2011 NYS P-12 Common Core Learning Standards for Mathematics

1. 2011 NYS P-12 Common Core Learning Standards for Mathematics Please visit www.engageNY.org for additional information regarding the Common Core Learning Standards

2. 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

3. Why Common Core State Standards? 3

4. 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

5. Common Core State Standards Design *Ready for first-year credit-bearing, postsecondary coursework in mathematics and English without the need for remediation. 5 • 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

6. Common Core State Standards Evidence Base 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 • England • Finland • Hong Kong • Ireland • Singapore 6 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.

7. Feedback and Review 7 • External and State Feedback teams included: • K-12 teachers • Postsecondary faculty • State curriculum and assessments experts • Researchers • National organizations (including, but not limited, to):

8. Process and Timeline • K-12 Common Standards: • Core writing teams in English Language Arts and Mathematics (Seewww.corestandards.orgfor 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

9. 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

10. NYS Common Core Learning Standards for Mathematics

11. Common Core Learning Standards

12. Instructional Shifts . . .

13. Instructional Shifts . . .

14. Instructional Shifts in Mathematics

15. 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. Shift 1 Focus

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

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

18. 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.

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

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

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

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

23. 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.”

24. 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.

25. 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.

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

27. 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)

28. The meaning of division with 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 fraction Concept of unit Meaning of addition Knowing and Teaching Elementary Mathematics – Liping Ma

29. 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.

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

31. World’s Easiest Math PuzzleListen and watch carefully…

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

33. 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.

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

35. 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.

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

37. Standards for Mathematical PracticeIntegrated 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.

38. 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.

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

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

41. 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

42. Standards for Mathematical Practice Jigsaw Activity

43. Standards for Mathematical PracticeTraditional 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.

44. 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.

46. 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.

47. 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.

48. 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

49. 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

50. 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.