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1. 1. Making Early Mathematics Visualizable with the AL Abacus Kathleen Cotter Lawler and Joan A. Cotter, Ph.D. Kathleen@rightstartmath.com, JoanCotter@rightstartmath.com QuickTimeª and a decompressor are needed to see this picture. NDCTM Saturday, March 31, 2012 Jamestown, North Dakota 7 3 7 31 © Joan A. Cotter, Ph.D., 2012
2. 2. Visualizing Enhances Standards K. Count to 100 by ones and by tens.2 © Joan A. Cotter, Ph.D., 2012
3. 3. Visualizing Enhances Standards K. Count to 100 by ones and by tens. K. Count forward beginning from a given number.3 © Joan A. Cotter, Ph.D., 2012
4. 4. Visualizing Enhances Standards K. Count to 100 by ones and by tens. K. Count forward beginning from a given number. K. Count to answer “how many”?4 © Joan A. Cotter, Ph.D., 2012
5. 5. Visualizing Enhances Standards K. Count to 100 by ones and by tens. K. Count forward beginning from a given number. K. Count to answer “how many”? K. Work with numbers 11 – 19 … for place value.5 © Joan A. Cotter, Ph.D., 2012
6. 6. Visualizing Enhances Standards K. Count to 100 by ones and by tens. K. Count forward beginning from a given number. K. Count to answer “how many”? K. Work with numbers 11 – 19 … for place value. 1. Count to 120, starting at any number less than 120.6 © Joan A. Cotter, Ph.D., 2012
7. 7. Visualizing Enhances Standards K. Count to 100 by ones and by tens. K. Count forward beginning from a given number. K. Count to answer “how many”? K. Work with numbers 11 – 19 … for place value. 1. Count to 120, starting at any number less than 120. 1. Place value: understand as special cases…7 © Joan A. Cotter, Ph.D., 2012
8. 8. Visualizing Enhances Standards K. Count to 100 by ones and by tens. K. Count forward beginning from a given number. K. Count to answer “how many”? K. Work with numbers 11 – 19 … for place value. 1. Count to 120, starting at any number less than 120. 1. Place value: understand as special cases… 2. Know from memory all sums of two 1-digit numbers.8 © Joan A. Cotter, Ph.D., 2012
9. 9. Visualizing Enhances Standards K. Count to 100 by ones and by tens. K. Count forward beginning from a given number. K. Count to answer “how many”? K. Work with numbers 11 – 19 … for place value. 1. Count to 120, starting at any number less than 120. 1. Place value: understand as special cases… 2. Know from memory all sums of two 1-digit numbers. 2. Find the number of objects in arrays up to 5 rows and 5 columns.9 © Joan A. Cotter, Ph.D., 2012
10. 10. Verbal Counting Model10 © Joan A. Cotter, Ph.D., 2012
11. 11. Verbal Counting Model From a childs perspective Because we’re so familiar with 1, 2, 3, we’ll use letters. A=1 B=2 C=3 D=4 E = 5, and so forth11 © Joan A. Cotter, Ph.D., 2012
12. 12. Verbal Counting Model From a childs perspective F +E12 © Joan A. Cotter, Ph.D., 2012
13. 13. Verbal Counting Model From a childs perspective F +E A13 © Joan A. Cotter, Ph.D., 2012
14. 14. Verbal Counting Model From a childs perspective F +E A B14 © Joan A. Cotter, Ph.D., 2012
15. 15. Verbal Counting Model From a childs perspective F +E A B C15 © Joan A. Cotter, Ph.D., 2012
16. 16. Verbal Counting Model From a childs perspective F +E A B C D E F16 © Joan A. Cotter, Ph.D., 2012
17. 17. Verbal Counting Model From a childs perspective F +E A B C D E F A17 © Joan A. Cotter, Ph.D., 2012
18. 18. Verbal Counting Model From a childs perspective F +E A B C D E F A B18 © Joan A. Cotter, Ph.D., 2012
19. 19. Verbal Counting Model From a childs perspective F +E A B C D E F A B C D E19 © Joan A. Cotter, Ph.D., 2012
20. 20. Verbal Counting Model From a childs perspective F +E A B C D E F A B C D E What is the sum? (It must be a letter.)20 © Joan A. Cotter, Ph.D., 2012
21. 21. Verbal Counting Model From a childs perspective F +E K A B C D E F G H I J K21 © Joan A. Cotter, Ph.D., 2012
22. 22. Verbal Counting Model From a childs perspective Now memorize the facts!! G +D22 © Joan A. Cotter, Ph.D., 2012
23. 23. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D23 © Joan A. Cotter, Ph.D., 2012
24. 24. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D +C24 © Joan A. Cotter, Ph.D., 2012
25. 25. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D C +C +G25 © Joan A. Cotter, Ph.D., 2012
26. 26. Verbal Counting Model From a childs perspective Now memorize the facts!! H E + G I F + +D D C +C +G26 © Joan A. Cotter, Ph.D., 2012
27. 27. Verbal Counting Model From a childs perspective Try subtracting H by “taking away” –E27 © Joan A. Cotter, Ph.D., 2012
28. 28. Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T.28 © Joan A. Cotter, Ph.D., 2012
29. 29. Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T. What is D × E?29 © Joan A. Cotter, Ph.D., 2012
30. 30. Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s30 © Joan A. Cotter, Ph.D., 2012
31. 31. Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s huh?31 © Joan A. Cotter, Ph.D., 2012
32. 32. Verbal Counting Model From a childs perspective L (twelve) is written AB because it is A J and B A’s32 © Joan A. Cotter, Ph.D., 2012
33. 33. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J and B A’s33 © Joan A. Cotter, Ph.D., 2012
34. 34. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s34 © Joan A. Cotter, Ph.D., 2012
35. 35. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s (two 1s).35 © Joan A. Cotter, Ph.D., 2012
36. 36. Verbal Counting Model Summary36 © Joan A. Cotter, Ph.D., 2012
37. 37. Verbal Counting Model Summary • Is not natural; it takes years of practice.37 © Joan A. Cotter, Ph.D., 2012
38. 38. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity.38 © Joan A. Cotter, Ph.D., 2012
39. 39. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value.39 © Joan A. Cotter, Ph.D., 2012
40. 40. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone.40 © Joan A. Cotter, Ph.D., 2012
41. 41. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. • Is tedious and time-consuming.41 © Joan A. Cotter, Ph.D., 2012
42. 42. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. • Is tedious and time-consuming. • Does not provide an efficient way to master the facts.42 © Joan A. Cotter, Ph.D., 2012
43. 43. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 201243
44. 44. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 201244
45. 45. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3145 © Joan A. Cotter, Ph.D., 2012
46. 46. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.Joan A. Cotter, Ph.D., 2012 ©46
47. 47. Calendar Math Septemb 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 10 11 12 13 14 5 6 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and 3. © Joan A. Cotter, Ph.D., 201247
48. 48. Calendar Math Septemb 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 5 10 11 12 13 14 6 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6A calendar is NOT a ruler. n a ruler the numbers are not in the spaces. © Joan A. Cotter, Ph.D., 201248
49. 49. Calendar Math August 1 2 3 4 5 6 7 8 9 10Always show the whole calendar. A child needs to see the wholebefore the parts. Children also need to learn to plan ahead. © Joan A. Cotter, Ph.D., 201249
50. 50. Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler.50 © Joan A. Cotter, Ph.D., 2012
51. 51. Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. • Purpose of calendar is to plan ahead. • Many ways to show the current date.51 © Joan A. Cotter, Ph.D., 2012
52. 52. Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. • Purpose of calendar is to plan ahead. • Many ways to show the current date. Calendars give a narrow view of patterning. • Patterns do not necessarily involve numbers. • Patterns rarely proceed row by row. • Patterns go on forever; they don’t stop at 31.52 © Joan A. Cotter, Ph.D., 2012
53. 53. Memorizing Math 9 +7 Flash cards:• Are often used to teach rote.• Are liked only by those who don’t need them.• Don’t work for those with learning disabilities.• Give the false impression that math isn’t aboutthinking.• Often produce stress – children under stressstop learning.• Are not concrete – use abstract symbols. © Joan A. Cotter, Ph.D., 2012
54. 54. Learning Arithmetic Compared to reading:• A child learns to read.• Later a child uses reading to learn.• A child learns to do arithmetic.• Later a child uses arithmetic to solve problems. © Joan A. Cotter, Ph.D., 2012
55. 55. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
56. 56. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
57. 57. Research on Counting Karen Wynn’s research57 © Joan A. Cotter, Ph.D., 2012
58. 58. Research on Counting Karen Wynn’s research58 © Joan A. Cotter, Ph.D., 2012
59. 59. Research on Counting Karen Wynn’s research59 © Joan A. Cotter, Ph.D., 2012
60. 60. Research on Counting Karen Wynn’s research60 © Joan A. Cotter, Ph.D., 2012
61. 61. Research on Counting Karen Wynn’s research61 © Joan A. Cotter, Ph.D., 2012
62. 62. Research on Counting Karen Wynn’s research62 © Joan A. Cotter, Ph.D., 2012
63. 63. Research on Counting Other research63 © Joan A. Cotter, Ph.D., 2012
64. 64. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.64 © Joan A. Cotter, Ph.D., 2012
65. 65. 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, 2008.65 © Joan A. Cotter, Ph.D., 2012
66. 66. 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, 2008.66 © Joan A. Cotter, Ph.D., 2012
67. 67. 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, 2009.67 © Joan A. Cotter, Ph.D., 2012
68. 68. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding.68 © Joan A. Cotter, Ph.D., 2012
69. 69. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. • They consistently group in 5s.69 © Joan A. Cotter, Ph.D., 2012
70. 70. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting.70 © Joan A. Cotter, Ph.D., 2012
71. 71. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth.71 © Joan A. Cotter, Ph.D., 2012
72. 72. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth72 © Joan A. Cotter, Ph.D., 2012
73. 73. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit73 © Joan A. Cotter, Ph.D., 2012
74. 74. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit • Subitizing seems to be a necessary skill for understanding what the counting process means.— Glasersfeld74 © Joan A. Cotter, Ph.D., 2012
75. 75. Visualizing Mathematics75 © Joan A. Cotter, Ph.D., 2012
76. 76. Visualizing Mathematics “Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 200976 © Joan A. Cotter, Ph.D., 2012
77. 77. Visualizing Mathematics “The role of physical manipulatives was to help the child form those visual images and thus to eliminate the need for the physical manipulatives.” Ginsberg and others77 © Joan A. Cotter, Ph.D., 2012
78. 78. Visualizing Mathematics 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
79. 79. Visualizing Mathematics Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
80. 80. Visualizing Mathematics Visualizing also needed in:• Reading • Architecture• Sports • Astronomy• Creativity • Archeology• Geography • Chemistry• Engineering • Physics• Construction • Surgery © Joan A. Cotter, Ph.D., 2012
81. 81. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
82. 82. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
83. 83. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
84. 84. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
85. 85. Visualizing MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
86. 86. Visualizing MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
87. 87. Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
88. 88. Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
89. 89. Visualizing Mathematics Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIII © Joan A. Cotter, Ph.D., 2012
90. 90. Visualizing Mathematics : Who could read the music?90 © Joan A. Cotter, Ph.D., 2012
91. 91. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
92. 92. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
93. 93. Naming Quantities Using fingers93 © Joan A. Cotter, Ph.D., 2012
94. 94. Naming Quantities Using fingers94 © Joan A. Cotter, Ph.D., 2012
95. 95. Naming Quantities Using fingers95 © Joan A. Cotter, Ph.D., 2012
96. 96. Naming Quantities Using fingers96 © Joan A. Cotter, Ph.D., 2012
97. 97. Naming Quantities 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
98. 98. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
99. 99. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
100. 100. Naming Quantities Recognizing 55 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
101. 101. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
102. 102. Naming Quantities Tally sticks102 © Joan A. Cotter, Ph.D., 2012
103. 103. Naming Quantities Tally sticks103 © Joan A. Cotter, Ph.D., 2012
104. 104. Naming Quantities Tally sticks104 © Joan A. Cotter, Ph.D., 2012
105. 105. Naming Quantities Tally sticks105 © Joan A. Cotter, Ph.D., 2012
106. 106. Naming Quantities Tally sticks106 © Joan A. Cotter, Ph.D., 2012
107. 107. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?107 © Joan A. Cotter, Ph.D., 2012
108. 108. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?108 © Joan A. Cotter, Ph.D., 2012
109. 109. Naming QuantitiesNumberChart 1 2 3 4 5 © Joan A. Cotter, Ph.D., 2012
110. 110. Naming Quantities Number Chart 1 2To help the 3child learnthe symbols 4 5 © Joan A. Cotter, Ph.D., 2012
111. 111. Naming Quantities Number Chart 1 6 2 7To help the 3 8child learnthe symbols 4 9 5 10 © Joan A. Cotter, Ph.D., 2012
112. 112. AL Abacus Cleared © Joan A. Cotter, Ph.D., 2012
113. 113. AL Abacus Entering quantities3 © Joan A. Cotter, Ph.D., 2012
114. 114. AL Abacus Entering quantities 5114 © Joan A. Cotter, Ph.D., 2012
115. 115. AL Abacus Entering quantities 7115 © Joan A. Cotter, Ph.D., 2012
116. 116. AL Abacus Entering quantities 10116 © Joan A. Cotter, Ph.D., 2012
117. 117. AL Abacus The stairs117 © Joan A. Cotter, Ph.D., 2012
124. 124. “Math” Way of Naming Numbers124 © Joan A. Cotter, Ph.D., 2012
125. 125. “Math” Way of Naming Numbers 11 = ten 1125 © Joan A. Cotter, Ph.D., 2012
126. 126. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2126 © Joan A. Cotter, Ph.D., 2012
127. 127. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3127 © Joan A. Cotter, Ph.D., 2012
128. 128. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4128 © Joan A. Cotter, Ph.D., 2012
129. 129. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9129 © Joan A. Cotter, Ph.D., 2012
130. 130. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9130 © Joan A. Cotter, Ph.D., 2012
131. 131. “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 9131 © Joan A. Cotter, Ph.D., 2012
132. 132. “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 9132 © Joan A. Cotter, Ph.D., 2012
133. 133. “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 9133 © Joan A. Cotter, Ph.D., 2012
134. 134. “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 9134 © Joan A. Cotter, Ph.D., 2012
135. 135. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7135 © Joan A. Cotter, Ph.D., 2012
136. 136. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7136 © Joan A. Cotter, Ph.D., 2012
137. 137. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 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, 319-332.137 © Joan A. Cotter, Ph.D., 2012
138. 138. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 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, 319-332.138 © Joan A. Cotter, Ph.D., 2012
139. 139. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 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, 319-332.139 © Joan A. Cotter, Ph.D., 2012
140. 140. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 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, 319-332.140 © Joan A. Cotter, Ph.D., 2012
141. 141. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 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, 319-332.141 © Joan A. Cotter, Ph.D., 2012
142. 142. 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.)142 © Joan A. Cotter, Ph.D., 2012
143. 143. 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.143 © Joan A. Cotter, Ph.D., 2012
144. 144. 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.144 © Joan A. Cotter, Ph.D., 2012
145. 145. 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.145 © Joan A. Cotter, Ph.D., 2012
146. 146. Math Way of Naming Numbers Compared to reading:146 © Joan A. Cotter, Ph.D., 2012
147. 147. Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic.147 © Joan A. Cotter, Ph.D., 2012
148. 148. 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 must first teach the name of the quantity (math way).148 © Joan A. Cotter, Ph.D., 2012
149. 149. Math Way of Naming Numbers “Rather, the increased gap between Chinese and U.S. students and that of Chinese Americans and Caucasian Americans may be due primarily to the nature of their initial gap prior to formal schooling, such as counting efficiency and base-ten number sense.” Jian Wang and Emily Lin, 2005 Researchers149 © Joan A. Cotter, Ph.D., 2012
150. 150. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
151. 151. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
152. 152. Math Way of Naming Numbers Traditional names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
153. 153. Math Way of Naming Numbers Traditional names3-ten = thirty“Thir” alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
154. 154. Math Way of Naming Numbers Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
155. 155. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
156. 156. Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
157. 157. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
158. 158. 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
159. 159. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
160. 160. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
161. 161. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
162. 162. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
163. 163. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
164. 164. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
165. 165. Math Way of Naming Numbers Traditional names two leftTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
166. 166. Math Way of Naming Numbers Traditional names two left twelveTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
167. 167. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
168. 168. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
169. 169. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
170. 170. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
171. 171. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
172. 172. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
173. 173. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
174. 174. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
175. 175. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
176. 176. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
177. 177. Composing Numbers 3-ten 7 30 7Notice the way we say the number, represent thenumber, and write the number all correspond. © Joan A. Cotter, Ph.D., 2012
178. 178. Composing Numbers7-ten70 Another example. © Joan A. Cotter, Ph.D., 2012
179. 179. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
180. 180. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
181. 181. Composing Numbers7-ten 870 8 © Joan A. Cotter, Ph.D., 2012
182. 182. Composing Numbers7-ten 878 © Joan A. Cotter, Ph.D., 2012
183. 183. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
184. 184. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
185. 185. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
186. 186. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
187. 187. Composing Numbers1 hundred © Joan A. Cotter, Ph.D., 2012
188. 188. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
189. 189. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
190. 190. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
191. 191. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
192. 192. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
193. 193. Composing Numbers2 hundred200 © Joan A. Cotter, Ph.D., 2012
194. 194. Fact Strategies194 © Joan A. Cotter, Ph.D., 2012
195. 195. Fact Strategies • A strategy is a way to learn a new fact or recall a forgotten fact.195 © Joan A. Cotter, Ph.D., 2012
196. 196. Fact Strategies • A strategy is a way to learn a new fact or recall a forgotten fact. • A visualizable representation makes a powerful strategy.196 © Joan A. Cotter, Ph.D., 2012
197. 197. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
198. 198. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
199. 199. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
200. 200. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
201. 201. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
202. 202. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
203. 203. Fact Strategies Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
204. 204. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
205. 205. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
206. 206. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
207. 207. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
208. 208. Fact Strategies Two Fives8+6=10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
209. 209. Fact Strategies Two Fives7+5= © Joan A. Cotter, Ph.D., 2012
210. 210. Fact Strategies Two Fives7+5= © Joan A. Cotter, Ph.D., 2012
211. 211. Fact Strategies Two Fives7 + 5 = 12 © Joan A. Cotter, Ph.D., 2012
212. 212. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
213. 213. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
214. 214. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
215. 215. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
216. 216. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
217. 217. Fact Strategies Going Down 15 – 9 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
218. 218. Fact Strategies Subtract from 1015 – 9 = © Joan A. Cotter, Ph.D., 2012
219. 219. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
220. 220. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
221. 221. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
222. 222. Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
223. 223. Fact Strategies Going Up13 – 9 = © Joan A. Cotter, Ph.D., 2012
224. 224. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
225. 225. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
226. 226. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
227. 227. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
228. 228. Fact Strategies Going Up 13 – 9 = 1+3=4Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
229. 229. MoneyPenny © Joan A. Cotter, Ph.D., 2012
230. 230. MoneyNickel © Joan A. Cotter, Ph.D., 2012
231. 231. Money Dime © Joan A. Cotter, Ph.D., 2012
232. 232. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
233. 233. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
234. 234. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
235. 235. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
237. 237. Trading Side Place Value: Two aspectsStatic © Joan A. Cotter, Ph.D., 2012
238. 238. Trading Side Place Value: Two aspectsStatic • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
239. 239. Trading Side 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
240. 240. Trading Side 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
241. 241. Trading Side 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. • Place value cards show this aspect. © Joan A. Cotter, Ph.D., 2012
242. 242. Trading Side 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. • Place value cards show this aspect.Dynamic © Joan A. Cotter, Ph.D., 2012
243. 243. Trading Side 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. • Place value cards show this aspect.Dynamic • Ten ones = 1 ten; ten tens = 1 hundred; ten hundreds = 1 thousand, …. © Joan A. Cotter, Ph.D., 2012
244. 244. Trading Side Cleared1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
245. 245. Trading Side Thousands1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
246. 246. Trading Side Hundreds1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
247. 247. Trading Side Tens1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
248. 248. Trading Side Ones1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
253. 253. Trading Side Adding1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
254. 254. Trading Side Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
255. 255. Trading Side Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
256. 256. Trading Side Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
258. 258. Trading Side Cleared1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
259. 259. Trading Side Bead Trading game1000 100 10 1 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
260. 260. Trading Side Bead Trading game1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
261. 261. Trading Side Bead Trading game1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
276. 276. Trading Side Bead Trading game• In the Bead Trading game trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. © Joan A. Cotter, Ph.D., 2012
277. 277. Trading Side Bead Trading game• In the Bead Trading game trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely.• Bead trading helps the child experience thegreater value of each column from left to right. © Joan A. Cotter, Ph.D., 2012
278. 278. Trading Side Bead Trading game• In the Bead Trading game trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely.• Bead trading helps the child experience thegreater value of each column from left to right.• To detect a pattern, there must be at least threeexamples in the sequence. Place value is a pattern. © Joan A. Cotter, Ph.D., 2012
279. 279. Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
280. 280. Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
281. 281. Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
282. 282. Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
283. 283. Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
284. 284. Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
285. 285. Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
286. 286. Trading Side 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
287. 287. Trading Side 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
288. 288. Trading Side 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
289. 289. Trading Side 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
290. 290. Trading Side 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
291. 291. Trading Side 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
292. 292. Trading Side 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
293. 293. Trading Side 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
294. 294. Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
295. 295. Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
296. 296. Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
297. 297. Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
298. 298. Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
299. 299. Trading Side Adding 4-digit numbers1000 100 10 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
300. 300. Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
301. 301. Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
302. 302. Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
303. 303. Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
304. 304. Trading Side Adding 4-digit numbers1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
305. 305. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
306. 306. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
307. 307. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
308. 308. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
309. 309. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
310. 310. Multiplication on the AL Abacus Basic facts 9× 3= © Joan A. Cotter, Ph.D., 2012
311. 311. Multiplication on the AL Abacus Basic facts 9× 3= © Joan A. Cotter, Ph.D., 2012
312. 312. Multiplication on the AL Abacus Basic facts 9× 3= 30 © Joan A. Cotter, Ph.D., 2012
313. 313. Multiplication on the AL Abacus Basic facts 9× 3= 30 – 3 = 27 © Joan A. Cotter, Ph.D., 2012
314. 314. Multiplication on the AL Abacus Basic facts 4× 8= © Joan A. Cotter, Ph.D., 2012
315. 315. Multiplication on the AL Abacus Basic facts 4× 8= © Joan A. Cotter, Ph.D., 2012
316. 316. Multiplication on the AL Abacus Basic facts 4× 8= © Joan A. Cotter, Ph.D., 2012
317. 317. Multiplication on the AL Abacus Basic facts 4× 8= 20 + 12 = 32 © Joan A. Cotter, Ph.D., 2012
318. 318. Multiplication on the AL Abacus Basic facts 7× 7= © Joan A. Cotter, Ph.D., 2012
319. 319. Multiplication on the AL Abacus Basic facts 7× 7= © Joan A. Cotter, Ph.D., 2012
320. 320. Multiplication on the AL Abacus Basic facts 7× 7= 25 + 10 + 10 + 4 = 49 © Joan A. Cotter, Ph.D., 2012
321. 321. Multiplication on the AL Abacus Commutative property 5×6= © Joan A. Cotter, Ph.D., 2012
322. 322. Multiplication on the AL Abacus Commutative property 5×6= © Joan A. Cotter, Ph.D., 2012
323. 323. Multiplication on the AL Abacus Commutative property 5×6= © Joan A. Cotter, Ph.D., 2012
324. 324. Multiplication on the AL Abacus Commutative property 5×6=6× 5 © Joan A. Cotter, Ph.D., 2012
325. 325. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20325 © Joan A. Cotter, Ph.D., 2012
326. 326. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 The ones repeat in the second row.326 © Joan A. Cotter, Ph.D., 2012
327. 327. Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40 The ones repeat in the second row.327 © Joan A. Cotter, Ph.D., 2012
328. 328. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80328 © Joan A. Cotter, Ph.D., 2012
329. 329. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80329 © Joan A. Cotter, Ph.D., 2012
330. 330. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80330 © Joan A. Cotter, Ph.D., 2012
331. 331. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.331 © Joan A. Cotter, Ph.D., 2012
332. 332. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.332 © Joan A. Cotter, Ph.D., 2012
333. 333. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.333 © Joan A. Cotter, Ph.D., 2012
334. 334. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.334 © Joan A. Cotter, Ph.D., 2012
335. 335. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 The ones in the 8s show the multiples of 2.335 © Joan A. Cotter, Ph.D., 2012
336. 336. Multiples Patterns Sixes and Eights 6 12 18 24 30 6× 4 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 6 × 4 is the fourth number (multiple).336 © Joan A. Cotter, Ph.D., 2012
337. 337. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 8× 7 8 × 7 is the seventh number (multiple).337 © Joan A. Cotter, Ph.D., 2012
338. 338. Multiples Patterns Nines 9 18 27 36 45 90 81 72 63 54 The second row is written in reverse order. Also the digits in each number add to 9.338 © Joan A. Cotter, Ph.D., 2012
339. 339. Multiples Patterns Threes 3 6 9 2 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.339 © Joan A. Cotter, Ph.D., 2012
340. 340. Multiples Patterns Threes 3 6 9 2 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.340 © Joan A. Cotter, Ph.D., 2012
341. 341. Multiples Patterns Threes 3 6 9 2 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.341 © Joan A. Cotter, Ph.D., 2012
342. 342. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.342 © Joan A. Cotter, Ph.D., 2012
343. 343. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.343 © Joan A. Cotter, Ph.D., 2012
344. 344. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.344 © Joan A. Cotter, Ph.D., 2012
345. 345. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.345 © Joan A. Cotter, Ph.D., 2012
346. 346. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.346 © Joan A. Cotter, Ph.D., 2012
347. 347. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.347 © Joan A. Cotter, Ph.D., 2012
348. 348. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.348 © Joan A. Cotter, Ph.D., 2012
349. 349. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Observe the ones.349 © Joan A. Cotter, Ph.D., 2012
350. 350. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: The tens are the same in each row.350 © Joan A. Cotter, Ph.D., 2012
351. 351. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.351 © Joan A. Cotter, Ph.D., 2012
352. 352. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.352 © Joan A. Cotter, Ph.D., 2012
353. 353. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.353 © Joan A. Cotter, Ph.D., 2012