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Supporting Teachers in Planning, Teaching, and Reflecting on Mathematics Instruction

Supporting Teachers in Planning, Teaching, and Reflecting on Mathematics Instruction. Modifying Tasks to Increase the Cognitive Demand . Tennessee Department of Education Mathematics Grade 1.

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Supporting Teachers in Planning, Teaching, and Reflecting on Mathematics Instruction

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  1. Supporting Teachers in Planning, Teaching, and Reflecting on Mathematics Instruction Modifying Tasks to Increase the Cognitive Demand Tennessee Department of Education Mathematics Grade 1

  2. 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 By determining the cognitive demands of tasks and being cognizant of those features of tasks that contribute to their cognitive demand, teachers will be able to create opportunities for students to engage in rigorous mathematics learning. Rationale

  3. Session Goals Participants will: • deepen understanding of the cognitive demand of a task; • analyze a set of original and modified tasks to learn strategies for increasing the cognitive demand of a task; and • recognize how increasing the cognitive demand of a task gives students access to the Common Core State Standards (CCSS) for Mathematical Practice.

  4. Overview of Activities Participants will: • discuss and compare the cognitive demand of mathematical tasks; • identify strategies for modifying tasks to increase their cognitive demand; and • modify tasks to increase the cognitive demand of the tasks.

  5. Mathematical Tasks:A Critical Starting Point for Instruction Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 3. New York: Teachers College Press. All tasks are not created equal−different tasks require different levels and kinds of student thinking.

  6. Mathematical Tasks:A Critical Starting Point for Instruction Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human, 1997 The level and kind of thinking in which students engage determines what they will learn.

  7. Mathematical Tasks:A Critical Starting Point for Instruction Stein & Lane, 1996 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.

  8. Revisiting the Characteristics of Cognitively Demanding Tasks

  9. Comparing the Cognitive Demand of Two Tasks Compare the two tasks in your participant handout. How are the tasks similar? How are the tasks different?

  10. Task #1: Comparing Cookies John has 2 chocolate cookies and he gets 4 more vanilla cookies. How many cookies does he have altogether? Jan has 4 chocolate cookies and she gets 2 more vanilla cookies. How many cookies does she have altogether? Write an equation and draw a picture for showing John and Jan’s cookies. Who has more cookies and how do you know?

  11. Task #2: John and Jan’s Cookies John has 2 chocolate cookies and he gets 4 more vanilla cookies. How many cookies does he have altogether? Jan has 5 vanilla cookies and she gets 2 more chocolate cookies. How many cookies does she have altogether? Write an equation and draw a picture for each problem.

  12. Linking to Research/Literature:The QUASAR Project • Low-Level Tasks • Memorization • Procedures Without Connections • High-Level Tasks • Procedures With Connections • Doing Mathematics

  13. The Mathematical Task Analysis Guide Stein and Smith, 1998; Stein, Smith, Henningsen, & Silver, 2000 and 2008.

  14. Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

  15. Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

  16. Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

  17. Common Core State Standards for Mathematics: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

  18. The CCSS for Mathematical Practice Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO • 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.

  19. Analyzing Modified Textbook Tasks

  20. Linking to Research/Literature: 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

  21. Comparing the Cognitive Demand of Tasks Compare the original version of a task with the modified version of the task. For each, determine: • How are the modified tasks the same and how are they different from the original? • In what ways was the original task modified, and for what purpose? • What is the “value added” by making the modification to the original task? • Which CCSS for Mathematical Practice will students use when solving each task? • Which CCSS for Mathematical Content are the foci of each task?

  22. Original Task: Missing Addend Problem Modified from EnVision Mathematics, Tennessee Version, Grade 1, 2012

  23. Modified Task: Missing Addend Problem You have two trays of cookies. Each tray fits 10 cookies. • Here is the first tray. How many cookies are needed on the tray to make 10 cookies? Write an equation to describe the cookies on the tray. If there was one less cookie on the second tray than the first tray, then how many new cookies would be needed to fill it? Write an equation to describe the cookies on the tray. Explain how the two situations are related. Modified from EnVision Mathematics, Tennessee Version, Grade 1, 2012

  24. Original Task: Using a Benchmark of Ten 7. 810 + 3 + 11 8. 9 10 + 3 + Modified from EnVision Mathematics, Tennessee Version, Grade 1, 2012

  25. Modified Task: Using a Benchmark of Ten to Make Problems Easier Show each number sentence with counters. What is the sum of 10 + 4? What is the sum of 10 + 6? What is the sum of 10 + 8? If you know 10 + 4 = 14, then how can this help you think about 9 + 4? Explain how the solution to one problem can be used to think of the solution to the other problem. 10 + 4 = ______ 9 + 4 = ______ If you know 10 + 6 = 16, then how can this help you think about 8 + 6? Explain how the solution to one problem can be used to think of the solution to the other problem. 10 + 6 = ______ 8 + 6 = ______ Modified from EnVision Mathematics, Tennessee Version, Grade 1, 2012

  26. The CCSS for Mathematical Practice Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO • 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.

  27. Strategies for Modifying Tasks Compare your list of task modifications with the list of task modification strategies identified by others. How is your list similar? Different?

  28. Strategies for Modifying Textbook Tasks Increasing the cognitive demands of tasks. • Ask students to create real-world stories for “naked number” problems. • Include a prompt that asks students to represent the information another way (with a picture, in a table, a graph, an equation, with a context). • Include a prompt that requires students to make a generalization. • Use a task “out of sequence” before students have memorized a rule or have practiced a procedure that can be routinely applied. • Eliminate components of the task that provide too much scaffolding.

  29. Strategies for Modifying Textbook Tasks (Continued) Increasing the cognitive demands of tasks. • Adapt a task so as to provide more opportunities for students to think and reason – let students figure things out for themselves. • Create a prompt that asks students to write about the meaning of the mathematics concept. • Add a prompt that asks students to make note of a pattern or to make a mathematical conjecture and to test their conjecture.  • Include a prompt that requires students to compare solution paths or mathematical relationships and write about the relationship between strategies or concepts. • Select numbers carefully so students are more inclined to note relationships between quantities (e.g., two tables can be used to think about the solutions to the four, six, or eight tables).

  30. Giving It a Go: Modifying Textbook Tasks to Increase the Cognitive Demand of Tasks

  31. Your Turn to Modify Tasks • Form groups of no more than three people. • Determine the current demand of the task. • Discuss briefly important NEW mathematical concepts, processes, or relationships you want students to uncover by the textbook page. Consult the CCSS. • Modify the textbook task by using one or more of the Textbook Modification Strategies. • Post your modified task for others to analyze and offer comments.

  32. EnVision Mathematics, Tennessee Version, Grade 1, 2012

  33. EnVision Mathematics, Tennessee Version, Grade 1, 2012

  34. EnVision Mathematics, Tennessee Version, Grade 1, 2012

  35. EnVision Mathematics, Tennessee Version, Grade 1, 2012

  36. Consider… What can you do if you want students to develop the capacity to think, reason, and problem solve, but your textbook doesn’t have many high-level, cognitively demanding tasks?

  37. Gallery Walk • Post the modified tasks. • Circulate, analyzing the modified tasks as you do so. On a Sticky-Note, describe ways in which the task was modified and the benefit of the modification to students. • If the task was not modified to increase the cognitive demand of the task, then ask a wondering about a way the task might be modified.

  38. The Cognitive Demand of Tasks Does the demand of the task matter? What are you now wondering about with respect to the task demands?

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