1. How the ideas and language of algebra K-5 set the stage for algebra 6–12 E. Paul Goldenberg 2008

2. With downloadable PowerPoint Ideas and approaches drawn from Think Math! a comprehensive K-5 program from Houghton Mifflin Harcourt School Publishers http://thinkmath.edc.org Before you scramble to take notes Go to marble bag trick Go to multiplication onions Go to Kindergarten sorting, CNPs Go to 3rd grade detectives Go to intersections Go to “Guess my number” (mental buffer)

3. Algebraic language & algebraic thinking

4. Algebraic thinking Is there anything interesting about addition and subtraction sentences? 2nd grade Math could be spark curiosity!

5. Write two number sentences… To 2nd graders: see if you can find some that don’t work! 4 + 2 = 6 3 + 1 = 4 10 += 7 3 How does this work?

6. Algebraic language Is there anything less sexy than memorizing multiplication facts? What helps people memorize? Something memorable! 4th grade Math could be fascinating! Go to “Mommy, give me…” Go to visual way to understandGo to index

7. Teaching without talking Wow! Will it always work? Big numbers? ? 3839404142 35 36 6789105432111213 80 81 1819202122 … … ? ? 1600 15 16 Go to visual way to understand Shhh… Students thinking!

8. Take it a step further What about two steps out?

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

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

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

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

13. “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

14. But nobody cares if kids can multiply 47  53 mentally!

15. 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 thinking Algebraic language Science

16. (7 – 1)  (7 + 1) = 7  7 – 1 n – 1 n + 1 n (n – 1)  (n + 1) = n  n – 1 (n – 1)  (n + 1)

17. (n – 3) (n – 3)  (n + 3) (7 – 3)  (7 + 3) = 7  7 – 9 n – 3 n + 3 n (n – 3)  (n + 3) = n  n – 9

18. Make a table 24 416 525 Distance awayWhat to subtract 11 39 d d  d

19. (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)

20. 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!

21. One way to look at it 5  5

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

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

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

25. How does it work? 473 50 53 47 3 50  50– 3  3 = 53  47

26. An important propaganda break…

27. “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…. Wrong! We gotta stop believing it’s all in the genes! Wrong! We gotta stop believing it’s all in the genes! We are equally endowed with most of it We are equally endowed with most of it Go to index

28. What could mathematics be like? Surprise! You’re good at algebra! 5th grade It could be surprising! Go to index

29. 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!

30. 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!

31. 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!

32. 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!

33. 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!

34. 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!

35. 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!

36. 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!

37. 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! Go to index

38. Kids need to do it themselves…

39. 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

40. 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

41. 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

42. 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. invent new tricks. invent new tricks. undo the trick. undo the trick. But most important, the idea that But most important, the idea that notation/representation is powerful! notation/representation is powerful!

43. 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

44. 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

45. Representing ideas and 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

46. Representing multiplication, itself

47. 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? Go to index

48. 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

49. Multiplication, coordinates, phonics? aisnt asas inin atat

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

51. 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? Go to Kindergarten sorting, CNPsGo to index

52. Representing 22  17 22 17

53. Representing the algorithm 20 10 2 7

54. Representing the algorithm 20 10 2 7 200 140 20 14

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

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

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

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

59. 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

60. 22 17 374 22  17 = 374

61. 22 17 374 22  17 = 374

62. Representing division (not the algorithm) “Oh! Division is just unmultipli- cation!” “Oh! Division is just unmultipli- cation!” 22 17 374 374 ÷ 17 = 22 22 17 374 Go to index

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

64. 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. Go to Multiplication algorithmGo to number adding sentencesGo to index

65. 6 4 7310 Back to the very beginnings Children can also summarize. “Data” from the buttons. bluegray large small

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

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

68. 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

69. Relating addition and subtraction 6 4 7310 42 31 6 4 73 42 31

70. 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

71. 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 25 + 38 = 6363 – 38 = 25 50 13

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

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

74. The algebra connection: algebra! 5x5x3y3y 2x2x3y3y 11 235x + 3y = 23 2x + 3y = 11 12 += 3x3x0 x = 4 3x3x0 12

75. All from sorting buttons 5x5x3y3y 2x2x3y3y 11 235x + 3y = 23 2x + 3y = 11 12 += 3x3x0 x = 4 3x3x0 12 Go to index

76. Thank you! E. Paul Goldenberg http://thinkmath.edc.org/ To see more of Think Math! visit the Houghton Mifflin Harcourt booth

77. Questions: Linguistics research in math? Building the mental buffer? Counting what we don’t see? E. Paul Goldenberg http://thinkmath.edc.org/ To see more of Think Math! visit the Houghton Mifflin Harcourt booth

78. Keeping things in one’s head 1 2 3 4 8 7 5 6 Go to indexGo to Kindergarten sorting, CNPshttp://thinkmath.edc.org/What’s_My_Number?

79. “Skill practice” in a second grade Video Video VideoVideo Go to index fingers