Common Core State Standards Initiative

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Common Core State Standards
Whats It All About?
Karen Kennedy, Ed.D. Mathematics Consultant
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A Balanced Mathematics Program
WHERE THE MATHEMATICS WORKS
Computational Procedural Skills
Problem Solving
DOING MATH
Conceptual Understanding
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Common Core State Standards Key Advances for
Mathematics

• Content Coherence
• Coherent progressions across grade levels.
• Balance of Concepts and Skills
• Content standards require both conceptual
  understanding and procedural fluency.
• Mathematical Practices
• Practices foster reasoning and sense-making in
  mathematics.

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Standards for Mathematical Practice
Common Core State Standards
CaCCSS pp.1-2, 47-48
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Common Core State Standards Components

• Standards for Mathematical Practice
• Carry across all grade levels.
• Describe habits of mind of a mathematically
  expert student.
• Describe varieties of expertise that educators
  should seek to develop in their students.
• Rest on important processes and proficiencies
  from the National Council of Teachers of
  Mathematics and National Research Council.
• Relate to mathematical proficiency as defined by
  the California Framework.

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Mathematical Proficiency
SMP
Framework

• Make sense of problems and persevere in solving
  them.
• Reason abstractly and quantitatively.
• Construct viable arguments and critique the
  reasoning of others.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated
  reasoning.

• Develop fluency in basic computational skills.
• Develop an understanding of mathematical
  concepts.
• Become mathematical problem solvers who can
  recognize and solve routine problems readily and
  can find ways to reach a solution or goal where
  no routine path is apparent.
• Communicate precisely about quantities, logical
  relationships, and unknown values through the use
  of signs, symbols, models, graphs, and
  mathematical terms.
• Reason mathematically by gathering data,
  analyzing evidence, and building arguments to
  support or refute hypotheses.
• Make connections among mathematical ideas and
  between mathematics and other disciplines.

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Mathematical Proficiency
SMP
Framework

• Make sense of problems and persevere in solving
  them.
• Reason abstractly and quantitatively.
• Construct viable arguments and critique the
  reasoning of others.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated
  reasoning.

• Develop fluency in basic computational skills.
• Develop an understanding of mathematical
  concepts.
• Become mathematical problem solvers who can
  recognize and solve routine problems readily and
  can find ways to reach a solution or goal where
  no routine path is apparent.
• Communicate precisely about quantities, logical
  relationships, and unknown values through the use
  of signs, symbols, models, graphs, and
  mathematical terms.
• Reason mathematically by gathering data,
  analyzing evidence, and building arguments to
  support or refute hypotheses.
• Make connections among mathematical ideas and
  between mathematics and other disciplines.

Not identified as an explicit practice in the
Framework.
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Standards for Mathematical Content

• Coherence
• Topics and performances are logical over time.
• Reflect hierarchical nature of the content.
• Are based upon learning progressions research
  on how students learn.
• Build upon learners schema (mental maps).
• Progressions
• Vertical articulation, i.e., domains progress
  over several grades.
• Organization
• K-8 standards are presented by grade level.
• High school standards are organized by
  conceptual themes (Number and Quantity,
  Algebra, Functions,
  Modeling, Geometry, Statistics and
  Probability).

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Common Core Content Standards Document Design
Grade Level Overviews
CaCCSS p. 12
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Common Core Content Standards Document Design
Grade Level Overviews
CaCCSS p. 12
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Standards for Mathematical Content

• Content standards define what students should
  understand and be able to do.
• Clusters are groups of related content standards.
• Domains are larger groups of related content
  standards that progress across grade levels.

Standard from Grade 3
CaCCSS p.14
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Standards for Mathematical Content

• K-5 Domains

• In grades K-5, students develop a solid
  foundation in whole numbers, addition,
  subtraction, multiplication, division, fractions,
  and decimals.

CaCCSS p. 3(K),6(1),9(2),12(3),16(4),20(5)
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Standards for Mathematical Content

• Middle Grades Domains

• With a strong foundation of content knowledge
  from grades K-5, middle school students are
  prepared for robust learning in geometry and
  statistics and probability.

CaCCSS p. 24(6),29(7),42(CC8)
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Standards for Mathematical Content

• K-7 Standards

• Focus on arithmetic and fluency with whole
  numbers at early grades.
• In grades K-5, students develop a solid
  foundation in whole numbers, addition,
  subtraction, multiplication, division, fractions,
  and decimals.
• Fluency with fractions and decimals.
• In grade three, students begin to develop an
  understanding of fractions as numbers and
  represent fractions on a number line diagram.
• Addition and subtraction of fractions are
  introduced in grade four and multiplication and
  division in grade five.
• The standards for grades six and seven extend
  work with fractions and develop concepts such as
  rational numbers and proportional relationships.

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Standards for Mathematical Content Fractions
Progression
Numbers and Operations NF to The Number
System NS
Domain

• The Number System 6.NS
• Apply and extend previous understandings of
  multiplication and division to divide fractions
  by fractions.
• Compute fluently with multi-digit numbers and
  find common factors and multiples.
• Apply and extend previous understandings of
  numbers to the system of rational numbers.

Cluster of Standards
CaCCSS p. 24
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Fractions Progression Numbers and Operations NF
to The Number System NS

• Grade 3
• Develop an understanding of fractions as numbers
  3.NF.
• Grade 4
• Extend understanding of fraction equivalence and
  ordering.
• Build fractions from unit fractions by applying
  and extending previous understandings of
  operations on whole numbers.
• Understand decimal notation for fractions and
  compare decimal fractions 4.NF.
• Grade 5
• Use equivalent fractions as a strategy to add and
  subtract fractions.
• Apply and extend previous understandings of
  multiplication and division to multiply and
  divide fractions 5.NF.
• Grade 6
• Apply and extend previous understandings of
  multiplication and division to divide fractions
  by fractions.
• Compute fluently with multi-digit numbers and
  find common factors and multiples 6.NS.

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Fractions Progression Numbers and Operations NF
to The Number System NS

• Grade 3
• Develop an understanding of fractions as numbers
  3.NF.
• Grade 4
• Extend understanding of fraction equivalence and
  ordering.
• Build fractions from unit fractions by applying
  and extending previous understandings of
  operations on whole numbers.
• Understand decimal notation for fractions and
  compare decimal fractions 4.NF.
• Grade 5
• Use equivalent fractions as a strategy to add and
  subtract fractions.
• Apply and extend previous understandings of
  multiplication and division to multiply and
  divide fractions 5.NF.
• Grade 6
• Apply and extend previous understandings of
  multiplication and division to divide fractions
  by fractions.
• Compute fluently with multi-digit numbers and
  find common factors and multiples 6.NS.

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Standards for Mathematical Content

• Grade 8 Standards

• Algebra readiness by grade eight or Algebra 1.
• The CCSS are consistent with Californias goal
  that all students succeed in Algebra 1.
• Students who master the content and skills
  through grade seven will be well-prepared for
  algebra in grade eight.
• With an expectation that all students continue
  their study of mathematics, the CCSS moves
  students forward with grade eight standards that
  prepare them for higher math (Algebra 1 and
  beyond).

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Standards for Mathematical Content

• SBEs Goal for Eighth Grade Students is Algebra 1

• After taking Algebra in eighth grade, students
  could, theoretically, take three or four years of
  high school math to assure their placement at a
  UC or CSU campus.
• Because not all students have the prerequisite
  skills for Algebra, the SBE adopted two sets of
  standards for grade 8.
• Existing CA Algebra 1 Standards
• Common Core Standards for eighth grade (published
  June 2, 2010), which spread the instruction of
  Algebra 1 over Grades 8 and 9 and introduce
  Geometry in Grade 8.

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Standards for Mathematical Content

• High School Standards Conceptual Categories

• Are arranged by conceptual cluster, not by
  course.

• Number and Quantity
• Algebra
• Functions

• Modeling
• Geometry
• Statistics and Probability

Eighth grade Algebra 1 standards are organized
around these themes as well.
Modeling Standard for Mathematical Practice is
emphasized at the high school level. Students
are expected to use mathematics to analyze
situations, understand them more fully, and make
decisions related to their everyday lives.
(pp.60-61)
CaCCSS p. 49(N),52(A),56(F),60(M),62(G),67(S)
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Standards for Mathematical Content
Theme
High School Conceptual Theme
Notation
Domain
Cluster of Standards
CaCCSS p. 63
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Standards for Mathematical Content

• Grades 9-12 Standards

• Real world applications using modeling.
• Throughout the standards, students apply the
  mathematics they have learned to solve problems
  that arise in everyday life, society, and the
  workplace.
• The Standards for Mathematical Practice
  emphasize this skill and provide specific
  suggestions for modeling real-world situations
  using mathematics.
• The high school standards include modeling
  standards throughout the other conceptual
  categories these standards are identified with a
  star (?) symbol.

Example CaCCSS p. 53(A)
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Standards for Mathematical Content

• High School Standards

• College and Career Readiness Preparation
• Standards specify the mathematics that all
  students should study in order to be college and
  career ready.
• Standards demand that students develop a depth
  of understanding and ability to apply mathematics
  outside of their classrooms.
• Standards are designated with a to indicate
  material that students should learn in order to
  take advanced courses such as calculus, advanced
  statistics, or discrete mathematics. (The
  standards may also be incorporated into courses
  that states require for all students.)

Example CaCCSS p. 53(A)
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Common Core Content Standards for California
Examples of Californias Additional 15
CA added standards to develop ideas in Grade 2
Operations and Algebraic Thinking

• Use repeated addition and counting by multiples
  to demonstrate multiplication.
• Use repeated subtraction and equal group sharing
  to demonstrate division.

High School Geometry Geometric Measurement and
Dimension
5. Determine how changes in dimensions affect the
perimeter, area, and volume of common geometric
figures and solids (CA Standard Geometry 11.0).
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Common Core State Standards for California

• Some Common Core Standards from Grade 8 were
  shifted to Grade 7, and some Common Core
  Standards from Grade 7 were shifted to Grade 6.
• CA added its own Algebra 1 Standards to achieve
  the states goal of Algebra 1 for all eighth
  grade students.
• Like the high school standards, Grade 8 Algebra 1
  standards are also organized by the high school
  conceptual themes.

Examples of Californias Additional 15
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Common Core State Standards for California

• CA also supplemented Common Core Standards in
  Algebra II, trigonometry, and geometry with key
  CA standards.
• CaCCSS include the addition of two courses
  Calculus and Advanced Statistics and Probability.
• Development of course descriptions will be done
  by CDE as part of their long-range implementation
  plan.

Examples of Californias Additional 15
Underlined standards in the CCSS document
represent Californias additional 15.
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Common Core Content Standards for California

• What Now?
• Recognize that there are more similarities than
  differences in the current state standards and
  CCSS.
• Implement a truly balanced math program as this
  will support the Standards for Mathematical
  Practice.
• Continue to use quality assessments to inform and
  drive effective instruction.
• Provide opportunities for teachers to collaborate
  and plan.

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Common Core Content Standards for California

• Resources
• California Department of Education
• http//www.cde.ca.gov/ci/cc/
• Sacramento County Office of Education
• http//scoe.net/castandards
• Common Core Standards
• http//corestandards.org
• Los Angeles County Office of Education
• http//cis.lacoe.edu/common_core/