THE COMMON CORE STATE
STANDARDS FOR MATHEMATICSThe U.S.A’s Quest for Greater Focus and Coherence
SEAMEO RECSAM-University of Tsukuba Joint Seminar

Gabriel Matney
Associate Professor of
Mathematics Education

2ND GRADE MATHEMATICS PROBLEM
What are some ways you think 2nd graders would solve thisproblem?
There were 48 kindergarteners and first graders all together in thecafeteria. The kindergarteners left and 26 second graders camein. Now there are 54 students in the cafeteria. How manykindergarteners were in the cafeteria? How many first graders
are in the cafeteria? 5
The creation of this problem was support by the CORES
Elementary grant funded by the Ohio Board of Regents (Matney,
Bostic, & Brahier, 2012).

AGENDA
Brief Viewing of Early US
Mathematics Education History
Brief Discussion of Learning from
Other Nations
Basics of the Common Core State
Standards
Curricular Research andConnections

INFLUENCE OF IDEAS
Archimedes, Socrates, Plato, Euclid
Rousseau
Emile pages 142 and 150, 1761
“Teach your scholar to observe the phenomenaof nature; you will soon rouse his curiosity, but if
you would have it grow, do not be in too great ahurry to satisfy this curiosity. Put the problems
before him and let him solve them himself. Let
him not be taught science, let him discover it. If
ever you substitute authority for reason he willcease to reason; he will be a mere plaything ofother people’s thoughts”

In 1842 the state ofConnecticut passed alaw that no child under
the age of 15 could be
employed without proofof first having attended
school that year.
By 1918 all state’s passed
compulsory education
laws requiring education
for children.
More than 120-190 years
ago mathematics
education for reasoning
was flourishing in the
United States.

By 1834, Joseph Ray of Ohio had alreadypublished two books with his method ofteaching through “analysis and induction”.
Ray and other early American Mathematics
Education Scholars like him believed thatmathematics had value as a mental
discipline.
Ray said, “[To] be able to reason correctly,
and to exercise, in all relations in life, theenergies of a cultivated and disciplined
mind is of more value that the mere
attainment of a branch of knowledge…thepupil should not be taught merely to performa certain routine of exercises mechanically,
but to understand the why and thewherefore” 7

The ideas about mathematics educationthat were developing in the United States
were being disseminated to other parts of
the world.
Warren Colburn’s “First Lessons inArithmetic” spread across England and wastranslated into many languages in Europe.1

Calvin Wilson Mateer’s book“Pen-calculation Arithmetic”
was being distributed
throughout China.1

Although difficult for students,
these approaches encouraged
both practical and theoreticalthinking about mathematics.
Explored mathematics
through the dual lenses of
the questions:

During this time Americanmathematics educators
tended to combine
arithmetic learning with areasoning system.

How do we accomplish thiscalculation?
Why is it mathematicallyjustified to do thecalculation this way?

After this time of flourishing a debate began in
the U.S.A about what kind of mathematicalknowledge school children need.
• Some favored the “How” and others favoredthe “Why”.
For complex reasons which include thisdebate about how and why, the overall
advance of mathematics classroom
instruction in the U.S.A stagnated in several
ways, while other countries, such as Japan,
Germany, and China, worked to improveteacher’s instruction of mathematics.
There have always been strong pockets ofresearch and innovation in mathematics andeducation, but systemically, teachers
pedagogical practice in the classroom had
not seen a substantial changed for many
decades.

Learning from the success of other countries, such as Japan andGermany, it was possible for mathematics educators to have arenewed conversation with parents, teachers, policy makers, andpoliticians.

*Image recreated from Schmidt, McKnight, & Raizen (1997)2

Another starkdifference between
the U.S.A. and more
successful nations
dealt with theamount of time a
topic spanned.

*Image recreated from Schmidt, McKnight, & Raizen (1997)2

Instead of taking a
topic and goingvery deep with it,
U.S.A curriculum’s
typically spread itout over many
grade levels.

We also learned that oursystems (educative,
governmental, and political)
contributed to the fragmentingof the curricula through thefact that textbooks were madeto be overly inclusive of many
topics so they could “cover”
the expectations of manystates.2*Image from NGAC (2010)4

Though many reformers had been discussing these notions as problematic
prior to these comparison, by the late 1990’s the U.S.A. was more broadly
informed that we suffered from unfocused curricula along with incoherent
and overly inclusive textbooks.

The U.S.A began a quest to find focus and coherencein the mathematics that was taught during grades K12, across the various states and territories.
In 2009 the NGA andCCSSO organized andworking groups began thedesign the Common Core
State Standards forMathematics (CCSSM)
based on research about
the world’s best education
systems, input fromteachers, and preexisting
US state standards. 4

Research about othernations and their results
on international exams
would served as a basis
for standards reform:
For example, TIMSS,PISA, and the The
Teaching Gap6 ensured
that ideas from aroundthe world would
influence the making of
the CCSSM.

After the CCSSM were designed “expert mathematics content analysts
conducted a side-by-side comparison” of the CCSSM and the Japanese
Mathematics Course of Study.3
Three Areas of Comparison
Rigor – “refers to the degree that sets ofstandards address key content that
prepares students for success beyond highschool”3

Major Findings
Comparable Rigor – Some
content given at different
grades but usually within ayear.3Coherence – “refers to whether the
standards reflect a meaningful structure,
revealing significant relationships among
topics and suggest a logical progression ofcontent and skills over the years.”3
Focus – “refers to whether the standards
suggest an appropriate balance inconceptual understanding, procedural skill,and problem solving with an emphasis on
application and modeling.”3Comparable Coherence –Share key traits. Both describe
coherent expectations throughgrade 12.3
Comparable Focus – Bothdescribe focused expectations.
At some points the CCSSM gives
more detail and specificity
about content balance.3

ADOPTION OF THE
COMMON CORE
STATE STANDARDSFOR MATHEMATICS*Image from NGAC (2010)4

TITLE LAYOUT
Subtitle*Image from NGAC (2010)4

STANDARDSCURRICULUM

Define what students should understandand be able to do and if possible specify
the grade level at which it should be
done.

May be thought of as a number ofdifferent ideas, but in this case, it isassociated with the content organizationand the pedagogical pathways plannedand enacted by districts and teachers.
These actions should be based on thecontent and processes given in thestandards.

In the USA, standards are developed anddecided upon at state level. The CCSSM
represents many states coming together
to make the same set of standards.

In the USA, curriculum decisions are
made at the district, school, and teacher
level. Though the CCSSM organizes
content by grades, it doesn’t dictate
order beyond that and it doesn’t state
how teachers are to implement.

THE CCSSM INFORMS CURRICULUM BUT DOES NOT DEFINE IT

CCSSM

Standards for Mathematics
Content

Standards forMathematical PracticeNational Research Council
Adaptive Reasoning
Strategic Competence
Conceptual Understanding
Procedural Fluency
Productive Disposition

National Council of Teachers ofMathematics
Communication
Connection
Problem Solving
Reasoning and Proof
Representation

Standards for Mathematical PracticeMake sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique thereasoning 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 repeatedreasoning.
1.

Standards for Mathematical PracticeMake sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique thereasoning 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 repeatedreasoning.
1.

Standards for Mathematical PracticeMake sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique thereasoning 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 repeatedreasoning.
1.

Standards for Mathematical PracticeMake sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique thereasoning 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 repeatedreasoning.
1.

Standards for Mathematical PracticeMake sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique thereasoning 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 repeatedreasoning.
1.

Standards for Mathematical PracticeMake sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique thereasoning 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 repeatedreasoning.
1.

Standards for Mathematical PracticeMake sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique thereasoning 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 repeatedreasoning.
1.

Standards for Mathematical PracticeMake sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique thereasoning 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 repeatedreasoning.
1.

Fluency and the CCSSM
In order to take a closer look at the curricular influence of
the CCSSM on one topic area that has been a struggle
for children in the U.S.A. I would like to share some of myresearch with teachers and children from the U.S.A. after
the CCSSM.
A Major Difficulty: Procedural Fluency
Procedural fluency is the skill incarrying out procedures flexibly,
accurately, efficiently andappropriately.4

Fluency and the CCSSM
NCTM Principals and Standards emphasize fluency as animportant aspect of mathematics learning, and its
included in both the Standards for Mathematics Content
and the Standards for Mathematical Practice.
Students who are mathematically fluent in theelementary grades give attention to accuracy, haveflexibility in thinking, and develop efficiencies for working
with numbers and arithmetic operations.5
How does the structure of CCSSM help with fluency?

Fluency and the CCSSM
The CCSSM mentions fluency 25 times
Fluency should not be confused with rote recall offacts from memorization, or dictating timed tests.
CCSSM lead author writes, “It is no accident that thestandard says ‘know from memory’ rather than“memorize”. The first describes an outcome, whereas thesecond might be seen as describing a method ofachieving that outcome. So no, the standards are notdictating timed tests.” (October 31, 2011)

Grade & Standard

Addition

Subtraction

Fluency K.OA.5

Numbers within 1 to 5

Solving K.OA.2 &Solve word problems by using objects or drawings anddecompose numbers less than or equal to 10

K K.OA.3
Modeling Ideas
K.NBT.1
Fluency 1.OA.6 &1.NBT.5

Use objects or drawing to compose and decompose
numbers from 11 to 19 into tens and ones
Numbers within 10

Solving 1.OA.1

Modeling Ideas
1.OA.2Use models, properties, andUse models, properties, andplace value strategies tostrategies to subtract one
add a two‐digit number to a
multiple of 10 from anotherone‐digit number or add a
multiple of ten in the range
two‐digit number to aof 10 to 90.
multiple of 10

Fluency 2.NBT.5 &2.OA.22 Solving 2.OA.1

Division

Fluency andthe CCSSM

Mentally find 10 more or 10 less than a two‐digit number
Solve word problems using
Solve word problems using
various strategies forvarious strategies fornumbers within 20,
including problems with 3 numbers within 20, by using
objects, drawings, andwhole numbers, by using
equationsobjects, drawings, andequations

1

Multiplication

Use strategies of place value and properties to add andsubtract numbers within 100
Mentally add and subtract numbers within 20, know frommemory all sums of two one‐digit numbers
Solve word problems involving one or two steps fornumbers within 100 by using drawings and equations

Use models, properties, and place value to add andUse addition to find theModeling Ideas
subtract numbers within 1000
number of objects arranged
2.NBT.7 & 2.NBT.8 &Mentally add or subtract 10 or 100 to a number between in rectangular arrays, 5x5 or
2.OA.4
smaller.
100 and 900

5Matney (2014).

Teaching Children
Mathematics, “Early
Mathematics Fluency with
the CCSSM”

GradeFluency 3.NBT.2 &3.OA.7
Solving 3.OA.3 &Addition

Subtraction

Numbers within 1,000 using strategies and algorithms
based on place value, properties, and the relationshipbetween addition and subtraction

Fluency 4.NBT.4

Solving 4.OA.2 &4.OA.3

Division

Numbers within 100
Know all products of two
one‐digit numbers

Solve two‐step word problems using addition, subtraction, multiplication and division

3 3.OA.8

Modeling Ideas
3.OA.9 & 3.OA.1 &3.OA.2Multiplication

Solve problems using various strategies for numbers
within 100 by using drawings and equations

Fluency andthe CCSSM

Identify arithmetic patterns and explain using properties of operations
Interpretations of products and quotients
Fluently add and subtract using the standard algorithm
for numbers less than or equal to 1,000,000
Solve multi‐step word problems using addition, subtraction, multiplication and division, including problems inwhich remainders must be interpreted
Solving problems involving multiplicative comparison

4
Modeling Ideas
4.OA.5 & 4.NBT.5 &4.NBT.6

Fluency 5.NBT.5

5
Modeling Ideas
5.NBT.6

Generate and analyze patterns
Use strategies of placevalue and properties tomultiply a four digit number
by a one‐digit number, andmultiply two two‐digit
numbers. Explain using
equations, arrays, and/or
area models
Multiply multi‐digit whole
numbers using the standard
algorithm

Use strategies of placevalue and properties todivide up to a four‐digit
number by a one‐digit
number. Explain using
equations, arrays, and/or
area models

5Matney (2014).
Use strategies andproperties to divide whole
numbers with up to four
digits by two‐digit numbers.
Explain using equations,arrays, and/or area models

Teaching Children
Mathematics, “Early
Mathematics Fluency with
the CCSSM”

Fluency and the CCSSM
Fluency through Focused Instruction
Number of the Day
Today’s number is 29. Write any number
sentence you want to start. Then come up
with another number sentence using thosesame numbers. 5

Fluency and the CCSSM
Fluency through Focused Instruction
Number of the Day
We have Marcia’s idea of 22 + 7 = 29. Can anyone besides Marcia share
one other number sentence she might have written?
[Denzell] She could write it 27 + 2 = 29. It’s easier to add 2 on 27 than 7 on 22.
[Marcia] I don’t think that will work. That would take part of the number andmove it instead of the whole thing. Like in 7 + 22 = 29?
[Mary] Hmm, I see Denzell’s idea works but like we were talking before, I
don’t know if it’s a strategy. 5

Fluency and the CCSSM
Fluency through Focused Instruction
Number of the Day
So Mary is bringing up another thing we have been considering. Does anidea work in all situations? In your partner pairs consider Denzell’s idea
and try to give reasons for why it will or will not always work. [After a few
minutes work] Let’s come together and share ideas.
[John] We found that Denzell is trading the two numbers and that works forothers. [Writes 23 + 6 = 29 and 26 + 3 = 29 on the white board][Mary] We also think it always works because he is not changing how much it
is all together. It’s just the ones. 5

Fluency and the CCSSM
Fluency through Focused Instruction
Number of the Day[Kendrae] We switched the numbers but we expanded first. Can we show?
[Teacher motions for Kendrae to write it on the board] So we wrote Denzell’sway. [He writes 20 + 2 + 7 = 20 +7 + 2] 22 is really 20 + 2. Then add 7 more.
Or we can switch when we add the 2 and 7. It will always work because he isadding in a different order but adding the same numbers.
[Marcia] Oh ya I can see that now. 5

Fluency and the CCSSM
Fluency through Problem Solving
There were 48 kindergarteners and first graders all together in thecafeteria. The kindergarteners left and 26 second graders came in.
Now there are 54 students in the cafeteria. How manykindergarteners were in the cafeteria? How many first graders are in
the cafeteria? 5
The creation of this problem was support by the CORES
Elementary grant funded by the Ohio Board of Regents (Matney,
Bostic, & Brahier, 2012).

Fluency and the
CCSSM
5Fluency throughProblem Solving

Fluency and the CCSSM
5Fluency through Problem Solving

Fluency and the
CCSSM
5Fluency throughProblem Solving

Fluency and the CCSSM
Fluency through Modeling andRepresenting New Ideas
5Mrs. Owens had been engaging her

students in working through double digit
addition problems using base 10 blocks asa model. When they were ready she
posed this question to them. “Is there a
way you can add the amounts without
the blocks?” 5

Fluency and the CCSSM
5Fluency through Modeling andRepresenting New Ideas

Fluency andthe CCSSM
5Fluency throughModeling andRepresenting
New Ideas

Fluency and the CCSSM – Focus and Coherence
1) Give students tasks like Number Talks, Problem of theDay, and modeling new thinking to develop number
relationships, accuracy, flexibility, and efficiency.5
2) Use Problem Solving within the fluency range in orderto practice and give more experiences with number andoperations. 53) Allow children the space to model mathematicalideas, and when its time, challenge them to use theirmodeling ideas with newer, more sophisticatedproblems. 5

QUESTIONS?
gmatney@bgsu.edu
Gabriel Matney
Associate Professor of
Mathematics Education

NOTES
1Ma, L. (2010). Knowing and Teaching Elementary Mathematics: Teachers’ Understandingof Fundamental Mathematics in China and the United States. New York, NY: Routledge.
2Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A Splintered Vision: An

Investigation of U.S. Science and Mathematics Education. Norwell, MA: Kluwer.
3Achieve (2010).

Comparing the Common Core State Standards and Japan’sMathematics Curriculum Course of Study. Retrieved on February 9th, 2016 at:
www.achieve.org
4National Governors Association Center for Best Practices (NGAC), Council of Chief State

School Officers (CCSSO). (2010). Common Core State Standards for Mathematics.
Washington, DC: Author. http://www.corestandards.org.

5Matney, G. (2014). Early Mathematics Fluency with the CCSSM. Teaching Children

Mathematics 21(1), 26-35. Reston, VA: National Council of Teachers of Mathematics.
6Stigler, J. W. and Hiebert, J. (1999) The Teaching Gap: Best Ideas from the World’s

Teachers for Improving Education in the Classroom. New York, NY: Free Press.

Kullman, D. (1997). Joseph Ray: The McGuffey of Mathematics. Paper submitted to the
Ohio Journal of Mathematics. Columbus, OH.