Selecting and Sequencing Students’ Solution Paths to Maximize Student Learning

1. Supporting Rigorous Mathematics Teaching and Learning Selecting and Sequencing Students’ Solution Paths to Maximize Student Learning Tennessee Department of Education High School Mathematics Geometry

2. Rationale Orchestrating discussions that build on students’ thinking places significant pedagogical demands on teachers and requires an extensive and interwoven network of knowledge. Teachers often feel that they should avoid telling students anything, but are not sure what they can do to encourage rigorous mathematical thinking and reasoning. (Stein, M.K., Engle, R., Smith, M., Hughes, E. 2008. Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell) In this session, we will focus on monitoring, selecting, and sequencing student work so you can assess and advance student learning during the Share, Discuss, and Analyze Phase of the lesson.

3. Session Goals Participants will: • learn what to monitor in student work when circulating during the Explore Phase of the lesson; • learn about guidelines or “rules of thumb” for selecting and sequencing student work that target essential understandings of the lesson; and • learn about focus questions that target the essential understandings.

4. Overview of Activities Participants will: • discuss the content standards and identify the related essential understandings of a lesson; • analyze samples of student work; • select and sequence student work for the Share, Discuss, and Analyze Phase of the lesson; • identify “rules of thumb” for selecting and sequencing student work; and • write focus questions that target essential understandings.

5. Midsegments Task A midsegment is a segment that connects the midpoints of two sides of a triangle. • Draw a triangle on the coordinate plane and label the coordinates of the vertices. Draw and label the coordinates of the midpoints of the sides and then draw the three midsegments.

6. Midsegments Task (continued) • Analyze the relationship between the midsegments and the sides of the triangles. What conjectures can you make? Support your conjectures with mathematical evidence and compare your findings with the findings of partners. • Marco’s group made the five conjectures listed below. • The three midsegments of a triangle always have the same length. • A midsegment is parallel to the side of the triangle that it does not intersect. • The three midsegments of a triangle form an acute triangle. • The length of the midsegment is half the length of the side of the triangle that it does not intersect. • The three midsegments create four congruent triangles. • Determine which conjectures are incorrect. For these conjectures, describe a triangle that Marco may have drawn for which this statement holds true. Then describe a counterexample for which the statement is false. • Determine which conjectures are true. Describe using words, diagrams, or symbols why the conjecture is a true statement.

7. The Task: Discussing Solution Paths • Solve the task in as many ways as you can. • Discuss the solution paths with colleagues at your table. • If only one solution path has been used, work together to create others. • Consider possible misconceptions or errors that we might see from students.

8. Linking the Standards to Student Solution Paths The task has been selected with specific Standards for Mathematical Content and Practice in mind. Where do you see the potential to work on these standards in the written task or the solution paths?

9. The CCSS for Mathematical ContentCCSS Conceptual Category – Geometry Common Core State Standards, 2010

10. Common Core Standards for Mathematical Practice What must happen in order for students to have opportunities to make use of the Standards for Mathematical Practice? • 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. Common Core State Standards, 2010

11. Pictures Manipulative Models Written Symbols Real-world Situations Oral & Written Language Five Representations of Mathematical Ideas Adapted from Van De Walle, 2004, p. 30

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

13. 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

14. Analyzing Student Work

15. Analyzing Student Work • Use the student work to further your understanding of the task. • Consider: • What do the students know? • How did the students solve the task? • How do their solution paths differ from each other?

16. Group A

17. Group B

18. Group C

19. Group D

20. Group E

21. Selecting and Sequencing Student Work

22. Monitoring Sheet

23. Selecting and Sequencing Student Work(Small Group Discussion) • Examine the students’ solution paths. • Determine which solution paths you want to share during the class discussion; keep track of your rationale for selecting the pieces of student work. • Determine the order in which work will be shared; keep track of your rationale for choosing a particular order for the sharing the work. Record the group’s decision on the chart in your participant handouts.

24. Standards and Essential Understandings • Using coordinates of a midsegment of a triangle justifies that the midsegment is parallel to the side that it does not intersect because the slope of the segment containing the midpoints is the same as the slope of the segment connecting the endpoints of the third side of the triangle. • Using coordinates of a midsegment of a triangle and the distance formula or Pythagorean Theorem justifies that the midsegment is half the length of the segment that it does not intersect. • Parallel lines have the same slope because they increase or decrease at the same rate per horizontal unit increment (they have the same rise per 1 run). Coordinate geometry can be used to support this understanding. • Coordinate Geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates.

25. Selecting and Sequencing Student Work(Small Group Discussion) • Each team should record their group’s sequence of solution paths on the chart. • Identify the student’s solution path that would be shared and discussed first, second, third, and so on. • Be prepared to justify your response.

26. Selecting and Sequencing Student Work(Group Discussion) • Listen to each group’s rationale for selecting and sequencing student work. • As you listen to the rationale, come up with a general “rule of thumb” that can be used to guide you when selecting and sequencing work for the Share, Discuss, and Analyze Phase of the lesson.

27. Reflecting On Essential Understandings Which of the sequences of student work were driven by the standards and essential understandings?

28. Reflecting on the Standards and the Essential Understandings • Using coordinates of a midsegment of a triangle justifies that the midsegment is parallel to the side that it does not intersect because the slope of the segment containing the midpoints is the same as the slope of the segment connecting the endpoints of the third side of the triangle. • Using coordinates of a midsegment of a triangle and the distance formula or Pythagorean Theorem justifies that the midsegment is half the length of the segment that it does not intersect. • Parallel lines have the same slope because they increase or decrease at the same rate per horizontal unit increment (they have the same rise per 1 run). Coordinate geometry can be used to support this understanding. • Coordinate Geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates.

29. Common Core Standards for Mathematical Practice • 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. Common Core State Standards, 2010

30. “Rules of Thumb” for Selecting and Sequencing Student Work What are the benefits of using the “rules of thumb” as a guide when selecting and sequencing student work for the Share, Discuss, and Analyze Phase of the lesson?

31. Pressing for Mathematical Understanding

32. Pressing for Mathematical Understanding Let’s focus on one piece of student work for the Share, Discuss, and Analyze Phase of the lesson. Assume that a student has explained the work and others in the class have repeated the ideas and asked questions. Now it is time to “FOCUS” the discussion on an important mathematical idea. What questions might you ask the class as a whole to focus the discussion? Write your questions on chart paper to be posted for a gallery walk.

33. Pressing for Mathematical Understanding • EU: Coordinate Geometry can be used to prove geometric theorems because it is possible to replace specific coordinates with variables to show that a relationship remains true regardless of the coordinates.

34. Pressing for Mathematical Understanding Do a gallery walk. Review other groups’ questions. • What are some similarities among the questions? • What are some differences between the questions?

35. Reflecting on Our Learning What have you learned today that you will think about and make use of next school year? Take a few minutes and jot your thoughts down.