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- 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. 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. Verbal Counting Model From a childs perspective © Joan A. Cotter, Ph.D., 2012
- 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. Verbal Counting Model From a childs perspective F +E © Joan A. Cotter, Ph.D., 2012
- 6. Verbal Counting Model From a childs perspective F +EA © Joan A. Cotter, Ph.D., 2012
- 7. Verbal Counting Model From a childs perspective F +EA B © Joan A. Cotter, Ph.D., 2012
- 8. Verbal Counting Model From a childs perspective F +EA B C © Joan A. Cotter, Ph.D., 2012
- 9. Verbal Counting Model From a childs perspective F +EA B C D E F © Joan A. Cotter, Ph.D., 2012
- 10. Verbal Counting Model From a childs perspective F +EA B C D E F A © Joan A. Cotter, Ph.D., 2012
- 11. Verbal Counting Model From a childs perspective F +EA B C D E F A B © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. Verbal Counting Model From a childs perspective Now memorize the facts!! G +D © Joan A. Cotter, Ph.D., 2012
- 16. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. Verbal Counting Model From a childs perspective H –ESubtract with your fingers by counting backward. © Joan A. Cotter, Ph.D., 2012
- 21. Verbal Counting Model From a childs perspective J –FSubtract without using your fingers. © Joan A. Cotter, Ph.D., 2012
- 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. 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. Verbal Counting Model From a childs perspectiveLis written ABbecause it is A Jand B A’s © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. 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. Calendar Math © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. 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. 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. Calendar Math Partial Calendar August1 2 3 4 5 6 78 9 10 © Joan A. Cotter, Ph.D., 2012
- 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. 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. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
- 41. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
- 42. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
- 43. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
- 44. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
- 45. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
- 46. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
- 47. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
- 48. Research on Counting Other research © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. Research on Counting In Japanese schools:• Children are discouraged from usingcounting for adding. © Joan A. Cotter, Ph.D., 2012
- 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. Subitizing Quantities(Identifying without counting) © Joan A. Cotter, Ph.D., 2012
- 56. Subitizing Quantities (Identifying without counting)• Five-month-old infants can subitize to 3. © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. 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. 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. 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. 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. Visualizing Quantities © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. Visualizing Quantities Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
- 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. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
- 72. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
- 73. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
- 74. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
- 75. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
- 76. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
- 77. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
- 78. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
- 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. Visualizing Quantities : Who could read the music? © Joan A. Cotter, Ph.D., 2012
- 81. Grouping in Fives © Joan A. Cotter, Ph.D., 2012
- 82. Grouping in Fives• Grouping in fives extends subitizing. © Joan A. Cotter, Ph.D., 2012
- 83. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
- 84. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
- 85. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
- 86. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
- 87. Grouping in Fives Using fingersGrouping in Fives is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
- 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. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
- 90. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
- 91. Grouping in Fives Recognizing 55 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
- 92. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
- 93. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
- 94. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
- 95. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
- 96. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
- 97. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. 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. Grouping in Fives Number Rods © Joan A. Cotter, Ph.D., 2012
- 104. Grouping in Fives Number Rods © Joan A. Cotter, Ph.D., 2012
- 105. Grouping in Fives Number Rods © Joan A. Cotter, Ph.D., 2012
- 106. Grouping in Fives Spindle Box © Joan A. Cotter, Ph.D., 2012
- 107. Grouping in Fives Spindle Box © Joan A. Cotter, Ph.D., 2012
- 108. Grouping in Fives Spindle Box0 1 2 3 4 © Joan A. Cotter, Ph.D., 2012
- 109. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
- 110. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
- 111. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
- 112. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
- 113. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
- 114. Grouping in Fives Spindle Box5 6 7 8 9 © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. 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. 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. Grouping in Fives Entering quantities © Joan A. Cotter, Ph.D., 2012
- 122. Grouping in Fives Entering quantities3 © Joan A. Cotter, Ph.D., 2012
- 123. Grouping in Fives Entering quantities5 © Joan A. Cotter, Ph.D., 2012
- 124. Grouping in Fives Entering quantities7 © Joan A. Cotter, Ph.D., 2012
- 125. Grouping in Fives Entering quantities10 © Joan A. Cotter, Ph.D., 2012
- 126. Grouping in Fives The stairs © Joan A. Cotter, Ph.D., 2012
- 127. Grouping in Fives Adding © Joan A. Cotter, Ph.D., 2012
- 128. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
- 129. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
- 130. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
- 131. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
- 132. Grouping in Fives Adding 4+3=7 © Joan A. Cotter, Ph.D., 2012
- 133. Math Card Games © Joan A. Cotter, Ph.D., 2012
- 134. Math Card Games• Provide repetition for learning the facts. © Joan A. Cotter, Ph.D., 2012
- 135. Math Card Games• Provide repetition for learning the facts.• Encourage autonomy. © Joan A. Cotter, Ph.D., 2012
- 136. Math Card Games• Provide repetition for learning the facts.• Encourage autonomy.• Promote social interaction. © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. “Math” Way of Naming Numbers © Joan A. Cotter, Ph.D., 2012
- 141. “Math” Way of Naming Numbers 11 = ten 1 © Joan A. Cotter, Ph.D., 2012
- 142. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 © Joan A. Cotter, Ph.D., 2012
- 143. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 © Joan A. Cotter, Ph.D., 2012
- 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. “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. “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. “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. “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. “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. “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. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 © Joan A. Cotter, Ph.D., 2012
- 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. “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. “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. “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. “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. “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. 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. 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. 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. 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. Math Way of Naming Numbers Compared to reading: © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
- 168. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
- 169. Math Way of Naming Numbers Traditional names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
- 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. Math Way of Naming Numbers Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
- 172. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
- 173. Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
- 174. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
- 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. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
- 177. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
- 178. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
- 179. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
- 180. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
- 181. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
- 182. Math Way of Naming Numbers Traditional names two leftTwo saidas “twoo.” © Joan A. Cotter, Ph.D., 2012
- 183. Math Way of Naming Numbers Traditional names two left twelveTwo saidas “twoo.” © Joan A. Cotter, Ph.D., 2012
- 184. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
- 185. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
- 186. Composing Numbers3-ten3030 © Joan A. Cotter, Ph.D., 2012
- 187. Composing Numbers3-ten3030 © Joan A. Cotter, Ph.D., 2012
- 188. Composing Numbers3-ten3030 © Joan A. Cotter, Ph.D., 2012
- 189. Composing Numbers3-ten 73030 © Joan A. Cotter, Ph.D., 2012
- 190. Composing Numbers3-ten 73030 © Joan A. Cotter, Ph.D., 2012
- 191. Composing Numbers3-ten 73030 7 7 © Joan A. Cotter, Ph.D., 2012
- 192. Composing Numbers3-ten 73037 0 7 © Joan A. Cotter, Ph.D., 2012
- 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. Composing Numbers1-ten1010 Another example. © Joan A. Cotter, Ph.D., 2012
- 195. Composing Numbers1-ten 81010 © Joan A. Cotter, Ph.D., 2012
- 196. Composing Numbers1-ten 81010 © Joan A. Cotter, Ph.D., 2012
- 197. Composing Numbers1-ten 81010 8 8 © Joan A. Cotter, Ph.D., 2012
- 198. Composing Numbers1-ten 81818 © Joan A. Cotter, Ph.D., 2012
- 199. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
- 200. Composing Numbers10-ten100100 © Joan A. Cotter, Ph.D., 2012
- 201. Composing Numbers10-ten100100 © Joan A. Cotter, Ph.D., 2012
- 202. Composing Numbers10-ten100100 © Joan A. Cotter, Ph.D., 2012
- 203. Composing Numbers1 hundred © Joan A. Cotter, Ph.D., 2012
- 204. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
- 205. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
- 206. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
- 207. Composing Numbers1 hundred100100 © Joan A. Cotter, Ph.D., 2012
- 208. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
- 209. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
- 210. Composing Numbers2 hundred200200 © Joan A. Cotter, Ph.D., 2012
- 211. Evens and Odds © Joan A. Cotter, Ph.D., 2012
- 212. Evens and Odds Evens © Joan A. Cotter, Ph.D., 2012
- 213. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 214. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 215. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 216. Evens and Odds Evens Use two fingers and touch each pair in succession. EVEN! © Joan A. Cotter, Ph.D., 2012
- 217. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 218. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 219. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 220. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 221. Evens and Odds Odds Use two fingers and touch each pair in succession. ODD! © Joan A. Cotter, Ph.D., 2012
- 222. Learning the Facts © Joan A. Cotter, Ph.D., 2012
- 223. Learning the FactsLimited success when:• Based on counting. Whether dots, fingers, number lines, or counting words. © Joan A. Cotter, Ph.D., 2012
- 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. 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. Fact Strategies © Joan A. Cotter, Ph.D., 2012
- 227. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
- 228. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
- 229. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
- 230. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
- 231. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
- 232. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
- 233. Fact Strategies Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
- 234. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
- 235. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
- 236. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
- 237. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
- 238. Fact Strategies Two Fives8+6=10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
- 239. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
- 240. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
- 241. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
- 242. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
- 243. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
- 244. Fact Strategies Going Down 15 – 9 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
- 245. Fact Strategies Subtract from 1015 – 9 = © Joan A. Cotter, Ph.D., 2012
- 246. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
- 247. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
- 248. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
- 249. Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
- 250. Fact Strategies Going Up15 – 9 = © Joan A. Cotter, Ph.D., 2012
- 251. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 252. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 253. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 254. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 255. Fact Strategies Going Up 15 – 9 = 1+5=6Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. 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. Rows and Columns Game 1 9 6 4 3 3 6 3 8 8 © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. Rows and Columns Game 1 6 4 3 3 1 5 1 3 8 8 © Joan A. Cotter, Ph.D., 2012
- 266. Rows and Columns Game © Joan A. Cotter, Ph.D., 2012
- 267. MoneyPenny © Joan A. Cotter, Ph.D., 2012
- 268. MoneyNickel © Joan A. Cotter, Ph.D., 2012
- 269. Money Dime © Joan A. Cotter, Ph.D., 2012
- 270. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
- 271. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
- 272. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
- 273. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
- 274. Place Value Two aspects © Joan A. Cotter, Ph.D., 2012
- 275. Place Value Two aspectsStatic © Joan A. Cotter, Ph.D., 2012
- 276. Place Value Two aspectsStatic • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. 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. 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. Exchanging1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 284. Exchanging Thousands1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 285. Exchanging Hundreds1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 286. Exchanging Tens1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 287. Exchanging Ones1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 288. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
- 289. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
- 290. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
- 291. Exchanging Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
- 292. Exchanging Adding1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
- 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. Exchanging Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
- 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. Exchanging Adding1000 100 10 1 8 +6 14 Same answer before and after exchanging. © Joan A. Cotter, Ph.D., 2012
- 297. Bead Frame 1 101001000 © Joan A. Cotter, Ph.D., 2012
- 298. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
- 299. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
- 300. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
- 301. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
- 302. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
- 303. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
- 304. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
- 305. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
- 306. Bead Frame 1 8 10 +61001000 © Joan A. Cotter, Ph.D., 2012
- 307. Bead Frame 1 8 10 +6100 141000 © Joan A. Cotter, Ph.D., 2012
- 308. 1 Bead Frame 10 100 1000Difficulties for the child © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. 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. 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. Exchanging Adding 4-digit numbers1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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