MathPLUS Content Day 1:

1. MathPLUSContent Day 1: Student-Centered Problem Solving Amy Jones Lewis Green River Regional Educational Cooperative

2. Day 1:Student-Centered Problem Solving • Solve, analyze, and discuss mathematical tasks. • Consider the effects of different levels of mathematical tasks on students’ achievement. • Work in a student-centered environment to complete a mathematical task.

3. Analyzing Mathematical Tasks “There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.” Lappan & Briars, 1995

4. What are Mathematical Tasks? We define mathematical tasks as a set of problems or a single complex problem the purpose of which is to focus students’ attention on a particular mathematical idea.

5. Why Focus on Mathematical Tasks? • Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it • Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information • The level and kind of thinking required by mathematical instructional tasks influences what students learn

6. Professional Practice Norms • Listening to and using others’ ideas. • Adopting a tentative stance toward practice -- wondering about the rationale/outcome for other’s professional decisions instead of espousing certainty and being judgmental about what the teacher was thinking or what you believed should have happened. • Backing up statements with evidence and providing reasoning. • Talking with respect yet engaging in critical analysis of teachers and students portrayed.

7. Comparing Two Mathematical Tasks

8. Comparing Two Mathematical Tasks Solve Two Tasks: • Martha’s Carpeting Task • The Fencing Task

9. Comparing Two Mathematical Tasks Martha’s Carpeting Task Martha was re-carpeting her bedroom, which was 15.25 feet long and 10.5 feet wide. How many square feet of carpeting will she need to purchase? Stein, Smith, Henningsen, & Silver, 2000, p. 1

10. Comparing Two Mathematical Tasks The (Modified) Fencing Task Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 2 Ms. Olson's 7th grade class at Roosevelt MS will raise rabbits for their spring science fair.  The class will use some portion of the school building as one of the sides of its rectangular rabbit pen and will use the fencing that was left over from the school play to enclose the other three sides of the pen. If Ms. Olson's class wants its rabbits to have as much room as possible, what would the dimensions of the pen be?  Try to organize your work so that someone else who reads it will understand.

11. How are Martha’s Carpeting Task and the Fencing Task the same and how are they different? (Consider your own experience in solving the tasks and the “mathematical possibilities” of the tasks.) Comparing Two Mathematical Tasks

13. Comparing Two Mathematical Tasks Martha’s Carpeting Task Martha was re-carpeting her bedroom which was 15.25 feet long and 10.5 feet wide. How many square feet of carpeting will she need to purchase? Stein, Smith, Henningsen, & Silver, 2000, p. 1

14. Comparing Two Mathematical Tasks The Fencing Task Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 2 Ms. Olson's 7th grade class at Roosevelt MS will raise rabbits for their spring science fair.  The class will use some portion of the school building as one of the sides of its rectangular rabbit pen and will use the fencing that was left over from the school play to enclose the other three sides of the pen. If Ms. Olson's class wants its rabbits to have as much room as possible, what would the dimensions of the pen be?  Try to organize your work so that someone else who reads it will understand.

15. How are Martha’s Carpeting Task and the Fencing Task the same and how are they different? (Consider your own experience in solving the tasks and the “mathematical possibilities” of the tasks.) Comparing Two Mathematical Tasks

16. Comparing Two Mathematical Tasks “Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” Stein, Smith, Henningsen, & Silver, 2000

17. Comparing Two Mathematical Tasks “The level and kind of thinking in which students engage determines what they will learn.” Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997

18. Characterizing Tasks

19. Characterizing Tasks Goals: Identify characteristics of high and low level mathematical tasks. • Martha’s Carpet: Low level • The Fencing Task: High level

20. Characterizing Tasks • Sort Tasks A – P into two categories: high level and low level. • Develop a list of criteria that describe in general the characteristics of low level and high level tasks.

22. Categorizing Tasks • Are all high-level tasks the same? [Is there an important difference between Tasks J and M?] • Are all low-level tasks the same? [Is there an important difference between Tasks E and O?]

23. Levels of Cognitive Demand & The Mathematical TasksFramework

24. Categorizing Tasks “If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks.” Stein & Lane, 1996

25. Linking to Research The QUASAR Project • Low-Level Tasks • Memorization • Procedures without Connections (e.g., Martha’s Carpeting Task) • High-Level Tasks • Procedures with Connections • Doing Mathematics (e.g., The Fencing Task)

26. Linking to Research The QUASAR Project The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

27. Linking to Research The QUASAR Project The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

28. Linking to Research The QUASAR Project The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

29. Linking to Research The QUASAR Project The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

30. Linking to Research The QUASAR Project The Mathematical Tasks Framework TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning Stein, Smith, Henningsen, & Silver, 2000, p. 4

31. Task Set-Up Task Implementation Student Learning High High High Low Low Low High Low Moderate Linking Task Level to Student Achievement Stein & Lane, 1996

32. Linking to Research The instructional tasks teachers’ select are crucial in helping students make connections and learn important mathematics concepts. Tasks that engage students in thinking about the defining characteristics of important mathematical concepts help students develop a deep understanding of core mathematical ideas that increase retention and transfer of knowledge to new situations.

33. Linking to Research “Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.” NCTM, 2000, p.16

34. Enacting Mathematical Tasks

35. The S-Pattern Task • Determine the number of tiles in the next 2 figures. • Describe the 20th figure in this pattern, including the number of tiles it contains and how they are arranged. Explain the reasoning that you used to determine this information.

36. The S-Pattern Task • Write an equation for the number of tiles needed for any figure in the pattern. • Explicitly connect your equation to the diagrams. • Generate as many different equations as you can for this relationship.

37. The S-Pattern Task • Make a graph that shows the relationship between the figure number and the number of tiles in the figure. • Share your solutions with your group.

38. The S-Pattern Task • Make a graph that shows the relationship between the figure number and the number of tiles in the figure. • Share your solutions with your group.

39. Group Poster • Show all the expressions generated by your group. • Explicitly connect your expressions to the diagrams.

40. Gallery Walk • One person from each group stays with the group’s poster to answer questions. • Rest of the group members view other posters. Look for: • The most common representations • The most unusual/surprising representations

41. A Graph of the Pattern The S-Pattern Task

42. Another Graph of the Pattern The S-Pattern Task

43. A Graph of All Values The S-Pattern Task

44. De-Briefing the S-Pattern Task • How would you characterize the level of this task: High or low cognitive demand? • What mathematical ideas are embedded in the task? • What makes this worthwhile mathematics?

45. De-Briefing the S-Pattern Task • What connections do you see to the Common Core Standards for Mathematical Practice?