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Reasoning and Sense Making Through the
Common Core State Standards for Mathematics (CCCSM)
Ed DickeyCollege of Education
University of South Carolina
SCCTM Annual ConferenceCharleston SC
October 28 2011
Mathematics for the 21st Century Classroom
bull US Secretary of Education Arne Duncan at NCTM Annual Meeting 15 April 2011
bull ldquoCurricular materials cover so much ground too superficially failing to provide students with an understanding of the concepts that are essential for successrdquo
bull ldquoTests donrsquot always measure whatrsquos important or provide information back to you to help you improverdquo
Mathematics in the 20th Century
bull Abbott and Costello Divisionbull 28 divided by 7 is 13 or 13 times 7 is 28bull Video
Mindless Application of Algorithms
bull Learning without understanding
bull Learning without sensitivity to culture
bull Learning without consideration of technology
bull Learning without student engagement
Mathematics for the 21st Century
Societal Need forbullCompetitiveness (Knowledge-based global economy)
bullFulfillment (sustainability in diverse society)
Competitiveness in the 21st Century
bull Technologybull New tools for understanding and
visualizing numerical ideasbull Web-based hand-held ubiquitous bull Hans Rosling bull ldquoThe Joy of Statsrdquo
Competitiveness in the 21st Century
bull Technology
bull New tools for understanding and visualizing numerical ideas
bull Computer Algebra Systems
bull On iPads and Smartphones
bull Wolfram Alpha
Wolfram Alpha
Competitiveness in the 21st Century
bull Computer Algebra Systems
bull Handheld Calculatorsbull Equation of a line ( y = mx + b) with sliders
using TI Nspire Calculator
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Mathematics for the 21st Century Classroom
bull US Secretary of Education Arne Duncan at NCTM Annual Meeting 15 April 2011
bull ldquoCurricular materials cover so much ground too superficially failing to provide students with an understanding of the concepts that are essential for successrdquo
bull ldquoTests donrsquot always measure whatrsquos important or provide information back to you to help you improverdquo
Mathematics in the 20th Century
bull Abbott and Costello Divisionbull 28 divided by 7 is 13 or 13 times 7 is 28bull Video
Mindless Application of Algorithms
bull Learning without understanding
bull Learning without sensitivity to culture
bull Learning without consideration of technology
bull Learning without student engagement
Mathematics for the 21st Century
Societal Need forbullCompetitiveness (Knowledge-based global economy)
bullFulfillment (sustainability in diverse society)
Competitiveness in the 21st Century
bull Technologybull New tools for understanding and
visualizing numerical ideasbull Web-based hand-held ubiquitous bull Hans Rosling bull ldquoThe Joy of Statsrdquo
Competitiveness in the 21st Century
bull Technology
bull New tools for understanding and visualizing numerical ideas
bull Computer Algebra Systems
bull On iPads and Smartphones
bull Wolfram Alpha
Wolfram Alpha
Competitiveness in the 21st Century
bull Computer Algebra Systems
bull Handheld Calculatorsbull Equation of a line ( y = mx + b) with sliders
using TI Nspire Calculator
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Mathematics in the 20th Century
bull Abbott and Costello Divisionbull 28 divided by 7 is 13 or 13 times 7 is 28bull Video
Mindless Application of Algorithms
bull Learning without understanding
bull Learning without sensitivity to culture
bull Learning without consideration of technology
bull Learning without student engagement
Mathematics for the 21st Century
Societal Need forbullCompetitiveness (Knowledge-based global economy)
bullFulfillment (sustainability in diverse society)
Competitiveness in the 21st Century
bull Technologybull New tools for understanding and
visualizing numerical ideasbull Web-based hand-held ubiquitous bull Hans Rosling bull ldquoThe Joy of Statsrdquo
Competitiveness in the 21st Century
bull Technology
bull New tools for understanding and visualizing numerical ideas
bull Computer Algebra Systems
bull On iPads and Smartphones
bull Wolfram Alpha
Wolfram Alpha
Competitiveness in the 21st Century
bull Computer Algebra Systems
bull Handheld Calculatorsbull Equation of a line ( y = mx + b) with sliders
using TI Nspire Calculator
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Mindless Application of Algorithms
bull Learning without understanding
bull Learning without sensitivity to culture
bull Learning without consideration of technology
bull Learning without student engagement
Mathematics for the 21st Century
Societal Need forbullCompetitiveness (Knowledge-based global economy)
bullFulfillment (sustainability in diverse society)
Competitiveness in the 21st Century
bull Technologybull New tools for understanding and
visualizing numerical ideasbull Web-based hand-held ubiquitous bull Hans Rosling bull ldquoThe Joy of Statsrdquo
Competitiveness in the 21st Century
bull Technology
bull New tools for understanding and visualizing numerical ideas
bull Computer Algebra Systems
bull On iPads and Smartphones
bull Wolfram Alpha
Wolfram Alpha
Competitiveness in the 21st Century
bull Computer Algebra Systems
bull Handheld Calculatorsbull Equation of a line ( y = mx + b) with sliders
using TI Nspire Calculator
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Mathematics for the 21st Century
Societal Need forbullCompetitiveness (Knowledge-based global economy)
bullFulfillment (sustainability in diverse society)
Competitiveness in the 21st Century
bull Technologybull New tools for understanding and
visualizing numerical ideasbull Web-based hand-held ubiquitous bull Hans Rosling bull ldquoThe Joy of Statsrdquo
Competitiveness in the 21st Century
bull Technology
bull New tools for understanding and visualizing numerical ideas
bull Computer Algebra Systems
bull On iPads and Smartphones
bull Wolfram Alpha
Wolfram Alpha
Competitiveness in the 21st Century
bull Computer Algebra Systems
bull Handheld Calculatorsbull Equation of a line ( y = mx + b) with sliders
using TI Nspire Calculator
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Competitiveness in the 21st Century
bull Technologybull New tools for understanding and
visualizing numerical ideasbull Web-based hand-held ubiquitous bull Hans Rosling bull ldquoThe Joy of Statsrdquo
Competitiveness in the 21st Century
bull Technology
bull New tools for understanding and visualizing numerical ideas
bull Computer Algebra Systems
bull On iPads and Smartphones
bull Wolfram Alpha
Wolfram Alpha
Competitiveness in the 21st Century
bull Computer Algebra Systems
bull Handheld Calculatorsbull Equation of a line ( y = mx + b) with sliders
using TI Nspire Calculator
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Competitiveness in the 21st Century
bull Technology
bull New tools for understanding and visualizing numerical ideas
bull Computer Algebra Systems
bull On iPads and Smartphones
bull Wolfram Alpha
Wolfram Alpha
Competitiveness in the 21st Century
bull Computer Algebra Systems
bull Handheld Calculatorsbull Equation of a line ( y = mx + b) with sliders
using TI Nspire Calculator
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Wolfram Alpha
Competitiveness in the 21st Century
bull Computer Algebra Systems
bull Handheld Calculatorsbull Equation of a line ( y = mx + b) with sliders
using TI Nspire Calculator
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Competitiveness in the 21st Century
bull Computer Algebra Systems
bull Handheld Calculatorsbull Equation of a line ( y = mx + b) with sliders
using TI Nspire Calculator
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Fulfillment in the 21st Century
bull Teaching IS a gratifying professionhellip
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Fulfillment the 21st Century
bull Cultural Diversity
Understanding mathematical content in a manner that ENGAGES the learners who populate our classrooms
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Cultural Diversity
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Cultural Diversity
ndash Vedic Multiplication 21 x 13 video
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Subtraction in Mexico
14
963-369
1
7
4
1
4
9
5
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
NCTM Affiliated Groups
bull TODOS advocating for equitable high quality mathematics education for all in particular HispanicLatino students
bull wwwtodos-mathorg
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
NCTM Affiliated Groups
bull Benjamin Banneker Association advocating for the mathematics education of African-American students
bull wwwbannekermathorg
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Mathematics for the 21st Century
bull Common Core State Standards for Mathematicsbull An opportunity to addresshellip
ndashCompetitiveness
ndashFulfillment
ndashReasoning
ndashSense Making
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Common Core Standards
bull Sponsored by the Council of Chief State School Officers (CCSS) and the National Governors Association (NGA)
bull First significant attempt to systematically align K-12 standards across the US
bull Building on NCTMrsquos standards documents from 1980 1989 2000 2006 and 2009
bull NCTM among groups providing feedback
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Common Core Standards
bull Different from most current state standardsbull Based on most recent research regarding studentsrsquo
learning trajectories related to mathematics contentbull Includes detailed description of the way
mathematics is learned and used by students (Mathematical Practice)
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Common Core Development
bull Initially 48 states and three territories signed on
bull Final Standards released June 2 2010 at wwwcorestandardsorg
bull Adoption required for Race to the Top funds
bull As of October 1 2011 44 states have officially adopted (plus DC US VI amp N Mariana I)
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Common Core Development
bull 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
bull 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
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Benefits for States and Districts
bull Allows collaborative professional development based on best practices
bull Allows development of common assessments and other tools (SC in SMARTER Balanced and PARCC)
bull Enables comparison of policies and achievement across states and districts
bull Creates potential for collaborative groups to get more economical mileage forndash Curriculum development assessment and
professional development
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
South Carolina
bull State Board of Education adopted the Common Core for SC on July 14 2010
bull 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
bull 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
bull 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
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
CCSSM
CCSSM stands for
Common Core State Standards for Mathematics
Arne Duncan at the NCTM Annual Meeting
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Arne Duncan at the NCTM Annual Meeting
bull ldquo todayrsquos tests donrsquot 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 statesrdquo
bull New assessments ldquohellip are the ones that yoursquove longed for They will measure critical thinking skills and complex student learningrdquo
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Common Core - Domain
bull Domains are overarching big ideas that connect topics across the grades
bull Descriptions of the mathematical content to be learned elaborated through clusters and standards
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Common Core - Standards
bull Standards are content statements An example content statement is ldquoUse properties of operations to generate equivalent expressionsrdquo
bull Progressions of increasing complexity from grade to grade
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Common Core - Clusters
bull May appear in multiple grade levels in the K-8 Common Core There is increasing development as the grade levels progress
bull What students should know and be able to do at each grade level
bull Reflect both mathematical understandings and skills which are equally important
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Characteristics
bull Fewer and more rigorous
bull Aligned with college and career expectations
bull Internationally benchmarked
bull Rigorous content and application of higher-order skills
bull Builds on strengths and lessons of current state standards
bull Research based
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Coherence
bull Articulated progressions of topics and performances that are developmental and connected to other progressions
bull Conceptual understanding AND procedural skills stressed equally
NCTM states coherence also means that instruction assessment and curriculum are aligned
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Focus
bull Key ideas understandings and skills are identified
bull Deep learning of concepts is emphasized
ndash That is time is spent on a topic and on learning the topic well This counters the ldquomile wide inch deeprdquo criticism leveled at most current US standards
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Math Common Core Resources
bull httpwwwnctmorgstandardsmathcommoncore
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Assessment
bull Partnership for the Assessment of Readiness for College and Career (PARCC)
bull Smarter-Balanced Assessment Consortium (SBAC)
bull South Carolina participates in BOTH
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
PARCC
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
SBAC
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
PARCC or SBAC
bull Mathematical Practices must be assessed
bull Assessments will formative and other assessments that address reasoning and sense making
bull and Secretary Duncanrsquos concern measuring critical thinking skills and complex student learning
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
bull Word Cloud
CCSSM
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Learning Trajectories
bull Descriptions of childrenrsquos thinking and learning in a specific mathematical domain and
bull 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
bull created with the intent of supporting childrenrsquos achievement of specific goals in that mathematical domain
bull Clements and Sarama 2004
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Learning Trajectories or Progressions
bull Teachers use ordered set of instructional experiences and tasks
bull Students think and learn through a developmental progression of levels to reach goal
bull Van Hiele Levels in Geometry (since 1970s)
bull Formative Assessments more recently
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Adding Fractions
bull As expressed by Hung-Hsi Wu in Fall 2011 American Educator
bull In the pasthellip
bull Memorize steps and mimic process without attention to understanding
bull Adding whole numbers is ldquocombiningrdquohellip how is adding of fractions combining things
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Adding Fractions Learning Trajectory
bull In CCSSM learning trajectories for adding fraction spans grades 3-5
bull In grade 3 students learn to think of a fraction as a point on a line
bull Unit fractions like 16 and copies of the unit like 56
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Adding Fractions Grade 3bull Equivalent fractionsbull is equal to or
bull When each of the 5 segments is divided into 3 equal segments the unit segment has 3 x 5 =15 equal segments
bull 25 is the same point as 615
5323
156
52
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Adding Fractions Grade 4
bull Adding fractions is joining two parts of same whole
bull Two segments joined end to end
bull 2 copies of the red segment followed by 5 copies so combining 2 + 5 copies gives you
352
35
32
37
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Adding Fractions Grade 5
bull Equivalent Fractions and
65
81
8616
6858
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Learning Trajectories
bull This 5th grade student got over 80 on a pre-assessment involving multiplication
bull Is she ready for new 5th or 6th grade concepts like multiplication of decimals (eg 25 x 078)
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
bull Mathematics curriculum in Japan has long used learning trajectories developed from lesson studies
Japan
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Distributive Property
bull Grade 3 Multiplicationbull Grade 4 Properties of Operations
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
FOIL
bull The verb ldquoto FOILrdquobull Memorize vs Understand
bull Trajectory for understanding whole number multiplication to algebra
bull x(a + b)bull x(a + b + c)bull (x + y)(a + b)bull (x + y)(a + b + c)
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Binomial Multiplication
bull Understand logic (multiply all possible pairs)bull Donrsquot memorize rules or tricks
UNDERSTANDbull Vedic Method bull Breaks down for 97 x 86bull Distributivebull Verticalbull Gridbull Tile
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
CCSSM Mathematical Practices
bull Common Core includes a set of Standards of Mathematical Practices that all teachers should develop in their students
bull Similar to NCTMrsquos Mathematical Processes from the Principles and Standards for School Mathematics
bull Practices MUST be assessed
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Mathematical Practices
bull Expectations that begin with ldquounderstandrdquo are especially good opportunities to connect practices to content
bull ldquoStudents who lack understanding of a topic may rely on procedures too heavilyrdquo
bull Understanding standards (intersection of content and practice) ldquoare intended to be weighted toward central and generative concepts That most merit time resources innovative energies and focushelliprdquo
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
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
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Geometry Lesson from Japan
Make Sense Critique Reasoning
Find the area of the shaded region
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Make Sense Critique Reasoning
Find the area of the shaded region
8 m 2 m 5 m
8 m
3 m 2m 10 m
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Reasoning
8 m 2 m 5 m
8 m
3 m 2m 10 m
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Learning Trajectories
bull Area is invariant under transformations
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Formulas
bull Memorize versus Understandbull Circumference and Area of a Circlebull All dependent on understanding πbull Ratio of Circumference and
Diameter
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Geometerrsquos Sketchpad
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Circumference
bullIf Cd = πbullC = d π or 2r π
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Area
bull Depends on circumference and radiusbull Area of a circle how to get the formulaflv
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
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
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Reasoning and Precision
bull Why is -2 x -3 = 2 x 3bull Inductive approach
bull Infer from the pattern that (-1)(-3) = 3 [ by adding 3 ] and thenbull (-2)(-3) = 6 again by adding 3bull Is this sound reasoning
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Multiplying Negative Numbers
bull It does not generalize hellip what is (-511)(-43)
bull How do we KNOW (-1)(-3) is 3 and not INFER
bull There is a deductive approach that can prove that (-1)(-1) = 1 and if so then all other cases follow
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Deductive Reasoning
bull (-1) (- 1) + (-1) =bull (-1) (-1) + 1(-1) =bull [ (-1) + 1 ] (-1) =bull [ 0 ] (-1) =bull 0bull If (-1)(-1) + (-1) = 0 then (-1)(-1) = 1
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Look for and make use of structure
bull What is the result when you addhellipbull Even + Even bull Odd + Odd
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Mod 2 Arithmetic
bull Modular arithmetic used in encryption codes
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Vi Hartrsquos Binary Hand Dance
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Mathematical Structures
Vi Hart Blog at vihartcombullBase 2bullModular arithmeticbullMathematical foodbullAnd other engaging ideas about mathematicsbullhellip BUT now back to the Common Core
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
High School Conceptual Categories
bull The big ideas that connect mathematics across high school ndash such as Functions or Probability and Statistics
bull A progression of increasing complexity
bull Description of mathematical content to be learned elaborated through domains clusters and standards
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
High School Pathways
bull The CCSSM Model Pathways are two models that organize the CCSSM into coherent rigorous courses
bull The CCSSM Model Pathways are NOT required The two sequences are examples not mandates
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
High School Pathways
bull Four years of mathematics ndash One course in each of the first two yearsndash Followed by two options for year three and
a variety of relevant courses for year fourbull Course descriptions
ndash Define what is covered in a course ndash Are not prescriptions for the curriculum or
pedagogy
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
High School Pathways
bull Pathway A Consists of two algebra courses and a geometry course with some data probability and statistics infused throughout each (traditional)
bull 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
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
NCTM President Michael Shaughnessy
bull An Opportune Time to Consider Integrated Mathematics March 2011
bull ldquoStudents need to see mathematics as an integrated whole with connections across the content domains
bull hellipthe 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 ldquolayer cakerdquo approach
bull hellip we have an unprecedented opportunityhellip to integrate the content of our secondary mathematicshelliprdquo
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Promising Opportunehellip but Perfect
Problem areas
bull CCSSM has never been field tested
bull Can the assessments address understanding and measure the Practices
bull How to accommodate exceptional learners
bull Learning trajectories require careful vertical articulation
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Not Perfect hellip
Problem areas
bull Too little technology particularly in K-8
bull No statistics in K-5
bull How can this be 21st Century competitive
bull Piling on in Grade 6
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Ideal (according to Ed)
bull CCSSM as standards and not mandated curriculum
bull Give districts choices for implementation (avoiding a lock-step approach)
bull Assessment includes parts addressed by teachers at the local level (as in Europe)
bull Reward success donrsquot punish non-successbull TRUST teachers to do the work we hire
them to do
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Others ldquoGet itrdquoNCTM Position on Teacher Evaluation (Oct 2011)bullldquoAlthough evidence of student learning can and should be considered in the evaluation of teachers it should be only one factor among many bullldquoComprehensive systems of evaluation of teachers of mathematics should focus primarily on the following domains of professional practice
ndash Lesson planning ndash Lesson implementation and instruction ndash Demonstration of content knowledge and pedagogical
content knowledge ndash Classroom culturendash Professionalismrdquo
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Others ldquoGet itrdquo
Calif Gov Jerry Brown October 2011 education bill veto
bullldquohellip nowhere mentions character or love of learninghellipIt does allude to student excitement and creativity but does not take these qualities seriously because they canrsquot be placed in a data streamrdquobullldquoteachers are forced to curb their own creativity and engagement with students as they focus on teaching to the testrdquo
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Ideal
bull We donrsquot buy a dog then bark for it
bull Invest in the best pre- and in-service teacher developmenthellip
bull Then get out of the way and let teachers do what these very intelligent professionals were educated to do
bull It works in Finlandhellip
bull hellip and it can work in South Carolina
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
SCCTM
2012 Institutes
High School Reasoning amp Sense Making July 24-26 Los Angeles California
K-8 Algebra Readiness InstituteJuly 31- August 2 Atlanta Georgia
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Additional Information
bull For grades preK-8 a model of implementation can be found in NCTMrsquos Curriculum Focal Points
bull For the secondary level please see NCTMrsquos Focus in High School Mathematics Reasoning and Sense Making
wwwnctmorgcfp
wwwnctmorgFHSM
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Citations
bull Brown Jr Edmund G (2011 Veto Message Letter for California State Senate Bill 547 Retrieved from httpgovcagovdocsSB_547_Veto_Messagepdf
bull CCSSONGA (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 httpcorestandardsorg
bull Clements D amp Sarama J (2004) Learning trajectories in mathematics education Mathematical Thinking and Learning 6(2) 81-89
bull Clements D amp Sarama J (2009) Learning and Teaching Early Math The learning Trajectories ApproachNew York Routledge
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Citation
bull Daro Phil Frederic Moser and Tom Corcoran( (2011) Learning Trajectories in Mathematics Retrieved from httpwwwcpreorgcciiimagesstoriesccii_pdfslearning20trajectories20in20math_ccii20reportpdf
bull Dojinsha Kyoiku Mathematics Workbook (Grade 1 to 6) Global Education Resources httpwwwglobaledresourcescom
bull National Council of Teachers of Mathematics Position Statement on Teacher Evaluation Retrieved from httpwwwnctmorgaboutcontentaspxid=31267
bull Shaughnessy J Michael An Opportune Time to Consider Integrated Mathematics NCTM Summing Up March 2011 Retrieved from httpwwwnctmorgaboutcontentaspxid=28655
bull Wu Hung-Hsi (2011) Phoenix Rising Bring the Common Core State Mathematics Standards to Life American Educator Fall 2011 Retrieved from httpwwwaftorgpdfsamericaneducatorfall2011Wupdf
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Web Resources
bull Wolfram Alpha httpwwwwolframalphacombull TODOS httpwwwtodos-mathorgbull Banneker httpwwwbannekermathorgbull Common Core httpwwwcorestandardsorgbull Math Common Core Resources
httpwwwnctmorgstandardsmathcommoncore bull PARRC httpwwwparcconlineorgbull SBAC httpwwwk12waussmarter bull South Carolina Common Core
httpedscgovagencyprstandards-and-curriculumSouth_Carolina_Common_Corecfm
bull Vi Hart Blog httpvihartcom
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Videosbull Abbott and Costello show 13 x 7 = 28
httpwwwyoutubecomwatchv=rLprXHbn19Ibull Hans Rosling and 200 Countries
httpwwwyoutubecomwatchv=jbkSRLYSojo bull TI Nspire Sliders for y = mx + b
httpwwwyoutubecomwatchv=fiD0vBjLN5E bull Vedic Multiplication httpwwwyoutubecomwatch
v=46oviWU-sQY bull Area of a Circle
httpwwwyoutubecomusermathematicsonlinepau1YokKp3pwVFc
bull Binary Hand Dance httpwwwyoutubecomwatchv=OCYZTg3jahU
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
Thank youhellip go forth to reason and make
sense
eddickeyscedu
wwwitescedudickeyhtml
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