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  • 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. Visualizing Enhances Standards K. Count to 100 by ones and by tens.2 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. 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. 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. 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. Verbal Counting Model10 © Joan A. Cotter, Ph.D., 2012
  • 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. Verbal Counting Model From a childs perspective F +E12 © Joan A. Cotter, Ph.D., 2012
  • 13. Verbal Counting Model From a childs perspective F +E A13 © Joan A. Cotter, Ph.D., 2012
  • 14. Verbal Counting Model From a childs perspective F +E A B14 © Joan A. Cotter, Ph.D., 2012
  • 15. Verbal Counting Model From a childs perspective F +E A B C15 © Joan A. Cotter, Ph.D., 2012
  • 16. Verbal Counting Model From a childs perspective F +E A B C D E F16 © Joan A. Cotter, Ph.D., 2012
  • 17. Verbal Counting Model From a childs perspective F +E A B C D E F A17 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. Verbal Counting Model From a childs perspective Now memorize the facts!! G +D22 © Joan A. Cotter, Ph.D., 2012
  • 23. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D23 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. Verbal Counting Model From a childs perspective Try subtracting H by “taking away” –E27 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. 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. 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. 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. 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. Verbal Counting Model Summary36 © Joan A. Cotter, Ph.D., 2012
  • 37. Verbal Counting Model Summary • Is not natural; it takes years of practice.37 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  • 56. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  • 57. Research on Counting Karen Wynn’s research57 © Joan A. Cotter, Ph.D., 2012
  • 58. Research on Counting Karen Wynn’s research58 © Joan A. Cotter, Ph.D., 2012
  • 59. Research on Counting Karen Wynn’s research59 © Joan A. Cotter, Ph.D., 2012
  • 60. Research on Counting Karen Wynn’s research60 © Joan A. Cotter, Ph.D., 2012
  • 61. Research on Counting Karen Wynn’s research61 © Joan A. Cotter, Ph.D., 2012
  • 62. Research on Counting Karen Wynn’s research62 © Joan A. Cotter, Ph.D., 2012
  • 63. Research on Counting Other research63 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding.68 © Joan A. Cotter, Ph.D., 2012
  • 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. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting.70 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. Visualizing Mathematics75 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. Visualizing Mathematics Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
  • 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. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  • 82. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  • 83. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  • 84. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  • 85. Visualizing MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  • 86. Visualizing MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  • 87. Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  • 88. Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  • 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. Visualizing Mathematics : Who could read the music?90 © Joan A. Cotter, Ph.D., 2012
  • 91. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  • 92. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  • 93. Naming Quantities Using fingers93 © Joan A. Cotter, Ph.D., 2012
  • 94. Naming Quantities Using fingers94 © Joan A. Cotter, Ph.D., 2012
  • 95. Naming Quantities Using fingers95 © Joan A. Cotter, Ph.D., 2012
  • 96. Naming Quantities Using fingers96 © Joan A. Cotter, Ph.D., 2012
  • 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. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  • 99. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  • 100. Naming Quantities Recognizing 55 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
  • 101. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  • 102. Naming Quantities Tally sticks102 © Joan A. Cotter, Ph.D., 2012
  • 103. Naming Quantities Tally sticks103 © Joan A. Cotter, Ph.D., 2012
  • 104. Naming Quantities Tally sticks104 © Joan A. Cotter, Ph.D., 2012
  • 105. Naming Quantities Tally sticks105 © Joan A. Cotter, Ph.D., 2012
  • 106. Naming Quantities Tally sticks106 © Joan A. Cotter, Ph.D., 2012
  • 107. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?107 © Joan A. Cotter, Ph.D., 2012
  • 108. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?108 © Joan A. Cotter, Ph.D., 2012
  • 109. Naming QuantitiesNumberChart 1 2 3 4 5 © Joan A. Cotter, Ph.D., 2012
  • 110. Naming Quantities Number Chart 1 2To help the 3child learnthe symbols 4 5 © Joan A. Cotter, Ph.D., 2012
  • 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. AL Abacus Cleared © Joan A. Cotter, Ph.D., 2012
  • 113. AL Abacus Entering quantities3 © Joan A. Cotter, Ph.D., 2012
  • 114. AL Abacus Entering quantities 5114 © Joan A. Cotter, Ph.D., 2012
  • 115. AL Abacus Entering quantities 7115 © Joan A. Cotter, Ph.D., 2012
  • 116. AL Abacus Entering quantities 10116 © Joan A. Cotter, Ph.D., 2012
  • 117. AL Abacus The stairs117 © Joan A. Cotter, Ph.D., 2012
  • 118. AL Abacus Adding © Joan A. Cotter, Ph.D., 2012
  • 119. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  • 120. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  • 121. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  • 122. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
  • 123. AL Abacus Adding4+3=7 © Joan A. Cotter, Ph.D., 2012
  • 124. “Math” Way of Naming Numbers124 © Joan A. Cotter, Ph.D., 2012
  • 125. “Math” Way of Naming Numbers 11 = ten 1125 © Joan A. Cotter, Ph.D., 2012
  • 126. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2126 © Joan A. Cotter, Ph.D., 2012
  • 127. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3127 © Joan A. Cotter, Ph.D., 2012
  • 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. “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. “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. “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. “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. “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. “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. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7135 © Joan A. Cotter, Ph.D., 2012
  • 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. “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. “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. “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. “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. “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. 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. 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. 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. 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. Math Way of Naming Numbers Compared to reading:146 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  • 151. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  • 152. Math Way of Naming Numbers Traditional names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  • 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. Math Way of Naming Numbers Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
  • 155. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  • 156. Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
  • 157. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
  • 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. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  • 160. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  • 161. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  • 162. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
  • 163. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
  • 164. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
  • 165. Math Way of Naming Numbers Traditional names two leftTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  • 166. Math Way of Naming Numbers Traditional names two left twelveTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  • 167. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  • 168. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  • 169. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
  • 170. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
  • 171. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
  • 172. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
  • 173. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
  • 174. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
  • 175. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
  • 176. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
  • 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. Composing Numbers7-ten70 Another example. © Joan A. Cotter, Ph.D., 2012
  • 179. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
  • 180. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
  • 181. Composing Numbers7-ten 870 8 © Joan A. Cotter, Ph.D., 2012
  • 182. Composing Numbers7-ten 878 © Joan A. Cotter, Ph.D., 2012
  • 183. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
  • 184. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  • 185. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  • 186. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
  • 187. Composing Numbers1 hundred © Joan A. Cotter, Ph.D., 2012
  • 188. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
  • 189. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
  • 190. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
  • 191. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
  • 192. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
  • 193. Composing Numbers2 hundred200 © Joan A. Cotter, Ph.D., 2012
  • 194. Fact Strategies194 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  • 198. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  • 199. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
  • 200. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  • 201. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  • 202. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  • 203. Fact Strategies Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
  • 204. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  • 205. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  • 206. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  • 207. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
  • 208. Fact Strategies Two Fives8+6=10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
  • 209. Fact Strategies Two Fives7+5= © Joan A. Cotter, Ph.D., 2012
  • 210. Fact Strategies Two Fives7+5= © Joan A. Cotter, Ph.D., 2012
  • 211. Fact Strategies Two Fives7 + 5 = 12 © Joan A. Cotter, Ph.D., 2012
  • 212. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 213. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 214. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  • 215. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  • 216. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  • 217. Fact Strategies Going Down 15 – 9 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
  • 218. Fact Strategies Subtract from 1015 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 219. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • 220. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • 221. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • 222. Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
  • 223. Fact Strategies Going Up13 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 224. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 225. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 226. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 227. Fact Strategies Going Up 13 – 9 =Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 228. Fact Strategies Going Up 13 – 9 = 1+3=4Start with 9;go up to 13. © Joan A. Cotter, Ph.D., 2012
  • 229. MoneyPenny © Joan A. Cotter, Ph.D., 2012
  • 230. MoneyNickel © Joan A. Cotter, Ph.D., 2012
  • 231. Money Dime © Joan A. Cotter, Ph.D., 2012
  • 232. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • 233. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • 234. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • 235. MoneyQuarter © Joan A. Cotter, Ph.D., 2012
  • 236. Trading Side © Joan A. Cotter, Ph.D., 2012
  • 237. Trading Side Place Value: Two aspectsStatic © Joan A. Cotter, Ph.D., 2012
  • 238. Trading Side Place Value: Two aspectsStatic • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. 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. Trading Side Cleared1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 245. Trading Side Thousands1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 246. Trading Side Hundreds1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 247. Trading Side Tens1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 248. Trading Side Ones1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 249. Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 250. Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 251. Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 252. Trading Side Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 253. Trading Side Adding1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. Trading Side Adding1000 100 10 1 8 +6 14 Same answer before and after trading. © Joan A. Cotter, Ph.D., 2012
  • 258. Trading Side Cleared1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 263. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 264. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 265. Trading Side Bead Trading game1000 100 10 1 6 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 266. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 267. Trading Side Bead Trading game1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 268. Trading Side Bead Trading game1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  • 269. Trading Side Bead Trading game1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  • 270. Trading Side Bead Trading game1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  • 271. Trading Side Bead Trading game1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  • 272. Trading Side Bead Trading game1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  • 273. Trading Side Bead Trading game1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. 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. Trading Side Adding 4-digit numbers1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 306. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 307. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 308. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 309. Multiplication on the AL Abacus Basic facts 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 310. Multiplication on the AL Abacus Basic facts 9× 3= © Joan A. Cotter, Ph.D., 2012
  • 311. Multiplication on the AL Abacus Basic facts 9× 3= © Joan A. Cotter, Ph.D., 2012
  • 312. Multiplication on the AL Abacus Basic facts 9× 3= 30 © Joan A. Cotter, Ph.D., 2012
  • 313. Multiplication on the AL Abacus Basic facts 9× 3= 30 – 3 = 27 © Joan A. Cotter, Ph.D., 2012
  • 314. Multiplication on the AL Abacus Basic facts 4× 8= © Joan A. Cotter, Ph.D., 2012
  • 315. Multiplication on the AL Abacus Basic facts 4× 8= © Joan A. Cotter, Ph.D., 2012
  • 316. Multiplication on the AL Abacus Basic facts 4× 8= © Joan A. Cotter, Ph.D., 2012
  • 317. Multiplication on the AL Abacus Basic facts 4× 8= 20 + 12 = 32 © Joan A. Cotter, Ph.D., 2012
  • 318. Multiplication on the AL Abacus Basic facts 7× 7= © Joan A. Cotter, Ph.D., 2012
  • 319. Multiplication on the AL Abacus Basic facts 7× 7= © Joan A. Cotter, Ph.D., 2012
  • 320. Multiplication on the AL Abacus Basic facts 7× 7= 25 + 10 + 10 + 4 = 49 © Joan A. Cotter, Ph.D., 2012
  • 321. Multiplication on the AL Abacus Commutative property 5×6= © Joan A. Cotter, Ph.D., 2012
  • 322. Multiplication on the AL Abacus Commutative property 5×6= © Joan A. Cotter, Ph.D., 2012
  • 323. Multiplication on the AL Abacus Commutative property 5×6= © Joan A. Cotter, Ph.D., 2012
  • 324. Multiplication on the AL Abacus Commutative property 5×6=6× 5 © Joan A. Cotter, Ph.D., 2012
  • 325. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20325 © Joan A. Cotter, Ph.D., 2012
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  • 354. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”354 © Joan A. Cotter, Ph.D., 2012
  • 355. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”355 © Joan A. Cotter, Ph.D., 2012
  • 356. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”356 © Joan A. Cotter, Ph.D., 2012
  • 357. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the “opposites.”357 © Joan A. Cotter, Ph.D., 2012
  • 358. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.358 © Joan A. Cotter, Ph.D., 2012
  • 359. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.359 © Joan A. Cotter, Ph.D., 2012
  • 360. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.360 © Joan A. Cotter, Ph.D., 2012
  • 361. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.361 © Joan A. Cotter, Ph.D., 2012
  • 362. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.362 © Joan A. Cotter, Ph.D., 2012
  • 363. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.363 © Joan A. Cotter, Ph.D., 2012
  • 364. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 Look at the tens.364 © Joan A. Cotter, Ph.D., 2012
  • 365. Multiples Memory © Joan A. Cotter, Ph.D., 2012
  • 366. Multiples MemoryAim: To help the players learn themultiples patterns. © Joan A. Cotter, Ph.D., 2012
  • 367. Multiples MemoryAim: To help the players learn themultiples patterns.Object of the game: To be the first player to collect all tencards of a multiple in order. © Joan A. Cotter, Ph.D., 2012
  • 368. Multiples Memory 7 14 21 28 35 42 49 56 63 70The 7s envelope contains 10 cards,each with one of the numbers listed. © Joan A. Cotter, Ph.D., 2012
  • 369. Multiples Memory 8 16 24 32 40 48 56 64 72 80The 8s envelope contains 10 cards,each with one of the numbers listed. © Joan A. Cotter, Ph.D., 2012
  • 370. Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 371. Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 372. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 373. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 63 1470 © Joan A. Cotter, Ph.D., 2012
  • 374. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 375. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 376. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 40 © Joan A. Cotter, Ph.D., 2012
  • 377. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 378. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 379. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • 380. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 381. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 382. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • 383. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • 384. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 63 5670 8 © Joan A. Cotter, Ph.D., 2012
  • 385. Multiples Memory 8 16 24 32 40 48 56 64 72 80 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • 386. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • 387. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 8 © Joan A. Cotter, Ph.D., 2012
  • 388. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 87 © Joan A. Cotter, Ph.D., 2012
  • 389. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 14 70 87 © Joan A. Cotter, Ph.D., 2012
  • 390. Multiples Memory 7 14 21 28 35 42 49 56 63 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 87 14 © Joan A. Cotter, Ph.D., 2012
  • 391. Multiples Memory 7 14 21 28 35 42 49 56 63 70 24 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 87 14 © Joan A. Cotter, Ph.D., 2012
  • 392. Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 40 28 35 42 48 56 64 72 80 49 56 63 70 87 14 © Joan A. Cotter, Ph.D., 2012
  • 393. Multiples Memory 7 14 21 28 35 42 8 16 24 32 40 49 56 63 48 56 64 72 80 70 7 14 21 8 16 24 32 4028 35 42 48 56 64 72 8049 56 6370 © Joan A. Cotter, Ph.D., 2012
  • 394. 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.394 © Joan A. Cotter, Ph.D., 2012
  • 395. 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 3395 © Joan A. Cotter, Ph.D., 2012