supporting teachers in planning, teaching, and reflecting on mathematics instruction

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER

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

Tennessee Department of EducationMathematicsGrade 1

Modifying Tasks to Increase the Cognitive Demand

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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

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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.

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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.

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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.

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

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

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Revisiting the Characteristics of Cognitively Demanding Tasks

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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?

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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?

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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.

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Linking to Research/Literature:The QUASAR Project

• Low-Level Tasks – Memorization – Procedures Without Connections

• High-Level Tasks– Procedures With Connections – Doing Mathematics

The Mathematical Task Analysis Guide

Stein and Smith, 1998; Stein, Smith, Henningsen, & Silver, 2000 and 2008.

Common Core State Standards for Mathematics: Grade 1

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Operations and Algebraic Thinking 1.OA

Represent and solve problems involving addition and subtraction.

1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

1.OA.A.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

Common Core State Standards for Mathematics: Grade 1

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Operations and Algebraic Thinking 1.OA

Understand and apply properties of operations and the relationship between addition and subtraction.

1.OA.B.3 Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

1.OA.B.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

Common Core State Standards for Mathematics: Grade 1

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Operations and Algebraic Thinking 1.OA

Add and subtract within 20.

1.OA.C.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

Common Core State Standards for Mathematics: Grade 1

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Operations and Algebraic Thinking 1.OA

Work with addition and subtraction equations.

1.OA.D.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = ? – 3, 6 + 6 = ?.

Common Core State Standards, 2010, p. 15, NGA Center/CCSSO

The CCSS for Mathematical Practice1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

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Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO

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Analyzing Modified Textbook Tasks

Linking to Research/Literature: The QUASAR Project

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

The Mathematical Tasks Framework

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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? 21

Original Task: Missing Addend Problem

Modified from EnVision Mathematics, Tennessee Version, Grade 1, 2012

Modified Task: Missing Addend Problem

23Modified from EnVision Mathematics, Tennessee Version, Grade 1, 2012

You have two trays of cookies. Each tray fits 10 cookies.

Explain how the two situations are related.

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.

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.

Original Task: Using a Benchmark of Ten

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Modified from EnVision Mathematics, Tennessee Version, Grade 1, 2012

7. 8 10

+ 3 +

11

8.9 10

+ 3 +

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

The CCSS for Mathematical Practice1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

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Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO

© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER

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?

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Strategies for Modifying Textbook TasksIncreasing 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.

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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).

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Giving It a Go:Modifying Textbook Tasks

to Increase the Cognitive Demand of Tasks

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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.

EnVision Mathematics, Tennessee Version, Grade 1, 2012

EnVision Mathematics, Tennessee Version, Grade 1, 2012

EnVision Mathematics, Tennessee Version, Grade 1, 2012

EnVision Mathematics, Tennessee Version, Grade 1, 2012

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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?

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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.

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The Cognitive Demand of Tasks

Does the demand of the task matter?

What are you now wondering about with respect to the task demands?