1. DEBORAH SCHIFTER EDUCATION DEVELOPMENT CENTER Where's Early Algebra in the Common Core State Standards?

2. DEBORAH SCHIFTER EDUCATION DEVELOPMENT CENTER Interpreting the Common Core State Standards

3. Promising Features of the CCSS A coordinated program from state to state A more focused, coherent curriculum Emphasis on mathematics practices

4. Dangers of a narrow reading Going from a curriculum that is a mile wide and an inch deep to one that is an INCH wide and an inch deep

5. Our role in interpreting the document We must all continue to investigate the possibility of deep, connected mathematical learning on the part of our students and we must work to figure out classroom practices that support such deep learning.

6. In this talk, I will 1. Present some specifics from the Core Standards, 2. Share some insights from work in early algebra, 3. Provide an extended example from a third-grade class. Together, we will 4. Reconsider those specific standards in light of the classroom examples.

7. Part 1 Some specifics from the Curriculum Core Standards

8. Mathematical Practices 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.

9. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.

10. Grade 3 Instructional time should focus on four critical areas (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.

11. Critical Area 1 Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal- sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size.

12. Critical Area 1 (cont.) Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

13. Grade 3 Domains Operations and Algebraic Thinking Number and Operations in Base Ten Number and Operations—Fractions Measurement and Data Geometry

14. Overview: Operations and Algebraic Thinking Represent and solve problems involving multiplication and division. Understand properties of multiplication and the relationship between multiplication and division. Multiply and divide within 100. Solve problems involving the four operations, and identify and explain patterns in arithmetic.

15. Understand properties of multiplication. 5. Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

16. Solve problems involving the four operations, and identify and explain patterns in arithmetic. 9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

17. Part 2: Some insights from work in early algebra Mathematical Practice 8: Look for and express regularity in repeated reasoning. Operations and Algebraic Thinking Standard 5: Understand properties of multiplication. Apply properties of operations as strategies to multiply and divide. Operations and Algebraic Thinking Standard 9: Identify arithmetic patterns and explain them using properties of operations.

18. Foundations of Algebra in the Elementary and Middle Grades Susan Jo Russell Deborah Schifter Virginia Bastable Funded by the National Science Foundation

19. Key aspects of integrating early algebra into arithmetic instruction Investigating, describing, and justifying general claims about how an operation behaves A shift in focus from solving individual problems to looking for regularities and patterns across problems Representations of the operations are the basis for proof The operations themselves become objects of study

20. 9 + 4 = 10 + 3 What is happening when you change the expression 9 + 4 to 10 + 3? Does this work with other numbers? Does this work with all numbers? How do you know?

21. 9 + 4 = 10 + 3 What is happening when you change the expression 9 + 4 to 10 + 3? Does this work with other numbers? Does this work with all numbers? How do you know?

22. 9 + 4 = 10 + 3 What is happening when you change the expression 9 + 4 to 10 + 3? a + b = (a + n) + (b – n)

23. What happens with subtraction? Is it true that 9 ­ 4 = 10 ­ 3? Why doesn’t it work? Is there another rule that works for subtraction? What happens with multiplication? Is it true that 9 x 4 = 10 x 3? Why doesn't it work? What does work?

24. Middle School Students’ Errors (3 + 5) 2 ≠ 3 2 + 5 2 but we see middle grade students write: (a + b) 2 = a 2 + b 2 2 x (3 x 5) ≠ (2 x 3) x (2 x 5) but we see middle grade students write: 2 (ab) = (2a)(2b)

25. Joy: an 8 th grader x 2 + 4x + 2x + 8 =

26. Joy: an 8 th grader x 2 + 4x + 2x + 8 = (x  x) + (2  4)

27. Joy: an 8 th grader x 2 + 4x + 2x + 8 = (4x) (2x)

28. I've been wondering for a while what the fundamental difficulties are that students encounter in mathematics at the middle school level. I used to think that the fundamental difficulty had to do with the introduction of variables, and with expressions and formulas. Mr. Lancaster

29. Mr. Lancaster (cont.) More recently, I've come to think that an even more fundamental difficulty has to do with the understanding of multiplication as distinct from addition. I'm not talking about whether students know their multiplication facts, and I'm not talking about whether students can memorize different sets of rules for things additive as for things multiplicative.

30. Mr. Lancaster (cont.) The conclusion I'm coming to is that the students who have the deepest trouble with middle school mathematics are those without a clear and rich set of models for what multiplication is and how it is different from addition. In the absence of such models, the students tend to rely increasingly on memorizing rules and how-to's, and their grasp is tentative and fragile. They have given up trying to make sense of what they are doing, and instead rely on their quite well-honed skills of guessing what the teacher wants.

31. Mr. Lancaster (cont.) To my surprise, this includes many of the students who have decent grades, and who can get by quite well with symbolic manipulation of formulas that may hold very little meaning for them.

32. Three pillars of elementary arithmetic Understanding numbers Computational fluency Examining the behavior of the operations

33. Three pillars of computation Understanding numbers written and oral counting the structure of the base ten system with whole numbers and decimals the meaning of fractions, zero, and negative numbers Computational fluency accurate, efficient, and flexible strategies for each operation knowing how and when to apply them

34. Three pillars of computation Examining the behavior of the operations modeling these operations learning about the properties of each operation describing and justifying the behaviors that are consequences of those properties comparing and contrasting the behaviors of different operations

35. Teachers’ realizations They used to think of the heart of their curriculum as understanding numbers and learning to compute. Now they’d identified a third objective of equal weight: investigating the behavior of the operations. They used to think they had to develop special lessons to engage students with generalizing about operations. Now they see that opportunities to work on this “third pillar” arise in all their work on number and operations.

36. Part 3: An extended example Mathematical Practice 8: Look for and express regularity in repeated reasoning. Operations and Algebraic Thinking Standard 5: Understand properties of multiplication. Apply properties of operations as strategies to multiply and divide. Operations and Algebraic Thinking Standard 9: Identify arithmetic patterns and explain them using properties of operations

37. Grade 3 example from Alice Kaye’s classroom 9 + 4 = 13 7 + 5 = 12 7 + 6 = __ 7 + 5 = 12 8 + 5 = __ 9 + 4 = 13 9 + 5 = __ 9 + 4 = 13 10 + 4 = __

38. Articulation of the claim In addition, if you increase one of the addends by 1, then the sum will also increase by 1. In addition, if you increase one of the addends by 1, and keep the other addend(s) the same, then the sum will also increase by 1.

39. Teacher’s challenge How would you go about convincing somebody that this statement is true? What would you do to convince yourself or someone else?

40. 9,000,000,000,000,000,000,000,000,000,000,000 + 9,000,000,000,000,000,000,000,000,000,000,001 = 18 decillion and 1 9 decillion and 2 + 9 decillion = 18 decillion and 2

41. Megan’s poster and explanation The picture could be used for ANY numbers, not just 3 and 4. I could have started with anything in one hand, and then anything else in the other hand, and put them together. If I got 1 more thing in either hand, the total would always only go up by 1.

42. Introduction of algebraic notation If a + b = c then (a + 1) + b = c + 1 and a + (b + 1) = c + 1 Some of the students were confused, and they objected. But most were able to see what they had just done reflected in these statements, saying, “Oh, that makes so much sense!”

43. Another version of the statement a + (b + 1) = (a + b) + 1

44. Multiplication, Grade 3 Writing prompt. In a multiplication problem, if you add 1 to a factor, I think this will happen to the product: 7 x 5 = 35 7 x 6 = 42 7 x 5 = 35 8 x 5 = 40 9 x 4 = 36 9 x 5 = 45 9 x 4 = 36 10 x 4 = 40

45. Students’ articulation of the claim The number that is not increased is the number that the answer goes up by. The number that is staying and not going up, increases by however many it is. I think that the factor you increase, it goes up by the other factor.

46. Choose which of the original equations you want to work with. Then do one of these… Draw a picture for the original equation; then change it just enough to match the new equations. Make an array for the original equation; then change it just enough to match the new equations. Write a story for the original equation; then change it just enough to match the new equations. 7 x 5 = 35 7 x 6 = 42 8 x 5 = 40

47. Frannie’s story context There are 7 jewelry boxes and each box has 5 pieces of jewelry. There are 35 pieces of jewelry altogether.

48. Jewelry boxes 7 x 5 Seven boxes with five pieces of jewelry in each box 35 pieces of jewelry

49. Jewelry boxes 7 x 5  8 x 5 Eight Seven boxes with five pieces of jewelry in each box 35 pieces of jewelry + 5 pieces of jewelry 40 pieces of jewelry

50. Jewelry boxes 7 x 5 Seven boxes with five pieces of jewelry in each box 35 pieces of jewelry

51. Jewelry boxes 7 x 5  7 x 6 six Seven boxes with five pieces of jewelry in each box 35 pieces of jewelry + 7 pieces of jewelry 42 pieces of jewelry

52. Articulating the claim about increasing a factor by 1 a x b Addison: If you increase the first number by 1, you increase the answer by the second number. Like if you increase a by 1, you increase the product by b. If you increase b by 1, you increase the answer by a.

53. Articulating the claim about increasing a factor by 1 a x b Addison: If you increase the first number by 1, you increase the answer by the second number. Like if you increase a by 1, you increase the product by b. If you increase b by 1, you increase the answer by a. a x b = ab a boxes with b pieces per box (a + 1) x b = ab + b1 more box  b additional pieces of jewelry a x (b + 1) = ab + a 1 more piece per box  a additional pieces of jewelry

54. Addison: If you increase the first number by 1, you increase the answer by the second number. Like if you increase a by 1, you increase the product by b. If you increase b by 1, you increase the answer by a. Ms. Kaye: I love this context! Because you can have any number of jewelry boxes, each one with a certain number of jewels in it. And the product is going to be that multiplication. And if you increase the number of boxes, you get an increase by the number in each box. If you increase by one jewel, the increase is the number of boxes because each box gets one jewel.

55. Other stories Baskets of bouncy balls Tanks with salmon eggs Baskets of mozzarella sticks Rows of chairs

58. Part 4 of this talk Together we will reconsider those specific standards in light of the classroom example.

59. Consider these standards Mathematical Practice 8: Look for and express regularity in repeated reasoning. Operations and Algebraic Thinking Standard 5: Understand properties of multiplication. Apply properties of operations as strategies to multiply and divide. Operations and Algebraic Thinking Standard 9: Identify arithmetic patterns and explain them using properties of operations.

60. Discuss at your tables: How do you see these three standards in the example from Ms. Kaye’s classroom? How does the example help you understand what the standards actually mean?

61. Concluding thoughts The term, "properties of the operations," is pervasive throughout the Core Curriculum State Standards. This should not be interpreted as simply learning the names, "associative property" or "distributive property." Rather, it is about being able to envision what the operations mean, to see regularities, and to investigate how the operations behave.

62. Concluding thoughts Similarly, we must reinterpret the word, "pattern." This involves seeing consistencies, exploring these, and investigating why they occur.

63. Concluding thoughts See how standards are interconnected.

64. Concluding thoughts Keep mathematical practices alive.

65. Resources Reasoning Algebraically about Operations, a module from the PD series, Developing Mathematical Ideas, published by Pearson. Connecting Arithmetic to Algebra, a book to appear in September 2011, published by Heinemann. The facilitators guide for Connecting Arithmetic to Algebra will appear in March 2012. An on-line course based on Connecting Arithmetic to Algebra will be offered by Virginia Bastable. Write to vbastabl@mtholyoke.edu. Put “CAA Fall 11” in the subject line vbastabl@mtholyoke.edu