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Building a Deeper
Understanding
of the
Common Core
State Standards
For
Mathematics
April 2011
The Common Core State Standards
Initiative (CCSSI) is a state-led effort
coordinated by the National Governors
Association (NGA) and the Council of Chief
State School Officers (CCSSO).
www.corestandards.org/the-standards
Mission Statement
The Common Core State Standards provide a
consistent, clear understanding of what students are
expected to learn, so teachers and parents know what
they need to do to help them. The standards are
designed to be robust and relevant to the real world,
reflecting the knowledge and skills that our young
people need for success in college and careers. With
American students fully prepared for the future, our
communities will be best positioned to compete
successfully in the global economy.
- Common Core State Standards Initiative
Common Core State
Standards for Mathematics
Standards for
Mathematical
Practices
CCSS
Mathematics
Standards for
Mathematical
Content
Mathematical Understanding
What does it mean
for a student to
have mathematical
understanding?
Building from Past Work
The Mathematical Practices rest on
“important processes and proficiencies
with longstanding importance in
mathematics education” (CCSS, 2010, p. 6).
– National Council of Teachers of Mathematics
(NCTM, 2000) Process Standards
– National Research Council (NRC, 2001)
Strands of Mathematical Proficiency (Adding It
Up)
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.
Standards for
Mathematical Practice
“Acquiring proficiency…involves much
more than memorizing.”
(Adding It Up, National Research Council, 2001)
What do the Mathematical
Practices look like in action?
What do the Mathematical
Practices look like in action?
Connecting Mathematical
Practices to Content Standards
“The Standards for Mathematical Content
are a balanced combination of procedure
and understanding. Expectations that
begin with the word ‘understand’ are
often especially good opportunities to
connect the practices to the content.”
- CCSS, Mathematics, p. 8
Standards for
Mathematical Content
Domain
Content strand across grades:
Operations & Algebraic Thinking
Cluster
“Big Idea” that groups together
a set of related standards.
Standards
Statements that define what
students should understand and
be able to do at a grade level.
Standards for Mathematical
Content: K-8 Domains
K
1
2
3
4
5
6
7
8
Operations and
Algebraic
Thinking
Operations and
Algebraic
Thinking
Operations and
Algebraic
Thinking
Operations and
Algebraic
Thinking
Operations and
Algebraic
Thinking
Operations and
Algebraic
Thinking
Ratios and
Proportional
Relationships
Ratios and
Proportional
Relationships
Number and
Operations in
Base Ten
Number and
Operations in
Base Ten
Number and
Operations –
Base Ten
Number and
Operations –
Base Ten
Number and
Operations –
Base Ten
Number and
Operations –
Base Ten
The Number
System
The Number
System
The Number
System
Number and
Operations Fractions
Number and
Operations Fractions
Number and
Operations Fractions
Expressions
and Equations
Expressions
and Equations
Expressions
and Equations
Counting and
cardinality
Functions
Measurement
and Data
Measurement
and Data
Measurement
and Data
Measurement
and Data
Measurement
and Data
Measurement
and Data
Statistics and
Probability
Statistics and
Probability
Statistics and
Probability
Geometry
Geometry
Geometry
Geometry
Geometry
Geometry
Geometry
Geometry
Geometry
Standards for Mathematical
Content: Learning Progressions
“…the development of these Standards
began with research-based learning
progressions detailing what is known
today about how students’ mathematical
knowledge, skill, and understanding
develop over time.”
- CCSS Mathematics, p. 4
Standards for Mathematical
Content: Learning Progressions
K
1
2
3
4
5
6
7
8
Counting
and
Cardinality
Number and Operations in Base Ten
Number and Operations Fractions
Ratios and
Proportional
Relationships
The Number System
Expressions and Equations
Operations and Algebraic Thinking
Functions
Measurement and Data
Statistics and Probability
Geometry
Geometry
Ohio Department of Education
Standards for Mathematical
Content: Learning Progressions
Know from
memory all sums
of two one-digit
numbers.
Fluently add and
subtract within 5.
K.OA.5
Add and subtract
within 20,
demonstrating
fluency for addition
and subtraction
within 10.
1.OA.6
(basic addition facts)
2.OA.2
Arkansas Crosswalk
The results of the analysis comparing the
Common Core State Standards with the
Arkansas Curriculum Frameworks for
English language arts and mathematics
are available at:
http://ccssarkansas.pbworks.com/w/page/
32131061/CCSS-Arkansas
Design for Mathematics
Grade Level Narrative
Grade Level Overview
Mathematical Practice (in gray box)
CCSS – Grade level standards
Standards for Mathematical
Content: Kindergarten
Standards for Mathematical
Content: Kindergarten
Focus Areas
(located in Narrative)
Domains and
Clusters
(located in Overview)
Representing and
comparing whole
numbers, initially
with sets of
objects
Describing
shapes and space
Standards
Mathematical
Practice
(located in Overview)
Grade K Overview
Standards for Mathematical
Content: Kindergarten
Focus Areas
(located in Narrative)
Domains and
Clusters
(located in Overview)
Representing and
comparing whole
numbers, initially
with sets of
objects
Counting &
Cardinality
* Know number
names and the
count sequence.
Describing
Geometry
shapes and space
•Identify and
describe shapes.
Standards
Mathematical
Practice
(located in Overview)
Standards for Mathematical
Content: Kindergarten
Standards for Mathematical
Content: Kindergarten
Focus Areas
(located in Narrative)
Domains and
Clusters
Standards
(located in Overview)
(located in Overview)
Representing and Counting &
comparing whole
Cardinality
numbers, initially
* Know number
with sets of objects
names and the
count sequence.
1. Count to 100 by ones and by tens.
2. Count forward beginning from a
given number within the known
sequence (instead of having to
begin at 1).
3. Write numbers from 0 to 20.
Represent a number of objects
with a written numeral 0-20 (with 0
representing a count of no
objects).
Describing shapes
and space
1. Describe objects in the
environment using names of
shapes, and describe the relative
positions of these objects using
terms such as above, below,
beside, in front of, behind, and
next to.
2. Correctly name shapes regardless
of their orientations or overall size.
3. Identify shapes as twodimensional (lying in a plane,
“flat”) or three-dimensional
(“solid”).
Geometry
•Identify and
describe shapes.
Mathematical
Practice
Changes in Grades K-5
More
•
•
•
•
•
•
•
Numeration and operation intensified
as well as introduced earlier
Early place value foundations in grade
K
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
Less
• Backed off of algebraic
patterns K‐5
• Backed off stats and
probability in K‐5
• Delayed content like
percent and ratio and
proportion
Changes in Grades 6-8
• Ratio and Proportion
focused on in grade 6
– ratio, unit rates, converting
measurement, tables of
– values, graphing, missing
value problems
• Percents introduced
grade 6
• Statistics is introduced in
grade 6
– statistical variability
(measures of central
tendency, distributions,
interquartile range, mean
and absolute deviation,
data shape)
• Rational numbers in
grade 7
• One‐third of algebra for
all students is in grade 8
Time to Reflect
Common Core State
Standards for Mathematics
Standards for
Mathematical
Practices
CCSS
Mathematics
Standards for
Mathematical
Content
(Traditional HS
Pathway)
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.
CCSS – Math
Learning Progressions
K
1
2
3
4
5
6
7
8
HS
Counting
and
Cardinality
Number and Operations in Base Ten
Number and Operations Fractions
Number and
Quantity
Ratios and
Proportional
Relationships
The Number System
Expressions and Equations
Algebra
Operations and Algebraic Thinking
Functions
Functions
Statistics and Probability
Statistics
and
Probability
Geometry
Geometry
Measurement and Data
Geometry
Ohio Department of Education
Mathematics HS Conceptual
Categories and Domains
Conceptual
Categories
Algebra I
Domains
Geometry
Domains
Algebra II
Domains
Number and
Quantity
•The Real Number
System
Algebra
•Seeing Structure in
Expressions
•Seeing Structure in
Expressions
•Creating Equations
* Arithmetic with
Polynomials & Rational
Expressions
Creating Equations
* The Complex Number
System
* Quantities
* Reasoning with
Equations and
Inequalities
4th Year
Courses
Domains
•The Complex
Number System
* Vector Quantities
and Matrices
* Creating Equations
* Reasoning with
Equations and
Inequalities
Functions
* Interpreting Functions
* Interpreting Functions
* Building Functions
* Building Functions
* Linear, Quadratic, and
Exponential Models
* Linear, Quadratic, and
Exponential Models
* Trigonometric
Functions
* Interpreting
Functions
* Building Functions
* Trigonometric
Functions
CCSS Mathematics
Traditional
Algebra I
Geometry
Algebra II
Domains
Domains
Domains
Pathway
Statistics and
Conceptual
Categories
Probability
•Interpreting
Categorical and
Quantitative Data
* Conditional
Probability and the
Rules of Probability
* Using Probability to
Make Decisions
Geometry
* Congruence
* Similarity, Right
Triangles, and
Trigonometry
* Circles
* Expressing
Geometric Properties
with Equations
* Geometric
Measurement and
Dimension
* Modeling with
Geometry
•Interpreting Categorical
and Quantitative Data
4th Year
Courses
Domains
* Using Probability to
Make Decisions
* Making Inferences and
Justifying Conclusions
* Using Probability to
Make Decisions
* Expressing
Geometric
Properties with
Equations
* Geometric
Measurement and
Dimension
Standards for Mathematical
Content: Learning Progressions
Apply and extend
previous
understandings of
numbers to the
system of rational
numbers.
(6th Grade)
Know that there are
numbers that not
rational, and
approximate them by
rational numbers.
(8th Grade)
Perform arithmetic
operations with
complex numbers.
(Algebra II)
http://ccssarkansas.pbworks.com/w/page/32131061/CCSS-Arkansas
The results of the analysis comparing the
Common Core State Standards with the
Arkansas Curriculum Frameworks for
English language arts and mathematics
are available at:
http://ccssarkansas.pbworks.com/w/page/
32131061/CCSS-Arkansas
Mathematics High School
Conceptual Categories
•
•
•
•
Number and Quantity –N
Algebra –A
Functions –F
Modeling – (across all conceptual categories)
Modeling standards are distributed under the five major headings
and are indicated with a () symbol.
• Geometry –G
• Statistics and Probability –S
CCSS Mathematics:
Design for Mathematics
Grade Level Narrative
Grade Level Overview
Mathematical Practice (in gray box)
CCSS – Grade level standards
Model Course Pathways
for Mathematics
Courses in higher level mathematics: Precalculus, Calculus (upon completion of Precalculus),
Advanced Statistics, Discrete Mathematics, Advanced Quantitative Reasoning, or other
courses to be designed at a later date, such as additional career technical courses.
Algebra II
Mathematics IIIs III
Geometry
Mathematics II
Algebra I
Mathematics I
Pathway A
Traditional in U.S.
Pathway B
International Integrated approach
(typical outside of U.S.)
CCSS Traditional Pathway
CCSS Traditional Pathway
CCSS Traditional Pathway
CCSS Traditional Pathway
High School
• All students must master topics
traditionally from Algebra II or beyond
– simple periodic functions
– polynomials,
– radicals
– more probability and statistics (correlation,
random processes)
– introduced to mathematical modeling
“Standards and assessments are only
the foundation upon which states will
construct high-quality curriculum,
professional development, and all the
other pieces that will support teachers
preparing to teach to these new
standards and students learning at
higher levels.”
- Arne Duncan, Education Secretary, September 2010
Resources
• ADE
http://ccssarkansas.pbworks.com/w/page/32131061/CCSS-Arkansas
http://arkansased.org/educators/curriculum/common_core.html
• CCSS website
http://www.corestandards.org/
• Achieve, Inc.
http://www.achieve.org/achievingcommoncore
• Education Service Centers’ state CCSS resource site
http://web.me.com/acaciatc/UACC/Intro.html