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Title: Reasoning and Sense Making Through the Common Core State Standards for Mathematics (CCCSM)


1
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
2
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.

3
Mathematics in the 20th Century
  • Abbott and Costello Division
  • 28 divided by 7 is 13 or 13 times 7 is 28
  • Video

4
Mindless Application of Algorithms
  • Learning without understanding
  • Learning without sensitivity to culture
  • Learning without consideration of technology
  • Learning without student engagement

5
Mathematics for the 21st Century
  • Societal Need for
  • Competitiveness (Knowledge-based global economy)
  • Fulfillment (sustainability in diverse society)

6
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

7
Competitiveness in the 21st Century
  • Technology
  • New tools for understanding and visualizing
    numerical ideas
  • Computer Algebra Systems
  • On iPads and Smartphones
  • Wolfram Alpha

8
Wolfram Alpha
9
Competitiveness in the 21st Century
  • Computer Algebra Systems
  • Handheld Calculators
  • Equation of a line
  • ( y mx b) with sliders using
  • TI Nspire Calculator

10
Fulfillment in the 21st Century
  • Teaching IS a gratifying profession

11
Fulfillment the 21st Century
  • Cultural Diversity

Understanding mathematical content in a manner
that ENGAGES the learners who populate our
classrooms
12
Cultural Diversity
13
Cultural Diversity
  • Vedic Multiplication 21 x 13 video

14
Subtraction in Mexico
963 -369
1 4 9
1 7 4
5
15
NCTM Affiliated Groups
  • TODOS advocating for equitable, high quality
    mathematics education for all, in particular,
    Hispanic/Latino students
  • www.todos-math.org

16
NCTM Affiliated Groups
  • Benjamin Banneker Association, advocating for the
    mathematics education of African-American
    students
  • www.bannekermath.org

17
Mathematics for the 21st Century
  • Common Core State Standards for Mathematics
  • An opportunity to address
  • Competitiveness
  • Fulfillment
  • Reasoning
  • Sense Making

18
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

19
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)

20
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)

21
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.

22
(No Transcript)
23
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

24
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

25
CCSSM
  • CCSSM stands for
  • Common Core State Standards for Mathematics

26
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.

27
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

28
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

29
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

30
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

31
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

32
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.

33
Math Common Core Resources
  • http//www.nctm.org/standards/mathcommoncore/

34
Assessment
  • Partnership for the Assessment of Readiness for
    College and Career (PARCC)
  • Smarter-Balanced Assessment Consortium (SBAC)
  • South Carolina participates in BOTH

35
PARCC
36
SBAC
37
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.

38
CCSSM
  • Word Cloud

39
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

40
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

41
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?

42
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

43
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

44
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

45
Adding Fractions, Grade 5
  • Equivalent Fractions and

46
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)?

47
Japan
  • Mathematics curriculum in Japan has long used
    learning trajectories developed from lesson
    studies.

48
Distributive Property
  • Grade 3 Multiplication
  • Grade 4 Properties of Operations

49
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)

50
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

51
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

52
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

53
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.

54
Geometry Lesson from Japan
55
Make Sense / Critique Reasoning
Find the area of the shaded region
56
Reasoning
57
Learning Trajectories
  • Area is invariant under transformations

58
Formulas
  • Memorize versus Understand
  • Circumference and Area of a Circle
  • All dependent on understanding p
  • Ratio of Circumference and Diameter

59
Geometers Sketchpad
60
Circumference
  • If C/d p
  • C d p or 2r p

61
Area
  • Depends on circumference and radius
  • Area of a circle, how to get the formula..flv

62
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.

63
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?

64
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.

65
Deductive Reasoning
  • (-1) (- 1) (-1)
  • (-1) (-1) 1(-1)
  • (-1) 1 (-1)
  • 0 (-1)
  • 0
  • If (-1)(-1) (-1) 0, then (-1)(-1) 1

66
Look for and make use of structure
  • What is the result when you add
  • Even Even ?
  • Odd Odd ?

67
Mod 2 Arithmetic
  • Modular arithmetic used in encryption codes..

68
Vi Harts Binary Hand Dance
69
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

70
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

71
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

72
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

73
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.

74
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

75
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

76
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

77
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

78
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 practice 
  • Lesson planning
  • Lesson implementation and instruction
  • Demonstration of content knowledge and
    pedagogical content knowledge
  • Classroom culture
  • Professionalism

79
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.

80
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!

81
2012 Institutes High School Reasoning Sense
Making July 24-26, Los Angeles,
California K-8 Algebra Readiness Institute July
31- August 2, Atlanta, Georgia
82
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
83
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.

84
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

85
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

86
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

87
Thank you go forth to reason and make
sense! ed.dickey_at_sc.edu www.ite.sc.edu/dickey.h
tml
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