1. Math Fellows January 22, 2014

2. 0. Setting Goals and Selecting Tasks 1.Anticipating 2.Monitoring 3.Selecting 4.Sequencing 5.Connecting The Five Practices (+)

3. Agenda Do some math! Review the five practices and apply to a new task Reflect on implementation of practices Using questioning to support the enactment of the practices/ or in efforts to engage students within discussion

4. 0 1 : Setting Goals Without explicit learning goals, it is difficult to know: What counts as evidence of student learning How students’ learning can be linked to particular instructional activities How to revise instruction to facilitate particular instructional activities How to revise instruction to facilitate students’ learning more effectively “Formulating clear, explicit learning goals sets the stage for everything else.” Hiebert

5. Our Task: The Sum of Two Odd Numbers Prove the following conjecture: The sum of any two odd numbers is an even number. 5

6. 0 1 : Setting Goals and 0 2 : Selecting Task Possible goals for instruction Explore the results when adding an odd and an even number. Make a conjecture about the sum of an odd and even number. Create a proof to justify that the sum of an odd and odd number is always an even number.

7. Our Goal for the Lesson 1. Realize that examples are not enough to show that a claim is always true. 2. Recognize that there are many different ways to prove that a claim is true and that it is not the form that matters but, rather, the consideration of all cases and creation of a clear and logical argument. 3. Understand that there are reasons WHY mathematics works the way it does that can be explored and explained. 7

8. Our Task: The Sum of Two Odd Numbers Prove the following conjecture: The sum of any two odd numbers is an even number. 8

9. 1: Anticipating Likely Responses Working individually, consider the correct and incorrect approaches that students might use to solve this task Working with a small group, share the approaches you have anticipated so far and see what other approaches you can come up with together Whole group: Share solutions

10. Anticipated Solution Methods Picture – odd numbers represented as a number of sticks, grouped by 2, showing one left over Sketch of a Rectangle – odd numbers represented as a 2-by-n rectangle with one extra square Logical Argument Algebra

11. 2: Monitoring Responses Which responses might you look for in your/and your teams student work?

12. Monitoring Actual Responses Imagine that the students in your class produced the solutions A-H. Individually, study the various solutions presented Record information on your monitoring sheet?

13. 3 & 4: Selecting and Sequencing Student Responses Individually, consider: Which solutions would you want to have shared during the group discussion? Why? In what order would the responses be shared? Why? As a table group Select four responses to be presented to the class Determine the order of the presentations

14. Selecting and Sequencing Possible Selections Picture (H) Loner Number (G) Algebra (A) Empirical Example (E)

15. 5: Connecting With your table group, discuss What connections will you help students make between solutions presented? What connections between key mathematical ideas can you help students construct?

16. Connecting Picture (H) Loner Number (G) Algebra (A) Empirical Example (E) Connect the “left over” dot in Student H’s response, the “loner number” in Student G’s response, and the +1 in the algebraic response. Connect the grouping by two in H with the 2x in A

17. Stepping Back Do you think that time spent engaging in practices 0 and 1 before the lesson begins would be worth the time and effort? Why or why not? How would planning a lesson using the five practices impact student learning?

18. Agenda Do some math! Review the five practices and apply to a new task Reflect on implementation of practices Using questioning to support the enactment of the practices

19. Networking Time

20. “ The five practices can help teachers manage classroom discussions productively. However, they cannot stand alone… In addition, teachers need to develop a range of ways of interacting with and engaging students as they work on tasks and share their thinking with other students.

21. This includes: having a repertoire of specific kinds of questions that can push students’ thinking toward core mathematical ideas methods for holding students accountable to rigorous, discipline-based norms for communicating their thinking.”

22. Questioning to Support Enactment of the Practices TypeDescription Exploring mathematical meanings and/or relationships Points to underlying mathematical relationships and meaning Makes links between mathematical ideas and representations Probing, getting students to explain their thinking Asks student to articulate, elaborate, or clarify ideas Generating discussionSolicits contributions from other members of the class “These questions do not take over the thinking for the students by providing too much information or by ‘giving away’ the answer or a quick route to the answer. Rather, they scaffold thinking to enable students to think harder and more deeply about the ideas at hand.”p.62

23. Exploring Questioning in Regina Quigley’s Classroom The background: 4 th grade classroom Geometry unit Before the lesson, students found areas of rectangles and squares Teacher’s goal for lesson: students will construct the formula for finding the area of a right triangle by manipulating premade cardboard right triangles against a backdrop of grid paper.

24. Regina Quigley’s Lesson Individually, read the vignette. With a partner: 1. Highlight questions 2. Categorize questions as to type Exploring, Probing, Generating

25. Questioning to Support Enactment of the Practices TypeDescription Exploring mathematical meanings and/or relationships Points to underlying mathematical relationships and meaning Makes links between mathematical ideas and representations Probing, getting students to explain their thinking Asks student to articulate, elaborate, or clarify ideas Generating discussionSolicits contributions from other members of the class “These questions do not take over the thinking for the students by providing too much information or by ‘giving away’ the answer or a quick route to the answer. Rather, they scaffold thinking to enable students to think harder and more deeply about the ideas at hand.”p.62

26. Examining the Explore Questions With a small group Identify Explore questions in the vignette For each Explore question, consider: What is the purpose of the question? What is the underlying mathematical idea or enduring understanding? Whole group share out

27. Reflection Individually, consider: Where does questioning fit within the Five Practices? Turn and Talk

28. Reflecting on Implementation in your Tutorial Individually, take a few minutes to write: What have you done to support the five practices in your class? What was the impact? Whole group: What new ideas did you get from the discussions?

29. Implications for your Work: How can you use today’s learning of strategies to help engage students in mathematical discussion within your tutorials?