Transcript Accountable Talk in the teaching profession: Building

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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LEARNING RESEARCH AND DEVELOPMENT CENTER
Rationale
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.
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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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LEARNING RESEARCH AND DEVELOPMENT CENTER
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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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LEARNING RESEARCH AND DEVELOPMENT CENTER
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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 standardsbased mathematics instruction: A casebook for professional development, p. 3.
New York: Teachers College Press.
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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
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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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LEARNING RESEARCH AND DEVELOPMENT CENTER
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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
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The Mathematical Task Analysis Guide
Stein and Smith, 1998; Stein, Smith, Henningsen, & Silver, 2000 and 2008.
Common Core State Standards for
Mathematics: Grade 1
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
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Common Core State Standards for
Mathematics: Grade 1
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
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Common Core State Standards for
Mathematics: Grade 1
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
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Common Core State Standards for
Mathematics: Grade 1
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
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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.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
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Analyzing Modified Textbook
Tasks
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Linking to Research/Literature:
The QUASAR Project
The Mathematical Tasks Framework
TASKS
TASKS
TASKS
as they
appear in
curricular/
instructional
materials
as set up by
the teachers
as
implemented
by students
Student
Learning
Stein, Smith, Henningsen, & Silver, 2000, p. 4
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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?
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Original Task: Missing Addend Problem
Modified from EnVision Mathematics, Tennessee Version, Grade 1, 2012
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
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Original Task: Using a Benchmark of Ten
7.
8.
8
+
3
10
+
9
+
3
10
+
11
Modified from EnVision Mathematics, Tennessee Version, Grade 1, 2012
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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
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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.
Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO
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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 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.
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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.
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EnVision Mathematics, Tennessee Version, Grade 1, 2012
EnVision Mathematics, Tennessee Version, Grade 1, 2012
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EnVision Mathematics, Tennessee Version, Grade 1, 2012
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EnVision Mathematics, Tennessee Version, Grade 1, 2012 35
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?
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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