1. © Joan A. Cotter, Ph.D., 2013
Teaching Primary Mathematics with More
Understanding and Less Counting
National Council of Supervisors of Mathematics
Monday, April 16, 2013
Denver, Colorado
Joan A. Cotter, Ph.D.
JoanCotter@RightStartMath.com
and
Tracy Mittleider, MESd
Tracy@RightStartMath.com
1

2. © Joan A. Cotter, Ph.D., 20132
Objectives
I. Review the traditional counting trajectory.

3. © Joan A. Cotter, Ph.D., 20133
Objectives
I. Review the traditional counting trajectory.
II. Experience traditional counting like a child.

4. © Joan A. Cotter, Ph.D., 20134
Objectives
I. Review the traditional counting trajectory.
II. Experience traditional counting like a child.
III. Group in 5s and 10s: an alternative to
counting.

5. © Joan A. Cotter, Ph.D., 20135
Objectives
I. Review the traditional counting trajectory.
II. Experience traditional counting like a child.
III. Group in 5s and 10s: an alternative to
counting.
IV. Meet CCSS without counting.

6. © Joan A. Cotter, Ph.D., 20136
Traditional Counting Model
1. Memorizing counting sequence.

7. © Joan A. Cotter, Ph.D., 20137
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.

8. © Joan A. Cotter, Ph.D., 20138
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.

9. © Joan A. Cotter, Ph.D., 20139
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.

10. © Joan A. Cotter, Ph.D., 201310
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.

11. © Joan A. Cotter, Ph.D., 201311
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
6. Adding by counting from the larger number.

12. © Joan A. Cotter, Ph.D., 201312
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
6. Adding by counting from larger number.
7. Subtracting by counting backward.

13. © Joan A. Cotter, Ph.D., 201313
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
6. Adding by counting from larger number.
7. Subtracting by counting backward.
8. Multiplying by skip counting.

14. © Joan A. Cotter, Ph.D., 201314
Traditional Counting Model
1. Memorizing counting sequence.
• String level
• Unbreakable list
• Breakable chain
• Numerable chain
• Bidirectional chain

15. © Joan A. Cotter, Ph.D., 2013
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
• Requires stable order for counting
words
• Common errors: double counting and
missed count
15

16. © Joan A. Cotter, Ph.D., 201316
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
• Unlike anything else in child‘s
experience (e.g. in naming family, baby
≠ all others).

17. © Joan A. Cotter, Ph.D., 201317
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
• Unlike anything else in child‘s
experience (e.g. in naming family, baby
≠ all others).
• ―How many‖ not a good test; take n is
better.

18. © Joan A. Cotter, Ph.D., 201318
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
• Focuses more on counting than adding.

19. © Joan A. Cotter, Ph.D., 201319
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
• Leads to counting words.

20. © Joan A. Cotter, Ph.D., 201320
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
• Leads to counting words.
• No need to learn strategies.

21. © Joan A. Cotter, Ph.D., 201321
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
• Leads to counting words.
• No need to learn strategies.
• Very difficult. (article in Nov. 2011, JRME)

22. © Joan A. Cotter, Ph.D., 201322
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
6. Adding by counting from larger number.
• First need to determine larger number.

23. © Joan A. Cotter, Ph.D., 201323
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
6. Adding by counting from the larger number.
7. Subtracting by counting backward.
• Extremely difficult. (Easier to go forward.)

24. © Joan A. Cotter, Ph.D., 201324
Traditional Counting Model
1. Memorizing counting sequence.
2. One-to-one correspondence.
3. Cardinality principal.
4. Adding by counting all.
5. Adding by counting on.
6. Adding by counting from larger number.
7. Subtracting by counting backward.
8. Multiplying by skip counting.
• Tedious for finding multiplication facts.

25. © Joan A. Cotter, Ph.D., 201325
Traditional Counting
From a child's perspective

26. © Joan A. Cotter, Ph.D., 201326
Traditional Counting
From a child's perspective
Because we're so familiar with 1, 2, 3, we‘ll use
letters.
A = 1
B = 2
C = 3
D = 4
E = 5, and so forth

27. © Joan A. Cotter, Ph.D., 201327
Traditional Counting
From a child's perspective
F + E =

28. © Joan A. Cotter, Ph.D., 201328
Traditional Counting
From a child's perspective
A
F + E =

29. © Joan A. Cotter, Ph.D., 201329
Traditional Counting
From a child's perspective
A B
F + E =

30. © Joan A. Cotter, Ph.D., 201330
Traditional Counting
From a child's perspective
A CB
F + E =

31. © Joan A. Cotter, Ph.D., 201331
Traditional Counting
From a child's perspective
A FC D EB
F + E =

32. © Joan A. Cotter, Ph.D., 201332
Traditional Counting
From a child's perspective
AA FC D EB
F + E =

33. © Joan A. Cotter, Ph.D., 201333
Traditional Counting
From a child's perspective
A BA FC D EB
F + E =

34. © Joan A. Cotter, Ph.D., 201334
Traditional Counting
From a child's perspective
A C D EBA FC D EB
F + E =

35. © Joan A. Cotter, Ph.D., 201335
Traditional Counting
From a child's perspective
A C D EBA FC D EB
What is the sum?
(It must be a letter.)
F + E =

36. © Joan A. Cotter, Ph.D., 201336
Traditional Counting
From a child's perspective
G I J KHA FC D EB
F + E =
K

37. © Joan A. Cotter, Ph.D., 201337
Traditional Counting
From a child's perspective
E + D =
Find the sum without counters.

38. © Joan A. Cotter, Ph.D., 201338
Traditional Counting
From a child's perspective
G + E =
Find the sum without fingers.

39. © Joan A. Cotter, Ph.D., 201339
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+ D

40. © Joan A. Cotter, Ph.D., 201340
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+ D

41. © Joan A. Cotter, Ph.D., 201341
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+ D
D
+ C

42. © Joan A. Cotter, Ph.D., 201342
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+ D
C
+ G
D
+ C

43. © Joan A. Cotter, Ph.D., 201343
Traditional Counting
From a child's perspective
Now memorize the facts!!
G
+ D
C
+ G
D
+ C

44. © Joan A. Cotter, Ph.D., 201344
Traditional Counting
From a child's perspective
Subtract counting backward by using your fingers.
H – C =

45. © Joan A. Cotter, Ph.D., 201345
Traditional Counting
From a child's perspective
Subtract by counting backward without fingers.
J – F =

46. © Joan A. Cotter, Ph.D., 201346
Traditional Counting
From a child's perspective
Try skip counting by B's to T:
B, D, . . . , T.

47. © Joan A. Cotter, Ph.D., 201347
Traditional Counting
From a child's perspective
Try skip counting by B's to T:
B, D, . . . , T.
What is D x E?

48. © Joan A. Cotter, Ph.D., 201348
Traditional Counting
―Special cases‖ of place value (1.NBT.2)
L
is a ―bundle‖ of J A‘s
and B A's.

49. © Joan A. Cotter, Ph.D., 201349
Traditional Counting
―Special cases‖ of place value (1.NBT.2)
L
is a ―bundle‖ of J A‘s
and B A's.
huh?

50. © Joan A. Cotter, Ph.D., 201350
Traditional Counting
―Special cases‖ of place value (1.NBT.2)
L
is a ―bundle‖ of J A‘s
and B A's.
(12)

51. © Joan A. Cotter, Ph.D., 201351
Traditional Counting
―Special cases‖ of place value (1.NBT.2)
L
is a ―bundle‖ of J A‘s
and B A's.
(ten ones)
(12)

52. © Joan A. Cotter, Ph.D., 201352
Traditional Counting
―Special cases‖ of place value (1.NBT.2)
L
is a ―bundle‖ of J A‘s
and B A's.
(ten ones)
(two ones)
(12)

53. © Joan A. Cotter, Ph.D., 2013
Grouping in Fives

54. © Joan A. Cotter, Ph.D., 2013
Grouping in Fives
Chinese abacus

55. © Joan A. Cotter, Ph.D., 2013
Grouping in Fives
I
II
III
IIII
V
VIII
1
2
3
4
5
8
Early Roman numerals

56. © Joan A. Cotter, Ph.D., 201356
Grouping in Fives
Musical staff

57. © Joan A. Cotter, Ph.D., 2013
Clocks and nickels
Grouping in Fives

58. © Joan A. Cotter, Ph.D., 2013
Grouping in Fives
Clocks and nickels

59. © Joan A. Cotter, Ph.D., 2013
Grouping in Fives
Tally marks

60. © Joan A. Cotter, Ph.D., 2013
Grouping in Fives
Subitizing
• Instant recognition of quantity is called subitizing.

61. © Joan A. Cotter, Ph.D., 2013
Grouping in Fives
Subitizing
• Instant recognition of quantity is called subitizing.
• Grouping in fives extends subitizing beyond five.

62. © Joan A. Cotter, Ph.D., 2013
Subitizing
• Five-month-old infants can subitize to 1–3.

63. © Joan A. Cotter, Ph.D., 2013
Subitizing
• Three-year-olds can subitize to 1–
5.
• Five-month-old infants can subitize to 1–3.

64. © Joan A. Cotter, Ph.D., 2013
Subitizing
• Three-year-olds can subitize to 1–
5.
• Four-year-olds can subitize 1–10 by
grouping with five.
• Five-month-old infants can subitize to 1–3.

65. © Joan A. Cotter, Ph.D., 2013
Subitizing
• Three-year-olds can subitize to 1–
5.
• Four-year-olds can subitize 1–10 by
grouping with five.
• Five-month-old infants can subitize to 1–3.
• Counting is analogous to sounding out a
word; subitizing, recognizing the word.

66. © Joan A. Cotter, Ph.D., 201366
Research on Subitizing

67. © Joan A. Cotter, Ph.D., 2013
Research on Subitizing
Karen Wynn's research

68. © Joan A. Cotter, Ph.D., 2013
Research on Subitizing
Karen Wynn's research

69. © Joan A. Cotter, Ph.D., 201369
Research on Subitizing
Karen Wynn's research

70. © Joan A. Cotter, Ph.D., 201370
Research on Subitizing
Karen Wynn's research

71. © Joan A. Cotter, Ph.D., 201371
Research on Subitizing
Karen Wynn's research

72. © Joan A. Cotter, Ph.D., 201372
Research on Subitizing
Karen Wynn's research

73. © Joan A. Cotter, Ph.D., 201373
Research on Subitizing
Karen Wynn's research

74. © Joan A. Cotter, Ph.D., 201374
Research on Subitizing
Karen Wynn's research

75. © Joan A. Cotter, Ph.D., 201375
Research on Subitizing
Other research

76. © Joan A. Cotter, Ph.D., 201376
Research on Subitizing
• Subitizing ―allows the child to grasp the
whole and the elements at the same time.‖—
Benoit
Other research

77. © Joan A. Cotter, Ph.D., 201377
Research on Subitizing
• 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. —Glasersfeld
Other research

78. © Joan A. Cotter, Ph.D., 201378
Research on Subitizing
• 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. —Glasersfeld
Other research

79. © Joan A. Cotter, Ph.D., 2013
Other research
Research on Subitizing
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London,
2008.
79

80. © Joan A. Cotter, Ph.D., 2013
Other research
Research on Subitizing
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London,
2008.
• Adult Pirahã from Amazon region.
Edward Gibson and Michael Frank, MIT, 2008.
80

81. © Joan A. Cotter, Ph.D., 2013
Other research
Research on Subitizing
• 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.
81

82. © Joan A. Cotter, Ph.D., 2013
Other research
Research on Subitizing
• 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.
82

83. © Joan A. Cotter, Ph.D., 2013
Research on Subitizing
In Japanese schools
• Children are discouraged from using
counting for adding.
83

84. © Joan A. Cotter, Ph.D., 2013
Research on Subitizing
In Japanese schools
• Children are discouraged from using
counting for adding.
• They consistently group in 5s.
84

85. © Joan A. Cotter, Ph.D., 2013
Research on Subitizing
Finger gnosia
• Finger gnosia is the ability to know which
fingers can been lightly touched without
looking.
85

86. © Joan A. Cotter, Ph.D., 2013
Research on Subitizing
Finger gnosia
• Finger gnosia is the ability to know which
fingers can been lightly touched without
looking.
• Part of the brain controlling fingers is
adjacent to math part of the brain.
86

87. © Joan A. Cotter, Ph.D., 2013
Research on Subitizing
Finger gnosia
• Finger gnosia is the ability to know which
fingers can been lightly touched without
looking.
• Part of the brain controlling fingers is
adjacent to math part of the brain.
• Children who use their fingers as
representational tools perform better in
mathematics.—Butterworth
87

88. © Joan A. Cotter, Ph.D., 2013
Research on Subitizing
Finger gnosia
• Finger gnosia is the ability to know which
fingers can been lightly touched without
looking.
• Part of the brain controlling fingers is
adjacent to math part of the brain.
• Children who use their fingers as
representational tools perform better in
mathematics.—Butterworth
88
• Children learn subitizing up to 5 before
counting.—Starkey & Cooper

89. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Using fingers

90. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Using fingers

91. © Joan A. Cotter, Ph.D., 201391
Learning 1–10
Using fingers

92. © Joan A. Cotter, Ph.D., 201392
Learning 1–10
Using fingers

93. © Joan A. Cotter, Ph.D., 201393
Learning 1–10
Using fingers

94. © Joan A. Cotter, Ph.D., 201394
Learning 1–10
Using fingers

95. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Subitizing 5

96. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Subitizing 5

97. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
5 has a middle; 4 does not.
Subitizing 5

98. © Joan A. Cotter, Ph.D., 201398
Learning 1–10
Tally sticks

99. © Joan A. Cotter, Ph.D., 201399
Learning 1–10
Tally sticks

100. © Joan A. Cotter, Ph.D., 2013100
Learning 1–10
Tally sticks

101. © Joan A. Cotter, Ph.D., 2013101
Learning 1–10
Tally sticks
Five as a group.

102. © Joan A. Cotter, Ph.D., 2013102
Learning 1–10
Tally sticks

103. © Joan A. Cotter, Ph.D., 2013103
Learning 1–10
Tally sticks

104. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Entering quantities

105. © Joan A. Cotter, Ph.D., 2013
3
Learning 1–10
Entering quantities

106. © Joan A. Cotter, Ph.D., 2013106
5
Learning 1–10
Entering quantities

107. © Joan A. Cotter, Ph.D., 2013107
7
Learning 1–10
Entering quantities

108. © Joan A. Cotter, Ph.D., 2013108
Learning 1–10
10
Entering quantities

109. © Joan A. Cotter, Ph.D., 2013109
Learning 1–10
The stairs

110. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Adding

111. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
4 + 3 =
Adding

112. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
4 + 3 =
Adding

113. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
4 + 3 =
Adding

114. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
4 + 3 =
Adding

115. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
4 + 3 = 7
Adding

116. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
4 + 3 = 7
Visualizing
Japanese children learn to do this mentally.

117. © Joan A. Cotter, Ph.D., 2013117
Visualizing
• Visual is related to seeing.
• Visualize is to form a mental image.

118. © Joan A. Cotter, Ph.D., 2013118
Visualizing
―Think in pictures, because the
brain remembers images better
than it does anything else.‖
—Ben Pridmore, World Memory Champion,
2009

119. © Joan A. Cotter, Ph.D., 2013119
Visualizing
―The role of physical manipulatives
was to help the child form those visual
images and thus to eliminate the need
for the physical manipulatives.‖
—Ginsberg and others

120. © Joan A. Cotter, Ph.D., 2013
Visualizing
Japanese criteria for manipulatives

121. © Joan A. Cotter, Ph.D., 2013
• Representative of structure of
numbers.
Visualizing
Japanese criteria for manipulatives

122. © Joan A. Cotter, Ph.D., 2013
• Representative of structure of
numbers.
• Easily manipulated by children.
Visualizing
Japanese criteria for manipulatives

123. © Joan A. Cotter, Ph.D., 2013
• Representative of structure of
numbers.
• Easily manipulated by children.
• Imaginable mentally.
Visualizing
Japanese criteria for manipulatives
—Japanese Council of
Mathematics Education

124. © Joan A. Cotter, Ph.D., 2013
Visualizing
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction
Necessary in:

125. © Joan A. Cotter, Ph.D., 2013
Visualizing
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction
• Architecture
• Astronomy
• Archeology
• Chemistry
• Physics
• Surgery
Necessary in:

126. © Joan A. Cotter, Ph.D., 2013
Visualizing
Try to visualize 8 identical apples without
grouping.

127. © Joan A. Cotter, Ph.D., 2013
Visualizing
Try to visualize 8 identical apples without
grouping.

128. © Joan A. Cotter, Ph.D., 2013
Visualizing
Now try to visualize 8 apples: 5 red and 3 green.

129. © Joan A. Cotter, Ph.D., 2013
Visualizing
Now try to visualize 8 apples: 5 red and 3 green.

130. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Partitioning

131. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
5 = +
Partitioning

132. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
5 = 4 + 1
Partitioning

133. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
5 = 3 + 2
Partitioning

134. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
5 = 2 + 3
Partitioning

135. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
5 = 1 + 4
Partitioning

136. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
5 = 5 + 0
Partitioning

137. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
5 = 0 + 5
Partitioning

138. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Place value
• Place value is the foundation of modern
arithmetic.

139. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Place value
• Place value is the foundation of modern
arithmetic.
• Critical for understanding algorithms.

140. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Place value
• Place value is the foundation of modern
arithmetic.
• Critical for understanding algorithms.
• Must be taught, not left for discovery.

141. © Joan A. Cotter, Ph.D., 2013
Learning 1–10
Place value
• Place value is the foundation of modern
arithmetic.
• Critical for understanding algorithms.
• Children need the big picture, not tiny
snapshots.
• Must be taught, not left for discovery.

142. © Joan A. Cotter, Ph.D., 2013
Place Value
CCSS (K.NBT.1, 1.NBT.2)
Does it make sense that students should:
• ―Work with numbers 11–19 to gain
foundations for place value.‖ (They are
the most difficult numbers we have in
English.)

143. © Joan A. Cotter, Ph.D., 2013
Place Value
CCSS (K.NBT.1, 1.NBT.2)
Does it make sense that students should:
• ―Work with numbers 11–19 to gain
foundations for place value.‖ (They are
the most difficult numbers we have in
English.)
Are these really ―special cases‖?
• ―10 can be thought of as a bundle of
ten ones — called a ‗ten.‘‖
• ―100 can be thought of as a bundle of
ten tens — called a ‗hundred.‘‖

144. © Joan A. Cotter, Ph.D., 2013
Place Value
Two aspects
Static
• 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.
• (Shown by the place-value cards.)
Dynamic (Trading)
• 10 ones = 1 ten; 10 tens = 1 hundred;
• 10 hundreds = 1 thousand, ….
• (Represented on the abacus and other

145. © Joan A. Cotter, Ph.D., 2013
Place Value
Asian number-naming
(Math way of number
naming)
• Asian children do not struggle with the
teens.

146. © Joan A. Cotter, Ph.D., 2013
Place Value
Asian number-naming
(Math way of number
naming)
• Their languages are completely ―ten-
based.‖
• Asian children do not struggle with the
teens.

147. © Joan A. Cotter, Ph.D., 2013
Place Value
Asian number-naming
(Math way of number
naming)
• Their languages are completely ―ten-
based.‖
• Asian children do not struggle with the
teens.
• Asian countries use the ten-based metric
system.

148. © Joan A. Cotter, Ph.D., 2013148
―Math‖ Way of Number Naming

149. © Joan A. Cotter, Ph.D., 2013149
―Math‖ Way of Number Naming
11 = ten 1

150. © Joan A. Cotter, Ph.D., 2013150
―Math‖ Way of Number Naming
11 = ten 1
12 = ten 2

151. © Joan A. Cotter, Ph.D., 2013151
―Math‖ Way of Number Naming
11 = ten 1
12 = ten 2
13 = ten 3

152. © Joan A. Cotter, Ph.D., 2013152
―Math‖ Way of Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4

153. © Joan A. Cotter, Ph.D., 2013153
―Math‖ Way of Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
. . . .
19 = ten 9

154. © Joan A. Cotter, Ph.D., 2013154
―Math‖ Way of Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
. . . .
19 = ten 9
20 = 2-ten

155. © Joan A. Cotter, Ph.D., 2013155
―Math‖ Way of Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
. . . .
19 = ten 9
20 = 2-ten
21 = 2-ten 1

156. © Joan A. Cotter, Ph.D., 2013156
―Math‖ Way of Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
. . . .
19 = ten 9
20 = 2-ten
21 = 2-ten 1
22 = 2-ten 2

157. © Joan A. Cotter, Ph.D., 2013157
―Math‖ Way of Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
. . . .
19 = ten 9
20 = 2-ten
21 = 2-ten 1
22 = 2-ten 2
23 = 2-ten 3

158. © Joan A. Cotter, Ph.D., 2013158
―Math‖ Way of Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
. . . .
19 = ten 9
20 = 2-ten
21 = 2-ten 1
22 = 2-ten 2
23 = 2-ten 3
. . . .
. . . .
99 = 9-ten 9

159. © Joan A. Cotter, Ph.D., 2013159
―Math‖ Way of Number Naming
137 = 1 hundred 3-ten 7

160. © Joan A. Cotter, Ph.D., 2013160
―Math‖ Way of Number Naming
137 = 1 hundred 3-ten 7
or
137 = 1 hundred and 3-ten 7

161. © Joan A. Cotter, Ph.D., 2013161
―Math‖ Way of Number Naming
0
10
20
30
40
50
60
70
80
90
100
4 5 6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Korean formal [math way]
Korean informal [not explicit]
Chinese
U.S.
AverageHighestNumberCounted

162. © Joan A. Cotter, Ph.D., 2013162
―Math‖ Way of Number Naming
0
10
20
30
40
50
60
70
80
90
100
4 5 6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Korean formal [math way]
Korean informal [not explicit]
Chinese
U.S.
AverageHighestNumberCounted

163. © Joan A. Cotter, Ph.D., 2013163
―Math‖ Way of Number Naming
0
10
20
30
40
50
60
70
80
90
100
4 5 6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Korean formal [math way]
Korean informal [not explicit]
Chinese
U.S.
AverageHighestNumberCounted

164. © Joan A. Cotter, Ph.D., 2013164
―Math‖ Way of Number Naming
0
10
20
30
40
50
60
70
80
90
100
4 5 6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Korean formal [math way]
Korean informal [not explicit]
Chinese
U.S.
AverageHighestNumberCounted

165. © Joan A. Cotter, Ph.D., 2013165
―Math‖ Way of Number Naming
0
10
20
30
40
50
60
70
80
90
100
4 5 6
Age (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Korean formal [math way]
Korean informal [not explicit]
Chinese
U.S.
AverageHighestNumberCounted

166. © Joan A. Cotter, Ph.D., 2013166
Math Way of Number Naming
• 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.)

167. © Joan A. Cotter, Ph.D., 2013167
Math Way of Number Naming
• 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.

168. © Joan A. Cotter, Ph.D., 2013168
Math Way of Number Naming
• 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.

169. © Joan A. Cotter, Ph.D., 2013169
Math Way of Number Naming
• 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.

170. © Joan A. Cotter, Ph.D., 2013170
Math Way of Number Naming
Compared to reading

171. © Joan A. Cotter, Ph.D., 2013171
Math Way of Number Naming
• Just as reciting the alphabet doesn‘t teach
reading, counting doesn‘t teach arithmetic.
Compared to reading

172. © Joan A. Cotter, Ph.D., 2013172
Math Way of Number Naming
• 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).
Compared to reading

173. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
4-ten = forty
The ―ty‖
means
tens.

174. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
4-ten = forty
The ―ty‖
means
tens.

175. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
6-ten = sixty
The ―ty‖
means
tens.

176. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
3-ten = thirty
―Thir‖ also
used in 1/3,
13 and 30.

177. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
5-ten = fifty
―Fif‖ also
used in 1/5,
15 and 50.

178. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
2-ten = twenty
Two used to
be
pronounced
―twoo.‖

179. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
A word game
fireplace place-fire

180. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
A word game
fireplace place-fire
paper-newsnewspaper

181. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
A word game
fireplace place-fire
paper-news
box-mail mailbox
newspaper

182. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
ten 4
Prefix -teen
means ten.

183. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
ten 4 teen 4
Prefix -teen
means ten.

184. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
ten 4 teen 4 fourteen
Prefix -teen
means ten.

185. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
a one left

186. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
a one left a left-one

187. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
a one left a left-one eleven

188. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
two left
Two said
as
―twoo.‖

189. © Joan A. Cotter, Ph.D., 2013
Math Way of Number Naming
Regular names
two left twelve
Two said
as
―twoo.‖

190. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
3-ten

191. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
3-ten

192. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
3-ten
3 0

193. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
3-ten
3 0

194. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
3-ten
3 0

195. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
3-ten
7
3 0

196. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
3-ten
7
3 0

197. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
3-ten
7
3 0
7

198. © Joan A. Cotter, Ph.D., 2013
3 0
Composing Numbers
3-ten
7
7

199. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
3-ten
7
Note the congruence in how we say the
number, represent the number, and write
the number.
3 07

200. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1-ten
1 0
Another example.

201. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1-ten 8
1 0

202. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1-ten
8
1 0

203. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1-ten
8
1 0
8

204. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1-ten
8
1 88

205. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
10-ten

206. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
10-ten
1 0 0

207. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
10-ten
1 0 0

208. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
10-ten
1 0 0

209. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1
hundred

210. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1
hundred
1 0 0

211. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1
hundred
1 0 0

212. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1
hundred
1 01 01 0 0

213. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
1
hundred
1 0 0

214. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
2
hundred

215. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
2
hundred

216. © Joan A. Cotter, Ph.D., 2013
Composing Numbers
2
hundred
2 0 0

217. © Joan A. Cotter, Ph.D., 2013217
Learning the Facts

218. © Joan A. Cotter, Ph.D., 2013218
Learning the Facts
Limited success, especially for struggling
children, when learning is:

219. © Joan A. Cotter, Ph.D., 2013219
Learning the Facts
• Based on counting: whether dots,
fingers, number lines, or counting
words.
Limited success, especially for struggling
children, when learning is:

220. © Joan A. Cotter, Ph.D., 2013220
Learning the Facts
• Based on counting: whether dots,
fingers, number lines, or counting
words.
Limited success, especially for struggling
children, when learning is:
• Based on rote memory: whether flash
cards, timed tests, or computer games.

221. © Joan A. Cotter, Ph.D., 2013221
Learning the Facts
• Based on counting: whether dots,
fingers, number lines, or counting
words.
Limited success, especially for struggling
children, when learning is:
• Based on rote memory: whether flash
cards, timed tests, or computer games.
• Based on skip counting: whether fingers or songs

222. © Joan A. Cotter, Ph.D., 2013222
Fact Strategies

223. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Complete the Ten
9 + 5 =

224. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Complete the Ten
9 + 5 =

225. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Complete the Ten
9 + 5 =

226. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Complete the Ten
9 + 5 =
Take 1 from
the 5 and give
it to the 9.

227. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Complete the Ten
9 + 5 =
Take 1 from
the 5 and give
it to the 9.

228. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Complete the Ten
9 + 5 =
Take 1 from
the 5 and give
it to the 9.

229. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Complete the Ten
9 + 5 = 14
Take 1 from
the 5 and give
it to the 9.

230. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Two Fives
8 + 6 =

231. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Two Fives
8 + 6 =

232. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Two Fives
8 + 6 =

233. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Two Fives
8 + 6 =

234. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Two Fives
8 + 6 =
10 + 4 = 14

235. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Down
15 – 9 =

236. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Down
15 – 9 =

237. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.

238. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.

239. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.

240. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Down
15 – 9 = 6
Subtract 5;
then 4.

241. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Subtract from 10
15 – 9 =

242. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.

243. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.

244. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.

245. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Subtract from 10
15 – 9 = 6
Subtract 9
from 10.

246. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Up
15 – 9 =

247. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.

248. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.

249. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.

250. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.

251. © Joan A. Cotter, Ph.D., 2013
Fact Strategies
Going Up
15 – 9 =
1 + 5 = 6
Start with 9;
go up to 15.

252. © Joan A. Cotter, Ph.D., 2013
Money
Penny

253. © Joan A. Cotter, Ph.D., 2013
Money
Nickel

254. © Joan A. Cotter, Ph.D., 2013
Money
Dime

255. © Joan A. Cotter, Ph.D., 2013
Money
Quarter

256. © Joan A. Cotter, Ph.D., 2013
Money
Quarter

257. © Joan A. Cotter, Ph.D., 2013
Money
Quarter

258. © Joan A. Cotter, Ph.D., 2013
Money
Quarter

259. © Joan A. Cotter, Ph.D., 2013
Trading
1000 10 1100

260. © Joan A. Cotter, Ph.D., 2013
Trading
Thousands
1000 10 1100

261. © Joan A. Cotter, Ph.D., 2013
Trading
Hundreds
1000 10 1100

262. © Joan A. Cotter, Ph.D., 2013
Trading
Tens
1000 10 1100

263. © Joan A. Cotter, Ph.D., 2013
Trading
Ones
1000 10 1100

264. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding
8
+ 6

265. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding
8
+ 6

266. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding
8
+ 6

267. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding
8
+ 6

268. © Joan A. Cotter, Ph.D., 2013
Trading
Adding
8
+ 6
14
1000 10 1100

269. © Joan A. Cotter, Ph.D., 2013
Trading
Adding
8
+ 6
14
Too many
ones; trade 10
ones for 1 ten.
1000 10 1100

270. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding
8
+ 6
14
Too many
ones; trade 10
ones for 1 ten.

271. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding
8
+ 6
14
Too many
ones; trade 10
ones for 1 ten.

272. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding
8
+ 6
14
Same answer
before and
after trading.

273. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738

274. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Enter the first
number from
left to right.

275. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Enter numbers
from left to right.

276. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Enter numbers
from left to right.

277. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Enter numbers
from left to right.

278. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Enter numbers
from left to right.

279. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Enter numbers
from left to right.

280. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Add starting at
the right. Write
results after
each step.

281. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Add starting at
the right. Write
results after each
step.

282. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Add starting at
the right. Write
results after each
step.

283. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
Add starting at
the right. Write
results after each
step.

284. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
6
Add starting at
the right. Write
results after
each step.

285. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
6
Add starting at
the right. Write
results after
each step.
1

286. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
6
Add starting at
the right. Write
results after
each step.
1

287. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
6
Add starting at
the right. Write
results after
each step.
1

288. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
96
Add starting at
the right. Write
results after
each step.
1

289. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
96
Add starting at
the right. Write
results after
each step.
1

290. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
96
Add starting at
the right. Write
results after
each step.
1

291. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
96
Add starting at
the right. Write
results after
each step.
1

292. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
96
Add starting at
the right. Write
results after
each step.
1

293. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
396
Add starting at
the right. Write
results after
each step.
1

294. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
396
Add starting at
the right. Write
results after
each step.
1 1

295. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
396
Add starting at
the right. Write
results after
each step.
1 1

296. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
396
Add starting at
the right. Write
results after each
step.
1 1

297. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
6396
Add starting at
the right. Write
results after each
step.
1 1

298. © Joan A. Cotter, Ph.D., 2013
1000 10 1100
Trading
Adding 4-digit numbers
3658
+ 2738
6396
Add starting at
the right. Write
results after each
step.
1 1

299. © Joan A. Cotter, Ph.D., 2013299
Meeting the Standards

300. © Joan A. Cotter, Ph.D., 2013300
Meeting the Standards
Page 5
―These Standards do not dictate curriculum or
teaching methods. For example, just because
topic A appears before topic B in the standards
for a given grade, it does not necessarily mean
that topic A must be taught before topic B. A
teacher might prefer to teach topic B before
topic A, or might choose to highlight
connections by teaching topic A and topic B at
the same time. Or, a teacher might prefer to
teach a topic of his or her own choosing that
leads, as a byproduct, to students reaching the
standards for topics A and B.‖ —CCSS

301. © Joan A. Cotter, Ph.D., 2013301
Meeting the Standards
Page 5 summary
• Standards do not dictate curriculum or
teaching methods.

302. © Joan A. Cotter, Ph.D., 2013302
Meeting the Standards
Page 5 summary
• Standards do not dictate curriculum or
teaching methods.
• Within a grade, topics may be taught in
any order or taught indirectly.

303. © Joan A. Cotter, Ph.D., 2013303
Meeting the Standards
Kindergarten (K.NBT)
Know number names and the count
sequence.
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given
number within the known sequence
(instead of having to begin at 1).
3. Write numbers from 0 to 20. Represent a
number of objects with a written numeral
0-20 (with 0 representing a count of no
objects).

304. © Joan A. Cotter, Ph.D., 2013304
Meeting the Standards
Kindergarten (K.CC)
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given
number.

305. © Joan A. Cotter, Ph.D., 2013305
Meeting the Standards
Kindergarten (K.CC)
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given
number.

306. © Joan A. Cotter, Ph.D., 2013306
Meeting the Standards
Kindergarten (K.CC)
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given
number.

307. © Joan A. Cotter, Ph.D., 2013
Meeting the Standards
61
72
83
94
105
Kindergarten (K.CC)
3. Write numbers from 0 to 20.
Number Chart

308. © Joan A. Cotter, Ph.D., 2013308
Meeting the Standards
Kindergarten (K.NBT)
Work with numbers 11–19.
1. Compose and partition numbers from 11 to
19 into ten ones and some further ones.

309. © Joan A. Cotter, Ph.D., 2013309
Meeting the Standards
Kindergarten (K.NBT)
Work with numbers 11–19.
1. Compose and partition numbers from 11 to
19 into ten ones and some further ones.

310. © Joan A. Cotter, Ph.D., 2013310
Meeting the Standards
Kindergarten (K.NBT)
Work with numbers 11–19.
1. Compose and partition numbers from 11 to
19 into ten ones and some further ones.
1 86
1 0
6

311. © Joan A. Cotter, Ph.D., 2013311
Meeting the Standards
Kindergarten (K.OA)
Understand addition and subtraction.
1. Represent addition and subtraction with
objects, fingers, . . . equations.
2. Solve addition and subtraction word
problems, and add and subtract within 10.
3. Partition numbers less than or equal to 10
into pairs.
4. For any number from 1 to 9, find the
number that makes 10.
5. Fluently add and subtract within 5.

312. © Joan A. Cotter, Ph.D., 2013312
Meeting the Standards
Kindergarten (K.OA)
2. Solve addition and subtraction word
problems, and add and subtract within 10.
Whole
Part Part
Part-whole
circles

313. © Joan A. Cotter, Ph.D., 2013313
Meeting the Standards
Using part-whole circles to solve problems
Lee received 3 goldfish as a gift. Now
Lee has 5. How many did Lee have to
start with?

314. © Joan A. Cotter, Ph.D., 2013314
Meeting the Standards
Using part-whole circles to solve problems
Lee received 3 goldfish as a gift. Now
Lee has 5. How many did Lee have to
start with?
Is 3 a part or whole?

315. © Joan A. Cotter, Ph.D., 2013315
Meeting the Standards
Using part-whole circles to solve problems
Lee received 3 goldfish as a gift. Now
Lee has 5. How many did Lee have to
start with?
Is 3 a part or whole?
3

316. © Joan A. Cotter, Ph.D., 2013316
Meeting the Standards
Using part-whole circles to solve problems
Lee received 3 goldfish as a gift. Now
Lee has 5. How many did Lee have to
start with?
Is 5 a part or whole?
3

317. © Joan A. Cotter, Ph.D., 2013317
Meeting the Standards
Using part-whole circles to solve problems
Lee received 3 goldfish as a gift. Now
Lee has 5. How many did Lee have to
start with?
Is 5 a part or whole?
3
5

318. © Joan A. Cotter, Ph.D., 2013318
Meeting the Standards
Using part-whole circles to solve problems
Lee received 3 goldfish as a gift. Now
Lee has 5. How many did Lee have to
start with?
What is the missing part?
3
5

319. © Joan A. Cotter, Ph.D., 2013319
Meeting the Standards
Using part-whole circles to solve problems
Lee received 3 goldfish as a gift. Now
Lee has 5. How many did Lee have to
start with?
What is the missing part?
3
5
2

320. © Joan A. Cotter, Ph.D., 2013320
Meeting the Standards
Kindergarten (K.OA)
4. For any number from 1 to 9, find the
number that makes 10.
10
7 3

321. © Joan A. Cotter, Ph.D., 2013321
Meeting the Standards
Grade 1 (1.OA)
Understand and apply properties of
operations and the relationship between
addition and subtraction.
1. Apply properties of operations as
strategies to add and subtract,
commutative property and associative
property of addition.
2. Understand subtraction as an unknown-
addend problem. [Subtract by going up.]

322. © Joan A. Cotter, Ph.D., 2013322
Meeting the Standards
Grade 1 (1.OA)
1. Apply properties of operations as
strategies to add and subtract,
commutative property and associative
property of addition.
6 + 3 = 9
3 + 6 = 9

323. © Joan A. Cotter, Ph.D., 2013323
Meeting the Standards
Grade 1 (1.OA)
Work with addition and subtraction
equations.
7. Understand the meaning of the equal sign.
8. Determine the unknown whole number in
an addition or subtraction equation.

324. © Joan A. Cotter, Ph.D., 2013324
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
Math balance

325. © Joan A. Cotter, Ph.D., 2013325
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
7 = 7

326. © Joan A. Cotter, Ph.D., 2013326
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 = 3 + 7
10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10

327. © Joan A. Cotter, Ph.D., 2013327
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
8 + 2 = 10

328. © Joan A. Cotter, Ph.D., 2013328
Meeting the Standards
Grade 1 (1.OA)
7. Understand the meaning of the equal sign.
10 9 8 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
7 + 7 = 14
7

329. © Joan A. Cotter, Ph.D., 2013329
Meeting the Standards
Grade 1 (1.OA)
8 + _ = 11
8. Determine the unknown whole number in
an addition or subtraction equation.

330. © Joan A. Cotter, Ph.D., 2013330
Meeting the Standards
Grade 1 (1.OA)
10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
8 + 3 = 11
8. Determine the unknown whole number in
an addition or subtraction equation.

331. © Joan A. Cotter, Ph.D., 2013331
Meeting the Standards
Grade 1 (1.OA)
Extend the counting sequence.
1. Count to 120, starting at any number less
than 120.

332. © Joan A. Cotter, Ph.D., 2013332
Meeting the Standards
Grade 1 (1.OA)
Extend the counting sequence.
1. Count to 120, starting at any number less
than 120.
1 0 0
1 0
9
1 0 01 09

333. © Joan A. Cotter, Ph.D., 2013333
Meeting the Standards
Grade 1 (1.NBT)
Understanding place value.
3. Compare two two-digit numbers, recording
the results of comparisons with symbols >,
=, <.
4. Add a two-digit number and a multiple of
10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
6. Subtract multiples of 10 in the range 10-90
from multiples of 10.

334. © Joan A. Cotter, Ph.D., 2013334
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording
the results of comparisons with symbols >,
=, <.
4 06 6 04

335. © Joan A. Cotter, Ph.D., 2013335
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording
the results of comparisons with symbols >,
=, <.
4 06 6 04
46 64Put two dots by greater
number.

336. © Joan A. Cotter, Ph.D., 2013336
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording
the results of comparisons with symbols >,
=, <.
4 06 6 04
46 64
.
.Put two dots by greater
number.

337. © Joan A. Cotter, Ph.D., 2013337
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording
the results of comparisons with symbols >,
=, <.
4 06 6 04
46 64
.
.Put two dots by greater
number.
Put one dot by lesser
number.

338. © Joan A. Cotter, Ph.D., 2013338
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording
the results of comparisons with symbols >,
=, <.
4 06 6 04
46 64
.
..
Put two dots by greater
number.
Put one dot by lesser
number.

339. © Joan A. Cotter, Ph.D., 2013339
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording
the results of comparisons with symbols >,
=, <.
4 06 6 04
46 64
.
..
Put two dots by greater
number.
Put one dot by lesser
number.

340. © Joan A. Cotter, Ph.D., 2013340
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording
the results of comparisons with symbols >,
=, <.
4 06 6 04
46 64
.
..
Put two dots by greater
number.
Put one dot by lesser
number.

341. © Joan A. Cotter, Ph.D., 2013341
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of
10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 + 10 = __

342. © Joan A. Cotter, Ph.D., 2013342
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of
10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 + 10 = 34

343. © Joan A. Cotter, Ph.D., 2013343
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of
10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 – 10 = __

344. © Joan A. Cotter, Ph.D., 2013344
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of
10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 – 10 = 14

345. © Joan A. Cotter, Ph.D., 2013345
Meeting the Standards
Grade 1 (1.NBT)
6. Subtract multiples of 10 in the range 10-90
from multiples of 10.
90 – 30 = __

346. © Joan A. Cotter, Ph.D., 2013346
Meeting the Standards
Grade 1 (1.NBT)
6. Subtract multiples of 10 in the range 10-90
from multiples of 10.
90 – 30 = 60

347. © Joan A. Cotter, Ph.D., 2013347
Meeting the Standards
Grade 2 (2.OA)
Work with equal groups of objects to gain
foundations for multiplication.
3. Determine whether a group of objects (up to
20) has an odd or even number of members.
4. Use addition to find the total number of
objects arranged in rectangular arrays.

348. © Joan A. Cotter, Ph.D., 2013348
Meeting the Standards
Grade 2 (2.OA)
3. Determine whether a group of objects (up to
20) has an odd or even number of members.
Is 17 even
or odd?

349. © Joan A. Cotter, Ph.D., 2013349
Meeting the Standards
Grade 2 (2.OA)
3. Determine whether a group of objects (up to
20) has an odd or even number of members.
Is 17 even
or odd?

350. © Joan A. Cotter, Ph.D., 2013350
Meeting the Standards
Grade 2 (2.OA)
4. Use addition to find the total number of
objects arranged in rectangular arrays.

351. © Joan A. Cotter, Ph.D., 2013351
Meeting the Standards
Grade 2 (2.OA)
4. Use addition to find the total number of
objects arranged in rectangular arrays.
5 + 5 + 5 + 5 = 20

352. © Joan A. Cotter, Ph.D., 2013352
Meeting the Standards
Grade 2 (2.OA)
4. Use addition to find the total number of
objects arranged in rectangular arrays.
10 9 8 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
5 + 5 + 5 + 5 = 20
7

353. © Joan A. Cotter, Ph.D., 2013353
Meeting the Standards
Grade 2 (2.NBT)
Number and Operations in Base Ten.
2. Count within 1000; skip-count by 2s, 5s,
10s, and 100s.
3. Read and write numbers to 1000 using
base-ten numerals, number names, and
expanded form.
4. Compare two three-digit numbers based
on meanings of the hundreds, tens, and
ones digits, using >, =, and <.

354. © Joan A. Cotter, Ph.D., 2013354
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.

355. © Joan A. Cotter, Ph.D., 2013355
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
5,

356. © Joan A. Cotter, Ph.D., 2013356
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
5, 10,

357. © Joan A. Cotter, Ph.D., 2013357
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
5, 10, 15, . . .

358. © Joan A. Cotter, Ph.D., 2013358
Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
1000 10 1100
100, 200, 300, . . .

359. © Joan A. Cotter, Ph.D., 2013359
Meeting the Standards
Grade 2 (2.NBT)
2. Count within 1000.
3. Read and write numbers to 1000 using
base-ten numerals, number names, and
expanded form.
3 0 0
7 08
378,

360. © Joan A. Cotter, Ph.D., 2013360
Meeting the Standards
Grade 2 (2.NBT)
2. Count within 1000.
3. Read and write numbers to 1000 using
base-ten numerals, number names, and
expanded form.
3 0 0
7 08
3 0 0
7 09
378, 379,

361. © Joan A. Cotter, Ph.D., 2013361
Meeting the Standards
Grade 2 (2.NBT)
2. Count within 1000.
3. Read and write numbers to 1000 using
base-ten numerals, number names, and
expanded form.
3 0 0
7 08
3 0 0
7 09
3 0 0
8 0
378, 379, 380

362. © Joan A. Cotter, Ph.D., 2013362
Meeting the Standards
Grade 2 (2.NBT)
4. Compare two three-digit numbers based
on meanings of the hundreds, tens, and
ones digits, using >, =, and <.
7 0 00 06 6 0 07 00
706 > 670

363. © Joan A. Cotter, Ph.D., 2013363
Objectives
I. Review the traditional counting trajectory.
II. Experience traditional counting like a child.
III. Group in 5s and 10s: an alternative to
counting.
IV. Meet CCSS without counting.

364. © Joan A. Cotter, Ph.D., 2013
Teaching Primary Mathematics with More
Understanding and Less Counting
National Council of Supervisors of Mathematics
Monday, April 16, 2013
Denver, Colorado
Joan A. Cotter, Ph.D.
JoanCotter@RightStartMath.com
and
Tracy Mittleider, MESd
Tracy@RightStartMath.com