2011 NYS P-12 Common Core
Learning Standards for
Mathematics
Please visit www.engageNY.org for additional
information regarding the Common Core Learning
Standards
The Common Core State
Standards Initiative
Beginning in the spring of 2009, Governors and state
commissioners of education from 48 states, 2 territories
and the District of Columbia committed to developing a
common core of state K-12 English-language arts (ELA) and
mathematics standards.
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
2
Why Common Core
State Standards?
3
Why Common Core State
Standards?
Preparation: The standards are college- and career-ready. They will help
prepare students with the knowledge and skills they need to succeed in
education and training after high school.
Competition: The standards are internationally benchmarked. Common
standards will help ensure our students are globally competitive.
Equity: Expectations are consistent for all – and not dependent on a
student’s zip code.
Clarity: The standards are focused, coherent, and clear. Clearer standards
help students (and parents and teachers) understand what is expected of
them.
Collaboration: The standards create a foundation to work collaboratively
across states and districts, pooling resources and expertise, to create
curricular tools, professional development, common assessments and
other materials.
4
Common Core State Standards
Design
•Building on the strength of current state
standards, the CCSS are designed to be:
– Focused, coherent, clear and rigorous
– Internationally benchmarked
– Anchored in college and career readiness*
– Evidence and research based
*Ready for first-year credit-bearing, postsecondary coursework in mathematics
and English without the need for remediation.
5
Common Core State Standards
Evidence Base
•For example: Standards from individual high-performing
countries and provinces were used to inform content, structure,
and language. Writing teams looked for examples of rigor,
coherence, and progression.
Mathematics
English language arts
1.Belgium
1.Australia
(Flemish)
2.Canada (Alberta)
3.China
4.Chinese Taipei
5.England
6.Finland
7.Hong Kong
8.India
9.Ireland
10.Japan
11.Korea
12.Singapore
New South Wales
•
Victoria
2.Canada
•
Alberta
•
British Columbia
•
Ontario
3.England
4.Finland
5.Hong Kong
6.Ireland
7.Singapore
•
6
Feedback and Review
•External and State Feedback teams included:
– K-12 teachers
– Postsecondary faculty
– State curriculum and assessments experts
– Researchers
– National organizations (including, but not limited, to):
 American Council on Education
(ACE)
 National Council of Teachers of
English (NCTE)
 American Federation of Teachers
(AFT)
 National Council of Teachers of
Mathematics (NCTM)
 Campaign for High School Equity
(CHSE)
 National Education Association
(NEA)
 Conference Board of the
Mathematical Sciences (CBMS)
 Modern Language Association (MLA)
7
Process and Timeline
K-12 Common Standards:
Core writing teams in English Language Arts and Mathematics
(See www.corestandards.org for list of team members)
External and state feedback teams provided on-going feedback
to writing teams throughout the process
Draft K-12 standards were released for public comment on
March 10, 2010; 9,600 comments received
Validation Committee of leading experts reviews standards
Final standards were released June 2, 2010 - NYS Board of
Regents Adopted July 20, 2010 (all but 5 states have adopted)
8
Common Core State Standards
for Mathematics
Grade-Level Standards
 K-8 grade-by-grade standards organized by domain
 9-12 high school standards organized by conceptual
categories
Standards for Mathematical Practice
 Describe mathematical “habits of mind”
 Standards for mathematical proficiency: reasoning,
problem solving, modeling, decision making, and
engagement
 Connect with content standards in each grade
9
NYS Common Core
Learning Standards for
Mathematics
Common Core Learning Standards
Instructional Shifts . . .
Instructional Shifts . . .
Instructional Shifts in Mathematics
Shift 1 Focus
Teachers use the power of the eraser and
significantly narrow and deepen the scope of
how time and energy is spent in the math
classroom. They do so in order to focus deeply
on only the concepts that are prioritized in
the standards so that students reach strong
foundational knowledge and deep conceptual
understanding and are able to transfer
mathematical skills and understanding across
concepts and grades.
Reflection
• Read the “Shift”
• What does the “Shift” mean to you?
• What does it look like in mathematics
classrooms (provide specific examples)?
Trends in International Mathematics
and Science Study (TIMSS)
Test your mathematics and science knowledge
by completing TIMSS items in the Dare to
Compare challenge!
Shift 2 Coherence
Principals and teachers carefully connect the
learning within and across grades so that, for
example, fractions or multiplication spiral
across grade levels and students can build
new understanding onto foundations built in
previous years. Teachers can begin to count on
deep conceptual understanding of core
content and build on it. Each standard is not a
new event, but an extension of previous
learning.
Reflection
• Read the “Shift”
• What does the “Shift” mean to you?
• What does it look like in mathematics
classrooms (provide specific examples)?
Dividing Fractions
• Imagine you are beginning to teach
students division with fractions. What
would you do to introduce this concept
to students?
Dividing Fractions
• How would you present the
following problem:
1¾ ÷ ½?
Knowing and Teaching Elementary
Mathematics – Liping Ma
What is the common phrase we hear
teachers say when teaching students
to divide fractions?
Knowing and Teaching Elementary
Mathematics – Liping Ma
Most of the Chinese teachers use the
phrase “dividing by a number is
equivalent to multiplying by its
reciprocal” instead of what many
U.S. teachers say “invert and
multiply.”
Knowing and Teaching Elementary
Mathematics – Liping Ma
Dividing by 2 is the same as
multiplying by ½, therefore dividing
by ½ is the same as multiplying by 2.
Knowing and Teaching Elementary
Mathematics – Liping Ma
• How many different ways can we
solve the problem 1 ¾ ÷ ½ ?
• Let’s share and record all the various
ways.
Knowing and Teaching Elementary
Mathematics – Liping Ma
• How could you put this problem in
context (possible model for
representing this problem)?
• Think/Pair/Share
Knowing and Teaching Elementary
Mathematics – Liping Ma
• Measurement Model – “How many
½s in 1 ¾?” (e.g., apples, graham crackers,
piece of wood)
• Partitive Model – “Finding a number such that
½ of it is 1 ¾” (e.g., box of candy, cake, pizza,
distance)
• Factors and Product – “Find a factor that when
multiplied by ½ will make 1 ¾” (e.g., area of a
rectangle)
Knowing and Teaching Elementary
Mathematics – Liping Ma
The meaning of
division with
fractions
Meaning of
division with whole
numbers
The concept of
inverse operations
Meaning of
multiplication with
fractions
Meaning of
multiplication with
whole numbers
Meaning of
addition
Concept of unit
Concept of fraction
Shift 3 Fluency
Students are expected to have speed and
accuracy with simple calculations; teachers
structure class time and/or homework time
for students to memorize, through repetition,
core functions (found in the attached list of
fluencies) such as multiplication tables so that
they are more able to understand and
manipulate more complex concepts.
Reflection
• Read the “Shift”
• What does the “Shift” mean to you?
• What does it look like in mathematics
classrooms (provide specific examples)?
World’s Easiest Math Puzzle
Listen and watch carefully…
World’s Easiest Math Puzzle
What’s your answer?
Shift 4 Deep Understanding
Teachers teach more than “how to get the
answer” and instead support students’ ability
to access concepts from a number of
perspectives so that students are able to see
math as more than a set of mnemonics or
discrete procedures. Students demonstrate
deep conceptual understanding of core math
concepts by applying them to new situations.
as well as writing and speaking about their
understanding.
Reflection
• Read the “Shift”
• What does the “Shift” mean to you?
• What does it look like in mathematics
classrooms (provide specific examples)?
Shift 5 Applications
Students are expected to use math and choose
the appropriate concept for application even
when they are not prompted to do so.
Teachers provide opportunities at all grade
levels for students to apply math concepts in
“real world” situations. Teachers in content
areas outside of math, particularly science,
ensure that students are using math – at all
grade levels – to make meaning of and access
content.
Reflection
• Read the “Shift”
• What does the “Shift” mean to you?
• What does it look like in mathematics
classrooms (provide specific examples)?
Standards for Mathematical Practice
Integrated into Instruction
McDonald’s Claim
Wikipedia reports that 8% of all Americans eat
as McDonalds every day. In the U.S., there
are approximately 310 million people and
12,800 McDonalds.
Do you believe the Wikipedia report to be true?
Create a mathematical argument to justify
your position.
Shift 6 Dual Intensity
Students are practicing and understanding. There is
more than a balance between these two things in
the classroom – both are occurring with intensity.
Teachers create opportunities for students to
participate in “drills” and make use of those skills
through extended application of math concepts. The
amount of time and energy spent practicing and
understanding learning environments is driven by
the specific mathematical concept and therefore,
varies throughout the given school year.
Reflection
• Read the “Shift”
• What does the “Shift” mean to you?
• What does it look like in mathematics
classrooms (provide specific examples)?
Thought for the Day
“It’s what you learn after
you know it all that
counts.”
- John Wooden
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
Jigsaw Activity
Standards for Mathematical Practice
Traditional U.S. Problem
Which fraction is closer to 1
4/5 or 5/4?
Same problem with Standards for
Mathematical Practice Integration
4/5 is closer to 1 than 5/4. Using a
number line, explain why this is true.
Content Standards
The content standards are organized by
domains across grade levels and each grade
level begins with a narrative description of
the grade level, followed by the standards
for mathematical practice, a list of the “Big
Ideas” for the specific grade level, and then
the content standards by domain.
How to read the grade level standards in Mathematics
Standards define what students should understand and be able to do.
Clusters summarize groups of related standards. Note that standards from different clusters may sometimes be
closely related, because mathematics is a connected subject.
Domains are larger groups of related standards. Standards from different domains may sometimes be closely
related.
These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before
topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A
teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A
and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a
byproduct, to students reaching the standards for topics A and B.
What students can learn at any particular grade level depends upon what they have learned before. Ideally then, each
standard in this document might have been phrased in the form, “Students who already know A should next come to
learn B.” But at present this approach is unrealistic—not least because existing education research cannot specify all
such learning pathways. Of necessity therefore, grade placements for specific topics have been made on the basis of
state and international comparisons and the collective experience and collective professional judgment of educators,
researchers and mathematicians. One promise of common state standards is that over time they will allow research
on learning progressions to inform and improve the design of standards to a much greater extent than is possible
today. Learning opportunities will continue to vary across schools and school systems, and educators should make
every effort to meet the needs of individual students based on their current understanding.
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.
Progressions
Because progressions are so important in the
Standards, suggestions for places to begin are
not a laundry list of topics but rather a menu
of progressions. Experts recommend
organizing implementation work according to
progressions because the instructional
approach to any given topic should be
informed by its place in an overall flow of
ideas.
Progressions
They emphasize the word menu. If a curriculum
provider delivers a single coherent progression
of materials to a district, then that provider
has added value. If a math coach helps
elementary school teachers in a district better
understand a single coherent progression,
then that coach has added value. The
quantum of improvement is not the textbook
series.
Progressions
• Counting and Cardinality and Operations and Algebraic
Thinking: grades K–2
• Operations and Algebraic Thinking: multiplication and division
in grades 3–5, tracing the evolving meaning of multiplication,
from equal-groups thinking with whole numbers in grade 3 to
scaling-oriented thinking with fractions in grade 5.
• Number and Operations—Base Ten: addition and subtraction
in grades 1–4
• Number and Operations—Base Ten: multiplication and
division in grades 3–6
• Number and Operations—Fractions: fraction addition and
subtraction in grades 4–5, including parallel development of
fraction equivalence in grades 3–5
Progressions
• Number and Operations—Fractions: fraction multiplication
and division in grades 4–6
• The Number System: grades 6–7
• Expressions and Equations: grades 6–8, including how this
extends prior work in arithmetic
• Ratio and Proportional Reasoning: its development in grades
6–7, its relationship to functional thinking in grades 6–8, and
its connection to lines and linear equations in grade 8
• Geometry: work with the coordinate plane in grades 5–8,
including connections to ratio, proportion, algebra and
functions in grades 6–HS
• Geometry: congruence and similarity of figures in grades 8–
HS, with emphasis on real-world and mathematical problems
involving scales and connections to ratio and proportion
Progressions
• Modeling with equations and inequalities in high school,
development from simple modeling tasks such as word
problems to richer more open-ended modeling tasks
• Seeing Structure in Expressions, from expressions appropriate
to 8th–9th grade to expressions appropriate to 10th–11th
grade
• Statistics and Probability: comparing populations and drawing
inferences in grades 6–HS.
• Additionally, one of the important ―invisible themes in the
Standards involves units as a cross-cutting theme in the areas
of measurement, geometric measurement, base-ten
arithmetic, unit fractions, and fraction arithmetic, including
the role of the number line.
When Not Knowing Math
Can Cost You $15,000
“Who wants to be a Millionaire?”
Question for $16,000
When Not Knowing Math
Can Cost You $15,000
Which of these square numbers also happens to
be the sum of two smaller square numbers?
a. 16
b. 25
c. 36
d. 49
List strategies to help students remember square numbers
Common Core State Standards K-12
Mathematics Progression of Domains
Summarized Objectives in Mathematics
for the Next Six Months are:
Materials:
– Focus
– Clear indication of fewer concepts at each grade level represented by
curriculum documents, district formative assessments
• Teachers:
– Identify focus areas and fluencies of grade level
– Shift in time spent on areas of in-depth instruction
• Students:
– Demonstrated fluency and understanding
– Display fluencies for the grade level and understand focus areas
Assessments . . .
• Spring 2012 NYS Grades 3-8 Assessments will focus on
the 2005 NYS Core Curriculums in ELA and mathematics
• Spring 2013 the NYS Grades 3-8 Assessments will focus
on the 2011 Common Core Learning Standards in ELA
and mathematics
• Spring 2015 PARCC Assessments (Grades 3-8)
administered for the first time
PARCC Assessments
• Partnership for Assessment of Readiness for
College and Careers (PARCC)
http://www.parcconline.org/
• PARCC is a 26-state consortium working
together to develop next-generation K-12
assessments in English and math.
• PARCC states collectively educate about 25 million public K-12
students in the United States.
• What brought all of these states together is a shared commitment
to develop an assessment system aligned to the Common Core
State Standards that is anchored in college and career readiness;
provides comparability across states; has the ability to assess and
measure higher-order skills such as critical thinking,
communications, and problem solving; and provides truly useful
information for educators, parents, and students alike.
• While each state has their own priorities and challenges, PARCC
provides the opportunity for participating states to come together
and collectively move the field forward and break new ground in
assessment design. In addition, many of the PARCC states are on
the leading edge of education reform, including 10 of the 12
winning Race to the Top states.
K-2 Formative Assessments
• To help states measure student knowledge and skills at the lower
grades, the Partnership will develop a bank of assessment resources
for teachers of grades K–2 that are aligned to the Common Core
State Standards, and vertically aligned to the PARCC assessment
system. The tasks will consist of developmentally-appropriate
assessment types, such as observations, checklists, classroom
activities and protocols, which reflect foundational aspects of the
Common Core State Standards. The K-2 formative assessments aim
to help set a foundation for students and put them on the track to
college and career readiness in the early years.
• These K-2 assessments will help educators prepare students for later
grades and provide information for educators about the knowledge
and skills of the students entering third grade, allowing classroom
teachers and administrators to adjust instruction as necessary. These
tools also will help states fully utilize the Common Core State
Standards across the entire K-12 spectrum.
The 3-8 PARCC assessments will be delivered at each grade
level and will be based directly on the Common Core State
Standards
• The distributed PARCC design includes through-course and
end-of-year components so that assessments are given closer
in time to when instruction happens.
• The 3-8 assessments will include a range of item types,
including innovative constructed response, extended
performance tasks, and selected response (all of which will be
computer based).
The high school PARCC assessments will be based directly on the
Common Core State Standards
• The distributed PARCC design includes through-course and endof-year components so that assessments are given closer in time
to when instruction happens. PARCC states have endorsed a
course-based design in math and a grade-based design in
ELA/Literacy.
• The high school assessments will include a range of item types,
including innovative constructed response, extended
performance tasks, and selected response (all of which will be
computer based). In addition, there will be college-ready cut
scores on high school tests in mathematics and ELA/Literacy,
which will signify whether students are ready for college-level
coursework. Earlier tests will be aligned vertically to ensure
students are on - and stay on - the track to graduating ready for
college and careers.
PARCC Content Frameworks
Fall 2011
Mathematics
The Model Content Frameworks for Mathematics are designed with
the following purposes in mind:
• identifying the big ideas in the Common Core State Standards
for each grade level,
• helping determine the focus for the various PARCC assessment
components, and
• supporting the development of the assessment blueprints.
Another Thought for the Day
“Coming together is a beginning,
keeping together is progress,
working together is success.”
- Henry Ford
Curriculum Unit as Defined by
Commissioner John B. King, Jr.
“This year I am asking every teacher to try at least
one Common Core-aligned unit each semester.
Math teachers should select one of the priority
concepts at the strategic expense of other, less
critical topics and go deep in a way they haven’t
before. ELA teachers will provide a thoughtful
learning experience around a particular text that
should result in students’ ability to make an
argument about that text. Content area teachers
can fulfill the “literacy” aspect of this
transformation by providing similar learning
experiences built around pivotal texts in their
subject area. “
Curriculum Unit Template
• Stage 1: Desired Results
- What students will know, do and understand
• Stage 2: Assessment Evidence
- Evidence that will be collected to determine whether
or not desired results are achieved
• Stage 3: Learning Plan
- Design learning activities to align with Stage 1 and 2
expectations
- What activities and instruction will engage students
and help them better grasp the essence and the value
of this topic/content?
Questions . . .
Teri Calabrese-Gray
Gray_teri@cves.org
http://www.cves.org/rttt