1. Common Core Mathematics Vertical Understanding District SIP Day January 27, 2016 Common Core Math Learning Progressions

2. Video: “The Mathematics Standards: How They Were Developed and Who Was Involved” 2

3. Video Reflection Discuss two findings from the video that were new to you or changed your perception of the CCSS. 3

4. Video: “The Mathematics Standards and the Shifts They Require” 4

5. Video Reflection How have/will these shifts changed your instruction? 5

6. Video: “The Importance of Coherence in Mathematics” 6

7. Learning Progressions What are the Common Core Learning Progressions? 7

8. Learning Progressions The CCSS for mathematics were built on progressions — narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics. 8

9. Video: “The Importance of Mathematical Progressions from the Student Perspective” 9

10. Progressions Documents Review the Questions Read the Progressions Document. (K-5) Operations and Algebraic Thinking Answer the questions. 10

11. Progression Documents Discuss the questions as a (K-4/5) group. 11

12. Learning Progression Represented through Tasks Look at the Illustrative Mathematics Tasks for K-8. How does the learning connect, build upon, and progress through the years? What are the Big Math Ideas represented in the tasks? 12

13. Learning Progression Represented through Tasks How can these tasks be used in your instruction? 13

14. The Operations and Algebraic Thinking Progression depends on a student’s ability to compute accurately. Computational Fluency Strategies vs. Algorithms Fluency Expectations “The US Standard Algorithm” 14

15. Computational Fluency 15

16. What does it mean to have Computational fluency? 16

17. Video: “Mathematics Fluency: A Balanced Approach” 17

18. Thoughts on the video related to your current instructional practices? 18

19. Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. NCTM, Principles and Standards for School Mathematics 19

20. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships. NCTM, Principles and Standards for School Mathematics 20

21. “Computational fluency entails bringing problem solving skills and conceptual understanding to computational problems.” NCTM, Principles and Standards for School Mathematics 21

22. Developing fluency requires a balance and connection between conceptual understanding and computation proficiency. Computational methods that are over- practiced without understanding are forgotten or remembered incorrectly. Understanding without fluency can inhibit the problem solving process. NCTM, Principles and Standards for School Mathematics 22

23. Conceptual Understanding is an… Important component of proficiency, along with factual knowledge and procedural facility Essential component of the knowledge needed to deal with novel problems and settings (Application) NCTM, Principles and Standards for School Mathematics 23

24. What are the computational fluency expectations for K-6? Grade Level Common Core StandardRequired Fluency K K.OA. 5Add/subtract within 5 1 1.OA.6Add/subtract within 10 2 2.OA.2 2.NBT.5 Add/subtract within 20 Add/subtract within 100 3 3.OA.7 3.NBT.2 Multiply/divide within 100 Add/subtract within 1000 4 4.NBT.4Add/subtract within 1,000,000 5 5.NBT.5Multi-digit multiplication 6 6.NS.2 6.NS.3 Multi-digit division Multi-digit decimal operations

25. Kindergarten Understand addition, and understand subtraction. K.OA.A.5 Fluently add and subtract within 5 25

26. First Grade Add and subtract within 20. 1.OA.C.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). 26

27. Second Grade Add and subtract within 20. 2.OA.B.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. Use place value understanding and properties of operations to add and subtract. 2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 27

28. Third Grade Multiply and divide within 100. 3.OA.C.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. Use place value understanding and properties of operations to perform multi-digit arithmetic.¹ 3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 28

29. Fourth Grade Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm. 4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 29

30. Fifth Grade Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm. 5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 30

31. Sixth Grade Compute fluently with multi-digit numbers 6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm. 6. NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 31

32. Strategies Vs. Algorithms 32

33. Why should we spend time teaching strategies instead of teaching only the standard algorithm?

34. The CCSSM distinguish strategies from algorithms: Computation strategy: purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another-Using numbers flexibly Computation algorithm: a set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly

35. Building Strategies “ Strategy” emphasizes that computation is being approached thoughtfully with an emphasis on student sense making.

36. Instruction Should Focus on: Strategies for computing with whole numbers so students develop flexibility, base-ten number sense, and fluency. Development and discussion of various strategies related to place value, properties, and relationships between operations. Making connections between strategies and representations to develop a deeper understanding of concepts.

37. Students who used invented strategies before they learned standard algorithms demonstrated a better knowledge of base-ten concepts and could better extend their knowledge to new situations. When students compute with strategies they invent or choose because they are meaningful, their learning tends to be robust—they are able to remember and apply their knowledge.

38. Common school practice has been to present a single algorithm for each operation. However, more than one efficient and accurate computational algorithm exists for each arithmetic operation. If given the opportunity, students naturally invent methods to compute that make sense to them-or they use strategies that they understand. Deep understanding is evident when students operate with the most efficient strategy for the situation.

39. Think about it….. “In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable.” (4 th and 5 th grade) CCSSM

40. Standard algorithms for base-ten computations with the four operations rely on decomposing numbers written in base-ten notation into base-ten units. The properties of operations then allow any multi-digit computation to be reduced to a collection of single- digit computations. These single-digit computations sometimes require the composition or decomposition of a base-ten unit. A solid understanding of base-ten units, decomposition, properties, and relationships is necessary prior to working with the standard US algorithm.

41. Students use strategies for addition and subtraction in grades K-3. Students are expected to fluently add and subtract whole numbers using the standard algorithm by the end of grade 4. Progressions for the Common Core State Standards in Mathematics, K-5 Number and Operations in Base Ten, pg. 3

42. For students to become fluent in arithmetic computation, they must have efficient and accurate methods that are supported by an understanding of numbers and operations. “Standard” algorithms for arithmetic computation are only one means of achieving this fluency. NCTM, Principles and Standards for School Mathematics

43. “Fluency refers to having efficient, accurate, and generalizable methods (algorithms) for computing that are based on well- understood properties and number relationships.” NCTM, Principles and Standards for School Mathematics

44. Solve the following problem mentally: 7 + 4 + 8 + 3 + 9 + 2 + 5 + 6 + 1 + 5 What habit of mind/relationship thinking would we hope our students would use?

45. 432 – 198 Why is the “standard algorithm” NOT the most efficient strategy? Can you mentally compute the problem? Share your strategy.

46. 4000 - 2368 Why is the “standard algorithm” NOT the most efficient strategy? How can one use number line thinking to solve this problem with more accuracy and understanding?

47. 12 x 12 After setting up and attempting the standard algorithm, a student answers “14”. This is a GREAT example of why we SHOULD NOT teach the standard algorithm prematurely…Explain why?

48. District Curriculum Guides The District Curriculum Guides for each grade level and unit are available on the District Mathematics website. There are multiple examples of strategies that work to develop place value, properties, and relationships.

49. Possible Next Step: Read through the Curriculum Guides for the units that focus on the NBT and OA standards for the grade level before and after the grade you teach. Read through the lens of computation strategies and expectations for reasoning. 49

50. The kinds of experiences teachers provide clearly play a major role in determining the extent and quality of students’ learning. Students’ understanding can be built by actively engaging in tasks and experiences designed to deepen and connect their knowledge. Procedural fluency and conceptual understanding can be developed through problem solving, reasoning, and constructing viable arguments. NCTM, Principles and Standards for School Mathematics

51. Meaningful practice is necessary to develop fluency with basic number combinations and strategies with multi-digit numbers. Practice should be purposeful and should focus on developing thinking strategies and a knowledge of number relationships rather than drill isolated facts and procedures. NCTM, Principles and Standards for School Mathematics, pg. 87

52. Students who memorize facts or procedures without understanding often are not sure when or how to use what they know, and such learning is often quite fragile. NCTM, Principles and Standards for School Mathematics

53. Students who understand the structure of numbers and the relationship among numbers can work with them flexibly and with consistent accuracy. NCTM, Principles and Standards for School Mathematics

54. Video: “Shifts in Math Practice: The Balance between Skills and Understanding” 54

55. Learning Progressions and Computational Fluency What are the big takeaways from today’s work with the Progression Document and Computational Fluency? 55

56. Next Steps? What are the next steps in using the Progression Documents and knowledge of computational fluency in your planning and instruction? 56

57. For all you do for the students of Peoria Public Schools!

58. What questions do you have? email: susan.gobeyn@psd150.org