2011 NYS P-12 Common Core Learning Standards for Mathematics Please visit www.engageNY.org for additional information regarding the Common Core Learning Standards

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). www.corestandards.org 2 2

Why Common Core State Standards? 3 3

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. 4 4

Common Core State Standards Design Building on the strength of current state standards, the CCSS are designed to be: Focused, coherent, clear and rigorous Internationally benchmarked Anchored in college and career readiness* Evidence and research based *Ready for first-year credit-bearing, postsecondary coursework in mathematics and English without the need for remediation. 5 5

Common Core State Standards Evidence Base For example: Standards from individual high-performing countries and provinces were used to inform content, structure, and language. Writing teams looked for examples of rigor, coherence, and progression. Mathematics Belgium (Flemish) Canada (Alberta) China Chinese Taipei England Finland Hong Kong India Ireland Japan Korea Singapore English language arts Australia New South Wales Victoria Canada Alberta British Columbia Ontario 6 6

Feedback and Review External and State Feedback teams included: K-12 teachers Postsecondary faculty State curriculum and assessments experts Researchers National organizations (including, but not limited, to): American Council on Education (ACE) American Federation of Teachers (AFT) Campaign for High School Equity (CHSE) Conference Board of the Mathematical Sciences (CBMS) Modern Language Association (MLA) National Council of Teachers of English (NCTE) National Council of Teachers of Mathematics (NCTM) National Education Association (NEA) 7 7

Process and Timeline K-12 Common Standards: Core writing teams in English Language Arts and Mathematics (See www.corestandards.org for list of team members) External and state feedback teams provided on-going feedback to writing teams throughout the process Draft K-12 standards were released for public comment on March 10, 2010; 9,600 comments received Validation Committee of leading experts reviews standards Final standards were released June 2, 2010 - NYS Board of Regents Adopted July 20, 2010 (all but 5 states have adopted) 8 8

Common Core State Standards for Mathematics Grade-Level Standards K-8 grade-by-grade standards organized by domain 9-12 high school standards organized by conceptual categories Standards for Mathematical Practice Describe mathematical “habits of mind” Standards for mathematical proficiency: reasoning, problem solving, modeling, decision making, and engagement Connect with content standards in each grade 9 9

Learning Standards for Mathematics NYS Common Core Learning Standards for Mathematics

Common Core Learning Standards

Instructional Shifts . . .

Instructional Shifts . . .

Instructional Shifts in Mathematics

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.

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

Trends in International Mathematics and Science Study (TIMSS) Test your mathematics and science knowledge by completing TIMSS items in the Dare to Compare challenge!

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.

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

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

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

Knowing and Teaching Elementary Mathematics – Liping Ma What is the common phrase we hear teachers say when teaching students to divide fractions?

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.”

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.

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.

Knowing and Teaching Elementary Mathematics – Liping Ma How could you put this problem in context (possible model for representing this problem)? Think/Pair/Share

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)

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

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 (found in the attached list of fluencies) such as multiplication tables so that they are more able to understand and manipulate more complex concepts.

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

World’s Easiest Math Puzzle Listen and watch carefully…

World’s Easiest Math Puzzle What’s your answer?

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.

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

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.

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

Standards for Mathematical Practice Integrated into Instruction McDonald’s Claim Wikipedia reports that 8% of all Americans eat as 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.

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.

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

“It’s what you learn after you know it all that counts.” Thought for the Day “It’s what you learn after you know it all that counts.” - John Wooden

Standards for Mathematical Practice 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 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

Standards for Mathematical Practice Jigsaw Activity

Standards for Mathematical Practice Traditional U.S. Problem Which fraction is closer to 1 4/5 or 5/4? Same problem with Standards for Mathematical Practice Integration 4/5 is closer to 1 than 5/4. Using a number line, explain why this is true.

Content Standards The content standards are organized by domains across grade levels and each grade level begins with a narrative description of the grade level, followed by the standards for mathematical practice, a list of the “Big Ideas” for the specific grade level, and then the content standards by domain.

Progressions Because progressions are so important in the Standards, suggestions for places to begin are not a laundry list of topics but rather a menu of progressions. Experts recommend organizing implementation work according to progressions because the instructional approach to any given topic should be informed by its place in an overall flow of ideas.

Progressions They emphasize the word menu. If a curriculum provider delivers a single coherent progression of materials to a district, then that provider has added value. If a math coach helps elementary school teachers in a district better understand a single coherent progression, then that coach has added value. The quantum of improvement is not the textbook series.

Progressions Counting and Cardinality and Operations and Algebraic Thinking: grades K–2 Operations and Algebraic Thinking: multiplication and division in grades 3–5, tracing the evolving meaning of multiplication, from equal-groups thinking with whole numbers in grade 3 to scaling-oriented thinking with fractions in grade 5. Number and Operations—Base Ten: addition and subtraction in grades 1–4 Number and Operations—Base Ten: multiplication and division in grades 3–6 Number and Operations—Fractions: fraction addition and subtraction in grades 4–5, including parallel development of fraction equivalence in grades 3–5

Progressions Number and Operations—Fractions: fraction multiplication and division in grades 4–6 The Number System: grades 6–7 Expressions and Equations: grades 6–8, including how this extends prior work in arithmetic Ratio and Proportional Reasoning: its development in grades 6–7, its relationship to functional thinking in grades 6–8, and its connection to lines and linear equations in grade 8 Geometry: work with the coordinate plane in grades 5–8, including connections to ratio, proportion, algebra and functions in grades 6–HS Geometry: congruence and similarity of figures in grades 8–HS, with emphasis on real-world and mathematical problems involving scales and connections to ratio and proportion

Progressions Modeling with equations and inequalities in high school, development from simple modeling tasks such as word problems to richer more open-ended modeling tasks Seeing Structure in Expressions, from expressions appropriate to 8th–9th grade to expressions appropriate to 10th–11th grade Statistics and Probability: comparing populations and drawing inferences in grades 6–HS. Additionally, one of the important ―invisible themes in the Standards involves units as a cross-cutting theme in the areas of measurement, geometric measurement, base-ten arithmetic, unit fractions, and fraction arithmetic, including the role of the number line.

When Not Knowing Math Can Cost You $15,000 “Who wants to be a Millionaire?” Question for $16,000

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? 16 b. 25 36 d. 49 List strategies to help students remember square numbers

Common Core State Standards K-12 Mathematics Progression of Domains

Summarized Objectives in Mathematics for the Next Six Months are: Materials: Focus Clear indication of fewer concepts at each grade level represented by curriculum documents, district formative assessments Teachers: Identify focus areas and fluencies of grade level Shift in time spent on areas of in-depth instruction Students: Demonstrated fluency and understanding Display fluencies for the grade level and understand focus areas

Assessments . . . Spring 2012 NYS Grades 3-8 Assessments will focus on the 2005 NYS Core Curriculums in ELA and mathematics Spring 2013 the NYS Grades 3-8 Assessments will focus on the 2011 Common Core Learning Standards in ELA and mathematics Spring 2015 PARCC Assessments (Grades 3-8) administered for the first time

PARCC Assessments Partnership for Assessment of Readiness for College and Careers (PARCC) http://www.parcconline.org/ PARCC is a 26-state consortium working together to develop next-generation K-12 assessments in English and math. 

PARCC states collectively educate about 25 million public K-12 students in the United States. What brought all of these states together is a shared commitment to develop an assessment system aligned to the Common Core State Standards that is anchored in college and career readiness; provides comparability across states; has the ability to assess and measure higher-order skills such as critical thinking, communications, and problem solving; and provides truly useful information for educators, parents, and students alike. While each state has their own priorities and challenges, PARCC provides the opportunity for participating states to come together and collectively move the field forward and break new ground in assessment design. In addition, many of the PARCC states are on the leading edge of education reform, including 10 of the 12 winning Race to the Top states.

K-2 Formative Assessments   To help states measure student knowledge and skills at the lower grades, the Partnership will develop a bank of assessment resources for teachers of grades K–2 that are aligned to the Common Core State Standards, and vertically aligned to the PARCC assessment system. The tasks will consist of developmentally-appropriate assessment types, such as observations, checklists, classroom activities and protocols, which reflect foundational aspects of the Common Core State Standards. The K-2 formative assessments aim to help set a foundation for students and put them on the track to college and career readiness in the early years. These K-2 assessments will help educators prepare students for later grades and provide information for educators about the knowledge and skills of the students entering third grade, allowing classroom teachers and administrators to adjust instruction as necessary. These tools also will help states fully utilize the Common Core State Standards across the entire K-12 spectrum.

The 3-8 PARCC assessments will be delivered at each grade level and will be based directly on the Common Core State Standards   The distributed PARCC design includes through-course and end-of-year components so that assessments are given closer in time to when instruction happens. The 3-8 assessments will include a range of item types, including innovative constructed response, extended performance tasks, and selected response (all of which will be computer based).

The high school PARCC assessments will be based directly on the Common Core State Standards   The distributed PARCC design includes through-course and end-of-year components so that assessments are given closer in time to when instruction happens. PARCC states have endorsed a course-based design in math and a grade-based design in ELA/Literacy. The high school assessments will include a range of item types, including innovative constructed response, extended performance tasks, and selected response (all of which will be computer based). In addition, there will be college-ready cut scores on high school tests in mathematics and ELA/Literacy, which will signify whether students are ready for college-level coursework. Earlier tests will be aligned vertically to ensure students are on - and stay on - the track to graduating ready for college and careers.

PARCC Content Frameworks Fall 2011   Mathematics The Model Content Frameworks for Mathematics are designed with the following purposes in mind: • identifying the big ideas in the Common Core State Standards for each grade level, • helping determine the focus for the various PARCC assessment components, and • supporting the development of the assessment blueprints.

Another Thought for the Day “Coming together is a beginning, keeping together is progress, working together is success.” - Henry Ford

Curriculum Unit as Defined by Commissioner John B. King, Jr. “This year I am asking every teacher to try at least one Common Core-aligned unit each semester. Math teachers should select one of the priority concepts at the strategic expense of other, less critical topics and go deep in a way they haven’t before. ELA teachers will provide a thoughtful learning experience around a particular text that should result in students’ ability to make an argument about that text. Content area teachers can fulfill the “literacy” aspect of this transformation by providing similar learning experiences built around pivotal texts in their subject area. “

Curriculum Unit Template Stage 1: Desired Results - What students will know, do and understand Stage 2: Assessment Evidence - Evidence that will be collected to determine whether or not desired results are achieved Stage 3: Learning Plan - Design learning activities to align with Stage 1 and 2 expectations - What activities and instruction will engage students and help them better grasp the essence and the value of this topic/content?

Questions. Teri Calabrese-Gray Gray_teri@cves. org http://www. cves