Focusing on Challenging Mathematical Tasks: A Strategy for Improving Teaching and Learning

1. Focusing on Challenging Mathematical Tasks:A Strategy for Improving Teaching and Learning Peg Smith University of Pittsburgh February 15, 2007 Teachers’ Development Group Leadership Seminar on Mathematics Professional Development

2. Overview • Argue for focusing on mathematical tasks • Discuss the components of the task-based model for Professional Development • Discuss the role of tools in the model • Present evidence of teacher learning

3. Why Focus on Tasks? • Classroom instruction is generally organized and orchestrated around mathematical tasks • The tasks with which students engage determines what they learn about mathematics and how they learn it • The inability to enact challenging tasks well is what distinguished teaching in the U. S. from teaching in other countries that had better student performance on TIMSS

4. The Importance of 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 and Briars, 1995

5. The Importance of 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

6. The Importance of 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

7. Task-Focused Activities • Distinguishing between high- and low-level mathematics tasks • Solvinghigh-level mathematical tasks • Analyzinghigh-level mathematics tasks and work produced by students on these tasks • Maintaining the cognitive demands of high-level tasks during instruction

8. Task-Focused Activities • Distinguishing between high- and low-level tasks • Develop teachers’ capacity to determine the kind and level of thinking required to solve a particular mathematics task • Comparing pairs of tasks that focus on the same mathematics content but different with respect to the thinking demands • Analyzing a set of tasks that differ with respect to their cognitive demands and task features (e.g., require an explanation, utilize a diagram, provide tools such as calculators)

9. Distinguishing • Martha’s Carpeting Task • The Fencing Task

10. Martha’s Carpeting Task Martha was recarpeting her bedroom which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase? Stein, Smith, Henningsen, & Silver, 2000, p. 1

11. The Fencing Task Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen in which to keep the rabbits. • If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be? • How long would each of the sides of the pen be if they had only 16 feet of fencing? • How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it. Stein, Smith, Henningsen, & Silver, 2000, p. 2

12. Both require prior knowledge of area Area problems Way in which the area formula is used The need to generalize The amount of thinking and reasoning required The number of ways the problem can be solved The range of ways to enter the problem Comparing Two Tasks

13. Importance of Distinguishing • Low-Level Tasks • High-Level Tasks

14. Importance of Distinguishing • Low-Level Tasks • memorization • procedures without connections • High-Level Tasks • procedures with connections • doing mathematics

15. Importance of Distinguishing • 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)

16. Task-Focused Activities • Solving high-level mathematical tasks • Develop teachers’ understanding of mathematical ideas, processes, and tools that support learning • Solving challenging mathematical tasks that focus on developing understanding of key ideas, that use a range of tools, that feature different representational forms, and that connect procedures with meaning • Multiplying binomials using algebra tiles • Using rectangular grids to make sense of the connections between fractions, decimals, and percents • Exploring visual patterns and determining the connection between the physical and symbolic representations

17. Task-Focused Activities • Analyzing high-level mathematics tasks and work produced by students on these tasks • Develop teachers’ ability to identify the mathematical potential of a task and to determine what students’ responses communicate about their current mathematical understandings • Specifying what mathematical ideas could be learned from engaging with a particular task and what standards could be addressed • Analyzing students’ written responses and determining what students appear to understand about mathematics, and developing questions to assess and advance student thinking

18. Task-Focused Activities • Maintaining the cognitive demands of high-level tasks during instruction • Develop teachers’ awareness of how high-level tasks “play out” in the classroom and the factors that support and inhibit students engagement at a high level • Solving tasks and reflecting on and discussing how the facilitator supported their learning • Analyzing narrative cases and identifying what the teacher featured in the case did to support or inhibit her students’ learning of mathematics

19. Consider and Discuss How do you help teachers apply the ideas that emerge in professional development sessions in their own classrooms?

20. Task-Focused Activities Distinguishing Solving Maintaining Using TOOLS Analyzing Practice-based Professional Development Classroom Teaching

21. Reflecting On Practice Planning Cycle of Teaching Classroom Practice Using

22. Frameworks and Tools • provide a focus for professional development; • bring coherence to PD across sessions; • provide a shared language for talking about teaching and learning; and • bridge the professional development and K-12 classroom environments.

23. Frameworks and Tools • Framework • Tools for: • Analyzing Cognitive Demands (purple) • Identifying Classroom Influences (gold) • Planning Lessons (salmon) • Conferencing after a Lesson • Talking about and Sharing Teaching Experiences

24. The Mathematics Task 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

25. The Mathematics Task 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

26. The Mathematics Task 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. The Mathematics Task 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. The Mathematics Task 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. SummaryTask-Based Activities in which Teachers Engage • Characterize mathematical tasks based on their cognitive demands • Solve, analyze,and discuss cognitively challenging mathematical tasks • Analyze narrative cases w/r/t the MTF and identify the factors that appear to support/inhibit students’ learning • Use narrative cases to generate issues which teachers can explore in their own practice • Plan, teach, and reflect on lessons based on cognitively challenging tasks

30. Summary • A task-based approach to professional development provides a focus for work with teachers • By influencing the tasks that teachers use during instruction and the ways in which they enact them, there is an opportunity to impact student learning • Tools help the ideas that emerge in professional development “travel” to the classroom and back • Tools help support enactment of tasks in teachers’ own classrooms and foster conservations about teaching between teachers

31. Pulling it All Together

32. Is there any evidence that suggests that a task-focused approach to professional development has an impact on teachers’ classroom practices? YES

33. ESP:Setting the Context • Workshop focused on selecting and enacting high level tasks • Intended for practicing mathematics teachers 7-12 with 3 or more years experience • Use practice-based materials with the Purple book (Implementing Standards-Based Reform) as the centerpiece • Assignments link to teachers’ practices in very specific ways and use tools (e.g., TTLP) to generalize ideas

34. ESP:Data Collected • Pre- and post task sorts • Pre- and post interviews • Videotapes of all workshop sessions • Teachers notebooks and all assignments • All artifacts generated during the course • Task packets • Student work packets • Classroom observations

35. ESP:What Teachers Learned • Significant increase in teachers’ ability to distinguish between high and low level tasks following their participation in the workshops • Significant increase from fall to spring the number of high-level tasks used per teacher over the 5-day data collection • Significant increase in the percent of high-level tasks that were maintained during implementation from fall to spring • The PD in which teachers engaged appeared to have an influence on teachers' practice, particularly with respect to their ability to use and maintain high-level tasks in their own classrooms. • Teachers who showed the most growth over time were those who consistently made connections between the PD and their own classroom practice. Boston, 2006

36. For More Information pegs@pitt.edu

37. Results from Data AnalysisThe Task Sort • Pre- and Post-Workshop Task Sort: • During the first (October) and last (May) session in the workshop, teachers were asked to: • classify a set of tasks as High-Level or Low-Level; • justify their classification of each task; and • provide a set of criteria for High-Level and Low-Level tasks.

38. Results from Data AnalysisThe Task Sort • Pre- and Post-Workshop Task Sort: • Highly significant increase between the pre- and post-workshop task sort scores • Teachers in the workshop scored significantly higher on the post-workshop task sort than a contrast group of secondary mathematics teachers who did not participate in the workshop • Teachers improved their ability to distinguish between high and low level tasksfollowing their participation in the workshop.

39. Results from Data AnalysisThe Task Sort • Pre- and Post-Workshop Task Sort: • Improvements in teachers’ justifications and criteria for high and low level tasks • No “inconsistent” criteria identified on the post • i.e., “Difficult is High-Level” or “Use of a diagram is Low-Level” • Criteria and justifications closely connected to our work in solving and distinguishingtasks in the sessions

40. Results from Data AnalysisTasks Used in Teachers’ Classroom • Teachers were asked to submit tasks used over 1-week period in Fall, Winter, and Spring. • In each data collection, 5 main instructional tasks scored using IQA Academic Rigor in Mathematics rubric Boston & Wolfe, 2004; Matsumura et al., 2004 • Score of 1 or 2 = Low-level cognitive demands as described on Task Analysis Guide • Score of 3 or 4 = High-level cognitive demands as described on Task Analysis guide Stein, Smith, Silver & Henningsen, 2000

41. Results from Data AnalysisTasks Used in Teachers’ Classroom Comparisons of Tasks Used From Fall to Spring: • Significant increases in Task Scores • Significant increase in overall % of H-L tasks used • Significant increase in the number of high-level tasks used per teacher over the 5-day data collection

42. Results from Data AnalysisStudent Work Collected from Teachers’ Classroom • Collections of Student work: • Teachers submitted 3 class-sets of student work in the Fall, Winter, and Spring. • Student-work was scored using the IQA rubric for student work Implementation • Scale of 1 to 4 • Score levels based on Task Analysis Guide • Low-Level < 2 • High-Level > 3

43. Results from Data AnalysisStudent Work Collected from Teachers’ Classroom • Comparisons of Student-Work Implementation Scores from Fall to Spring: • Significant increase in mean scores • Significant increase in number of high-level student work implementations • Significantly less occurrences of decline of high-level cognitive demands

44. Results from Data AnalysisStudent Work Collected from Teachers’ Classroom In all data collections, • Implementation scores were lower than task scores. • The number of high-level implementations per teacher is lower than the number of high-level tasks used per teacher. These findings indicate a persistent trend of decline in the level of cognitive demands.

45. Results from Data AnalysisStudent Work Collected from Teachers’ Classroom Was the decline in the level of cognitive demands significant? • Fall and Winter: Implementation scores were highly significantly lower than task scores • Spring: Implementation scores were not significantly lower than Task scores • Increase in # of high-level implementations per teacher • Increase in % of high-level tasks that were maintained during implementation

46. Results from Data AnalysisObservation of Teachers’ Classroom • Eleven teachers participated in 1 classroom observation per data collection • Marginally significant increase in lesson implementation from Fall to Spring • Teachers significantly more likely to maintain high-level cognitive demands during implementation in Spring than in Fall or Winter. • In Spring, teachers implemented tasks at a significantly higher level than teachers in a contrast group who did not participate in the workshop

47. Connecting Professional Development to Teacher’s Practice • Focusing on a particular factor (light gold sheet) they wanted to work on and use the factor as a lens for reflecting on classroom instruction • Expectations • Teachers would teach a lesson based on a high-level mathematical task of their choice • Teachers would select several pieces of work produced by students during the lesson that they felt accurately reflect the lesson • Teachers would bring blinded copies of the student work to share

48. Rick Carson’sCase Story • Teacher: • 9 years of experience • First time student work has been shared in this format • School Setting: • 10th grade • Integrated curriculum in a probability unit

49. Case Stories:Storytelling through Student Work • Storyteller will distribute a complete set of student work to each team member without comment. • The team members will individually review the work in silence. • 5 minutes allowed

50. Case Stories:Storytelling through Student Work • The team should share what they saw in their review of the student work. • Only factual statements can be made. Do not share your evaluation of the work, or statements of personal preference. • Start comments with, “I noticed that…” • The storyteller is quietly listening and making note of statements. • 5 minutes is allowed