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

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH

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

Modifying Tasks to Increase the Cognitive Demand

Tennessee Department of EducationElementary School MathematicsGrade 2

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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 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

LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH 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.


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.

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.


The level and kind of thinking in which students engage determines what they will learn.

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, & Human, 1997


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


Revisiting the Characteristics of Cognitively Demanding

Tasks


Comparing the Cognitive Demand of Two Tasks

• Compare the two tasks.

• How are the tasks similar? How are the tasks different?


Task #1: A Place Value Task

Identify the place value for each of the underlined digits.

a. 351

b. 76

c. 4,789


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?

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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, 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.

The CCSS for Mathematics: Grade 2Number and Operations in Base Ten 2.NBT

Understand place value.2.NBT.A.1 Understand that the three digits of a three-digit number represent

amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. 100 can be thought of as a bundle of ten tens—called a “hundred.”b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to

one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

Use place value understanding and properties of operations to add and subtract. 2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and

strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Common Core State Standards, 2010, p. 19, NGA Center/CCSSO

The CCSS for Mathematical Practice

1. 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.

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


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


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?

Original Guided Practice Task

Modified from EnVision, Tennessee version, Grade 1, 2012


Modified Guided Practice Task

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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.

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Original Addition Story Problem Task

This 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.

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Modified Addition Story Problem Task

Eliminate 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.

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.)

23Modified from EnVision, Tennessee version, Grade 1, 2012


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?

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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?

The CCSS for Mathematics: Grade 2Operations and Algebraic Thinking 2.OARepresent and solve problems involving addition and subtraction.

2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Add and subtract within 20.

2.OA.B.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.


The CCSS for Mathematics: Grade 2Operations and Algebraic Thinking 2.OAWork with equal groups of objects to gain foundations for multiplication.

2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

2.OA.C.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.



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?


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.


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).

The CCSS for Mathematical Practice

1. 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.

31Common Core State Standards, NGA Center/CCSSO, 2010


Giving it a Go:Modifying Textbook Tasks to

Increase the Cognitive Demand of Tasks


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.


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.


The Cognitive Demand of Tasks

• Does the demand of the task matter?

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

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

Documents

lowlevel tasks

highlevel tasks

features of tasks

types of tasks

cognitive demands of

set of modified tasks

development center

students opportunities