CCSSI FOR MATHEMATICS
“STANDARDS OF PRACTICE”
Collegial Conversations
HIGH SCHOOL
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 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: “Use the structure of an expression to identify ways to
rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus
recognizing it as a difference of squares that can be factored
as (x2 – y2)(x2 + y2)”.
•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
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
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.
Statistics and Probability, Grade 6
Develop understanding of statistical variability
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Recognize a statistical question as one that anticipates variability in the
data related to the question and accounts for it in the answers. For
example, “How old am I?” is not a statistical question, but “How old are
the students in my school?” is a statistical question because one
anticipates variability in students’ ages.
Understand that a set of data collected to answer a statistical question
has a distribution which can be described by its center, spread, and
overall shape.
Recognize that a measure of center for a numerical data set summarizes
all of its values with a single number, while a measure of variation
describes how its values vary with a single number.
Algebra, Grade 8
Graded ramp up to Algebra in Grade 8
• Properties of operations, similarity, ratio and proportional
relationships, rational number system.
Focus on linear equations and functions in Grade 8
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Expressions and Equations
– Work with radicals and integer exponents.
– Understand the connections between proportional relationships, lines, and
linear equations.
– Analyze and solve linear equations and pairs of simultaneous linear
equations.
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Functions
– Define, evaluate, and compare functions.
– Use functions to model relationships between quantities.
High School
Conceptual themes in high school
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Number and Quantity
Algebra
Functions
Modeling
Geometry
Statistics and Probability
College and career readiness threshold
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(+) standards indicate material beyond the threshold; can
be in courses required for all students.
Format of High School
Domain
Cluster
Standard
Format of High School Standards
Regular
Standard
Modeling
STEM
Common Core - Domain
• Overarching “big ideas” that connect topics
across the grades
• Descriptions of the mathematical content to
be learned, elaborated through clusters and
standards
Common Core - Clusters
• May appear in multiple grade levels with
increasing developmental standards as the
grade levels progress
• Indicate 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 - Standards
• Content statements
• Progressions of increasing complexity from
grade to grade
– In high school, this may occur over the course
of one year or through several years
HS Pathways
1.) Traditional (US) – 2 Algebra, Geometry and Data, probability
and statistics included in each course
2.) International (integrated) three courses including number ,
algebra, geometry, probability and statistics each year
3.) Compacted version of traditional – grade 7/8 and algebra
completed by end of 8th grade
4.) Compacted integrated model, allowing students to reach
Calculus or other college level courses
High School Pathways
• The CCSS Model Pathways are NOT required. The two
sequences are examples, not mandates
• Two models that organize the CCSS into coherent,
rigorous courses
• Four years of mathematics:
– One course in each of the first two years
– Followed by two options for year 3 and a variety of relevant courses for
year 4
• Course descriptions
– Define what is covered in a course
– Are not prescriptions for the curriculum or pedagogy
High School Pathways
• Four years of mathematics:
– One course in each of the first two years
– Followed by two options for year three and a
variety of relevant courses for year four
• Course descriptions
– Define what is covered in a course
– Are not prescriptions for the curriculum or
pedagogy
High School Pathways
• Pathway A: Consists of two algebra courses and
a geometry course, with some data, probability
and statistics infused throughout each (traditional)
• Pathway B: Typically seen internationally that
consists of a sequence of 3 courses each of which
treats aspects of algebra, geometry and data,
probability, and statistics.
Interrelationships
• Algebra and Functions
– Expressions can define functions
– Determining the output of a function can
involve evaluating an expression
• Algebra and Geometry
– Algebraically describing geometric shapes
– Proving geometric theorems algebraically
Numbers and Quantity
• Extend the Real Numbers to include work
with rational exponents and study of the
properties of rational and irrational
numbers
• Use quantities and quantitative reasoning
to solve problems.
Numbers and Quantity
Required for higher math and/or STEM
• Compute with and use the Complex
Numbers, use the Complex Number plane
to represent numbers and operations
• Represent and use vectors
• Compute with matrices
• Use vector and matrices in modeling
Algebra and Functions
• Two separate conceptual categories
• Algebra category contains most of the
typical “symbol manipulation” standards
• Functions category is more conceptual
• The two categories are interrelated
Algebra
• Creating, reading, and manipulating
expressions
– Understanding the structure of expressions
– Includes operating with polynomials and
simplifying rational expressions
• Solving equations and inequalities
– Symbolically and graphically
Algebra
Required for higher math and/or STEM
• Expand a binomial using the Binomial
Theorem
• Represent a system of linear equations as
a matrix equation
• Find the inverse if it exists and use it to
solve a system of equations
Functions
• Understanding, interpreting, and building
functions
– Includes multiple representations
• Emphasis is on linear and exponential
models
• Extends trigonometric functions to
functions defined in the unit circle and
modeling periodic phenomena
Functions
Required for higher math and/or STEM
• Graph rational functions and identify zeros
and asymptotes
• Compose functions
• Prove the addition and subtraction
formulas for trigonometric functions and
use them to solve problems
Functions
Required for higher math and/or STEM
• Inverse functions
– Verify functions are inverses by composition
– Find inverse values from a graph or table
– Create an invertible function by restricting the
domain
– Use the inverse relationship between
exponents and logarithms and in
trigonometric functions
High School - Modeling
• Linking mathematics and statistics to everyday
life, work, etc.
• Process of choosing and using appropriate
mathematics and statistics
• Examples: pg 72
Modeling
Modeling has no specific domains, clusters
or standards. Modeling is included in the
other conceptual categories and marked
with a asterisk.
Modeling
Modeling links classroom mathematics and
statistics to everyday life, work, and
decision-making. Technology is valuable in
modeling.
A model can be very simple, such as writing
total cost as a product of unit price and
number bought, or using a geometric
shape to describe a physical object.
Modeling
• Planning a table tennis tournament for 7
players at a club with 4 tables, where each
player plays against each other player.
• Analyzing stopping distance for a car.
• Modeling savings account balance,
bacterial colony growth, or investment
growth.
Geometry, High School
Middle school foundations
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Hands-on experience with transformations.
Low tech (transparencies) or high tech (dynamic
geometry software).
High school rigor and applications
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Properties of rotations, reflections, translations, and
dilations are assumed, proofs start from there.
Connections with algebra and modeling
Geometry
• Understanding congruence
• Using similarity, right triangles, and
trigonometry to solve problems
Congruence, similarity, and symmetry are
approached through geometric
transformations
Geometry
• Circles
• Expressing geometric properties with
equations
– Includes proving theorems and describing
conic sections algebraically
• Geometric measurement and dimension
• Modeling with geometry
Geometry
Required for higher math and/or STEM
• Non-right triangle trigonometry
• Derive equations of hyperbolas and
ellipses given foci and directrices
• Give an informal argument using
Cavalieri’s Principal for the formulas for
the volume of solid figures
Statistics and Probability
• Analyze single a two variable data
• Understand the role of randomization in
experiments
• Make decisions, use inference and justify
conclusions from statistical studies
• Use the rules of probability
Key Advances
Focus and coherence
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Focus on key topics at each grade level.
Coherent progressions across grade levels.
Balance of concepts and skills
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Content standards require both conceptual
understanding and procedural fluency.
Mathematical practices
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Foster reasoning and sense-making in mathematics.
College and career readiness
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Level is ambitious but achievable.
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?
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Have a great day!