Journey to the Core
Focus, Coherence, and
Understanding in the
Common Core State
Standards for Mathematics
Dr. DeAnn Huinker
University of Wisconsin-Milwaukee
huinker@uwm.edu
Wisconsin Mathematics Council
Green Lake, Wisconsin
4 May 2012
Journey to the Core
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Focus
Coherence
Understanding
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Common Core State
Standards
Shared,
the
same for
everyone
Essential,
fundamental
knowledge
and skills
necessary for
student
success
Adopted
and
maintained
by States;
not a
federal
policy
Benchmarks
of what
students are
expected to
learn in a
content area
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
We are learning to...

Understand “Focus” and “Coherence”

Consider how the standards detail or specify
“Ways of Knowing” mathematics

Embrace “Shifts”



content topics
curriculum & assessment
instructional approaches
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
How much of a shift is the
Math Common Core for …

District

School

Curriculum


Great
Major
Strong
Moderate
Small
Minor
Teaching
Students
Not Felt
Magnitude
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
A Long Overdue Shifting of the Foundation
For as long as most of us can remember, the K-12
mathematics program in the U.S. has been aptly
characterized in many rather uncomplimentary
ways: underperforming, incoherent, fragmented,
poorly aligned, narrow in focus, skill-based, and, of
course, “a mile wide and an inch deep.”
---Steve Leinwand, Principal Research Analyst
American Institutes for Research in Washington, D.C
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
But hope and change have arrived!
Like the long awaited cavalry, the new Common Core
State Standards for Mathematics (CCSS) presents us
a once in a lifetime opportunity to rescue ourselves
and our students from the myriad curriculum problems
we’ve faced for years.
---Steve Leinwand, Principal Research Analyst
American Institutes for Research in Washington, D.C
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Standards for
Mathematical Practice


Standards for
Mathematics Content
Make sense of problems
Reason quantitatively

Viable arguments & critique

Model with mathematics

Strategic use of tools

Attend to precision

Look for and use structure

Look for regularity in reasoning
 K-8
Grade Levels
 HS
Conceptual Categories
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
1
2
3
4
5
Operations & Algebraic Thinking
6
7
8
HS
Algebra
Expressions and
Equations
Number & Operations in Base Ten
The Number System
Number &
Operations
Fractions
Ratios &
Proportional
Relationships
Measurement & Data
Number
and
Quantity
Modeling
K
Counting
&
Cardinality
Functions
Statistics & Probability
Geometry
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Digging in…
Begin to unearth some discoveries:

Mathematics content

Teaching of mathematics

Student “knowing” of mathematics
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Reflecting…
2NBT9. Explain why addition and subtraction
strategies work, using place value and the
properties of operations.
3OA3. Use multiplication and division within 100
to solve word problems in situations involving
equal groups, arrays, and measurement
quantities, e.g., by using drawings and
equations with a symbol for the unknown
number to represent the problem.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Reflecting…
4NF2. Compare two fractions with different
numerators and different denominators,
e.g., by creating common denominators or
numerators, or by comparing to a benchmark
fraction such as 1/2. Recognize that
comparisons are valid only when the two
fractions refer to the same whole. Record the
results of comparisons with symbols >, =, or <,
and justify the conclusions, e.g., by using a
visual fraction model.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Reflecting…
4NF2. Compare two fractions with different
numerators and different denominators,
e.g., by creating common denominators or
numerators, or by comparing to a benchmark
fraction such as 1/2. Recognize that
comparisons are valid only when the two
fractions refer to the same whole. Record the
results of comparisons with symbols >, =, or <,
and justify the conclusions, e.g., by using a
visual fraction model.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Which is larger?
Find a
common
numerator!
3
6
or
4
7
Rename
6
or
8
6
7
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Focus
and
Coherence
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Focus
Coherence
CCSS “design principles”
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Helping Teachers: Coherence and Focus
Dr. William McCallum
Professor of Mathematics, University of Arizona
Lead Writer, Common Core Standards for Mathematics
The Hunt Institute Video Series
Common Core State Standards: A New Foundation for Student Success
www.youtube.com/user/TheHuntInstitute#p
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Features of Focus and Coherence
Fewer Topics
“Free up time” to do fewer
things more deeply.
Progressions
Show how ideas fit with
subsequent or previous
grade levels.
More Detail
“Give more detail than
teachers were used to
seeing in standards.”
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Unifying Themes
Domains
Clusters
Details
Standards
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Unifying Themes
Details
Grade
Domains
Clusters
Standards
K
5
9
22
1
4
11
21
2
4
10
26
3
5
11
25
4
5
12
28
5
5
11
26
6
5
10
29
7
5
9
24
8
5
10
28
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Unifying Themes
Conceptual
Category
Domains
Clusters
Details
Standards Standards
All
Advanced
Number &
Quantity
Algebra
Functions
Geometry
Statistics &
Probability
Modeling
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Unifying Themes
Conceptual
Category
Details
Domains
Clusters
Standards Standards
All
Advanced
Number &
Quantity
4
9
9
18
Algebra
4
11
23
4
Functions
4
10
22
6
Geometry
6
15
37
6
Statistics &
Probability
4
9
22
9
Modeling
*
*
*
*
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Coherence
Discipline of
mathematics
Research on students’
mathematics learning
Content Standards:
Reflect hierarchical nature
& structure of the discipline.
– Progressions
– Ways of Knowing
Reflects what we know
about how students develop
mathematical knowledge.
Practice Standards:
Reflect how knowledge is
generated within the
discipline.
Reflects the needs of
learners to organize and
connect ideas in their minds
(e.g., brain research).
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
CCSSM Progression Documents (draft)
by The Common Core Standards Writing Team
Comprehensive discussions on:
• Intent of specific standards.
• Development within and across grades.
• Connections across domains.
• Suggested instructional approaches.
ime.math.arizona.edu/progressions
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
K
1
2
3
4
5
6
7
8
HS
Counting
&
Cardinality
Algebra
Expressions and Equations
Number & Operations in Base Ten
The Number System
Number &
Operations
Fractions
Ratios &
Proportional
Relationships
Number and
Quantity
Modeling
Operations & Algebraic Thinking
Functions
Measurement & Data
Statistics & Probability
Geometry
Geometry
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Focus and Coherence
Domains and Clusters
as unifying themes
within & across grades.
Detail in the standards
give guidance on
“ways of knowing”
the mathematics
Embedded
progressions of
mathematical ideas.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
“Ways of Knowing”
the mathematics
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Operations and Algebraic Thinking
Dr. Jason Zimba
Professor of Physics and Mathematics
Bennington College, Vermont
Lead Writer, Common Core Standards for Mathematics
The Hunt Institute Video Series
Common Core State Standards: A New Foundation for Student Success
www.youtube.com/user/TheHuntInstitute#p
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
The number strand “has often been a single strand in
elementary school, but in CCSS it is three domains.”
Operations and Algebraic
Thinking (OA)
Expressions
and Equations
(EE)
Algebra
Number and Operations in
Base Ten (NBT)
Number
System (NS)
Number and
Operations –
Fractions
(NF)
K
1
2
3
4
5
6
7
8
High School
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Operations & Algebraic Thinking (OA)
‘“Addition, subtraction, multiplication, & division have
meanings, mathematical properties, and uses that
transcend the particular sorts of objects that one is
operating on, whether those be multi-digit numbers or
fractions or variables or variables expressions.”
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Meanings of the
Operations
Properties of the
Operations
Contextual
Situations
The foundation for algebra!
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
72 – 29 = ?
24 x 25 = ?
Mental Math
Solve in your head.
No pencil or paper!
Nor calculators, cell phones
computers, or iPads or ....
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
72 – 29 = ?
24 x 25 = ?
Turn and share your reasoning.
Discuss how you:
“Decomposed and composed the quantities.”
(a.k.a. properties of the operations)
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
24 x 25 = ?
I thought 25 x 25 = 625 and then I
subtracted 25. 625 – 25 = 600.
I figured that there are
I thought
4 twenty-fives in 100,
24 x 100 = 2400,
and there are 6 fours in
and 2400 ÷ 4 = 600.
24, so 100 x 6 = 600.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
24 x 25 = ?
25 x 4 = 100,
6 x 100 = 600,
600 + 100 = 700.
Well,10 x 25 = 250,
2(10 x 25) = 500,
500 x 4 = 2000.
“I would try to
multiply in my head,
but I can't do that.”
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
The properties of operations.
Associative property of addition
(a + b) + c = a + (b + c)
Commutative property of addition
a+b=b+a
Additive identity property of 0
a+0=0+a=a
Existence of additive inverses
For every a there exists –a so
that a + (–a) = (–a) + a = 0
Associative property of multiplication
(a × b) × c = a × (b × c)
Commutative property of
multiplication
a×b=b×a
Multiplicative identity property of 1
a×1=1×a=a
Existence of multiplicative inverses
For every a ≠ 0 there exists
1/a so that a × 1/a = 1/a × a = 1
Distributive property of multiplication
over addition
a × (b + c) = a × b + a × c
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
And in the domain of Operations and Algebraic
Thinking, it is those meanings, properties, and uses
which are the focus; and it is those meanings,
properties, and uses that will remain when students
begin doing algebra in middle grades [and beyond].
--Jason Zimba
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
In Grades K-8, how many standards
reference “properties of the operations”?
Grade 1: OA, NBT
Grade 2: NBT
28
standards
Grade 3: OA, NBT
Grade 4: NBT, NF
Grade 5: NBT
Grade 6: NS, EE
Grade 7: NS, EE
Grade 8: NS
12% of K-8 standards
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Using properties of operations

1OA3. Apply properties of operations as strategies to
add and subtract.

3OA5. Apply properties of operations as strategies to
multiply and divide.

4NBT5. Multiply two two-digit numbers using strategies
based on place value and the properties of operations.

5NBT6. Find whole-number quotients and remainders with
… using strategies based on place value, properties of
operations ….

5NBT7. Add, subtract, multiply, and divide decimals to
hundredths, using concrete models or drawings and
strategies based on place value, properties of operations….
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee

6EE3. Apply the properties of operations to
generate equivalent expressions.

7NS2c: Apply properties of operations as
strategies to multiply and divide rational numbers.

7EE1. Apply properties of operations as
strategies to add, subtract, factor, and expand
linear expressions with rational coefficients.

and into high school……
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Develop and
use strategies
based on properties
of the operations
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
CCSS Glossary
Computation strategy
Purposeful manipulations that may be chosen for
specific problems, may not have a fixed order,
and may be aimed at converting one problem
into another.
Computation algorithm
A set of predefined steps applicable to a class of
problems that gives the correct result in every
case when the steps are carried out correctly.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
In Grades K-8, how many standards
reference using “strategies”?
Grade K: CC
Grade 1: OA, NBT
26
standards
Grade 2: OA, NBT
Grade 3: OA, NBT
Grade 4: NBT, NF
11% of K-8 standards
Grade 5: NBT
Grade 7: NS, EE
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Standard 1OA6: “Basic Facts”
Add and subtract within 20, demonstrating fluency for
addition and subtraction within 10.
Use strategies such as counting on; making ten (e.g.,
8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a
number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10
– 1 = 9); using the relationship between addition and
subtraction (e.g., knowing that 8 + 4 = 12, one knows
12 – 8 = 4); and creating equivalent but easier or
known sums (e.g., adding 6 + 7 by creating the known
equivalent 6 + 6 + 1 = 12 + 1 = 13).
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Standard 3OA5: Basic Facts
Apply properties of operations as
strategies to multiply and divide.
Examples:
If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known.
(Commutative property of multiplication.)
3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or
by 5 × 2 = 10, then 3 × 10 = 30.
(Associative property of multiplication.)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7
as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56.
(Distributive property.)
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
In Grades K-8, how many standards
reference using “algorithms”?
Grade 3: NBT2
5
standards
Grade 4: NBT4
Grade 5: NBT5
Grade 6: NS2, NS3
2% of K-8 standards
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Algorithms
Grade 3 “use strategies and algorithms” to add and
subtract within 1000. (Footnote: A range of algorithms
may be used.) (3NBT2)
Grade 4 “use the standard algorithm” to add and
subtract multi-digit whole numbers. (4NBT4)
Grade 5 “use the standard algorithm” to multiply multidigit whole numbers. (5NBT4)
Grade 6 “use the standard algorithm” to divide multidigit numbers and to divide multi-digit decimals.
(6NS2, 6NS3)
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Algorithms
Grade 3 “use strategies and algorithms” to add and
subtract within 1000. (Footnote: A range of algorithms
may be used.) (3NBT2)
Grade 4 “use the standard algorithm” to add and
subtract multi-digit whole numbers. (4NBT4)
Grade 5 “use the standard algorithm” to multiply multidigit whole numbers. (5NBT4)
Grade 6 “use the standard algorithm” to divide multidigit numbers and to divide multi-digit decimals.
(6NS2, 6NS3)
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Strategies first!
Develop and use strategies
for learning basic facts
before any expectation of
knowing facts from memory.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Strategies first!
Develop and use strategies to
compute with whole numbers,
fractions, decimals …. before
use of standard algorithms.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Meanings of the
Operations
Properties of the
Operations
Contextual
Situations
The foundation for algebra!
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
In Grades K-8, how many standards reference
“real-world contexts” or “word problems”?
Grade K: OA
Grade 1: OA
Grade 2: OA, MD
54
standards
Grade 3: OA, MD
Grade 4: OA, NF, MD
Grade 5: NF, MD, G
Grade 6: RP, EE, NS, G
24% of K-8 standards
Grade 7: RP, EE, NS, G
Grade 8: EE, G
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Lots of real-world contexts!
Proficient students make sense of
quantities and their relationships
in problem situations. (MP2)
decontexualize & contextualize
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Properties of the
Operations
Strategies
Real-world
Contexts
Algorithms
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Great
Shifts in
Classroom Practice
Major
Strong
Moderate
Small
Minor
Not Felt
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Shifts . . . Content
Less data analysis and probability in K-5
 More statistics in 6-8 and lots more in HS
 Much more emphasis on statistical variability
Less algebraic patterns in K-5
 Much more algebraic thinking in K-5
 More algebra in 7-8 and functions in 8th
More focus on Ratio and Proportion beginning in 6th
 Percents in 6-7, not in K-5
More geometry in K-HS
 Much more transformational geometry in HS.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Shifts… Curriculum & Assessment
Real-world applications, contexts, and problem solving
 Strong emphasis on contexts and word problems from K-HS
 Use of measurement contexts across domains,
especially “linear” and “liquid” contexts
 Multi-step Word Problems beginning in Grade 2
 Mathematical modeling interwoven throughout HS
HS standards as “conceptual categories” not courses ….
 supports either integrated or traditional approach
or new models that synthesize both approaches.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Shifts . . . Teaching
Strategies and sense-making before algorithms
 Strategies based on properties of the operations
 Algorithms culminate years of prior work
Increased emphasis on visual models
 Number line model
 Area model
Using a “unit fraction” approach
 Understand and use unit fraction reasoning and
language and expect it of our students
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
And so in closing …
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Focus: Unifying themes
and guidance on “ways of
knowing” the mathematics.
Coherence: Progressions
across grades based on
discipline of mathematics
and on student learning.
Understanding: Deep, genuine
understanding of mathematics
and ability to use that knowledge
in real-world situations.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Please keep digging, there are many more
discoveries in the Core to unearth and we
know that the work we are all doing is
important for Wisconsin students, for their
learning and understanding of mathematics,
and for their futures.
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Thank you!
Focus
Coherence
Understanding
Dr. DeAnn Huinker
University of Wisconsin-Milwaukee
huinker@uwm.edu
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Resources
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
CCSSM Resources

www.dpi.wi.gov/standards/ccss.html

www.mmp.uwm.edu
 Quick link: CCSS Resources
 www.tinyurl.com/CCSSresources

commoncoretools.wordpress.com

ime.math.arizona.edu/progressions

www.youtube.com/user/TheHuntInstitute#p

www.corestandards.org
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee
Video Series: William McCallum and Jason Zimba
lead writers of the CCSSM (The Hunt Institute)

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


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
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The Mathematics Standards: How They Were Developed and Who Was Involved
The Mathematics Standards: Key Changes in Their Evidence
The Importance of Coherence in Mathematics
The Importance of Focus in Mathematics
The Importance of Mathematical Practices
Mathematical Practices, Focus and Coherence in the Classroom
Whole Numbers to Fractions in Grades 3-6
Operations and Algebraic Thinking
The Importance of Mathematics Progressions
The Importance of Mathematics Progressions from the Student Perspective
Gathering Momentum for Algebra
Mathematics Fluency: A Balanced Approach
Ratio and Proportion in Grades 6-8
Shifts in Math Practice: The Balance Between Skills and Understanding
The Mathematics Standards and the Shifts They Require
Helping Teachers: Coherence and Focus
High School Math Courses
www.youtube.com/user/TheHuntInstitute#p
Dr. DeAnn Huinker, University of Wisconsin-Milwaukee