supporting rigorous mathematics teaching and learning
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Supporting Rigorous Mathematics Teaching and Learning. The Instructional Tasks Matter: Analyzing the Demand of Instructional T asks. Tennessee Department of Education Elementary School Mathematics Grade 1. Rationale – Comparing Two Mathematical Tasks. - PowerPoint PPT PresentationTRANSCRIPT
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© 2013 UNIVERSITY OF PITTSBURGH LEARNING RESEARCH AND DEVELOPMENT CENTER
Supporting Rigorous Mathematics Teaching and Learning
Tennessee Department of EducationElementary School MathematicsGrade 1
The Instructional Tasks Matter:Analyzing the Demand of Instructional Tasks
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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 TaskMary, 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 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 + __ = 15What pattern do you see? ____________________________________________
Mary
Nick
Jean
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Nine Plus a Number TaskSolve each addition problem. Use the blocks when solving each problem.
9 + 5 = ___ 5 + 9 = ____ 10 + 4 = ___
Solve the problems below.
9 + 8 = ___ 7 + 7 = ___ 6 + 6 = ___ 8 + 8 = ___ 7 + 9 = ___ 6 + 9 = ___ 9 + 7 = ___ 8 + 6 = ___ 5 + 8 = ___ 6 + 7 = ___ 8 + 5 = ___ 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 1Operations 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 1Operations 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 1Operations 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 1Operations 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 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
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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 4 + (5 + 9)
9 + 8 10 + 8
8 + 6 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 DManipulatives/Tools available: Counters, cubes, grid paper, base ten blocksWrite 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 EManipulatives/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 = ____ 16 – 8 = ___ 12 – 8 = ____ 18 – 9 = ___ 14 – 7 = ____ 16 – 8 = ___ 12 – 8 = ____ 18 – 9 = ___ 14 – 7 = ____ 16 – 8 = ___ 12 – 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 = ___ 83 – 37 = ___65 – 25 = 40 83 – 33 = 50
40 – 1 = 39 50 – 4 = 46 45 – 26 = ____ 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.
348 349 345 384 336
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Place Value Task JManipulatives/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
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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