More Related Content More from rightstartmath (20) 1. Enriching Montessori
Mathematics with Visualization
by Joan A. Cotter, Ph.D.
JoanCotter@rightstartmath.com
1000 3 2
5 5
100
10
7 x7 1
NJMAC Conference
March 2, 2012
Edison, New Jersey
Presentations available: rightstartmath.com © Joan A. Cotter, Ph.D., 2012
3. 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
3 © Joan A. Cotter, Ph.D., 2012
10. Verbal Counting Model
From a child's perspective
F
+E
A B C D E F A B
10 © Joan A. Cotter, Ph.D., 2012
11. Verbal Counting Model
From a child's perspective
F
+E
A B C D E F A B C D E
11 © 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
What is the sum?
(It must be a letter.)
12 © Joan A. Cotter, Ph.D., 2012
13. Verbal Counting Model
From a child's perspective
F
+E
K
A B C D E F G H I J K
13 © Joan A. Cotter, Ph.D., 2012
14. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
G
+D
14 © Joan A. Cotter, Ph.D., 2012
15. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
+
G
F
+D
15 © Joan A. Cotter, Ph.D., 2012
16. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
+
G
F
+D
D
+C
16 © 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
+C +G
17 © Joan A. Cotter, Ph.D., 2012
18. Verbal Counting Model
From a child's perspective
Now memorize the facts!!
H
E
+
G
I
F
+
+D
D C
+C +G
18 © Joan A. Cotter, Ph.D., 2012
19. Verbal Counting Model
From a child's perspective
Try subtracting H
by “taking away” –E
19 © Joan A. Cotter, Ph.D., 2012
20. Verbal Counting Model
From a child's perspective
Try skip counting by B’s to T:
B, D, . . . T.
20 © Joan A. Cotter, Ph.D., 2012
21. Verbal Counting Model
From a child's perspective
Try skip counting by B’s to T:
B, D, . . . T.
What is D × E?
21 © Joan A. Cotter, Ph.D., 2012
22. Verbal Counting Model
From a child's perspective
L
is written AB
because it is A J
and B A’s
22 © Joan A. Cotter, Ph.D., 2012
23. Verbal Counting Model
From a child's perspective
L
is written AB
because it is A J
and B A’s
huh?
23 © Joan A. Cotter, Ph.D., 2012
24. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB
because it is A J
and B A’s
24 © Joan A. Cotter, Ph.D., 2012
25. Verbal Counting Model
From a child's perspective
L (twelve)
is written AB (12)
because it is A J
and B A’s
25 © Joan A. Cotter, Ph.D., 2012
26. 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
26 © 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 (one 10)
and B A’s (two 1s).
27 © Joan A. Cotter, Ph.D., 2012
29. Verbal Counting Model
Summary
• Is not natural; it takes years of practice.
29 © Joan A. Cotter, Ph.D., 2012
30. Verbal Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
30 © Joan A. Cotter, Ph.D., 2012
31. Verbal Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
31 © Joan A. Cotter, Ph.D., 2012
32. Verbal Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
• Is very error prone.
32 © Joan A. Cotter, Ph.D., 2012
33. Verbal Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
• Is very error prone.
• Is tedious and time-consuming.
33 © Joan A. Cotter, Ph.D., 2012
34. Verbal Counting Model
Summary
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
• Is very error prone.
• Is tedious and time-consuming.
• Does not provide an efficient way
to master the facts.
34 © Joan A. Cotter, Ph.D., 2012
35. 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
Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 2012
35
36. 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
Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 2012
36
37. 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
37 © Joan A. Cotter, Ph.D., 2012
38. 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
This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.Joan A. Cotter, Ph.D., 2012
©
38
39. Calendar Math
Septemb
1234567
August
89101214
1
113
11921
2
15112628
122820
8
67527
9
3 4
10 11 12 13 14
5 6 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, not cardinal counting. The 4 doesn’t include 1, 2 and 3. © Joan A. Cotter, Ph.D., 2012
39
40. Calendar Math
Septemb
1234567
August
89101214
1
113
11921
2
15112628
122820
8
67527
9
3 4 5
10 11 12 13 14
6 7
2234
20
15 16 17 18 19 20 21
29
3
22 23 24 25 26 27 28
29 30 31
1 2 3 4 5 6
A calendar is NOT a ruler. On a ruler the numbers are not in the spaces. © Joan A. Cotter, Ph.D., 2012
40
41. Calendar Math
August
1 2 3 4 5 6 7
8 9 10
Always show the whole calendar. A child needs to see the whole
before the parts. Children also need to learn to plan ahead. © Joan A. Cotter, Ph.D., 2012
41
42. Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines like a ruler.
42 © Joan A. Cotter, Ph.D., 2012
43. Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines like a ruler.
Children need to see the whole month, not just part.
• Purpose of calendar is to plan ahead.
• Many ways to show the current date.
43 © Joan A. Cotter, Ph.D., 2012
44. Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines like a ruler.
Children need to see the whole month, not just part.
• Purpose of calendar is to plan ahead.
• Many ways to show the current date.
Calendars give a narrow view of patterning.
• Patterns do not necessarily involve numbers.
• Patterns rarely proceed row by row.
• Patterns go on forever; they don’t stop at 31.
44 © Joan A. Cotter, Ph.D., 2012
45. Memorizing Math 9
+7
Flash cards:
• Are often used to teach rote.
• Are liked only by those who don’t need them.
• Don’t work for those with learning disabilities.
• Give the false impression that math isn’t about
thinking.
• Often produce stress – children under stress
stop learning.
• Are not concrete – use abstract symbols.
© Joan A. Cotter, Ph.D., 2012
46. Learning Arithmetic
Compared to reading:
• A child learns to read.
• Later a child uses reading to learn.
• A child learns to do arithmetic.
• Later a child uses arithmetic to solve problems.
Show the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
47. Research on Counting
Karen Wynn’s research
Show the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
48. Research on Counting
Karen Wynn’s research
Show the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
49. Research on Counting
Karen Wynn’s research
Then hide them with a screen. © Joan A. Cotter, Ph.D., 2012
49
50. Research on Counting
Karen Wynn’s research
Show the baby a third teddy bear and put it behind the screen. © Joan A. Cotter, Ph.D., 2012
50
51. Research on Counting
Karen Wynn’s research
Show the baby a third teddy bear and put it behind the screen. © Joan A. Cotter, Ph.D., 2012
51
52. Research on Counting
Karen Wynn’s research
Raise screen. Baby seeing 3 won’t look long because it is expected. © Joan A. Cotter, Ph.D., 2012
52
53. Research on Counting
Karen Wynn’s research
Researcher can change the number of teddy bears behind the screen. © Joan A. Cotter, Ph.D., 2012
53
54. Research on Counting
Karen Wynn’s research
A baby seeing 1 teddy bear will look much longer, because it’s unexpected.Joan A. Cotter, Ph.D., 2012
©
54
56. Research on Counting
Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.
These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 2012
56
57. 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.
These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 2012
57
58. 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.
These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 2012
58
59. 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.
These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 2012
59
60. Research on Counting
In Japanese schools:
• Children are discouraged from using
counting for adding.
60 © Joan A. Cotter, Ph.D., 2012
61. Research on Counting
In Japanese schools:
• Children are discouraged from using
counting for adding.
• They consistently group in 5s.
61 © Joan A. Cotter, Ph.D., 2012
62. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
62 © Joan A. Cotter, Ph.D., 2012
63. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
• Human babies and some animals can subitize
small quantities at birth.
63 © Joan A. Cotter, Ph.D., 2012
64. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
• Human babies and some animals can subitize
small quantities at birth.
• Children who can subitize perform better in
mathematics long term.—Butterworth
64 © Joan A. Cotter, Ph.D., 2012
65. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
• Human babies and some animals can subitize
small quantities at birth.
• Children who can subitize perform better in
mathematics long term.—Butterworth
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit
65 © Joan A. Cotter, Ph.D., 2012
66. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
• Human babies and some animals can subitize
small quantities at birth.
• Children who can subitize perform better in
mathematics long term.—Butterworth
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit
• Subitizing seems to be a necessary skill for
understanding what the counting process means.—
Glasersfeld
66 © Joan A. Cotter, Ph.D., 2012
68. Visualizing Mathematics
“In our concern about the memorization of
math facts or solving problems, we must not
forget that the root of mathematical study is
the creation of mental pictures in the
imagination and manipulating those images
and relationships using the power of reason
and logic.”
Mindy Holte (E I)
68 © Joan A. Cotter, Ph.D., 2012
69. Visualizing Mathematics
“Think in pictures, because the
brain remembers images better
than it does anything else.”
Ben Pridmore, World Memory Champion, 2009
69 © Joan A. Cotter, Ph.D., 2012
70. Visualizing Mathematics
“The role of physical manipulatives
was to help the child form those
visual images and thus to eliminate
the need for the physical
manipulatives.”
Ginsberg and others
70 © Joan A. Cotter, Ph.D., 2012
71. Visualizing Mathematics
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
72. Visualizing Mathematics
Visualizing also needed in:
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction
© Joan A. Cotter, Ph.D., 2012
73. Visualizing Mathematics
Visualizing also needed in:
• Reading • Architecture
• Sports • Astronomy
• Creativity • Archeology
• Geography • Chemistry
• Engineering • Physics
• Construction • Surgery
© Joan A. Cotter, Ph.D., 2012
85. Visualizing Mathematics
Early Roman numerals
1 I
2 II
3 III
4 IIII
5 V
8 VIII
Romans grouped in fives. Notice 8 is 5 and 3. © Joan A. Cotter, Ph.D., 2012
86. Visualizing Mathematics
:
Who could read the music?
Music needs 10 lines, two groups of five. © Joan A. Cotter, Ph.D., 2012
86
87. Very Early Computation
Numerals
In English there are two ways of writing numbers:
Numerals: 3578
87 © Joan A. Cotter, Ph.D., 2012
88. Very Early Computation
Numerals
In English there are two ways of writing numbers:
Numerals: 3578
Words: Three thousand five hundred seventy-eight
88 © Joan A. Cotter, Ph.D., 2012
89. Very Early Computation
Numerals
In English there are two ways of writing numbers:
Numerals: 3578
Words: Three thousand five hundred seventy-eight
In ancient Chinese there was only one way of writing
numbers:
3 Th 5 H 7 T 8 U
(8 characters)
89 © Joan A. Cotter, Ph.D., 2012
90. Very Early Computation
Calculating rods
Because their characters are cumbersome
to use for computing, the Chinese used
calculating rods, beginning in the 4th
century BC.
90 © Joan A. Cotter, Ph.D., 2012
92. Very Early Computation
Calculating rods
Numerals for Ones and Hundreds (Even Powers of Ten)
92 © Joan A. Cotter, Ph.D., 2012
93. Very Early Computation
Calculating rods
Numerals for Ones and Hundreds (Even Powers of Ten)
93 © Joan A. Cotter, Ph.D., 2012
94. Very Early Computation
Calculating rods
Numerals for Ones and Hundreds (Even Powers of Ten)
Numerals for Tens and Thousands (Odd Powers of Ten)
94 © Joan A. Cotter, Ph.D., 2012
96. Very Early Computation
Calculating rods
3578
3578,3578
They grouped, not in thousands, but ten-thousands!
96 © Joan A. Cotter, Ph.D., 2012
98. Naming Quantities
Using fingers
Naming quantities is a three-period lesson.
© Joan A. Cotter, Ph.D., 2012
99. Naming Quantities
Using fingers
Use left hand for 1-5 because we read from left to right. © Joan A. Cotter, Ph.D., 2012
102. Naming Quantities
Using fingers
Always show 7 as 5 and 2, not for example, as 4 and 3. © Joan A. Cotter, Ph.D., 2012
102
104. Naming Quantities
Yellow is the Sun
Yellow is the sun.
Six is five and one.
Why is the sky so blue?
Seven is five and two.
Salty is the sea.
Eight is five and three.
Hear the thunder roar.
Nine is five and four.
Ducks will swim and dive.
Ten is five and five.
–Joan A. Cotter
Also set to music. Listen and download sheet music from Web site. © Joan A. Cotter, Ph.D., 2012
107. Naming Quantities
Recognizing 5
5 has a middle; 4 does not.
Look at your hand; your middle finger is longer to remind you 5 has a middle. A. Cotter, Ph.D., 2012
© Joan
108. Naming Quantities
Tally sticks
Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart. © Joan A. Cotter, Ph.D., 2012
111. Naming Quantities
Tally sticks
Stick is horizontal, because it won’t fit diagonally and young children have
problems with diagonals.
111 © Joan A. Cotter, Ph.D., 2012
113. Naming Quantities
Tally sticks
Start a new row for every ten. © Joan A. Cotter, Ph.D., 2012
113
114. Naming Quantities
Solving a problem without counting
What is 4 apples plus 3 more apples?
How would you find the answer without counting? © Joan A. Cotter, Ph.D., 2012
114
115. Naming Quantities
Solving a problem without counting
What is 4 apples plus 3 more apples?
To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Then
take 1 from the 3 and give it to the 4 to make 5 and 2. © Joan A. Cotter, Ph.D., 2012
115
118. Naming Quantities
Number
Chart 1 6
2 7
To help the 3 8
child learn
the symbols
4 9
5 10
© Joan A. Cotter, Ph.D., 2012
119. Naming Quantities
Pairing Finger Cards
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areTIFF (LZW) decompressor
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Use two sets of finger cards and match them. © Joan A. Cotter, Ph.D., 2012
119
120. Naming Quantities
Ordering Finger Cards
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are needed to see this picture. QuickTimeª and a
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Putting the finger cards in order. © Joan A. Cotter, Ph.D., 2012
120
121. Naming Quantities
Matching Numbers to Finger Cards
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Match the number to the finger card. © Joan A. Cotter, Ph.D., 2012
121
122. Naming Quantities
Matching Fingers to Number Cards
9 1 10 4 6
QuickTimeª and a
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2 3 7 8 5
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Match the finger card to the number. © Joan A. Cotter, Ph.D., 2012
122
123. Naming Quantities
Finger Card Memory game
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Use two sets of finger cards and play Memory. © Joan A. Cotter, Ph.D., 2012
123
126. Naming Quantities
Number Rods
Using different colors. © Joan A. Cotter, Ph.D., 2012
126
127. Naming Quantities
Spindle Box
45 dark-colored and 10 light-colored spindles. Could be in separate containers. Cotter, Ph.D., 2012
© Joan A.
127
128. Naming Quantities
Spindle Box
45 dark-colored and 10 light-colored spindles in two containers. © Joan A. Cotter, Ph.D., 2012
128
129. Naming Quantities
Spindle Box
0 1 2 3 4
The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
129
130. Naming Quantities
Spindle Box
5 6 7 8 9
The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
130
131. Naming Quantities
Spindle Box
5 6 7 8 9
The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
131
132. Naming Quantities
Spindle Box
5 6 7 8 9
The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
132
133. Naming Quantities
Spindle Box
5 6 7 8 9
The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
133
134. Naming Quantities
Spindle Box
5 6 7 8 9
The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
134
135. Naming Quantities
Spindle Box
5 6 7 8 9
The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012
135
136. Naming Quantities
Black and White Bead Stairs
“Grouped in fives so the child does not
need to count.” A. M. Joosten
This was the inspiration to group in 5s. © Joan A. Cotter, Ph.D., 2012
136
137. AL Abacus
Cleared
© Joan A. Cotter, Ph.D., 2012
138. AL Abacus
Entering quantities
3
Quantities are entered all at once, not counted. © Joan A. Cotter, Ph.D., 2012
139. AL Abacus
Entering quantities
5
Relate quantities to hands. © Joan A. Cotter, Ph.D., 2012
139
140. AL Abacus
Entering quantities
7
140 © Joan A. Cotter, Ph.D., 2012
141. AL Abacus
Entering quantities
10
141 © Joan A. Cotter, Ph.D., 2012
142. AL Abacus
The stairs
Can use to “count” 1 to 10. Also read quantities on the right side. © Joan A. Cotter, Ph.D., 2012
142
143. AL Abacus
Adding
© Joan A. Cotter, Ph.D., 2012
144. AL Abacus
Adding
4+3=
© Joan A. Cotter, Ph.D., 2012
145. AL Abacus
Adding
4+3=
© Joan A. Cotter, Ph.D., 2012
146. AL Abacus
Adding
4+3=
© Joan A. Cotter, Ph.D., 2012
147. AL Abacus
Adding
4+3=
© Joan A. Cotter, Ph.D., 2012
148. AL Abacus
Adding
4+3=7
Answer is seen immediately, no counting needed. © Joan A. Cotter, Ph.D., 2012
149. Go to the Dump Game
Aim:
To learn the facts that total 10:
1+9
2+8
3+7
4+6
5+5
Children use the abacus while playing this “Go Fish” type game. © Joan A. Cotter, Ph.D., 2012
149
150. Go to the Dump Game
Aim:
To learn the facts that total 10:
1+9
2+8
3+7
4+6
5+5
Object of the game:
To collect the most pairs that equal ten.
Children use the abacus while playing this “Go Fish” type game. © Joan A. Cotter, Ph.D., 2012
150
151. Go to the Dump Game
The ways to partition 10. © Joan A. Cotter, Ph.D., 2012
151
152. Go to the Dump Game
Starting
A game viewed from above. © Joan A. Cotter, Ph.D., 2012
152
153. Go to the Dump Game
72 7 9 5
72 1 3 8 4 6 34 9
Starting
Each player takes 5 cards. © Joan A. Cotter, Ph.D., 2012
153
154. Go to the Dump Game
72 7 9 5
72 1 3 8 4 6 34 9
Finding pairs
Does YellowCap have any pairs? [no] © Joan A. Cotter, Ph.D., 2012
154
155. Go to the Dump Game
72 7 9 5
72 1 3 8 4 6 34 9
Finding pairs
Does BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012
155
156. Go to the Dump Game
72 7 9 5
72 1 3 8 4 6 34 9
Finding pairs
Does BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012
156
157. Go to the Dump Game
72 7 9 5
4 6
72 1 3 8 34 9
Finding pairs
Does BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012
157
158. Go to the Dump Game
72 7 9 5
4 6
72 1 3 8 34 9
Finding pairs
Does PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012
158
159. Go to the Dump Game
72 7 9 5
4 6
72 1 3 8 34 9
Finding pairs
Does PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012
159
160. Go to the Dump Game
72 7 9 5
7 3 4 6
2 1 8 34 9
Finding pairs
Does PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012
160
161. Go to the Dump Game
72 7 9 5
2 8 4 6
1 34 9
Finding pairs
Does PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012
161
162. Go to the Dump Game
72 7 9 5
2 8 4 6
1 34 9
Playing
The player asks the player on her left. © Joan A. Cotter, Ph.D., 2012
162
163. Go to the Dump Game
BlueCap, do you
have an3?
have a 3?
72 7 9 5
2 8 4 6
1 34 9
Playing
The player asks the player on her left. © Joan A. Cotter, Ph.D., 2012
163
164. Go to the Dump Game
BlueCap, do you
have an3?
have a 3?
72 7 9 5 3
2 8 4 6
1 4 9
Playing
The player asks the player on her left. © Joan A. Cotter, Ph.D., 2012
164
165. Go to the Dump Game
7 3 BlueCap, do you
have an3?
have a 3?
2 7 9 5
2 8 4 6
1 4 9
Playing
165 © Joan A. Cotter, Ph.D., 2012
166. Go to the Dump Game
7 3 BlueCap, do you
have an3?
have a 8?
2 7 9 5
2 8 4 6
1 4 9
Playing
YellowCap gets another turn. © Joan A. Cotter, Ph.D., 2012
166
167. Go to the Dump Game
7 3 BlueCap, do you
have an3?
have a 8?
2 7 9 5
2 8 4 6
1 4 9
Go to the dump.
Playing
YellowCap gets another turn. © Joan A. Cotter, Ph.D., 2012
167
168. Go to the Dump Game
7 3 BlueCap, do you
have an3?
have a 8?
2 2 7 9 5
2 8 4 6
1 4 9
Go to the dump.
Playing
168 © Joan A. Cotter, Ph.D., 2012
169. Go to the Dump Game
7 3
2 2 7 9 5
2 8 4 6
1 4 9
Playing
169 © Joan A. Cotter, Ph.D., 2012
170. Go to the Dump Game
7 3
2 2 7 9 5
2 8 4 6
1 4 9
PinkCap, do you
Playing have a 6?
170 © Joan A. Cotter, Ph.D., 2012
171. Go to the Dump Game
7 3
2 2 7 9 5
2 8 4 6
1 4 9
PinkCap, do you
Go to the dump.
Playing have a 6?
171 © Joan A. Cotter, Ph.D., 2012
172. Go to the Dump Game
7 3
2 2 7 9 5
2 8 4 6
1 5 4 9
Playing
172 © Joan A. Cotter, Ph.D., 2012
173. Go to the Dump Game
7 3
2 2 7 9 5
2 8 4 6
1 5 4 9
Playing
173 © Joan A. Cotter, Ph.D., 2012
174. Go to the Dump Game
7 3
2 2 7 9 5
2 8 4 6
1 5 4 9
YellowCap, do
you have a 9? Playing
174 © Joan A. Cotter, Ph.D., 2012
175. Go to the Dump Game
7 3
2 2 7 5
2 8 4 6
1 5 4 9
YellowCap, do
you have a 9? Playing
175 © Joan A. Cotter, Ph.D., 2012
176. Go to the Dump Game
7 3
2 2 7 5
2 8 4 6
19 5 4 9
YellowCap, do
you have a 9? Playing
176 © Joan A. Cotter, Ph.D., 2012
177. Go to the Dump Game
7 3
2 2 7 5
2
1 8
9 4 6
5 4 9
Playing
177 © Joan A. Cotter, Ph.D., 2012
178. Go to the Dump Game
7 3
2 2 7 5
2
1 8
9 4 6
2 9 1 7 7 5 4 9
Playing
PinkCap is not out of the game. Her turn ends, but she takes 5 more cards.Joan A. Cotter, Ph.D., 2012
©
178
179. Go to the Dump Game
9 1
4 6 5 5
Winner?
179 © Joan A. Cotter, Ph.D., 2012
180. Go to the Dump Game
9
1
4
6 5
Winner?
No counting. Combine both stacks. © Joan A. Cotter, Ph.D., 2012
180
181. Go to the Dump Game
9
1
4
6 5
Winner?
Whose stack is the highest? © Joan A. Cotter, Ph.D., 2012
181
182. Go to the Dump Game
Next game
No shuffling needed for next game. © Joan A. Cotter, Ph.D., 2012
182
183. “Math” Way of Naming Numbers
183 © Joan A. Cotter, Ph.D., 2012
184. “Math” Way of Naming Numbers
11 = ten 1
184 © Joan A. Cotter, Ph.D., 2012
185. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
185 © Joan A. Cotter, Ph.D., 2012
186. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
186 © Joan A. Cotter, Ph.D., 2012
187. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
187 © Joan A. Cotter, Ph.D., 2012
188. “Math” Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
188 © Joan A. Cotter, Ph.D., 2012
189. “Math” Way of Naming Numbers
11 = ten 1 20 = 2-ten
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012
189
190. “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
Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012
190
191. “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
Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012
191
192. “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
Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012
192
193. “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
193 © Joan A. Cotter, Ph.D., 2012
194. “Math” Way of Naming Numbers
137 = 1 hundred 3-ten 7
Only numbers under 100 need to be said the “math” way. © Joan A. Cotter, Ph.D., 2012
194
195. “Math” Way of Naming Numbers
137 = 1 hundred 3-ten 7
or
137 = 1 hundred and 3-ten 7
Only numbers under 100 need to be said the “math” way. © Joan A. Cotter, Ph.D., 2012
195
196. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Shows how far children from 3 countries can count at ages 4, 5, and 6. © Joan A. Cotter, Ph.D., 2012
196
197. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Purple is Chinese. Note jump between ages 5 and 6. © Joan A. Cotter, Ph.D., 2012
197
198. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Dark green is Korean “math” way. © Joan A. Cotter, Ph.D., 2012
198
199. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Dotted green is everyday Korean; notice smaller jump between ages 5 and Joan A. Cotter, Ph.D., 2012
©
6.
199
200. “Math” Way of Naming Numbers
100 Chinese
Average Highest Number Counted
U.S.
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.
Red is English speakers. They learn same amount between ages 4-5 and©5-6. Cotter, Ph.D., 2012
Joan A.
200
201. 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.)
201 © Joan A. Cotter, Ph.D., 2012
202. 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.
202 © Joan A. Cotter, Ph.D., 2012
203. 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.
203 © Joan A. Cotter, Ph.D., 2012
204. 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.
204 © Joan A. Cotter, Ph.D., 2012
205. Math Way of Naming Numbers
Compared to reading:
205 © Joan A. Cotter, Ph.D., 2012
206. Math Way of Naming Numbers
Compared to reading:
• Just as reciting the alphabet doesn’t teach reading,
counting doesn’t teach arithmetic.
206 © Joan A. Cotter, Ph.D., 2012
207. 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).
207 © Joan A. Cotter, Ph.D., 2012
208. 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 do use the math way of naming
numbers but are too quick to switch to traditional
names. Use the math way for a longer period of time.
208 © Joan A. Cotter, Ph.D., 2012
209. 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
209 © Joan A. Cotter, Ph.D., 2012
210. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
210 © Joan A. Cotter, Ph.D., 2012
211. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
211 © Joan A. Cotter, Ph.D., 2012
212. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones count 14.
212 © Joan A. Cotter, Ph.D., 2012
213. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
213 © Joan A. Cotter, Ph.D., 2012
214. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
214 © Joan A. Cotter, Ph.D., 2012
215. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
215 © Joan A. Cotter, Ph.D., 2012
216. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
216 © Joan A. Cotter, Ph.D., 2012
217. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
217 © Joan A. Cotter, Ph.D., 2012
218. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
218 © Joan A. Cotter, Ph.D., 2012
219. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children thinking of 14 as 14 ones counted 14.
219 © Joan A. Cotter, Ph.D., 2012
220. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children who understand tens remove a ten and 4 ones.
220 © Joan A. Cotter, Ph.D., 2012
221. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children who understand tens remove a ten and 4 ones.
221 © Joan A. Cotter, Ph.D., 2012
222. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask the
child to construct 48.
Then ask the child to
subtract 14.
Children who understand tens remove a ten and 4 ones.
222 © Joan A. Cotter, Ph.D., 2012
223. Math Way of Naming Numbers
Traditional names
4-ten =
forty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2012
224. Math Way of Naming Numbers
Traditional names
4-ten =
forty
The “ty”
means tens.
The traditional names for 40, 60, 70, 80, and 90 follow a pattern. © Joan A. Cotter, Ph.D., 2012
225. Math Way of Naming Numbers
Traditional names
6-ten = sixty
The “ty”
means tens.
© Joan A. Cotter, Ph.D., 2012
226. 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
227. 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
228. Math Way of Naming Numbers
Traditional names
2-ten = twenty
Two used to be
pronounced
“twoo.”
© Joan A. Cotter, Ph.D., 2012
229. Math Way of Naming Numbers
Traditional names
A word game
fireplace place-fire
Say the syllables backward. This is how we say the teen numbers. © Joan A. Cotter, Ph.D., 2012
230. Math Way of Naming Numbers
Traditional names
A word game
fireplace place-fire
newspaper paper-news
Say the syllables backward. This is how we say the teen numbers. © Joan A. Cotter, Ph.D., 2012
231. Math Way of Naming Numbers
Traditional names
A word game
fireplace place-fire
newspaper paper-news
box-mail mailbox
Say the syllables backward. This is how we say the teen numbers. © Joan A. Cotter, Ph.D., 2012
232. Math Way of Naming Numbers
Traditional names
ten 4
“Teen” also
means ten.
© Joan A. Cotter, Ph.D., 2012
233. Math Way of Naming Numbers
Traditional names
ten 4 teen 4
“Teen” also
means ten.
© Joan A. Cotter, Ph.D., 2012
234. Math Way of Naming Numbers
Traditional names
ten 4 teen 4 fourtee
n
“Teen” also
means ten.
© Joan A. Cotter, Ph.D., 2012
235. Math Way of Naming Numbers
Traditional names
a one left
© Joan A. Cotter, Ph.D., 2012
236. Math Way of Naming Numbers
Traditional names
a one left a left-one
© Joan A. Cotter, Ph.D., 2012
237. Math Way of Naming Numbers
Traditional names
a one left a left-one eleven
© Joan A. Cotter, Ph.D., 2012
238. Math Way of Naming Numbers
Traditional names
two left
Two
pronounced
“twoo.”
© Joan A. Cotter, Ph.D., 2012
239. Math Way of Naming Numbers
Traditional names
two left twelve
Two
pronounced
“twoo.”
© Joan A. Cotter, Ph.D., 2012
243. Composing Numbers
3-ten
30
Point to the 3 and say 3. © Joan A. Cotter, Ph.D., 2012
244. Composing Numbers
3-ten
30
Point to 0 and say 10. The 0 makes 3 a ten. © Joan A. Cotter, Ph.D., 2012
248. Composing Numbers
3-ten 7
30
7
Place the 7 on top of the 0 of the 30. © Joan A. Cotter, Ph.D., 2012
249. Composing Numbers
3-ten 7
30
7
Notice the way we say the number, represent the
number, and write the number all correspond.
© Joan A. Cotter, Ph.D., 2012
251. Composing Numbers
7-ten 6
78
6
In the UK, pupils are expected to know the
amount remaining: 24, that is 100 – 76.
Another example. © Joan A. Cotter, Ph.D., 2012
258. Composing Numbers
1 hundred
100
Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
259. Composing Numbers
1 hundred
100
Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
260. Composing Numbers
1 hundred
100
Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
261. Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading
of numbers and text:
4258
© Joan A. Cotter, Ph.D., 2012
262. Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading
of numbers and text:
4258
262 © Joan A. Cotter, Ph.D., 2012
263. Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading
of numbers and text:
4258
263 © Joan A. Cotter, Ph.D., 2012
264. Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading
of numbers and text:
4258
264 © Joan A. Cotter, Ph.D., 2012
265. Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading
of numbers and text:
4258
The Decimal Cards encourage reading numbers
in the normal order.
265 © Joan A. Cotter, Ph.D., 2012
266. Composing Numbers
Scientific Notation
3
4000 = 4 x 10
In scientific notation, we “stand” on
the left digit and note the number of
digits to the right. (That’s why we
shouldn’t refer to the 4 as the 4th
column.)
266 © Joan A. Cotter, Ph.D., 2012
Editor's Notes Montessori math materials ingeniously introduces children to our decimal system, but current research suggests that mathematical mastery can be better facilitated with simple enhancements in teaching techniques and material extensions. In this workshop, learn about research-based math discoveries, and explore ideas for Montessori math refinements, such as grouping the materials in fives to reduce counting and help the child in forming abstract images. Show the baby 2 bears. Show the baby 2 bears. Show the baby 2 bears. Show the baby 2 bears. Show the baby 2 bears. Stairs Montessori math materials ingeniously introduces children to our decimal system, but current research suggests that mathematical mastery can be better facilitated with simple enhancements in teaching techniques and material extensions. In this workshop, learn about research-based math discoveries, and explore ideas for Montessori math refinements, such as grouping the materials in fives to reduce counting and help the child in forming abstract images.
