1. Algebra and thinking: Kaput Center, February 9, 2011 — E. Paul Goldenberg Education Development Center © 2011 E. Paul Goldenberg This presentation was created in PowerPoint 2008; some parts may not play perfectly on earlier versions. Fonts may align differently on different machines. what babies and little kids tell us about algebra

2. Brain science’s influence on standards curriculum assessment teaching

3. Brain science’s influence on standards curriculum assessment teaching

4. Babies know things at birth! How can we know that? How can we know that? Well, we ask them! Well, we ask them! Same or different? Which do you prefer? Same or different? Which do you prefer? HOW DO THEY ANSWER?! HOW DO THEY ANSWER?!

5. What do they know? Mommy’s voice Mommy’s voice Facial expressions Facial expressions Which expression fits a voice Which expression fits a voice How to match a face How to match a face Foreign languages Foreign languages Perfect pitch Perfect pitch Math… (to be continued) Math… (to be continued)

6. And they think about things! And they remember! And they remember! And they communicate. And they communicate. And they perform experiments! And they perform experiments!

7. The miracle of communication By about 1, some words. By about 1, some words. By about 5 or 6, a full half of our adult vocabulary! And virtually all of our grammar! By about 5 or 6, a full half of our adult vocabulary! And virtually all of our grammar! We start out not speaking at all.

8. Math and musical ears We all have perfectly working equipment. We all have perfectly working equipment. Only the training is different! Only the training is different! Equally adept at learning our language. Equally adept at learning our language. No reason to believe we aren’t equally adept at learning mathematics. No reason to believe we aren’t equally adept at learning mathematics. Our use of pitch.

9. Babies already know some math! They recognize symmetry They recognize symmetry They know quantity They know quantity What about probability? What about probability? We all have the “math gene”! We all have the “math gene”!

10. What’s this have to do w/ algebra? Knowing what children already know, how they learn, and how much they can learn lets us be appropriately ambitious! Knowing what children already know, how they learn, and how much they can learn lets us be appropriately ambitious! We can design to take advantage of the way children think and learn… (distributed learning, use of language) We can design to take advantage of the way children think and learn… (distributed learning, use of language) …and to enhance, extend, refine what they do naturally. …and to enhance, extend, refine what they do naturally.

11. Enough generalities! Examples!

12. Distributive property… …of multiplication over addition. …of multiplication over addition. But built into cognition before children multiply! But built into cognition before children multiply!

13. A number trick-- a multi-step arithmetic process 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! 4th grade

14. How did it work? Think of a number. Think of a number.

15. How did it work? -- building a symbolic system Think of a number. Think of a number.

16. How did it work? Think of a number. Think of a number. Add 3. Add 3.

17. How did it work? Think of a number. Think of a number. Add 3. Add 3. Double the result. Double the result.

18. 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.

19. 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.

20. 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.

21. 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. Kids need to do it themselves…

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

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

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

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

27. 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! Algebra Is Downright Useful

28. Children are language learners… a story from 2nd grade They are pattern-finders, abstracters… They are pattern-finders, abstracters… …natural sponges for language in context. …natural sponges for language in context. Important algebraic idea: 10 – 8, vs. n – 8 n 10 n – 8 2 8 0 28 20 1817 34 5857 910 11 12 50 49

29. hundreds digit > 7 tens digit is 7, 8, or 9 the number is a multiple of 5 the ones digit is greater than the tens digit ones digit > 4 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 Language in context

30. 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

31. Distilling algebra: three fractions Describing what we know — learnable early Describing what we know — learnable early  Notation is never “obvious” or discoverable — needs “native speaker” and use in context Deducing what we don’t know — comes later Deducing what we don’t know — comes later No need to be taught as a package deal. No need to be taught as a package deal. Properties: Distributive property already there. Properties: Distributive property already there.

32. 4 + 5 = 9 3 + 1 = 4 7 Addition and Subtraction: From buttons to algebra Challenge: can you find some that don’t work? Challenge: can you find some that don’t work? 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! 13 += 6 5 1 4

33. Children classify things Words are a kind of sorting! And babies learn these words from the most chaotic data. Words are a kind of sorting! And babies learn these words from the most chaotic data.

34. 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.

35. 6 4 7310 Back to the very beginnings Children can also summarize. “Data” from the buttons. bluegray large small

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

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

38. Detour: How’d they learn the facts? One hand, plus One hand, plus All about 10 All about 10  Making it (How far from 10?)  Adding, subtracting it  Remembering to use it! Using it Using it  Adjusting with it (+ 8)  Four-hand addition

39. A video from 2 nd grade: mental math

40. The addition 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

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

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

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

44. The eighth-grade look 5x5x3y3y 2x2x3y3y 11 235x + 3y = 23 2x + 3y = 11 12 += 3x3x0 x = 4 3x3x0 12 All from buttons!

45. Mental math vs. visual vs. print Different! Different! The pairs to 10, 20, 30 were mental (probably visual) The pairs to 10, 20, 30 were mental (probably visual) The “negative a million” was visual (partly also linguistic) The “negative a million” was visual (partly also linguistic) “two fifths + three fifths” is linguistic “two fifths + three fifths” is linguistic 2 / 5 + 3 / 5 is print 2 / 5 + 3 / 5 is print

46. Vision… Can babies focus? Yes and no… Especially yes. Can babies focus? Yes and no… Especially yes.

47. Can they focus? Yes and no… Especially yes. Can they focus? Yes and no… Especially yes. Seeing is built in, but they also “learn” about seeing. Seeing is built in, but they also “learn” about seeing. Totally different image on the retina, but baby quickly learns to recognize objects in any position. Totally different image on the retina, but baby quickly learns to recognize objects in any position. Vision…

48. So these are all the same object in different positions: So these are all the same object in different positions: We don’t start out knowing that, but learn it. We don’t start out knowing that, but learn it. Vision…

49. So these are all the same object in different positions: So these are all the same object in different positions: We don’t start out knowing that, but learn it. We don’t start out knowing that, but learn it. Then, in order to read, we must unlearn it! Then, in order to read, we must unlearn it! It’s not a defect that children reverse letters! It’s not a defect that children reverse letters! pqdb Vision and reading… was ≠ saw

50. Reading requires special ways of Orientation matters: d ≠ b Orientation matters: d ≠ b Order matters: was ≠ saw Order matters: was ≠ saw (In English) We read left to right. Dog bites boy ≠ Boy bites dog. (In English) We read left to right. Dog bites boy ≠ Boy bites dog. We need to monitor our seeing! EXECUTIVE FUNCTION We need to monitor our seeing! EXECUTIVE FUNCTION seeing

51. Math-reading uses different visual conventions

52. Think of a number. Double it. Add 6. 5 10 16 DanaCoryWords 4 8 14 Pictures 3 2 ≠ 32 9 + 6 = __ + 5 3 + 4 × 2 1 + g (x + 1)3( 1 / 3 + 1 / 6 )

53. Actually even regular text reading Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae. The rset can be a toatl mses and you can sitll raed it wouthit porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe. isn’t strictly left to right.

54. Conjecture… Because we don’t need to attend to the insides of words in order to be able to read, some of kids’ spelling difficulties arises from not having activities in which they do need to attend to the insides of words. Because we don’t need to attend to the insides of words in order to be able to read, some of kids’ spelling difficulties arises from not having activities in which they do need to attend to the insides of words.

55. Reading different, language same

56. Language and mathematics Michelle’s strategy for 24 – 7: Well, 24 – 4 is easy! Well, 24 – 4 is easy! Now, 20 minus another 3… Now, 20 minus another 3… Well, I know 10 – 3 is 7, and 20 is 10 + 10, so, 20 – 3 is 17. Well, I know 10 – 3 is 7, and 20 is 10 + 10, so, 20 – 3 is 17. So, 24 – 7 = 17. So, 24 – 7 = 17. A linguistic idea (mostly) Algebraic ideas (breaking it up) Arithmetic knowledge

57. What are the “linguistic” ideas? 24 – 7 on her fingers… Fingers are counters, for grasping the idea, and (initially) for finding or verifying answers to problems like 24 – 7… but then let language take over! “Twenty four” as a name

58. Five hundred plus two hundred Our language makes this “obvious” to children. Ask first graders about five ninths plus two ninths. “Five hundred” and “five sevenths” as names

59. “Conservation” of number Commutative and associative properties: What WW Sawyer called the “any order any grouping” property Commutative and associative properties: What WW Sawyer called the “any order any grouping” property Remember, babies can count. And even add! A little.

60. “Conservation” of number A 4-year-old may see “more” here A 4-year-old may see “more” here than here, even after counting correctly. than here, even after counting correctly. While that’s true, 2 + 3 = 5 can’t make sense. While that’s true, 2 + 3 = 5 can’t make sense.

61. They already know that They already know that but they also know that print is weird! but they also know that print is weird! so maybe so maybe They have to unlearn again! But only about print! Not math. They have to unlearn again! But only about print! Not math. When quantity is stable… = was ≠ saw2+5 ≠ 5+2

62. Why puzzles? Preschoolers’ play — the lever toy Preschoolers’ play — the lever toy Because we’re not cats! Because we’re not cats! Puzzles grab attention! Puzzles grab attention! Puzzles give permission to think! Puzzles give permission to think!

63. “Executive function” Monitoring one’s thinking Monitoring one’s thinking Extending one’s short term memory Extending one’s short term memory Picturing and manipulating ideas and objects in one’s head Picturing and manipulating ideas and objects in one’s head Counting what we can’t see! (how many fingers don’t you see, “spilled ink,” four hand addition) Counting what we can’t see! (how many fingers don’t you see, “spilled ink,” four hand addition) What can I do? vs. Whatamispoztado?! What can I do? vs. Whatamispoztado?!

64. Distributive property, half of 48 Monitoring one’s thinking Monitoring one’s thinking Extending one’s short term memory Extending one’s short term memory

65. Associative property, 5 × 42 Monitoring one’s thinking Monitoring one’s thinking Extending one’s short term memory Extending one’s short term memory

66. Go forth and teach!

67. What’s that got to do w/ algebra? Left as an exercise for the reader. Left as an exercise for the reader.

68. Or research!

69. Attention and silence “ When we say preschoolers can’t pay attention, we really mean that They can’t not pay attention.” “ When we say preschoolers can’t pay attention, we really mean that They can’t not pay attention.” If it’s quiet, it can sneak up on me. I’d better watch closely! If it’s quiet, it can sneak up on me. I’d better watch closely!

70. Putting things in one’s head 1 2 3 4 8 7 5 6 2nd grade