Transcript Mathematical Tasks: The Study of Equivalence November 18

Supporting Rigorous Mathematics
Teaching and Learning
Engaging In and Analyzing Teaching and
Learning
Tennessee Department of Education
Elementary School Mathematics
Grade 2
© 2013 UNIVERSITY OF PITTSBURGH
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.
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Overview of Activities
Participants will:
• engage in a lesson; and
• reflect on learning in relationship to the CCSS.
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Looking Over the Standards
• Read the task. Before you solve the task, look over
the second grade standards for Operations and
Algebraic Thinking and Number Operations in Base
Ten.
• 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
standards?
 What gets “counted” as learning?
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Eduardo’s and Katrina’s Strategies
Eduardo solves the story problem below by using
subtraction. Show Eduardo’s equation.
You have 100 stickers. You put 48 of the
stickers into your sticker album. How many
stickers do you still need to put in an album?
When Eduardo compares his work to Katrina’s, he sees
that she used addition to solve the problem.
Explain to Eduardo why Katrina can use addition to
solve this problem.
© 2013 UNIVERSITY OF PITTSBURGH
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The CCSS for Mathematics: Grade 2
Operations and Algebraic Thinking
2.OA
Represent 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.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO
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The CCSS for Mathematics: Grade 2
Number 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).
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO
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The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten
2.NBT
Understand place value.
2.NBT.A.2 Count within 1000; skip-count by 5s, 10s, and 100s.
2.NBT.A.3 Read and write numbers to 1000 using base-ten numerals, number
names, and expanded form.
2.NBT.A.4 Compare two three-digit numbers based on meanings of the
hundreds, tens, and ones digits, using >, =, and < symbols to record
the results of comparisons.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO
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The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten
2.NBT
Use place value understanding and properties of operations to add and
subtract.
2.NBT.B.5 Fluently add and subtract within 100 using strategies based on
place value, properties of operations, and/or the relationship
between addition and subtraction.
2.NBT.B.6 Add up to four two-digit numbers using strategies based on place
value and properties of operations.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO
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The CCSS for Mathematics: Grade 2
Number and Operations in Base Ten
2.NBT
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.
2.NBT.B.8 Mentally add 10 or 100 to a given number 100–900, and mentally
subtract 10 or 100 from a given number 100–900.
2.NBT.B.9 Explain why addition and subtraction strategies work, using place
value and the properties of operations.
Common Core State Standards, 2010, p. 19, NGA Center/CCSSO
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Table 1: Common Addition and
Subtraction Situations
Common Core State Standards, 2010
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Engage In and Reflect on a Lesson
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The Structures and Routines of a Lesson
Set
Task
Set Up
Up the
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
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
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
similarities and difference between
solution paths.
FOCUS: Discuss the meaning of
mathematical ideas in each
representation.
REFLECT by engaging students in a
quick write or a discussion of the
process.
Solve the Task
(Private Think Time)
• Work privately on the Eduardo’s and Katrina’s
Strategies Task.
• Work with others at your table. Compare your
solution paths.
• Make observations about relationships that you
notice.
© 2013 UNIVERSITY OF PITTSBURGH
Eduardo’s and Katrina’s Strategies
Eduardo solves the story problem below by using
subtraction. Show Eduardo’s equation.
You have 100 stickers. You put 48 of the
stickers into your sticker album. How many
stickers do you still need to put in an album?
When Eduardo compares his work to Katrina’s, he sees
that she used addition to solve the problem.
Explain to Eduardo why Katrina can use addition to
solve this problem.
© 2013 UNIVERSITY OF PITTSBURGH
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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.
© 2013 UNIVERSITY OF PITTSBURGH
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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?
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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
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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
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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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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 students’
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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