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- 1. The Future of Primary Math: More Understanding/Less Counting by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com MCTM Saturday, May 5, 2012 Duluth, Minnesota 1000 100 10 1 30 7 30 7 PowerPoint PresentationRightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012
- 2. Verbal Counting Model2 © Joan A. Cotter, Ph.D., 2012
- 3. 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 forth3 © Joan A. Cotter, Ph.D., 2012
- 4. Verbal Counting Model From a childs perspective F +E4 © Joan A. Cotter, Ph.D., 2012
- 5. Verbal Counting Model From a childs perspective F +E A5 © Joan A. Cotter, Ph.D., 2012
- 6. Verbal Counting Model From a childs perspective F +E A B6 © Joan A. Cotter, Ph.D., 2012
- 7. Verbal Counting Model From a childs perspective F +E A B C7 © Joan A. Cotter, Ph.D., 2012
- 8. Verbal Counting Model From a childs perspective F +E A B C D E F8 © Joan A. Cotter, Ph.D., 2012
- 9. Verbal Counting Model From a childs perspective F +E A B C D E F A9 © Joan A. Cotter, Ph.D., 2012
- 10. Verbal Counting Model From a childs perspective F +E A B C D E F A B10 © Joan A. Cotter, Ph.D., 2012
- 11. Verbal Counting Model From a childs perspective F +E A B C D E F A B C D E11 © Joan A. Cotter, Ph.D., 2012
- 12. 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.)12 © Joan A. Cotter, Ph.D., 2012
- 13. Verbal Counting Model From a childs perspective F +E K A B C D E F G H I J K13 © Joan A. Cotter, Ph.D., 2012
- 14. Verbal Counting Model From a childs perspective Now memorize the facts!! G +D14 © Joan A. Cotter, Ph.D., 2012
- 15. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D15 © Joan A. Cotter, Ph.D., 2012
- 16. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D +C16 © Joan A. Cotter, Ph.D., 2012
- 17. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D C +C +G17 © Joan A. Cotter, Ph.D., 2012
- 18. Verbal Counting Model From a childs perspective Now memorize the facts!! H E + G I F + +D D C +C +G18 © Joan A. Cotter, Ph.D., 2012
- 19. Verbal Counting Model From a childs perspective H –E Subtract with your fingers by counting backward.19 © Joan A. Cotter, Ph.D., 2012
- 20. Verbal Counting Model From a childs perspective J –F Subtract without using your fingers.20 © Joan A. Cotter, Ph.D., 2012
- 21. Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T.21 © Joan A. Cotter, Ph.D., 2012
- 22. Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T. What is D × E?22 © Joan A. Cotter, Ph.D., 2012
- 23. Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s23 © Joan A. Cotter, Ph.D., 2012
- 24. Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s huh?24 © Joan A. Cotter, Ph.D., 2012
- 25. Verbal Counting Model From a childs perspective L (twelve) is written AB because it is A J and B A’s25 © Joan A. Cotter, Ph.D., 2012
- 26. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J and B A’s26 © Joan A. Cotter, Ph.D., 2012
- 27. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s27 © Joan A. Cotter, Ph.D., 2012
- 28. 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).28 © Joan A. Cotter, Ph.D., 2012
- 29. 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 3129 © Joan A. Cotter, Ph.D., 2012
- 30. Calendar Math Calendar Counting 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 3130 © Joan A. Cotter, Ph.D., 2012
- 31. Calendar Math Calendar Counting 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 3131 © Joan A. Cotter, Ph.D., 2012
- 32. Calendar Math Calendar Counting 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 3132 © Joan A. Cotter, Ph.D., 2012
- 33. Calendar Math Septemb Calendar Counting 1234567 August 89101214 1 2 113 11921 15112628 8 122820 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 3133 © Joan A. Cotter, Ph.D., 2012
- 34. Calendar Math Septemb Calendar Counting 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 This is ordinal counting, not cardinal counting.34 © Joan A. Cotter, Ph.D., 2012
- 35. Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 1035 © Joan A. Cotter, Ph.D., 2012
- 36. Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 10 Children need the whole month to plan ahead.36 © Joan A. Cotter, Ph.D., 2012
- 37. Calendar Math Septemb Calendar Patterning 1234567 August 89101214 1 2 113 11921 15112628 8 122820 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 Patterns are rarely based on 7s or proceed row by row. Patterns go on forever; they don’t stop at 31.37 © Joan A. Cotter, Ph.D., 2012
- 38. Minnesota Standards Number Sense K: Represent quantities using whole numbers and understand relationships among whole numbers. 1–2: Understand place value and relationships among whole numbers. With the counting model, how difficult are the associated benchmarks for children to master?38 © Joan A. Cotter, Ph.D., 2012
- 39. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers.39 © Joan A. Cotter, Ph.D., 2012
- 40. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10.40 © Joan A. Cotter, Ph.D., 2012
- 41. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity.41 © Joan A. Cotter, Ph.D., 2012
- 42. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets.42 © Joan A. Cotter, Ph.D., 2012
- 43. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less.43 © Joan A. Cotter, Ph.D., 2012
- 44. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting.44 © Joan A. Cotter, Ph.D., 2012
- 45. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. • Add and subtract whole numbers up to 6, using objects.45 © Joan A. Cotter, Ph.D., 2012
- 46. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers.46 © Joan A. Cotter, Ph.D., 2012
- 47. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120.47 © Joan A. Cotter, Ph.D., 2012
- 48. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120.48 © Joan A. Cotter, Ph.D., 2012
- 49. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30.49 © Joan A. Cotter, Ph.D., 2012
- 50. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12.50 © Joan A. Cotter, Ph.D., 2012
- 51. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. • Represent whole numbers up to 20 in various ways.51 © Joan A. Cotter, Ph.D., 2012
- 52. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers.52 © Joan A. Cotter, Ph.D., 2012
- 53. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999.53 © Joan A. Cotter, Ph.D., 2012
- 54. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number.54 © Joan A. Cotter, Ph.D., 2012
- 55. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten.55 © Joan A. Cotter, Ph.D., 2012
- 56. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12.56 © Joan A. Cotter, Ph.D., 2012
- 57. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. • Represent whole numbers up to 20 in various ways.57 © Joan A. Cotter, Ph.D., 2012
- 58. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
- 59. Research on Counting Karen Wynn’s research © 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 Karen Wynn’s research63 © Joan A. Cotter, Ph.D., 2012
- 64. Research on Counting Karen Wynn’s research64 © Joan A. Cotter, Ph.D., 2012
- 65. Research on Counting Karen Wynn’s research65 © Joan A. Cotter, Ph.D., 2012
- 66. Research on Counting Other research66 © Joan A. Cotter, Ph.D., 2012
- 67. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.67 © Joan A. Cotter, Ph.D., 2012
- 68. 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, 2008.68 © Joan A. Cotter, Ph.D., 2012
- 69. 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, 2008.69 © Joan A. Cotter, Ph.D., 2012
- 70. 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, 2009.70 © Joan A. Cotter, Ph.D., 2012
- 71. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding.71 © Joan A. Cotter, Ph.D., 2012
- 72. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. • They consistently group in 5s.72 © Joan A. Cotter, Ph.D., 2012
- 73. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting.73 © 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.74 © Joan A. Cotter, Ph.D., 2012
- 75. 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.—Butterworth75 © Joan A. Cotter, Ph.D., 2012
- 76. 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.”—Benoit76 © Joan A. Cotter, Ph.D., 2012
- 77. 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.— Glasersfeld77 © Joan A. Cotter, Ph.D., 2012
- 78. Visualizing Quantities78 © Joan A. Cotter, Ph.D., 2012
- 79. Visualizing Quantities “Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 200979 © Joan A. Cotter, Ph.D., 2012
- 80. Visualizing Quantities “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 others80 © Joan A. Cotter, Ph.D., 2012
- 81. Visualizing Quantities 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
- 82. Visualizing Quantities Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
- 83. Visualizing Quantities Visualizing also needed in:• Reading • Architecture• Sports • Astronomy• Creativity • Archeology• Geography • Chemistry• Engineering • Physics• Construction • Surgery © Joan A. Cotter, Ph.D., 2012
- 84. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
- 85. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
- 86. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
- 87. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
- 88. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
- 89. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
- 90. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
- 91. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
- 92. Visualizing Quantities Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIII © Joan A. Cotter, Ph.D., 2012
- 93. Visualizing Quantities : Who could read the music?93 © Joan A. Cotter, Ph.D., 2012
- 94. AN ALTERNATIVE to learning place value: Subitizing (groups of five) Math Way (of number naming) Place Value Cards Trading (with 4-digit numbers)94 © Joan A. Cotter, Ph.D., 2012
- 95. Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
- 96. Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
- 97. Grouping in Fives Using fingers97 © Joan A. Cotter, Ph.D., 2012
- 98. Grouping in Fives Using fingers98 © Joan A. Cotter, Ph.D., 2012
- 99. Grouping in Fives Using fingers99 © Joan A. Cotter, Ph.D., 2012
- 100. Grouping in Fives Using fingers100 © Joan A. Cotter, Ph.D., 2012
- 101. Grouping in Fives 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
- 102. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
- 103. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
- 104. Grouping in Fives Recognizing 55 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
- 105. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
- 106. Grouping in Fives Tally sticks106 © Joan A. Cotter, Ph.D., 2012
- 107. Grouping in Fives Tally sticks107 © Joan A. Cotter, Ph.D., 2012
- 108. Grouping in Fives Tally sticks108 © Joan A. Cotter, Ph.D., 2012
- 109. Grouping in Fives Tally sticks109 © Joan A. Cotter, Ph.D., 2012
- 110. Grouping in Fives Tally sticks110 © Joan A. Cotter, Ph.D., 2012
- 111. Grouping in Fives Entering quantities © Joan A. Cotter, Ph.D., 2012
- 112. Grouping in Fives Entering quantities3 © Joan A. Cotter, Ph.D., 2012
- 113. Grouping in Fives Entering quantities 5113 © Joan A. Cotter, Ph.D., 2012
- 114. Grouping in Fives Entering quantities 7114 © Joan A. Cotter, Ph.D., 2012
- 115. Grouping in Fives Entering quantities 10115 © Joan A. Cotter, Ph.D., 2012
- 116. Grouping in Fives The stairs116 © Joan A. Cotter, Ph.D., 2012
- 117. Grouping in Fives Adding © Joan A. Cotter, Ph.D., 2012
- 118. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
- 119. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
- 120. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
- 121. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
- 122. Grouping in Fives Adding 4+3=7 © Joan A. Cotter, Ph.D., 2012
- 123. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
- 124. Go to the Dump Game Objective: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5124 © Joan A. Cotter, Ph.D., 2012
- 125. 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.125 © Joan A. Cotter, Ph.D., 2012
- 126. Go to the Dump Game 6+ = 10126 © Joan A. Cotter, Ph.D., 2012
- 127. “Math” Way of Naming Numbers127 © Joan A. Cotter, Ph.D., 2012
- 128. “Math” Way of Naming Numbers 11 = ten 1128 © Joan A. Cotter, Ph.D., 2012
- 129. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2129 © Joan A. Cotter, Ph.D., 2012
- 130. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3130 © Joan A. Cotter, Ph.D., 2012
- 131. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4131 © Joan A. Cotter, Ph.D., 2012
- 132. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 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 13 = ten 3 14 = ten 4 .... 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 14 = ten 4 .... 19 = ten 9134 © Joan A. Cotter, Ph.D., 2012
- 135. “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 9135 © Joan A. Cotter, Ph.D., 2012
- 136. “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 9136 © Joan A. Cotter, Ph.D., 2012
- 137. “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 9137 © Joan A. Cotter, Ph.D., 2012
- 138. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7138 © Joan A. Cotter, Ph.D., 2012
- 139. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7139 © 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 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, 319-332.142 © Joan A. Cotter, Ph.D., 2012
- 143. “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, 319-332.143 © Joan A. Cotter, Ph.D., 2012
- 144. “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, 319-332.144 © 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.)145 © Joan A. Cotter, Ph.D., 2012
- 146. 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.146 © Joan A. Cotter, Ph.D., 2012
- 147. 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.147 © Joan A. Cotter, Ph.D., 2012
- 148. 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.148 © Joan A. Cotter, Ph.D., 2012
- 149. Math Way of Naming Numbers Compared to reading:149 © Joan A. Cotter, Ph.D., 2012
- 150. Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic.150 © Joan A. Cotter, Ph.D., 2012
- 151. 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).151 © Joan A. Cotter, Ph.D., 2012
- 152. 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 Researchers152 © Joan A. Cotter, Ph.D., 2012
- 153. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
- 154. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
- 155. Math Way of Naming Numbers Traditional names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
- 156. Math Way of Naming Numbers Traditional names3-ten = thirty“Thir” alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
- 157. Math Way of Naming Numbers Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
- 158. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
- 159. Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
- 160. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
- 161. 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
- 162. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
- 163. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
- 164. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
- 165. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
- 166. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
- 167. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
- 168. Math Way of Naming Numbers Traditional names two leftTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
- 169. Math Way of Naming Numbers Traditional names two left twelveTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
- 170. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
- 171. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
- 172. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
- 173. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
- 174. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
- 175. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
- 176. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
- 177. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
- 178. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
- 179. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
- 180. 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
- 181. Composing Numbers7-ten70 Another example. © Joan A. Cotter, Ph.D., 2012
- 182. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
- 183. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
- 184. Composing Numbers7-ten 870 8 © Joan A. Cotter, Ph.D., 2012
- 185. Composing Numbers7-ten 878 © Joan A. Cotter, Ph.D., 2012
- 186. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
- 187. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
- 188. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
- 189. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
- 190. Composing Numbers1 hundred © Joan A. Cotter, Ph.D., 2012
- 191. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
- 192. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
- 193. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
- 194. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
- 195. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
- 196. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
- 197. Composing Numbers2 hundred200 © Joan A. Cotter, Ph.D., 2012
- 198. Counting by 2s and 5s © Joan A. Cotter, Ph.D., 2012
- 199. Counting by 2s and 5s Counting by 2s © Joan A. Cotter, Ph.D., 2012
- 200. Counting by 2s and 5s Counting by 2s2 © Joan A. Cotter, Ph.D., 2012
- 201. Counting by 2s and 5s Counting by 2s2 4 © Joan A. Cotter, Ph.D., 2012
- 202. Counting by 2s and 5s Counting by 2s2 4 6 © Joan A. Cotter, Ph.D., 2012
- 203. Counting by 2s and 5s Counting by 2s2 4 6 8 © Joan A. Cotter, Ph.D., 2012
- 204. Counting by 2s and 5s Counting by 2s2 4 6 8 10 © Joan A. Cotter, Ph.D., 2012
- 205. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 © Joan A. Cotter, Ph.D., 2012
- 206. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 © Joan A. Cotter, Ph.D., 2012
- 207. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 16 © Joan A. Cotter, Ph.D., 2012
- 208. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 16 18 © Joan A. Cotter, Ph.D., 2012
- 209. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 16 18 20 © Joan A. Cotter, Ph.D., 2012
- 210. Counting by 2s and 5s Counting by 5s © Joan A. Cotter, Ph.D., 2012
- 211. Counting by 2s and 5s Counting by 5s5 © Joan A. Cotter, Ph.D., 2012
- 212. Counting by 2s and 5s Counting by 5s5 10 © Joan A. Cotter, Ph.D., 2012
- 213. Counting by 2s and 5s Counting by 5s 5 1015 © Joan A. Cotter, Ph.D., 2012
- 214. Counting by 2s and 5s Counting by 5s 5 1015 20 © Joan A. Cotter, Ph.D., 2012
- 215. Counting by 2s and 5s Counting by 5s 5 1015 2025 © Joan A. Cotter, Ph.D., 2012
- 216. Counting by 2s and 5s Counting by 5s 5 1015 2025 30 © Joan A. Cotter, Ph.D., 2012
- 217. Evens and Odds Evens © Joan A. Cotter, Ph.D., 2012
- 218. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 219. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 220. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 221. Evens and Odds Evens Use two fingers and touch each pair in succession. EVEN! © Joan A. Cotter, Ph.D., 2012
- 222. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 223. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 224. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 225. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
- 226. Evens and Odds Odds Use two fingers and touch each pair in succession. ODD! © Joan A. Cotter, Ph.D., 2012
- 227. Fact Strategies227 © Joan A. Cotter, Ph.D., 2012
- 228. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
- 229. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
- 230. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
- 231. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
- 232. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
- 233. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
- 234. Fact Strategies Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
- 235. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
- 236. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
- 237. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
- 238. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
- 239. Fact Strategies Two Fives8+6=10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
- 240. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
- 241. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
- 242. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
- 243. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
- 244. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
- 245. Fact Strategies Going Down 15 – 9 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
- 246. Fact Strategies Subtract from 1015 – 9 = © Joan A. Cotter, Ph.D., 2012
- 247. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
- 248. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
- 249. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
- 250. Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
- 251. Fact Strategies Going Up15 – 9 = © Joan A. Cotter, Ph.D., 2012
- 252. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 253. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 254. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 255. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 256. Fact Strategies Going Up 15 – 9 = 1+5=6Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
- 257. Fact Strategies Multiplication6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
- 258. Fact Strategies Multiplication6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
- 259. Place Value Two aspects © Joan A. Cotter, Ph.D., 2012
- 260. Place Value Two aspectsStatic © Joan A. Cotter, Ph.D., 2012
- 261. Place Value Two aspectsStatic • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
- 262. 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
- 263. 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
- 264. 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
- 265. 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
- 266. 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
- 267. Trading1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 268. Trading Thousands1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 269. Trading Hundreds1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 270. Trading Tens1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 271. Trading Ones1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 272. Trading Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
- 273. Trading Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
- 274. Trading Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
- 275. Trading Adding1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
- 276. Trading Adding1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
- 277. Trading Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
- 278. Trading Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
- 279. Trading Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
- 280. Trading Adding1000 100 10 1 8 +6 14 Same answer before and after trading. © Joan A. Cotter, Ph.D., 2012
- 281. Trading Bead Trading Activity1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
- 282. Trading Bead Trading Activity1000 100 10 1 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
- 283. Trading Bead Trading Activity1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
- 284. Trading Bead Trading Activity1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
- 285. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
- 286. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
- 287. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
- 288. Trading Bead Trading Activity1000 100 10 1 6 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
- 289. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
- 290. Trading Bead Trading Activity1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
- 291. Trading Bead Trading Activity1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
- 292. Trading Bead Trading Activity1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
- 293. Trading Bead Trading Activity1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
- 294. Trading Bead Trading Activity1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
- 295. Trading Bead Trading Activity1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
- 296. Trading Bead Trading Activity1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
- 297. Trading Bead Trading Activity• In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; © Joan A. Cotter, Ph.D., 2012
- 298. Trading Bead Trading Activity• In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; © Joan A. Cotter, Ph.D., 2012
- 299. Trading Bead Trading Activity• In the Bead Trading activity 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
- 300. Trading Bead Trading Activity• In the Bead Trading activity 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
- 301. Trading Bead Trading Activity• In the Bead Trading activity 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
- 302. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
- 303. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
- 304. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
- 305. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
- 306. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
- 307. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
- 308. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
- 309. Trading 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
- 310. Trading 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
- 311. Trading 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
- 312. Trading 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
- 313. Trading 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
- 314. Trading 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
- 315. Trading 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
- 316. Trading 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
- 317. Trading 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
- 318. Trading 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
- 319. Trading 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
- 320. Trading 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
- 321. Trading 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
- 322. Trading 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
- 323. Trading 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
- 324. Trading 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
- 325. Trading 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
- 326. Trading 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
- 327. Trading 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
- 328. Minnesota Standards Number Sense K: Represent quantities using whole numbers and understand relationships among whole numbers. 1–2: Understand place value and relationships among whole numbers. With this alternate model, how difficult are the associated benchmarks for children to master?328 © Joan A. Cotter, Ph.D., 2012
- 329. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10.329 © Joan A. Cotter, Ph.D., 2012
- 330. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity.330 © Joan A. Cotter, Ph.D., 2012
- 331. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets.331 © Joan A. Cotter, Ph.D., 2012
- 332. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less.332 © Joan A. Cotter, Ph.D., 2012
- 333. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting.333 © Joan A. Cotter, Ph.D., 2012
- 334. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. • Add and subtract whole numbers up to 6, using objects.334 © Joan A. Cotter, Ph.D., 2012
- 335. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120.335 © Joan A. Cotter, Ph.D., 2012
- 336. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120.336 © Joan A. Cotter, Ph.D., 2012
- 337. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30.337 © Joan A. Cotter, Ph.D., 2012
- 338. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12.338 © Joan A. Cotter, Ph.D., 2012
- 339. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. • Represent whole numbers up to 20 in various ways.339 © Joan A. Cotter, Ph.D., 2012
- 340. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999.340 © Joan A. Cotter, Ph.D., 2012
- 341. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number.341 © Joan A. Cotter, Ph.D., 2012
- 342. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten.342 © Joan A. Cotter, Ph.D., 2012
- 343. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12.343 © Joan A. Cotter, Ph.D., 2012
- 344. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. • Represent whole numbers up to 20 in various ways.344 © Joan A. Cotter, Ph.D., 2012
- 345. Research Highlights345 © Joan A. Cotter, Ph.D., 2012

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