1. Mathematics (for all) for Tomorrow: How to start a job we can’t finish from and Harcourt School Publishers With support from Vail Symposium, 2006 Some ideas from Think Math!, a new NSF program

2. Curiosity, understanding, and skill No trade-off We can’t afford a trade-off. We can’t afford to think there is one. Fortunately, there doesn’t have to be one.

3. My personal goal… … as a teacher, is to get kids to notice how smart they really are. … as a curriculum developer, is to get teachers to notice how smart they are… …and to notice how smart their kids are!

4. One tiny part of Think Math! Some ways to play algebraically There’s more to the program, of course, including fractions, geometry, statistics, logic and reasoning, word problems… Building algebra: the ideas and the language

5. Just to get you curious: Part I 4 + 2 = 6 3 + 1 = 4 10 += 7 3

6. Just to get you curious: Part II Name a two-digit number…

7. A number trick A number trick, near the end of 4th grade 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!

8. How did it work?

9. 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!

10. 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!

11. 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!

12. 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!

13. 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!

14. 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!

15. 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. Aha! Your answer is 1!

16. Of course, they’ll need to do it themselves!

17. Using the notation: following the 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

18. Using the notation: undoing the steps Think of a number. Double it. Add 6. Divide by 2. What did you get? 5 10 16 8 4 8 14 7320 Hard to undo using the words. Much easier to undo using the notation. Dan a CorySand y ChrisWordsPictures

19. Using the notation: simplifying the steps Think of a number. Double it. Add 6. Divide by 2. What did you get? 5 10 16 8 4 8 14 7320 Dan a CorySand y ChrisWordsPictures

20. What are the kids getting? Computational practice and more. Notation helps them understand the trick. Notation lets them invent new tricks. Notation helps them undo the trick. And a most important idea… Notation/representation is powerful!

21. A bit corny, but… soon becomes

22. A bit corny, but… soon becomes x

23. Kids are already used to it! Focused arithmetic practice with a pattern (function) and a “pattern indicator” Since 1 st grade n 10 n – 8 2 8 0 28 20 1817 34 5857

24. A game, from 3rd grade 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

25. Puzzles, from 3rd grade 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

26. What’s My Number? 1 2 3 4 8 7 5 6

27. 1 2 3 4 8 7 5 6 9 10 11 12 16 15 13 14

28. Combinatorics 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?

29. Early in 2 nd grade

30. Combinatorics or Phonics? aisnt How many two-letter combinations can you make starting with one of these two vowels, and ending with one of these three consonants? asasininatat…

31. Or coordinates? aisnt asas inin atat

32. Or Multiplication in an unfamiliar context? wsil l itin k bp

33. Or Multiplication in an unfamiliar context? wsil l itin k bp stic k ac k in g brtr

34. Speaking of multiplication…

35. …what could be less sexy than memorizing 4th grade multiplication facts?

36. Just the facts Kids already know 4  4, 5  5, 6  6, 7  7, … Have most others and easily work out what they don’t have memorized. Goal now is to consolidate!

37. What helps kids memorize multiplication facts? Something memorable!

38. One way to look at it 5  5

39. One way to look at it 5  4 Removing a column leaves

40. One way to look at it 6  4 Replacing as a row leaves with one left over.

41. One way to look at it 6  4 Removing the leftover leaves showing that it is one less than 5  5.

42. Where does this lead? To do… 53  47

43. Where does this lead? To do……I think… 533 more than 50  47

44. Where does this lead? To do……I think… 533 more than 50  473 less than 50 50  50 (well, 5  5 and …) …2500 Minus 3  3 – 9

45. Where does this lead? To do……I think… 533 more than 50  473 less than 50 50  50 (well, 5  5 and …) …2500 Minus 3  3 – 9 2491

46. Why does it work? 473 50 53

47. aisnt Another view of multiplication: from 2 x 3 = 6 to 22 x 17 = 374

48. Another view of multiplicatio n: 22 x 17 22 17

49. Another view of multiplicatio n: 22 x 17 20 10 2 7

50. Another view of multiplicatio n: 22 x 17 20 10 2 7

51. 20 10 2 7 20 x 10 20 x 7 2 x 10 2 x 7 200 140 20 14

52. 20 10 2 7 200 20 14 140 220 154 374 34 340

53. 20 10 2 7 200 20 14 140 220 154 374 34 340 22 17 154 220 374 x 1

54. 20 10 2 7 200 20 14 140 220 154 374 34 340 17 22 34 340 374 x 1

55. 22 x 17 = 374 22 17 374

56. 22 x 17 = 374 22 17 374

57. 374 ÷ 17 = 22 22 17 374 22 17 374

58. 20 10 2 7 200 20 14 140 220 154 374 34 340 A kindergarten look at…

59. 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.

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

61. large small bluegray 6 4 7310 42 31 If we substitute numbers for the original objects… 10 4 6 3 7 Abstraction

62. Puzzling 5 Don’t always start with the question! 21 8 13 9 12 76 3

63. Building the addition algorithm Only multiples of 10 in yellow. Only less than 10 in blue. 63 38 25 13 50 20 5 8 30

64. Relating addition and subtraction 42 31 10 4 6 3 7 73 31 6 4 2 4

65. …the subtraction algorithm Only multiples of 10 in yellow. Only less than 10 in blue. 63 38 25 13 50 20 5 8 30 25 38 63 -5 30 60 3 8 30 25 + 38 = 6363 – 38 = 25

66. …the subtraction algorithm Only multiples of 10 in yellow. Only less than 10 in blue. 63 38 25 13 50 20 5 8 30 25 38 63 5 20 60 3 8 30 50 13 25 + 38 = 6363 – 38 = 25

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

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

69. The algebra connection: subtracting “number” sentences 5x5x3y3y 2x2x3y3y 11 235x + 3y = 23 2x + 3y = 11 12 += 3x3x0 x = 4 3x3x0 12

70. All from sorting buttons… 5x5x3y3y 2x2x3y3y 23 12 += 3x3x0 x = 4 5x + 3y = 23 2x + 3y = 1111 3x3x0 12

71. Think Math ! www2.edc.org/thinkmath Thanks!

72. www2.edc.org/thinkmath Bye! Thanks! Think Math !

73. Think Math! What is Think Math! ? NSF / EDC / Harcourt comprehensive K–5 6 Not a supplement to other programs; no need to supplement it. Built to conform with NCTM Standards, and to meet and exceed the high-stakes state tests Fully researched in classrooms with children and teachers Easy to teach from; easy to learn from Skills, understanding, problem solving, algebraic focus from K