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How the ideas and language of algebra K-5 set the stage for Algebra 8-12

This presentation explores how the ideas and language of algebra developed in K-5 mathematics can set the stage for algebraic thinking in later grades. The speaker highlights the role of algebraic notation as a language to describe processes, computations, patterns, and existing knowledge. The presentation also examines the connection between algebraic language and computational tools, as well as the developmental nature of algebraic ideas in relation to arithmetic. The importance of linguistic approaches to teaching algebraic concepts is emphasized, utilizing strategies such as algebra as a second language, number tricks, pattern indicators, and understanding two-dimensional information.

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How the ideas and language of algebra K-5 set the stage for Algebra 8-12

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  1. How the ideas and language of algebra K-5 set the stage for Algebra 8-12 MSRI, May 15, 2008 E. Paul Goldenberg To save note-taking, http://thinkmath.edc.orgClick download presentations link(next week)

  2. We can also use such notation as language (not manipulated) • to describe a process or computation or pattern, or • to express what we already know, e.g., (n – d)(n + d) = n2 – d2 Language vs. computational tool • To us, expressions like (n – d)(n + d) can be manipulated • to derive things we don’t yet know, or • to prove things that we conjectured from experiment. Claim: While most elementary school children cannot use algebraic notation the first two ways, as a computational tool,most can use it the last two ways, as language.

  3. Great built-in apparatus • Abstraction (categories, words, pictures) • Syntax, structure, sensitivity to order • Phenomenal language-learning ability • Quantification (limited, but there) • Logic (evolving, but there) • Theory-making about the world irrelevance of orientationIn learning math, little differentiation

  4. Some algebraic ideas precede arithmetic Developmental • w/o rearrangeability 3 + 5 = 8 can’t make sense • Nourishment to extend/apply/refine built-ins • breaking numbers and rearranging parts (any-order-any-grouping, commutativity/associativity), • breaking arrays; describing whole & parts (linearity, distributive property) • But many of the basic intuitions are built in, developmental, not “learned” in math class.

  5. Algebraic language, like any language, is Convention • Children are phenomenal language-learners • Build it from language spoken around them • Infer meaning and structure from use: not explicit definitions and lessons, but from language used in context • Where “math is spoken at home” (not drill, lessons, but conversation that makes salient logical puzzle, quantity, etc.) kids learn it

  6. Demand “does it work with kids?”

  7. Algebraic language & algebraic thinking • Linguistics and mathematics • Algebra as abbreviated speech (Algebra as a Second Language) • A number trick • “Pattern indicators” • Difference of squares • Systems of equations in kindergarten? • Understanding two dimensional information

  8. Algebraic ideas A linguistic idea (mostly) Arithmetic knowledge Linguistics and mathematics Michelle’s strategy for 24 – 8: • Well, 24 –4 is easy! • Now, 20 minus another 4… • Well, I know 10– 4 is 6, and 20 is 10 + 10,so, 20– 4 is 16. • So, 24 – 8 = 16. (breaking it up)

  9. What is the “linguistic” idea? 28 – 8 on her fingers… Fingers are counters,good for grasping the idea, and good (initially) for finding or verifying answers to problems like 28 –4, but…

  10. Algebraic language & algebraic thinking • Linguistics and mathematics • Algebra as abbreviated speech (Algebra as a Second Language) • A number trick • “Pattern indicators” • Difference of squares • Systems of equations in kindergarten? • Understanding two dimensional information

  11. Algebra as abbreviated speech (Algebra as a second Language) • A number trick • “Pattern indicators” • Difference of squares Surprise! You speak algebra! 5th grade

  12. 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!

  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. • Your answer is 1!

  16. 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!

  17. 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!

  18. 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!

  19. 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!

  20. 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!

  21. Kids need to do it themselves…

  22. Using notation: following steps Words Pictures Dana Cory Sandy Chris 5 Think of a number. 10 Double it. 16 Add 6. Divide by 2.What did you get? 8 7 3 20

  23. Pictures Using notation: undoing steps Words Dana Cory Sandy Chris 5 Think of a number. 10 Double it. 14 16 Add 6. Divide by 2.What did you get? 8 7 3 20 Hard to undo using the words. Much easier to undo using the notation.

  24. Using notation: simplifying steps Words Pictures Dana Cory Sandy Chris 4 5 Think of a number. 10 Double it. 16 Add 6. Divide by 2.What did you get? 8 7 3 20

  25. Abbreviated speech: simplifying pictures Words Pictures Dana Cory Sandy Chris b 4 5 Think of a number. 2b 10 Double it. 2b+6 16 Add 6. Divide by 2.What did you get? b+3 8 7 3 20

  26. Notation is powerful! • Computational practice, but much more • Notation helps them understand the trick. • Notation helps them invent new tricks. • Notation helps them undo the trick. • Algebra is a favor, not just “another thing to learn.”

  27. Algebra as abbreviated speech (Algebra as a second Language) • A number trick • “Pattern indicators” • Difference of squares

  28. Children are language learners… • They are pattern-finders, abstracters… • …natural sponges for language in context. n 10 8 28 18 17 58 57 n – 8 2 0 20 3 4 Go to index

  29. Algebra as abbreviated speech (Algebra as a second Language) • A number trick • “Pattern indicators” • Difference of squares Math could be fascinating! • Is there anything less sexy than memorizing multiplication facts? • What helps people memorize? Something memorable! • 4th grade

  30. 2 3 4 5 6 7 8 9 10 11 12 13 38 18 39 19 20 40 41 21 42 22 Teaching without talking Shhh… Students thinking! Wow! Will it always work? Big numbers? 35 80 15 36 81 16 ? ? 1600 ? … …

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

  32. 16 2 3 4 5 6 7 8 58 28 9 59 29 10 30 60 11 12 31 61 13 32 62 Teaching without talking Shhh… Students thinking! Again?! Always? Find some bigger examples. 12 60 64 ? ? ? ? … …

  33. 100 4 5 6 7 8 9 10 11 12 13 14 15 Take it even further What about three steps out? What about four? What about five? 75

  34. 29 30 31 32 33 34 35 36 37 38 39 40 Take it even further What about three steps out? What about four? What about five? 1200 1225

  35. 29 30 31 32 33 34 35 36 37 38 39 40 Take it even further What about two steps out? 1221 1225

  36. 47 48 49 50 51 52 53 “Mommy!Give me a 2-digit number!” about 50 2500 • “OK, um, 53” • “Hmm, well… • …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

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

  38. Algebraic/arithmeticthinking Science Algebraic language What do we care about, then? • 50  50 (well, 5  5 and place value) • Keeping 2500 in mind while thinking 3  3 • Subtracting 2500 – 9 • Finding the pattern • Describing the pattern

  39. n n – 3 n + 3 Q? (7 – 3)  (7 + 3) = 7  7–9 (50 – 3)  (50 + 3) = 50  50–9 (n– 3)  (n+ 3) = nn– 9 (n– 3)  (n+ 3) Nicolina Malara, Italy: “algebraic babble”

  40. Distance away What to subtract 1 1 2 4 3 9 4 16 5 25 ddd Make a table; use pattern indicator.

  41. n n – d n + d (7 –d)  (7 +d) = 7  7–dd (n– d)  (n+ d) = nn– (n– d)  (n+ d) = nn–dd (n– d)  (n+ d) (n– d)

  42. We also care about thinking! • Kids feel smart!Why silent teaching? • Teachers feel smart! • Practice.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? It matters!

  43. One way to look at it 5  5

  44. One way to look at it Removing a column leaves 5  4 Not “concrete vs. abstract”semantic (spatial) vs. syntacticKids don’t derive/prove with algebra.

  45. One way to look at it Replacing as a row leaves 6  4 with one left over. Not “concrete vs. abstract”semantic (spatial) vs. syntacticKids don’t derive/prove with algebra.

  46. One way to look at it Removing the leftover leaves 6  4 showing that it is one less than 5 5. Not “concrete vs. abstract”semantic (spatial) vs. syntacticKids don’t derive/prove with algebra.

  47. Algebraic language & algebraic thinking • Linguistics and mathematics • Algebra as abbreviated speech (Algebra as a Second Language) • A number trick • “Pattern indicators” • Difference of squares • Systems of equations in kindergarten? • Understanding two dimensional information

  48. 4 + 2 = 6 5x + 3y = 23 3 + 1 = 4 2x + 3y = 11 Systems of equations in Kindergarten?! Challenge: can you find some that don’t work? + = 3 10 7 Math could be spark curiosity! • Is there anything interesting about addition and subtraction sentences? • Start with 2nd grade

  49. 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.

  50. Back to the very beginnings Children can also summarize. “Data” from the buttons. blue gray 6 small 4 large 7 3 10

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