Transcript PowerPoint Presentation - Supporting Student Learning of

Supporting Rigorous Mathematics
Teaching and Learning
Engaging In and Analyzing Teaching and
Learning
Tennessee Department of Education
Elementary School Mathematics
Grade 1
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
Rationale
Asking a student to understand something means asking a
teacher to assess whether the student has understood it.
But what does mathematical understanding look like? One
hallmark of mathematical understanding is the ability to
justify, in a way appropriate to the student’s mathematical
maturity, why a particular mathematical statement is true mathematical understanding and procedural skill are equally
important, and both are assessable using mathematical
tasks of sufficient richness.
Common Core State Standards for Mathematics, 2010
By engaging in a task, teachers will have the opportunity to
consider the potential of the task and engagement in the
task for helping learners develop the facility for expressing a
relationship between quantities in different representational
forms, and for making connections between those forms.
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Session Goals
Participants will:
• develop a shared understanding of teaching and
learning; and
• deepen content and pedagogical knowledge of
mathematics as it relates to the Common Core State
Standards (CCSS) for Mathematics.
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
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Overview of Activities
Participants will:
• engage in a lesson; and
• reflect on learning in relationship to the CCSS.
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
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Looking Over the Standards
• Look over the first grade standards.
• We will return to the standards at the end of the
lesson and consider what it means to say:
 In what ways did we have opportunities to
learn about the concepts underlying the
standard?
 What gets “counted” as learning?
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
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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
1.OA.A.2
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.
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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Table 1: Common Addition and Subtraction
Situations
Common Core State Standards, 2010, p. 88, 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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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
task below.
9 + 8 = ___
9 + 7 = ___
9 + 6 = ___
8 + 9 = ___
7 + 9 = ___
6 + 9 = ___
10 + 7 = ___
10 + __ = 16
10 + __ = 15
What patterns do you see? ____________________________________________
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
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The Structure and Routines of a Lesson
Set
Task
SetUp
Upthe
of the
Task
The Explore Phase/Private Work Time
Generate Solutions
The Explore Phase/
Small-Group Problem Solving
1. Generate and Compare Solutions
2. Assess and Advance Student Learning
MONITOR: Teacher selects
examples for the Share, Discuss,
and Analyze Phase based on:
• Different solution paths to the
same task
• Different representations
• Errors
• Misconceptions
SHARE: Students explain their
methods, repeat others’ ideas,
put ideas into their own words,
add on to ideas and ask
for clarification.
REPEAT THE CYCLE FOR EACH
SOLUTION PATH
COMPARE: Students discuss
Share, Discuss, and Analyze Phase of the Lesson
1. Share and Model
2. Compare Solutions
3. Focus the Discussion on Key
Mathematical Ideas
4. Engage in a Quick Write
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
similarities and difference
between solution paths.
FOCUS: Discuss the meaning
of mathematical ideas in each
representation
REFLECT: Engage students
in a Quick Write or a discussion
of the process.
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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
task below.
9 + 8 = ___
9 + 7 = ___
9 + 6 = ___
8 + 9 = ___
7 + 9 = ___
6 + 9 = ___
10 + 7 = ___
10 + __ = 16
10 + __ = 15
What patterns do you see? ____________________________________________
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
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Solve the Task
(Private Think Time)
• Work privately on the Counting Houses task.
• Work with others at your table. Compare your
solution paths.
• Make observations about relationships that you
notice.
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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Expectations for Group Discussion
• Solution paths will be shared.
• Listen with the goals of:
– putting the ideas into your own words;
– adding on to the ideas of others;
– making connections between solution paths;
and
– asking questions about the ideas shared.
• The goal is to understand the mathematical
relationships and to make connections among the
various strategies used when solving the problems
in the task.
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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Discuss the Task
(Whole Group Discussion)
• What relationships did you discover between the
set of related expressions?
• How can one expression help you think about
another similar expression?
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LEARNING RESEARCH AND DEVELOPMENT CENTER
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Reflecting on Our Learning
• What supported your learning?
• Which of the supports listed will EL students benefit
from during instruction?
• Which CCSS for Mathematical Content did we
discuss?
• Which CCSS for Mathematical Practice did you use
when solving the task?
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
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Linking to Research/Literature
Connections between Representations
Pictures
Manipulative
Models
Written
Symbols
Real-world
Situations
Oral
Language
Adapted from Lesh, Post, & Behr, 1987
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Reflecting on Our Learning
• What supported your learning?
• Which of the supports listed will EL students benefit
from during instruction?
• Which CCSS for Mathematical Content did we
discuss?
• Which CCSS for Mathematical Practice did you use
when solving the task?
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
20
Reflecting on Our Learning
• What supported your learning?
• Which of the supports listed will EL students benefit
from during instruction?
• Which CCSS for Mathematical Content did we
discuss?
• Which CCSS for Mathematical Practice did you use
when solving the task?
© 2013 UNIVERSITY OF PITTSBURGH
LEARNING RESEARCH AND DEVELOPMENT CENTER
21
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
1.OA.A.2
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.
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
22
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
23
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
24
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
25
Which CCSS for Mathematical Practice
made it possible for us to learn?
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
26
Research Connection: Findings from
Tharp and Gallimore
• For teaching to have occurred - Teachers must “be
aware of the students’ ever-changing relationships to
the subject matter.”
• They [teachers] can assist because, while the learning
process is alive and unfolding, they see and feel the
student's progression through the zone, as well as the
stumbles and errors that call for support.
• For the development of thinking skills—the [students’]
ability to form, express, and exchange ideas in speech
and writing—the critical form of assisting learners is
dialogue—the questioning and sharing of ideas and
knowledge that happen in conversation.
Tharp & Gallimore, 1991
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