1. K–12 Session 4.2 Standards for Mathematical Practices Part 2: Student Dispositions & Teacher Actions Module 1: A Closer Look at the Common Core State Standards for Mathematics

2. Expected Outcomes Build understanding of the standards for mathematical practice. Enhance skills in identifying the extent to which students exhibit the standards for mathematical practice. Generate ideas for how teachers can integrate the standards for mathematical practice with instruction to support student proficiency.

3. Principle #1: Increases in student learning occur only as a consequence of improvements in the level of content, teachers’ knowledge and skill, and student engagement. Richard Elmore, Ph.D., Harvard Graduate School of Education Principle #2: If you change one element of the instructional core, you have to change the other two. The Instructional Core

4. Adapted from the Public Education Leadership Project at Harvard University STRUCTURES POLICIES, PROCESSES & PROCEDURES RESOURCES HUMAN, MATERIAL, MONEY STAKEHOLDERS CULTURE Organizational Elements

5. Multiple Representations: Connecting Concrete, Pictorial, and Symbolic Representations Interpreting Distance–Time Graphs Activity 3A Materials for this activity were obtained through the Mathematics Assessment Project (MAP). Original materials for this lesson can be found at: http://map.mathshell.org/materials/http://map.mathshell.org/materials/

6. Matching Cards 6 Take turns at matching pairs of cards. You may want to take a graph and find a story that matches it. Alternatively, you may prefer to take a story and find a graph that matches it. Each time you do this, explain your thinking clearly and carefully. If you think there is no suitable card that matches, write one of your own. Place your cards side by side on your large sheet of paper, not on top of one another, so that everyone can see them. Write your reasons for the match on the cards or the poster just as we did with the example in class. Give explanations for each line segment. Make sure you leave plenty of space around the cards, as eventually you will be adding another card to each matched pair.

7. Sharing Work 7 One student from each group is to visit another group's poster. If you are staying at your desk, be ready to explain the reasons for your group's matches. If you are visiting another group: Write your card placements on a piece of paper. Go to another group's desk and check to see which matches are different from your own. If there are differences, ask for an explanation. If you still don't agree, explain your own thinking. When you return to your own desk, you need to consider as a group whether to make any changes to your own poster.

8. Connection to the Practices 8 Attending to the Student (Practice Standards) What opportunities were there to engage in the practice standards in this activity? Did the extension (explaining) give opportunities to engage in additional practices? Or deepen the ones identified above? Attending to the Student (Practice Standards) What opportunities were there to engage in the practice standards in this activity? Did the extension (explaining) give opportunities to engage in additional practices? Or deepen the ones identified above? Attending to Instruction How can using multiple representations [e.g.. pictorial (graphs), symbolic (tables or equations)] help students engage in the practices? How can using multiple representations of mathematical ideas facilitate deeper levels of thinking?

9. Activity 3B: Identifying Student Outcomes & Dispositions Introduction to the CCSSM Standards for Mathematical Practices

10. Grouping of Math Practices Reasoning and Explaining 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others Modeling and Using Tools 4. Model with mathematics 5. Use appropriate tools strategically Seeing Structure and Generalizing 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Overarching Habits of Mind of a Productive Mathematical Thinker 1. Make sense of problems and persevere in solving them 6. Attend to precision Adapted from (McCallum, 2011) 10

11. Identify Student Outcomes Form groups around pairs of math practices How did the activities from session 4.1 & 4.2 engage in the practices? – Identify 1 or 2 student outcomes and/or dispositions that would convince you that your students were engaged in the given practice – How does using multiple representations help in show that students are able to engage in the math practices? Be ready to share with the group 11

12. Jigsaw Discussion Each group will share their student outcomes, as you listen be sure to think about the following questions: Are the student outcomes clearly articulated? How might you change this for students in you class? Are there additional outcomes that should be identified? 12

13. Attending to Precision: Communicating Precisely to Others Interpreting Distance–Time Graphs Activity 4A Materials for this activity were obtained through the Mathematics Assessment Project (MAP). Original materials for this lesson can be found at: http://map.mathshell.org/materials/http://map.mathshell.org/materials/

14. Making Up Data for a Graph 14 TimeDistance 0 2 4 6 8 10

15. Extension: Matching Tables You are now going to match tables with the cards already on your desk. In your group take a graph and try to find a table that matches it, or take a table and find a graph that matches it. Again take turns at matching cards you think belong together. Each time you do this, explain your thinking clearly and carefully. Write your reasons for the match on the poster. 15

16. Connection to the Practices 16 Attending to the Student (Practice Standards) What opportunities were there to engage in the practice standards in this activity? Did the extension (explaining) give opportunities to engage in additional practices? Or deepen the ones identified above? Attending to the Student (Practice Standards) What opportunities were there to engage in the practice standards in this activity? Did the extension (explaining) give opportunities to engage in additional practices? Or deepen the ones identified above? Attending to Instruction Why is attending to precision critical consideration as students engage in the mathematical practices? How can directly attending to issues of precision facilitate deeper levels of thinking?

17. Activity 4B: Identifying Teacher Activities & Strategies Introduction to the CCSSM Standards for Mathematical Practices

18. Grouping of Math Practices Reasoning and Explaining 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others Modeling and Using Tools 4. Model with mathematics 5. Use appropriate tools strategically Seeing Structure and Generalizing 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Overarching Habits of Mind of a Productive Mathematical Thinker 1. Make sense of problems and persevere in solving them 6. Attend to precision Adapted from (McCallum, 2011) 18

19. Identify Teaching Strategies Form groups around pairs of math practices How did the activities from session 4.1 & 4.2 engage in the practices? – Identify 1 or 2 teaching strategies that were used to engage you in the practices – How do precision activities help support the practices? Be ready to share with the group 19

20. Jigsaw Discussion Each group will share their student outcomes, as you listen be sure to think about the following questions: Are the student outcomes clearly articulated? How might you change this for students in you class? Are there additional outcomes that should be identified? 20

21. Reflection How does the use of multiple representations help engage students in the mathematical practices? How does engaging students in precision activities help support the practices? What are important student dispositions you need to look for as evidence that a student is engaging in the practices? How can you use these ideas to help your students engage in the mathematical practices? How has this activity increased your understanding of the instructional core? 21