Learning Arithmetic as a Foundation for Learning Algebra Developing relational thinking Adapted from… Thomas Carpenter University of Wisconsin-Madison

Defining Algebra Many adults equate school algebra with symbol manipulation– solving complicated equations and simplifying algebraic expressions. Indeed, the algebraic symbols and the procedures for working with them are a towering, historic mathematical accomplishment and are critical in mathematical work. But algebra is more than moving symbols around. Students need to understand the concepts of algebra, the structures and principles that govern the manipulation of the symbols, and how the symbols themselves can be used for recording ideas and gaining insights into situations. (NCTM, 2000, p. 37) Many adults equate school algebra with symbol manipulation– solving complicated equations and simplifying algebraic expressions. Indeed, the algebraic symbols and the procedures for working with them are a towering, historic mathematical accomplishment and are critical in mathematical work. But algebra is more than moving symbols around. Students need to understand the concepts of algebra, the structures and principles that govern the manipulation of the symbols, and how the symbols themselves can be used for recording ideas and gaining insights into situations. (NCTM, 2000, p. 37)

Never the twain shall meet The artificial separation of arithmetic and algebra deprives students of powerful ways of thinking about mathematics in the early grades and makes it more difficult for them to learn algebra in the later grades. The artificial separation of arithmetic and algebra deprives students of powerful ways of thinking about mathematics in the early grades and makes it more difficult for them to learn algebra in the later grades.

Arithmetic vs Algebra Arithmetic Calculating answers Calculating answers = signifies the answer is next = signifies the answer is nextAlgebra Transforming expressions Transforming expressions = as a relation = as a relation

Arithmetic U Algebra Arithmetic Transforming expressions Transforming expressions = as a relation = as a relationAlgebra Transforming expressions Transforming expressions = as a relation = as a relation

Developing Algebraic Reasoning in Elementary School Rather than teaching algebra procedures to elementary school children, our goal is to support them to develop ways of thinking about arithmetic that are more consistent with the ways that students have to think to learn algebra successfully. Rather than teaching algebra procedures to elementary school children, our goal is to support them to develop ways of thinking about arithmetic that are more consistent with the ways that students have to think to learn algebra successfully.

Developing Algebraic Reasoning in Elementary School Enhances the learning of arithmetic in the elementary grades. Enhances the learning of arithmetic in the elementary grades. Smoothes the transition to learning algebra in middle school and high school. Smoothes the transition to learning algebra in middle school and high school.

Relational Thinking Focusing on relations rather than only on calculating answers Focusing on relations rather than only on calculating answers Looking at expressions and equations in their entirety rather than as procedures to be carried out step by step Looking at expressions and equations in their entirety rather than as procedures to be carried out step by step Engaging in anticipatory thinking Engaging in anticipatory thinking Using fundamental properties of arithmetic to relate or transform quantities and expressions Using fundamental properties of arithmetic to relate or transform quantities and expressions Recomposing numbers and expressions Recomposing numbers and expressions Flexible use of operations and relations Flexible use of operations and relations

6 + 2 = □ + 3

David: It’s 5. David: It’s 5. Ms. F: How do you know it’s 5, David? Ms. F: How do you know it’s 5, David? David: It’s there. There’s a 3 there. I couldn’t decide between 5 and 7. Three was one more than 2, and 5 was one less than 6. So it was 5 David: It’s there. There’s a 3 there. I couldn’t decide between 5 and 7. Three was one more than 2, and 5 was one less than 6. So it was 5

= David: I know it’s true, because it’s like the other one I did, is the same as David: I know it’s true, because it’s like the other one I did, is the same as Ms. F. It’s the same. How is it the same? Ms. F. It’s the same. How is it the same? David: 57 is right there, and 56 is there, and 6 is there and 5 is there, and there is 38 there and 39 there. David: 57 is right there, and 56 is there, and 6 is there and 5 is there, and there is 38 there and 39 there. Ms. F. I’m a little confused. You said the 57 is like the 5 and the 56 is like the 6. Why? Ms. F. I’m a little confused. You said the 57 is like the 5 and the 56 is like the 6. Why? David: Because the 5 and the 56, they both are one number lower than the other number. The one by the higher number is lowest, and the one by the lowest number up there would be more. So it’s true. David: Because the 5 and the 56, they both are one number lower than the other number. The one by the higher number is lowest, and the one by the lowest number up there would be more. So it’s true.

= (56 + 1) + 38 = 56 + (1 + 38)

Recomposing numbers = □ 8 + (2 + 5) = □ (8 + 2) + 5 = □

Recomposing numbers ½ + ¾ = □ ½ + (½ + ¼) = □

Using basic properties Relating arithmetic and algebra = 7 X X 10 = (7 + 4) X 10 = (7 + 4) X 10 = 110 = 110 7/12 + 4/12 = 7(1/12) + 4(1/12) = (7 + 4) X 1/12 = (7 + 4) X 1/12 7a + 4a = 7(a) + 4(a) = (7 + 4)a = (7 + 4)a = 11a = 11a

Using basic properties (not) 7a + 4 b = 11ab

X 2 - X - 2 = 0 (X – 2)(X + 1) = 0 X + 1 = 0X – 2 = 0 X = -1X = 2

X 2 - X - 2 = 0 (X + 1)(X - 2) = 0 X + 1 = 0X – 2 = 0 X = -1X = 2 (X + 1)(X - 2) = 6

X 2 - X - 2 = 0 (X + 1)(X - 2) = 0 X + 1 = 0X – 2 = 0 X = -1X = 2 (X + 1)(X - 2) = 6 X + 1 = 6X – 2 = 6 X = 5X = 8

X 2 - X - 2 = 0 (X + 1)(X - 2) = 0 X + 1 = 0X – 2 = 0 X = -1X = 2 (X + 1)(X - 2) = 6 X + 1 = 6X – 2 = 6 X = 5X = 8 (5 + 1)(5 - 2) = 18 (8 + 1)(8 -2) = 54

Multiplication properties of zero ax0 = 0 ax0 = 0 axb = 0 implies a = 0 or b = 0 axb = 0 implies a = 0 or b = 0

Equality as a relation = □ = □ + 5

Percent of Students Offering Various Solutions to =  + 5 Response/ Grade & 17 1 and 2 ???? 3 and 4 ???? 5 and 6 ???? 24

Percent of Students Offering Various Solutions to =  + 5 Response/ Grade & 17 1 and and and

Challenge--Try this! What are the different responses that students may give to the following open number sentence: What are the different responses that students may give to the following open number sentence: =  = 

Challenging students’ conceptions of equality = = = = = = = =

Challenging students’ conceptions of equality = = = = = = = = = = 4 + 7

Correct solutions to =  + 5 before and after instruction Grade Grade Before Inst. Before Inst. After Inst. After Inst. 1 and 2 1 and and 4 3 and and 6 5 and

Learning to think relationally, thinking relationally to learn Using true/false and open number sentences (equations) to engage students in thinking more flexibly and more deeply about arithmetic Using true/false and open number sentences (equations) to engage students in thinking more flexibly and more deeply about arithmetic

Learning to think relationally, thinking relationally to learn Using true/false and open number sentences (equations) to engage students in thinking more flexibly and more deeply about arithmetic Using true/false and open number sentences (equations) to engage students in thinking more flexibly and more deeply about arithmetic in ways that are consistent with the ways that they need to think about algebra. in ways that are consistent with the ways that they need to think about algebra.

True and false number sentences = = = = = = /16 – 2 17/18 = 4 11/ /16 – 2 17/18 = 4 11/15 12÷0 = 0 12÷0 = 0

Challenge--Try this! Construct a series of true/false sentences that might be used to elicit one of the conjectures in Table 4.1 Construct a series of true/false sentences that might be used to elicit one of the conjectures in Table 4.1 (on p ). (on p ). 34

Learning to think relationally =  =  =  = 

Learning to think relationally =  =  =  = 

More challenging problems A =  B. 82 – 39 = 85 – 37 -  C. 45 – 28 =  - 24

True or False = = × 47 = 37 × × 47 = 37 × 45

= True =

35 × 47 = 37 × 45 False

35 × 47 = 37 × 45 False 35 × 47 = (30 + 5) × (40 + 7) = (30 + 5) ×40 + (30 + 5) × 7 = (30×40 + 5×40) + (30×7 + 5×7) 37 × 45 = (30 + 7) × (40 + 5) = (30 + 7) ×40 + (30 + 7) × 5 = (30×40 + 7×40) + (30×5 + 7×5)

35 × 47 = 37 × 45 False 35 × 47 = (30 + 5) × (40 + 7) = (30 + 5) ×40 + (30 + 5) × 7 = (30×40 + 5×40) + (30×7 + 5×7) 37 × 45 = (30 + 7) × (40 + 5) = (30 + 7) ×40 + (30 + 7) × 5 = (30×40 + 7×40) + (30×5 + 7×5)

Parallels with multiplying binomials (X + 7)(X + 5) = (X + 7)(X + 5) = (X + 7)X + (X + 7) 5 = (X + 7)X + (X + 7) 5 = X 2 + 7X +5X + 35 = X 2 + 7X +5X + 35 = X 2 +(7 +5)X + 35 = X 2 +(7 +5)X + 35 = X 2 +12X + 35 X 2 +12X + 35

Thinking relationally to learn Learning number facts with understanding Learning number facts with understanding Constructing algorithms and procedures for operating on whole numbers and fractions Constructing algorithms and procedures for operating on whole numbers and fractions

Number sentences to develop Relational Thinking (Large numbers are used to discourage calculation) (Large numbers are used to discourage calculation) Rank from easiest to most difficult Rank from easiest to most difficult a) = 71 + d a) = 71 + d b) 92 – 57 = g – 56 b) 92 – 57 = g – 56 c) 68 + b = c) 68 + b = d) 56 – 23 = f – 25 d) 56 – 23 = f – 25 e) = 67 + p e) = 67 + p f) = y + 46 f) = y + 46 g) 74 – 37 = 75 - q g) 74 – 37 = 75 - q

Learning Multiplication facts using relational thinking Julie Koehler Zeringue Julie Koehler Zeringue

A learning trajectory for thinking relationally Starting to think relationally Starting to think relationally The equal sign as a relational symbol The equal sign as a relational symbol Using relational thinking to learn multiplication Using relational thinking to learn multiplication Multiplication as repeated addition Multiplication as repeated addition Beginning to use the distributive property Beginning to use the distributive property Recognizing relations involving doubles, fives, and tens Recognizing relations involving doubles, fives, and tens Appropriating relational strategies to derive number facts Appropriating relational strategies to derive number facts

Multiplication as repeated addition 3  7 =  7 = b = 2  6 2  9 = h + h

Beginning to use the distributive property 3  = 4  6 3  = 4  6 3  = 4  6 3  = 4  6 5  4 = 2   4 = 2   6 = 3  6 + g 5  6 = 3  6 + g 6  7 = a  7 + b  7 6  7 = a  7 + b  7 6  7 = h  7 + h  7 6  7 = h  7 + h  7

Multiplication facts x

Generating number facts based on doubles 3 × 8 = 2 × × 8 = 2 × × 8 = × 8 = × 8 = 8 × 3 3 × 8 = 8 × 3 4 × 9 = 2 × × 9 4 × 9 = 2 × × 9

Multiplication facts x

Generating number facts for nines and fives 9 × 7 = 10 × 7 – 7 9 × 7 = 10 × 7 – 7 9 × 7 = 10 × 7 – 9 9 × 7 = 10 × 7 – 9 9 × 7 = 10 × 7 - □ 9 × 7 = 10 × 7 - □ 7 × 5 = × 5 =

Multiplication facts x

Appropriating relational strategies to derive number facts 6 × 6 = 6 × × 6 = 6 × × 8 = 7 × 9 – 7 7 × 8 = 7 × 9 – 7

A learning trajectory for thinking relationally Starting to think relationally Starting to think relationally The equal sign as a relational symbol The equal sign as a relational symbol Using relational thinking to learn multiplication Using relational thinking to learn multiplication Multiplication as repeated addition Multiplication as repeated addition Beginning to use the distributive property Beginning to use the distributive property Recognizing relations involving doubles, fives, and tens Recognizing relations involving doubles, fives, and tens Appropriating relational strategies to derive number facts Appropriating relational strategies to derive number facts

3  7 = Ms L: “Could you read that number sentence for me and tell me if it is true or false”? Kelly: “Three times 7 is the same as 7 plus 7 plus 7. That’s true, because times means groups of and there are 3 groups of 7, 3 times 7 just says it in a shorter way”. Ms L: “Ok, nice explanation”.

3  7 = Ms L: “How about this, 3  7 = , is that true or false”? Kelly: “It’s true”. Ms L: “Wow, that was quick, how do you know that is true”? Kelly: “Can we go back up here [pointing to 3  7 = ]”? Ms L: “Sure”. Kelly: “Seven and 7 is 14, that is right here [drawing a line connecting two 7s in the first number sentence and writing 14 under them]. Fourteen went right into here [pointing to the 14 in the second number sentence]. Then there is one 7 left pointing to the third 7 in the first number sentence], and that went right here [pointing to the last 7 in the second number sentence]”.

4  6 = Ms L: “Ok, I have another one for you 4  6 = , true or false”? Kelly: “That is true”. Ms L: “Ok, how did you get that one so quickly”? Kelly: “Six plus 6 is 12, in this case, there are 4 groups of 6, so it is like this [writing ]. Six and 6 is 12, that leaves another 6 and 6, and that equals 12. So one 12 is here and one 12 went here [indicating the two 12s in the problem]. What I’m trying to say is there are four 6s and you broke them in half and made them into two 12s”.

4  6 = Continued Ms L: “Nice! Kelly, do you know right away what 4 times 6 is”? Kelly: “Yes”. Ms L: “What is it?” Kelly: “It’s [pause] thirty- [long pause] two.” Ms L: “Ok, do you know what 12 plus 12 is”? Kelly: “Yeah. That is the same thing, 32”. Ms L:” Do you have a way of doing 12 plus 12, to check it”? Kelly: “Well, there are two 10s, 20- oh wait, I was thinking of a different one”! Ms L: “You were thinking of a different multiplication problem”? Kelly: “Yes. 4 times 6 is 24, because 10 and 10 is 20, and 2 and 2 is 4, put those together and its 24”.

4  7 = □ Ms L: “Ok, here is another one. Four times 7 equals box. I want you tell me what you would put in the box to make this a true number sentence”. Kelly: “That would be [short pause] 28”. Ms L: “Ok, how did you get 28”? Kelly: “Well, I kinda had other problems… that went into this problem. If you go up here [pointing to 3  7 = ] 3 times 7 is the same as 7 plus 7 plus 7. That problem helped me and I used it with this problem, [pointing to 3  7 = ] 3 sevens is the same as 14 and 7… You add one more seven and that goes right to here. [Then she points to 4  6 = ] This problem also helped me because 4  7 is like… My mind went back up to here [pointing to 3  7 = ], and I said, there is another 7 so I could put those two 7s together, that’s 14, and there are two 14s, 10 and 10 is 20, 4 and 4 is 8, 28”.

Problem sequence 3  7 =  7 =  6 =  7 = □

Inventing algorithms

/8 6-5/8 5 3/8 5 3/8

5 ½ ÷ __ =  Number choices ½, ¼, ¾, 3/8 Choose one of the numbers depending on the level of your students. Choose one of the numbers depending on the level of your students. Change the problem to add additional challenge as needed. Change the problem to add additional challenge as needed. Allow students to choose one to vary the problem and allow choice. Allow students to choose one to vary the problem and allow choice.

5 ½ ÷ __ =  Number choices ½, ¼, ¾, 3/8 Put the problem into a context. Put the problem into a context. It takes __ of a cup of sugar to make a batch of cookies. I have 5 ½ cups of sugar. How many batches of cookies can I make? It takes __ of a cup of sugar to make a batch of cookies. I have 5 ½ cups of sugar. How many batches of cookies can I make? Solve for 3/8 cup of sugar for a batch. Solve for 3/8 cup of sugar for a batch.

It takes 3/8 of a cup of sugar to make a batch of cookies. I have 5 ½ cups of sugar. How many batches of cookies can I make?

5 ½ ÷ 3/8 =  5 ½ ÷ 3/8 =   × 3/8 = 5 ½  × 3/8 = 5 ½

 × 3/8 = 5 ½ 8 × 3/8 = 3 8 × 3/8 = 3 4 × 3/8 would be ½ of 3 or 1 ½ 4 × 3/8 would be ½ of 3 or 1 ½ 12 × 3/8 = 4 ½ => 4 ½ cups makes 12 batches 12 × 3/8 = 4 ½ => 4 ½ cups makes 12 batches Need to use 1 more cup of sugar Need to use 1 more cup of sugar Because 8 × 3/8 = 3, a third as much would be 1 Because 8 × 3/8 = 3, a third as much would be 1 i.e. 1/3 × (8 × 3/8) = 1 i.e. 1/3 × (8 × 3/8) = 1 So you need 1/3 of 8, which is 8/3 So you need 1/3 of 8, which is 8/3 i.e. 1 cup makes 8/3 batches. i.e. 1 cup makes 8/3 batches. So altogether you get a total of /3 or 14 2/3 batches So altogether you get a total of /3 or 14 2/3 batches

 × 3/8 = 5 ½ 8 × 3/8 = 3 8 × 3/8 = 3 ½(8 × 3/8) = ½ × 3 ½(8 × 3/8) = ½ × 3 (½×8)×3/8 = 1 ½ (½×8)×3/8 = 1 ½ 4 ×3/8 = 1 ½ 4 ×3/8 = 1 ½

Next subgoal: How many 3/8 cups to use the remaining cup? 8 × 3/8 = 3 8 × 3/8 = 3 1/3 × (8 × 3/8) = 1/3 × 3 1/3 × (8 × 3/8) = 1/3 × 3 (1/3×8)×3/8 = 1 (1/3×8)×3/8 = 1 8/3 ×3/8 = 1 8/3 ×3/8 = 1

Putting the parts together (8 ×3/8) + (4 ×3/8) + (8/3 ×3/8) = (8 ×3/8) + (4 ×3/8) + (8/3 ×3/8) = ½ + 1 = 5 ½ And And (8 ×3/8) + (4 ×3/8) + (8/3 ×3/8) = (8 ×3/8) + (4 ×3/8) + (8/3 ×3/8) = ( /3) ×3/8 = 14 2/3 ×3/8 ( /3) ×3/8 = 14 2/3 ×3/8 So 5 ½ cups of sugar makes 14 2/3 batches of 3/8 cups of sugar

Solving equations k + k + 13 = k + 20 k + k + 13 = k + 20

From arithmetic to algebraic reasoning Attend to relations rather than teaching only step by step procedures Attend to relations rather than teaching only step by step procedures Align the teaching of arithmetic with the concepts and skills students need to learn algebra Align the teaching of arithmetic with the concepts and skills students need to learn algebra Enhance the learning of arithmetic Enhance the learning of arithmetic Provide a foundation for and smooth the transition to learning algebra Provide a foundation for and smooth the transition to learning algebra

Learning arithmetic and algebra with understanding Algebra for all Algebra for all Not watering down algebra to teach isolated procedures Not watering down algebra to teach isolated procedures Develop algebraic reasoning rather than teaching meaningless algebraic procedures Develop algebraic reasoning rather than teaching meaningless algebraic procedures Learning arithmetic and algebra grounded in fundamental properties of number and number operations Learning arithmetic and algebra grounded in fundamental properties of number and number operations

Assignment What you will do before the next meeting You will be writing a series of problems that you might use with your students to encourage them to begin to look for relations. Write a problem to assess student thinking Write a problem to assess student thinking Predict student responses Predict student responses Write a series of problems to address your students…back up…extend??? Write a series of problems to address your students…back up…extend??? Try these with your students Try these with your students

Assignment cont’d You will be planning a lesson with your colleagues for a lesson study cycle You will be planning a lesson with your colleagues for a lesson study cycle As a team choose a topic to be taught on December 6 th As a team choose a topic to be taught on December 6 th Choose a topic that is typically difficult for students Choose a topic that is typically difficult for students Bring planning materials to the October session Bring planning materials to the October session

Challenge Addition is associative, but subtraction is not. How about the following: Addition is associative, but subtraction is not. How about the following: a) Is (a + b) - c = a + (b - c) true for all numbers? b) Is (a - b) + c = a - (b + c) true for all numbers? Thinking Mathematically p. 120 #4 Thinking Mathematically p. 120 #4 81

Challenge What kind of number do you get when you add three odd numbers? What kind of number do you get when you add three odd numbers? Can you justify your response? Can you justify your response? Thinking Mathematically p.103 #1 Thinking Mathematically p.103 #1 82

Challenge--Try this! Design a sequence of true/false and/or open sentences that you might use to engage your students in thinking about the equal sign. Design a sequence of true/false and/or open sentences that you might use to engage your students in thinking about the equal sign. Thinking Mathematically p. 24 #4 Thinking Mathematically p. 24 #4 83

References Carpenter, T.P., Franke, M.L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in the elementary school. Portsmouth, NH: Heinemann. Carpenter, T.P., Franke, M.L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in the elementary school. Portsmouth, NH: Heinemann. Carpenter, T. P., Franke, M.L., & Levi, L. (2005). Algebra in Elementary School. ZDM. 37(1), 1-7. Carpenter, T. P., Franke, M.L., & Levi, L. (2005). Algebra in Elementary School. ZDM. 37(1), 1-7. Carpenter, T. P., Levi, L., Berman, P., & Pligge, M. (2005). Developing algebraic reasoning in the elementary school. In T. A. Romberg, T. P. Carpenter, & F. Dremock (Eds). Understanding mathematics and science matters. Mahwah, NJ: Erlbaum. Carpenter, T. P., Levi, L., Berman, P., & Pligge, M. (2005). Developing algebraic reasoning in the elementary school. In T. A. Romberg, T. P. Carpenter, & F. Dremock (Eds). Understanding mathematics and science matters. Mahwah, NJ: Erlbaum. Jacobs, V.J., Franke, M.L., Carpenter, T. P., Levi, L., & Battey, D. (2007) A large-scale study of professional development focused on children’s algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38, Jacobs, V.J., Franke, M.L., Carpenter, T. P., Levi, L., & Battey, D. (2007) A large-scale study of professional development focused on children’s algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38,