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Implementing the NYS P-12 Common Core Learning Standards for Mathematics Please visit for additional information regarding the Common Core Learning Standardswww.engageNY.org 1 New York State Education Department. (2011). EngageNY. Retrieved from

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The Common Core State Standards Initiative 2 Beginning in the spring of 2009, Governors and State Commissioners of Education from 48 states, 2 territories and the District of Columbia committed to developing a common core of state K-12 English language arts (ELA) and mathematics standards. The Common Core State Standards Initiative (CCSSI) is a state-led effort coordinated by the National Governors Association (NGA) and the Council of Chief State School Officers (CCSSO). Common Core State Standards Initiative. (2011). Common Core State Standards. Retrieved from

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Why Common Core State Standards? 3

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Why Common Core State Standards? 4 Preparation: The standards are college- and career-ready. They will help prepare students with the knowledge and skills they need to succeed in education and training after high school. Competition: The standards are internationally benchmarked. Common standards will help ensure our students are globally competitive. Equity: Expectations are consistent for all – and not dependent on a student’s zip code. Clarity: The standards are focused, coherent, and clear. Clearer standards help students (and parents and teachers) understand what is expected of them. Collaboration: The standards create a foundation to work collaboratively across states and districts, pooling resources and expertise, to create curricular tools, professional development, common assessments and other materials.

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The Mathematics Standards: How They Were Developed and Who Was Involved Key Points from Video General discussion of mathematics standards Aspirations for mathematics instruction at higher levels Greater mastery through focus and coherence Review of groups involved General discussion of mathematics progressions What is and is not included at the elementary level What happens at middle school Discussion of migration away from strands and into domains of mathematics 5 McCallum, W. (2011). Mathematic Standards: How They Were Developed and Who Was Involved [Video file]. Retrieved from

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Underlying Frameworks Strands of Mathematical Proficiency Strategic Competence Adaptive Reasoning Conceptual Understanding Productive Disposition Procedural Fluency NRC (2001). Adding It Up. Washington, D.C.:Adding It Up National Academies Press. 6 Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it Up: Helping Children Learn Mathematics. Washington, DC: National Academies Press.

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7 Improving K-12 Mathematics is an URGENT Matter!

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Key Questions What is the current state of mathematics performance… In the United States compared to other nations? In the United States? In New York State compared to other states? In your school/district compared to other districts in New York State? 8

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What is TIMSS? The Trends in International Mathematics and Science Study (TIMSS) compares math and science achievement of 4th and 8th graders internationally.Trends in International Mathematics and Science Study TIMSS is closely linked to the curricula of the participating countries, providing an indication of the degree to which students have learned concepts in mathematics and science they have encountered in school To date, more than 50 countries have participated. 9 Trends in International Mathematics and Science Study(TIMSS). (n.d.). Retrieved from Institute of Educational Science website:

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TIMSS 2007 Assessment 8 th Grade Math 10 Gonzales, P. (2009). Highlights From TIMSS 2007:Mathematics and Science Achievement of U.S. Fourth and Eighth-Grade Students in an International Context. Retrieved from National Center for Education Statistics website:

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TIMSS: Countries Behind U.S. Armenia, Australia, Sweden, Malta, Scotland, Serbia, Italy, Malaysia, Norway, Cyprus, Bulgaria, Israel, Ukraine, Romania, Bosnia and Herzegovina, Lebanon, Thailand, Turkey, Jordan, Tunisia, Georgia, Iran, Islamic Rep of, Bahrain, Indonesia, Syrian Arab Republic, Egypt, Algeria, Colombia, Oman, Palestinian Nat'l Auth., Botswana, Kuwait, El Salvador, Saudi Arabia, Ghana, Qatar, Morocco 11

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What is PISA PISA (Programme for International Student Assessment) PISA (Programme for International Student Assessment) is an international study which began in the year It aims to evaluate education systems worldwide by testing the skills and knowledge of 15- year-old students in participating countries/economies. Since the year 2000 over 70 countries and economies have participated in PISA. 12 Programme for International Student Assessment (PISA). (n.d.). Retrieved from Organisation for Economic Co-operation and Development website: 0,3746,en_ _ _ _1_1_1_1,00.htmlhttp://www.oecd.org/document/61/ 0,3746,en_ _ _ _1_1_1_1,00.html

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PISA PISA assesses how far students near the end of compulsory education have acquired some of the knowledge and skills that are essential for full participation in society. In all cycles, the domains of reading, mathematical and scientific literacy are covered not merely in terms of mastery of the school curriculum, but in terms of important knowledge and skills needed in adult life. In the PISA 2003 cycle, an additional domain of problem solving was introduced to continue the examination of cross-curriculum competencies. Take the Test 13

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PISA Programme for International Student Assessment. (n.d.). First Results from PISA 2003 [Executive Summary]. Retrieved from Organisation for Economic Co-operation and Development website: pdfhttp://www.oecd.org/dataoecd/1/63/ pdf

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PISA

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The “Thin Elite Layer” 16

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NAEPNAEP 2011 NY State 17 National Center for Education Statistics. (n.d.). National Assessment of Educational Progress (NAEP). Retrieved from U.S. Department of Education website:

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NAEP 4 th Grade Math National Center for Education Statistics. (n.d.). National Assessment of Educational Progress (NAEP). Retrieved from U.S. Department of Education website: /http://nces.ed.gov/nationsreportcard / 18

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19 NAEP 8 th Grade Math National Center for Education Statistics. (n.d.). National Assessment of Educational Progress (NAEP). Retrieved from U.S. Department of Education website: /http://nces.ed.gov/nationsreportcard /

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NYS P-12 Common Core Learning Standards for Mathematics 20 New York State Education Department. (2011). Common Core in Mathematics: Overview. Retrieved from common-core-in-mathematics-overview/http://engageny.org/resource/ common-core-in-mathematics-overview/

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Common Core Learning Standards 21

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Instructional Shifts... 22

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Shift 1 Focus Teachers use the power of the eraser and significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades. 23

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Reflection Read the “Shift” What does the “Shift” mean to you? What does it look like in mathematics classrooms (provide specific examples)? 24

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Trends in International Mathematics and Science Study (TIMSS) Test your mathematics and science knowledge by completing test items in the Dare to Compare challenge! 25 NCES Kids' Zone. (n.d.). Dare to Compare. Retrieved from U.S. Department of Education website:

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The Importance of Focus in Mathematics First-year college remediation challenges Mismatch between higher education and K more mastery of fewer topics vs. covering more Focus as it relates to teachers' needs to build a solid foundation in early grades Solid early foundation enables greater success later 26 Zimba, J., & McCallum, W. (n.d.). The Importance of Focus in Mathematics [Motion picture]. Retrieved from watch?v=2rje1NOgHWs&list=UUF0pa3nE3aZAfBMT8pqM5PA&index=18&feature= plcphttp://www.youtube.com/ watch?v=2rje1NOgHWs&list=UUF0pa3nE3aZAfBMT8pqM5PA&index=18&feature= plcp

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Shift 2 Coherence Principals and teachers carefully connect the learning within and across grades so that, for example, fractions or multiplication spiral across grade levels and students can build new understanding onto foundations built in previous years. Teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning. 27

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Reflection Read the “Shift” What does the “Shift” mean to you? What does it look like in mathematics classrooms (provide specific examples)? 28

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The Importance of Coherence in Mathematics Mathematics consists of pieces that make sense; they are not just independent manipulation/skills to be practiced and memorized – as perceived by many students. These individual pieces progress through different grades (in organized structures we called “flows”) and can/should be unified together into a coherent whole. Algebra as an example 29 Zimba, J., & McCallum, W. (n.d.). The Importance of Coherence In Mathematics [Motion picture]. Retrieved from watch?v=2rje1NOgHWs&feature=autoplay&list=UUF0pa3nE3aZAfBMT8pqM5PA&playne xt=1http://www.youtube.com/ watch?v=2rje1NOgHWs&feature=autoplay&list=UUF0pa3nE3aZAfBMT8pqM5PA&playne xt=1

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The Structure is the Standards The Structure is the Standards by Phil Daro, Bill McCallum, Jason Zimba A Grecian Urn You have just purchased an expensive Grecian urn and asked the dealer to ship it to your house. He picks up a hammer, shatters it into pieces, and explains that he will send one piece a day in an envelope for the next year. You object; he says “don’t worry, I’ll make sure that you get every single piece, and the markings are clear, so you’ll be able to glue them all back together. I’ve got it covered.” Absurd, no? But this is the way many school systems require teachers to deliver mathematics to their students; one piece (i.e. one standard) at a time. They promise their customers (the taxpayers) that by the end of the year they will have “covered” the standards … 30

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The Structure is the Standards by Phil Daro, Bill McCallum, Jason Zimba Once you have read the article, please answer the questions below with an elbow partner. 1.How do the authors describe the standards? Provide evidence in the text. 2.How were the Common Core State Standards developed? Point to evidence in the text. 3.Why did they use the word “structure” in the title? Discuss with your elbow partner. 31 Daro, P., McCallum, B., & Zimba, J. (2012, February 16). The Structure is the Standards [Web log post]. Retrieved from 16/the-structure-is-the-standards/http://commoncoretools.me/2012/02/ 16/the-structure-is-the-standards/

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Dividing Fractions Imagine you are beginning to teach students division with fractions. What would you do to introduce this concept to students? 32

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Dividing Fractions How would you present the following problem: 1 ¾ ÷ ½ ? 33

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Knowing and Teaching Elementary Mathematics – Liping Ma What is the common phrase we hear teachers say when teaching students to divide fractions? 34 Ma, L. (2010). Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series) (2nd ed.). Routledge.

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Knowing and Teaching Elementary Mathematics – Liping Ma Most of the Chinese teachers use the phrase “dividing by a number is equivalent to multiplying by its reciprocal” instead of what many U.S. teachers say “invert and multiply.” 35

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Knowing and Teaching Elementary Mathematics – Liping Ma Dividing by 2 is the same as multiplying by ½, therefore dividing by ½ is the same as multiplying by 2. 36

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Knowing and Teaching Elementary Mathematics – Liping Ma How many different ways can we solve the problem 1 ¾ ÷ ½ ? Let’s share and record all the various ways. 37

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Knowing and Teaching Elementary Mathematics – Liping Ma Measurement Model – “How many ½s in 1 ¾?” (e.g., apples, graham crackers, piece of wood) Partitive Model – “Finding a number such that ½ of it is 1 ¾” (e.g., box of candy, cake, pizza, distance) Factors and Product – “Find a factor that when multiplied by ½ will make 1 ¾” (e.g., area of a rectangle) 38

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Knowing and Teaching Elementary Mathematics – Liping Ma The meaning of division by fractions Meaning of division with whole numbers The concept of inverse operations Meaning of multiplication with whole numbers Meaning of multiplication with fractions Concept of fractionConcept of unit Meaning of addition 39

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Shift 3 Fluency Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions such as multiplication tables so that they are more able to understand and manipulate more complex concepts. 40

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Reflection Read the “Shift” What does the “Shift” mean to you? What does it look like in mathematics classrooms (provide specific examples)? 41

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42 Granny Prix Oliver, N. (n.d.). Granny Prix [Math Game]. Retrieved from multiplication.com website:

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Key Fluencies 43 GradeRequired Fluency KAdd/subtract within 5 1Add/subtract within 10 2 Add/subtract within 20 Add/subtract within 100 (pencil and paper) 3 Multiply/divide within 100 Add/subtract within Add/subtract within 1,000,000 5Multi-digit multiplication 6 Multi-digit division Multi-digit decimal operations 7Solve px + q = r, p(x + q) = r 8Solve simple 2 2 systems by inspection

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Mathematical Fluency: A Balanced Approach Balanced between procedural fluency and conceptual understanding, with examples Building on required fluencies 44 McCallum, B., & Zimba, J. (n.d.). Mathematics Fluency A Balance Approach [Video file]. Retrieved from

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Shift 4 Deep Understanding Teachers teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations. as well as writing and speaking about their understanding. 45

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Reflection Read the “Shift” What does the “Shift” mean to you? What does it look like in mathematics classrooms (provide specific examples)? 46

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Deep Conceptual Understanding McDonald’s Claim Wikipedia reports that 8% of all Americans eat at McDonalds every day. In the U.S., there are approximately 310 million people and 12,800 McDonalds. Do you believe the Wikipedia report to be true? Create a mathematical argument to justify your position. 47

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Shift 5 Applications Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. Teachers in content areas outside of math, particularly science, ensure that students are using math – at all grade levels – to make meaning of and access content. 48

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Reflection Read the “Shift” What does the “Shift” mean to you? What does it look like in mathematics classrooms (provide specific examples)? 49

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Shift 6 Dual Intensity Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year. 50

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Reflection Read the “Shift” What does the “Shift” mean to you? What does it look like in mathematics classrooms (provide specific examples)? 51

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When Not Knowing Math Can Cost You $15,000 “Who wants to be a Millionaire?” Question for $16, Mathclips. (2007, July 26). When Not Knowing Math Can Cost You $15,000 [Motion picture].

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When Not Knowing Math Can Cost You $15,000 Which of these square numbers also happens to be the sum of two smaller square numbers? A.16B. 25 C. 36D. 49 List strategies to help students remember square numbers 53

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The Mathematics Standards and the Shifts They Require Calls for Conceptual Understanding, Procedural Fluency, and Problem Solving Content and skills need to be connected with the Standards of Mathematical Practices Greater focus and better coherence 54 Zimba, J. (n.d.). The Mathematics Standards and the Shifts They Require [Video file]. Retrieved from watch?v=5pBOnvzC_Yw&list=UUF0pa3nE3aZAfBMT8pqM5PA&index=29&feature= plcphttp://www.youtube.com/ watch?v=5pBOnvzC_Yw&list=UUF0pa3nE3aZAfBMT8pqM5PA&index=29&feature= plcp

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Assessment of Learning The Next Shift 55

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New York State Assessment Transition Plan: ELA and Mathematics 56 1 The PARCC assessments are scheduled to be operational in and are subject to adoption by the New York State Board of Regents. The PARCC assessments are still in development. All PARCC assessments will be aligned to the Common Core. 2 Funding Pending. 3 The PARCC consortium is developing ELA and mathematics assessments that will cover grades New York State will continue to monitor the development of these assessments to determine how the PARCC assessments might intersect with the Regents Exams. Note that all new Regents Exams and PARCC assessments will be implemented starting with the end-of-year administration, rather than the winter or summer administrations. 4 The names of New York State’s Mathematics Regents Exams are expected to change to reflect the new alignment of these assessments to the Common Core. For additional information about the upper-level mathematics course sequence and related standards, see the “Traditional Pathway” section of Common Core Mathematics Appendix A (http://engageny.org/news/traditional-course- pathway-for-high-school-mathematics-courses-approved/).http://engageny.org/news/traditional-course- pathway-for-high-school-mathematics-courses-approved/ 5 This transition plan is specific to the NYSAA in ELA and mathematics. 6 New York State is a member of the NCSC national alternate assessment consortium that is engaged in research and development of new alternate assessments for alternate achievement standards. The NCSC assessments are scheduled to be operational in and are subject to adoption by the New York State Board of Regents. As of October 12, 2012 (Subject to Revision)

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What will assessment look like in relation to the Common Core Shifts? 57 *Recent update for Shift 3 In a memo dated March 2012, Deputy Commission Slentz stated the following: Grade 6 students must have the use of a four-function calculator with a square root key or a scientific calculator for the entire test. (Schools may choose which type they purchase.)

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Resources 58

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Common Core Resources on EngageNY 59

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collaboratively built tools informed and approved by the authors of the CCSS, which evaluate the Common Core alignment of curricular materials collaboratively built tools informed and approved by the authors of the CCSS, which evaluate the Common Core alignment of curricular materials 60

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as of 3/5/12; subject to revision NYSED is delivering Curriculum Modules aligned to the Core in ELA & Math Scope and Sequence available in July 2012 P-5 Modules begin arriving in quarter-length chunks beginning in August Modules begin arriving in quarter-length chunks beginning in the Fall. Modules: ELA & Math ModulesSummer 2012Fall 2012Spring 2013Summer 2013 P-5 Scope & Sequence 1/6 Module 3/6 Modules4/6 Modules6/6 Modules 6-12Scope & Sequence2/6 Modules4/6 Modules6/6 Modules PD for NTs, Teachers, Principals Week Long Summer Intensive P-5 Ongoing 61

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Video Exemplars MaterialsJuly 2012Fall/Winter 2012 Spring/Summer 2013 Winter 2013/14 Common Core Shifts Videos Total Teacher Practice Videos Videos Total Principal Practice Videos Videos Total Data Driven Instruction Videos Total Studio Videos24 Videos Total NYSED is delivering more than 500 videos to: Exemplify the New York State CCLS aligned instruction from NYS teachers Use for calibration of teacher and principal evaluation Model data driven instruction cycles Instruct and Inform regarding CCLS, DDI, TLE as of 3/5/12; subject to revision 62

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The End 63

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