1. 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) from and Harcourt School Publishers ASSM, NCSM, Atlanta, 2007 Some ideas from the newest NSF program, Think Math!

2. What could mathematics be like? What helps people memorize? Something memorable! Is there anything less sexy than memorizing multiplication facts?

3. © EDC, Inc., ThinkMath! 2007 Teaching without talking Shhh… Students thinking! Wow! Will it always work? Big numbers? ? 3839404142 35 36 6789105432111213 80 81 1819202122 … … ? ? 1600 15 16

4. © EDC, Inc., ThinkMath! 2007 Take it a step further What about two steps out?

5. © EDC, Inc., ThinkMath! 2007 Shhh… Students thinking! Again?! Always? Find some bigger examples. Teaching without talking 12 16 6789105432111213 60 64 ? 58596061622829303132 … … ? ? ?

6. © EDC, Inc., ThinkMath! 2007 Take it even further What about three steps out? What about four? What about five?

7. © EDC, Inc., ThinkMath! 2007 “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

8. © EDC, Inc., ThinkMath! 2007 Why bother? Kids feel smart! Kids feel smart! 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!

9. © EDC, Inc., ThinkMath! 2007 One way to look at it 5  5

10. © EDC, Inc., ThinkMath! 2007 One way to look at it 5  4 Removing a column leaves

11. © EDC, Inc., ThinkMath! 2007 One way to look at it 6  4 Replacing as a row leaves with one left over.

12. © EDC, Inc., ThinkMath! 2007 One way to look at it 6  4 Removing the leftover leaves showing that it is one less than 5  5.

13. © EDC, Inc., ThinkMath! 2007 How does it work? 473 50 53 47 3 50  50– 3  3 = 53  47

14. © EDC, Inc., ThinkMath! 2007 An important propaganda break…

15. © EDC, Inc., ThinkMath! 2007 “Math talent” is made, not found We all “know” that some people have… 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! We gotta stop believing it’s all in the genes! And we are equally endowed with much of it And we are equally endowed with much of it

16. © EDC, Inc., ThinkMath! 2007 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!

17. © EDC, Inc., ThinkMath! 2007 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!

18. © EDC, Inc., ThinkMath! 2007 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!

19. © EDC, Inc., ThinkMath! 2007 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!

20. © EDC, Inc., ThinkMath! 2007 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!

21. © EDC, Inc., ThinkMath! 2007 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!

22. © EDC, Inc., ThinkMath! 2007 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!

23. © EDC, Inc., ThinkMath! 2007 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!

24. © EDC, Inc., ThinkMath! 2007 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!

25. © EDC, Inc., ThinkMath! 2007 Kids need to do it themselves…

26. © EDC, Inc., ThinkMath! 2007 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

27. © EDC, Inc., ThinkMath! 2007 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 4 8 14 Hard to undo using the words. Much easier to undo using the notation. Pictures

28. © EDC, Inc., ThinkMath! 2007 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

29. © EDC, Inc., ThinkMath! 2007 Why a number trick? Why bags? 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. But most important, the idea that But most important, the idea that notation/representation is powerful! notation/representation is powerful!

30. © EDC, Inc., ThinkMath! 2007 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

31. © EDC, Inc., ThinkMath! 2007 Representing processes Bags and letters can represent numbers. Bags and letters can represent numbers. We need also to represent… We need also to represent…  ideas — multiplication  processes — the multiplication algorithm

32. © EDC, Inc., ThinkMath! 2007 Representing multiplication, itself

33. © 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?

34. © EDC, Inc., ThinkMath! 2007 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

35. © EDC, Inc., ThinkMath! 2007 Multiplication, coordinates, phonics? aisnt asas inin atat

36. © EDC, Inc., ThinkMath! 2007 Multiplication, coordinates, phonics? wsil l itin k bp stic k ac k in g brtr

37. © 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?

38. © EDC, Inc., ThinkMath! 2007 Representing 22  17 22 17

39. © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 10 2 7

40. © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 10 2 7 200 140 20 14

41. © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 10 2 7 200 140 20 14 220 154 374 34 340

42. © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 10 2 7 200 140 20 14 220 154 374 34 340 22 17 154 220 374 x 1

43. © EDC, Inc., ThinkMath! 2007 Representing the algorithm 20 10 2 7 200 140 20 14 220 154 374 34 340 17 22 34 340 374 x 1

44. © EDC, Inc., ThinkMath! 2007 22 17 374 22  17 = 374

45. © EDC, Inc., ThinkMath! 2007 22 17 374 22  17 = 374

46. © EDC, Inc., ThinkMath! 2007 Representing division (not the algorithm) 22 17 374 374 ÷ 17 = 22 22 17 374

47. © EDC, Inc., ThinkMath! 2007 hundreds digit > 6 tens digit is 7, 8, or 9 the number is a multiple of 5 the tens digit is greater than the hundreds digit ones digit < 5 the number is even tens digit < ones digit the ones digit is twice the tens digit the number is divisible by 3 A game in grade 3

48. © EDC, Inc., ThinkMath! 2007 3rd grade detectives! I. I am even. htu 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 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. 432 342 234 324 144 414 1 4 4

49. © EDC, Inc., ThinkMath! 2007 Is it all puzzles and tricks? No. (And that’s too bad, by the way!) No. (And that’s too bad, by the way!) Curiosity. How to start what we can’t finish. Curiosity. How to start what we can’t finish. We’ve evolved fancy brains. We’ve evolved fancy brains.

50. © EDC, Inc., ThinkMath! 2007 Learning by doing, for teachers Professional development of 1.6M teachers Professional development of 1.6M teachers To take advantage of time they already have, a curriculum must be… To take advantage of time they already have, a curriculum must be…  Easy to start (well, as easy as it can ge)  Appealing to adult minds (obviously to kids, too!)  Comforting (covering the bases, the tests)  Solid math, solid pedagogy (brain science, Montessori, Singapore, language)

51. © EDC, Inc., ThinkMath! 2007 “Skill practice” in a second grade Video Video VideoVideo

52. © EDC, Inc., ThinkMath! 2007 Keeping things in one’s head 1 2 3 4 8 7 5 6

53. © EDC, Inc., ThinkMath! 2007 Thank you! E. Paul Goldenberg E. Paul Goldenberg http://www.edc.org/thinkmath http://www.edc.org/thinkmath