Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

NDCTM

4,196 views

Published on

Published in: Education, Technology, Business
  • Be the first to comment

  • Be the first to like this

NDCTM

  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
  118. 118. AL Abacus Adding © Joan A. Cotter, Ph.D., 2012
  119. 119. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  120. 120. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  121. 121. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  122. 122. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  123. 123. AL Abacus Adding4+3=7 © 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
  236. 236. Trading Side © 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
  249. 249. Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  250. 250. Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  251. 251. Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  252. 252. Trading Side Adding1000 100 10 1 8 +6 © 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
  257. 257. Trading Side Adding1000 100 10 1 8 +6 14 Same answer before and after trading. © 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
  262. 262. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  263. 263. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  264. 264. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  265. 265. Trading Side Bead Trading game1000 100 10 1 6 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  266. 266. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  267. 267. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  268. 268. Trading Side Bead Trading game1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  269. 269. Trading Side Bead Trading game1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  270. 270. Trading Side Bead Trading game1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  271. 271. Trading Side Bead Trading game1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  272. 272. Trading Side Bead Trading game1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  273. 273. Trading Side Bead Trading game1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  274. 274. Trading Side Bead Trading game• In the Bead Trading game trading 10 ones for 1 ten occurs frequently; © Joan A. Cotter, Ph.D., 2012
  275. 275. 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; © 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

×