1. Math Common Core State Standards A Closer Look at the CCSS Instructional Shifts in Math and their Implications on Mathematical Instruction

2. Link to Video: http://engageny.org/resource/common-core-in- mathematics-overview/ This video explains, in greater detail, the shifts we see in Math with the Common Core State Standards. The speaker is one of the writers of the Common Core State Standards. Introduction to the Common Core State Standards Mathematical Shifts

3. Shift 1: FOCUS

4. The Importance of Focus in Mathematics Link to Video: http://www.youtube.com/watch?v=2rje1NOgHWs&feature=BFa&list=EC913348 FFD75155C6 To further aid states as they continue to implement the Common Core State Standards (Standards), the Hunt Institute and the Council of Chief State School Officers have commissioned a series of video vignettes that explain the Standards in far greater depth. Several of the key Standards writers were asked, in their own words, to talk about how the Standards were developed and the goals they set for all students.

5. Information for each grade level begins with a focus page that delineates two to four critical areas for instruction with additional information about each critical area. These critical areas bring focus to the standards at each grade by grouping and summarizing the big ideas that educators will use to develop their curriculum and to guide instruction. Critical Areas

6. Example of a K-8 Focus Page

7. Connect Mathematical Content Standards to Critical Areas in Third Grade

8. Choose a critical area for your grade level. Using the standards page, list numbers for all standards on the recording sheet. Write “yes” or “no” beside each standard number to indicate whether or not it is connected to your chosen critical area. Discuss the results of your analysis. Are there connections among critical areas and big ideas? Why? Extension…Connecting Standards to Critical Areas

9. Consider and discuss how focus on a few critical areas impacts teaching and learning in your grade level and/or course. Extension: Critical Areas Small Group Discussion

10. Example of a Course Focus Page

11. The high school standards- Consist of 6 conceptual categories. Specify the mathematics that all students should study in order to be college and career ready. Portray a coherent view of high school mathematics Conceptual Categories for 9-12

12. Extension: Connect Mathematical Content Standards to Conceptual Areas Conceptual Areas Course Number and QuantityAlgebra 1, Algebra 2, 4 th Courses Algebra Functions Geometry Statistics and Probability

13. Choose a conceptual category for your course. Using Appendix A, locate standards that connect to your chosen conceptual category. Discuss how this grouping defines the big ideas in math for your course. Extension Activity for grades 9-12: Conceptual Categories

14. Shift 2: COHERENCE

15. The Importance of Coherence in Mathematics Link to Video: http://www.youtube.com/watch?v=83Ieur9qy5k&list=EC913348FFD751 55C6&index=13&feature=plpp_video To further aid states as they continue to implement the Common Core State Standards (Standards), the Hunt Institute and the Council of Chief State School Officers have commissioned a series of video vignettes that explain the Standards in far greater depth. Several of the key Standards writers were asked, in their own words, to talk about how the Standards were developed and the goals they set for all students.

19. Note for each domain at your grade level whether it is a BEGINNING POINT, BUILDING POINT, OR END POINT.

20. A Coherent Curriculum is…. ●Is organized around the big ideas that connect mathematics across grade levels and into high school ●Builds concepts through logical progressions with increased complexity ●Description of the mathematical content to be learned, elaborated through domains, clusters, and standards

21. Shift 3: FLUENCY

22. The Importance of Fluency in Mathematics Link to Video: http://www.youtube.com/watch?v=ZFUAV00bTwA&feature=BFa&list=EC91334 8FFD75155C6 To further aid states as they continue to implement the Common Core State Standards (Standards), the Hunt Institute and the Council of Chief State School Officers have commissioned a series of video vignettes that explain the Standards in far greater depth. Several of the key Standards writers were asked, in their own words, to talk about how the Standards were developed and the goals they set for all students.

23. Fluent in the Standards means “fast and accurate.” It might also help to think of fluency as meaning the same thing as when we say that somebody is fluent in a foreign language: when you’re fluent, you flow. Fluent isn’t halting, stumbling, or reversing oneself. “Fluency”

24. Students are efficient and accurate in performing foundational computational procedures without always having to refer to tables and other aids. Teachers help students to study algorithms as “general procedures” so they can gain insights to the structure of mathematics (e.g. organization, patterns, predictability). Students are able to apply a variety of appropriate procedures flexibly as they solve problems. Helping students master key procedures will help them understand and manipulate more complex concepts in later grades. (NRC, 2001, p. 121; CCSSM, 2010, p.6) Procedural Fluency

25. Key Fluencies for K-8 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 1000 4 Add/subtract within 1,000,000 Multi-digit multiplication 5 Addition/subtraction of fractions Whole number and decimal operations 6 Multi-digit division Multi-digit decimal operations 7 Operations with rational numbers Solve equations px + q = r, p(x + q) = r 8 Solve linear equations in one variable Solve pairs of simultaneous linear equations

26. Evidence concerning college and career readiness shows clearly that the knowledge, skills, and practices important for readiness include a great deal of mathematics prior to the boundary defined by (+) symbols in the high school standards. Some of the highest priority content for college and career readiness comes from Grades 6–8. Powerfully useful proficiencies are developed in middle school and applied in high school courses, such as:  Applying ratio reasoning in real-world and mathematical problems  Computing fluently with positive and negative fractions and decimals, and  Solving real-world and mathematical problems involving angle measure, area, surface area, and volume. Procedural Fluency in High School

27. Writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems in Algebra 1 Extend and apply reasoning to other types of equations in courses following Algebra 1. For example, students solve exponential equations with logarithms in Algebra 2. Fluencies in High School

28. Look at the fluencies for your grade level. Now examine your grade level standards and previous grade’s standards. List any standards from your grade level or the previous grade level that will support the fluencies for your grade level. Activity 1

29. Shift 4: DEEP UNDERSTANDING

30. The Importance of Deep Understanding in Mathematics Link to Video: http://www.youtube.com/watch?v=m1rxkW8ucAI&feature=BFa&list=EC913348 FFD75155C6 To further aid states as they continue to implement the Common Core State Standards (Standards), the Hunt Institute and the Council of Chief State School Officers have commissioned a series of video vignettes that explain the Standards in far greater depth. Several of the key Standards writers were asked, in their own words, to talk about how the Standards were developed and the goals they set for all students.

31. The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. - Common Core State Standards for Mathematics, page 6 The Standards for Mathematical Practice

32. 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. The Standards for Mathematical Practice

33. Grouping of Math Practices Reasoning and Explaining 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others Modeling and Using Tools 4. Model with mathematics 5. Use appropriate tools strategically Seeing Structure and Generalizing 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Overarching Habits of Mind of a Productive Mathematical Thinker 1. Make sense of problems and persevere in solving them 6. Attend to precision Adapted from (McCallum, 2011)

34. Standards for Mathematical Practice (SMP) in a Classroom Traditional U.S. Problem Which fraction is closer to 1: 1/2 or 1/4? Same Problem with SMP integration 1/2 is closer to 1 than is 1/4. Using a number line, explain why this is so. (Daro, Feb 2011)

35. Integration of Standards for Mathematical Practice Not “Problem Solving Fridays” Not “enrichment” for advanced students Most lie in the process of arriving at an answer, not necessarily in the answer itself Every lesson should seek to build student expertise in Content and Practice standards

36. Refer to the Standards for Mathematical Practice Read Standards for Mathematical Practice Highlight the key points in each description Discuss the key points you noted for each standard Activity

37. Make sense of problems and persevere in solving them. Understands the meaning of the problem and looks for entry points to its solution Monitors and evaluates the progress and changes course as necessary Analyzes information (givens, constraints, relationships, goals) Checks their answers to problems and ask, “Does this make sense?” Designs a plan Mathematical Practice 1

38. Reason Abstractly and Quantitatively Makes sense of quantities and relationships Represents a problem symbolically Considers the units involved Understands and uses properties of operations Mathematical Practice 2

39. Construct viable arguments and critique the reasoning of others. Uses definitions and previously established causes/effects (results) in constructing arguments Makes conjectures and attempts to prove or disprove through examples and counterexamples Communicates and defends their mathematical reasoning using objects, drawings, diagrams, actions Listens or reads the arguments of others Decide if the arguments of others make sense Ask useful questions to clarify or improve the arguments Mathematical Practice 3

40. Model with mathematics. Apply reasoning to create a plan or analyze a real world problem Applies formulas/equations Makes assumptions and approximations to make a problem simpler Checks to see if an answer makes sense and changes a model when necessary Mathematical Practice 4

41. Use appropriate tools strategically. Identifies relevant external math resources (digital content on a website) and uses them to pose or solve problems Makes sound decisions about the use of specific tools. Examples may include: Calculator Concrete models Digital Technology Pencil/paper Ruler, compass, protractor Uses technological tools to explore and deepen understanding of concepts Mathematical Practice 5

42. Attend to precision. Communicates precisely using clear definitions Provides carefully formulated explanations States the meaning of symbols, calculates accurately and efficiently Labels accurately when measuring and graphing Mathematical Practice 6

43. Look for and make use of structure. Looks for patterns or structure Recognize the significance in concepts and models and can apply strategies for solving related problems Looks for the big picture or overview Mathematical Practice 7

44. Look for and express regularity in repeated reasoning. Notices repeated calculations and looks for general methods and shortcuts Continually evaluates the reasonableness of their results while attending to details and makes generalizations based on findings Solves problems arising in everyday life Mathematical Practice 8

45. As you review the mathematical practices for your grade band consider: What instructional decisions will the teacher have to make so that students are successful? How will the teacher ensure the tasks done in class support the development of Standards for Mathematical Practice? Activity 2

46. Shift 5 and 6: APPLICATION and DUAL INTENSITY

47. The Importance of Dual Intensity in Mathematics Link to Video: http://www.youtube.com/watch?v=5dUQtIXoptY&feature=BFa&list=EC913348F FD75155C6 To further aid states as they continue to implement the Common Core State Standards (Standards), the Hunt Institute and the Council of Chief State School Officers have commissioned a series of video vignettes that explain the Standards in far greater depth. Several of the key Standards writers were asked, in their own words, to talk about how the Standards were developed and the goals they set for all students.

48. 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. Application

49. This is an end to the false dichotomy of the “math wars.” For years people have fought over which is most important: computational skill or conceptual understanding. The Common Core State Standards for Mathematics have ended the war. They are equally important and both crucial for mathematical understanding. Balanced Emphasis

50. Emphasis on both conceptual understanding and procedural fluency starting in the early grades. Focus on mastery of complex concepts in higher math using more hands-on learning Emphasis on mathematical modeling in the upper grades Balanced Emphasis