Supporting Rigorous Mathematics Teaching and Learning

1. Supporting Rigorous Mathematics Teaching and Learning Engaging In and Analyzing Teaching and Learning Tennessee Department of Education High School Mathematics Algebra 2

2. Rationale Common Core State Standards for Mathematics, 2010 Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true….…Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. By engaging in a task, teachers will have the opportunity to consider the potential of the task and engagement in the task for helping learners develop the facility for expressing a relationship between quantities in different representational forms, and for making connections between those forms.

3. Session Goals Participants will: • develop a shared understanding of teaching and learning; and • deepen content and pedagogical knowledge of mathematics as it relates to the Common Core State Standards (CCSS) for Mathematics.

4. Overview of Activities Participants will: • engage in a lesson; and • reflect on learning in relationship to the CCSS.

5. Looking Over the Standards • Look over the focus cluster standards. • Briefly Turn and Talk with a partner about the meaning of the standards. • We will return to the standards at the end of the lesson and consider: • What focus cluster standards were addressed in the lesson? • What gets “counted” as learning?

6. Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

7. The Structures and Routines of a Lesson MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: • Different solution paths to the • same task • Different representations • Errors • Misconceptions Set Up of the Task The Explore Phase/Private Work Time Generate Solutions The Explore Phase/Small Group Problem Solving Generate and Compare Solutions Assess and Advance Student Learning SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT: By engaging students in a quick write or a discussion of the process. Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

8. Solve the Task(Private Think Time and Small Group Time) • Work privately on theMissing Function Task. • Work with others at your table. Compare your solution paths. If everyone used the same method to solve the task, see if you can come up with a different way. • Consider what each person determined about g(x).

9. Expectations for Group Discussion • Solution paths will be shared. • Listen with the goals of: • putting the ideas into your own words; • adding on to the ideas of others; • making connections between solution paths; and • asking questions about the ideas shared. • The goal is to understand the mathematics and to make connections among the various solution paths.

10. Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

11. Discuss the Task(Whole Group Discussion) What do we know about g(x)? How did you use the information in the table and graph and the knowledge that h(x) = f(x) · g(x) to determine the equation of g(x)? How can you use what you know about the graphs of f(x) and g(x) to help you think about the graph of h(x)? Predict the shape of the graph of a function that is the product of two linear functions. Explain from the graphs of the two functions why you have made your prediction.

12. Reflecting on Our Learning What supported your learning? Which of the supports listed will EL students benefit from during instruction? Which CCSS for Mathematical Content did we discuss? Which CCSS for Mathematical Practice did you use when solving the task?

13. Pictures Manipulative Models Written Symbols Real-world Situations Oral Language Linking to Research/LiteratureConnections between Representations Adapted from Lesh, Post, & Behr, 1987

14. Language Context Table Graph Equation Five Different Representations of a Function Van De Walle, 2004, p. 440

15. Reflecting on Our Learning What supported your learning? Which of the supports listed will EL students benefit from during instruction? Which CCSS for Mathematical Content did we discuss? Which CCSS for Mathematical Practice did you use when solving the task?

16. The CCSS for Mathematical ContentCCSS Conceptual Category – Number and Quantity Common Core State Standards, 2010, p. 60, NGA Center/CCSSO

17. The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

18. The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 64, NGA Center/CCSSO

19. The CCSS for Mathematical ContentCCSS Conceptual Category – Functions Common Core State Standards, 2010, p. 70, NGA Center/CCSSO

20. Reflecting on Our Learning What supported your learning? Which of the supports listed would EL students benefit from during instruction? Which CCSS for Mathematical Content did we discuss? Which CCSS for Mathematical Practice did you use when solving the task?

21. What math practices made it possible for us to learn? Common Core State Standards for Mathematics, 2010 • 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.

22. Research Connection: Findings by Tharp and Gallimore Tharp & Gallimore, 1991 For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.” They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the student's progression through the zone, as well as the stumbles and errors that call for support. For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue -- the questioning and sharing of ideas and knowledge that happen in conversation.

23. Underlying Mathematical Ideas Related to the Lesson (Essential Understandings) • The product of two or more linear functions is a polynomial function.  The resulting function will have the same x-intercepts as the original functions because the original functions are factors of the polynomial. • Two or more polynomial functions can be multiplied using the algebraic representations by applying the distributive property and combining like terms. • Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x1, the point (x1, f(x1)+g(x1)) will be on the graph of the sum f(x)+g(x). (This is true for subtraction and multiplication as well.)

24. Essential Understandings

25. Essential Understandings