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© Joan A. Cotter, Ph.D., 2012
The Future of Primary Math: More Understanding/Less Counting
MCTMSaturday, May 5, 2012
Duluth, Minnesota
by Joan A. Cotter, Ph.D.JoanCotter@RightStartMath.com
3 03 077
3 03 0
77
1000 10 1100
PowerPoint PresentationRightStartMath.com >Resources
© Joan A. Cotter, Ph.D., 20122
Verbal Counting Model
© Joan A. Cotter, Ph.D., 20123
Verbal Counting ModelFrom a child's perspective
Because we’re so familiar with 1, 2, 3, we’ll use letters.
A = 1B = 2C = 3D = 4E = 5, and so forth
© Joan A. Cotter, Ph.D., 20124
Verbal Counting Model From a child's perspective
F + E
© Joan A. Cotter, Ph.D., 20125
Verbal Counting Model From a child's perspective
A
F + E
© Joan A. Cotter, Ph.D., 20126
Verbal Counting Model From a child's perspective
A B
F + E
© Joan A. Cotter, Ph.D., 20127
Verbal Counting Model From a child's perspective
A CB
F + E
© Joan A. Cotter, Ph.D., 20128
Verbal Counting Model From a child's perspective
A FC D EB
F + E
© Joan A. Cotter, Ph.D., 20129
Verbal Counting Model From a child's perspective
AA FC D EB
F + E
© Joan A. Cotter, Ph.D., 201210
Verbal Counting Model From a child's perspective
A BA FC D EB
F + E
© Joan A. Cotter, Ph.D., 201211
Verbal Counting Model From a child's perspective
A C D EBA FC D EB
F + E
© Joan A. Cotter, Ph.D., 201212
Verbal Counting Model From a child's perspective
A C D EBA FC D EB
F + E
What is the sum?(It must be a letter.)
© Joan A. Cotter, Ph.D., 201213
Verbal Counting Model From a child's perspective
K
G I J KHA FC D EB
F + E
© Joan A. Cotter, Ph.D., 201214
Verbal Counting Model From a child's perspective
Now memorize the facts!!
G + D
© Joan A. Cotter, Ph.D., 201215
Verbal Counting Model From a child's perspective
Now memorize the facts!!
G + D
H + F
© Joan A. Cotter, Ph.D., 201216
Verbal Counting Model From a child's perspective
Now memorize the facts!!
G + D
H + F
D + C
© Joan A. Cotter, Ph.D., 201217
Verbal Counting Model From a child's perspective
Now memorize the facts!!
G + D
H + F
C + G
D + C
© Joan A. Cotter, Ph.D., 201218
Verbal Counting Model From a child's perspective
E
+ I
Now memorize the facts!!
G + D
H + F
C + G
D + C
© Joan A. Cotter, Ph.D., 201219
Verbal Counting Model From a child's perspective
H – E
Subtract with your fingers by counting backward.
© Joan A. Cotter, Ph.D., 201220
Verbal Counting Model From a child's perspective
J – F
Subtract without using your fingers.
© Joan A. Cotter, Ph.D., 201221
Verbal Counting Model From a child's perspective
Try skip counting by B’s to T: B, D, . . . T.
© Joan A. Cotter, Ph.D., 201222
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., 201223
Verbal Counting Model From a child's perspective
Lis written ABbecause it is A J and B A’s
© Joan A. Cotter, Ph.D., 201224
Verbal Counting Model From a child's perspective
Lis written ABbecause it is A J and B A’s
huh?
© Joan A. Cotter, Ph.D., 201225
Verbal Counting Model From a child's perspective
Lis written ABbecause it is A J and B A’s
(twelve)
© Joan A. Cotter, Ph.D., 201226
Verbal Counting Model From a child's perspective
Lis written ABbecause it is A J and B A’s
(12)(twelve)
© Joan A. Cotter, Ph.D., 201227
Verbal Counting Model From a child's perspective
Lis written ABbecause it is A J and B A’s
(12)(one 10)
(twelve)
© Joan A. Cotter, Ph.D., 201228
Verbal Counting Model From a child's perspective
Lis written ABbecause it is A J and B A’s
(12)(one 10)
(two 1s).
(twelve)
© Joan A. Cotter, Ph.D., 201229
Calendar Math
August
29
22
15
8
1
30
23
16
9
2
24
17
10
3
25
18
11
4
26
19
12
5
27
20
13
6
28
21
14
7
31
© Joan A. Cotter, Ph.D., 201230
Calendar Math
August
29
22
15
8
1
30
23
16
9
2
24
17
10
3
25
18
11
4
26
19
12
5
27
20
13
6
28
21
14
7
31
Calendar Counting
© Joan A. Cotter, Ph.D., 201231
Calendar Math
August
29
22
15
8
1
30
23
16
9
2
24
17
10
3
25
18
11
4
26
19
12
5
27
20
13
6
28
21
14
7
31
Calendar Counting
© Joan A. Cotter, Ph.D., 201232
Calendar Math
August
29
22
15
8
1
30
23
16
9
2
24
17
10
3
25
18
11
4
26
19
12
5
27
20
13
6
28
21
14
7
31
Calendar Counting
© Joan A. Cotter, Ph.D., 201233
Calendar Math
September123489101115161718222324252930
567121314192021262728
August
29
22
15
8
1
30
23
16
9
2
24
17
10
3
25
18
11
4
26
19
12
5
27
20
13
6
28
21
14
7
31
Calendar Counting
© Joan A. Cotter, Ph.D., 201234
Calendar Math
September123489101115161718222324252930
567121314192021262728
August
29
22
15
8
1
30
23
16
9
2
24
17
10
3
25
18
11
4
26
19
12
5
27
20
13
6
28
21
14
7
31
This is ordinal counting, not cardinal counting.
Calendar Counting
© Joan A. Cotter, Ph.D., 201235
Calendar Math
August
8
1
9
2
10
3 4 5 6 7
Partial Calendar
© Joan A. Cotter, Ph.D., 201236
Calendar Math
August
8
1
9
2
10
3 4 5 6 7
Partial Calendar
Children need the whole month to plan ahead.
© Joan A. Cotter, Ph.D., 201237
Calendar Math
September123489101115161718222324252930
567121314192021262728
August
29
22
15
8
1
30
23
16
9
2
24
17
10
3
25
18
11
4
26
19
12
5
27
20
13
6
28
21
14
7
31
Patterns are rarely based on 7s or proceed row by row.Patterns go on forever; they don’t stop at 31.
Calendar Patterning
© Joan A. Cotter, Ph.D., 201238
Minnesota Standards
K: Represent quantities using whole numbers and understand relationships among whole numbers.
1–2: Understand place value and relationships among whole numbers.
Number Sense
With the counting model, how difficult are the associated benchmarks for children to master?
© Joan A. Cotter, Ph.D., 201239
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
Kindergarten
© Joan A. Cotter, Ph.D., 201240
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
Kindergarten
© Joan A. Cotter, Ph.D., 201241
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
Kindergarten
© Joan A. Cotter, Ph.D., 201242
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
Kindergarten
© Joan A. Cotter, Ph.D., 201243
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
Kindergarten
© Joan A. Cotter, Ph.D., 201244
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
• Recognize number of objects up to 6, without counting.
Kindergarten
© Joan A. Cotter, Ph.D., 201245
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
• Recognize number of objects up to 6, without counting.
• Add and subtract whole numbers up to 6, using objects.
Kindergarten
© Joan A. Cotter, Ph.D., 201246
Minnesota Standards
Understand place value and relationships among whole numbers.
Grade 1
© Joan A. Cotter, Ph.D., 201247
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
Grade 1
© Joan A. Cotter, Ph.D., 201248
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
Grade 1
© Joan A. Cotter, Ph.D., 201249
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
Grade 1
© Joan A. Cotter, Ph.D., 201250
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
• Demonstrate understanding of odd and even to 12.
Grade 1
© Joan A. Cotter, Ph.D., 201251
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
• Demonstrate understanding of odd and even to 12.
• Represent whole numbers up to 20 in various ways.
Grade 1
© Joan A. Cotter, Ph.D., 201252
Minnesota Standards
Understand place value and relationships among whole numbers.
Grade 2
© Joan A. Cotter, Ph.D., 201253
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
Grade 2
© Joan A. Cotter, Ph.D., 201254
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
Grade 2
© Joan A. Cotter, Ph.D., 201255
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
Grade 2
© Joan A. Cotter, Ph.D., 201256
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
• Demonstrate understanding of odd and even up to 12.
Grade 2
© Joan A. Cotter, Ph.D., 201257
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
• Demonstrate understanding of odd and even up to 12.
• Represent whole numbers up to 20 in various ways.
Grade 2
© Joan A. Cotter, Ph.D., 2012
Research on CountingKaren Wynn’s research
© Joan A. Cotter, Ph.D., 2012
Research on CountingKaren Wynn’s research
© Joan A. Cotter, Ph.D., 201260
Research on Counting
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 201261
Research on Counting
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 201262
Research on Counting
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 201263
Research on CountingKaren Wynn’s research
© Joan A. Cotter, Ph.D., 201264
Research on Counting
Karen Wynn’s research
© Joan A. Cotter, Ph.D., 201265
Research on CountingKaren Wynn’s research
© Joan A. Cotter, Ph.D., 201266
Research on CountingOther research
© Joan A. Cotter, Ph.D., 201267
Research on Counting
• Australian Aboriginal children from two tribes.Brian Butterworth, University College London, 2008.
Other research
© Joan A. Cotter, Ph.D., 201268
Research on Counting
• Australian Aboriginal children from two tribes.Brian Butterworth, University College London, 2008.
• Adult Pirahã from Amazon region.Edward Gibson and Michael Frank, MIT, 2008.
Other research
© Joan A. Cotter, Ph.D., 201269
Research on Counting
• 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.
Other research
© Joan A. Cotter, Ph.D., 201270
Research on Counting
• 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.
Other research
© Joan A. Cotter, Ph.D., 201271
Research on CountingIn Japanese schools:
• Children are discouraged from using counting for adding.
© Joan A. Cotter, Ph.D., 201272
Research on CountingIn Japanese schools:
• Children are discouraged from using counting for adding.
• They consistently group in 5s.
© Joan A. Cotter, Ph.D., 201273
Research on CountingSubitizing
• Subitizing is quick recognition of quantity without counting.
© Joan A. Cotter, Ph.D., 201274
Research on CountingSubitizing
• Subitizing is quick recognition of quantity without counting.
• Human babies and some animals can subitize small quantities at birth.
© Joan A. Cotter, Ph.D., 201275
Research on CountingSubitizing
• 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
© Joan A. Cotter, Ph.D., 201276
Research on CountingSubitizing
• 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
© Joan A. Cotter, Ph.D., 201277
Research on CountingSubitizing
• 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
© Joan A. Cotter, Ph.D., 201278
Visualizing Quantities
© Joan A. Cotter, Ph.D., 201279
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., 201280
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
• Representative of structure of numbers.• Easily manipulated by children.• Imaginable mentally.
Visualizing QuantitiesJapanese criteria for manipulatives
Japanese Council ofMathematics Education
© Joan A. Cotter, Ph.D., 2012
Visualizing Quantities
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction
Visualizing also needed in:
© Joan A. Cotter, Ph.D., 2012
Visualizing Quantities
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction
• Architecture
• Astronomy
• Archeology
• Chemistry
• Physics
• Surgery
Visualizing also needed in:
© Joan A. Cotter, Ph.D., 2012
Visualizing QuantitiesReady: How many?
© Joan A. Cotter, Ph.D., 2012
Visualizing QuantitiesReady: How many?
© Joan A. Cotter, Ph.D., 2012
Visualizing QuantitiesTry again: How many?
© Joan A. Cotter, Ph.D., 2012
Visualizing QuantitiesTry again: How many?
© Joan A. Cotter, Ph.D., 2012
Visualizing QuantitiesTry to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2012
Visualizing QuantitiesTry to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2012
Visualizing QuantitiesNow try to visualize 5 as red and 3 as green.
© Joan A. Cotter, Ph.D., 2012
Visualizing QuantitiesNow try to visualize 5 as red and 3 as green.
© Joan A. Cotter, Ph.D., 2012
Visualizing Quantities
I II III IIII V VIII
1 23458
Early Roman numerals
© Joan A. Cotter, Ph.D., 201293
Visualizing Quantities
Who could read the music?
:
© Joan A. Cotter, Ph.D., 201294
Subitizing (groups of five)
Math Way (of number naming)
Place Value Cards
Trading (with 4-digit numbers)
AN ALTERNATIVE to learning place value:
© Joan A. Cotter, Ph.D., 2012
Grouping in FivesUsing fingers
© Joan A. Cotter, Ph.D., 2012
Grouping in FivesUsing fingers
© Joan A. Cotter, Ph.D., 201297
Grouping in FivesUsing fingers
© Joan A. Cotter, Ph.D., 201298
Grouping in FivesUsing fingers
© Joan A. Cotter, Ph.D., 201299
Grouping in FivesUsing fingers
© Joan A. Cotter, Ph.D., 2012100
Grouping in FivesUsing fingers
© Joan A. Cotter, Ph.D., 2012
Grouping in Fives
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
Yellow is the Sun
© Joan A. Cotter, Ph.D., 2012
Grouping in FivesRecognizing 5
© Joan A. Cotter, Ph.D., 2012
Grouping in FivesRecognizing 5
© Joan A. Cotter, Ph.D., 2012
Grouping in Fives
5 has a middle; 4 does not.
Recognizing 5
© Joan A. Cotter, Ph.D., 2012
Grouping in FivesTally sticks
© Joan A. Cotter, Ph.D., 2012106
Grouping in FivesTally sticks
© Joan A. Cotter, Ph.D., 2012107
Grouping in FivesTally sticks
© Joan A. Cotter, Ph.D., 2012108
Grouping in FivesTally sticks
© Joan A. Cotter, Ph.D., 2012109
Grouping in FivesTally sticks
© Joan A. Cotter, Ph.D., 2012110
Grouping in FivesTally sticks
© Joan A. Cotter, Ph.D., 2012
Grouping in FivesEntering quantities
© Joan A. Cotter, Ph.D., 2012
3
Grouping in FivesEntering quantities
© Joan A. Cotter, Ph.D., 2012113
5
Grouping in FivesEntering quantities
© Joan A. Cotter, Ph.D., 2012114
7
Grouping in FivesEntering quantities
© Joan A. Cotter, Ph.D., 2012115
Grouping in Fives
10
Entering quantities
© Joan A. Cotter, Ph.D., 2012116
Grouping in FivesThe stairs
© Joan A. Cotter, Ph.D., 2012
Grouping in FivesAdding
© Joan A. Cotter, Ph.D., 2012
Grouping in FivesAdding
4 + 3 =
© Joan A. Cotter, Ph.D., 2012
Grouping in Fives
4 + 3 = Adding
© Joan A. Cotter, Ph.D., 2012
Grouping in Fives
4 + 3 = Adding
© Joan A. Cotter, Ph.D., 2012
Grouping in Fives
4 + 3 = Adding
© Joan A. Cotter, Ph.D., 2012
Grouping in Fives
4 + 3 = 7 Adding
© Joan A. Cotter, Ph.D., 2012
Grouping in Fives
4 + 3 = Adding
© Joan A. Cotter, Ph.D., 2012124
Go to the Dump GameObjective: To learn the facts that total 10:
1 + 92 + 83 + 74 + 65 + 5
© Joan A. Cotter, Ph.D., 2012125
Go to the Dump GameObjective: To learn the facts that total 10:
1 + 92 + 83 + 74 + 65 + 5
Object of the game: To collect the most pairs that equal ten.
© Joan A. Cotter, Ph.D., 2012126
Go to the Dump Game
6 + = 10
© Joan A. Cotter, Ph.D., 2012127
“Math” Way of Naming Numbers
© Joan A. Cotter, Ph.D., 2012128
“Math” Way of Naming Numbers
11 = ten 1
© Joan A. Cotter, Ph.D., 2012129
“Math” Way of Naming Numbers
11 = ten 112 = ten 2
© Joan A. Cotter, Ph.D., 2012130
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 3
© Joan A. Cotter, Ph.D., 2012131
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4
© Joan A. Cotter, Ph.D., 2012132
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9
© Joan A. Cotter, Ph.D., 2012133
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9
20 = 2-ten
© Joan A. Cotter, Ph.D., 2012134
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9
20 = 2-ten 21 = 2-ten 1
© Joan A. Cotter, Ph.D., 2012135
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9
20 = 2-ten 21 = 2-ten 122 = 2-ten 2
© Joan A. Cotter, Ph.D., 2012136
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9
20 = 2-ten 21 = 2-ten 122 = 2-ten 223 = 2-ten 3
© Joan A. Cotter, Ph.D., 2012137
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9
20 = 2-ten 21 = 2-ten 122 = 2-ten 223 = 2-ten 3 . . . . . . . .99 = 9-ten 9
© Joan A. Cotter, Ph.D., 2012138
“Math” Way of Naming Numbers
137 = 1 hundred 3-ten 7
© Joan A. Cotter, Ph.D., 2012139
“Math” Way of Naming Numbers
137 = 1 hundred 3-ten 7or
137 = 1 hundred and 3-ten 7
© Joan A. Cotter, Ph.D., 2012140
“Math” Way of Naming Numbers
0
10
20
30
40
50
60
70
80
90
100
4 5 6Ages (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.
Ave
rage
Hig
hest
Num
ber
Cou
nted
© Joan A. Cotter, Ph.D., 2012141
“Math” Way of Naming Numbers
0
10
20
30
40
50
60
70
80
90
100
4 5 6Ages (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.
Ave
rage
Hig
hest
Num
ber
Cou
nted
© Joan A. Cotter, Ph.D., 2012142
“Math” Way of Naming Numbers
0
10
20
30
40
50
60
70
80
90
100
4 5 6Ages (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.
Ave
rage
Hig
hest
Num
ber
Cou
nted
© Joan A. Cotter, Ph.D., 2012143
“Math” Way of Naming Numbers
0
10
20
30
40
50
60
70
80
90
100
4 5 6Ages (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.
Ave
rage
Hig
hest
Num
ber
Cou
nted
© Joan A. Cotter, Ph.D., 2012144
“Math” Way of Naming Numbers
0
10
20
30
40
50
60
70
80
90
100
4 5 6Ages (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.
Ave
rage
Hig
hest
Num
ber
Cou
nted
© Joan A. Cotter, Ph.D., 2012145
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., 2012146
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., 2012147
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., 2012148
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., 2012149
Math Way of Naming NumbersCompared to reading:
© Joan A. Cotter, Ph.D., 2012150
Math Way of Naming Numbers
• Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic.
Compared to reading:
© Joan A. Cotter, Ph.D., 2012151
Math Way of Naming Numbers
• 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:
© Joan A. Cotter, Ph.D., 2012152
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, 2005Researchers
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
4-ten = forty
The “ty” means tens.
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
4-ten = forty
The “ty” means tens.
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
6-ten = sixty
The “ty” means tens.
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
3-ten = thirty
“Thir” also used in 1/3, 13 and 30.
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
5-ten = fifty
“Fif” also used in 1/5, 15 and 50.
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
2-ten = twenty
Two used to be pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
A word game
fireplace place-fire
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
A word game
fireplace place-fire
paper-newsnewspaper
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
A word game
fireplace place-fire
paper-news
box-mail mailbox
newspaper
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
ten 4
“Teen” also means ten.
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
ten 4 teen 4
“Teen” also means ten.
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
ten 4 teen 4 fourteen
“Teen” also means ten.
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
a one left
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
a one left a left-one
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
a one left a left-one eleven
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
two left
Two pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2012
Math Way of Naming NumbersTraditional names
two left twelve
Two pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten
3 03 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten
3 03 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten
3 03 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten 7
3 03 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten 7
3 03 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten 7
3 03 0
77
© Joan A. Cotter, Ph.D., 2012
3 03 0
Composing Numbers
3-ten 7
77
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
3-ten 7
Notice the way we say the number, represent the number, and write the number all correspond.
3 03 077
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
7-ten
7 07 0
Another example.
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
7-ten 8
7 07 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
7-ten 8
7 07 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
7-ten 8
7 07 0
88
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
7-ten 8
7 87 888
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
10-ten
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
10-ten
1 0 01 0 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
10-ten
1 0 01 0 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
10-ten
1 0 01 0 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
1 hundred
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
1 hundred
1 0 01 0 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
1 hundred
1 0 01 0 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
1 hundred
11 001 01 0 01 0 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
1 hundred
1 0 01 0 0
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
2 hundred
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
2 hundred
© Joan A. Cotter, Ph.D., 2012
Composing Numbers
2 hundred
2 0 02 0 0
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2 4
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2 4 6
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2 4 6 8
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2 4 6 8 10
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2 4 6 8 10
12
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2 4 6 8 10
12 14
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2 4 6 8 10
12 14 16
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2 4 6 8 10
12 14 16 18
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 2s
2 4 6 8 10
12 14 16 18 20
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5sCounting by 5s
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s
5
Counting by 5s
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s
5 10
Counting by 5s
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s
5 10
15
Counting by 5s
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s
5 10
15 20
Counting by 5s
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s
5 10
15 20
25
Counting by 5s
© Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s
5 10
15 20
25 30
Counting by 5s
© Joan A. Cotter, Ph.D., 2012
Evens and OddsEvens
© Joan A. Cotter, Ph.D., 2012
Evens and OddsEvens
Use two fingers and touch each pair in succession.
© Joan A. Cotter, Ph.D., 2012
Evens and OddsEvens
Use two fingers and touch each pair in succession.
© Joan A. Cotter, Ph.D., 2012
Evens and OddsEvens
Use two fingers and touch each pair in succession.
© Joan A. Cotter, Ph.D., 2012
Evens and OddsEvens
Use two fingers and touch each pair in succession.
EVEN!
© Joan A. Cotter, Ph.D., 2012
Evens and OddsOdds
Use two fingers and touch each pair in succession.
© Joan A. Cotter, Ph.D., 2012
Evens and OddsOdds
Use two fingers and touch each pair in succession.
© Joan A. Cotter, Ph.D., 2012
Evens and OddsOdds
Use two fingers and touch each pair in succession.
© Joan A. Cotter, Ph.D., 2012
Evens and OddsOdds
Use two fingers and touch each pair in succession.
© Joan A. Cotter, Ph.D., 2012
Evens and OddsOdds
Use two fingers and touch each pair in succession.
ODD!
© Joan A. Cotter, Ph.D., 2012227
Fact Strategies
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 =
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesComplete the Ten
9 + 5 = 14
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesTwo Fives
8 + 6 =
10 + 4 = 14
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 =
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Down
15 – 9 = 6
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesSubtract from 10
15 – 9 = 6
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Fact StrategiesGoing Up
15 – 9 =
1 + 5 = 6
Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2012
Fact Strategies
6 4 =(6 taken 4 times)
Multiplication
© Joan A. Cotter, Ph.D., 2012
Fact Strategies
6 4 =(6 taken 4 times)
Multiplication
© Joan A. Cotter, Ph.D., 2012
Place ValueTwo aspects
© Joan A. Cotter, Ph.D., 2012
Place ValueTwo aspects
Static
© Joan A. Cotter, Ph.D., 2012
Place ValueTwo aspects
Static • Value of a digit is determined by position
© Joan A. Cotter, Ph.D., 2012
Place ValueTwo aspects
Static • Value of a digit is determined by position.• No position may have more than nine.
© Joan A. Cotter, Ph.D., 2012
Place ValueTwo 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
Place ValueTwo 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.• Place value cards show this aspect.
© Joan A. Cotter, Ph.D., 2012
Place ValueTwo 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.• Place value cards show this aspect.
Dynamic
© Joan A. Cotter, Ph.D., 2012
Place ValueTwo 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.• Place value cards show this aspect.
Dynamic • Ten ones = 1 ten; ten tens = 1 hundred; ten hundreds = 1 thousand, ….
© Joan A. Cotter, Ph.D., 2012
Trading
1000 10 1100
© Joan A. Cotter, Ph.D., 2012
TradingThousands
1000 10 1100
© Joan A. Cotter, Ph.D., 2012
TradingHundreds
1000 10 1100
© Joan A. Cotter, Ph.D., 2012
TradingTens
1000 10 1100
© Joan A. Cotter, Ph.D., 2012
TradingOnes
1000 10 1100
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding
8+ 6
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding
8+ 6
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding
8+ 6
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding
8+ 6
© Joan A. Cotter, Ph.D., 2012
TradingAdding
8+ 614
1000 10 1100
© Joan A. Cotter, Ph.D., 2012
TradingAdding
8+ 614
Too many ones; trade 10 ones for 1 ten.
1000 10 1100
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding
8+ 614
Too many ones; trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding
8+ 614
Too many ones; trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding
8+ 614
Same answer before and after trading.
© Joan A. Cotter, Ph.D., 2012
TradingBead Trading Activity
1000 10 1100
© Joan A. Cotter, Ph.D., 2012
TradingBead Trading Activity
Object: To get a high score by adding numbers on the green cards.
1000 10 1100
© Joan A. Cotter, Ph.D., 2012
TradingBead Trading Activity
Object: To get a high score by adding numbers on the green cards.
71000 10 1100
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
Object: To get a high score by adding numbers on the green cards.
7
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
6
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
6
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
6
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
Trade 10 ones for 1 ten.
6
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
6
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
6
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
9
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
9
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
Another trade.
9
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
Another trade.
9
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
3
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingBead Trading Activity
3
© Joan A. Cotter, Ph.D., 2012
Trading
• In the Bead Trading activity trading10 ones for 1 ten occurs frequently;
Bead Trading Activity
© Joan A. Cotter, Ph.D., 2012
TradingBead Trading Activity
• In the Bead Trading activity trading10 ones for 1 ten occurs frequently;10 tens for 1 hundred, less often;
© Joan A. Cotter, Ph.D., 2012
TradingBead Trading Activity
• In the Bead Trading activity trading10 ones for 1 ten occurs frequently;10 tens for 1 hundred, less often;10 hundreds for 1 thousand, rarely.
© Joan A. Cotter, Ph.D., 2012
TradingBead Trading Activity
• In the Bead Trading activity trading10 ones for 1 ten occurs frequently;10 tens for 1 hundred, less often;10 hundreds for 1 thousand, rarely.
• Bead trading helps the child experience the greater value of each column from left to right.
© Joan A. Cotter, Ph.D., 2012
Trading
• In the Bead Trading activity trading10 ones for 1 ten occurs frequently;10 tens for 1 hundred, less often;10 hundreds for 1 thousand, rarely.
• Bead trading helps the child experience the greater value of each column from left to right.
• To detect a pattern, there must be at least three examples in the sequence. Place value is a pattern.
Bead Trading Activity
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
6
Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
6
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
6
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
6
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
396
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
6396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2012
1000 10 1100
TradingAdding 4-digit numbers
3658+ 2738
6396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2012328
Minnesota Standards
K: Represent quantities using whole numbers and understand relationships among whole numbers.
1–2: Understand place value and relationships among whole numbers.
Number Sense
With this alternate model, how difficult are the associated benchmarks for children to master?
© Joan A. Cotter, Ph.D., 2012329
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
Kindergarten
© Joan A. Cotter, Ph.D., 2012330
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
Kindergarten
© Joan A. Cotter, Ph.D., 2012331
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
Kindergarten
© Joan A. Cotter, Ph.D., 2012332
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
Kindergarten
© Joan A. Cotter, Ph.D., 2012333
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
• Recognize number of objects up to 6, without counting.
Kindergarten
© Joan A. Cotter, Ph.D., 2012334
Minnesota Standards
Represent quantities using whole numbers and understand relationships among whole numbers.
• Count forward to 31, backward from 10.
• Count number of objects and identify the quantity.
• Compare the number of objects in two or more sets.
• Given a number, identify one more or one less.
• Recognize number of objects up to 6, without counting.
• Add and subtract whole numbers up to 6, using objects.
Kindergarten
© Joan A. Cotter, Ph.D., 2012335
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
Grade 1
© Joan A. Cotter, Ph.D., 2012336
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
Grade 1
© Joan A. Cotter, Ph.D., 2012337
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
Grade 1
© Joan A. Cotter, Ph.D., 2012338
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
• Demonstrate understanding of odd and even to 12.
Grade 1
© Joan A. Cotter, Ph.D., 2012339
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 120.
• Count by 2s to 30 and by 5s to 120.
• Count backwards from 30.
• Demonstrate understanding of odd and even to 12.
• Represent whole numbers up to 20 in various ways.
Grade 1
© Joan A. Cotter, Ph.D., 2012340
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
Grade 2
© Joan A. Cotter, Ph.D., 2012341
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
Grade 2
© Joan A. Cotter, Ph.D., 2012342
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
Grade 2
© Joan A. Cotter, Ph.D., 2012343
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
• Demonstrate understanding of odd and even up to 12.
Grade 2
© Joan A. Cotter, Ph.D., 2012344
Minnesota Standards
Understand place value and relationships among whole numbers.
• Read, write, compare and order numbers to 999.
• Count by 2s, 5s, 10s from any given whole number.
• Understand the significance of groups of ten.
• Demonstrate understanding of odd and even up to 12.
• Represent whole numbers up to 20 in various ways.
Grade 2
© Joan A. Cotter, Ph.D., 2012345
Research Highlights
© Joan A. Cotter, Ph.D., 2012346
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
© Joan A. Cotter, Ph.D., 2012347
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
© Joan A. Cotter, Ph.D., 2012348
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
© Joan A. Cotter, Ph.D., 2012349
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
© Joan A. Cotter, Ph.D., 2012350
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012351
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012352
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012353
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012354
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012355
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012356
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012357
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children thinking of 14 as 14 ones counted 14.
© Joan A. Cotter, Ph.D., 2012358
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children who understand tens remove a ten and 4 ones.
© Joan A. Cotter, Ph.D., 2012359
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children who understand tens remove a ten and 4 ones.
© Joan A. Cotter, Ph.D., 2012360
Research Highlights
Using 10s and 1s, ask the child to construct 48.
Research task:
Then ask the child to subtract 14.
Children who understand tens remove a ten and 4 ones.
© Joan A. Cotter, Ph.D., 2012
Research HighlightsTASK EXPER CTRL
14 as 10 & 4 48 – 14 81% 33%
© Joan A. Cotter, Ph.D., 2012
Research HighlightsTASK EXPER CTRL
Teens 10 + 3 94% 47%14 as 10 & 4 48 – 14 81% 33%
© Joan A. Cotter, Ph.D., 2012
Research HighlightsTASK EXPER CTRL
Teens 10 + 3 94% 47%6 + 10 88% 33%
14 as 10 & 4 48 – 14 81% 33%
© Joan A. Cotter, Ph.D., 2012
Research HighlightsTASK EXPER CTRL
Teens 10 + 3 94% 47%6 + 10 88% 33%
Circle TensPlace
78 75% 67%
14 as 10 & 4 48 – 14 81% 33%
© Joan A. Cotter, Ph.D., 2012
Research HighlightsTASK EXPER CTRL
Teens 10 + 3 94% 47%6 + 10 88% 33%
Circle TensPlace
78 75% 67%
14 as 10 & 4 48 – 14 81% 33%
3924 44% 7%
© Joan A. Cotter, Ph.D., 2012
Research HighlightsTASK EXPER CTRL
Teens 10 + 3 94% 47%6 + 10 88% 33%
Circle TensPlace
78 75% 67%
3924 44% 7%
14 as 10 & 4 48 – 14 81% 33%
Mental Computation
85 – 70 31% 0%
© Joan A. Cotter, Ph.D., 2012
Research HighlightsTASK EXPER CTRL
Teens 10 + 3 94% 47%6 + 10 88% 33%
Circle TensPlace
78 75% 67%
3924 44% 7%
14 as 10 & 4 48 – 14 81% 33%
Mental Computation
85 – 70 31% 0% 2nd Graders in US (Reys): 9%
© Joan A. Cotter, Ph.D., 2012
Research HighlightsTASK EXPER CTRL
Teens 10 + 3 94% 47%6 + 10 88% 33%
Circle TensPlace
78 75% 67%
3924 44% 7%
14 as 10 & 4 48 – 14 81% 33%
Mental Computation
85 – 70 31% 0% 2nd Graders in US (Reys): 9%
38 + 24 = 512 or 0% 40%
57 + 35 = 812
© Joan A. Cotter, Ph.D., 2012
Framing the Future of Mathematics in Minnesota
Place value, not counting, is the key to under-standing numbers.
Place value is best taught by:
• Subitizing (with groups of fives),• Initially using Math Way of number naming,• Incorporating Place Value Cards,• Patterning (trading with 4-digit numbers).
© Joan A. Cotter, Ph.D., 2012
The Future of Primary Math: More Understanding/Less Counting
MCTMSaturday, May 5, 2012
Duluth, Minnesota
by Joan A. Cotter, Ph.D.JoanCotter@RightStartMath.com
3 03 077
3 03 0
77
1000 10 1100
PowerPoint PresentationRightStartMath.com >Resources
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