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How the ideas and language of algebra K-5 set the stage for Algebra 8-12 MSRI, May 15, 2008 E. Paul Goldenberg To save note-taking, http://thinkmath.edc.org Click download presentations link (next week)

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Language vs. computational tool (n – d)(n + d) To us, expressions like (n – d)(n + d) can be manipulated to derive things we don’t yet know, or to prove things that we conjectured from experiment. most can use it the last two ways, as language. Claim: While most elementary school children cannot use algebraic notation the first two ways, as a computational tool, most can use it the last two ways, as language. (n – d)(n + d) = n 2 – d 2 We can also use such notation as language (not manipulated) to describe a process or computation or pattern, or to express what we already know, e.g.,

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Great built-in apparatus Abstraction (categories, words, pictures) Abstraction (categories, words, pictures) Syntax, structure, sensitivity to order Syntax, structure, sensitivity to order Phenomenal language-learning ability Phenomenal language-learning ability Quantification (limited, but there) Quantification (limited, but there) Logic (evolving, but there) Logic (evolving, but there) Theory-making about the world irrelevance of orientation In learning math, little differentiation Theory-making about the world irrelevance of orientation In learning math, little differentiation

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Some algebraic ideas precede arithmetic w/o rearrangeability 3 + 5 = 8 can’t make sense w/o rearrangeability 3 + 5 = 8 can’t make sense Nourishment to extend/apply/refine built-ins Nourishment to extend/apply/refine built-ins breaking numbers and rearranging parts (any-order-any-grouping, commutativity/associativity), breaking arrays; describing whole & parts (linearity, distributive property) But many of the basic intuitions are built in, developmental, not “learned” in math class. But many of the basic intuitions are built in, developmental, not “learned” in math class. Developmental

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Algebraic language, like any language, is Children are phenomenal language-learners Children are phenomenal language-learners Build it from language spoken around them Build it from language spoken around them Infer meaning and structure from use: not explicit definitions and lessons, but from language used in context Infer meaning and structure from use: not explicit definitions and lessons, but from language used in context Where “math is spoken at home” (not drill, lessons, but conversation that makes salient logical puzzle, quantity, etc.) kids learn it Where “math is spoken at home” (not drill, lessons, but conversation that makes salient logical puzzle, quantity, etc.) kids learn it Convention

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Demand “does it work with kids?”

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Algebraic language & algebraic thinking Linguistics and mathematics Linguistics and mathematics Algebra as abbreviated speech (Algebra as a Second Language) Algebra as abbreviated speech (Algebra as a Second Language) A number trick “Pattern indicators” Difference of squares Systems of equations in kindergarten? Systems of equations in kindergarten? Understanding two dimensional information Understanding two dimensional information

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Linguistics and mathematics Michelle’s strategy for 24 – 8: Well, 24 – 4 is easy! Well, 24 – 4 is easy! Now, 20 minus another 4… Now, 20 minus another 4… Well, I know 10 – 4 is 6, and 20 is 10 + 10, so, 20 – 4 is 16. Well, I know 10 – 4 is 6, and 20 is 10 + 10, so, 20 – 4 is 16. So, 24 – 8 = 16. So, 24 – 8 = 16. A linguistic idea (mostly) Algebraic ideas (breaking it up) Arithmetic knowledge

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What is the “linguistic” idea? 28 – 8 on her fingers… Fingers are counters, good for grasping the idea, and good (initially) for finding or verifying answers to problems like 28 – 4, but…

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Algebraic language & algebraic thinking Linguistics and mathematics Linguistics and mathematics Algebra as abbreviated speech (Algebra as a Second Language) Algebra as abbreviated speech (Algebra as a Second Language) A number trick “Pattern indicators” Difference of squares Systems of equations in kindergarten? Systems of equations in kindergarten? Understanding two dimensional information Understanding two dimensional information

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Algebra as abbreviated speech (Algebra as a second Language) A number trick A number trick “Pattern indicators” “Pattern indicators” Difference of squares Difference of squares Surprise! You speak algebra! 5th grade

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A number trick Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result. Subtract 4. Subtract 4. Divide the result by 2. Divide the result by 2. Subtract the number you first thought of. Subtract the number you first thought of. Your answer is 1! Your answer is 1!

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How did it work? Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result. Subtract 4. Subtract 4. Divide the result by 2. Divide the result by 2. Subtract the number you first thought of. Subtract the number you first thought of. Your answer is 1! Your answer is 1!

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How did it work? Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result. Subtract 4. Subtract 4. Divide the result by 2. Divide the result by 2. Subtract the number you first thought of. Subtract the number you first thought of. Your answer is 1! Your answer is 1!

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How did it work? Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result. Subtract 4. Subtract 4. Divide the result by 2. Divide the result by 2. Subtract the number you first thought of. Subtract the number you first thought of. Your answer is 1! Your answer is 1!

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How did it work? Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result. Subtract 4. Subtract 4. Divide the result by 2. Divide the result by 2. Subtract the number you first thought of. Subtract the number you first thought of. Your answer is 1! Your answer is 1!

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How did it work? Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result. Subtract 4. Subtract 4. Divide the result by 2. Divide the result by 2. Subtract the number you first thought of. Subtract the number you first thought of. Your answer is 1! Your answer is 1!

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How did it work? Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result. Subtract 4. Subtract 4. Divide the result by 2. Divide the result by 2. Subtract the number you first thought of. Subtract the number you first thought of. Your answer is 1! Your answer is 1!

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How did it work? Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result. Subtract 4. Subtract 4. Divide the result by 2. Divide the result by 2. Subtract the number you first thought of. Subtract the number you first thought of. Your answer is 1! Your answer is 1!

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How did it work? Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result. Subtract 4. Subtract 4. Divide the result by 2. Divide the result by 2. Subtract the number you first thought of. Subtract the number you first thought of. Your answer is 1! Your answer is 1!

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Kids need to do it themselves…

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Using notation: following steps Think of a number. Double it. Add 6. Divide by 2. What did you get? 5 10 16 87320 Dan a CorySand y ChrisWordsPictures

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Using notation: undoing steps Think of a number. Double it. Add 6. Divide by 2. What did you get? 5 10 16 87320 Dan a CorySand y ChrisWords 14 Hard to undo using the words. Much easier to undo using the notation. Pictures

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Using notation: simplifying steps Think of a number. Double it. Add 6. Divide by 2. What did you get? 5 10 16 87320 Dan a CorySand y ChrisWordsPictures 4

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Abbreviated speech: simplifying pictures Think of a number. Double it. Add 6. Divide by 2. What did you get? 5 10 16 87320 Dan a CorySand y ChrisWordsPictures 4 b 2b2b 2b + 62b + 6 b + 3b + 3

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Notation is powerful! Computational practice, but much more Computational practice, but much more Notation helps them understand the trick. Notation helps them understand the trick. Notation helps them invent new tricks. Notation helps them invent new tricks. Notation helps them undo the trick. Notation helps them undo the trick. Algebra is a favor, not just “another thing to learn.” Algebra is a favor, not just “another thing to learn.”

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Algebra as abbreviated speech (Algebra as a second Language) A number trick A number trick “Pattern indicators” “Pattern indicators” Difference of squares Difference of squares

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Children are language learners… They are pattern-finders, abstracters… They are pattern-finders, abstracters… …natural sponges for language in context. …natural sponges for language in context. n 10 n – 8 2 8 0 28 20 1817 34 5857 Go to index

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Algebra as abbreviated speech (Algebra as a second Language) A number trick A number trick “Pattern indicators” “Pattern indicators” Difference of squares Difference of squares Is there anything less sexy than memorizing multiplication facts? Is there anything less sexy than memorizing multiplication facts? What helps people memorize? Something memorable! What helps people memorize? Something memorable! 4th grade 4th grade Math could be fascinating!

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Teaching without talking Wow! Will it always work? Big numbers? ? 3839404142 35 36 6789105432111213 80 81 1819202122 … … ? ? 1600 15 16 Shhh… Students thinking!

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Take it a step further What about two steps out?

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Shhh… Students thinking! Again?! Always? Find some bigger examples. Teaching without talking 12 16 6789105432111213 60 64 ? 58596061622829303132 … … ? ? ?

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Take it even further What about three steps out? What about four? What about five? 100 678910541514111213 75

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Take it even further What about three steps out? What about four? What about five? 1200 313233343530294039363738 1225

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Take it even further What about two steps out? 1221 313233343530294039363738 1225

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“OK, um, 53” “OK, um, 53” “Hmm, well… “Hmm, well… …OK, I’ll pick 47, and I can multiply those numbers faster than you can!” …OK, I’ll pick 47, and I can multiply those numbers faster than you can!” To do… 53 47 53 47 I think… 50 50 (well, 5 5 and …) … 2500 Minus 3 3 – 9 2491 2491 “Mommy! Give me a 2-digit number!” 2500 47484950515253 about 50

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But nobody cares if kids can multiply 47 53 mentally!

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What do we care about, then? 50 50 (well, 5 5 and place value) 50 50 (well, 5 5 and place value) Keeping 2500 in mind while thinking 3 3 Keeping 2500 in mind while thinking 3 3 Subtracting 2500 – 9 Subtracting 2500 – 9 Finding the pattern Finding the pattern Describing the pattern Describing the pattern Algebraic language Algebraic/arithmetic thinking Science

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(7 – 3) (7 + 3) = 7 7 – 9 n – 3 n + 3 n (n – 3) (n + 3) = n n – 9 (n – 3) (n + 3) Q? Nicolina Malara, Italy: “algebraic babble” (50 – 3) (50 + 3) = 50 50 – 9

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Make a table; use pattern indicator. 24 416 525 Distance awayWhat to subtract 11 39 d d d

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(n – d) (n + d) = n n – (n – d) (n + d) = n n – d d (7 – d) (7 + d) = 7 7 – d d n – d n + d n (n – d) (n + d) (n – d)

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We also care about thinking! Kids feel smart! Why silent teaching? Kids feel smart! Why silent teaching? Teachers feel smart! Teachers feel smart! Practice. Gives practice. Helps me memorize, because it’s memorable! Practice. Gives practice. Helps me memorize, because it’s memorable! Something new. Foreshadows algebra. In fact, kids record it with algebraic language! Something new. Foreshadows algebra. In fact, kids record it with algebraic language! And something to wonder about: How does it work? And something to wonder about: How does it work? It matters!

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One way to look at it 5 5

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One way to look at it 5 4 Removing a column leaves Not “concrete vs. abstract” semantic (spatial) vs. syntactic Kids don’t derive/prove with algebra.

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One way to look at it 6 4 Replacing as a row leaves with one left over. Not “concrete vs. abstract” semantic (spatial) vs. syntactic Kids don’t derive/prove with algebra.

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One way to look at it 6 4 Removing the leftover leaves showing that it is one less than 5 5. Not “concrete vs. abstract” semantic (spatial) vs. syntactic Kids don’t derive/prove with algebra.

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Algebraic language & algebraic thinking Linguistics and mathematics Linguistics and mathematics Algebra as abbreviated speech (Algebra as a Second Language) Algebra as abbreviated speech (Algebra as a Second Language) A number trick “Pattern indicators” Difference of squares Systems of equations in kindergarten? Systems of equations in kindergarten? Understanding two dimensional information Understanding two dimensional information

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Systems of equations Challenge: can you find some that don’t work? Challenge: can you find some that don’t work? in Kindergarten?! 5x + 3y = 23 2x + 3y = 11 Is there anything interesting about addition and subtraction sentences? Is there anything interesting about addition and subtraction sentences? Start with 2nd grade Start with 2nd grade Math could be spark curiosity! 4 + 2 = 6 3 + 1 = 4 10 += 7 3

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Back to the very beginnings Picture a young child with a small pile of buttons. Natural to sort. We help children refine and extend what is already natural.

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6 4 7310 Back to the very beginnings Children can also summarize. “Data” from the buttons. bluegray large small

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large small bluegray If we substitute numbers for the original objects… Abstraction 6 4 7310 6 4 73 42 31

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A Cross Number Puzzle 5 Don’t always start with the question! 21 8 13 9 12 76 3

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Relating addition and subtraction 6 4 7310 42 31 6 4 73 42 31 Ultimately, building the addition and subtraction algorithms

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The algebra connection: adding 42 31 10 4 6 3 7 4 + 2 = 6 3 + 1 = 4 10 += 7 3

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The algebra connection: subtracting 73 31 6 4 10 2 4 7 + 3 = 10 3 + 1 = 4 6 += 4 2

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The eighth-grade look 5x5x3y3y 2x2x3y3y 11 235x + 3y = 23 2x + 3y = 11 12 += 3x3x0 x = 4 3x3x0 12

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Algebraic language & algebraic thinking Linguistics and mathematics Linguistics and mathematics Algebra as abbreviated speech (Algebra as a Second Language) Algebra as abbreviated speech (Algebra as a Second Language) A number trick “Pattern indicators” Difference of squares Systems of equations in kindergarten? Systems of equations in kindergarten? Understanding two dimensional information Understanding two dimensional information

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Two-dimensional information Think of a number. Double it. Add 6. 5 10 16 DanaCoryWords 4 8 14 Pictures

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Naming intersections, first grade Put a red house at the intersection of A street and N avenue. Where is the green house? How do we go from the green house to the school?

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Combinatorics, beginning of 2nd How many two-letter words can you make, starting with a red letter and ending with a purple letter? How many two-letter words can you make, starting with a red letter and ending with a purple letter? aisnt

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Multiplication, coordinates, phonics? aisnt asas inin atat

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wsil l itin k bp stic k ac k in g brtr

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Similar questions, similar image Four skirts and three shirts: how many outfits? Five flavors of ice cream and four toppings: how many sundaes? (one scoop, one topping) With four different bottom blocks and three different top blocks, how many 2-block Lego towers can you make? Quick bail out!

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Thank you! E. Paul Goldenberg E. Paul Goldenberg http://thinkmath.edc.org/ http://thinkmath.edc.org/ Quick recover

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The idea of a word problem… An attempt at reality An attempt at reality A situation rather than a “naked” calculation The goal is the problem, not the words The goal is the problem, not the words Necessarily bizarre dialect: low redundancy or very wordy Necessarily bizarre dialect: low redundancy or very wordy

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The goal is the problem, not the words Necessarily bizarre dialect: low redundancy or very wordy State ELA tests test ELA State ELA tests test ELA State Math tests test Math State Math tests test Math The idea of a word problem… An attempt at reality A situation rather than a “naked” calculation “Clothing the naked” with words makes it linguistically hard without improving the mathematics. In tests it is discriminatory! and ELA

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Attempts to be efficient (spare) Stereotyped wordingkey words Stereotyped wording key words Stereotyped structure Stereotyped structure autopilot strategies

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Key words Ben and his sister were eating pretzels. Ben left 7 of his pretzels. His sister left 4 of hers. How many pretzels were left? We rail against key word strategies. So writers do cartwheels to subvert them. But, frankly, it is smart to look for clues! This is how language works!

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Autopilot strategies We make fun of thought-free “strategies.” Writers create bizarre wordings with irrelevant numbers, just to confuse kids. Many numbers: + Two numbers close together: – or Two numbers, one large, one small: ÷

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But, if the goal is mathematics and to teach children to think and communicate clearly… …deliberately perverting our wording to make it unclear is not a good model! So what can we do to help students learn to read and interpret story-based problems correctly?

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“Headline Stories” Ben and his sister were eating pretzels. Ben left 7 of his pretzels. His sister left 4 of hers. Less is more! What questions can we ask? Children learn the anatomy of problems by creating them. (Neonatal problem posing!)

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“Headline Stories” Do it yourself! Use any word problem you like. Do it yourself! Use any word problem you like. What can I do? What can I figure out? What can I do? What can I figure out?

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Representing 22 17 22 17

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Representing the algorithm 20 10 2 7

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Representing the algorithm 20 10 2 7 200 140 20 14

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Representing the algorithm 20 10 2 7 200 140 20 14 220 154 374 34 340

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Representing the algorithm 20 10 2 7 200 140 20 14 220 154 374 34 340 22 17 154 220 374 x 1

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Representing the algorithm 20 10 2 7 200 140 20 14 220 154 374 34 340 17 22 34 340 374 x 1

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More generally, (d+2) (r+7) = d r 2 7 dr 7d7d 2r2r 14 2r + dr 7d + 14 2r + 14 dr + 7d

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More generally, (d+2) (r+7) = d r 2 7 dr 7d7d 2r2r 14 dr + 2r + 7d + 14 150 37 25 600 35 925 x 140

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