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

1. Supporting Teachers in Planning, Teaching, and Reflecting on Mathematics Instruction Modifying Tasks to Increase the Cognitive Demand Tennessee Department of Education Elementary School Mathematics Grade 2

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 the features of tasks that make them high-level or low-level tasks, teachers will be prepared to select or modify tasks that create opportunities for students to engage with more tasks that are high-level tasks. 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; and • modify tasks to increase the cognitive demand of the tasks.

5. Mathematical Tasks:A Critical Starting Point for Instruction All tasks are not created equal−different tasks require different levels and kinds of student thinking. 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.

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

7. Mathematical Tasks:A Critical Starting Point for Instruction 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

8. Revisiting the Characteristics of Cognitively Demanding Tasks

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

10. Task #1: A Place Value Task Identify the place value for each of the underlined digits. 351 76 4,789

11. Task #2: What is Changing? Solve each equation. 234 + 10 = ___ 379 + 10 = ___ 389 + 10 = ___ 399 + 10 = ___ 489 + 10 = ___ 499 + 10 = ___ When ten is added to each of the numbers above, how is the sum changing from one equation to the next? Sometimes the tens place changes and sometimes the hundreds place change when ten is added to the number. Why does this happen and when does it happen? Look at the number 2,399. Which numbers will change when ten is added to this number and why?

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, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press.

14. The CCSS for Mathematics: Grade 2 Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

15. The CCSS for Mathematical Practice Common Core State Standards, NGA Center/CCSSO, 2010 • 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.

16. Analyzing Modified Textbook Tasks

17. 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

18. Comparing the Cognitive Demand of Tasks Compare the original versions of the tasks with the modified versions of the tasks. The mathematical objective, the mathematical understandings, and the prior knowledge that students should have acquired are listed for each lesson. 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?

19. Original Guided Practice Task Modified from EnVision, Tennessee version, Grade 1, 2012

20. Modified Guided Practice Task • Solve the problem two different ways. Write two equations showing the two different approaches that can be taken when solving this problem. • Alex bought 16 green peppers. He bought 11 red peppers. How many peppers did Alex buy in all? • Alex uses 14 of the peppers. How many peppers are left? • Compare the two different ways of solving the problem and explain how the two ways differ from each other. Explain what remains the same with each way of solving the problem.

21. Original Addition Story Problem TaskThis page appears at the end of several days of work on solving two-digit addition problems in which students are directed to regroup when the amount is over ten. (Page 230) Journal: Write an addition story about something you collect. Add. Regroup if you need to. Answers will vary. Modified from EnVision, Tennessee version, Grade 1, 2012

22. Modified Addition Story Problem TaskEliminate the picture and the problem template. Write a story problem for the expression 43 + 38. Make sure you ask a question. Solve the problem. Journal: Write an addition story about something you collect. Add. Regroup if you need to. Answers will vary. Modified from EnVision, Tennessee version, Grade 1, 2012

23. Original Relating Addition and Subtraction Task Model/Demonstrate: How can you use addition to check subtraction? Draw an arrow from the difference (30) to the place where the first addend of your vertical addition problem will be. Help children to understand that you write the difference as the first addend and the number being subtracted as the second addend. Guide children to complete the addition. What is the sum? (50) Is it the same as the number that you subtracted from in the first problem? (Yes) That means you subtracted correctly. Use a part-part-whole model like the one on page 271, with 50 as the whole and 20 and 30 as the parts to make sure that children see the relationship between the pairs of problems. What addition problem does this model? (20 + 30 = 50) Does it also model the subtraction problem 50 – 30 = 20? (Yes, the parts and the whole are the same.) Modified from EnVision, Tennessee version, Grade 1, 2012

24. Modified Relating Addition and Subtraction Task • Explain how you can use addition to solve 5 – 2. ________________________________________ • Explain how you can use addition to solve 50 – 20.__________________________________________ • Solve the subtraction problem 31 – 13 = _____________. • Solve the subtraction problem 31 – 13 by using addition. • Explain how addition can be used to solve the subtraction problem. • Show 31 – 13 on the part-part-whole. Do you know the parts or the whole in this problem?

25. 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?

26. The CCSS for Mathematics: Grade 2 Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

27. The CCSS for Mathematics: Grade 2 Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

28. 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?

29. 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.

30. 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).

31. The CCSS for Mathematical Practice Common Core State Standards, NGA Center/CCSSO, 2010 • 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.

32. Giving it a Go: Modifying Textbook Tasks to Increase the Cognitive Demand of Tasks

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

34. Gallery Walk • Post the modified tasks. • Circulate, analyzing the modified tasks. On a “Stickie-Note,” describe ways in which the tasks were modified and the benefit 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.

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