Reasoning and Sense Making
Through the
Common Core State Standards
for Mathematics (CCCSM)
SCCTM Annual Conference
Charleston, SC
October 28, 2011
Ed Dickey
College of Education
University of South Carolina
Mathematics for the 21st Century
• U.S. Secretary of Education Arne Duncan at
NCTM Annual Meeting, 15 April 2011
• “Curricular materials cover so much ground too
superficially, failing to provide students with an
understanding of the concepts that are essential
for success.”
• “Tests don’t always measure what’s important,
or provide information back to you to help you
Mathematics in the 20th Century
• Abbott and Costello Division
• 28 divided by 7 is 13 or 13 times 7 is 28
• Video
Mindless Application of Algorithms
• Learning without understanding
• Learning without sensitivity to culture
• Learning without consideration of
• Learning without student engagement
Mathematics for the 21st Century
Societal Need for
• Competitiveness
(Knowledge-based global
• Fulfillment (sustainability
in diverse society)
Competitiveness in the 21st
• Technology
• New tools for understanding and
visualizing numerical ideas
• Web-based, hand-held, ubiquitous
• Hans Rosling
• “The Joy of Stats”
Competitiveness in the 21st
• Technology
• New tools for understanding and visualizing
numerical ideas
• Computer Algebra Systems
• On iPads and Smartphones
• Wolfram Alpha
Wolfram Alpha
Competitiveness in the 21st
• Computer Algebra Systems
• Handheld Calculators
• Equation of a line
( y = mx + b) with sliders using
TI Nspire Calculator
Fulfillment in the 21st Century
• Teaching IS a gratifying profession…
Fulfillment the 21st Century
• Cultural Diversity
mathematical content in
a manner that
ENGAGES the learners
who populate our
Cultural Diversity
– Vedic Multiplication 21 x 13 video
Subtraction in Mexico
5 94
NCTM Affiliated Groups
• TODOS advocating for equitable, high quality
mathematics education for all, in particular,
Hispanic/Latino students
NCTM Affiliated Groups
• Benjamin Banneker Association, advocating for the
mathematics education of African-American
Mathematics for the 21st Century
• Common Core State Standards for Mathematics
• An opportunity to address…
–Sense Making
Common Core Standards
• Sponsored by the Council of Chief State School
Officers (CCSS) and the National Governors
Association (NGA)
• First significant attempt to systematically align
K-12 standards across the U.S.
• Building on NCTM’s standards documents from
1980, 1989, 2000, 2006, and 2009
• NCTM among groups providing feedback
Common Core Standards
• Different from most current state standards
• Based on most recent research regarding students’
learning trajectories related to mathematics content
• Includes detailed description of the way
mathematics is learned and used by students
(Mathematical Practice)
Common Core Development
• Initially 48 states and three territories
signed on
• Final Standards released June 2, 2010, at
• Adoption required for Race to the Top
• As of October 1, 2011, 44 states have
officially adopted (plus DC, US VI, & N.
Mariana I)
Common Core Development
• Each state adopting the common core either
directly or by fully aligning its state standards
may do so in accordance with current state
timelines for standards adoption not to
exceed three (3) years.
• States that choose to align their standards to
the common core standards accept 100% of
the core. States may add additional standards.
Benefits for States and Districts
Allows collaborative professional development based
on best practices
Allows development of common assessments and other
tools (SC in SMARTER Balanced and PARCC)
Enables comparison of policies and achievement
across states and districts
Creates potential for collaborative groups to get more
economical mileage for:
– Curriculum development, assessment, and
professional development
South Carolina
• State Board of Education adopted the Common Core for
SC on July 14, 2010
• In November 2010, Mick Zais was elected
Superintendent of Education and with Governor Haley
has chose to not apply for Race to the Top funds
• In May 2011 Senator Mike Fair introduced a Proviso in
the SC Budget to prohibit the SC Department of
Education from spending related to the Common Core
• In October 2011, the SC Board and Department of
Education continue to move toward implementation
CCSSM stands for
Common Core State Standards
for Mathematics
Arne Duncan at the
NCTM Annual Meeting
• “... today’s tests don’t measure higher-order
thinking skills or deep understanding of subject
material. American students deserve better than
the fill-in-the-bubble tests that are now common
across states.”
• New assessments “… are the ones that you’ve
longed for. They will measure critical thinking skills
and complex student learning.”
Common Core - Domain
• Domains are overarching big ideas that
connect topics across the grades
• Descriptions of the mathematical content to
be learned elaborated through clusters and
Common Core - Standards
• Standards are content statements. An example
content statement is: “Use properties of operations
to generate equivalent expressions.”
• Progressions of increasing complexity from grade
to grade
Common Core - Clusters
• May appear in multiple grade levels in the K-8
Common Core. There is increasing development
as the grade levels progress
• What students should know and be able to do at
each grade level
• Reflect both mathematical understandings and
skills, which are equally important
• Fewer and more rigorous
• Aligned with college and career expectations
• Internationally benchmarked
• Rigorous content and application of higherorder skills.
• Builds on strengths and lessons of current
state standards.
• Research based
• Articulated progressions of topics and
performances that are developmental and
connected to other progressions
• Conceptual understanding AND procedural
skills stressed equally
NCTM states coherence also means that
instruction, assessment, and curriculum are
• Key ideas, understandings, and skills are
• Deep learning of concepts is emphasized
– That is, time is spent on a topic and on
learning the topic well. This counters
the “mile wide, inch deep” criticism
leveled at most current U.S. standards.
Math Common Core Resources
• Partnership for the Assessment of Readiness for
College and Career (PARCC)
• Smarter-Balanced Assessment Consortium (SBAC)
• South Carolina participates in BOTH
• Mathematical Practices must be assessed
• Assessments will formative and other assessments
that address reasoning and sense making
• ... and Secretary Duncan’s concern measuring
critical thinking skills and complex student
• Word Cloud
Learning Trajectories
• Descriptions of children’s thinking and learning in a
specific mathematical domain, and
• a related conjectured route through a set of instructional
tasks designed to engender those mental processes or
actions hypothesized to move children through a
developmental progression of levels of thinking,
• created with the intent of supporting children’s
achievement of specific goals in that mathematical
• Clements and Sarama, 2004
Learning Trajectories or Progressions
• Teachers use ordered set of instructional
experiences and tasks
• Students think and learn through a developmental
progression of levels to reach goal
• Van Hiele Levels in Geometry (since 1970s)
• Formative Assessments more recently
Adding Fractions
• As expressed by Hung-Hsi Wu, in Fall 2011, American
• In the past…
• Memorize steps and mimic process without attention to
• Adding whole numbers is “combining”… how is adding of
fractions combining things?
Adding Fractions Learning Trajectory
• In CCSSM, learning trajectories for adding fraction spans
grades 3-5
• In grade 3, students learn to think of a fraction as a point
on a line.
• Unit fractions, like 1/6 and copies of the unit like 5/6
Adding Fractions, Grade 3
• Equivalent fractions
is equal to
3 5
• When each of the 5 segments is divided into 3
equal segments, the unit segment has 3 x 5 =15
equal segments
• 2/5 is the same point as 6/15
Adding Fractions, Grade 4
• Adding fractions is joining two parts of same whole
• Two segments joined end to end 2 5 2  5
 
3 3
• 2 copies of the red segment followed by 5 copies,
so combining 2 + 5 copies, gives you 7
Adding Fractions, Grade 5
1 5
8 6
• Equivalent Fractions:
8 6
Learning Trajectories
• This 5th grade student
got over 80% on a preassessment involving
• Is s/he ready for new 5th
or 6th grade concepts
like multiplication of
decimals (e.g. 2.5 x
• Mathematics curriculum in Japan has long
used learning trajectories developed
from lesson studies.
Distributive Property
• Grade 3 Multiplication
• Grade 4 Properties of Operations
• The verb “to FOIL”
• Memorize vs Understand
• Trajectory for understanding whole number
multiplication to algebra:
• x(a + b)
• x(a + b + c)
• (x + y)(a + b)
• (x + y)(a + b + c)
Binomial Multiplication
Understand logic (multiply all possible pairs)
Don’t memorize rules or tricks: UNDERSTAND
Vedic Method
Breaks down for 97 x 86
CCSSM Mathematical Practices
• Common Core includes a set of Standards
of Mathematical Practices that all teachers
should develop in their students.
• Similar to NCTM’s Mathematical Processes
from the Principles and Standards for
School Mathematics.
• Practices MUST be assessed
Mathematical Practices
• Expectations that begin with “understand” are
especially good opportunities to connect practices to
• “Students who lack understanding of a topic may rely
on procedures too heavily.”
• Understanding standards (intersection of content and
practice) “are intended to be weighted toward
central and generative concepts.. That most merit
time, resources, innovative energies, and focus…”
8 CCSSM Mathematical Practices
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.
Geometry Lesson from Japan
Make Sense / Critique Reasoning
Find the area of the shaded region:
10 m
10 m
Learning Trajectories
• Area is invariant under transformations
Memorize versus Understand
Circumference and Area of a Circle
All dependent on understanding π
Ratio of Circumference and Diameter
Geometer’s Sketchpad
• If C/d = π
• C = d π or 2r π
• Depends on circumference and radius
• Area of a circle, how to get the formula..flv
8 CCSSM Mathematical Practices
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 and Precision
• Why is -2 x -3 = 2 x 3?
• Inductive approach:
• Infer from the pattern that (-1)(-3) = 3
[ by adding 3 ] and then
• (-2)(-3) = 6, again by adding 3
• Is this sound reasoning?
Multiplying Negative Numbers
• It does not generalize … what is (-5/11)(-4/3)?
• How do we KNOW (-1)(-3) is 3 and not INFER?
• There is a deductive approach that can prove that
(-1)(-1) = 1 and if so, then all other cases follow.
Deductive Reasoning
(-1) (- 1) + (-1) =
(-1) (-1) + 1(-1) =
[ (-1) + 1 ] (-1) =
[ 0 ] (-1) =
If (-1)(-1) + (-1) = 0, then (-1)(-1) = 1
Look for and make use of structure
• What is the result when you add…
• Even + Even ?
• Odd + Odd ?
Mod 2 Arithmetic
• Modular arithmetic used in encryption codes..
Vi Hart’s Binary Hand Dance
Mathematical Structures
Vi Hart Blog at
• Base 2
• Modular arithmetic
• Mathematical food
• And other engaging ideas about mathematics
• … BUT now back to the Common Core
High School
Conceptual Categories
• The big ideas that connect mathematics
across high school – such as Functions or
Probability and Statistics
• A progression of increasing complexity
• Description of mathematical content to be
learned elaborated through domains,
clusters, and standards
High School Pathways
• The CCSSM Model Pathways are two
models that organize the CCSSM into
coherent, rigorous courses
• The CCSSM Model Pathways are NOT
required. The two sequences are examples,
not mandates
High School Pathways
• Four years of mathematics:
– One course in each of the first two years
– Followed by two options for year three
and a variety of relevant courses for year
• Course descriptions
– Define what is covered in a course
– Are not prescriptions for the curriculum or
High School Pathways
• Pathway A: Consists of two algebra courses
and a geometry course, with some data,
probability and statistics infused throughout
each (traditional)
• Pathway B: Typically seen internationally that
consists of a sequence of 3 courses each of
which treats aspects of algebra, geometry and
data, probability, and statistics.
NCTM President
Michael Shaughnessy
• An Opportune Time to Consider Integrated
Mathematics March, 2011
• “Students need to see mathematics as an integrated whole,
with connections across the content domains...
• …the United States will never show well in international
comparisons of mathematics performance as long as other
countries have an integrated mathematics, and we take a
“layer cake” approach.
• … we have an unprecedented opportunity… to integrate the
content of our secondary mathematics…”
Promising, Opportune… but Perfect?
Problem areas:
• CCSSM has never been field tested
• Can the assessments address
understanding and measure the Practices?
• How to accommodate exceptional learners?
• Learning trajectories require careful vertical
Not Perfect …
Problem areas:
• Too little technology particularly in K-8
• No statistics in K-5
• How can this be 21st Century
• Piling on in Grade 6
Ideal (according to Ed)
• CCSSM as standards and not mandated
• Give districts choices for implementation
(avoiding a lock-step approach)
• Assessment includes parts addressed by
teachers at the local level (as in Europe)
• Reward success, don’t punish non-success
• TRUST teachers to do the work we hire
them to do
Others “Get it”
NCTM Position on Teacher Evaluation (Oct 2011):
• “Although evidence of student learning can and should be considered
in the evaluation of teachers, it should be only one factor among
• “Comprehensive systems of evaluation of teachers of mathematics
should focus primarily on the following domains of professional
– Lesson planning
– Lesson implementation and instruction
– Demonstration of content knowledge and pedagogical content
– Classroom culture
– Professionalism”
Others “Get it”
Calif. Gov. Jerry Brown October 2011 education bill veto:
• “… nowhere mentions character or love of
learning….It does allude to student excitement
and creativity, but does not take these qualities
seriously because they can’t be placed in a data
• “teachers are forced to curb their own creativity
and engagement with students as they focus on
teaching to the test.”
• We don’t buy a dog, then bark for it.
• Invest in the best pre- and in-service teacher
• Then get out of the way and let teachers do
what these very intelligent professionals
were educated to do
• It works in Finland…
• … and it can work in South Carolina!
2012 Institutes
High School Reasoning & Sense Making:
July 24-26, Los Angeles, California
K-8 Algebra Readiness Institute:
July 31- August 2, Atlanta, Georgia
Additional Information
• For grades preK-8, a model of
implementation can be found in
NCTM’s Curriculum Focal Points
• For the secondary level, please see
NCTM’s Focus in High School
Mathematics: Reasoning and Sense
• Brown, Jr., Edmund G (2011. Veto Message Letter for California
State Senate Bill 547. Retrieved from
• CCSSO/NGA. (2010). Common core state standards for
mathematics. Washington, DC: Council of Chief State School
Officers and the National Governors Association Center for Best
Practices. Retrieved from
• Clements, D., & Sarama, J. (2004). Learning trajectories in
mathematics education. Mathematical Thinking and Learning, 6(2),
• Clements, D., & Sarama, J. (2009). Learning and Teaching Early
Math: The learning Trajectories Approach.New York: Routledge.
• Daro, Phil, Frederic Moser, and Tom Corcoran( (2011) Learning Trajectories in
Mathematics. Retrieved from
• Dojinsha, Kyoiku. Mathematics Workbook (Grade 1 to 6). Global Education
• National Council of Teachers of Mathematics. Position Statement on Teacher
Evaluation. Retrieved from
• Shaughnessy, J. Michael. An Opportune Time to Consider Integrated
Mathematics. NCTM Summing Up, March 2011. Retrieved from
• Wu, Hung-Hsi (2011) Phoenix Rising: Bring the Common Core State Mathematics
Standards to Life. American Educator, Fall 2011. Retrieved from
Web Resources
Wolfram Alpha:
Common Core:
Math Common Core Resources:
South Carolina Common Core:
Vi Hart Blog:
• Abbott and Costello show 13 x 7 = 28:
• Hans Rosling and 200 Countries:
• TI Nspire Sliders for y = mx + b:
• Vedic Multiplication:
• Area of a Circle:
• Binary Hand Dance:
Thank you… go forth to
reason and make
[email protected]

Powerpoint - College of Education