NDCTM

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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
    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

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