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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 Standards Presenter Notes: This PowerPoint toolkit provides a brief history on the Common Core State Standards (CCSS) for Mathematics, the rationale behind the development of the CCSS, and an overview of the six instructional shifts needed to effectively implement the NYS P-12 CCLS for Mathematics. Hyperlink: Presenter Directions: Throughout the PowerPoint there are a number of hyperlinks (identified in bright blue lettering) that can be accessed by right clicking to “open hyperlink.” These video clips, resources and activities can be used to support the various topics addressed in this presentation Suggested Slide Timing for Presentation: 2 minutes New York State Education Department. (2011). EngageNY. Retrieved from http://engageny.org/ 1 1

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**The Common Core State Standards Initiative**

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). Presenter Notes: This slide provides information of how and when the CCSS conversation began and the two organizations that oversaw the development of the Common Core State Standards (CCSS). Presenter Directions: Open the hyperlink and show participants where the Common Core Mathematics Standards are found. Review them briefly. Hyperlink: This is link to the National Common Core Standards website where information can be found about the development of the standards and all of the Common Core Standards in PDF format Suggested Slide Timing for Presentation: 5 minutes Common Core State Standards Initiative. (2011). Common Core State Standards. Retrieved from 2 2 2

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**Why Common Core State Standards?**

Presenter Notes: “Why Common Core State Standards?” Presenter Directions: Pose this question to your audience and engage in a conversation (do not advance to the next slide or you may choose to hide the next slide of the PPT). What to Expect: Discussion usually focuses on College and Career Readiness (CCR), global competition, states would be able to compare themselves to others states for the first time, equity for all students regardless of zip code, and focus and coherence (reduce “the mile wide, inch deep” curriculum in the United States). What to Expect: Discussion usually focuses on College and Career Readiness (CCR), global competition, states would be able to compare themselves to others states for the first time, equity for all students regardless of zip code, and focus and coherence (reduce “the mile wide, inch deep” curriculum in the United States). Suggested Slide Timing for Presentation: 5 minute activity 3 3 3

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**Why Common Core State Standards?**

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. Presenter Notes: This slide lists rationale as to why states agreed to collaborate and develop the CCSS. Suggested Slide Timing for Presentation: 2 minutes 4 4 4

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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 Presenter Notes: The hyperlink contained is this slide is a video that explains the process on the development of the CCSS for Mathematics. Bill McCallum and Jason Zimba, two of the principal writers, share how they worked as a team with 60 writers and various professional organizations to develop the mathematics standards. There are many other Hunt Institute videos on You Tube. Presenter Directions: Click on the hyperlink in the title to view a video. Hyperlink: The Hunt Institute The Mathematics Standards: How They Were Developed and Who Was Involved by Bill McCallum, PhD, Math Team Coordinator and Jason Zimba, PhD, Math Team Coordinator Video Timing = 8:11 Suggested Slide Timing for Presentation: 10 minutes McCallum, W. (2011). Mathematic Standards: How They Were Developed and Who Was Involved [Video file]. Retrieved from 5 5

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**Underlying Frameworks**

Strands of Mathematical Proficiency Conceptual Understanding Productive Disposition Strategic Competence Procedural Fluency Adaptive Reasoning Presenter Notes: Adding It Up provides a historical perspective in mathematics education and the presenter can refer participants to this document for advancing their own knowledge about how students in pre-K through 8th grade learn mathematics and it recommends how teaching, curricula, and teacher education should change to improve mathematics learning and teaching. Hyperlink: Visit to read it online, free of cost. Suggested Slide Timing for Presentation: 2 minutes NRC (2001). Adding It Up. Washington, D.C.: National Academies Press. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it Up: Helping Children Learn Mathematics. Washington, DC: National Academies Press. 6 6

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**Improving K-12 Mathematics is an**

URGENT Matter! Presenter Notes: This matter is URGENT! Presenter Directions: Emphasize that Improving K-12 Mathematics Education is an urgent matter! Suggested Slide Timing for Presentation: 10 seconds 7 7

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**What is the current state of mathematics performance…**

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? Presenter Notes: “What is the current state of mathematical performance?” Presenter Directions: The key questions for this activity are outlined on this slide Pose these questions one at a time: Ask participants to discuss at their tables. Another option is to ask tables to count off by fours and discuss one of the four questions and report out. Continue to the next slide and present the data Suggested Slide Timing for Presentation: 3-4 minutes 8 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. 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. Presenter Notes: This slide highlights the TIMMS study. Presenter Directions: Ask participants if they are familiar with this study. Review the information contained on the slide and refer participants to the hyperlink for more information about this important study. Hyperlink: The Trends in International Mathematics and Science Study (TIMSS) provides reliable and timely data on the mathematics and science achievement of U.S. 4th- and 8th-grade students compared to that of students in other countries. TIMSS data have been collected in 1995, 1999, 2003, and In 2011, more than 60 countries and jurisdictions, including the United States, participated in TIMSS. More than 20,000 students in more than 1,000 schools across the United States took the assessment in spring 2011, joining almost 500,000 other students around the world taking part in TIMSS. Suggested Slide Timing for Presentation: 1 Minute Trends in International Mathematics and Science Study(TIMSS). (n.d.). Retrieved from Institute of Educational Science website: 9 9

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**TIMSS 2007 Assessment 8th Grade Math**

Presenter Notes: The 2007 Trends in International Mathematics and Science Study (TIMSS) is the fourth administration since 1995 of this international comparison. Developed and implemented at the international level by the International Association for the Evaluation of Educational Achievement (IEA)—an international organization of national research institutions and governmental research agencies—TIMSS is used to measure over time the mathematics and science knowledge and skills of fourth- and eighth-graders. This report focuses on the performance of U.S. students relative to that of their peers in other countries in 2007, and on changes in mathematics and science achievement since Thirty-six countries or educational jurisdictions participated at grade four in 2007, while 48 participated at grade eight. This report also describes additional details about the achievement of U.S. student subpopulations. All differences described in this report are statistically significant at the .05 level. No statistical adjustments to account for multiple comparisons were used. Note: This is 8th grade math data. Hyperlink: Key findings from the report include the following: • In 2007, the average mathematics scores of both U.S. fourth-graders (529) and eighth-graders (508) were higher than the TIMSS scale average (500 at both grades). The average U.S. fourth-grade mathematics score was higher than those of students in 23 of the 35 other countries, lower than those in 8 countries (all located in Asia or Europe), and not measurably different from those in the remaining 4 countries. At eighth grade, the average U.S. mathematics score was higher than those of students in 37 of the 47 other countries, lower than those in 5 countries (all of them located in Asia), and not measurably different from those in the other 5 countries. • Compared to 1995, the average mathematics scores for both U.S. fourth- and eighth-grade students were higher in At fourth grade, the U.S. average score in 2007 was 529, 11 points higher than the 1995 average of 518. At eighth grade, the U.S. average mathematics score in 2007 was 508, 16 points higher than the 1995 average of 492. • In 2007, 10 percent of U.S. fourth-graders and 6 percent of U.S. eighth-graders scored at or above the advanced international benchmark in mathematics. At grade four, seven countries had higher percentages of students performing at or above the advanced international mathematics benchmark than the United States: Singapore, Hong Kong SAR, Chinese Taipei, Japan, Kazakhstan, England, and the Russian Federation. Fourth-graders in these seven countries were also found to outperform U.S. fourth-graders, on average, on the overall mathematics scale. At grade eight, a slightly different set of seven countries had higher percentages of students performing at or above the advanced mathematics benchmark than the United States: Chinese Taipei, Korea, Singapore, Hong Kong SAR, Japan, Hungary, and the Russian Federation. These seven countries include the five countries that had higher average overall mathematics scores than the United States, as well as Hungary and the Russian Federation. Suggested Slide Timing for Presentation: 6-8 minutes 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: 10 10

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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 Presenter Notes: Does this give us cause for alarm? Suggested Slide Timing for Presentation: 10 seconds 11 11

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What is PISA 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. Presenter Notes: PISA - All students take pencil-and-paper tests, with assessments lasting a total of two hours for each student. For the PISA 2009 assessment, some participating countries/economies have also opted for an assessment of the reading of electronic texts. The PISA test is a very rigorous test for 15 year olds, and focused on mathematical application. Hyperlink: Presenter Directions: Test items are a mixture of multiple-choice items and questions requiring students to construct their own responses. The items are organized in groups based on a passage setting out a real-life situation. Suggested Slide Timing for Presentation: 30 seconds Programme for International Student Assessment (PISA). (n.d.). Retrieved from Organisation for Economic Co-operation and Development website: http://www.oecd.org/document/61/ 0,3746,en_ _ _ _1_1_1_1,00.html 12 12

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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 Presenter Notes: PISA (Programme for International Student Assessment) Please note that this is math data comparing 15 year olds. Take an interactive version of the tests online at You can also consult the publication 'Take the Test ‘ which lists all the publicly released items from the first three assessments (PISA 2000, 2003 and 2006). The PISA test is a very rigorous test for 15 year olds and focused on mathematical application. Hyperlink: Suggested Slide Timing for Presentation: 2-3 minutes 13 13

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PISA 2003 Presenter Notes: Only 4 percent of students in the combined Organization for Economic Cooperation and Development (OECD) area, but more than 8 per cent in Belgium, Japan, Korea and the partner country Hong Kong-China – can perform the highly complex tasks required to reach Level 6. • About a third of OECD students can perform relatively difficult tasks at Levels 4, 5 or 6, but over 49 percent of students in Finland, Korea and the partner country Hong Kong-China can perform at least at Level 4. • About three-quarters of OECD students can perform at least mathematical tasks at Level 2 (shown above the central line in the graph). However, over a quarter of students are not proficient beyond Level 1 in Italy and Portugal, over a third in Greece and over half in Mexico and Turkey. A number of partner countries also have high numbers at Level 1 or below. • Eleven percent of students in OECD countries are not capable even of Level1 tasks. These students may still be able to perform basic mathematical operations, but were unable to utilize mathematical skills in a given situation, as required by the easiest PISA tasks. In some countries, over 20 percent are in this category. Hyperlink: First Results from PISA 2003 Executive Summary Suggested Slide Timing for Presentation: 2 minutes Programme for International Student Assessment. (n.d.). First Results from PISA 2003 [Executive Summary]. Retrieved from Organisation for Economic Co-operation and Development website: pdf 14 14

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PISA 2006 Presenter Notes: In 2006, the average U.S. score in mathematics literacy was 474, lower than the OECD average score of 498. Thirty-one jurisdictions (23 OECD jurisdictions and 8 non-OECD jurisdictions) scored higher, on average, than the United States in mathematics literacy in In contrast, 20 jurisdictions (4 OECD jurisdictions and 16 non-OECD jurisdictions) scored lower than the United States in mathematics literacy in When comparing the performance of the highest achieving students—those at the 90th percentile—U.S. students scored lower (593) than the OECD average (615) on the mathematics literacy scale. Twenty-nine jurisdictions (23 OECD jurisdictions and 6 non-OECD jurisdictions) had students at the 90th percentile with higher scores than the United States on the mathematics literacy scale. Suggested Slide Timing for Presentation: 1 minute 15 15

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The “Thin Elite Layer” Presenter Notes: This graph shows the percentage of students achieving at an advanced level and how various states performed on the PISA 2006 in comparison to other states and countries. The red bars represents some of our leading states. Even our best states (Massachusetts) performed considerably lower than the leading countries. The cutting edge defines a country’s ability to remain competitive. Suggested Slide Timing: 30 seconds 16 16

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NAEP 2011 NY State Presenter Notes: The National Assessment of Educational Progress (NAEP) is the largest nationally representative and continuing assessment of what America's students know and can do in various subject areas. Assessments are conducted periodically in mathematics, reading, science, writing, the arts, civics, economics, geography, and U.S. history. In 2011, New York State is the only state whose 4th graders scored lower than in 2009. Hyperlink: National Assessment of Educational Progress (NAEP) Suggested Slide Timing: 30 seconds National Center for Education Statistics. (n.d.). National Assessment of Educational Progress (NAEP). Retrieved from U.S. Department of Education website: 17 17 17

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NAEP 4th Grade Math Presenter Notes: This slide depicts the 4th grade NAEP Math Results. Presenter Directions: Review these results with participants Suggested Slide Timing: 2 minutes National Center for Education Statistics. (n.d.). National Assessment of Educational Progress (NAEP). Retrieved from U.S. Department of Education website: 18

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NAEP 8th Grade Math Presenter Notes: This slide depicts the 8th grade NAEP Math Results. Presenter Directions: Review these results with participants Suggested Slide Timing: 2 minutes National Center for Education Statistics. (n.d.). National Assessment of Educational Progress (NAEP). Retrieved from U.S. Department of Education website: 19

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**Learning Standards for Mathematics**

NYS P-12 Common Core Learning Standards for Mathematics Presenter Notes: This 14-minute video provides an overview of the Common Core State Standards in Mathematics. NYS Commissioner of Education John B. King Jr. and contributing author David Coleman discuss the background of the Common Core State Standards, their value in the state, the principles of their development, and the changes required of schools during this transition. Based on the video where there any “ah-ha” moments for you? Hyperlink: Common Core Math EngageNY.org Suggested Slide Timing for Presentation: *If video is viewed timing could be as much as 16 minutes New York State Education Department. (2011). Common Core in Mathematics: Overview. Retrieved from common-core-in-mathematics-overview/ 20 20

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**Common Core Learning Standards**

Presenter Notes: Presenter Notes: There are six instructional shifts necessary to implement the NYS P-12 New York State P-12 Common Core Learning Standards for Mathematics. Suggested Slide Timing for Presentation: 5 seconds 21 21

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Instructional Shifts . . . Presenter Notes: The six shifts represent key areas of focus as teachers and administrators work to implement the NYS P-12 New York State P-12 Common Core Learning Standards for Mathematics. Educators are likely to be at different stages in practicing these shifts, however, focusing on these areas can help schools and districts develop a common understanding of what is needed in mathematics instruction as they move forward with implementation. Presenter Directions: Ask participants to read the shifts, think about what the shifts mean, and what would these look like in a mathematics classroom? Suggested Slide Timing for Presentation: 2 minutes 22 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. Presenter Notes: Ask participants to read the shift, think about what the shift means, and what it would look like in a mathematics classroom. Suggested Slide Timing for Presentation: 3-4 minutes 23 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)? Presenter Notes: Ask participants to paraphrase the shift. Focus involves depth, not breadth. The intent of the NYS P-12 New York State P-12 Common Core Learning Standards for Mathematics is to use the power of the eraser and get rid of a curriculum that is “a mile wide and an inch deep.” Another key word in the definition is “prioritized”. Educators need to be sure they are spending more time on those concepts critical to mathematical proficiency. Students need to have deep conceptual understanding to be able to transfer skills across concepts and grades. Suggested Slide Timing for Presentation: 3-4 minutes 24 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! Presenter Notes: The purpose of the National Center for Education Statistics' Kids' Zone is to provide information to help visitors learn about schools; decide on a college; find a public library; engage in several games, quizzes and skill building about math, probability, graphing, and mathematicians; and to learn many interesting facts about education. Of course, all of these things have been designed to be fun too, so jump in! Dare to Compare, a part of the Kids’ Zone website, contains questions from the Trends in International Mathematics and Science Study (TIMSS), the Civic Education Study (CivEd) and National Assessment of Education Progress (NAEP). Presenter Directions: Ask participants to engage in a math problem on the Dare to Compare site and discuss possible application in the mathematics classroom. You could use the math questions on the Dare to Compare site as “Do Nows” or Bell Ringers, peer to peer, learning centers, quizzes for students to work on in teams or pairs, review, or share with parents in your newsletter. You could use the math questions on the Dare to Compare site as Do Nows or Bell Ringers, peer to peer, learning centers, quizzes for students to work on in teams or pairs, review, or share with parents in your newsletter. Hyperlink: Dare to Compare – Suggested Slide Timing for Presentation: 5-8 minutes for problem and discussion. NCES Kids' Zone. (n.d.). Dare to Compare. Retrieved from U.S. Department of Education website: 25 25

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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 Presenter Notes: First-year college remediation problem: Mismatch of definitions of College Readiness between higher education and K-12 [Teachers should] teach to greater mastery of fewer topics vs. covering more [Teachers need to] build a solid foundation for students in early grades on key topics Solid foundation in early grades enables greater success later Hyperlink: The Hunt Institute The Importance of Focus in Mathematics by Bill McCallum and Jason Zimba, PhD, Math Team Coordinator Video Timing = 2:42 Presenter Directions: Click on the hyperlink in the title. Jason Zimba, one of the authors of the CCSSM, reviews how important focus is at all levels in mathematics to insure students are College and Career Ready. Suggested Slide Timing for Presentation: 5 minutes Zimba, J., & McCallum, W. (n.d.). The Importance of Focus in Mathematics [Motion picture]. Retrieved from watch?v=2rje1NOgHWs&list=UUF0pa3nE3aZAfBMT8pqM5PA&index=18&feature=plcp 26 26

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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. Presenter Notes: Ask participants to read the shift, think about what the shift means, and what it would look like in a mathematics classroom. Suggested Slide Timing for Presentation: 1 minute 27 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)? Presenter Notes: Ask participants to paraphrase the shift. Instead of treating math in each grade as a series of disconnected topics, principals and teachers insure coherence by carefully connecting the learning within and across grades so that, for example, fractions or multiplication develop across grade levels and students can build new understanding onto foundations built in previous years. Each standard is not a new event, but an extension of previous learning. Suggested Slide Timing for Presentation: 3-4 minutes 28 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 Presenter Notes: Bill McCallum, one of the authors of the CCSSM, discusses the importance of coherence and how we should instill in students an understanding why it all fits together (mathematics is not a subject that keeps branching out – it is based on underlying principles that help mathematics come together under one umbrella). Hyperlink: Video Timing: 4:38 Suggested Slide Timing for Presentation: 6-8 minutes Zimba, J., & McCallum, W. (n.d.). The Importance of Coherence In Mathematics [Motion picture]. Retrieved from watch?v=2rje1NOgHWs&feature=autoplay&list=UUF0pa3nE3aZAfBMT8pqM5PA&playnext=1 29 29

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**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… Presenter Notes: The article, The Structure is the Standards, was posted in the winter of 2012 on Bill McCallum’s blog and written by Bill McCallum, Phil Daro and Jason Zimba. The article allows for powerful conversations around the CCSS and the philosophy behind the development of those standards. Presenter Directions: Distribute copies of the entire article (only two pages), The Structure is the Standards to participants. Prior to asking participants to read the article, advance to the next slide Hyperlink: The Structure is the Standards by Phil Daro Senior Fellow America’s Choice & Pearson, Bill McCallum, PhD, Math Team Coordinator and Jason Zimba, PhD, Math Team CoordinatorPresenter Directions: Suggested Timing for Presentation: 2-3 minutes 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. How do the authors describe the standards? Provide evidence in the text. How were the Common Core State Standards developed? Point to evidence in the text. Why did they use the word “structure” in the title? Discuss with your elbow partner. Presenter Directions: As participants to engage in mathematical discussion and answer the questions on the slide with an elbow partner after reading the article. Select participants to report out to the group Suggested Timing for Presentation: minutes 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/ 31

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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? Presenter Notes: The example that follows is from the book “Knowing and Teaching Elementary Mathematics” by Liping Ma. This book is a must read for mathematics educators. The presenter may want to seek assistance from a mathematics content specialist if the presenter is not familiar or comfortable with the mathematical content. Usually the answer to this question is procedural and the response is, “multiply by the reciprocal.” Others share a mnemonic device, such as “keep, switch, flip.” Suggested Slide Timing for Presentation: 1 minute 32 32

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**How would you present the following problem:**

Dividing Fractions How would you present the following problem: 1 ¾ ÷ ½ ? Presenter Notes: At your tables, discuss with others how they would present this problem to students. This is just an opportunity to discuss possible solutions that will be shared later. (The presenter may want to seek assistance from a mathematics content specialist if the presenter is not familiar or comfortable with the mathematical content.) . Suggested Slide Timing for Presentation: 2-3 minutes 33 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? Presenter Notes: What is the most common phrase we hear teachers say when teaching students to divide fractions? Presenter Directions: (The presenter may want to seek assistance from a mathematics content specialist if the presenter is not familiar or comfortable with the mathematical content.) The most common phrase is to invert and multiply. Ask them to tell you why this works. Do they have a deep conceptual understanding themselves? Suggested Slide Timing for Presentation: 30 seconds 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. 34 34

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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.” Presenter Notes: Based on the responses of Chinese teachers, we need to start using the same language and focus on multiplication being the inverse of division. Presenter Notes: (The presenter may want to seek assistance from a mathematics content specialist if the presenter is not familiar or comfortable with the mathematical content.) Discuss vocabulary terms: division, number, multiplication, equivalent, and reciprocal . Suggested Slide Timing for Presentation: 10 seconds 35 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. Presenter Notes: (Extend) “dividing by a number is equivalent to multiplying by its reciprocal.” (The presenter may want to seek assistance from a mathematics content specialist if the presenter is not familiar or comfortable with the mathematical content.) Suggested Slide Timing for Presentation: 10 seconds 36 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. Presenter Notes: How many different ways can we solve the problem? Presenter Directions Ask each table to come up with as many ways to solve/model this problem as possible. Provide them plenty of time to complete the task. Most will start out with the typical solutions/models, but the more time they have, the more creative they become using various supplies at their tables. Examples could include (how many halves are in one and three-quarters?): a number line, measuring cups, manipulatives (i.e., candy, money, Legos, fraction bars etc.) (The presenter may want to seek assistance from a mathematics content specialist if the presenter is not familiar or comfortable with the mathematical content.) *Use chart paper to chart responses – have participants do a gallery walk so they can see multiple solutions. Suggested Slide Timing for Presentation: minutes (including gallery walk) 37 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) Presenter Notes: Ask the participants to determine which method they used (as outlined above) to represent the problem. Be specific (Measurement, Partitive, or Factors and Product) to ensure that participants are using content specific vocabulary. (The presenter may want to seek assistance from a mathematics content specialist if the presenter is not familiar or comfortable with the mathematical content.) . Suggested Slide Timing for Presentation: 2-3 minutes . 38 38

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**Knowing and Teaching Elementary Mathematics – Liping Ma**

The meaning of division by fractions Meaning of multiplication with fractions Concept of unit Meaning of division with whole numbers Meaning of multiplication with whole numbers Presenter Notes: Presenter poses question: What does it take to understand division of fractions? This concept map lists all the concepts students need to understand in order to solve the problem presented. This shows the importance of coherence required within and across grades. (The presenter may want to seek assistance from a mathematics content specialist if the presenter is not familiar or comfortable with the mathematical content.) Suggested Slide Timing for Presentation: 1-2 minutes Concept of fraction The concept of inverse operations Meaning of addition 39 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. Presenter Notes: Ask participants to read the shift, think about what the shift means, and what it would look like in a mathematics classroom. Suggested Slide Timing for Presentation: 1 minute 40 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)? Presenter Notes: Ask participants to paraphrase the shift. Students must have the skill to carry out procedures flexibly, accurately, efficiently and appropriately. Ask participants how they can assist with fluency, even if they are not the classroom teacher? Who else inside and outside the school can help with fluency in your district/school? Share ideas - physical education teachers, music teachers, cafeteria staff and other support personnel, parents, family members, day care providers (parents drop off their children prior to the start of school or they go there after school until the parent finishes work), local libraries and other after school providers/organizations Suggested Slide Timing for Presentation: 3-4 minutes 41 41

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Granny Prix Presenter Notes: Granny Prix is a fun activity for students to work on fluency Presenter Directions: Demonstrate how Granny Prix can be used in a classroom as a tool to helping students become more fluent. Hyperlink: Granny Prix Suggested Slide Timing for Presentation: 3-4 minutes for activity Oliver, N. (n.d.). Granny Prix [Math Game]. Retrieved from multiplication.com website: 42 42

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**Key Fluencies K Add/subtract within 5 1 Add/subtract within 10 2**

Grade Required Fluency K Add/subtract within 5 1 Add/subtract within 10 2 Add/subtract within 20 Add/subtract within 100 (pencil and paper) 3 Multiply/divide within 100 Add/subtract within 1000 4 Add/subtract within 1,000,000 5 Multi-digit multiplication 6 Multi-digit division Multi-digit decimal operations 7 Solve px + q = r, p(x + q) = r 8 Solve simple 22 systems by inspection Presenter Notes: Create school and classroom contests around the fluencies; make these a fun school project Suggested Slide Timing for Presentation: 2 minutes 43 43 43

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**Mathematical Fluency: A Balanced Approach**

Balanced between procedural fluency and conceptual understanding, with examples Building on required fluencies Presenter Notes: We must strike a balance between the two. Each grade has one or two fluencies teachers should focus on. Presenter Directions: Click on the hyperlink in the title. Bill McCallum and Jason Zimba, explain the importance of fluency but not at the expense of conceptual understanding. Hyperlink: The Hunt Institute Mathematical Fluency: A Balanced Approach by Bill McCallum and Jason Zimba Video Timing = 1:57 Suggested Slide Timing for Presentation: 3 minutes McCallum, B., & Zimba, J. (n.d.). Mathematics Fluency A Balance Approach [Video file]. Retrieved from 44 44

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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. Presenter Notes: Ask participants to read the shift, think about what the shift means, and what it would look like in a mathematics classroom. Students with conceptual understanding know more than isolated facts and methods - they understand why a mathematical idea is important and the contexts in which it is useful. Suggested Slide Timing for Presentation: 1 minute 45 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)? Presenter Notes: Ask participants to paraphrase the shift. Teachers take time to understand not only the content standards, but the Standards for Mathematical Practice describe the student expertise needed to develop a deep conceptual understanding of mathematics. Suggested Slide Timing for Presentation: 3-4 minutes 46 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. Presenter Directions: Present this problem to the group. Ask them to work on the problem and then be prepared to share a justification. This leads to a great discussion and demonstrates how students could determine if this could possibly be true through collection of data. (The presenter may want to seek assistance from a mathematics content specialist if the presenter is not familiar or comfortable with the mathematical content.) Suggested Slide Timing for Presentation: 6-8 minutes Solution: 8% of 310 million = 24,800,000 24,800,000 /12,800 = people served per day Some participants may divide by 24 hours, 18 hours, etc. and justify their reasoning. 47 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. Presenter Directions: Ask participants to read the shift, think about what the shift means, and what it would look like in a mathematics classroom. Focus on some of the key vocabulary. Identify opportunities for students to apply math concepts in “real world” situations. Presenter Notes: The process of modeling that includes choosing and using appropriate mathematics and statistics to analyze and understand situations, is key in improving decisions as well as linking classroom mathematics and statistics to everyday life, work, and decision-making. Suggested Slide Timing for Presentation: 1 minute 48 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)? Presenter Directions: Ask participants to paraphrase the shift. Presenter Notes: Students are expected to use math and choose the appropriate mathematical models even when they are not prompted to do so. Suggested Slide Timing for Presentation: 3-4 minutes 49 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. Presenter Directions: Ask participants to read the shift, think about what the shift means, and what it would look like in a mathematics classroom. Presenter Notes: Teachers struggle on balancing how much time they spend on procedural skills and the amount of time they are spending on deep conceptual understanding/application. They must find that balance. Students need to practice and understand mathematics Suggested Slide Timing for Presentation: 1 minute 50 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)? Presenter Directions: Ask participants to paraphrase the shift. Presenter Notes: Teachers create opportunities for students to participate in authentic practice and make use of those skills through extended application of math concepts. . Suggested Slide Timing for Presentation: 3-4 minutes 51 51

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**When Not Knowing Math Can Cost You $15,000**

“Who wants to be a Millionaire?” Question for $16,000 Presenter Directions: This video is 2:38 (total) Click on the hyperlink and play the video. Pause at 1:00 when the contestant asks to poll the audience. Presenter Notes: Ask participants: “To answer this question, what mathematics does the contestant need to know? (Chart the responses). Then say: “Now that you know this, can you now solve the problem?” Hyperlink: Not Knowing Math Can Cost You $15,000 Video Timing = 2:38 Suggested Slide Timing for Presentation: 3 minutes Mathclips. (2007, July 26). When Not Knowing Math Can Cost You $15,000 [Motion picture]. 52 52

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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? B. 25 C D. 49 List strategies to help students remember square numbers Presenter Directions: Discuss the necessary vocabulary and why square numbers are called square numbers. List the square numbers (1x1=1, 2x2=4, 3x3=9, 4x4=16, 5x5=25, etc.). Now ask them to solve the problem. Continue the video from where it was paused. Discuss the shifts addressed in the problem and provide specifics. Hyperlink: When Not Knowing Math Can Cost You $15,000 Video Timing = 2:38 Suggested Slide Timing for Presentation: 3 minutes 53 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 Presenter Directions: Click on the hyperlink in the title. Presenter Notes: Jason Zimba solidifies the need for the shifts and how educators need to focus on these shifts to help all students be college and career ready Hyperlink: The Hunt Institute The Mathematics Standards and the Shifts They Require by Bill McCallum and Jason Zimba, PhD, Math Team Coordinator Video Timing = 1:15 Suggested Slide Timing for Presentation: 2-3 minutes 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=plcp 54 54

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**Assessment of Learning The Next Shift**

Presenter Notes: What’s Next? Suggested Slide Timing for Presentation: 10 seconds 55 55

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**New York State Assessment Transition Plan: ELA and Mathematics**

As of October 12, 2012 (Subject to Revision) 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/). 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. 56

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**What will assessment look like in relation to the Common Core Shifts?**

Presenter Notes: Assessments in relation to the Common Core Shifts? Suggested Slide Timing for Presentation: 2-3 minutes *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.) 57 57

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**http://engageny.org/resource/math-toolkit/ Resources**

Presenter Notes: Use this toolkit to support your work. Suggested Slide Timing for Presentation: 3-4 minutes to review links etc. 58 58

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**Common Core Resources on EngageNY**

Presenter Notes: There are also resources available on EngageNY.org Bookmark it on your desktop and visit it often for updates. Suggested Slide Timing for Presentation: 30 seconds 59 59

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**which evaluate the Common Core alignment of curricular materials **

informed and approved by the authors of the CCSS, collaboratively built tools Presenter Notes: Collaborations between New York State and other states have resulted in the creation of additional tools and resources. The Tri-State Rubric Quality Review Rubric for Mathematics produced by a collaboration between New York State, Rhode Island and Massachusetts is available currently on EngageNY.org Suggested Slide Timing for Presentation 1 minute 60 60

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**as of 3/5/12; subject to revision**

Modules: ELA & Math as of 3/5/12; subject to revision Modules Summer 2012 Fall 2012 Spring 2013 Summer 2013 P-5 Scope & Sequence 1/6 Module 3/6 Modules 4/6 Modules 6/6 Modules 6-12 2/6 Modules PD for NTs, Teachers, Principals Week Long Summer Intensive P-5 Ongoing 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. 6-12 Modules begin arriving in quarter-length chunks beginning in the Fall. Presenter Notes: New York State is due to release Mathematics Curriculum Modules in July of Week long intensive professional development sessions are planned for teachers leaders, principals and BOCES Network Teams to learn how to roll out these modules in classrooms beginning in September of 2012. Suggested Slide Timing for Presentation 2 minutes 61 61

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**Video Exemplars NYSED is delivering more than 500 videos to:**

as of 3/5/12; subject to revision Materials July 2012 Fall/Winter 2012 Spring/Summer Winter 2013/14 Common Core Shifts 2 7 11 100 Videos Total Teacher Practice Videos 45 114 227 Videos Total Principal Practice Videos 27 68 135 Videos Total Data Driven Instruction 6 15 30 Videos Total Studio Videos 24 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 Presenter Notes: New York State is also in the process of developing more than 500 videos to support quality teaching and learning. The first videos will be available in July of 2012. Suggested Slide Timing for Presentation 20 seconds 62 62

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**The End Presenter Notes: Questions?**

Suggested Slide Timing for Presentation 63 63

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