNoyce Foundationfor their generous grant to NCSM that made this series possible! http://www.noycefdn.org/

37. 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