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What could mathematics be like? Think Math! Using (and building) mathematical curiosity and the spirit of puzzlement to develop algebraic ideas and computation skill Or, Who needs another math program? (especially if there are other good ones to choose among) Some ideas from the newest NSF program, Think Math! from and Harcourt School Publishers ASSM, NCSM, Atlanta, 2007 What could mathematics be like? What helps people memorize? Something memorable! Is there anything less sexy than memorizing multiplication facts? © EDC, Inc., ThinkMath! 2007 Teaching without talking Shhh… Students thinking! 35 36 15 16 2 3 4 5 6 80 81 7 8 9 10 11 12 13 Wow! Will it always work? Big numbers? ? ? ? … 18 19 20 1600 21 22 … 38 39 40 41 42 © EDC, Inc., ThinkMath! 2007 Take it a step further What about two steps out? © EDC, Inc., ThinkMath! 2007 Teaching without talking Shhh… Students thinking! 12 16 2 3 4 60 64 5 6 7 8 9 10 11 12 13 Again?! Always? Find some bigger examples. ? ? ? ? … 28 29 30 31 32 … 58 59 60 61 62 © EDC, Inc., ThinkMath! 2007 Take it even further What about three steps out? What about four? What about five? © EDC, Inc., ThinkMath! 2007 “Mommy! Give me a 2-digit number!” “OK, um, 53” “Hmm, well… 47 2500 about 50 48 49 50 51 52 53 …OK, I’ll pick 47, and I can multiply those numbers faster than you can!” To do… 53 47 I think… 50 50 (well, 5 5 and …)… 2500 Minus 3 3 –9 2491 © EDC, Inc., ThinkMath! 2007 Why bother? Kids feel smart! Teachers feel smart! Practice. It matters! Gives practice. Helps me memorize, because it’s memorable! Something new. Foreshadows algebra. In fact, kids record it with algebraic language! And something to wonder about: How does it work? © EDC, Inc., ThinkMath! 2007 One way to look at it 55 © EDC, Inc., ThinkMath! 2007 One way to look at it Removing a column leaves 54 © EDC, Inc., ThinkMath! 2007 One way to look at it Replacing as a row leaves 64 with one left over. © EDC, Inc., ThinkMath! 2007 One way to look at it Removing the leftover leaves 64 showing that it is one less than 5 5. © EDC, Inc., ThinkMath! 2007 How does it work? 47 50 53 3 47 3 50 50 – 3 3 = 53 47 © EDC, Inc., ThinkMath! 2007 An important propaganda break… © EDC, Inc., ThinkMath! 2007 “Math talent” is made, not found We all “know” that some people have… musical ears, mathematical minds, a natural aptitude for languages…. We gotta stop believing it’s all in the genes! And we are equally endowed with much of it © EDC, Inc., ThinkMath! 2007 A number trick Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number you first thought of. Your answer is 1! © EDC, Inc., ThinkMath! 2007 How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number you first thought of. Your answer is 1! © EDC, Inc., ThinkMath! 2007 How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number you first thought of. Your answer is 1! © EDC, Inc., ThinkMath! 2007 How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number you first thought of. Your answer is 1! © EDC, Inc., ThinkMath! 2007 How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number you first thought of. Your answer is 1! © EDC, Inc., ThinkMath! 2007 How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number you first thought of. Your answer is 1! © EDC, Inc., ThinkMath! 2007 How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number you first thought of. Your answer is 1! © EDC, Inc., ThinkMath! 2007 How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number you first thought of. Your answer is 1! © EDC, Inc., ThinkMath! 2007 How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number you first thought of. Your answer is 1! © EDC, Inc., ThinkMath! 2007 Kids need to do it themselves… © EDC, Inc., ThinkMath! 2007 Using notation: following steps Words Think of a number. Double it. Add 6. Divide by 2. What did you get? Pictures Dan Cory Sand Chris y 5a 10 16 8 7 3 20 © EDC, Inc., ThinkMath! 2007 Using notation: undoing steps Words Think of a number. Double it. Add 6. Divide by 2. What did you get? Pictures Dan Cory Sand Chris 5a 4 y 10 8 16 14 8 7 3 20 Hard to undo using the words. Much easier to undo using the notation. © EDC, Inc., ThinkMath! 2007 Using notation: simplifying steps Words Think of a number. Double it. Add 6. Divide by 2. What did you get? Pictures Dan Cory Sand Chris 5a 4 y 10 16 8 7 3 20 © EDC, Inc., ThinkMath! 2007 Why a number trick? Why bags? Computational practice, but much more Notation helps them understand the trick. Notation helps them invent new tricks. Notation helps them undo the trick. But most important, the idea that notation/representation is powerful! © EDC, Inc., ThinkMath! 2007 Children are language learners… They are pattern-finders, abstracters… …natural sponges for language in context. n 10 8 28 18 17 n–8 2 0 20 58 57 3 4 © EDC, Inc., ThinkMath! 2007 Representing processes Bags and letters can represent numbers. We need also to represent… ideas — multiplication processes — the multiplication algorithm © EDC, Inc., ThinkMath! 2007 Representing multiplication, itself © EDC, Inc., ThinkMath! 2007 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? © EDC, Inc., ThinkMath! 2007 Combinatorics, beginning of 2nd a i s n t How many two-letter words can you make, starting with a red letter and ending with a purple letter? © EDC, Inc., ThinkMath! 2007 Multiplication, coordinates, phonics? a i s n t in as at © EDC, Inc., ThinkMath! 2007 Multiplication, coordinates, phonics? b pw s il it in l k br tr st ic ac in k k g © EDC, Inc., ThinkMath! 2007 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) How many 2-block towers can you make from four differently-colored Lego blocks? © EDC, Inc., ThinkMath! 2007 Representing 22 17 22 17 © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 10 7 2 © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 2 10 200 20 7 140 14 © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 2 10 200 20 220 7 140 14 154 340 34 374 © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 2 1 10 200 20 220 7 140 14 154 340 34 374 22 x17 154 220 374 © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 2 1 10 200 20 220 7 140 14 154 340 34 374 17 x22 34 340 374 © EDC, Inc., ThinkMath! 2007 22 17 374 22 17 = 374 © EDC, Inc., ThinkMath! 2007 22 17 374 22 17 = 374 © EDC, Inc., ThinkMath! 2007 Representing division (not the algorithm) 22 17 374 22 17 374 374 ÷ 17 = 22 © EDC, Inc., ThinkMath! 2007 A game in grade 3 Who Am I? © EDC, Inc., ThinkMath! 2007 I. I am even II. All of my digits < 5 3rd grade detectives! III. h + t + u = 9 IV. I am less than 400 V. Exactly two of my digits are the same. I. I am even. II. All of my digits < 5 III. h+t+u=9 IV. I am less than 400. V. Exactly two of my digits are the same. ht u h t u 1 4 4 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 432 342 234 324 144 414 © EDC, Inc., ThinkMath! 2007 Is it all puzzles and tricks? No. (And that’s too bad, by the way!) Curiosity. How to start what we can’t finish. We’ve evolved fancy brains. © EDC, Inc., ThinkMath! 2007 Learning by doing, for teachers Professional development of 1.6M teachers To take advantage of time they already have, a curriculum must be… Easy to start Appealing to adult minds Comforting Solid math, solid pedagogy (well, as easy as it can ge) (obviously to kids, too!) (covering the bases, the tests) (brain science, Montessori, Singapore, language) © EDC, Inc., ThinkMath! 2007 “Skill practice” in a second grade Video V i d e o © EDC, Inc., ThinkMath! 2007 Keeping things in one’s head 8 6 2 5 7 1 3 4 © EDC, Inc., ThinkMath! 2007 Thank you! E. Paul Goldenberg http://www.edc.org/thinkmath