How to help students deepen their learning of K-5 math now while your better prepare them for the future Monica Neagoy, Ph.D. 2013 Outline Algebra in the Elementary Grades? (A little history) What is Algebra? What is Early Algebra? Concrete Examples for grades Pre-K through 5 These examples will be the essence of my talk 1
Algebra in Elementary School Geometry back in the 1960s Algebra in the 1980s By 2000, the PSSM (NCTM) established algebra as a content strand that cuts across all grades In 2006, the CFP (NCTM), and in 2010 CCSSM (NGACBP/CCSSO) reiterated the connection of algebra to the critical ideas that young children need to learn Early Algebra Algebra in Elementary School Geometry back in the 1960s Algebra in the 1980s By 2000, the PSSM (NCTM) established algebra as a content strand that cuts across all grades In 2006, the CFP (NCTM), and in 2010 CCSSM (NGACBP/CCSSO) reiterated the connection of algebra to the critical ideas that young children need to learn Early Algebra 2
Algebra in Elementary School Geometry back in the 1960s Algebra in the 1980s By 2000, the PSSM (NCTM) established algebra as a content strand that cuts across all grades In 2006, the CFP (NCTM), and in 2010 CCSSM (NGACBP/CCSSO) reiterated the connection of algebra to the critical ideas that young children need to learn Early Algebra Algebra in Elementary School Geometry back in the 1960s Algebra in the 1980s By 2000, the PSSM (NCTM) established algebra as a content strand that cuts across all grades In 2006, the CFP (NCTM), and in 2010 CCSSM (NGACBP/CCSSO) reiterated the connection of algebra to the critical ideas that young children need to learn The phrase Early Algebra was coined 3
But What is Algebra? Just like the definition of mathematics, the definition of algebra has evolved through the ages The evolution of Algebra s notation and the development of its subject matter have gone hand in hand since its earliest traces in Babylonian mathematics about 4000 years ago But What is Algebra? Just like the definition of mathematics, the definition of algebra has evolved through the ages The evolution of both Algebra s notation and subject matter have gone hand in hand since its earliest traces in Babylonian mathematics (about 4000 years ago) 4
Algebra Subject Matter (Conceptual Stages) 1 st stage: The geometric stage, where most of the concepts of algebra are geometric 1 3 6 10 15 x x 2 x xy x x x y x x 3 x Algebra Subject Matter (Conceptual Stages) 1 st stage: The geometric stage, where most of the concepts of algebra are geometric 1 3 6 10 15 x x 2 x x xy x x y x x 3 x 5
Algebra Subject Matter (Conceptual Stages) 2 nd stage: The static equation-solving stage: the goal is to find a specific number that satisfies a relationship This stage began with Al-Khwarizmi in the 9 th century and lasted about 800 years Hisab al-jabr w al-muqabala, Kitab al-jabr wa-l-muqabala (820 C.E.) which means The Compendious Book on Calculation by Completion and Balancing al-jabr Algebra Al-Khwarithmi Algorithm (Classical Arabic pronunciation) Algebra Subject Matter (Conceptual Stages) 2 nd stage: The static equation-solving stage: the goal is to find a specific number that satisfies a relationship This stage began with Al-Khwarizmi in the 9 th century and lasted about 800 years Hisab al-jabr w al-muqabala, Kitab al-jabr wa-l-muqabala (820 C.E.) which means The Compendious Book on Calculation by Completion and Balancing al-jabr Algebra Al-Khwarithmi Algorithm (Classical Arabic pronunciation) Persian Mathematician 780-850 C.E. Scientist and Scholar in the House of Wisdom Baghdad 6
Algebra Subject Matter (Conceptual Stages) 2 nd stage: The static equation-solving stage: the goal is to find a specific number that satisfies a relationship This stage began with Al-Khwarizmi in the 9 th century and lasted about 800 years Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa l-muqābala (820 C.E.) which means The Compendious Book on Calculation by Restoring and Balancing al-ğabr Algebra (An example: 5x = 45 4x 9x = 45) Al-Khwarithmi Algorithm (Classical Arabic pronunciation) Persian Mathematician 780-850 C.E. Scientist and Scholar in the House of Wisdom Baghdad Equations were born from problems in everyday life Quadratic equations (x 2 ) x 2 20x + 50 = 0 (I) Solution: x is about 3 Linear equations (x) 20 x 10 An inheritance problem regarding equal sharing of a plot of land 10x = 6 (II) Solution: x is 1/2 + 1/10 (6/10 or 3/5) Equal sharing of 10 breads among 6 7
Algebra Subject Matter (Conceptual Stages) 3 rd stage: The dynamic function stage, where motion was an underlying idea (around the 17 th century) Johann Kepler path of planets Galileo Galilei path of projectiles Algebra Subject Matter (Conceptual Stages) 4 th stage: By the beginning of the 20 th century, algebra was less about finding solutions of equations and more about looking for common structures in different mathematical objects or situations ( ) The abstract stage, where mathematical structure plays the central role. 8
Algebra Subject Matter (Conceptual Stages) 4 th stage: By the beginning of the 20 th century, algebra was less about finding solutions of equations and more about looking for common structures in different mathematical objects or situations ( ) The abstract stage, where mathematical structure plays the central role. School algebra today is all this and more: Express or model geometric and other real-world situations Solve equations. Ex: 2x = 10. Here x is an unknown Study relationships between quantities that change Ex: y = 2x. Here x and y are variables and the relationship is called a function. (Curves or graphs) Examine mathematical structure arithmetic, functional thinking, mathematical modeling, and quantitative reasoning. (NCTM, 2011) 9
Algebra s notation: x, y, z, w, t, v a, b, c, d, e x + y, x y, xy, =,,,, x y ( ), [ ], { } x 2, x 3, x 4 x, x! (For example: 5! = 5 x 4 x 3 x 2 x 1 The story about x Questions: So What is Early Algebra? Are elementary teachers expected to teach high school algebra, early? 10
Answer: An emphatic NO! Early Algebra is about: Fostering ways of thinking, doing, and communicating about mathematics with understanding; Making connections, analyzing relationships, noticing structure, studying change, and solving problems; Conjecturing, justifying, symbolizing, mathematizing and most importantly, generalizing! 11
Early Algebra is about: Fostering ways of thinking, doing, and communicating about mathematics with understanding; Making connections, analyzing relationships, noticing structure, studying change, and solving problems; Analyzing regularities, conjecturing, justifying, mathematizing, symbolizing and most importantly, generalizing! Early Algebra is about: Fostering ways of thinking, doing, and communicating about mathematics with understanding; Making connections, analyzing relationships, noticing structure, studying change, and solving problems; Analyzing regularities, conjecturing, justifying, mathematizing, symbolizing and most importantly, generalizing! 12
Early Algebra is about cultivating habits of mind that instill in teachers and students a very different view of algebra from the popular one captured in this cartoon: 13
Quotes If I had to explain algebra to a student, I would say: think of all that you know about mathematics. Algebra is about making it richer, more connected, more general, and more explicit Ricardo Nemirovsky (Early Algebra Research Team) The real voyage of discovery consists not in seeking new landscapes but in seeing the old ones in new ways Marcel Proust (French novelist) The concrete examples Planting the Seeds of Algebra: Explorations for the Early Grades Corwin Press 2 Grade Bands: Grades Pre-K 2 Grades 3 5 (2012) (2013) 14
A Pre-K 1 Example: The Concept of Odd and Even as a Bridge from Pre-K to High School Do you think young children would know the answers to these questions? EVEN + EVEN =? ODD + ODD =? 2 minutes EVEN + ODD =? Guiding Questions (1) Would they know one, two, three, all three, or none of these? (2) If they knew something about even or odd numbers, what would it be? (3) How might they express what they know? 15
Build Double Deckers with In this discussion, you are examining the structure of even numbers You re not simply looking at the ending (ones) digit and checking if it is a 0, 2, 4, 6, or 8 16
Over time All even numbers can be written as 2 times 2 times something 2 x a quantity 2 x a number 2 x n 2n Examples: 8 = 2 x 4 20 = 2 x 10 86 = 2 x 43 For the Odd Numbers think of double-decker car carriers 17
Over time All odd numbers can be written as 2 times + 2 times something + 1 2 x a quantity + 1 2 x a number + 1 2 x n + 1 2n + 1 Examples: 9 = 2 x 4 + 1 21 = 2 x 10 + 1 86 = 2 x 43 + 1 At the end of the Bridge from Kindergarten to High School Students make sense of formal proofs such as this one: (2n+1) + (2m+1) = 2n + 1 + 2m + 1 = 2n + 2m + 1 + 1 = 2n + 2m + 2 = 2(n + m + 1) Renaming n + m + 1 as N, we get: = 2N This number is even! Q.E.D. 18
In what ways is this algebraic? We re treating numbers algebraically (not numerically) We re looking at the common structure of all even numbers; and of all odd numbers We re looking at relationships between even and odd: We re defining odd in terms of even (In many languages, the word for odd is not even ); We re making a connection We re combining even + even, odd + odd, and odd + even We re generalizing: we re saying something about all even numbers (2n); about all odd numbers (2n + 1); about all possible combinations In what ways is this algebraic? We re treating numbers algebraically (not numerically) We re looking at the common structure of all even numbers; and of all odd numbers We re looking at relationships between even and odd: We re defining odd in terms of even (In many languages, the word for odd is not even ); We re making a connection We re combining even + even, odd + odd, and odd + even We re generalizing: we re saying something about all even numbers (2n); about all odd numbers (2n + 1); about all possible combinations 19
In what ways is this algebraic? We re treating numbers algebraically (not numerically) We re looking at the common structure of all even numbers; and of all odd numbers We re looking at relationships between even and odd: We re defining odd in terms of even (In many languages, the word for odd is not even ); We re making a connection We re combining even + even, odd + odd, and odd + even We re generalizing: we re saying something about all even numbers (2n); about all odd numbers (2n + 1); about all possible combinations In what ways is this algebraic? We re treating numbers algebraically (not numerically) We re looking at the common structure of all even numbers; and of all odd numbers We re looking at relationships between even and odd: We re defining odd in terms of even (In many languages, the word for odd is not even ); We re making a connection We re combining even + even, odd + odd, and odd + even We re generalizing: we re saying something about all even numbers (2n); about all odd numbers (2n + 1); about all possible combinations 20
In what ways is this algebraic? We re treating numbers algebraically (not numerically) We re looking at the common structure of all even numbers; and of all odd numbers We re looking at relationships between even and odd: We re defining odd in terms of even (In many languages, the word for odd is not even ); We re making a connection We re combining even + even, odd + odd, and odd + even We re generalizing: we re saying something about all even numbers (2n); about all odd numbers (2n + 1); about all possible combinations Quote I do not suggest avoiding the last digit rule, this would be impossible, but rather postponing it in order to achieve odd/even classification by essence and not by superficial features of the common representation [of numbers]. Rina Zazkis Relearning Mathematics: A Challenge for Prospective Elementary School Teachers 21
How was that for a first exploration in early algebra? How many of you think you might use it with your students? Pointers: Use one color for even numbers, and another color for odd You will need 30 cubes to build numbers 2, 4, 6, 8, 10 You will need 25 cubes to build numbers 1, 3, 5, 7, 9 22
A Grade 1 4 Example: A Model for Subtraction as a Bridge from Elementary School to High School and College Models for Subtraction Think of the mental image, metaphor, or model for Subtraction that you use in the Early Grades to help students understand the concept 2 minutes for discussion Guiding Questions: What s an example of a subtraction story you might give your students? What s an example of a subtraction story your students might come up with? What s an example of a subtraction story a parent might give his/her child? 23
A Common Model for Subtraction LeAnh has a jar with 6 cookies She eats 2 and 4 are left That s an example of the take-away model 24
Or Carlos has 7 balloons He pops 2 and 5 are left Another example of the take-away model for subtraction 25
Take-away is an example of a subtraction story (model, metaphor) Change minus or Change to less This is the most common model or metaphor we offer students in elementary school. A less common model: The comparison model Place 10 multi-link cubes in one pile Place 6 multi-link cubes in another pile Ask a young child, Which pile has more? Ask, How many more? and How do you know? 26
Some children will Make 2 rods: One with the 10 cubes, another with the 6 cubes then Place them side by side to visualize the difference Geometric Model: The Difference between 10 and 6 is the Distance between the two points 0 6 10 Make a 4-cube rod to symbolize the difference 27
Make a 4-cube difference rod. Use Handout 1 to shift the 4-cube rod up and down the x-axis. Write the corresponding subtraction equations that equal 4 Handout 1: Exploration on Subtraction Model -4-3 -2-1 0 1 2 3 4 5 6 7 3 minutes to write equations Using connecting, multilink, or unifix cubes, make a 4-cube rod and then wait for instructions from Dr. Neagoy given during the Webinar 10 6 = 4 7 3 = 4 6 2 = 4 5 1 = 4 4 0 = 4 3 (-1) = 4 2 (-2) = 4 1 (-3) = 4 0 (-4) = 4 Observe the relationship between any two equivalent equations 28
10 6 = 4 7 3 = 4 6 2 = 4 5 1 = 4 4 0 = 4 3 (-1) = 4 2 (-2) = 4 1 (-3) = 4 0 (-4) = 4 Observe the relationship between any two equivalent equations The distance between two numbers on a number path begins in Grades Pre-K and K! 7 6 5 4 3 2 1 0-1 -2 29
Let s make some connections: Consider this child-unfriendly subtraction problem 3 0 0-1 9 9 100 200 300 3 0 0-1 9 9 1 + 100 = 101 100 200 300 30
3 0 0 + 1 3 0 1-1 9 9 + 1-2 0 0 1 0 1 Thinking geometrically The distance on the number line between 199 and 300 is the same as the distance between 200 and 301 We ve shifted the difference rod to the right by 1 3 0 0 + 1 3 0 1-1 9 9 + 1-2 0 0 1 0 1 Thinking algebraically: x y = (x + c) (y + c) = x + c y c = x y + c c ( +c and c are additive inverses) = x y + 0 ( +c c = 0, the additive identity) = x y 31
+12 3 3 4 5 3 3 5 7 +12 8 8 1 0 0??? 3 2 5 7 What is the connection to HS algebra? Slope: m= y x 2 2 y1 x 1 y 2 P (x 2, y 2 ) y 1 Q (x 1, y 1 ) x 1 x 2 32
The distance formula also contains differences The distance from Point P to Point Q is: One final connection (back to Elementary Mathematics) 0 1 2 3 4 5 6 7 8 9 10 0 0-1 -2-3 -4-5 -6-7 -8-9 -10 1 1 0-1 -2-3 -4-5 -6-7 -8-9 2 2 1 0-1 -2-3 -4-5 -6-7 -8 3 3 2 1 0-1 -2-3 -4-5 -6-7 4 4 3 2 1 0-1 -2-3 -4-5 -6 5 5 4 3 2 1 0-1 -2-3 -4-5 6 6 5 4 3 2 1 0-1 -2-3 -4 7 7 6 5 4 3 2 1 0-1 -2-3 8 8 7 6 5 4 3 2 1 0-1 -2 9 9 8 7 6 5 4 3 2 1 0-1 10 10 9 8 7 6 5 4 3 2 1 0 33
I hope you got something out of the subtraction exploration. Let me know if it was helpful For All Grades: Exploring the equals sign (=) 34
Warm-up: What number would you put in the? 200 - = 98-45 3 minutes After you found the number: Share your thinking/strategy with your neighbor or group Compare strategies If you are watching alone, think of another to solve this equation 200 = 98 45 35
8 + 4 = + 5 What do you think = signals to young children? Why do you suppose this is the case? Why is relational thinking helpful? Solve using relational thinking Handout 3: Exploration on Number Sentences (or Equations) 1.NumberSentencesorNumericalEquationsinvolvingSubtraction 7 4 = 5 (I) 18 9 = 11 (II) 78 = 82 59 (III) 351 = 724 346 (IV) 2.NumberSentencesorNumericalEquationsinvolvingAddition 7 + 12 = 10 + (I) 3 + 25 = + 10 (II) 39 + = 77 + 40 (III) + 99 = 255 + 104 (IV) 3.NumberSentencesorNumericalEquationsinvolvingmultiplication AftertheWebinar,explorethesequestions: What would they look like? Make up a few How would you solve them? How would you guide your students toward discovering relationships? 36
Next Exploration: Algorithms An Example: 2 1 x 1 3 1. We work from right to left 2. We betray early place-value sense 3. We say, Put a smiley face and move to the left 37
Partial Products Algorithm: 2 1 (20 + 1) x 1 3 (10 + 3) 2 0 0 1 0 6 0 3 2 7 3 Let s look at this Geometrically Handout 2: For Exploration on Multiplication Place 13 along this edge P l a c e 21 a l o n g t h i s Using base-ten blocks, ask students to construct this product as a surface area e d g e 38
10 3 20 200 60 1 10 3 A Bridge to Algebra x y A common error in Algebra: (x + y) 2 = x 2 + y 2??? (x + y) 2 = (x + y)(x + y) x x 2 xy = x 2 + xy + yx + y 2 = x 2 + 2xy + y 2 y yx y 2 39
Another Example of a Bridge from K to 12 In the U.S., there is an emphasis on the Function Approach to Algebra. This begins early with the study of simple relations In the past HS Students almost exclusively worked with the symbolic representations of algebra. We should explore multiple representations, early on 40
Algebra in the early grades A function is a relationship between quantities that change. For example: The number of eyes in this classroom is equal to twice the number of students OR There are twice as many eyes as students in this classroom Algebra in the early grades The Multiple Representations of this simple relationship between the two quantities that change or variables: e (# of eyes) and s (# of students): -- Verbal -- Tabular (numerical) -- Concrete -- Graphical -- Pictorial -- Symbolic 41
Verbal: There are twice as many eyes as students in the classroom There are two times as many Concrete 42
Pictorial # of student s # of eyes Tabular S E 1 // 2 //// 3 //// / 4 //// /// 5 //// Etc. //// Etc. 43
S E 1 2 2 4 3 6 4 8 5 10 Tabular/Numeric (Continued) S 1 2 3 4 5 E 2 4 6 8 10 Why the emphasis on tables? Graphical 14 E = 2 x C 12 E = # of Eyes 10 8 6 4 2 0 0 1 2 3 4 5 6 7 C = # of Children 44
And Finally Symbolic e = # of eyes s = # of students 2e = s or 2s = e There are twice as many eyes as students in this classroom Symbolic (Cont) By the time students get to high school y = 2x or f (x) = 2x dependent independent This is an example of a linear function 45
Final Challenge: Even if I know all my math facts, all the vocabulary, and all the algorithms, can I answer this: 62 x 48 = 52 x 58? 52 is 10 less than 62 and 58 is 10 more than 48, so these two products must be equal, right? How would you respond? Thank you very much!!! Monica Neagoy MonicaNeagoy@earthlink.net www.monicaneagoy.com Planting the Seeds of Algebra Books Corwin Press for Grades 3-5 and 6-8 coming soon! 46