K–12 Session 4.1 Standards for Mathematical Practices Part 1: Making Sense of the Mathematics Module 1: A Closer Look at the Common Core State Standards for Mathematics

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.

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

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

Making Sense of the Mathematics: Every Problem has a Story Interpreting Distance–Time Graphs Activity 1A Materials for this activity were obtained through the Mathematics Assessment Project (MAP). Original materials for this lesson can be found at:

Assessment Task: Tom’s journey to the Bus Stop Question 1: Write a narrative (describe) what is happening Question 2: Is the graph realistic? Explain Question 1: Write a narrative (describe) what is happening Question 2: Is the graph realistic? Explain

Activity 1B: History and Overview of the CCSSM Practices Introduction to the CCSSM Standards for Mathematical Practices

Session Informed by NCSM Webinar 8

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. - Common Core State Standards for Mathematics, page 6 9

Draws from Two Sources Principals and Standards for School Mathematics Adding It Up 10

NCTM Process Standards Problem Solving Reasoning and Proof Communication Representation Connections 11

Adding It Up: Strands of Mathematical Proficiency 12

The 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. 13

Connection to the Practices 14 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 What are common misconceptions associated with the given task? How does anticipating and addressing common misconceptions help facilitate deeper levels of thinking?

Contextualization: Context Matters in Understanding the Problem Interpreting Distance–Time Graphs Activity 2A Materials for this activity were obtained through the Mathematics Assessment Project (MAP). Original materials for this lesson can be found at:

Matching a Graph to a Story 16 A. Tom took his dog for a walk to the park. He set off slowly and then increased his pace. At the park Tom turned around and walked slowly back home. C. Tom went for a jog. At the end of his road he bumped into a friend and his pace slowed. When Tom left his friend he walked quickly back home. B. Tom rode his bike east from his home up a steep hill. After a while the slope eased off. At the top he raced down the other side.

Explaining the match A. Tom took his dog for a walk to the park. He set off slowly and then increased his pace. At the park Tom turned around and walked slowly back home. B. Tom rode his bike east from his home up a steep hill. After a while the slope eased off. At the top he raced down the other side. C. Tom went for a jog. At the end of his road he bumped into a friend and his pace slowed. When Tom left his friend he walked quickly back home.

Extension: Hurdles Race 18 The rough sketch graph shown above describes what happens when 3 athletes A, B, and C enter a 400 meters hurdles race. Imagine that you are the race commentator. Describe what is happening as carefully as you can. You need not measure anything accurately. The rough sketch graph shown above describes what happens when 3 athletes A, B, and C enter a 400 meters hurdles race. Imagine that you are the race commentator. Describe what is happening as carefully as you can. You need not measure anything accurately.

Connection to the Practices 19 Attending to the Content Standards How could something like this be used in the class you teach? Why is it important for students to understand this content? Attending to the Content Standards How could something like this be used in the class you teach? Why is it important for students to understand this content? 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?

Activity 2B: A Closer look at the CCSSM Practices Introduction to the CCSSM Standards for Mathematical Practices

Exercise Read the first three words for each mathematical practice. What do you notice?

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) 22

Small Group Discussion Form groups around pairs of math practices Identify verbs used in the practice Questions: – What is this practice standard asking of students? – What would this practice look like in your classroom? 23

Whole Group Discussion On a scale of 1 (low) to 5 (high), with a show of fingers, to what extent is your school or district promoting students’ proficiency for this practice? To what extent is this practice being explicitly addressed in every classroom? How does understanding the context of the problem help engage students in the practice? What is challenging about helping students develop proficiency in this standard? 24

Reflection How do the standards for mathematical practices connect to what you know of past standards (e.g. NCTM & Adding it up) as well as what you know about best practices? What do you anticipate to be challenges in supporting students in developing proficiency in these practice standards? What actions will you take to address and overcome these challenges? How does addressing misconceptions and context help students engage in the mathematical practices? How has this activity increased your understanding of the instructional core? 25