IDA-UMB: Visualizing with the AL Abacus March 2011

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  • IDA-UMB: Visualizing with the AL Abacus March 2011

    1. 1. Overcoming Obstacles Learning Arithmetic through Visualizing with the AL Abacus IDA-UMB Conference March 12, 2011 Saint Paul, Minnesota by Joan A. Cotter, Ph.D. [email_address] 7 5 2 VII
    2. 2. Children with MD (Math Difficulties) Often experience difficulties with:
    3. 3. Children with MD (Math Difficulties) Often experience difficulties with: <ul><li>Counting in its various forms. </li></ul>
    4. 4. Children with MD (Math Difficulties) Often experience difficulties with: <ul><li>Counting in its various forms. </li></ul><ul><li>Composing numbers. </li></ul>
    5. 5. Children with MD (Math Difficulties) Often experience difficulties with: <ul><li>Counting in its various forms. </li></ul><ul><li>Composing numbers. </li></ul><ul><li>Memorizing the facts. </li></ul>
    6. 6. Children with MD (Math Difficulties) Often experience difficulties with: <ul><li>Counting in its various forms. </li></ul><ul><li>Composing numbers. </li></ul><ul><li>Understanding and applying math symbols. </li></ul><ul><li>Memorizing the facts. </li></ul>
    7. 7. Children with MD (Math Difficulties) Often experience difficulties with: <ul><li>Counting in its various forms. </li></ul><ul><li>Composing numbers. </li></ul><ul><li>Understanding and applying math symbols. </li></ul><ul><li>Learning algorithms. </li></ul><ul><li>Memorizing the facts. </li></ul>
    8. 8. Children with MD (Math Difficulties) Often learn best when:
    9. 9. Children with MD (Math Difficulties) Often learn best when: <ul><li>They are taught visually, not orally. </li></ul>
    10. 10. Children with MD (Math Difficulties) Often learn best when: <ul><li>They are taught visually, not orally. </li></ul><ul><li>They use the “math way” of counting initially. </li></ul>
    11. 11. Children with MD (Math Difficulties) Often learn best when: <ul><li>They are taught visually, not orally. </li></ul><ul><li>They use the “math way” of counting initially. </li></ul><ul><li>They truly understand math concepts. </li></ul>
    12. 12. Children with MD (Math Difficulties) Often learn best when: <ul><li>They are taught visually, not orally. </li></ul><ul><li>They use the “math way” of counting initially. </li></ul><ul><li>They are given the “big picture” before details. </li></ul><ul><li>They truly understand math concepts. </li></ul>
    13. 13. Children with MD (Math Difficulties) Often learn best when: <ul><li>They are taught visually, not orally. </li></ul><ul><li>They use the “math way” of counting initially. </li></ul><ul><li>They are given the “big picture” before details. </li></ul><ul><li>They use part/whole circles for solving problems </li></ul><ul><li>They truly understand math concepts. </li></ul>
    14. 14. Children with MD (Math Difficulties) Often learn best when: <ul><li>They are taught visually, not orally. </li></ul><ul><li>They use the “math way” of counting initially. </li></ul><ul><li>They are given the “big picture” before details. </li></ul><ul><li>They use part/whole circles for solving problems </li></ul><ul><li>They are provided with references as needed. </li></ul><ul><li>They truly understand math concepts. </li></ul>
    15. 15. Learning Arithmetic Traditionally Counting
    16. 16. Learning Arithmetic Traditionally Counting Memorizing 390 Facts
    17. 17. Learning Arithmetic Traditionally Counting Memorizing 390 Facts Learning Procedures
    18. 18. Learning Arithmetic Traditionally Counting Memorizing 390 Facts Learning Procedures Solving Problems
    19. 19. Learning Arithmetic Traditionally Counting Memorizing 390 Facts Learning Procedures Solving Problems Place Value
    20. 20. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic.
    21. 21. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic. Naming Quantities
    22. 22. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic. Naming Quantities Visualizing 390 Facts
    23. 23. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic. Naming Quantities Visualizing 390 Facts Learning Procedures
    24. 24. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic. Naming Quantities Visualizing 390 Facts Learning Procedures Solving Problems
    25. 25. Counting Based-Arithmetic Arithmetic is deemed to be based on counting.
    26. 26. Counting Based-Arithmetic <ul><li>Rote counting to 100 in kindergarten. </li></ul>Arithmetic is deemed to be based on counting.
    27. 27. Counting Based-Arithmetic <ul><li>Rote counting to 100 in kindergarten. </li></ul><ul><li>Calendars (mis)used to teach counting. </li></ul>Arithmetic is deemed to be based on counting.
    28. 28. Counting Based-Arithmetic <ul><li>Rote counting to 100 in kindergarten. </li></ul><ul><li>Calendars (mis)used to teach counting. </li></ul>Arithmetic is deemed to be based on counting. <ul><li>Addition and subtraction taught with counting. </li></ul>
    29. 29. Counting Based-Arithmetic <ul><li>Rote counting to 100 in kindergarten. </li></ul><ul><li>Calendars (mis)used to teach counting. </li></ul><ul><li>Number lines, a counting artifact. </li></ul>Arithmetic is deemed to be based on counting. <ul><li>Addition and subtraction taught with counting. </li></ul>
    30. 30. Counting Based-Arithmetic <ul><li>Rote counting to 100 in kindergarten. </li></ul><ul><li>Calendars (mis)used to teach counting. </li></ul><ul><li>Skip counting used for multiplication facts. </li></ul><ul><li>Number lines, a counting artifact. </li></ul>Arithmetic is deemed to be based on counting. <ul><li>Addition and subtraction taught with counting. </li></ul>
    31. 31. Counting Based-Arithmetic <ul><li>Rote counting to 100 in kindergarten. </li></ul><ul><li>Calendars (mis)used to teach counting. </li></ul><ul><li>Skip counting used for multiplication facts. </li></ul><ul><li>Graphing primarily a counting activity. </li></ul><ul><li>Number lines, a counting artifact. </li></ul>Arithmetic is deemed to be based on counting. <ul><li>Addition and subtraction taught with counting. </li></ul>
    32. 32. Counting Based-Arithmetic <ul><li>Rote counting to 100 in kindergarten. </li></ul><ul><li>Calendars (mis)used to teach counting. </li></ul><ul><li>Skip counting used for multiplication facts. </li></ul><ul><li>Graphing primarily a counting activity. </li></ul><ul><li>Number lines, a counting artifact. </li></ul>Arithmetic is deemed to be based on counting. <ul><li>Addition and subtraction taught with counting. </li></ul><ul><li>Doesn’t work well for fractions or algebra. </li></ul>
    33. 33. Counting Model From a child's 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 forth
    34. 34. Counting Model From a child's perspective F + E
    35. 35. Counting Model From a child's perspective F + E A
    36. 36. Counting Model From a child's perspective F + E A B
    37. 37. Counting Model From a child's perspective F + E A C B
    38. 38. Counting Model From a child's perspective F + E A F C D E B
    39. 39. Counting Model From a child's perspective F + E A A F C D E B
    40. 40. Counting Model From a child's perspective F + E A B A F C D E B
    41. 41. Counting Model From a child's perspective F + E A C D E B A F C D E B
    42. 42. Counting Model From a child's perspective F + E What is the sum? (It must be a letter.) A C D E B A F C D E B
    43. 43. Counting Model From a child's perspective K G I J K H A F C D E B F + E
    44. 44. Counting Model From a child's perspective Now memorize the facts!! G + D
    45. 45. Counting Model From a child's perspective Now memorize the facts!! G + D H + F
    46. 46. Counting Model From a child's perspective Now memorize the facts!! G + D H + F D + C
    47. 47. Counting Model From a child's perspective Now memorize the facts!! G + D H + F C + G D + C
    48. 48. Counting Model From a child's perspective Now memorize the facts!! E + I G + D H + F C + G D + C
    49. 49. Counting Model From a child's perspective Try subtracting by “taking away” H – E
    50. 50. Counting Model From a child's perspective Try skip counting by B’s to T : B , D , . . . T .
    51. 51. Counting Model From a child's perspective Try skip counting by B’s to T : B , D , . . . T . What is D  E ?
    52. 52. Counting Model From a child's perspective L is written AB because it is A J and B A’s
    53. 53. Counting Model From a child's perspective L is written AB because it is A J and B A’s huh?
    54. 54. Counting Model From a child's perspective L is written AB because it is A J and B A’s (twelve)
    55. 55. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (twelve)
    56. 56. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (one 10) (twelve)
    57. 57. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (one 10) (two 1s). (twelve)
    58. 58. Counting Model Counting:
    59. 59. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul>Counting:
    60. 60. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul>Counting:
    61. 61. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul><ul><li>Ignores place value. </li></ul>Counting:
    62. 62. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul><ul><li>Ignores place value. </li></ul><ul><li>Is very error prone. </li></ul>Counting:
    63. 63. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul><ul><li>Ignores place value. </li></ul><ul><li>Is very error prone. </li></ul><ul><li>Is tedious and time-consuming. </li></ul>Counting:
    64. 64. Counting Model <ul><li>Is not natural; it takes years of practice. </li></ul><ul><li>Provides poor concept of quantity. </li></ul><ul><li>Ignores place value. </li></ul><ul><li>Is very error prone. </li></ul><ul><li>Is tedious and time-consuming. </li></ul>Counting: <ul><li>Does not provide an efficient way to master the facts. </li></ul>
    65. 65. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.
    66. 66. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.
    67. 67. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31
    68. 68. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.
    69. 69. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.
    70. 70. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.
    71. 71. Calendar Math August 8 1 9 2 10 3 4 5 6 7 Always show the whole calendar. A child needs to see the whole before the parts. Children also need to learn to plan ahead.
    72. 72. Calendar Math <ul><li>The calendar is not a number line. </li></ul><ul><ul><li>No quantity is involved. </li></ul></ul><ul><ul><li>Numbers are in spaces, not at lines like a ruler. </li></ul></ul>
    73. 73. Calendar Math <ul><li>The calendar is not a number line. </li></ul><ul><ul><li>No quantity is involved. </li></ul></ul><ul><ul><li>Numbers are in spaces, not at lines like a ruler. </li></ul></ul><ul><li>Children need to see the whole month, not just part. </li></ul><ul><ul><li>Purpose of calendar is to plan ahead. </li></ul></ul><ul><ul><li>Many ways to show the current date. </li></ul></ul>
    74. 74. Calendar Math <ul><li>The calendar is not a number line. </li></ul><ul><ul><li>No quantity is involved. </li></ul></ul><ul><ul><li>Numbers are in spaces, not at lines like a ruler. </li></ul></ul><ul><li>Children need to see the whole month, not just part. </li></ul><ul><ul><li>Purpose of calendar is to plan ahead. </li></ul></ul><ul><ul><li>Many ways to show the current date. </li></ul></ul><ul><li>Calendars give a narrow view of patterning. </li></ul><ul><ul><li>Patterns do not necessarily involve numbers. </li></ul></ul><ul><ul><li>Patterns rarely proceed row by row. </li></ul></ul><ul><ul><li>Patterns go on forever; they don’t stop at 31. </li></ul></ul>
    75. 75. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    76. 76. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    77. 77. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    78. 78. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
    79. 79. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall Even worse if you have MD.
    80. 80. Memorizing Math Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall Even worse if you have MD.
    81. 81. Memorizing Math Flash Cards 9 + 7
    82. 82. Memorizing Math Flash Cards <ul><li>Are often used to teach rote. </li></ul>9 + 7
    83. 83. Memorizing Math Flash Cards <ul><li>Are often used to teach rote. </li></ul><ul><li>Are liked only by those who don’t need them. </li></ul>9 + 7
    84. 84. Memorizing Math Flash Cards <ul><li>Are often used to teach rote. </li></ul><ul><li>Are liked only by those who don’t need them. </li></ul><ul><li>Give the false impression that math isn’t about thinking. </li></ul>9 + 7
    85. 85. Memorizing Math Flash Cards <ul><li>Are often used to teach rote. </li></ul><ul><li>Are liked only by those who don’t need them. </li></ul><ul><li>Give the false impression that math isn’t about thinking. </li></ul><ul><li>Often produce stress – children under stress stop learning. </li></ul>Even worse if you have MD. 9 + 7
    86. 86. Memorizing Math Flash Cards <ul><li>Are often used to teach rote. </li></ul><ul><li>Are liked only by those who don’t need them. </li></ul><ul><li>Give the false impression that math isn’t about thinking. </li></ul><ul><li>Often produce stress – children under stress stop learning. </li></ul><ul><li>Are not concrete – use abstract symbols. </li></ul>Even worse if you have MD. 9 + 7
    87. 87. Research on Counting Karen Wynn’s research Show the baby two teddy bears.
    88. 88. Research on Counting Karen Wynn’s research Then hide them with a screen.
    89. 89. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.
    90. 90. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.
    91. 91. Research on Counting Karen Wynn’s research Raise screen. Baby seeing 3 won’t look long because it is expected.
    92. 92. Research on Counting Karen Wynn’s research Researcher can change the number of teddy bears behind the screen.
    93. 93. Research on Counting Karen Wynn’s research A baby seeing 1 teddy bear will look much longer, because it’s unexpected.
    94. 94. Research on Counting <ul><li>Other research </li></ul>
    95. 95. Research on Counting <ul><li>Australian Aboriginal children from two tribes. </li></ul><ul><ul><li>Brian Butterworth, University College London, 2008 . </li></ul></ul>Other research These groups matched quantities without using counting words.
    96. 96. Research on Counting <ul><li>Australian Aboriginal children from two tribes. </li></ul><ul><ul><li>Brian Butterworth, University College London, 2008 . </li></ul></ul><ul><li>Adult Pirahã from Amazon region. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul>Other research These groups matched quantities without using counting words.
    97. 97. Research on Counting <ul><li>Australian Aboriginal children from two tribes. </li></ul><ul><ul><li>Brian Butterworth, University College London, 2008 . </li></ul></ul><ul><li>Adult Pirahã from Amazon region. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul><ul><li>Adults, ages 18-50, from Boston. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul>Other research These groups matched quantities without using counting words.
    98. 98. Research on Counting <ul><li>Australian Aboriginal children from two tribes. </li></ul><ul><ul><li>Brian Butterworth, University College London, 2008 . </li></ul></ul><ul><li>Adult Pirahã from Amazon region. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul><ul><li>Adults, ages 18-50, from Boston. </li></ul><ul><ul><li>Edward Gibson and Michael Frank, MIT, 2008. </li></ul></ul><ul><li>Baby chicks from Italy. </li></ul><ul><ul><li>Lucia Regolin, University of Padova, 2009. </li></ul></ul>Other research These groups matched quantities without using counting words.
    99. 99. Research on Counting In Japanese schools: <ul><li>Children are discouraged from using counting for adding. </li></ul><ul><li>They consistently group in 5s. </li></ul>
    100. 100. Visualizing Mathematics Visualizing is an alternative to copious counting and mind-numbing memorization.
    101. 101. Visualizing Mathematics “ Think in pictures, because the brain remembers images better than it does anything else.”   Ben Pridmore, World Memory Champion, 2009
    102. 102. Visualizing Mathematics “ In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.” Mindy Holte (E I)
    103. 103. Visualizing Mathematics “ The process of connecting symbols to imagery is at the heart of mathematics learning.” Dienes
    104. 104. Visualizing Mathematics “ Mathematics is the activity of creating relationships, many of which are based in visual imagery. ” Wheatley and Cobb
    105. 105. 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 others
    106. 106. Visualizing Mathematics Japanese criteria for manipulatives Japanese Council of Mathematics Education <ul><li>Representative of structure of numbers. </li></ul><ul><li>Easily manipulated by children. </li></ul><ul><li>Imaginable mentally. </li></ul>
    107. 107. Visualizing Mathematics Visualizing also needed in: <ul><li>Reading </li></ul><ul><li>Sports </li></ul><ul><li>Creativity </li></ul><ul><li>Geography </li></ul><ul><li>Engineering </li></ul><ul><li>Construction </li></ul>
    108. 108. Visualizing Mathematics Visualizing also needed in: <ul><li>Reading </li></ul><ul><li>Sports </li></ul><ul><li>Creativity </li></ul><ul><li>Geography </li></ul><ul><li>Engineering </li></ul><ul><li>Construction </li></ul><ul><li>Architecture </li></ul><ul><li>Astronomy </li></ul><ul><li>Archeology </li></ul><ul><li>Chemistry </li></ul><ul><li>Physics </li></ul><ul><li>Surgery </li></ul>
    109. 109. Visualizing Mathematics Ready: How many?
    110. 110. Visualizing Mathematics Ready: How many?
    111. 111. Visualizing Mathematics Try again: How many?
    112. 112. Visualizing Mathematics Try again: How many?
    113. 113. Visualizing Mathematics Try again: How many?
    114. 114. Visualizing Mathematics Ready: How many?
    115. 115. Visualizing Mathematics Try again: How many?
    116. 116. Visualizing Mathematics Try to visualize 8 identical apples without grouping.
    117. 117. Visualizing Mathematics Try to visualize 8 identical apples without grouping.
    118. 118. Visualizing Mathematics Now try to visualize 5 as red and 3 as green.
    119. 119. Visualizing Mathematics Now try to visualize 5 as red and 3 as green.
    120. 120. Visualizing Mathematics Early Roman numerals I II III IIII V VIII 1 2 3 4 5 8 Romans grouped in fives. Notice 8 is 5 and 3.
    121. 121. Visualizing Mathematics Who could read the music? : Music needs 10 lines, two groups of five.
    122. 122. Naming Quantities Using fingers
    123. 123. Naming Quantities Using fingers Use left hand for 1-5 because we read from left to right.
    124. 124. Naming Quantities Using fingers
    125. 125. Naming Quantities Using fingers
    126. 126. Naming Quantities Using fingers Always show 7 as 5 and 2, not for example, as 4 and 3.
    127. 127. Naming Quantities Using fingers
    128. 128. 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 Also set to music. Listen and download sheet music from Web site.
    129. 129. Naming Quantities Recognizing 5
    130. 130. Naming Quantities Recognizing 5
    131. 131. Naming Quantities Recognizing 5 5 has a middle; 4 does not. Look at your hand; your middle finger is longer to remind you 5 has a middle.
    132. 132. Naming Quantities Tally sticks Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.
    133. 133. Naming Quantities Tally sticks
    134. 134. Naming Quantities Tally sticks
    135. 135. Naming Quantities Tally sticks Stick is horizontal, because it won’t fit diagonally and young children have problems with diagonals.
    136. 136. Naming Quantities Tally sticks
    137. 137. Naming Quantities Tally sticks Start a new row for every ten.
    138. 138. Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting How would you find the answer without counting?
    139. 139. Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Then take 1 from the 3 and give it to the 4 to make 5 and 2.
    140. 140. Naming Quantities A typical worksheet The child counts all the horsies and forgets the fact before turning the page.
    141. 141. Naming Quantities Number Chart 1 2 3 4 5
    142. 142. Naming Quantities Number Chart 1 2 3 4 5 To help the child learn the symbols
    143. 143. Naming Quantities Number Chart 6 1 7 2 8 3 9 4 10 5 To help the child learn the symbols
    144. 144. AL Abacus Double-sided AL abacus. Side 1 is grouped in 5s. Trading Side introduces algorithms with trading. 1000 10 1 100
    145. 145. AL Abacus Cleared
    146. 146. 3 Entering quantities AL Abacus Quantities are entered all at once, not counted.
    147. 147. 3 Entering quantities AL Abacus Quantities are entered all at once, not counted.
    148. 148. 5 AL Abacus Entering quantities Relate quantities to hands.
    149. 149. 5 AL Abacus Entering quantities Relate quantities to hands.
    150. 150. 7 AL Abacus Entering quantities
    151. 151. 7 AL Abacus Entering quantities
    152. 152. AL Abacus 10 Entering quantities
    153. 153. AL Abacus 10 Entering quantities
    154. 154. AL Abacus The stairs Can use to “count” 1 to 10. Also read quantities on the right side.
    155. 155. AL Abacus Adding
    156. 156. AL Abacus Adding 4 + 3 =
    157. 157. AL Abacus Adding 4 + 3 =
    158. 158. AL Abacus Adding 4 + 3 =
    159. 159. AL Abacus Adding 4 + 3 =
    160. 160. AL Abacus Adding 4 + 3 = 7 Answer is seen immediately, no counting needed.
    161. 161. Go to the Dump Game Objective: To learn the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 Children use the abacus while playing this “Go Fish” type game.
    162. 162. Go to the Dump Game Objective: To learn the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 Object of the game: To collect the most pairs that equal ten. Children use the abacus while playing this “Go Fish” type game.
    163. 163. Go to the Dump Game Children use the abacus while playing this “Go Fish” type game.
    164. 164. Go to the Dump Game A game viewed from above. Starting
    165. 165. Go to the Dump Game Each player takes 5 cards. 7 2 7 9 5 7 4 2 6 1 3 8 3 4 9 Starting
    166. 166. Go to the Dump Game Does YellowCap have any pairs? [no] 7 2 7 9 5 7 2 4 6 1 3 8 3 4 9 Finding pairs
    167. 167. Go to the Dump Game Does BlueCap have any pairs? [yes, 1] 7 2 7 9 5 7 2 4 6 1 3 8 3 4 9 Finding pairs
    168. 168. Go to the Dump Game Does BlueCap have any pairs? [yes, 1] 7 2 7 9 5 7 2 1 3 8 Finding pairs 4 6 3 4 9
    169. 169. Go to the Dump Game Does BlueCap have any pairs? [yes, 1] 4 6 7 2 7 9 5 7 2 1 3 8 3 4 9 Finding pairs
    170. 170. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 7 2 1 3 8 3 4 9 Finding pairs
    171. 171. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 3 4 9 Finding pairs 7 2 1 3 8
    172. 172. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 2 1 8 3 4 9 Finding pairs 7 3
    173. 173. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 1 3 4 9 Finding pairs 7 3 2 8
    174. 174. Go to the Dump Game The player asks the player on her left. 2 4 6 7 2 7 9 5 1 3 4 9 2 8 Playing
    175. 175. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? The player asks the player on her left. 2 4 6 7 2 7 9 5 1 3 4 9 2 8 Playing
    176. 176. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? The player asks the player on her left. 7 4 6 2 7 9 5 1 4 9 2 8 Playing
    177. 177. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? The player asks the player on her left. 7 4 6 2 7 9 5 1 4 9 2 8 Playing 3
    178. 178. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? 4 6 2 7 9 5 1 4 9 2 8 Playing 7 3
    179. 179. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? YellowCap gets another turn. 4 6 2 7 9 5 1 4 9 2 8 Playing 7 3
    180. 180. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. YellowCap gets another turn. 4 6 2 7 9 5 1 4 9 2 8 Playing 7 3
    181. 181. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. 2 4 6 7 3 2 7 9 5 1 4 9 2 8 Playing
    182. 182. Go to the Dump Game 2 8 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1
    183. 183. Go to the Dump Game PinkCap, do you have a 6? 2 8 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1
    184. 184. Go to the Dump Game PinkCap, do you have a 6? Go to the dump. 2 8 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1
    185. 185. Go to the Dump Game 5 2 8 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1
    186. 186. Go to the Dump Game 2 8 5 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1
    187. 187. Go to the Dump Game YellowCap, do you have a 9? 1 9 2 8 5 4 6 7 3 2 2 7 9 5 4 9 Playing 1
    188. 188. Go to the Dump Game YellowCap, do you have a 9? 1 9 2 8 5 4 6 7 3 2 2 7 5 4 9 Playing 1
    189. 189. Go to the Dump Game YellowCap, do you have a 9? 1 9 2 8 5 4 6 7 3 2 2 7 5 4 9 Playing 1 9
    190. 190. Go to the Dump Game 1 9 1 9 5 4 6 7 3 2 2 7 5 4 9 Playing
    191. 191. Go to the Dump Game PinkCap is not out of the game. Her turn ends, but she takes 5 more cards. 1 9 5 4 6 7 3 2 2 7 5 4 9 Playing 2 9 1 7 7
    192. 192. Go to the Dump Game 6 5 1 Winner? 4 5 9 5
    193. 193. Go to the Dump Game No counting. Combine both stacks. Winner? 4 5 9 6 5 1
    194. 194. Go to the Dump Game Whose pile is the highest? Winner? 4 6 5 5 9 1
    195. 195. Go to the Dump Game No shuffling needed for next game. Next game
    196. 196. Part-Whole Circles Part-whole circles help children see relationships and solve problems.
    197. 197. Part-Whole Circles Whole Part-whole circles help children see relationships and solve problems.
    198. 198. Part-Whole Circles Whole Part-whole circles help children see relationships and solve problems. Part Part
    199. 199. Part-Whole Circles 10 If 10 is the whole
    200. 200. Part-Whole Circles 10 4 and 4 is one part,
    201. 201. Part-Whole Circles 10 4 What is the other part?
    202. 202. Part-Whole Circles 10 4 6 What is the other part?
    203. 203. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? A missing addend problem, considered very difficult for first graders. They can do it with Part-Whole Circles.
    204. 204. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole?
    205. 205. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole? 3
    206. 206. Part-Whole Circles 3 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole?
    207. 207. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole ? 5 3
    208. 208. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 5 3 What is the missing part?
    209. 209. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? What is the missing part? 5 3 2
    210. 210. Part-Whole Circles 5 3 2 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Write the equation.
    211. 211. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 Write the equation.
    212. 212. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 3 + 2 = 5 Write the equation.
    213. 213. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 3 + 2 = 5 5 – 3 = 2 Write the equation. Is this an addition or subtraction problem?
    214. 214. Part-Whole Circles Part-whole circles help young children solve problems. Writing equations do not.
    215. 215. Part-Whole Circles Do not try to help children solve story problems by teaching “key” words.
    216. 216. “ Math” Way of Naming Numbers
    217. 217. “ Math” Way of Naming Numbers 11 = ten 1
    218. 218. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2
    219. 219. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3
    220. 220. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4
    221. 221. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9
    222. 222. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.
    223. 223. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.
    224. 224. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.
    225. 225. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.
    226. 226. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 . . . . . . . . 99 = 9-ten 9
    227. 227. “ Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 Only numbers under 100 need to be said the “math” way.
    228. 228. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Shows how far children from 3 countries can count at ages 4, 5, and 6.
    229. 229. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Purple is Chinese. Note jump between ages 5 and 6.
    230. 230. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Dark green is Korean “math” way.
    231. 231. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Average Highest Number Counted Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Dotted green is everyday Korean; notice smaller jump between ages 5 and 6.
    232. 232. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Red is English speakers. They learn same amount between ages 4-5 and 5-6.
    233. 233. Math Way of Naming Numbers <ul><li>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.) </li></ul>
    234. 234. Math Way of Naming Numbers <ul><li>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.) </li></ul><ul><li>Asian children learn mathematics using the math way of counting. </li></ul>
    235. 235. Math Way of Naming Numbers <ul><li>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.) </li></ul><ul><li>Asian children learn mathematics using the math way of counting. </li></ul><ul><li>They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. </li></ul>
    236. 236. Math Way of Naming Numbers <ul><li>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.) </li></ul><ul><li>Asian children learn mathematics using the math way of counting. </li></ul><ul><li>They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. </li></ul><ul><li>Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense. </li></ul>
    237. 237. Math Way of Naming Numbers Compared to reading:
    238. 238. Math Way of Naming Numbers <ul><li>Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. </li></ul>Compared to reading:
    239. 239. Math Way of Naming Numbers <ul><li>Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. </li></ul><ul><li>Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way). </li></ul>Compared to reading:
    240. 240. 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 Researchers
    241. 241. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task:
    242. 242. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14.
    243. 243. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    244. 244. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    245. 245. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    246. 246. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    247. 247. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    248. 248. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    249. 249. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    250. 250. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.
    251. 251. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
    252. 252. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
    253. 253. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.
    254. 254. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens.
    255. 255. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens. The traditional names for 40, 60, 70, 80, and 90 follow a pattern.
    256. 256. Math Way of Naming Numbers Traditional names 6-ten = sixty The “ty” means tens.
    257. 257. Math Way of Naming Numbers Traditional names 3-ten = thirty “ Thir” also used in 1/3, 13 and 30.
    258. 258. Math Way of Naming Numbers Traditional names 5-ten = fifty “ Fif” also used in 1/5, 15 and 50.
    259. 259. Math Way of Naming Numbers Traditional names 2-ten = twenty Two used to be pronounced “twoo.”
    260. 260. Math Way of Naming Numbers Traditional names A word game fireplace place-fire Say the syllables backward. This is how we say the teen numbers.
    261. 261. Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news newspaper Say the syllables backward. This is how we say the teen numbers.
    262. 262. Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news box-mail mailbox newspaper Say the syllables backward. This is how we say the teen numbers.
    263. 263. Math Way of Naming Numbers Traditional names ten 4 “ Teen” also means ten.
    264. 264. Math Way of Naming Numbers Traditional names ten 4 teen 4 “ Teen” also means ten.
    265. 265. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen “ Teen” also means ten.
    266. 266. Math Way of Naming Numbers Traditional names a one left
    267. 267. Math Way of Naming Numbers Traditional names a one left a left-one
    268. 268. Math Way of Naming Numbers Traditional names a one left a left-one eleven
    269. 269. Math Way of Naming Numbers Traditional names two left Two pronounced “twoo.”
    270. 270. Math Way of Naming Numbers Traditional names two left twelve Two pronounced “twoo.”
    271. 271. Composing Numbers 3-ten
    272. 272. Composing Numbers 3-ten
    273. 273. Composing Numbers 3-ten 3 0
    274. 274. Composing Numbers 3-ten 3 0 Point to the 3 and say 3.
    275. 275. Composing Numbers 3-ten 3 0 Point to 0 and say 10. The 0 makes 3 a ten.
    276. 276. Composing Numbers 3-ten 7 3 0
    277. 277. Composing Numbers 3-ten 7 3 0
    278. 278. Composing Numbers 3-ten 7 3 0 7
    279. 279. Composing Numbers 3-ten 7 3 0 7 Place the 7 on top of the 0 of the 30.
    280. 280. Composing Numbers 3-ten 7 Notice the way we say the number, represent the number, and write the number all correspond. 3 0 7
    281. 281. Composing Numbers 7-ten
    282. 282. Composing Numbers 7-ten 70 is visualizable—again because of the fives’ grouping.
    283. 283. Composing Numbers 7-ten 7 0 70 is visualizable—again because of the fives’ grouping.
    284. 284. Composing Numbers 7-ten 8 7 0 70 is visualizable—again because of the fives’ grouping.
    285. 285. Composing Numbers 7-ten 8 7 0 70 is visualizable—again because of the fives’ grouping.
    286. 286. Composing Numbers 7-ten 8 7 0 8 70 is visualizable—again because of the fives grouping.
    287. 287. Composing Numbers 7-ten 8 7 8 8 Place the 8 on top of the 0 of the 70.
    288. 288. Composing Numbers 10-ten
    289. 289. Composing Numbers 10-ten 1 0 0
    290. 290. Composing Numbers 10-ten 1 0 0
    291. 291. Composing Numbers 10-ten 1 0 0
    292. 292. Composing Numbers 1 hundred
    293. 293. Composing Numbers 1 hundred 1 0 0
    294. 294. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred.
    295. 295. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred. 1 0 1 0
    296. 296. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred.
    297. 297. Composing Numbers 2 hundred
    298. 298. Composing Numbers 2 hundred 2 0 0 Just the edges of the abacuses are shown.
    299. 299. Composing Numbers 2 hundred 2 0 0 Just the edges of the abacuses are shown.
    300. 300. Composing Numbers 6 hundred 6 0 0 Maintaining the fives’ grouping.
    301. 301. Composing Numbers 10 hundred 1 0 0 0
    302. 302. Composing Numbers 10 hundred 1 0 0 0
    303. 303. Composing Numbers 1 thousand 1 0 0 0 Of course, we can also read it as one th-ou-sand.
    304. 304. Composing Numbers 1 thousand 1 0 0 0 Of course, we can also read it as one th-ou-sand.
    305. 305. Composing Numbers 1 thousand 1 0 0 0 Of course, we can also read it as one th-ou-sand.
    306. 306. Composing Numbers 1 thousand 1 0 0 0 Of course, we can also read it as one th-ou-sand.
    307. 307. Composing Numbers Reading numbers backward 2 5 8 4 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
    308. 308. Composing Numbers 2 5 8 4 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
    309. 309. Composing Numbers 2 5 8 4 2 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
    310. 310. Composing Numbers 2 5 8 4 2 5 8 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward
    311. 311. Fact Strategies
    312. 312. Fact Strategies <ul><li>A strategy is a way to learn a new fact or recall a forgotten fact. </li></ul>
    313. 313. Fact Strategies <ul><li>A strategy is a way to learn a new fact or recall a forgotten fact. </li></ul><ul><li>Powerful strategies are often visualizable representations. </li></ul>
    314. 314. Fact Strategies Complete the Ten 9 + 5 =
    315. 315. Fact Strategies Complete the Ten 9 + 5 =
    316. 316. Fact Strategies Complete the Ten 9 + 5 =
    317. 317. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
    318. 318. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9. Use two hands and move the bead simultaneously.
    319. 319. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
    320. 320. Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9.
    321. 321. Fact Strategies Two Fives 8 + 6 =
    322. 322. Fact Strategies Two Fives 8 + 6 =
    323. 323. Fact Strategies Two Fives 8 + 6 = Two fives make 10.
    324. 324. Fact Strategies Two Fives 8 + 6 = Just add the “leftovers.”
    325. 325. Fact Strategies Two Fives 8 + 6 = 10 + 4 = 14 Just add the “leftovers.”
    326. 326. Fact Strategies Two Fives 7 + 5 = Another example.
    327. 327. Fact Strategies Two Fives 7 + 5 =
    328. 328. Fact Strategies Two Fives 7 + 5 = 12
    329. 329. Fact Strategies Going Down 15 – 9 =
    330. 330. Fact Strategies Going Down 15 – 9 =
    331. 331. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
    332. 332. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
    333. 333. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
    334. 334. Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4.
    335. 335. Fact Strategies Subtract from 10 15 – 9 =
    336. 336. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
    337. 337. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
    338. 338. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
    339. 339. Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10.
    340. 340. Fact Strategies Going Up 13 – 9 =
    341. 341. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
    342. 342. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
    343. 343. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
    344. 344. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.
    345. 345. Fact Strategies Going Up 13 – 9 = 1 + 3 = 4 Start with 9; go up to 13.
    346. 346. Money Penny
    347. 347. Money Nickel
    348. 348. Money Dime
    349. 349. Money Quarter
    350. 350. Money Quarter
    351. 351. Money Quarter
    352. 352. Money Quarter
    353. 353. Trading Side Cleared 1000 10 1 100
    354. 354. Trading Side Thousands 1000 10 1 100
    355. 355. Trading Side Hundreds 1000 10 1 100 The third wire from each end is not used.
    356. 356. Trading Side Tens 1000 10 1 100 The third wire from each end is not used.
    357. 357. Trading Side Ones 1000 10 1 100 The third wire from each end is not used.
    358. 358. Trading Side Adding 8 + 6 1000 10 1 100
    359. 359. Trading Side Adding 8 + 6 1000 10 1 100
    360. 360. Trading Side Adding 8 + 6 1000 10 1 100
    361. 361. Trading Side Adding 8 + 6 1000 10 1 100
    362. 362. Trading Side Adding 8 + 6 14 1000 10 1 100
    363. 363. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. You can see the 10 ones (yellow). 1000 10 1 100
    364. 364. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1 100
    365. 365. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1 100
    366. 366. Trading Side Adding 8 + 6 14 Same answer before and after trading. 1000 10 1 100
    367. 367. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 1000 10 1 100
    368. 368. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7 1000 10 1 100
    369. 369. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7 1000 10 1 100
    370. 370. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100
    371. 371. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100
    372. 372. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100
    373. 373. Trading Side Bead Trading game 6 Trade 10 ones for 1 ten. 1000 10 1 100
    374. 374. Trading Side Bead Trading game 6 1000 10 1 100
    375. 375. Trading Side Bead Trading game 6 1000 10 1 100
    376. 376. Trading Side Bead Trading game 9 1000 10 1 100
    377. 377. Trading Side Bead Trading game 9 1000 10 1 100
    378. 378. Trading Side Bead Trading game 9 Another trade. 1000 10 1 100
    379. 379. Trading Side Bead Trading game 9 Another trade. 1000 10 1 100
    380. 380. Trading Side Bead Trading game 3 1000 10 1 100
    381. 381. Trading Side Bead Trading game 3 1000 10 1 100
    382. 382. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul>
    383. 383. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul><ul><ul><li>10 tens for 1 hundred, less often; </li></ul></ul>
    384. 384. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul><ul><ul><li>10 tens for 1 hundred, less often; </li></ul></ul><ul><ul><li>10 hundreds for 1 thousand, rarely. </li></ul></ul>
    385. 385. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul><ul><ul><li>10 tens for 1 hundred, less often; </li></ul></ul><ul><ul><li>10 hundreds for 1 thousand, rarely. </li></ul></ul><ul><li>Bead trading helps the child experience the greater value of each column from left to right. </li></ul>
    386. 386. Trading Side Bead Trading game <ul><li>In the Bead Trading game </li></ul><ul><ul><li>10 ones for 1 ten occurs frequently; </li></ul></ul><ul><ul><li>10 tens for 1 hundred, less often; </li></ul></ul><ul><ul><li>10 hundreds for 1 thousand, rarely. </li></ul></ul><ul><li>Bead trading helps the child experience the greater value of each column from left to right. </li></ul><ul><li>To detect a pattern, there must be at least three examples in the sequence. (Place value is a pattern.) </li></ul>
    387. 387. Trading Side Adding 4-digit numbers 3658 + 2738 1000 10 1 100
    388. 388. Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right. 1000 10 1 100
    389. 389. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100
    390. 390. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100
    391. 391. Trading Side Adding 4-digit numbers 3 6 58 + 2738 Enter the first number from left to right. 1000 10 1 100
    392. 392. Trading Side Adding 4-digit numbers 36 5 8 + 2738 Enter the first number from left to right. 1000 10 1 100
    393. 393. Trading Side Adding 4-digit numbers 365 8 + 2738 Enter the first number from left to right. 1000 10 1 100
    394. 394. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
    395. 395. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
    396. 396. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
    397. 397. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
    398. 398. Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. . . . 6 ones. Did anything else happen? 1000 10 1 100
    399. 399. Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1 Is it okay to show the extra ten by writing a 1 above the tens column? 1000 10 1 100
    400. 400. Trading Side Adding 4-digit numbers 3658 + 27 3 8 6 Add starting at the right. Write results after each step. 1 1000 10 1 100
    401. 401. Trading Side Adding 4-digit numbers 3658 + 27 3 8 6 Add starting at the right. Write results after each step. 1 Do we need to trade? [no] 1000 10 1 100
    402. 402. Trading Side Adding 4-digit numbers 3658 + 2738 9 6 Add starting at the right. Write results after each step. 1 1000 10 1 100
    403. 403. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 1000 10 1 100
    404. 404. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 Do we need to trade? [yes] 1000 10 1 100
    405. 405. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 Notice the number of yellow beads. [3] Notice the number of blue beads left. [3] Coincidence? No, because 13 – 10 = 3. 1000 10 1 100
    406. 406. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 1000 10 1 100
    407. 407. Trading Side Adding 4-digit numbers 3658 + 2738 3 96 Add starting at the right. Write results after each step. 1 1000 10 1 100
    408. 408. Trading Side Adding 4-digit numbers 3658 + 2738 3 96 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    409. 409. Trading Side Adding 4-digit numbers 3658 + 2 738 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    410. 410. Trading Side Adding 4-digit numbers 3658 + 2 738 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    411. 411. Trading Side Adding 4-digit numbers 3658 + 2738 6 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    412. 412. Trading Side Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100
    413. 413. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    414. 414. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    415. 415. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    416. 416. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    417. 417. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)
    418. 418. Multiplication on the AL Abacus Basic facts 9  3 =
    419. 419. Multiplication on the AL Abacus Basic facts 9  3 =
    420. 420. Multiplication on the AL Abacus Basic facts 9  3 = 30
    421. 421. Multiplication on the AL Abacus Basic facts 9  3 = 30 – 3 = 27
    422. 422. Multiplication on the AL Abacus Basic facts 4  8 =
    423. 423. Multiplication on the AL Abacus Basic facts 4  8 =
    424. 424. Multiplication on the AL Abacus Basic facts 4  8 =
    425. 425. Multiplication on the AL Abacus Basic facts 4  8 = 20 + 12 = 32
    426. 426. Multiplication on the AL Abacus Basic facts 7  7 =
    427. 427. Multiplication on the AL Abacus Basic facts 7  7 =
    428. 428. Multiplication on the AL Abacus Basic facts 7  7 = 25 + 10 + 10 + 4 = 49
    429. 429. Multiplication on the AL Abacus Commutative property 5  6 =
    430. 430. Multiplication on the AL Abacus Commutative property 5  6 =
    431. 431. Multiplication on the AL Abacus Commutative property 5  6 = 6  5
    432. 432. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 Recognizing multiples needed for fractions and algebra.
    433. 433. Multiples Patterns Twos 2 4 6 8 10 1 2 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    434. 434. Multiples Patterns Twos 2 4 6 8 10 12 1 4 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    435. 435. Multiples Patterns Twos 2 4 6 8 10 12 14 1 6 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    436. 436. Multiples Patterns Twos 2 4 6 8 10 12 14 16 1 8 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    437. 437. Multiples Patterns Twos 2 4 6 8 1 0 12 14 16 18 2 0 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.
    438. 438. Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40
    439. 439. Multiples Patterns Fours 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 The ones repeat in the second row.
    440. 440. 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
    441. 441. Multiples Patterns Sixes and Eights 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4 6 0 8 16 24 32 40 48 56 64 72 80
    442. 442. Multiples Patterns Sixes and Eights 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4 6 0 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4 7 2 8 0 Again the ones repeat in the second row.
    443. 443. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 4 0 48 56 64 72 8 0 The ones in the 8s show the multiples of 2.
    444. 444. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 3 2 4 0 48 56 64 7 2 8 0 The ones in the 8s show the multiples of 2.
    445. 445. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 2 4 3 2 4 0 48 56 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.
    446. 446. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 1 6 2 4 3 2 4 0 48 5 6 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.
    447. 447. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.
    448. 448. 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 6  4 6  4 is the fourth number (multiple).
    449. 449. 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).
    450. 450. 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.
    451. 451. Multiples Patterns Threes <ul><li>3 6 9 </li></ul><ul><li>15 18 </li></ul><ul><li>21 24 27 </li></ul><ul><li>30 </li></ul>The 3s have several patterns: Observe the ones.
    452. 452. Multiples Patterns Threes <ul><li>3 6 9 </li></ul><ul><li>15 18 </li></ul><ul><li>21 24 27 </li></ul><ul><li>3 0 </li></ul>The 3s have several patterns: Observe the ones.
    453. 453. Multiples Patterns Threes <ul><li>3 6 9 </li></ul><ul><li>15 18 </li></ul><ul><li>2 1 24 27 </li></ul><ul><li>3 0 </li></ul>The 3s have several patterns: Observe the ones.
    454. 454. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 24 27 3 0 The 3s have several patterns: Observe the ones.
    455. 455. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 24 27 3 0 The 3s have several patterns: Observe the ones.
    456. 456. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.
    457. 457. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.
    458. 458. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.
    459. 459. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.
    460. 460. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.
    461. 461. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.
    462. 462. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 30 The 3s have several patterns: The tens are the same in each row.
    463. 463. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
    464. 464. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
    465. 465. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.
    466. 466. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
    467. 467. Multiples Patterns Sevens 7 14 2 1 28 35 4 2 49 56 6 3 70 The 7s have the 1, 2, 3… pattern.
    468. 468. Multiples Patterns Sevens 7 1 4 2 1 28 3 5 4 2 49 5 6 6 3 70 The 7s have the 1, 2, 3… pattern.
    469. 469. Multiples Patterns Sevens 7 1 4 2 1 2 8 3 5 4 2 4 9 5 6 6 3 7 0 The 7s have the 1, 2, 3… pattern.
    470. 470. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.
    471. 471. Multiples Patterns Sevens 7 14 21 2 8 35 42 4 9 56 63 7 0 The 7s have the 1, 2, 3… pattern.
    472. 472. Research Highlights TASK EXPER CTRL TEENS 10 + 3 94% 47% 6 + 10 88% 33% CIRCLE TENS 78 75% 67% 3924 44% 7% 14 as 10 & 4 48 – 14 81% 33%
    473. 473. Research Highlights TASK EXPER CTRL 26-TASK (tens) 6 (ones) 94% 100% Other research questions asked. 2 (tens) 63% 13% MENTAL COMP: 85 – 70 31% 0% 2nd Graders in U.S. (Reys): 9% 38 + 24 = 512 or 0% 40% 57 + 35 = 812
    474. 474. Some Important Conclusions
    475. 475. Some Important Conclusions <ul><li>We need to use quantity, not counting words, and place value as the foundation of arithmetic. </li></ul>
    476. 476. Some Important Conclusions <ul><li>We need to use quantity, not counting words, and place value as the foundation of arithmetic. </li></ul><ul><li>We need to introduce the thousands much sooner to give children the big picture. </li></ul>
    477. 477. Some Important Conclusions <ul><li>We need to use quantity, not counting words, and place value as the foundation of arithmetic. </li></ul><ul><li>We need to introduce the thousands much sooner to give children the big picture. </li></ul><ul><li>Fostering visualization reduces the heavy memory load, allowing our disadvantaged youngsters to succeed. </li></ul>
    478. 478. Some Important Conclusions <ul><li>We need to use quantity, not counting words, and place value as the foundation of arithmetic. </li></ul><ul><li>We need to introduce the thousands much sooner to give children the big picture. </li></ul><ul><li>Fostering visualization reduces the heavy memory load, allowing our disadvantaged youngsters to succeed. </li></ul><ul><li>Games, not flash cards, not timed tests, are the best way to help our students understand, master, apply, and enjoy mathematics. </li></ul>
    479. 479. References <ul><li>Cotter, Joan. “Using Language and Visualization to Teach Place Value.” Teaching Children Mathematics 7 (October, 2000): 108-114. </li></ul><ul><li>Also reprinted in NCTM (National Council of Teachers of Mathematics) On-Math Journal and in Growing Professionally: Readings from NCTM Publications for Grades K-8 , in 2008. </li></ul>
    480. 480. Overcoming Obstacles Learning Arithmetic through Visualizing with the AL Abacus IDA-UMB Conference March 12, 2011 Saint Paul, Minnesota by Joan A. Cotter, Ph.D. [email_address] 7 5 2 VII
    481. 481. Overcoming Obstacles Learning Arithmetic through Visualizing with the AL Abacus IDA-UMB Conference March 12, 2011 Saint Paul, Minnesota by Joan A. Cotter, Ph.D. [email_address] 7 5 2 (PowerPoint is available on Alabacus.com under Resources.) VII
    482. 482. ABACUS WORK PAGE 1000 10 1 100

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