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IMF: Visualizing and Montessori Math PART 1

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IMF: Visualizing and Montessori Math PART 1

  1. 1. How Visualization Enhances Montessori Mathematics PART 1 by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com Montessori Foundation 30 30 Conference 77 Friday, Nov 2, 2012 Sarasota, Florida 30 370 7 1000 100 10 1 7 7 7 3 3 3 PowerPoint PresentationRightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012
  2. 2. Counting ModelIn Montessori, counting is pervasive: • Number Rods • Spindle Boxes • Decimal materials • Snake Game • Dot Game • Stamp Game • Multiplication Board • Bead Frame © Joan A. Cotter, Ph.D., 2012
  3. 3. Verbal Counting Model From a childs perspective © Joan A. Cotter, Ph.D., 2012
  4. 4. Verbal Counting Model From a childs perspectiveBecause we’re so familiar with 1, 2, 3, we’ll use letters. A=1 B=2 C=3 D=4 E = 5, and so forth © Joan A. Cotter, Ph.D., 2012
  5. 5. Verbal Counting Model From a childs perspective F +E © Joan A. Cotter, Ph.D., 2012
  6. 6. Verbal Counting Model From a childs perspective F +EA © Joan A. Cotter, Ph.D., 2012
  7. 7. Verbal Counting Model From a childs perspective F +EA B © Joan A. Cotter, Ph.D., 2012
  8. 8. Verbal Counting Model From a childs perspective F +EA B C © Joan A. Cotter, Ph.D., 2012
  9. 9. Verbal Counting Model From a childs perspective F +EA B C D E F © Joan A. Cotter, Ph.D., 2012
  10. 10. Verbal Counting Model From a childs perspective F +EA B C D E F A © Joan A. Cotter, Ph.D., 2012
  11. 11. Verbal Counting Model From a childs perspective F +EA B C D E F A B © Joan A. Cotter, Ph.D., 2012
  12. 12. Verbal Counting Model From a childs perspective F +EA B C D E F A B C D E © Joan A. Cotter, Ph.D., 2012
  13. 13. Verbal Counting Model From a childs perspective F +EA B C D E F A B C D E What is the sum? (It must be a letter.) © Joan A. Cotter, Ph.D., 2012
  14. 14. Verbal Counting Model From a childs perspective F +E KA B C D E F G H I J K © Joan A. Cotter, Ph.D., 2012
  15. 15. Verbal Counting Model From a childs perspective Now memorize the facts!! G +D © Joan A. Cotter, Ph.D., 2012
  16. 16. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D © Joan A. Cotter, Ph.D., 2012
  17. 17. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D +C © Joan A. Cotter, Ph.D., 2012
  18. 18. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D C +C +G © Joan A. Cotter, Ph.D., 2012
  19. 19. Verbal Counting Model From a childs perspective Now memorize the facts!! H +E G F I+ +D D C +C +G © Joan A. Cotter, Ph.D., 2012
  20. 20. Verbal Counting Model From a childs perspective H –ESubtract with your fingers by counting backward. © Joan A. Cotter, Ph.D., 2012
  21. 21. Verbal Counting Model From a childs perspective J –FSubtract without using your fingers. © Joan A. Cotter, Ph.D., 2012
  22. 22. Verbal Counting Model From a childs perspectiveTry skip counting by B’s to T: B, D, . . . T. © Joan A. Cotter, Ph.D., 2012
  23. 23. Verbal Counting Model From a childs perspectiveTry skip counting by B’s to T: B, D, . . . T.What is D × E? © Joan A. Cotter, Ph.D., 2012
  24. 24. Verbal Counting Model From a childs perspectiveLis written ABbecause it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
  25. 25. Verbal Counting Model From a childs perspectiveLis written ABbecause it is A Jand B A’s huh? © Joan A. Cotter, Ph.D., 2012
  26. 26. Verbal Counting Model From a childs perspectiveL (twelve)is written ABbecause it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
  27. 27. Verbal Counting Model From a childs perspectiveL (twelve)is written AB (12)because it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
  28. 28. Verbal Counting Model From a childs perspectiveL (twelve)is written AB (12)because it is A J (one 10)and B A’s © Joan A. Cotter, Ph.D., 2012
  29. 29. Verbal Counting Model From a childs perspectiveL (twelve)is written AB (12)because it is A J (one 10)and B A’s (two 1s). © Joan A. Cotter, Ph.D., 2012
  30. 30. Calendar Math © Joan A. Cotter, Ph.D., 2012
  31. 31. Calendar Math August1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
  32. 32. Calendar Math Calendar Counting August1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
  33. 33. Calendar Math Calendar Counting August1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
  34. 34. Calendar Math Calendar Counting August1 2 3 4 5 6 78 9 10 11 12 13 1415 16 17 18 19 20 2122 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
  35. 35. Calendar Math Septemb Calendar Counting1234567 August891012141 2 113 11921151126288 122820 67527 9 3 4 5 6 10 11 12 13 14 72234 2015 16 17 18 19 20 2129 322 23 24 25 26 27 2829 30 31 © Joan A. Cotter, Ph.D., 2012
  36. 36. Calendar Math Septemb Calendar Counting 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31This is ordinal counting, not cardinal counting. © Joan A. Cotter, Ph.D., 2012
  37. 37. Calendar Math Partial Calendar August1 2 3 4 5 6 78 9 10 © Joan A. Cotter, Ph.D., 2012
  38. 38. Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 10Children need the whole month to plan ahead. © Joan A. Cotter, Ph.D., 2012
  39. 39. Calendar Math Septemb Calendar patterning 1234567 August 89101214 1 2 113 11921 15112628 8 122820 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31Patterns are rarely based on 7s or proceed row by row.Patterns go on forever; they don’t stop at 31. © Joan A. Cotter, Ph.D., 2012
  40. 40. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  41. 41. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  42. 42. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  43. 43. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  44. 44. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  45. 45. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  46. 46. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  47. 47. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  48. 48. Research on Counting Other research © Joan A. Cotter, Ph.D., 2012
  49. 49. Research on Counting Other research• Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. © Joan A. Cotter, Ph.D., 2012
  50. 50. Research on Counting Other research• Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.• Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. © Joan A. Cotter, Ph.D., 2012
  51. 51. Research on Counting Other research• Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.• Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008.• Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. © Joan A. Cotter, Ph.D., 2012
  52. 52. Research on Counting Other research• Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.• Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008.• Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008.• Baby chicks from Italy. Lucia Regolin, University of Padova, 2009. © Joan A. Cotter, Ph.D., 2012
  53. 53. Research on Counting In Japanese schools:• Children are discouraged from usingcounting for adding. © Joan A. Cotter, Ph.D., 2012
  54. 54. Research on Counting In Japanese schools:• Children are discouraged from usingcounting for adding.• They consistently group in 5s. © Joan A. Cotter, Ph.D., 2012
  55. 55. Subitizing Quantities(Identifying without counting) © Joan A. Cotter, Ph.D., 2012
  56. 56. Subitizing Quantities (Identifying without counting)• Five-month-old infants can subitize to 3. © Joan A. Cotter, Ph.D., 2012
  57. 57. Subitizing Quantities (Identifying without counting)• Five-month-old infants can subitize to 3.• Three-year-olds can subitize to 5. © Joan A. Cotter, Ph.D., 2012
  58. 58. Subitizing Quantities (Identifying without counting)• Five-month-old infants can subitize to 3.• Three-year-olds can subitize to 5.• Four-year-olds can subitize 6 to 10 byusing five as a subbase. © Joan A. Cotter, Ph.D., 2012
  59. 59. Subitizing Quantities (Identifying without counting)• Five-month-old infants can subitize to 3.• Three-year-olds can subitize to 5.• Four-year-olds can subitize 6 to 10 byusing five as a subbase.• Counting is like sounding out each letter;subitizing is recognizing the quantity. © Joan A. Cotter, Ph.D., 2012
  60. 60. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit © Joan A. Cotter, Ph.D., 2012
  61. 61. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld © Joan A. Cotter, Ph.D., 2012
  62. 62. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld• Children who can subitize perform better inmathematics long term.—Butterworth © Joan A. Cotter, Ph.D., 2012
  63. 63. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld• Children who can subitize perform better inmathematics long term.—Butterworth• Counting-on is a difficult skill for many children.—Journal for Res. in Math Ed. Nov. 2011 © Joan A. Cotter, Ph.D., 2012
  64. 64. Research on Counting Subitizing• Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld• Children who can subitize perform better inmathematics long term.—Butterworth• Counting-on is a difficult skill for many children.—Journal for Res. in Math Ed. Nov. 2011• Math anxiety affects counting ability, butnot subitizing ability. © Joan A. Cotter, Ph.D., 2012
  65. 65. Visualizing Quantities © Joan A. Cotter, Ph.D., 2012
  66. 66. Visualizing Quantities“Think in pictures, because thebrain remembers images betterthan it does anything else.” Ben Pridmore, World Memory Champion, 2009 © Joan A. Cotter, Ph.D., 2012
  67. 67. Visualizing Quantities“The role of physical manipulativeswas to help the child form thosevisual images and thus to eliminatethe need for the physicalmanipulatives.” Ginsberg and others © Joan A. Cotter, Ph.D., 2012
  68. 68. Visualizing Quantities Japanese criteria for manipulatives• Representative of structure of numbers.• Easily manipulated by children.• Imaginable mentally. Japanese Council of Mathematics Education © Joan A. Cotter, Ph.D., 2012
  69. 69. Visualizing Quantities Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
  70. 70. Visualizing Quantities Visualizing also needed in:• Reading • Architecture• Sports • Astronomy• Creativity • Archeology• Geography • Chemistry• Engineering • Physics• Construction • Surgery © Joan A. Cotter, Ph.D., 2012
  71. 71. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
  72. 72. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
  73. 73. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
  74. 74. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
  75. 75. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  76. 76. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  77. 77. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  78. 78. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  79. 79. Visualizing Quantities Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIII © Joan A. Cotter, Ph.D., 2012
  80. 80. Visualizing Quantities : Who could read the music? © Joan A. Cotter, Ph.D., 2012
  81. 81. Grouping in Fives © Joan A. Cotter, Ph.D., 2012
  82. 82. Grouping in Fives• Grouping in fives extends subitizing. © Joan A. Cotter, Ph.D., 2012
  83. 83. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
  84. 84. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
  85. 85. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
  86. 86. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
  87. 87. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
  88. 88. Grouping in Fives Yellow is the Sun Yellow is the sun. Six is five and one. Why is the sky so blue? Seven is five and two. Salty is the sea. Eight is five and three. Hear the thunder roar. Nine is five and four. Ducks will swim and dive. Ten is five and five. –Joan A. Cotter © Joan A. Cotter, Ph.D., 2012
  89. 89. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  90. 90. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  91. 91. Grouping in Fives Recognizing 55 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
  92. 92. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
  93. 93. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
  94. 94. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
  95. 95. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
  96. 96. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
  97. 97. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
  98. 98. Grouping in Fives Pairing Finger Cards QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and aa TIFFQuickTime™and aa QuickTime™ and QuickTime™ and TIFF(LZW) decompressor areTIFF (LZW) decompressor TIFF (LZW) decompressor are needed toto seethisa picture. needed(LZW)seedecompressor see this (LZW) and QuickTime™ are needed toseedecompressorpicture. are neededto seethis picture. TIFF to are needed this picture. picture. this © Joan A. Cotter, Ph.D., 2012
  99. 99. Grouping in Fives Ordering Finger Cards QuickTime™ and a TIFF (LZW) decompressorare needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor QuickTime™ and a are needed to see this picture. TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressorare needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. © Joan A. Cotter, Ph.D., 2012
  100. 100. Grouping in Fives Matching Number Cards to Finger Cards QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. 5 1 QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. 10 © Joan A. Cotter, Ph.D., 2012
  101. 101. Grouping in FivesMatching Finger Cards to Number Cards9 1 10 4 6 QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.2 3 7 8 5 QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture. QuickTime™ and aa QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor a QuickTime™ and TIFF (LZW) decompressor QuickTime™ and are needed (LZW)this picture. a TIFF (LZW)decompressor QuickTime™ and are neededtotosee this picture. TIFF tosee decompressor are needed (LZW)decompressor TIFF (LZW)this picture. are needed tosee this picture. TIFF see decompressor are needed to see this picture. are needed to see this picture. © Joan A. Cotter, Ph.D., 2012
  102. 102. Grouping in Fives Finger Card Memory game QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressorare needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. © Joan A. Cotter, Ph.D., 2012
  103. 103. Grouping in Fives Number Rods © Joan A. Cotter, Ph.D., 2012
  104. 104. Grouping in Fives Number Rods © Joan A. Cotter, Ph.D., 2012
  105. 105. Grouping in Fives Number Rods © Joan A. Cotter, Ph.D., 2012
  106. 106. Grouping in Fives Spindle Box © Joan A. Cotter, Ph.D., 2012
  107. 107. Grouping in Fives Spindle Box © Joan A. Cotter, Ph.D., 2012
  108. 108. Grouping in Fives Spindle Box0 1 2 3 4 © Joan A. Cotter, Ph.D., 2012
  109. 109. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
  110. 110. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
  111. 111. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
  112. 112. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
  113. 113. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
  114. 114. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
  115. 115. Grouping in Fives 1000 100 10 1 1000 100 10 1 1000 100 10 1 1000 100 10 1 100 10 1 100 10 1 100 10 1 100 1Stamp Game © Joan A. Cotter, Ph.D., 2012
  116. 116. Grouping in Fives 1000 100 10 1 1000 100 10 1 1000 100 10 1 1000 100 10 1 100 10 1 100 10 1 100 10 1 100 1Stamp Game © Joan A. Cotter, Ph.D., 2012
  117. 117. Grouping in Fives 1000 1000 100 100 10 10 1 1 1000 1000 100 100 10 10 1 1 100 100 10 10 1 1 100 100 10 10 1 1 100 100 10 10 1 1 100 100 10 100 100 100 100Stamp Game © Joan A. Cotter, Ph.D., 2012
  118. 118. Grouping in Fives 1000 1000 100 100 10 1 1 1000 1000 100 100 1 1 10 10 100 100 1 1 10 10 100 100 1 1 10 10 100 100 1 1 10 10 100 100 10 10 100 100 100 100Stamp Game © Joan A. Cotter, Ph.D., 2012
  119. 119. Grouping in Fives 1000 1000 100 100 10 1 1 1000 1000 100 100 1 1 10 10 100 100 1 1 10 10 1 1 100 100 10 10 1 1 100 100 10 10 100 100 10 10 100 100Stamp Game 100 100 © Joan A. Cotter, Ph.D., 2012
  120. 120. Grouping in Fives Black and White Bead Stairs“Grouped in fives so the child does notneed to count.” A. M. Joosten © Joan A. Cotter, Ph.D., 2012
  121. 121. Grouping in Fives Entering quantities © Joan A. Cotter, Ph.D., 2012
  122. 122. Grouping in Fives Entering quantities3 © Joan A. Cotter, Ph.D., 2012
  123. 123. Grouping in Fives Entering quantities5 © Joan A. Cotter, Ph.D., 2012
  124. 124. Grouping in Fives Entering quantities7 © Joan A. Cotter, Ph.D., 2012
  125. 125. Grouping in Fives Entering quantities10 © Joan A. Cotter, Ph.D., 2012
  126. 126. Grouping in Fives The stairs © Joan A. Cotter, Ph.D., 2012
  127. 127. Grouping in Fives Adding © Joan A. Cotter, Ph.D., 2012
  128. 128. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  129. 129. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  130. 130. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  131. 131. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  132. 132. Grouping in Fives Adding 4+3=7 © Joan A. Cotter, Ph.D., 2012
  133. 133. Math Card Games © Joan A. Cotter, Ph.D., 2012
  134. 134. Math Card Games• Provide repetition for learning the facts. © Joan A. Cotter, Ph.D., 2012
  135. 135. Math Card Games• Provide repetition for learning the facts.• Encourage autonomy. © Joan A. Cotter, Ph.D., 2012
  136. 136. Math Card Games• Provide repetition for learning the facts.• Encourage autonomy.• Promote social interaction. © Joan A. Cotter, Ph.D., 2012
  137. 137. Math Card Games• Provide repetition for learning the facts.• Encourage autonomy.• Promote social interaction.• Are enjoyed by the children. © Joan A. Cotter, Ph.D., 2012
  138. 138. Go to the Dump GameObjective: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5 © Joan A. Cotter, Ph.D., 2012
  139. 139. Go to the Dump GameObjective: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5Object of the game: To collect the most pairs that equal ten. © Joan A. Cotter, Ph.D., 2012
  140. 140. “Math” Way of Naming Numbers © Joan A. Cotter, Ph.D., 2012
  141. 141. “Math” Way of Naming Numbers 11 = ten 1 © Joan A. Cotter, Ph.D., 2012
  142. 142. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 © Joan A. Cotter, Ph.D., 2012
  143. 143. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 © Joan A. Cotter, Ph.D., 2012
  144. 144. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 © Joan A. Cotter, Ph.D., 2012
  145. 145. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9 © Joan A. Cotter, Ph.D., 2012
  146. 146. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9 © Joan A. Cotter, Ph.D., 2012
  147. 147. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 14 = ten 4 .... 19 = ten 9 © Joan A. Cotter, Ph.D., 2012
  148. 148. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 .... 19 = ten 9 © Joan A. Cotter, Ph.D., 2012
  149. 149. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... 19 = ten 9 © Joan A. Cotter, Ph.D., 2012
  150. 150. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... .... 19 = ten 9 .... 99 = 9-ten 9 © Joan A. Cotter, Ph.D., 2012
  151. 151. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 © Joan A. Cotter, Ph.D., 2012
  152. 152. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7 © Joan A. Cotter, Ph.D., 2012
  153. 153. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. © Joan A. Cotter, Ph.D., 2012
  154. 154. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. © Joan A. Cotter, Ph.D., 2012
  155. 155. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. © Joan A. Cotter, Ph.D., 2012
  156. 156. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. © Joan A. Cotter, Ph.D., 2012
  157. 157. “Math” Way of Naming Numbers 100 Chinese U.S. Average Highest Number Counted 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. © Joan A. Cotter, Ph.D., 2012
  158. 158. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) © Joan A. Cotter, Ph.D., 2012
  159. 159. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. © Joan A. Cotter, Ph.D., 2012
  160. 160. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. © Joan A. Cotter, Ph.D., 2012
  161. 161. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. • Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense. © Joan A. Cotter, Ph.D., 2012
  162. 162. Math Way of Naming Numbers Compared to reading: © Joan A. Cotter, Ph.D., 2012
  163. 163. Math Way of Naming Numbers Compared to reading:• Just as reciting the alphabet doesn’t teach reading,counting doesn’t teach arithmetic. © Joan A. Cotter, Ph.D., 2012
  164. 164. Math Way of Naming Numbers Compared to reading:• Just as reciting the alphabet doesn’t teach reading,counting doesn’t teach arithmetic.• Just as we first teach the sound of the letters, we mustfirst teach the name of the quantity (math way). © Joan A. Cotter, Ph.D., 2012
  165. 165. Math Way of Naming Numbers Compared to reading:• Just as reciting the alphabet doesn’t teach reading,counting doesn’t teach arithmetic.• Just as we first teach the sound of the letters, we mustfirst teach the name of the quantity (math way).• Montessorians need to use the math way of namingnumbers for a longer period of time. © Joan A. Cotter, Ph.D., 2012
  166. 166. Math Way of Naming Numbers“Rather, the increased gap between Chinese andU.S. students and that of Chinese Americans andCaucasian Americans may be due primarily to thenature of their initial gap prior to formal schooling,such as counting efficiency and base-ten numbersense.” Jian Wang and Emily Lin, 2005 Researchers © Joan A. Cotter, Ph.D., 2012
  167. 167. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  168. 168. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  169. 169. Math Way of Naming Numbers Traditional names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  170. 170. Math Way of Naming Numbers Traditional names3-ten = thirty“Thir” alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
  171. 171. Math Way of Naming Numbers Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
  172. 172. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  173. 173. Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
  174. 174. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
  175. 175. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news box-mail mailbox © Joan A. Cotter, Ph.D., 2012
  176. 176. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  177. 177. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  178. 178. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  179. 179. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
  180. 180. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
  181. 181. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
  182. 182. Math Way of Naming Numbers Traditional names two leftTwo saidas “twoo.” © Joan A. Cotter, Ph.D., 2012
  183. 183. Math Way of Naming Numbers Traditional names two left twelveTwo saidas “twoo.” © Joan A. Cotter, Ph.D., 2012
  184. 184. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  185. 185. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  186. 186. Composing Numbers3-ten3030 © Joan A. Cotter, Ph.D., 2012
  187. 187. Composing Numbers3-ten3030 © Joan A. Cotter, Ph.D., 2012
  188. 188. Composing Numbers3-ten3030 © Joan A. Cotter, Ph.D., 2012
  189. 189. Composing Numbers3-ten 73030 © Joan A. Cotter, Ph.D., 2012
  190. 190. Composing Numbers3-ten 73030 © Joan A. Cotter, Ph.D., 2012
  191. 191. Composing Numbers3-ten 73030 7 7 © Joan A. Cotter, Ph.D., 2012
  192. 192. Composing Numbers3-ten 73037 0 7 © Joan A. Cotter, Ph.D., 2012
  193. 193. Composing Numbers 3-ten 7 30 37 0 7Note the congruence in how we say the number,represent the number, and write the number. © Joan A. Cotter, Ph.D., 2012
  194. 194. Composing Numbers1-ten1010 Another example. © Joan A. Cotter, Ph.D., 2012
  195. 195. Composing Numbers1-ten 81010 © Joan A. Cotter, Ph.D., 2012
  196. 196. Composing Numbers1-ten 81010 © Joan A. Cotter, Ph.D., 2012
  197. 197. Composing Numbers1-ten 81010 8 8 © Joan A. Cotter, Ph.D., 2012
  198. 198. Composing Numbers1-ten 81818 © Joan A. Cotter, Ph.D., 2012
  199. 199. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
  200. 200. Composing Numbers10-ten100100 © Joan A. Cotter, Ph.D., 2012
  201. 201. Composing Numbers10-ten100100 © Joan A. Cotter, Ph.D., 2012
  202. 202. Composing Numbers10-ten100100 © Joan A. Cotter, Ph.D., 2012
  203. 203. Composing Numbers1 hundred © Joan A. Cotter, Ph.D., 2012
  204. 204. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
  205. 205. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
  206. 206. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
  207. 207. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
  208. 208. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
  209. 209. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
  210. 210. Composing Numbers2 hundred200200 © Joan A. Cotter, Ph.D., 2012
  211. 211. Evens and Odds © Joan A. Cotter, Ph.D., 2012
  212. 212. Evens and Odds Evens © Joan A. Cotter, Ph.D., 2012
  213. 213. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  214. 214. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  215. 215. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  216. 216. Evens and Odds Evens Use two fingers and touch each pair in succession. EVEN! © Joan A. Cotter, Ph.D., 2012
  217. 217. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  218. 218. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  219. 219. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  220. 220. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  221. 221. Evens and Odds Odds Use two fingers and touch each pair in succession. ODD! © Joan A. Cotter, Ph.D., 2012
  222. 222. Learning the Facts © Joan A. Cotter, Ph.D., 2012
  223. 223. Learning the FactsLimited success when:• Based on counting. Whether dots, fingers, number lines, or counting words. © Joan A. Cotter, Ph.D., 2012
  224. 224. Learning the FactsLimited success when:• Based on counting. Whether dots, fingers, number lines, or counting words.• Based on rote memory. Whether by flash cards or timed tests. © Joan A. Cotter, Ph.D., 2012
  225. 225. Learning the FactsLimited success when:• Based on counting. Whether dots, fingers, number lines, or counting words.• Based on rote memory. Whether by flash cards or timed tests.• Based on skip counting for multiplication facts. © Joan A. Cotter, Ph.D., 2012
  226. 226. Fact Strategies © Joan A. Cotter, Ph.D., 2012
  227. 227. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  228. 228. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  229. 229. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  230. 230. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  231. 231. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  232. 232. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  233. 233. Fact Strategies Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  234. 234. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  235. 235. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  236. 236. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  237. 237. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  238. 238. Fact Strategies Two Fives8+6=10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
  239. 239. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
  240. 240. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
  241. 241. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  242. 242. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  243. 243. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  244. 244. Fact Strategies Going Down 15 – 9 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  245. 245. Fact Strategies Subtract from 1015 – 9 = © Joan A. Cotter, Ph.D., 2012
  246. 246. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  247. 247. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  248. 248. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  249. 249. Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  250. 250. Fact Strategies Going Up15 – 9 = © Joan A. Cotter, Ph.D., 2012
  251. 251. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
  252. 252. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
  253. 253. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
  254. 254. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
  255. 255. Fact Strategies Going Up 15 – 9 = 1+5=6Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
  256. 256. Rows and Columns GameObjective: To find a total of 15 by adding 2, 3, or 4cards in a row or in a column. © Joan A. Cotter, Ph.D., 2012
  257. 257. Rows and Columns GameObjective: To find a total of 15 by adding 2, 3, or 4cards in a row or in a column.Object of the game: To collect the most cards. © Joan A. Cotter, Ph.D., 2012
  258. 258. Rows and Columns Game 8 7 1 9 6 4 3 3 2 2 5 6 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
  259. 259. Rows and Columns Game 8 7 1 9 6 4 3 3 2 2 5 6 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
  260. 260. Rows and Columns Game 8 7 1 9 6 4 3 3 2 2 5 6 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
  261. 261. Rows and Columns Game 1 9 6 4 3 3 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
  262. 262. Rows and Columns Game 7 6 1 9 6 4 3 3 2 1 5 1 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
  263. 263. Rows and Columns Game 7 6 1 9 6 4 3 3 2 1 5 1 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
  264. 264. Rows and Columns Game 7 6 1 9 6 4 3 3 2 1 5 1 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
  265. 265. Rows and Columns Game 1 6 4 3 3 1 5 1 3 8 8 © Joan A. Cotter, Ph.D., 2012
  266. 266. Rows and Columns Game © Joan A. Cotter, Ph.D., 2012
  267. 267. MoneyPenny © Joan A. Cotter, Ph.D., 2012
  268. 268. MoneyNickel © Joan A. Cotter, Ph.D., 2012
  269. 269. Money Dime © Joan A. Cotter, Ph.D., 2012
  270. 270. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  271. 271. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  272. 272. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  273. 273. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  274. 274. Place Value Two aspects © Joan A. Cotter, Ph.D., 2012
  275. 275. Place Value Two aspectsStatic © Joan A. Cotter, Ph.D., 2012
  276. 276. Place Value Two aspectsStatic • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
  277. 277. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. © Joan A. Cotter, Ph.D., 2012
  278. 278. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. © Joan A. Cotter, Ph.D., 2012
  279. 279. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position.(Shown by the Decimal Cards.) © Joan A. Cotter, Ph.D., 2012
  280. 280. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position.(Shown by the Decimal Cards.)Dynamic © Joan A. Cotter, Ph.D., 2012
  281. 281. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position.(Shown by the Decimal Cards.)Dynamic • 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand, …. © Joan A. Cotter, Ph.D., 2012
  282. 282. Place Value Two aspectsStatic • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position.(Shown by the Decimal Cards.)Dynamic • 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand, ….(Represented on the Abacus and other materials.) © Joan A. Cotter, Ph.D., 2012
  283. 283. Exchanging1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  284. 284. Exchanging Thousands1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  285. 285. Exchanging Hundreds1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  286. 286. Exchanging Tens1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  287. 287. Exchanging Ones1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  288. 288. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  289. 289. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  290. 290. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  291. 291. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  292. 292. Exchanging Adding1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
  293. 293. Exchanging Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  294. 294. Exchanging Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  295. 295. Exchanging Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  296. 296. Exchanging Adding1000 100 10 1 8 +6 14 Same answer before and after exchanging. © Joan A. Cotter, Ph.D., 2012
  297. 297. Bead Frame 1 101001000 © Joan A. Cotter, Ph.D., 2012
  298. 298. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
  299. 299. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
  300. 300. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
  301. 301. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
  302. 302. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
  303. 303. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
  304. 304. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
  305. 305. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
  306. 306. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
  307. 307. Bead Frame 1 8 10 +6100 141000 © Joan A. Cotter, Ph.D., 2012
  308. 308. 1 Bead Frame 10 100 1000Difficulties for the child © Joan A. Cotter, Ph.D., 2012
  309. 309. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives. © Joan A. Cotter, Ph.D., 2012
  310. 310. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left. © Joan A. Cotter, Ph.D., 2012
  311. 311. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left.• Hierarchies of numbers represented sideways:They need to be in vertical columns. © Joan A. Cotter, Ph.D., 2012
  312. 312. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left.• Hierarchies of numbers represented sideways:They need to be in vertical columns.• Exchanging done before second number iscompletely added: Addends need to be combined beforeexchanging. © Joan A. Cotter, Ph.D., 2012
  313. 313. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left.• Hierarchies of numbers represented sideways:They need to be in vertical columns.• Exchanging done before second number iscompletely added: Addends need to be combined beforeexchanging.• Answer is read going up: We read top to bottom. © Joan A. Cotter, Ph.D., 2012
  314. 314. 1 Bead Frame 10 100 1000 Difficulties for the child• Not visualizable: Beads need to be grouped in fives.• When beads are moved right, inconsistent withequation order: Beads need to be moved left.• Hierarchies of numbers represented sideways:They need to be in vertical columns.• Exchanging before second number is completelyadded: Addends need to be combined before exchanging.• Answer is read going up: We read top to bottom.• Distracting: Room is visible through the frame. © Joan A. Cotter, Ph.D., 2012
  315. 315. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
  316. 316. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  317. 317. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  318. 318. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  319. 319. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  320. 320. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  321. 321. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  322. 322. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  323. 323. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  324. 324. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  325. 325. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  326. 326. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  327. 327. Exchanging Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  328. 328. Exchanging Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  329. 329. Exchanging Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012

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