04.05.12 module 2: content and curriculum, k-8 mathematics
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04.0
5.12
Module 2:Content and Curriculum, K-8Mathematics
Agenda• Common Core State Standards Overview• Standards Progression: A Closer Look
– Number Systems:• K-5 Base Ten Operations • 3-5 Fractions• 6-8 Rational Numbers
• Mathematical Practices in Action• Closing
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Math Class Needs a Makeover
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Common Core State Standards“These standards define what students should understand and be able to do in their study of mathematics.”
National Governors Association Center for Best Practicesand Council of Chief State School Officers (2010)Common Core State Standards for Mathematics
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Two Types of Standards• Standards for Mathematical Practice
Eight standards common to all grade levels K-12
• Mathematical Content StandardsStandards that vary by grade level and conceptual category
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Mathematical Habits of Mind“Mathematical habits of mind allow one to bring coherence to many different parts of the curriculum… Building coherence in the development of mathematical ideas across grades is key to improving students’ mathematical learning.”
Cuoco, Goldenberg, & Mark (2010)Organizing a Curriculum around Mathematical Habits of Mind
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Pedagogical Shifts with Common Core State StandardsFocus• Identifies key ideas,
understandings, and skills for each grade or course
• Stresses deep learning, applying concepts and skills within the same grade or course
Coherence• Articulates a progression
of topics across grades• Connects ideas with
other ideas, enabling students to learn with understanding, develop skill proficiency, and solve problems
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Grade-Level Standards• Locate the CCSS for Mathematics handout.• Read the introduction for your grade level.• Highlight critical focus areas and mark as follows:
Expectations similar to CA 1997 Standards! New expectations? Questions I have regarding these expectations
• Be ready to discuss.
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Format of the OverviewDomains:
Overarching ideas that
connect topics across the
grades
Clusters:Illustrate the
progression of increasing
complexity from grade to grade
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
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Format of the StandardsDomain:
Cluster:
Standards: Define what
students should
know and be able to
do for each grade level or course.
© Copyright 2010. National Governors Association Center for Best Practicesand Council of Chief State School Officers. All rights reserved.
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Domains and Conceptual Categories
Distribution
Findell & Foughty (2011)College and Career-Readiness Through the
Common Core State Standards for Mathematics
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Example of Domain Flows
Elementary to High School
Calahan (2011)Delving Deeper into the Common Core
Operations & Algebraic Thinking
Expressions & Equations
Algebra
Number: Base Ten
Number & Quantity
Number:Fractions
Geometry Geometry Geometry
Measurement & Data
Statistics &Probability
Statistics &Probability
Functions Functions
The NumberSystem
K - 5 6 - 8 9 - 12
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Base Ten Place Value (K-5)
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Base Ten Place Value (K-5) Chart
On
es
Ten
s
Hu
nd
red
s
Th
ou
san
ds
x 10x 10x 10
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Base Ten Place Value (K-5) Chart
On
es
Ten
s
Hu
nd
red
s
Th
ou
san
ds
x 10x 10x 10 x 10
Ten
ths
x 10
Hu
nd
red
ths
x 10
Th
ou
san
dth
s
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Symmetry with
Respect to the Ones Place
On
es
Ten
s
Hu
nd
red
s
Th
ou
san
ds
Ten
ths
Hu
nd
red
ths
Th
ou
san
dth
s
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Base Ten Place Value (K-5):
On
es
Ten
s
Hu
nd
red
s
Th
ou
san
ds
Ten
ths
Hu
nd
red
ths
Th
ou
san
dth
s
100
30
10
3
Fractional Notation
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Base Ten Place Value (K-5):
On
es
Ten
s
Hu
nd
red
s
Th
ou
san
ds
Ten
ths
Hu
nd
red
ths
Th
ou
san
dth
s
Decimal Notation
1000 100 10 1 0.1 0.01 0.001
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Base Ten Place Value (K-5):
On
es
Ten
s
Hu
nd
red
s
Th
ou
san
ds
Ten
ths
Hu
nd
red
ths
Th
ou
san
dth
s
103 102 101 100
Exponential Notation
1000 = 103 0.001 = 1.0 x 10-3
10-1 10-2 10-3
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Part 1: Number and Operations
Base Ten Progression• Grade K: Work with numbers 11-19 to gain foundation for
place value. (K.NBT.1)• Grade 1: Extend the counting sequence; understand place
value. (1.NBT.2• Grade 2: Understand place value and use this
understanding and properties of operations to add and subtract. (2.NBT.1)
• Grade 3: Use place value understanding and properties of operations to perform multi-digit arithmetic. (
• Grade 4: Generalize place value understanding for multi-digit whole numbers; use this understanding to perform multi-digit arithmetic. (4.NBT.1 – 4.NBT.3)
• Grade 5: Understand the place value system and perform operations with multi-digit whole numbers and with decimals to hundredths. (5.NBT.1 through 5.NBT.4)
What part of the activity depends on the expectations of the K-2 domains?
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Connecting to Current PracticeThink/Pair/Share• How does the place value demonstration with money
highlight the progression of expectations in the K-5 Number and Operations in Base Ten Domain?
• How does the progression of standards within the Base Ten domain correspond to your current practice?
• What are some things you might do now to help students transition to the CCSS?
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Fractions Domain (3-5)Solve this problem on your own.
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Fraction Misconceptions
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Video Debrief
Fractions MisconceptionsThink/Pair/Share• What were this student’s strengths?• What were his misunderstandings regarding
the division of fractions?• How might you address these misunderstandings?
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“Represent a fraction a/b on a number line diagram by marking off the lengths 1/b from 0…”
Represent 2/3 on this number line:
Number and Operations: Fractions Progression (3-5)
Standard 3.NF.2a
0 1
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Number and Operations: Fractions Progression (3-5)
Standard 4.NF.2“Compare two fractions… by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.”
0 1½
64?
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Number and Operations: Fractions Progression (3-5)
Standard 5.NF.2“Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.”
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Determine whether the number sentence is correct without using common denominators. Explain how you know.
Number Systems Progression (6-8)
Standard 6.NS.6c“Find and position integers and other rational numbers on a horizontalor vertical number line diagram…”
-1 10
-1
1
0
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Number Systems Progression (6-8)
Standards 7.NS.5 and 8.NS.2“Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram…”
-1 4
Where does this belong?
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Number Systems Progression
Area = 1
Area = 424 11
3
Where does this belong?
11
24
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Number Systems Progression (6-8)
Standards 7.NS.5 and 8.NS.2“Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram…”
-1 4
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Common Core State Standards for
Mathematical Practice21st Century Skills• Communication• Collaboration• Creativity• Critical Thinking• Construction of
Knowledge
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Part 1: Standards for
Mathematical Practice in Action • Locate the MARS Tasks and Student
Work handout.• Complete the task on your own.• Once task is completed, discuss the prompts:
– What mathematical skills and understanding are needed to successfully complete the task?
– Which Standards for Mathematical Practice did you use to solve the task?
• Use the Standards of Mathematical Practice Look-For Tool to identify practices used.
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Part 2
Mathematical Practices in Action • Using the MARS Tasks and Student Work
handout, examine the student work provided for the task you completed.
• Determine which Standards for Mathematical Practice are represented in the student work.
• Compare your work with the student work. – How are the practices similar? – How are the practices different?
• More tasks available: – www.insidemathematics.org
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Connecting to Current PracticeThink/Pair/Share• How does the progression of standards within the
Number and Operations in Base Ten, Number and Operations in Fractions, and Number Systems domains correspond to your current practice?
• What are some things you might do now to help students transition to the CCSS?
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Next Steps• Think about what you learned today.• Decide on one thing you will do differently
to start transitioning to the CCSS.• Share your ideas with a partner.
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For Further Investigation• California’s Common Core State Standards
http://www.cde.ca.gov/ci/cc• Common Core State Standards Initiative
http://www.corestandards.org• Tools for the Common Core
(includes the Progressions)http://www.commoncoretools.wordpress.com
• Inside Mathematics (MARS Tasks & Resources) www.insidemathematics.org
• K-5 Math Teaching Resources www.k-5mathteachingresources.com
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