Common Core State Standards
for Mathematics:
Shifts and Implications for
Mathematics Instruction
K-5 Common Core Lead Teachers
6-8 Math Department Heads
Spring 2012
1
The Three Shifts in Mathematics
• Focus strongly where
the standards focus
• Coherence: Think
across grades and link
to major topics within
grades
• Rigor: Require
conceptual
understanding, fluency,
and application
2
Shift One: Focus
strongly where the Standards focus
• Significantly narrow the scope of content and
deepen how time and energy is spent in the
math classroom
• Focus deeply only on what is emphasized in
the standards, so that students gain strong
foundations
3
Focus in International Comparisons
• U.S. curriculum is “a mile wide and an inch deep”
• TIMSS – Trends in International Mathematics and
Science Study
• Highest performing countries omit more material
– U.S. omits 17% of TIMSS items through grade 4, and 2%
through grade 8
– Hong Kong omits 48% of TIMSS items through grade 4,
and 18% through grade 8
– Average omission rate is 40% for 11 comparison countries
• Less topic coverage is associated with higher scores
• Students have more time to master the content that
is taught
– Ginsburg et al., 2005
4
The shape of math in A+ countries
Mathematics topics intended at
each grade by at least
two-thirds of A+ countries
Mathematics topics intended
at each grade by at least twothirds of 21 U.S. states
1 Schmidt,
Houang, & Cogan, “A Coherent Curriculum: The Case of
Mathematics.” (2002).
Higher Demands for Similar Content…
But Much Sharper Focus
Traditional U.S. Approach
K
12
Number and
Operations
Measurement
and Geometry
Algebra and
Functions
Statistics and
Probability
8
Focusing attention within Number and
Operations
Operations and Algebraic
Thinking
Expressions
 and
Equations
Number and Operations—
Base Ten

K
1
2
3
4
Algebra
The Number
System
Number and
Operations—
Fractions



5
6
7
8
High School
9
Shift One: Focus
Find the Fib – Which is not true?
A. Focus means we will narrow the scope of
content in each grade level.
B. Focus means we will deepen how time and
energy is spent in the math classroom.
C. Focus means some standards will be
emphasized more than others.
D. Focus means we will not teach the less
important standards in the common core.
This is the Fib!
10
Shift Two: Coherence
Think across grades, and link to major
topics within grades
• Carefully connect the learning within and across
grades so that students can build new understanding
onto foundations built in previous years.
• Begin to count on solid conceptual understanding of
core content and build on it. Each standard is not a
new event, but an extension of previous learning.
11
Coherence example: Progression across grades
“The coherence and sequential nature of mathematics dictate
the foundational skills that are necessary for the learning of
algebra. The most important foundational skill not presently
developed appears to be proficiency with fractions (including
decimals, percents, and negative fractions). The teaching of
fractions must be acknowledged as critically important and
improved before an increase in student achievement in
algebra can be expected.”
Final Report of the National Mathematics Advisory Panel (2008, p. 18)
Coherence example: Progression across grades
Hong Kong
CCSS
Grade 4
4.NF.4. Apply and extend
previous understandings of
multiplication to multiply a
fraction by a whole number.
5.NF.4. Apply and extend
previous understandings of
multiplication to multiply a
fraction or whole number by a
fraction.
Grade 5
5.NF.7. Apply and extend
previous understandings of
division to divide unit fractions
by whole numbers and whole
numbers by unit fractions.
6.NS. Apply and extend
previous understandings of
multiplication and division to
divide fractions by fractions.
Grade 6
Informing Grades 1-6 Mathematics Standards Development: What Can Be
Learned from High-Performing Hong Kong, Singapore, and Korea?
American Institutes for Research (2009, p. 13)
6.NS.1. Interpret and compute
quotients of fractions, and solve
word problems involving
division of fractions by
fractions, e.g., by using visual
fraction models and equations
to represent the problem.
13
Coherence example: Grade 3
The standards make explicit connections at a single grade
Multiplication and
Division
3.OA.5
Properties
of
Operations
3.MD.7a
3.MD.7c
Area
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Shift Two: Coherence
Find the Fib – Which is not true?
A. Coherence means the standards within a
grade level are related and the relationships
are used to deepen understanding.
B. Coherence means the standards are the
same at different grade levels.
C. Coherence means the standards at one grade
level are built upon in the next grade level.
D. Coherence means authors made great efforts
to illustrate the connectedness of math ideas
within and between grade levels.
This is the Fib!
15
Shift Three: Rigor
Equal intensity in conceptual understanding,
procedural skill/fluency, and application
• The CCSSM require a balance of:
– Solid conceptual understanding
– Procedural skill and fluency
– Application of skills in problem solving situations
• This requires equal intensity in time, activities,
and resources in pursuit of all three
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(a) Solid Conceptual Understanding
• Teach more than “how to get the answer” and
instead support students’ ability to access
concepts from a number of perspectives
• Students are able to see math as more than a
set of mnemonics or discrete procedures
• Conceptual understanding supports the other
aspects of rigor (fluency and application)
17
(b) Procedural Skill and Fluency
• Fluency Standards: The standards require
speed and accuracy in calculation.
• Teachers structure class time and/or
homework time for students to practice core
functions such as single-digit multiplication so
that they are more able to understand and
manipulate more complex concepts
18
Required Fluencies in K-6
Grade
Standard
K
K.OA.5
Add/subtract within 5
1
1.OA.6
Add/subtract within 10
2.OA.2
Add/subtract within 20 (know single-digit sums from memory)
2.NBT.5
3.OA.7
Add/subtract within 100
Multiply/divide within 100 (know single-digit products from memory)
3.NBT.2
Add/subtract within 1000
4
4.NBT.4
Add/subtract within 1,000,000
5
5.NBT.5
Multi-digit multiplication
6
6.NS.2,3
2
3
Required Fluency
Multi-digit division
Multi-digit decimal operations
19
(c) Application
• Students can use appropriate concepts and
procedures for application even when not
prompted to do so
• Provide opportunities at all grade levels for
students to apply math concepts in “real world”
situations, recognizing this means different things
in K-5, 6-8, and HS
• Teachers in content areas outside of math,
particularly science, ensure that students are
using grade-level-appropriate math to make
meaning of and access science content
20
Shift Three: Rigor
Find the Fib – Which is not true?
A. Rigor means teachers should emphasize
conceptual understanding, application, and
procedural skills/fluency equally.
B. Rigor calls for solid conceptual
understanding.
C. Rigor means teachers should no longer
emphasize procedural skills.
D. Rigor calls for problem solving and explaining
skills.
This is the Fib!
21
Next Steps
in our transition
• Incremental changes to the pacing each year for the
next 3 years
– Gradually increase time for major work of each grade
(focus)
– Begin to include problem solving and conceptual
development lessons (focus)
• Modify SBA’s to include open ended item(s) which
require explanation (rigor)
• Development teams (grades 3-5, and 6-7) will create
instructional materials emphasizing problem solving
and conceptual development (rigor)
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And finally…
Can you name and describe the 3 shifts?
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