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IMF: Visualizing and Montessori Math PART 1
- 1. How Visualization Enhances
Montessori Mathematics PART 1
by Joan A. Cotter, Ph.D.
JoanCotter@RightStartMath.com
Montessori Foundation 30
30
Conference 77
Friday, Nov 2, 2012
Sarasota, Florida 30
370
7
1000 100 10 1
7
7
7 3
3
3
PowerPoint Presentation
RightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012
- 2. Counting Model
In Montessori, counting is pervasive:
• Number Rods
• Spindle Boxes
• Decimal materials
• Snake Game
• Dot Game
• Stamp Game
• Multiplication Board
• Bead Frame
© Joan A. Cotter, Ph.D., 2012
- 4. Verbal Counting Model
From a child's perspective
Because we’re so familiar with 1, 2, 3, we’ll use letters.
A=1
B=2
C=3
D=4
E = 5, and so forth
© Joan A. Cotter, Ph.D., 2012
- 12. Verbal Counting Model
From a child's perspective
F
+E
A B C D E F A B C D E
© Joan A. Cotter, Ph.D., 2012
- 13. Verbal Counting Model
From a child's perspective
F
+E
A B C D E F A B C D E
What is the sum?
(It must be a letter.)
© Joan A. Cotter, Ph.D., 2012
- 14. Verbal Counting Model
From a child's perspective
F
+E
K
A B C D E F G H I J K
© Joan A. Cotter, Ph.D., 2012
- 15. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
G
+D
© Joan A. Cotter, Ph.D., 2012
- 16. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
+
G F
+D
© Joan A. Cotter, Ph.D., 2012
- 17. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
+
G F
+D
D
+C
© Joan A. Cotter, Ph.D., 2012
- 18. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
+
G F
+D
D C
+C +G
© Joan A. Cotter, Ph.D., 2012
- 19. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
+
E
G F
I
+
+D
D C
+C +G
© Joan A. Cotter, Ph.D., 2012
- 20. Verbal Counting Model
From a child's perspective
H
–E
Subtract with your fingers by counting backward.
© Joan A. Cotter, Ph.D., 2012
- 21. Verbal Counting Model
From a child's perspective
J
–F
Subtract without using your fingers.
© Joan A. Cotter, Ph.D., 2012
- 22. Verbal Counting Model
From a child's perspective
Try skip counting by B’s to T:
B, D, . . . T.
© Joan A. Cotter, Ph.D., 2012
- 23. Verbal Counting Model
From a child's perspective
Try skip counting by B’s to T:
B, D, . . . T.
What is D × E?
© Joan A. Cotter, Ph.D., 2012
- 24. Verbal Counting Model
From a child's perspective
L
is written AB
because it is A J
and B A’s
© Joan A. Cotter, Ph.D., 2012
- 25. Verbal Counting Model
From a child's perspective
L
is written AB
because it is A J
and B A’s
huh?
© Joan A. Cotter, Ph.D., 2012
- 26. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB
because it is A J
and B A’s
© Joan A. Cotter, Ph.D., 2012
- 27. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB (12)
because it is A J
and B A’s
© Joan A. Cotter, Ph.D., 2012
- 28. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB (12)
because it is A J (one 10)
and B A’s
© Joan A. Cotter, Ph.D., 2012
- 29. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB (12)
because it is A J (one 10)
and B A’s (two 1s).
© Joan A. Cotter, Ph.D., 2012
- 31. 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 31
© Joan A. Cotter, Ph.D., 2012
- 32. Calendar Math
Calendar Counting
August
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31
© Joan A. Cotter, Ph.D., 2012
- 33. 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 31
© Joan A. Cotter, Ph.D., 2012
- 34. 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 31
© Joan A. Cotter, Ph.D., 2012
- 35. 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 31
© Joan A. Cotter, Ph.D., 2012
- 36. 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.
© Joan A. Cotter, Ph.D., 2012
- 37. Calendar Math
Partial Calendar
August
1 2 3 4 5 6 7
8 9 10
© Joan A. Cotter, Ph.D., 2012
- 38. Calendar Math
Partial Calendar
August
1 2 3 4 5 6 7
8 9 10
Children need the whole month to plan ahead.
© Joan A. Cotter, Ph.D., 2012
- 39. 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.
© Joan A. Cotter, Ph.D., 2012
- 49. Research on Counting
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
© Joan A. Cotter, Ph.D., 2012
- 50. 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.
© Joan A. Cotter, Ph.D., 2012
- 51. 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.
© Joan A. Cotter, Ph.D., 2012
- 52. 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.
© Joan A. Cotter, Ph.D., 2012
- 53. Research on Counting
In Japanese schools:
• Children are discouraged from using
counting for adding.
© Joan A. Cotter, Ph.D., 2012
- 54. Research on Counting
In Japanese schools:
• Children are discouraged from using
counting for adding.
• They consistently group in 5s.
© Joan A. Cotter, Ph.D., 2012
- 56. Subitizing Quantities
(Identifying without counting)
• Five-month-old infants can subitize to 3.
© Joan A. Cotter, Ph.D., 2012
- 57. Subitizing Quantities
(Identifying without counting)
• Five-month-old infants can subitize to 3.
• Three-year-olds can subitize to 5.
© Joan A. Cotter, Ph.D., 2012
- 58. Subitizing Quantities
(Identifying without counting)
• Five-month-old infants can subitize to 3.
• Three-year-olds can subitize to 5.
• Four-year-olds can subitize 6 to 10 by
using five as a subbase.
© Joan A. Cotter, Ph.D., 2012
- 59. Subitizing Quantities
(Identifying without counting)
• Five-month-old infants can subitize to 3.
• Three-year-olds can subitize to 5.
• Four-year-olds can subitize 6 to 10 by
using five as a subbase.
• Counting is like sounding out each letter;
subitizing is recognizing the quantity.
© Joan A. Cotter, Ph.D., 2012
- 60. Research on Counting
Subitizing
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit
© Joan A. Cotter, Ph.D., 2012
- 61. Research on Counting
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
© Joan A. Cotter, Ph.D., 2012
- 62. Research on Counting
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
• Children who can subitize perform better in
mathematics long term.—Butterworth
© Joan A. Cotter, Ph.D., 2012
- 63. Research on Counting
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
• Children who can subitize perform better in
mathematics long term.—Butterworth
• Counting-on is a difficult skill for many children.
—Journal for Res. in Math Ed. Nov. 2011
© Joan A. Cotter, Ph.D., 2012
- 64. Research on Counting
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
• Children who can subitize perform better in
mathematics long term.—Butterworth
• Counting-on is a difficult skill for many children.
—Journal for Res. in Math Ed. Nov. 2011
• Math anxiety affects counting ability, but
not subitizing ability.
© Joan A. Cotter, Ph.D., 2012
- 66. Visualizing Quantities
“Think in pictures, because the
brain remembers images better
than it does anything else.”
Ben Pridmore, World Memory Champion, 2009
© Joan A. Cotter, Ph.D., 2012
- 67. 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 others
© Joan A. Cotter, Ph.D., 2012
- 68. 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
- 69. Visualizing Quantities
Visualizing also needed in:
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction
© Joan A. Cotter, Ph.D., 2012
- 70. Visualizing Quantities
Visualizing also needed in:
• Reading • Architecture
• Sports • Astronomy
• Creativity • Archeology
• Geography • Chemistry
• Engineering • Physics
• Construction • Surgery
© Joan A. Cotter, Ph.D., 2012
- 79. Visualizing Quantities
Early Roman numerals
1 I
2 II
3 III
4 IIII
5 V
8 VIII
© Joan A. Cotter, Ph.D., 2012
- 83. Grouping in Fives
Using fingers
Grouping in Fives is a three-period lesson.
© Joan A. Cotter, Ph.D., 2012
- 84. Grouping in Fives
Using fingers
Grouping in Fives is a three-period lesson.
© Joan A. Cotter, Ph.D., 2012
- 85. Grouping in Fives
Using fingers
Grouping in Fives is a three-period lesson.
© Joan A. Cotter, Ph.D., 2012
- 86. Grouping in Fives
Using fingers
Grouping in Fives is a three-period lesson.
© Joan A. Cotter, Ph.D., 2012
- 87. Grouping in Fives
Using fingers
Grouping in Fives is a three-period lesson.
© Joan A. Cotter, Ph.D., 2012
- 88. 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
- 91. Grouping in Fives
Recognizing 5
5 has a middle; 4 does not.
© Joan A. Cotter, Ph.D., 2012
- 98. Grouping in Fives
Pairing Finger Cards
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QuickTime™ and a
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© Joan A. Cotter, Ph.D., 2012
- 99. Grouping in Fives
Ordering Finger Cards
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
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TIFF (LZW) decompressor
are needed to see this picture.
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TIFF (LZW) decompressor
are needed to see this picture. QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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TIFF (LZW) decompressor
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TIFF (LZW) decompressor
are needed to see this picture.
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TIFF (LZW) decompressor
are needed to see this picture.
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TIFF (LZW) decompressor
are needed to see this picture.
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TIFF (LZW) decompressor
are needed to see this picture.
© Joan A. Cotter, Ph.D., 2012
- 100. Grouping in Fives
Matching Number Cards to Finger Cards
QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a
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5 1
QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a
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© Joan A. Cotter, Ph.D., 2012
- 101. Grouping in Fives
Matching Finger Cards to Number Cards
9 1 10 4 6
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
2 3 7 8 5
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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QuickTime™ and a
TIFF (LZW) decompressor a
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are needed (LZW)this picture. a
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are needed (LZW)decompressor
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© Joan A. Cotter, Ph.D., 2012
- 102. Grouping in Fives
Finger Card Memory game
QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a
TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor
are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture.
QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a
TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor
are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture.
QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a
TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor
are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture.
QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a
TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor
are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture.
QuickTime™ and a QuickTime™ and a QuickTime™ and a QuickTime™ and a
TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor
are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture.
© Joan A. Cotter, Ph.D., 2012
- 115. Grouping in Fives
1000 100 10 1
1000 100 10 1
1000 100 10 1
1000 100 10 1
100 10 1
100 10 1
100 10 1
100 1
Stamp Game
© Joan A. Cotter, Ph.D., 2012
- 116. Grouping in Fives
1000 100 10 1
1000 100 10 1
1000 100 10 1
1000 100 10 1
100 10 1
100 10 1
100 10 1
100 1
Stamp Game
© Joan A. Cotter, Ph.D., 2012
- 117. Grouping in Fives
1000 1000 100 100 10 10 1 1
1000 1000 100 100 10 10 1 1
100 100 10 10 1 1
100 100 10 10 1 1
100 100 10 10 1 1
100 100 10
100 100
100 100
Stamp Game
© Joan A. Cotter, Ph.D., 2012
- 118. Grouping in Fives
1000 1000 100 100 10 1 1
1000 1000 100 100 1 1
10 10
100 100 1 1
10 10
100 100 1 1
10 10
100 100 1 1
10 10
100 100
10 10
100 100
100 100
Stamp Game
© Joan A. Cotter, Ph.D., 2012
- 119. Grouping in Fives
1000 1000 100 100 10 1 1
1000 1000 100 100 1 1
10 10
100 100 1 1
10 10
1 1
100 100 10 10
1 1
100 100 10 10
100 100 10 10
100 100
Stamp Game 100 100
© Joan A. Cotter, Ph.D., 2012
- 120. Grouping in Fives
Black and White Bead Stairs
“Grouped in fives so the child does not
need to count.” A. M. Joosten
© Joan A. Cotter, Ph.D., 2012
- 134. Math Card Games
• Provide repetition for learning the facts.
© Joan A. Cotter, Ph.D., 2012
- 135. Math Card Games
• Provide repetition for learning the facts.
• Encourage autonomy.
© Joan A. Cotter, Ph.D., 2012
- 136. Math Card Games
• Provide repetition for learning the facts.
• Encourage autonomy.
• Promote social interaction.
© Joan A. Cotter, Ph.D., 2012
- 137. Math Card Games
• Provide repetition for learning the facts.
• Encourage autonomy.
• Promote social interaction.
• Are enjoyed by the children.
© Joan A. Cotter, Ph.D., 2012
- 138. Go to the Dump Game
Objective:
To learn the facts that total 10:
1+9
2+8
3+7
4+6
5+5
© Joan A. Cotter, Ph.D., 2012
- 139. 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.
© Joan A. Cotter, Ph.D., 2012
- 141. “Math” Way of Naming Numbers
11 = ten 1
© Joan A. Cotter, Ph.D., 2012
- 142. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
© Joan A. Cotter, Ph.D., 2012
- 143. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
© Joan A. Cotter, Ph.D., 2012
- 144. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
© Joan A. Cotter, Ph.D., 2012
- 145. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
© Joan A. Cotter, Ph.D., 2012
- 146. “Math” Way of Naming Numbers
11 = ten 1 20 = 2-ten
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
© Joan A. Cotter, Ph.D., 2012
- 147. “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 9
© Joan A. Cotter, Ph.D., 2012
- 148. “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 9
© Joan A. Cotter, Ph.D., 2012
- 149. “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
© Joan A. Cotter, Ph.D., 2012
- 150. “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 9
© Joan A. Cotter, Ph.D., 2012
- 151. “Math” Way of Naming Numbers
137 = 1 hundred 3-ten 7
© Joan A. Cotter, Ph.D., 2012
- 152. “Math” Way of Naming Numbers
137 = 1 hundred 3-ten 7
or
137 = 1 hundred and 3-ten 7
© Joan A. Cotter, Ph.D., 2012
- 153. “Math” Way of Naming Numbers
100 Chinese
U.S.
Average Highest Number Counted
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
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2012
- 154. “Math” Way of Naming Numbers
100 Chinese
U.S.
Average Highest Number Counted
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
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2012
- 155. “Math” Way of Naming Numbers
100 Chinese
U.S.
Average Highest Number Counted
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
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2012
- 156. “Math” Way of Naming Numbers
100 Chinese
U.S.
Average Highest Number Counted
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
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2012
- 157. “Math” Way of Naming Numbers
100 Chinese
U.S.
Average Highest Number Counted
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
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
© Joan A. Cotter, Ph.D., 2012
- 158. 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.)
© Joan A. Cotter, Ph.D., 2012
- 159. 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.
© Joan A. Cotter, Ph.D., 2012
- 160. 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.
© Joan A. Cotter, Ph.D., 2012
- 161. 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.
© Joan A. Cotter, Ph.D., 2012
- 162. Math Way of Naming Numbers
Compared to reading:
© Joan A. Cotter, Ph.D., 2012
- 163. Math Way of Naming Numbers
Compared to reading:
• Just as reciting the alphabet doesn’t teach reading,
counting doesn’t teach arithmetic.
© Joan A. Cotter, Ph.D., 2012
- 164. 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).
© Joan A. Cotter, Ph.D., 2012
- 165. 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).
• Montessorians need to use the math way of naming
numbers for a longer period of time.
© Joan A. Cotter, Ph.D., 2012
- 166. Math Way of Naming Numbers
“Rather, the increased gap between Chinese and
U.S. students and that of Chinese Americans and
Caucasian Americans may be due primarily to the
nature of their initial gap prior to formal schooling,
such as counting efficiency and base-ten number
sense.”
Jian Wang and Emily Lin, 2005
Researchers
© Joan A. Cotter, Ph.D., 2012
- 167. Math Way of Naming Numbers
Traditional names
4-ten =
forty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2012
- 168. Math Way of Naming Numbers
Traditional names
4-ten =
forty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2012
- 169. Math Way of Naming Numbers
Traditional names
6-ten = sixty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2012
- 170. Math Way of Naming Numbers
Traditional names
3-ten = thirty
“Thir” also
used in 1/3,
13 and 30.
© Joan A. Cotter, Ph.D., 2012
- 171. Math Way of Naming Numbers
Traditional names
5-ten = fifty
“Fif” also
used in 1/5,
15 and 50.
© Joan A. Cotter, Ph.D., 2012
- 172. Math Way of Naming Numbers
Traditional names
2-ten = twenty
Two used to be
pronounced
“twoo.”
© Joan A. Cotter, Ph.D., 2012
- 173. Math Way of Naming Numbers
Traditional names
A word game
fireplace place-fire
© Joan A. Cotter, Ph.D., 2012
- 174. Math Way of Naming Numbers
Traditional names
A word game
fireplace place-fire
newspaper paper-news
© Joan A. Cotter, Ph.D., 2012
- 175. 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
- 176. Math Way of Naming Numbers
Traditional names
ten 4
“Teen” also
means ten.
© Joan A. Cotter, Ph.D., 2012
- 177. Math Way of Naming Numbers
Traditional names
ten 4 teen 4
“Teen” also
means ten.
© Joan A. Cotter, Ph.D., 2012
- 178. Math Way of Naming Numbers
Traditional names
ten 4 teen 4 fourtee
n
“Teen” also
means ten.
© Joan A. Cotter, Ph.D., 2012
- 179. Math Way of Naming Numbers
Traditional names
a one left
© Joan A. Cotter, Ph.D., 2012
- 180. Math Way of Naming Numbers
Traditional names
a one left a left-one
© Joan A. Cotter, Ph.D., 2012
- 181. Math Way of Naming Numbers
Traditional names
a one left a left-one eleven
© Joan A. Cotter, Ph.D., 2012
- 182. Math Way of Naming Numbers
Traditional names
two left
Two said
as “twoo.”
© Joan A. Cotter, Ph.D., 2012
- 183. Math Way of Naming Numbers
Traditional names
two left twelve
Two said
as “twoo.”
© Joan A. Cotter, Ph.D., 2012
- 193. Composing Numbers
3-ten 7
30
37
0
7
Note the congruence in how we say the number,
represent the number, and write the number.
© Joan A. Cotter, Ph.D., 2012
- 213. Evens and Odds
Evens
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 214. Evens and Odds
Evens
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 215. Evens and Odds
Evens
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 216. Evens and Odds
Evens
Use two fingers
and touch each
pair in succession.
EVEN!
© Joan A. Cotter, Ph.D., 2012
- 217. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 218. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 219. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 220. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
© Joan A. Cotter, Ph.D., 2012
- 221. Evens and Odds
Odds
Use two fingers
and touch each
pair in succession.
ODD!
© Joan A. Cotter, Ph.D., 2012
- 223. Learning the Facts
Limited success when:
• Based on counting.
Whether dots, fingers, number lines, or
counting words.
© Joan A. Cotter, Ph.D., 2012
- 224. Learning the Facts
Limited success when:
• Based on counting.
Whether dots, fingers, number lines, or
counting words.
• Based on rote memory.
Whether by flash cards or timed tests.
© Joan A. Cotter, Ph.D., 2012
- 225. Learning the Facts
Limited success when:
• Based on counting.
Whether dots, fingers, number lines, or
counting words.
• Based on rote memory.
Whether by flash cards or timed tests.
• Based on skip counting for multiplication facts.
© Joan A. Cotter, Ph.D., 2012
- 230. Fact Strategies
Complete the Ten
9+5=
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 231. Fact Strategies
Complete the Ten
9+5=
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 232. Fact Strategies
Complete the Ten
9+5=
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 233. Fact Strategies
Complete the Ten
9 + 5 = 14
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 238. Fact Strategies
Two Fives
8+6=
10 + 4 = 14
© Joan A. Cotter, Ph.D., 2012
- 241. Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 242. Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 243. Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 244. Fact Strategies
Going Down
15 – 9 = 6
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 245. Fact Strategies
Subtract from 10
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
- 246. Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 247. Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 248. Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 249. Fact Strategies
Subtract from 10
15 – 9 = 6
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 251. Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 252. Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 253. Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 254. Fact Strategies
Going Up
15 – 9 =
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 255. Fact Strategies
Going Up
15 – 9 =
1+5=6
Start with 9;
go up to 15.
© Joan A. Cotter, Ph.D., 2012
- 256. Rows and Columns Game
Objective:
To find a total of 15 by adding 2, 3, or 4
cards in a row or in a column.
© Joan A. Cotter, Ph.D., 2012
- 257. Rows and Columns Game
Objective:
To find a total of 15 by adding 2, 3, or 4
cards in a row or in a column.
Object of the game:
To collect the most cards.
© Joan A. Cotter, Ph.D., 2012
- 258. Rows and Columns Game
8 7 1 9
6 4 3 3
2 2 5 6
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
- 259. Rows and Columns Game
8 7 1 9
6 4 3 3
2 2 5 6
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
- 260. Rows and Columns Game
8 7 1 9
6 4 3 3
2 2 5 6
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
- 262. Rows and Columns Game
7 6 1 9
6 4 3 3
2 1 5 1
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
- 263. Rows and Columns Game
7 6 1 9
6 4 3 3
2 1 5 1
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
- 264. Rows and Columns Game
7 6 1 9
6 4 3 3
2 1 5 1
6 3 8 8
© Joan A. Cotter, Ph.D., 2012
- 275. Place Value
Two aspects
Static
© Joan A. Cotter, Ph.D., 2012
- 276. Place Value
Two aspects
Static
• Value of a digit is determined by position
© Joan A. Cotter, Ph.D., 2012
- 277. Place Value
Two aspects
Static
• Value of a digit is determined by position.
• No position may have more than nine.
© Joan A. Cotter, Ph.D., 2012
- 278. 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.
© Joan A. Cotter, Ph.D., 2012
- 279. 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 Decimal Cards.)
© Joan A. Cotter, Ph.D., 2012
- 280. 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 Decimal Cards.)
Dynamic
© Joan A. Cotter, Ph.D., 2012
- 281. 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 Decimal Cards.)
Dynamic
• 10 ones = 1 ten; 10 tens = 1 hundred;
10 hundreds = 1 thousand, ….
© Joan A. Cotter, Ph.D., 2012
- 282. 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 Decimal Cards.)
Dynamic
• 10 ones = 1 ten; 10 tens = 1 hundred;
10 hundreds = 1 thousand, ….
(Represented on the Abacus and other materials.)
© Joan A. Cotter, Ph.D., 2012
- 284. Exchanging
Thousands
1000 100 10 1
© Joan A. Cotter, Ph.D., 2012
- 285. Exchanging
Hundreds
1000 100 10 1
© Joan A. Cotter, Ph.D., 2012
- 286. Exchanging
Tens
1000 100 10 1
© Joan A. Cotter, Ph.D., 2012
- 287. Exchanging
Ones
1000 100 10 1
© Joan A. Cotter, Ph.D., 2012
- 288. Exchanging
Adding
1000 100 10 1
8
+6
© Joan A. Cotter, Ph.D., 2012
- 289. Exchanging
Adding
1000 100 10 1
8
+6
© Joan A. Cotter, Ph.D., 2012
- 290. Exchanging
Adding
1000 100 10 1
8
+6
© Joan A. Cotter, Ph.D., 2012
- 291. Exchanging
Adding
1000 100 10 1
8
+6
© Joan A. Cotter, Ph.D., 2012
- 292. Exchanging
Adding
1000 100 10 1
8
+6
14
© Joan A. Cotter, Ph.D., 2012
- 293. Exchanging
Adding
1000 100 10 1
8
+6
14
Too many ones;
trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2012
- 294. Exchanging
Adding
1000 100 10 1
8
+6
14
Too many ones;
trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2012
- 295. Exchanging
Adding
1000 100 10 1
8
+6
14
Too many ones;
trade 10 ones for
1 ten.
© Joan A. Cotter, Ph.D., 2012
- 296. Exchanging
Adding
1000 100 10 1
8
+6
14
Same answer
before and after
exchanging.
© Joan A. Cotter, Ph.D., 2012
- 297. Bead Frame
1
10
100
1000
© Joan A. Cotter, Ph.D., 2012
- 298. Bead Frame
1 8
10 +6
100
1000
© Joan A. Cotter, Ph.D., 2012
- 299. Bead Frame
1 8
10 +6
100
1000
© Joan A. Cotter, Ph.D., 2012
- 300. Bead Frame
1 8
10 +6
100
1000
© Joan A. Cotter, Ph.D., 2012
- 301. Bead Frame
1 8
10 +6
100
1000
© Joan A. Cotter, Ph.D., 2012
- 302. Bead Frame
1 8
10 +6
100
1000
© Joan A. Cotter, Ph.D., 2012
- 303. Bead Frame
1 8
10 +6
100
1000
© Joan A. Cotter, Ph.D., 2012
- 304. Bead Frame
1 8
10 +6
100
1000
© Joan A. Cotter, Ph.D., 2012
- 305. Bead Frame
1 8
10 +6
100
1000
© Joan A. Cotter, Ph.D., 2012
- 306. Bead Frame
1 8
10 +6
100
1000
© Joan A. Cotter, Ph.D., 2012
- 307. Bead Frame
1 8
10 +6
100 14
1000
© Joan A. Cotter, Ph.D., 2012
- 308. 1
Bead Frame
10
100
1000
Difficulties for the child
© Joan A. Cotter, Ph.D., 2012
- 309. 1
Bead Frame
10
100
1000
Difficulties for the child
• Not visualizable: Beads need to be grouped in fives.
© Joan A. Cotter, Ph.D., 2012
- 310. 1
Bead Frame
10
100
1000
Difficulties for the child
• Not visualizable: Beads need to be grouped in fives.
• When beads are moved right, inconsistent with
equation order: Beads need to be moved left.
© Joan A. Cotter, Ph.D., 2012
- 311. 1
Bead Frame
10
100
1000
Difficulties for the child
• Not visualizable: Beads need to be grouped in fives.
• When beads are moved right, inconsistent with
equation order: Beads need to be moved left.
• Hierarchies of numbers represented sideways:
They need to be in vertical columns.
© Joan A. Cotter, Ph.D., 2012
- 312. 1
Bead Frame
10
100
1000
Difficulties for the child
• Not visualizable: Beads need to be grouped in fives.
• When beads are moved right, inconsistent with
equation order: Beads need to be moved left.
• Hierarchies of numbers represented sideways:
They need to be in vertical columns.
• Exchanging done before second number is
completely added: Addends need to be combined before
exchanging.
© Joan A. Cotter, Ph.D., 2012
- 313. 1
Bead Frame
10
100
1000
Difficulties for the child
• Not visualizable: Beads need to be grouped in fives.
• When beads are moved right, inconsistent with
equation order: Beads need to be moved left.
• Hierarchies of numbers represented sideways:
They need to be in vertical columns.
• Exchanging done before second number is
completely added: Addends need to be combined before
exchanging.
• Answer is read going up: We read top to bottom.
© Joan A. Cotter, Ph.D., 2012
- 314. 1
Bead Frame
10
100
1000
Difficulties for the child
• Not visualizable: Beads need to be grouped in fives.
• When beads are moved right, inconsistent with
equation order: Beads need to be moved left.
• Hierarchies of numbers represented sideways:
They need to be in vertical columns.
• Exchanging before second number is completely
added: Addends need to be combined before exchanging.
• Answer is read going up: We read top to bottom.
• Distracting: Room is visible through the frame.
© Joan A. Cotter, Ph.D., 2012
- 315. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
© Joan A. Cotter, Ph.D., 2012
- 316. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 317. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 318. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 319. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 320. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 321. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Enter the first
number from left
to right.
© Joan A. Cotter, Ph.D., 2012
- 322. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 323. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 324. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 325. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 326. Exchanging
Adding 4-digit numbers
1000 100 10 1
3658
+ 2738
6
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 327. Exchanging
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
6
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 328. Exchanging
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
6
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 329. Exchanging
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
6
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 330. Exchanging
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 331. Exchanging
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 332. Exchanging
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 333. Exchanging
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 334. Exchanging
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
96
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 335. Exchanging
Adding 4-digit numbers
1000 100 10 1 1
3658
+ 2738
396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 336. Exchanging
Adding 4-digit numbers
1000 100 10 1 1 1
3658
+ 2738
396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 337. Exchanging
Adding 4-digit numbers
1000 100 10 1 1 1
3658
+ 2738
396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
- 338. Exchanging
Adding 4-digit numbers
1000 100 10 1 1 1
3658
+ 2738
396
Add starting at
the right. Write
results after each
step.
© Joan A. Cotter, Ph.D., 2012
Notas do Editor
- Show the baby 2 bears.
- Show the baby 2 bears.
- Show the baby 2 bears.
- Show the baby 2 bears.
- Stairs