An Introduction to
The Common Core State Standards
for Mathematics
Presented at the Hawaii Department of Education
Common Core State Standards Training Sessions
January – March 2011
Dewey Gottlieb
Educational Specialist for Mathematics
Office of Curriculum, Instruction and Student Support
Desired Outcomes
Increased understanding of the
• CCSS domain progressions
• Standards for Mathematical Practice
• CCSS-HCPS III Crosswalk documents
A Shift in Perspective
The CCSS for Mathematics compel a
change in the culture of traditional
mathematics classroom.
In the typical mathematics classroom
students are “too busy covering content”
to be engaged with mathematics.
CCSS for Mathematics
The emphasis on teaching and learning
The CCSS attempts to tell teachers when to slow
down and emphasize student understanding of
significant mathematical ideas.
“To say,
‘It was a good lesson but the students didn’t get it’,
is like saying,
‘The operation was a success, but the patient died.’”
(Lewis, 2002)
But what does “higher
standards” mean?
More topics?
No. The U.S. curriculum is already cluttered with too many topics
Teaching topics in earlier grades?
No. Analyses of the standards of high-performing countries suggest
otherwise.
In Singapore, division of fractions is a 6th grade expectation; in the
U.S. it is typically a 4th or 5th grade expectation.
In Japan, probability is introduced in the 7th grade; in the U.S., it can
be found anywhere throughout grades 3-6, depending on the state.
Standards are “high” for what students take
away from cumulative learning experiences
A Shift in Perspective
Current U.S. curricula (“mile wide, inch deep”)
coupled with high-stakes testing pressures
teachers to
“cover” at “pace”
turn the page regardless of student needs
However, the study of mathematics should not
be reduced to merely “a list of topics to cover”
Singapore preaches, “Teach less, learn more”
The Domains in the CCSS
• Groups of related standards are organized into
domains. Domains are overarching big ideas that
connect topics across grades.
• Standards from different domains may be closely
related, conveying an internal coherence among the
domains.
• In HCPS III, the benchmarks were organized into
“strands.” With the transition to CCSS, what we used
to call a strand will now be referred to as a domain.
Grades K – 5 Mathematics
Each grade level focuses upon particular DOMAINS
• Counting and Cardinality (K)
• Operations and Algebraic Thinking (K-5)
• Number and Operations in Base Ten (K-5)
• Number and Operations: Fractions (3-5)
• Measurement and Data (K-5)
• Geometry (K-5)
Grades 6 – 8 Mathematics
Each grade level focuses upon particular DOMAINS
• Ratios & Proportional Relationships (6 - 7)
• The Number System (6-8)
• Expressions and Equations (6-8)
• Functions (8)
• Geometry (6-8)
• Statistics and Probability (6-8)
High School Mathematics
Learning Expectations are Organized by
Conceptual Categories
• Number and Quantity
• Algebra
• Functions
• Modeling*
• Geometry
• Statistics and Probability
The Clusters in the CCSS
• Within a domain, smaller groups of related
standards are organized into clusters.
• The clusters help to inform teachers’ decisionmaking regarding their instructional design and
the learning and assessment opportunities
provided to students in the mathematics
classroom.
The Clusters in the CCSS
For example, in grade 4, the standards in the
Fractions domain are organized into three clusters:
• 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.
The Clusters in the CCSS
For example, in grade 6, the standards in “The
Number System” domain are organized into three
clusters:
• 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.
The Clusters in the CCSS
• We don’t want to simply teach to the standards
(i.e., as if checking off a to-do list).
• Rather, we want to teach THROUGH the standards,
using the specific learning expectations (i.e., the
standards) as building blocks for student
understanding of significant mathematical ideas
(i.e., the clusters) that will prepare them for the
mathematics they will be engaging with in
subsequent grades.
Getting to the Clusters: Teaching
THROUGH the Standards
Grade 2 Cluster: Use place value understanding and
properties of operations to add and subtract.
2.NBT.5: Fluently add and subtract within 100 using strategies based on place
value, properties of operations, and/or the relationship between addition
and subtraction.
2.NBT.6: Add up to four two-digit numbers using strategies based on place
value and properties of operations.
2.NBT.7: Add and subtract within 1000, using … strategies based on place
value, properties of operations, ….
2.NBT.8: Mentally add 10 or 100 to a given number 100–900, and mentally
subtract 10 or 100 from a given number 100–900.
2.NBT.9: Explain why addition and subtraction strategies work, ….
Getting to the Clusters: Teaching
THROUGH the Standards
Grade 6 Cluster: Apply and extend previous
understandings of numbers to the system of rational
numbers.
6.NS.5: Understand that positive and negative numbers are used
together to describe quantities having opposite directions or
values …
6.NS.6: Understand a rational number as a point on the number line …
6.NS.7: Understand ordering and absolute value of rational numbers …
6.NS.8: Solve real-world and mathematical problems by graphing points
in all four quadrants of the coordinate plane …
Learning Progressions:
Developing Expertise
The brain is a sense-making machine ... it does not store
what doesn’t make sense.
If we want to make information meaningful to students,
we have two options:
Find the prior experience they’ve had and hook the
new information to it.
OR
Create the experience with them (i.e., build a new
network).
CCSS for Mathematics
The “understand” standards
The “understand” standards interact with the “skills”
standards to support the development of expertise
Students who understand a concept can
(a) Explain it
(b) Demonstrate or illustrate it
(c) Use it into their own arguments and critique someone
else’s explanation of it
(d) Show an example of how to apply it (make connections to
other mathematical ideas and/or to real-world contexts)
Domain Progressions
Small Group Task:
• Select one domain that goes across grades
6-8.
• Review the clusters and standards for that
domain for each grade level.
• Discuss how the clusters and standards are
organized into learning progressions that
develop student expertise over time.
A Shift in Perspective
Too often, students view mathematics as a trivial
exercise because they are rarely given the
opportunity to grapple with and come to appreciate
the intrinsic complexity of the mathematics.
Despite our instincts and best intentions, we need to
stop “GPS-ing” our students to death.
Source: Shannon, A. (2010). Common Core: Two Perspectives on Tasks and Assessments.
Presentation at the Urban Mathematics Leadership Network Retreat, June 2010.
The 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)
The Standards for
Mathematical Practice
Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards.
A National Council of Supervisors of Mathematics webinar. November 2010.
The Standards for
Mathematical Practice
Conceptual
Understanding
Strategic
Competence
Adaptive
Reasoning
Productive
Disposition
Procedural
Fluency
Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards.
A National Council of Supervisors of Mathematics webinar. November 2010.
The Standards for
Mathematical Practice
“Encouraging these practices in students of all ages
should be as much a goal of the mathematics curriculum
as the learning of specific content” (CCSS, 2010).
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.
The Standards for
Mathematical Practice
The description of each Mathematical
Practice begins with the same first three
words:
Mathematically proficient students …
Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards.
A National Council of Supervisors of Mathematics webinar. November 2010.
The Standards for
Mathematical Practice
The Mathematical Practices “describe the
thinking processes, habits of mind and
dispositions that students need to develop a
deep, flexible, and enduring understanding of
mathematics; in this sense they are also a
means to an end.”
Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards.
A National Council of Supervisors of Mathematics webinar. November 2010.
The Standards for
Mathematical Practice
MP #1: Make sense of problems and
persevere in solving them.
Mathematically proficient students …
analyze givens, constraints, relationships
and goals … they monitor and evaluate
their progress and change course if
necessary … and continually ask
themselves, “Does this make sense?”
Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards.
A National Council of Supervisors of Mathematics webinar. November 2010.
Points of Intersection:
Content and Practices
MP #3: Construct viable arguments and
critique the reasoning of others
Consider the following subtraction algorithm:
• How could I demonstrate the idea that the
algorithm always works?
400 – 139  399 – 138
43 – 17  46 – 20
Points of Intersection:
Content and Practices
MP #7: Look for and make use of structure
Partitioning
• 8x7
• 33 + 58
Points of Intersection:
Content and Practices
MP #7: Look for and make use of structure
Example:
Understanding and interpreting the equation
of a line expressed in “Point-Slope Form”
y – y1 = m(x – x1)
Points of Intersection:
Content and Practices
MP #4: Model with Mathematics
• Model Drawing (“Singapore Math”)
– Mrs. Obama has 28 students in her fifth grade
class and 2/7 of the class is girls. How many of
her students are boys?
Points of Intersection:
Content and Practices
MP #4: Model with mathematics
• Double number lines
Today 40% of the 370 sixth graders at Barack
Obama Middle School are on a field trip.
0%
0
sixth graders
100%
370
sixth graders
Points of Intersection:
Content and Practices
MP #4: Model with mathematics
MP #5: Use appropriate tools strategically
• Compare and contrast directly and
inversely proportional relationships
Points of Intersection:
Content and Practices
MP #2: Reason abstractly and quantitatively.
Consider :
• x2 – 1 = (x + 1)(x – 1)
• (a + b)2 = a2 + 2ab + b2
The Standards for
Mathematical Practice
Small Group Task:
• Select two of the Standards for
Mathematical Practice
• Identify apparent “points of intersection”
between the content standards (in CCSS)
and the Standards for Mathematical
Practice.
CCSS-HCPS III Crosswalks
Crosswalk Documents posted at
http://standardstoolkit.k12.hi.us/index.html
(Click on the “Document Library” link near
the top of the webpage)
Crosswalk Documents posted at http://standardstoolkit.k12.hi.us/index.html
(Click on the “Document Library” link near the top of the webpage)
37
CCSS-HCPS III Crosswalks
• Grade level overview
• Mapping of CCSS to HCPS III benchmarks
Standards matched to benchmarks
Degree of Match
Comments
• Mapping of HCPS III benchmarks to CCSS
CCSS-HCPS III Crosswalks
Small Group Task:
• Gather in groups according to your grade
level of interest
• Analyze the crosswalk document for your
selected grade level to respond to the
prompts in the handout
Final Thoughts
"These Standards are not intended to be new
names for old ways of doing business. They
are a call to take the next step. It is time for
states to work together to build on lessons
learned from two decades of standards based
reforms. It is time to recognize that these
standards are not just promises to our
children, but promises we intend to keep."
A Shift in Perspective
Video: “Math Class Needs a Makeover”
Dan Meyer (a high school mathematics teacher)
http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html
After the video: “Think-Pair-Share”
•One idea that resonated with you