CCSSI FOR MATHEMATICS
“STANDARDS OF PRACTICE”
Collegial Conversations
GRADES 4 – 5
Today’s Goal
 To explore the Standards for Content and
Practice for Mathematics
 Begin to consider how these new CCSS
Standards are likely to impact your classroom
practices
What are the Common Core State
Standards?
 Aligned with college and work expectations
 Focused and coherent
 Included rigorous content and application of
knowledge through high-order skills
 Build upon strengths and lessons of current state
standards
 Internationally benchmarked so that all students are
prepared to succeed in our global economy and society
 Research and evidence based
 State led- coordinated by NGA Center and CCSSO
Focus
• Key ideas, understandings, and skills are
identified
• Deep learning of concepts is emphasized
– That is, time is spent on a topic and on
learning it well. This counters the “mile wide,
inch deep” criticism leveled at most current
U.S. standards.
Benefits for States and Districts
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Allows collaborative professional development based
on best practices
Allows development of common assessments and other
tools
Enables comparison of policies and achievement
across states and districts
Creates potential for collaborative groups to get more
economical mileage for:
– Curriculum development, assessment, and
professional development
Common Core Development
• Initially 48 states and three territories
signed on
• As of November 29, 2010, 42 states have
officially adopted
• Final Standards released June 2, 2010, at
www.corestandards.org
• Adoption required for Race to the Top
funds
Michigan’s Implementation Timeline
• Held October and November of 2010 rollouts
• District curricula and assessments that provide a
K-12 progression for meeting the MMC
requirements will require minimal adjustments to
meet CCSS
• Curriculum and assessment alignment in SY10-11
• Implementation SY11-12
• New assessment 2014-15 (Smarter Balanced
Assessment Consortium or SBAC – replaces MEAP
and MME)
Background
Responsibilities of States in the Consortium
Each State that is a member of the Consortium in 2014–
2015 also agrees to do the following:
 Adopt common achievement standards no later than the 2014–2015 school
year,
 Fully implement the Consortium summative assessment in grades 3–8 and
high school for both mathematics and English language arts no later than
the 2014–2015 school year,
 Adhere to the governance requirements,
 Agree to support the decisions of the Consortium,
 Agree to follow agreed-upon timelines,
 Be willing to participate in the decision-making process and, if a Governing
State, final decisions, and
 Identify and implement a plan to address barriers in State law, statute,
regulation, or policy to implementing the proposed assessment system and
address any such barriers prior to full implementation of the summative
assessment components of the system.
Technology Approach
SBAC Item Bank
• Partitioned into a secure item bank for
summative assessments and a non-secure
bank for the interim/benchmark assessments:
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Traditional selected-response items
Constructed-response items
Curriculum-embedded performance events
Technology-enhanced items (modeled after
assessments in use by the U.S. military, the
architecture licensure exam, and NAEP)
HOW TO READ THE GRADE LEVEL
STANDARDS
Domains are large groups of related
standards. Standards from different
domains may sometimes be closely
related. Look for the name with the code
number on it for a Domain.
Common Core Format
Clusters are groups of related standards.
Standards from different clusters may
sometimes be closely related, because
mathematics is a connected subject.
• Clusters appear inside domains.
Standards
define what
students
should be
Common
Core
Format
able to understand and be able to do –
part of a cluster.
They are content statements. An example content statement
is: “Find all factor pairs for a whole number in the range 1 –
100. Recognize that a whole number is a multiple of each of
its factors. Determine whether a given whole number in the
range 1 – 100 is a multiple of a given one-digit number.
Determine whether a given whole number in the range 1 –
100 is prime or composite,” 4.OA.4. The “OA” stands for
“Operations and Algebraic Thinking”. Please refer to page
three in your grade level appropriate Common Core document.
•Progressions of increasing complexity from grade to grade
Common Core - Clusters
• May appear in multiple grade levels in the K-8
Common Core. There is increasing development
as the grade levels progress
• What students should know and be able to do
at each grade level
• Reflect both mathematical understandings and
skills, which are equally important
Common Core Format
K-8
High School
Grade
Conceptual Category
Domain
Domain
Cluster
Cluster
Standards
(There are no preK Common Core Standards)
Standards
Format of K-8 Standards
Grade Level
Domain
Format of K-8 Standards
Standard
Cluster
Standard
Cluster
Mathematics » Grade 4 » Introduction
In Grade 4, instructional time should focus on three critical areas: (1) developing understanding and fluency with multi-digit
multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an
understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of
fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified based on their
properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.
1.
Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in
each place. They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place
value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient,
accurate, and generalizable methods to compute products of multi-digit whole numbers. Depending on the numbers
and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products. They
develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures
work based on place value and properties of operations; and use them to solve problems. Students apply their
understanding of models for division, place value, properties of operations, and the relationship of division to
multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients
involving multi-digit dividends. They select and accurately apply appropriate methods to estimate and mentally
calculate quotients, and interpret remainders based upon the context.
2.
Students develop understanding of fraction equivalence and operations with fractions. They recognize that two
different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent
fractions. Students extend previous understandings about how fractions are built from unit fractions, composing
fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the
meaning of multiplication to multiply a fraction by a whole number.
3.
Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing
two-dimensional shapes, students deepen their understanding of properties of two-dimensional objects and the use of
them to solve problems involving symmetry.
Mathematics » Grade 5 » Introduction
In Grade 5, instructional time should focus on three critical areas: (1) developing fluency with addition and subtraction of fractions, and
developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole
numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the
place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole
number and decimal operations; and (3) developing understanding of volume.
1. Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with
unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of
fractions, and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and
the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing
fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit
fractions.)
2. Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of
operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings
of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop
fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and
fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate
power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make
sense. They compute products and quotients of decimals to hundredths efficiently and accurately.
3. Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding
the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by
1-unit by 1-unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving
problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right
rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in
order to determine volumes to solve real world and mathematical problems.
K – 5 DOMAINS
Domains
Grade Levels
Counting and Cardinality
K only
Operations and Algebraic
Thinking
1-5
Number and Operations in
Base Ten
1-5
Number and Operations Fractions
3-5
Measurement and Data
1-5
Geometry
1-5
MIDDLE GRADES DOMAINS
Domains
Grade Levels
Ratio and Proportional
Relationships
6-7
The Number System
6-8
Expressions and Equations
6-8
Functions
8
Geometry
6-8
Statistics and Probability
6-8
Michigan GLCE vs. CCSS
Grade
Topic
Whole Number: Meaning
Whole Number: Operations
Measurement Units
Common Fractions
Equations & Formulas
Data Representation & Analysis
2-D Geometry: Basics
2-D Geometry: Polygons & Circles
Measurement: Perimeter, Area & Volume
Rounding & Significant Figures
Estimating Computations
Whole Numbers: Properties of Operations
Estimating Quantity & Size
Decimal Fractions
Relation of Common & Decimal Fractions
Properties of Common & Decimal Fractions
Percentages
Proportionality Concepts
Proportionality Problems
2-D Geometry: Coordinate Geometry
Geometry: Transformations
Negative Numbers, Integers, & Their Properties
Number Theory
Exponents, Roots & Radicals
Exponents and Orders of Magnitude
Measurement: Estimation & Errors
Constructions Using Straightedge & Compass
3-D Geometry
Geometry: Congruence & Similarity
Rational Numbers & Their Properties
Patterns, Relations & Functions
Proportionality: Slope & Trigonometry
Uncertainty & Probability
Real Numbers: Their Subsets & Properties
Topic intended in Michigan GLCE
Topic intended in CCSS
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MAJOR SHIFTS K - 5
Numeration and operation intensified, and introduced
earlier
•Early place value foundations in Kindergarten
•Regrouping as composing/decomposing in Grade 2
•Decimals to hundredths in Grade 4
All three types of measurement simultaneously
•Non-standard, English and metric
Emphasis on fractions as numbers
Emphasis on number line as visualization/structure
HOW IS THERE LESS?
•Backed off of algebraic patterns K – 5
•Backed off of statistics and probability in
K–5
•Delayed content like percent and ratios
and proportions
Fractions, Grades 3–6
 3. Develop an understanding of fractions as numbers.
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4. Extend understanding of fraction equivalence and ordering.
4. Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.
4. Understand decimal notation for fractions, and compare
decimal fractions.
5. Use equivalent fractions as a strategy to add and subtract
fractions.
5. Apply and extend previous understandings of multiplication
and division to multiply and divide fractions.
6. Apply and extend previous understandings of multiplication
and division to divide fractions by fractions.
THE REASON
WHY WE ARE HERE
TODAY!
CCSSM
Mathematical
Practices
The Common Core proposes a set of
Mathematical Practices that all teachers should
develop in their students. These practices are
similar to NCTM’s Mathematical Processes from
the Principles and Standards for School
Mathematics.
Design and Organization
Mathematical Practice – expertise students
should acquire: (Processes & proficiencies)
• NCTM five process standards:
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Problem solving
Reasoning and Proof
Communication
Connections
Representations
NCTM Process Standards and the
CCSS Mathematical Practice Standards
NCTM Process Standards
CCSS Mathematical Practices
Problem Solving
Make sense of problems and persevere
in solving them.
Use appropriate tools strategically
Reasoning and Proof
Reason abstractly and quantitatively.
Critique the reasoning of others.
Look for and express regularity in
repeated reasoning
Communication
Construct viable arguments
Connections
Attend to precision.
Look for and make use of structure
Representations
Model with mathematics.
Design and Organization
• Mathematical proficiency (National Research
Council’s report Adding It Up)
– Adaptive reasoning
– Strategic competence
– Conceptual understanding (comprehension of
mathematical concepts, operations, relations)
– Procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently, and appropriately)
– Productive disposition (ability to see mathematics as
sensible, useful, and worthwhile
Mathematics/Standards 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
Mathematics/Standards for Mathematical Practice
“The Standards for Mathematical Practice describe varieties of expertise that
mathematics educators at all levels should seek to develop in their students.
These practices rest on important “processes and proficiencies” with
longstanding importance in mathematics education.” CCSS, 2010
Standards for Mathematical Practice
• Carry across all grade levels
• Describe habits of a mathematically expert student
Standards for Mathematical Content
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K-8 presented by grade level
Organized into domains that progress over several grades
Grade introductions give 2-4 focal points at each grade level
High school standards presented by conceptual theme (Number &
Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability
Standards of Mathematical Practice
1. Choose a partner at your table and “Pair Share” the
Standards of Practice between you and your partner.
2. When you and your partner feel you understand
generally each of the standards, discuss the following
question:
What implications might the standards
of practice have on your classroom?
Transition from Current State Standards & Assessments
to New Common Core Standards
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Develop Awareness
Needs Assessment/Gap Analysis
Planning
Capacity Building
Job-embedded Professional Development
Transition Planning
Next Steps:
• Alignment of CCSS with curriculum
• Gap analysis (content and skills that vary from
the MEAP and MME)
• What instructional practices will facilitate the
transition?
• What new assessment strategies will be
needed?
• Professional development needs?
Transition Planning
• Gather in teams from your schools and discuss
– What are your immediate needs as a classroom teacher
being asked to implement the CCSS?
– What professional development is needed?
– What initial gaps come to mind and how do you address
these gaps?
– As a school team, look at the eight Standards for
Mathematical Practice. What do they look like? Sound
like? What will students need in order to implement them?
What will teachers need? What are the implications for
assessment and grading?
Select a recorder, time keeper and someone to report out for
your group.
Questions?
Please contact:
PUT YOUR
INFORMATION HERE!
Have a great day!