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# MCTM Future Primary Math

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### MCTM Future Primary Math

1. 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. 2. Verbal Counting Model2 © Joan A. Cotter, Ph.D., 2012
3. 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. 4. Verbal Counting Model From a childs perspective F +E4 © Joan A. Cotter, Ph.D., 2012
5. 5. Verbal Counting Model From a childs perspective F +E A5 © Joan A. Cotter, Ph.D., 2012
6. 6. Verbal Counting Model From a childs perspective F +E A B6 © Joan A. Cotter, Ph.D., 2012
7. 7. Verbal Counting Model From a childs perspective F +E A B C7 © Joan A. Cotter, Ph.D., 2012
8. 8. Verbal Counting Model From a childs perspective F +E A B C D E F8 © Joan A. Cotter, Ph.D., 2012
9. 9. Verbal Counting Model From a childs perspective F +E A B C D E F A9 © Joan A. Cotter, Ph.D., 2012
10. 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. 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. 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. 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. 14. Verbal Counting Model From a childs perspective Now memorize the facts!! G +D14 © Joan A. Cotter, Ph.D., 2012
15. 15. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D15 © Joan A. Cotter, Ph.D., 2012
16. 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. 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. 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. 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. 20. Verbal Counting Model From a childs perspective J –F Subtract without using your fingers.20 © Joan A. Cotter, Ph.D., 2012
21. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 35. Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 1035 © Joan A. Cotter, Ph.D., 2012
36. 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. 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. 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. 39. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers.39 © Joan A. Cotter, Ph.D., 2012
40. 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. 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. 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. 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. 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. 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. 46. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers.46 © Joan A. Cotter, Ph.D., 2012
47. 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. 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. 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. 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. 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. 52. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers.52 © Joan A. Cotter, Ph.D., 2012
53. 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. 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. 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. 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. 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. 58. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
59. 59. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
60. 60. Research on Counting Karen Wynn’s research60 © Joan A. Cotter, Ph.D., 2012
61. 61. Research on Counting Karen Wynn’s research61 © Joan A. Cotter, Ph.D., 2012
62. 62. Research on Counting Karen Wynn’s research62 © Joan A. Cotter, Ph.D., 2012
63. 63. Research on Counting Karen Wynn’s research63 © Joan A. Cotter, Ph.D., 2012
64. 64. Research on Counting Karen Wynn’s research64 © Joan A. Cotter, Ph.D., 2012
65. 65. Research on Counting Karen Wynn’s research65 © Joan A. Cotter, Ph.D., 2012
66. 66. Research on Counting Other research66 © Joan A. Cotter, Ph.D., 2012
67. 67. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.67 © Joan A. Cotter, Ph.D., 2012
68. 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. 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. 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. 71. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding.71 © Joan A. Cotter, Ph.D., 2012
72. 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. 73. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting.73 © Joan A. Cotter, Ph.D., 2012
74. 74. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth.74 © Joan A. Cotter, Ph.D., 2012
75. 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. 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. 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. 78. Visualizing Quantities78 © Joan A. Cotter, Ph.D., 2012
79. 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. 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. 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. 82. Visualizing Quantities Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
83. 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. 84. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
85. 85. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
86. 86. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
87. 87. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
88. 88. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
89. 89. Visualizing QuantitiesTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
90. 90. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
91. 91. Visualizing QuantitiesNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
92. 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. 93. Visualizing Quantities : Who could read the music?93 © Joan A. Cotter, Ph.D., 2012
94. 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. 95. Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
96. 96. Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
97. 97. Grouping in Fives Using fingers97 © Joan A. Cotter, Ph.D., 2012
98. 98. Grouping in Fives Using fingers98 © Joan A. Cotter, Ph.D., 2012
99. 99. Grouping in Fives Using fingers99 © Joan A. Cotter, Ph.D., 2012
100. 100. Grouping in Fives Using fingers100 © Joan A. Cotter, Ph.D., 2012
101. 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. 102. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
103. 103. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
104. 104. Grouping in Fives Recognizing 55 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
105. 105. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
106. 106. Grouping in Fives Tally sticks106 © Joan A. Cotter, Ph.D., 2012
107. 107. Grouping in Fives Tally sticks107 © Joan A. Cotter, Ph.D., 2012
108. 108. Grouping in Fives Tally sticks108 © Joan A. Cotter, Ph.D., 2012
109. 109. Grouping in Fives Tally sticks109 © Joan A. Cotter, Ph.D., 2012
110. 110. Grouping in Fives Tally sticks110 © Joan A. Cotter, Ph.D., 2012
111. 111. Grouping in Fives Entering quantities © Joan A. Cotter, Ph.D., 2012
112. 112. Grouping in Fives Entering quantities3 © Joan A. Cotter, Ph.D., 2012
113. 113. Grouping in Fives Entering quantities 5113 © Joan A. Cotter, Ph.D., 2012
114. 114. Grouping in Fives Entering quantities 7114 © Joan A. Cotter, Ph.D., 2012
115. 115. Grouping in Fives Entering quantities 10115 © Joan A. Cotter, Ph.D., 2012
116. 116. Grouping in Fives The stairs116 © Joan A. Cotter, Ph.D., 2012
117. 117. Grouping in Fives Adding © Joan A. Cotter, Ph.D., 2012
118. 118. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
119. 119. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
120. 120. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
121. 121. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
122. 122. Grouping in Fives Adding 4+3=7 © Joan A. Cotter, Ph.D., 2012
123. 123. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
124. 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. 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. 126. Go to the Dump Game 6+ = 10126 © Joan A. Cotter, Ph.D., 2012
127. 127. “Math” Way of Naming Numbers127 © Joan A. Cotter, Ph.D., 2012
128. 128. “Math” Way of Naming Numbers 11 = ten 1128 © Joan A. Cotter, Ph.D., 2012
129. 129. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2129 © Joan A. Cotter, Ph.D., 2012
130. 130. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3130 © Joan A. Cotter, Ph.D., 2012
131. 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. 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. 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. 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. 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. 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. 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. 138. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7138 © Joan A. Cotter, Ph.D., 2012
139. 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. 140. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.140 © Joan A. Cotter, Ph.D., 2012
141. 141. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.141 © Joan A. Cotter, Ph.D., 2012
142. 142. “Math” Way of Naming Numbers 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. 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. 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. 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. 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. 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. 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. 149. Math Way of Naming Numbers Compared to reading:149 © Joan A. Cotter, Ph.D., 2012
150. 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. 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. 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. 153. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
154. 154. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
155. 155. Math Way of Naming Numbers Traditional names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
156. 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. 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. 158. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
159. 159. Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
160. 160. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
161. 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. 162. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
163. 163. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
164. 164. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
165. 165. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
166. 166. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
167. 167. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
168. 168. Math Way of Naming Numbers Traditional names two leftTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
169. 169. Math Way of Naming Numbers Traditional names two left twelveTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
170. 170. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
171. 171. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
172. 172. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
173. 173. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
174. 174. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
175. 175. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
176. 176. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
177. 177. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
178. 178. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
179. 179. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
180. 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. 181. Composing Numbers7-ten70 Another example. © Joan A. Cotter, Ph.D., 2012
182. 182. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
183. 183. Composing Numbers7-ten 870 © Joan A. Cotter, Ph.D., 2012
184. 184. Composing Numbers7-ten 870 8 © Joan A. Cotter, Ph.D., 2012
185. 185. Composing Numbers7-ten 878 © Joan A. Cotter, Ph.D., 2012
186. 186. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
187. 187. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
188. 188. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
189. 189. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
190. 190. Composing Numbers1 hundred © Joan A. Cotter, Ph.D., 2012
191. 191. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
192. 192. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
193. 193. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
194. 194. Composing Numbers1 hundred100 © Joan A. Cotter, Ph.D., 2012
195. 195. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
196. 196. Composing Numbers2 hundred © Joan A. Cotter, Ph.D., 2012
197. 197. Composing Numbers2 hundred200 © Joan A. Cotter, Ph.D., 2012
198. 198. Counting by 2s and 5s © Joan A. Cotter, Ph.D., 2012
199. 199. Counting by 2s and 5s Counting by 2s © Joan A. Cotter, Ph.D., 2012
200. 200. Counting by 2s and 5s Counting by 2s2 © Joan A. Cotter, Ph.D., 2012
201. 201. Counting by 2s and 5s Counting by 2s2 4 © Joan A. Cotter, Ph.D., 2012
202. 202. Counting by 2s and 5s Counting by 2s2 4 6 © Joan A. Cotter, Ph.D., 2012
203. 203. Counting by 2s and 5s Counting by 2s2 4 6 8 © Joan A. Cotter, Ph.D., 2012
204. 204. Counting by 2s and 5s Counting by 2s2 4 6 8 10 © Joan A. Cotter, Ph.D., 2012
205. 205. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 © Joan A. Cotter, Ph.D., 2012
206. 206. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 © Joan A. Cotter, Ph.D., 2012
207. 207. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 16 © Joan A. Cotter, Ph.D., 2012
208. 208. Counting by 2s and 5s Counting by 2s 2 4 6 8 1012 14 16 18 © Joan A. Cotter, Ph.D., 2012
209. 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. 210. Counting by 2s and 5s Counting by 5s © Joan A. Cotter, Ph.D., 2012
211. 211. Counting by 2s and 5s Counting by 5s5 © Joan A. Cotter, Ph.D., 2012
212. 212. Counting by 2s and 5s Counting by 5s5 10 © Joan A. Cotter, Ph.D., 2012
213. 213. Counting by 2s and 5s Counting by 5s 5 1015 © Joan A. Cotter, Ph.D., 2012
214. 214. Counting by 2s and 5s Counting by 5s 5 1015 20 © Joan A. Cotter, Ph.D., 2012
215. 215. Counting by 2s and 5s Counting by 5s 5 1015 2025 © Joan A. Cotter, Ph.D., 2012
216. 216. Counting by 2s and 5s Counting by 5s 5 1015 2025 30 © Joan A. Cotter, Ph.D., 2012
217. 217. Evens and Odds Evens © Joan A. Cotter, Ph.D., 2012
218. 218. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
219. 219. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
220. 220. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
221. 221. Evens and Odds Evens Use two fingers and touch each pair in succession. EVEN! © Joan A. Cotter, Ph.D., 2012
222. 222. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
223. 223. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
224. 224. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
225. 225. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
226. 226. Evens and Odds Odds Use two fingers and touch each pair in succession. ODD! © Joan A. Cotter, Ph.D., 2012
227. 227. Fact Strategies227 © Joan A. Cotter, Ph.D., 2012
228. 228. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
229. 229. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
230. 230. Fact Strategies Complete the Ten9+5= © Joan A. Cotter, Ph.D., 2012
231. 231. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
232. 232. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
233. 233. Fact Strategies Complete the Ten 9+5=Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
234. 234. Fact Strategies Complete the Ten 9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. © Joan A. Cotter, Ph.D., 2012
235. 235. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
236. 236. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
237. 237. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
238. 238. Fact Strategies Two Fives8+6= © Joan A. Cotter, Ph.D., 2012
239. 239. Fact Strategies Two Fives8+6=10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
240. 240. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
241. 241. Fact Strategies Going Down15 – 9 = © Joan A. Cotter, Ph.D., 2012
242. 242. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
243. 243. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
244. 244. Fact Strategies Going Down 15 – 9 =Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
245. 245. Fact Strategies Going Down 15 – 9 = 6Subtract 5;then 4. © Joan A. Cotter, Ph.D., 2012
246. 246. Fact Strategies Subtract from 1015 – 9 = © Joan A. Cotter, Ph.D., 2012
247. 247. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
248. 248. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
249. 249. Fact Strategies Subtract from 10 15 – 9 =Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
250. 250. Fact Strategies Subtract from 10 15 – 9 = 6Subtract 9from 10. © Joan A. Cotter, Ph.D., 2012
251. 251. Fact Strategies Going Up15 – 9 = © Joan A. Cotter, Ph.D., 2012
252. 252. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
253. 253. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
254. 254. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
255. 255. Fact Strategies Going Up 15 – 9 =Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
256. 256. Fact Strategies Going Up 15 – 9 = 1+5=6Start with 9;go up to 15. © Joan A. Cotter, Ph.D., 2012
257. 257. Fact Strategies Multiplication6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
258. 258. Fact Strategies Multiplication6× 4=(6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
259. 259. Place Value Two aspects © Joan A. Cotter, Ph.D., 2012
260. 260. Place Value Two aspectsStatic © Joan A. Cotter, Ph.D., 2012
261. 261. Place Value Two aspectsStatic • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
262. 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. 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. 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. 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. 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. 267. Trading1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
268. 268. Trading Thousands1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
269. 269. Trading Hundreds1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
270. 270. Trading Tens1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
271. 271. Trading Ones1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
277. 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. 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. 279. Trading Adding1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
282. 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. 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. 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
300. 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. 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. 302. Trading Adding 4-digit numbers1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
303. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 345. Research Highlights345 © Joan A. Cotter, Ph.D., 2012