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# IMF: Visualizing and Montessori Math PART 1

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• ### Transcript of "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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