Transcript No Slide Title

Supporting Rigorous Mathematics
Teaching and Learning
The Instructional Tasks Matter:
Analyzing the Demand of Instructional Tasks
Tennessee Department of Education
Elementary School Mathematics
Grade 1
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Rationale –
Comparing Two Mathematical Tasks
Tasks form the basis for students’ opportunities to learn
what mathematics is and how one does it, yet not all
tasks afford the same levels and opportunities for student
thinking. [They] are central to students’ learning, shaping
not only their opportunity to learn but also their view of
the subject matter.
Adding It Up, National Research Council, p. 335, 2001
By analyzing two tasks that are mathematically similar,
teachers will begin to differentiate between tasks that
require thinking and reasoning and those that require the
application of previously learned rules and procedures.
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Learning Goals and Activities
Participants will:
• compare mathematical tasks to determine the
demand of the tasks; and
• identify the Common Core State Standards (CCSS)
for Mathematical Content and the Standards for
Mathematical Practice addressed by each of the
tasks.
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Comparing the Cognitive Demand of
Two Mathematical Tasks
What are the similarities and differences between the
two tasks?
• Counting Houses Task
• Nine Plus a Number Task
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Counting Houses Task
Mary, Nick, and Jean are collecting donations to support homeless people.
Each student starts on a different path. The houses are side-by-side. Which student will visit
the most houses and how do you know? Write an equation that describes each part of the
students’ paths and explain which student visited the most houses and how you know.
Mary
Nick
Jean
Mary claims she sees a pattern in the Counting Houses Task that she can use to solve the
tasks below.
9 + 8 = ___
9 + 7 = ___
9 + 6 = ___
8 + 9 = ___
7 + 9 = ___
6 + 9 = ___
10 + 7 = ___
10 + __ = 16
10 + __ = 15
What pattern do you see? ____________________________________________
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Nine Plus a Number Task
Solve each addition problem. Use the blocks when solving each
problem.
9 + 5 = ___
5 + 9 = ____
10 + 4 = ___
Solve the problems below.
9 + 8 = ___
8 + 8 = ___
9 + 7 = ___
6 + 7 = ___
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7 + 7 = ___
7 + 9 = ___
8 + 6 = ___
8 + 5 = ___
6 + 6 = ___
6 + 9 = ___
5 + 8 = ___
8 + 5 = ___
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The Common Core State Standards
Examine the CCSS
− for Mathematical Content
− for Mathematical Practice
• Will first grade students have opportunities to use
the standards within the domain of Operations and
Algebraic Thinking?
• What kind of student engagement will be possible
with each task?
• Which Standards for Mathematical Practice will
students have opportunities to use?
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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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Comparing Two Mathematical Tasks
How do the differences between the Counting Houses
Task and the Nine Plus a Number Task impact
students’ opportunities to learn the Standards for
Mathematical Content and to use the Standards for
Mathematical Practice?
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Linking to Research/Literature:
The QUASAR Project
…Not all tasks are created equal - different tasks will
provoke 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
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Linking to Research/Literature
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
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Instructional Tasks: The Cognitive
Demand of Tasks Matters
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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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Linking to Research/Literature:
The QUASAR Project (continued)
• Low-Level Tasks
– Nine Plus a Number Task
• High-Level Tasks
– Counting Houses Task
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Linking to Research/Literature:
The QUASAR Project (continued)
• Low-Level Tasks
– Memorization
– Procedures Without Connections (e.g., Nine
Plus a Number Task)
• High-Level Tasks
– Doing Mathematics (e.g., Counting Houses
Task)
– Procedures With Connections
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The Mathematical Task Analysis Guide
Research has identified characteristics related to
each of the categories on the Mathematical Task
Analysis Guide (TAG).
How do the characteristics that we identified when
discussing the Counting Houses Task relate to those
on the TAG? Which characteristics describe the Nine
Plus a Number Task?
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The Cognitive Demand of Tasks
(Small Group Work)
•
Working individually, use the TAG to determine if
tasks A – L are high- or low-level tasks.
•
Identify and record the characteristics on the TAG
that best describe the cognitive demand of each
task.
•
Identify the CCSS for Mathematical Practice that
the written task requires students to use.
•
Share your categorization in pairs or trios. Be
prepared to justify your conclusions using the TAG
and the Standards for Mathematical Practice.
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Identifying High-level Tasks
(Whole Group Discussion)
Compare and contrast the four tasks.
Which of the four tasks are considered to have a high
level of cognitive demand and why?
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Relating the Cognitive Demand of Tasks
to the Standards for Mathematical
Practice
What relationships do you notice between the
cognitive demand of the written tasks and the
Standards for Mathematical Practice?
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Addition Task A
Determine the sum of each addition problem.
5 + 6 = ___
6 + 4 = ____
7 + 9 = ___
5 + 5 = ____
8 + 9 = ___
7 + 6 = ____
8 + 9 = ___
8 + 5 = ____
6 + 8 = ___
8 + 4 = ____
7 + 7 = ___
6 + 5 = ____
8 + 8 = ___
9 + 9 = ____
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Addition Task B
Tell if the scale will balance or tilt. If the scale does not
balance, write which side will tilt down and why and indicate
what would have to change to make the scale balance.
4+5+9
9+8
8+6
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4 + (5 + 9)
10 + 8
4+4+6
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Addition Task C
Use your solution to one problem to solve the second problem. The first
problem is given as an example.
6 + 6 = 12
6 + 7 = ___
Solve each set of problems by using the first problem to solve the second
problem.
7 + 7 = ____
8 + 8 = ____
5 + 5 = ___
7 + 8 = ____
8 + 9 = ____
5 + 6 = ___
The problems below work in the opposite way as the ones above. How can
you use the first problem to solve the second problem in each set of problems?
7 + 7 = ____
8 + 8 = ____
5 + 5 = ___
7 + 6 = ____
8 + 7 = ____
5 + 4 = ___
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Addition Task D
Manipulatives/Tools available: Counters, cubes, grid paper, base ten
blocks
Write a word problem for the number sentence.
8 + 6 = ___
Ask a question with your story problem so we know what we are
supposed to figure out.
Write a word problem for the number sentence.
14 – 5 = ___
Ask a question with your story problem so we know what we are
supposed to figure out.
Compare the two word problems. How do they differ from each other?
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Subtraction Task E
Manipulatives/Tools available: base ten blocks
Solve this problem in two different ways:
32 - 17
After each way, write about how you did it. Be sure to include:
• what materials, if any, you used to solve this problem;
• how you solved it; and
• an explanation of your thinking as you solved it.
First Way:
Second Way:
Adapted from Investigations in Number, Data, and Space, Dale Seymour, Menlo Park, CA, 1998.
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Subtraction Task F
Manipulatives/Tools available: base ten blocks
Use base ten blocks to model the situations below. Write a number sentence
for each problem.
1. Jim has 23 red pencils and 8 pencils are not sharpened. How many pencils
are sharpened?
2. Jamie has 48 cookies and some of them are chocolate and some are
vanilla. 26 cookies are chocolate. How many cookies are vanilla?
3. 32 cookies are in the box and you ate some of them. Now there are 26
cookies left. How many cookies did you eat?
Explain how the problems are similar to each other. Explain how the problems
differ from each other.
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Subtraction Task G
Manipulatives/Tools available: none
Solve the subtraction number sentences.
14 – 7 = ____
12 – 8 = ____
14 – 7 = ____
12 – 8 = ____
14 – 7 = ____
12 – 8 = ____
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16 – 8 = ___
18 – 9 = ___
16 – 8 = ___
18 – 9 = ___
16 – 8 = ___
18 – 9 = ___
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Subtraction Task H
Manipulatives/Tools available: none
Study the strategy of rounding the subtrahend in order to
subtract all of the ones available and doing mental
subtraction.
65 – 26 = ___
65 – 25 = 40
40 – 1 = 39
45 – 26 = ____
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83 – 37 = ___
83 – 33 = 50
50 – 4 = 46
62 – 28 = ____
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Place Value Task I
Manipulatives/Tools available: base ten blocks
Order the amounts from smallest to largest.
234
243
284
254
233
Order the amounts from smallest to largest.
348
349
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345
384
336
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Place Value Task J
Manipulatives/Tools available: base ten blocks
Identify the number of tens possible in each of the amounts.
236__________________
368__________________
589__________________
2389_________________
3458_________________
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Place Value Task K
Manipulatives/Tools available: base ten blocks
Study each set of addition problems and write about what
keeps changing with each sum.
345 + 10 = ______________
355 + 10 = ______________
365 + 10 = ______________
375 + 10 = ______________
Which numbers stayed the same? Which numbers
changed? Explain why only one number kept changing.
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Place Value Task L
Manipulatives/Tools available: base ten blocks
Circle the number in the ones place in each of the numbers
below.
45
56
67
78
89
Circle the number in the tens place in each of the numbers
below.
45
345 567 678 689
Circle the number in the hundreds place in each of the
numbers below.
3,459
459
5,679
3,457
2,349
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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
36
Linking to Research/Literature:
The QUASAR Project
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, M. K. & Lane, S. (1996). Instructional tasks and the development of student capacity to think and
reason: An analysis of the relationship between teaching and learning in a reform mathematics project.
Educational Research and Evaluation, 2 (4), 50-80.
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Linking to Research/Literature
Tasks are central to students’ learning, shaping not only
their opportunity to learn but also their view of the subject
matter.
Adding It Up, National Research Council, p. 335, 2001
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