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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 Classroom

- 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 dont always measure whats important, or

provide information back to you to help you

improve.

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 technology
- Learning without student engagement

Mathematics for the 21st Century

- Societal Need for
- Competitiveness (Knowledge-based global economy)
- Fulfillment (sustainability in diverse society)

Competitiveness in the 21st Century

- Technology
- New tools for understanding and visualizing

numerical ideas - Web-based, hand-held, ubiquitous
- Hans Rosling
- The Joy of Stats

Competitiveness in the 21st Century

- Technology
- New tools for understanding and visualizing

numerical ideas - Computer Algebra Systems
- On iPads and Smartphones
- Wolfram Alpha

Wolfram Alpha

Competitiveness in the 21st Century

- 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

Understanding mathematical content in a manner

that ENGAGES the learners who populate our

classrooms

Cultural Diversity

Cultural Diversity

- Vedic Multiplication 21 x 13 video

Subtraction in Mexico

963 -369

1 4 9

1 7 4

5

NCTM Affiliated Groups

- TODOS advocating for equitable, high quality

mathematics education for all, in particular,

Hispanic/Latino students - www.todos-math.org

NCTM Affiliated Groups

- Benjamin Banneker Association, advocating for the

mathematics education of African-American

students - www.bannekermath.org

Mathematics for the 21st Century

- Common Core State Standards for Mathematics
- An opportunity to address
- Competitiveness
- Fulfillment
- Reasoning
- 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 NCTMs 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

www.corestandards.org - Adoption required for Race to the Top funds
- 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.

(No Transcript)

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

- CCSSM stands for
- Common Core State Standards for Mathematics

Arne Duncan at the NCTM Annual Meeting

- ... todays tests dont 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 youve

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 standards

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

Characteristics

- Fewer and more rigorous
- Aligned with college and career expectations
- Internationally benchmarked
- Rigorous content and application of higher-order

skills. - Builds on strengths and lessons of current state

standards. - Research based

Coherence

- 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

aligned

Focus

- Key ideas, understandings, and skills are

identified - 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

- http//www.nctm.org/standards/mathcommoncore/

Assessment

- Partnership for the Assessment of Readiness for

College and Career (PARCC) - Smarter-Balanced Assessment Consortium (SBAC)
- South Carolina participates in BOTH

PARCC

SBAC

PARCC or SBAC

- Mathematical Practices must be assessed
- Assessments will formative and other assessments

that address reasoning and sense making - ... and Secretary Duncans concern measuring

critical thinking skills and complex student

learning.

CCSSM

- Word Cloud

Learning Trajectories

- Descriptions of childrens 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 childrens

achievement of specific goals in that

mathematical domain. - 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 Educator - In the past
- Memorize steps and mimic process without

attention to understanding - 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 or
- 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 copies of the red segment followed by 5 copies,

so combining 2 5 copies, gives you

Adding Fractions, Grade 5

- Equivalent Fractions and

Learning Trajectories

- This 5th grade student got over 80 on a

pre-assessment involving multiplication - Is s/he ready for new 5th or 6th grade concepts

like multiplication of decimals (e.g. 2.5 x 0.78)?

Japan

- Mathematics curriculum in Japan has long used

learning trajectories developed from lesson

studies.

Distributive Property

- Grade 3 Multiplication
- Grade 4 Properties of Operations

FOIL

- 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)
- Dont memorize rules or tricks UNDERSTAND
- Vedic Method
- Breaks down for 97 x 86
- Distributive
- Vertical
- Grid
- Tile

CCSSM Mathematical Practices

- Common Core includes a set of Standards of

Mathematical Practices that all teachers should

develop in their students. - Similar to NCTMs 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 content. - 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

- Make sense of problems and persevere in solving

them. - Reason abstractly and quantitatively.
- Construct viable arguments and critique the

reasoning of others. - Model with mathematics.

Geometry Lesson from Japan

Make Sense / Critique Reasoning

Find the area of the shaded region

Reasoning

Learning Trajectories

- Area is invariant under transformations

Formulas

- Memorize versus Understand
- Circumference and Area of a Circle
- All dependent on understanding p
- Ratio of Circumference and Diameter

Geometers Sketchpad

Circumference

- If C/d p
- C d p or 2r p

Area

- 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.

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)
- 0
- 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 Harts Binary Hand Dance

Mathematical Structures

- Vi Hart Blog at vihart.com
- 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 four - Course descriptions
- Define what is covered in a course
- Are not prescriptions for the curriculum or

pedagogy

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

articulation

Not Perfect

- Problem areas
- Too little technology particularly in K-8
- No statistics in K-5
- How can this be 21st Century competitive?
- Piling on in Grade 6

Ideal (according to Ed)

- CCSSM as standards and not mandated curriculum
- 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, dont 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

many. - Comprehensive systems of evaluation of teachers

of mathematics should focus primarily on the

following domains of professional practiceÂ - Lesson planning
- Lesson implementation and instruction
- Demonstration of content knowledge and

pedagogical content knowledge - 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 cant be placed in a data

stream. - teachers are forced to curb their own creativity

and engagement with students as they focus on

teaching to the test.

Ideal

- We dont buy a dog, then bark for it.
- Invest in the best pre- and in-service teacher

development. - 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 NCTMs Curriculum Focal Points - For the secondary level, please see NCTMs Focus

in High School Mathematics Reasoning and Sense

Making

www.nctm.org/cfp

www.nctm.org/FHSM

Citations

- Brown, Jr., Edmund G (2011. Veto Message Letter

for California State Senate Bill 547. Retrieved

from http//gov.ca.gov/docs/SB_547_Veto_Message.pd

f - 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 http//corestandards.org/ - Clements, D., Sarama, J. (2004). Learning

trajectories in mathematics education.

Mathematical Thinking and Learning, 6(2), 81-89. - Clements, D., Sarama, J. (2009). Learning and

Teaching Early Math The learning Trajectories

Approach.New York Routledge.

Citation

- Daro, Phil, Frederic Moser, and Tom Corcoran(

(2011) Learning Trajectories in Mathematics.

Retrieved from http//www.cpre.org/ccii/images/sto

ries/ccii_pdfs/learning20trajectories20in20math

_ccii20report.pdf - Dojinsha, Kyoiku. Mathematics Workbook (Grade 1

to 6). Global Education Resources,

http//www.globaledresources.com/ - National Council of Teachers of Mathematics.

Position Statement on Teacher Evaluation.

Retrieved from http//www.nctm.org/about/content.a

spx?id31267 - Shaughnessy, J. Michael. An Opportune Time to

Consider Integrated Mathematics. NCTM Summing

Up, March 2011. Retrieved from http//www.nctm.org

/about/content.aspx?id28655 - Wu, Hung-Hsi (2011) Phoenix Rising Bring the

Common Core State Mathematics Standards to Life.

American Educator, Fall 2011. Retrieved from

http//www.aft.org/pdfs/americaneducator/fall2011/

Wu.pdf

Web Resources

- Wolfram Alpha http//www.wolframalpha.com
- TODOS http//www.todos-math.org
- Banneker http//www.bannekermath.org
- Common Core http//www.corestandards.org
- Math Common Core Resources http//www.nctm.org/st

andards/mathcommoncore/ - PARRC http//www.parcconline.org/
- SBAC http//www.k12.wa.us/smarter/
- South Carolina Common Core http//ed.sc.gov/agenc

y/pr/standards-and-curriculum/South_Carolina_Commo

n_Core.cfm - Vi Hart Blog http//vihart.com

Videos

- Abbott and Costello show 13 x 7 28

http//www.youtube.com/watch?vrLprXHbn19I - Hans Rosling and 200 Countries

http//www.youtube.com/watch?vjbkSRLYSojo - TI Nspire Sliders for y mx b

http//www.youtube.com/watch?vfiD0vBjLN5E - Vedic Multiplication http//www.youtube.com/watc

h?v46oviWU-sQY - Area of a Circle http//www.youtube.com/user/mat

hematicsonlinep/a/u/1/YokKp3pwVFc - Binary Hand Dance http//www.youtube.com/watch?v

OCYZTg3jahU

Thank you go forth to reason and make

sense! ed.dickey_at_sc.edu www.ite.sc.edu/dickey.h

tml