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NDMA 2012
- 1. RightStart™ Mathematics in a
Montessori Environment
by Joan A. Cotter, Ph.D.
JoanCotter@RightStartMath.com
7x7 3 2
5 5
New Discoveries
Montessori Academy
August 31, 2012
Hutchinson, Minnesota
Other presentations available: rightstartmath.com Cotter, Ph.D., 2012
© Joan A.
- 4. National Math Crisis
• 25% of college freshmen take remedial math.
• In 2009, of the 1.5 million students who took
the ACT test, only 42% are ready for college
algebra.
© Joan A. Cotter, Ph.D., 2012
- 5. National Math Crisis
• 25% of college freshmen take remedial math.
• In 2009, of the 1.5 million students who took
the ACT test, only 42% are ready for college
algebra.
• A generation ago, the US produced 30% of the
world’s college grads; today it’s 14%. (CSM 2006)
© Joan A. Cotter, Ph.D., 2012
- 6. National Math Crisis
• 25% of college freshmen take remedial math.
• In 2009, of the 1.5 million students who took
the ACT test, only 42% are ready for college
algebra.
• A generation ago, the US produced 30% of the
world’s college grads; today it’s 14%. (CSM 2006)
• Two-thirds of 4-year degrees in Japan and
China are in science and engineering; one-third
in the U.S.
© Joan A. Cotter, Ph.D., 2012
- 7. National Math Crisis
• 25% of college freshmen take remedial math.
• In 2009, of the 1.5 million students who took
the ACT test, only 42% are ready for college
algebra.
• A generation ago, the US produced 30% of the
world’s college grads; today it’s 14%. (CSM 2006)
• Two-thirds of 4-year degrees in Japan and
China are in science and engineering; one-third
in the U.S.
• U.S. students, compared to the world, score
high at 4th grade, average at 8th, and near
bottom at 12th.
© Joan A. Cotter, Ph.D., 2012
- 8. National Math Crisis
• 25% of college freshmen take remedial math.
• In 2009, of the 1.5 million students who took
the ACT test, only 42% are ready for college
algebra.
• A generation ago, the US produced 30% of the
world’s college grads; today it’s 14%. (CSM 2006)
• Two-thirds of 4-year degrees in Japan and
China are in science and engineering; one-third
in the U.S.
• U.S. students, compared to the world, score
high at 4th grade, average at 8th, and near
bottom at 12th.
• Ready, Willing, and Unable to Serve says that
75% of 17 to 24 year-olds are unfit for military
service. (2010)
© Joan A. Cotter, Ph.D., 2012
- 10. Math Education is Changing
• The field of mathematics is doubling every 7
years.
© Joan A. Cotter, Ph.D., 2012
- 11. Math Education is Changing
• The field of mathematics is doubling every 7
years.
• Math is used in many new ways. The
workplace needs analytical thinkers and
problem solvers.
© Joan A. Cotter, Ph.D., 2012
- 12. Math Education is Changing
• The field of mathematics is doubling every 7
years.
• Math is used in many new ways. The
workplace needs analytical thinkers and
problem solvers.
• State exams require more than arithmetic:
including geometry, algebra, probability, and
statistics.
© Joan A. Cotter, Ph.D., 2012
- 13. Math Education is Changing
• The field of mathematics is doubling every 7
years.
• Math is used in many new ways. The
workplace needs analytical thinkers and
problem solvers.
• State exams require more than arithmetic:
including geometry, algebra, probability, and
statistics.
• Brain research is providing clues on how to
better facilitate learning, including math.
© Joan A. Cotter, Ph.D., 2012
- 14. Math Education is Changing
• The field of mathematics is doubling every 7
years.
• Math is used in many new ways. The
workplace needs analytical thinkers and
problem solvers.
• State exams require more than arithmetic:
including geometry, algebra, probability, and
statistics.
• Brain research is providing clues on how to
better facilitate learning, including math.
• Calculators and computers have made
computation with many digits an unneeded skill.
© Joan A. Cotter, Ph.D., 2012
- 15. Math Education is Changing
• The field of mathematics is doubling every 7
years.
• Math is used in many new ways. The
workplace needs analytical thinkers and
problem solvers.
• State exams require more than arithmetic:
including geometry, algebra, probability, and
statistics.
• Brain research is providing clues on how to
better facilitate learning, including math.
• Calculators and computers have made
computation with many digits an unneeded skill.
© Joan A. Cotter, Ph.D., 2012
- 17. 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
- 19. Counting Model
From a child's perspective
F
+E
A
© Joan A. Cotter, Ph.D., 2012
- 20. Counting Model
From a child's perspective
F
+E
A B
© Joan A. Cotter, Ph.D., 2012
- 21. Counting Model
From a child's perspective
F
+E
A B C
© Joan A. Cotter, Ph.D., 2012
- 22. Counting Model
From a child's perspective
F
+E
A B C D E F
© Joan A. Cotter, Ph.D., 2012
- 23. Counting Model
From a child's perspective
F
+E
A B C D E F A
© Joan A. Cotter, Ph.D., 2012
- 24. Counting Model
From a child's perspective
F
+E
A B C D E F A B
© Joan A. Cotter, Ph.D., 2012
- 25. 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
- 26. 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
- 27. 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
- 28. Counting Model
From a child's perspective
Now memorize the facts!!
G
+D
© Joan A. Cotter, Ph.D., 2012
- 29. Counting Model
From a child's perspective
Now memorize the facts!!
G
+D
© Joan A. Cotter, Ph.D., 2012
- 30. Counting Model
From a child's perspective
Now memorize the facts!!
G
+D
D
+C
© Joan A. Cotter, Ph.D., 2012
- 31. Counting Model
From a child's perspective
Now memorize the facts!!
G
+D
D C
+C +G
© Joan A. Cotter, Ph.D., 2012
- 32. Counting Model
From a child's perspective
Now memorize the facts!!
G
+D
D C
+C +G
© Joan A. Cotter, Ph.D., 2012
- 33. Counting Model
From a child's perspective
Try subtracting H
by ―taking away‖ – E
© Joan A. Cotter, Ph.D., 2012
- 34. Counting Model
From a child's perspective
Try skip counting by B’s to T:
B, D, . . . T.
© Joan A. Cotter, Ph.D., 2012
- 35. 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
- 36. 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
- 37. 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
- 38. 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
- 39. 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
- 40. Counting Model
From a child's perspective
L (twelve)
is written AB (12)
(one 10)
because it is A J
and B A’s
© Joan A. Cotter, Ph.D., 2012
- 41. Counting Model
From a child's perspective
L (twelve)
is written AB (12)
(one 10)
because it is A J
and B A’s (two 1s).
© Joan A. Cotter, Ph.D., 2012
- 42. 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
- 44. Counting Model
Summary
• Is not natural; it takes years of
practice.
© Joan A. Cotter, Ph.D., 2012
- 45. Counting Model
Summary
• Is not natural; it takes years of
practice.
• Provides poor concept of quantity.
© Joan A. Cotter, Ph.D., 2012
- 46. Counting Model
Summary
• Is not natural; it takes years of
practice.
• Provides poor concept of quantity.
• Ignores place value.
© Joan A. Cotter, Ph.D., 2012
- 47. Counting Model
Summary
• Is not natural; it takes years of
practice.
• Provides poor concept of quantity.
• Ignores place value.
• Is very error prone.
© Joan A. Cotter, Ph.D., 2012
- 48. 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.
© Joan A. Cotter, Ph.D., 2012
- 49. 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.
© Joan A. Cotter, Ph.D., 2012
- 50. 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
- 51. 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
- 52. 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
- 53. 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. A. Cotter, Ph.D., 2012
Joan
- 54. 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 4 doesn’t include 1, 2 and 3. © Joan A. Cotter, Ph.D., 2012
- 55. 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
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
- 56. 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
- 57. Calendar Math
The calendar is not a number line.
• No quantity is involved.
• Numbers are in spaces, not at lines like a
ruler.
© Joan A. Cotter, Ph.D., 2012
- 58. 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.
© Joan A. Cotter, Ph.D., 2012
- 59. 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. © Joan A. Cotter, Ph.D., 2012
- 60. Memorizing Math
Percentage Recall
Immediatel After 1 day After 4 wks
y
Rote 32 23 8
69 69 58
Concept
© Joan A. Cotter, Ph.D., 2012
- 61. Memorizing Math
Percentage Recall
Immediatel After 1 day After 4 wks
y
Rote 32 23 8
69 69 58
Concept
© Joan A. Cotter, Ph.D., 2012
- 62. Memorizing Math
Percentage Recall
Immediatel After 1 day After 4 wks
y
Rote 32 23 8
69 69 58
Concept
© Joan A. Cotter, Ph.D., 2012
- 63. Memorizing Math
Percentage Recall
Immediatel After 1 day After 4 wks
y
Rote 32 23 8
69 69 58
Concept
© Joan A. Cotter, Ph.D., 2012
- 64. Memorizing Math
Percentage Recall
Immediatel After 1 day After 4 wks
y
Rote 32 23 8
69 69 58
Concept
© Joan A. Cotter, Ph.D., 2012
- 65. Memorizing Math
Percentage Recall
Immediatel After 1 day After 4 wks
y
Rote 32 23 8
69 69 58
Concept
© Joan A. Cotter, Ph.D., 2012
- 66. Memorizing Math
Percentage Recall
Immediatel After 1 day After 4 wks
y
Rote 32 23 8
69 69 58
Concept
Math needs to be taught so 95%
is understood and only 5%
memorized.
Richard Skemp
© Joan A. Cotter, Ph.D., 2012
- 68. 9
Memorizing Math +7
Flash cards:
• Are often used to teach rote.
© Joan A. Cotter, Ph.D., 2012
- 69. 9
Memorizing Math +7
Flash cards:
• Are often used to teach rote.
• Are liked only by those who don’t need
them.
© Joan A. Cotter, Ph.D., 2012
- 70. 9
Memorizing Math +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.
© Joan A. Cotter, Ph.D., 2012
- 71. 9
Memorizing Math +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.
© Joan A. Cotter, Ph.D., 2012
- 72. 9
Memorizing Math +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.
© Joan A. Cotter, Ph.D., 2012
- 73. 9
Memorizing Math +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.
© Joan A. Cotter, Ph.D., 2012
- 74. Research on Counting
Karen Wynn’s research
Show the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
- 75. Research on Counting
Karen Wynn’s research
Then hide them with a screen. © Joan A. Cotter, Ph.D., 2012
- 76. 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
- 77. 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
- 78. 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
- 79. Research on Counting
Karen Wynn’s research
Researcher can change the number of teddy bears behind the screen. © Joan A. Cotter, Ph.D., 2012
- 80. Research on Counting
Karen Wynn’s research
A baby seeing 1 teddy bear will look much longer, because it’s unexpected. A. Cotter, Ph.D., 2012
© Joan
- 82. 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
- 83. 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
- 84. 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
- 85. 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
- 86. Research on Counting
In Japanese schools:
• Children are discouraged from using
counting for adding.
© Joan A. Cotter, Ph.D., 2012
- 87. 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
- 88. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
© Joan A. Cotter, Ph.D., 2012
- 89. Research on Counting
Subitizing
• Subitizing is quick recognition of quantity
without counting.
• Human babies and some animals can
subitize small quantities at birth.
© Joan A. Cotter, Ph.D., 2012
- 90. 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.—Butterworth
© Joan A. Cotter, Ph.D., 2012
- 91. 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.—Butterworth
• Subitizing ―allows the child to grasp the
whole and the elements at the same time.‖—
Benoit
© Joan A. Cotter, Ph.D., 2012
- 92. 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.—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., 2012
- 93. Research on Counting
Finger gnosia
• Finger gnosia is the ability to know which
fingers can been lightly touched without
looking.
© Joan A. Cotter, Ph.D., 2012
- 94. Research on Counting
Finger gnosia
• Finger gnosia is the ability to know which
fingers can been lightly touched without
looking.
• Part of the brain controlling fingers is
adjacent to math part of the brain.
© Joan A. Cotter, Ph.D., 2012
- 95. Research on Counting
Finger gnosia
• Finger gnosia is the ability to know which
fingers can been lightly touched without
looking.
• Part of the brain controlling fingers is
adjacent to math part of the brain.
• Children who use their fingers as
representational tools perform better in
mathematics—Butterworth
© Joan A. Cotter, Ph.D., 2012
- 97. 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 (E1)
© Joan A. Cotter, Ph.D., 2012
- 98. Visualizing Mathematics
―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
- 101. 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
© Joan A. Cotter, Ph.D., 2012
- 102. 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
- 103. Visualizing Mathematics
Visualizing also needed in:
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction
© Joan A. Cotter, Ph.D., 2012
- 104. Visualizing Mathematics
Visualizing also needed in:
• Reading • Architecture
• Sports • Astronomy
• Creativity • Archeology
• Geography • Chemistry
• Engineering • Physics
• Construction • Surgery
© Joan A. Cotter, Ph.D., 2012
- 116. 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
- 117. Visualizing Mathematics
:
Who could read the music?
Music needs 10 lines, two groups of five. © Joan A. Cotter, Ph.D., 2012
- 118. Research on Counting
Teach Counting
• Finger gnosia is the ability to know which
fingers can been lightly touched without
looking.
• Part of the brain controlling fingers is
adjacent to math part of the brain.
• Children who use their fingers as
representational tools perform better in
mathematics—Butterworth
© Joan A. Cotter, Ph.D., 2012
- 119. Very Early Computation
Numerals
In English there are two ways of writing numbers:
3578
Three thousand five hundred seventy eight
© Joan A. Cotter, Ph.D., 2012
- 120. Very Early Computation
Numerals
In English there are two ways of writing numbers:
3578
Three thousand five hundred seventy eight
In Chinese there is only one way of writing numbers:
3 Th 5 H 7 T 8 U
(8 characters)
© Joan A. Cotter, Ph.D., 2012
- 121. 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.
© Joan A. Cotter, Ph.D., 2012
- 123. Very Early Computation
Calculating rods
Numerals for Ones and Hundreds (Even Powers of
Ten)
© Joan A. Cotter, Ph.D., 2012
- 124. Very Early Computation
Calculating rods
Numerals for Ones and Hundreds (Even Powers of
Ten)
© Joan A. Cotter, Ph.D., 2012
- 125. Very Early Computation
Calculating rods
Numerals for Ones and Hundreds (Odd Powers of
Ten)
Numerals for Tens and Thousands (Odd Powers of
Ten)
© Joan A. Cotter, Ph.D., 2012
- 127. Very Early Computation
Calculating rods
3578
3578,3578
They grouped, not in thousands, but ten-thousands!
© Joan A. Cotter, Ph.D., 2012
- 129. Naming Quantities
Using fingers
Naming quantities is a three-period
lesson.
© Joan A. Cotter, Ph.D., 2012
- 130. Naming Quantities
Using fingers
Use left hand for 1-5 because we read from left to right. © Joan A. Cotter, Ph.D., 2012
- 133. Naming Quantities
Using fingers
Always show 7 as 5 and 2, not for example, as 4 and 3. © Joan A. Cotter, Ph.D., 2012
- 135. 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
- 138. 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
- 139. Naming Quantities
Tally sticks
Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart. © Joan A. Cotter, Ph.D., 2012
- 142. Naming Quantities
Tally sticks
Stick is horizontal, because it won’t fit diagonally and young children have
problems with diagonals.
© Joan A. Cotter, Ph.D., 2012
- 144. Naming Quantities
Tally sticks
Start a new row for every ten. © Joan A. Cotter, Ph.D., 2012
- 145. 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
- 146. 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
- 149. Naming Quantities
Numbe
r 1 6
Chart
2 7
To help the 3 8
child learn
the
symbols 4 9
5 10
© Joan A. Cotter, Ph.D., 2012
- 150. Naming Quantities
Pairing Finger Cards
Use two sets of finger cards and match them. © Joan A. Cotter, Ph.D., 2012
- 151. Naming Quantities
Ordering Finger Cards
Putting the finger cards in order. © Joan A. Cotter, Ph.D., 2012
- 152. Naming Quantities
Matching Numbers to Finger Cards
5 1
10
Match the number to the finger card. © Joan A. Cotter, Ph.D., 2012
- 153. Naming Quantities
Matching Fingers to Number Cards
9 1 10 4 6
2 3 7 8 5
Match the finger card to the number. © Joan A. Cotter, Ph.D., 2012
- 154. Naming Quantities
Finger Card Memory game
Use two sets of finger cards and play Memory. © Joan A. Cotter, Ph.D., 2012
- 157. Naming Quantities
Number Rods
Using different colors. © Joan A. Cotter, Ph.D., 2012
- 158. Naming Quantities
Spindle Box
45 dark-colored and 10 light-colored spindles. Could be in separate containers. Ph.D., 2012
© Joan A. Cotter,
- 159. Naming Quantities
Spindle Box
45 dark-colored and 10 light-colored spindles in two containers. © Joan A. Cotter, Ph.D., 2012
- 160. 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
- 161. 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
- 162. 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
- 163. 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
- 164. 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
- 165. 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
- 166. 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
- 167. 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
- 168. AL Abacus
1000 100 10 1
Double-sided AL abacus. Side 1 is grouped in 5s.
Trading Side introduces algorithms with trading.
© Joan A. Cotter, Ph.D., 2012
- 169. AL Abacus
Cleared
© Joan A. Cotter, Ph.D., 2012
- 170. AL Abacus
Entering quantities
3
Quantities are entered all at once, not counted. © Joan A. Cotter, Ph.D., 2012
- 171. AL Abacus
Entering quantities
5
Relate quantities to hands. © Joan A. Cotter, Ph.D., 2012
- 172. AL Abacus
Entering quantities
7
© Joan A. Cotter, Ph.D., 2012
- 173. AL Abacus
Entering quantities
10
© Joan A. Cotter, Ph.D., 2012
- 174. AL Abacus
The stairs
Can use to ―count‖ 1 to 10. Also read quantities on the right side. © Joan A. Cotter, Ph.D., 2012
- 175. AL Abacus
Adding
© Joan A. Cotter, Ph.D., 2012
- 176. AL Abacus
Adding
4+3=
© Joan A. Cotter, Ph.D., 2012
- 177. AL Abacus
Adding
4+3=
© Joan A. Cotter, Ph.D., 2012
- 178. AL Abacus
Adding
4+3=
© Joan A. Cotter, Ph.D., 2012
- 179. AL Abacus
Adding
4+3=
© Joan A. Cotter, Ph.D., 2012
- 180. AL Abacus
Adding
4+3=7
Answer is seen immediately, no counting needed. © Joan A. Cotter, Ph.D., 2012
- 181. 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
- 182. 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
- 183. Go to the Dump Game
The ways to partition 10. © Joan A. Cotter, Ph.D., 2012
- 184. Go to the Dump Game
Starting
A game viewed from above. © Joan A. Cotter, Ph.D., 2012
- 185. 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
- 186. 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
- 187. 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
- 188. 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
- 189. 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
- 190. 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
- 191. 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
- 192. 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
- 193. 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
- 194. 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
- 195. 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
- 196. 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
- 197. 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
© Joan A. Cotter, Ph.D., 2012
- 198. 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
- 199. 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
- 200. 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
© Joan A. Cotter, Ph.D., 2012
- 201. Go to the Dump Game
7 3
2 2 7 9 5
2 8 4 6
1 4 9
Playing
© Joan A. Cotter, Ph.D., 2012
- 202. 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?
© Joan A. Cotter, Ph.D., 2012
- 203. 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?
© Joan A. Cotter, Ph.D., 2012
- 204. Go to the Dump Game
7 3
2 2 7 9 5
2 8 4 6
1 5 4 9
Playing
© Joan A. Cotter, Ph.D., 2012
- 205. Go to the Dump Game
7 3
2 2 7 9 5
2 8 4 6
1 5 4 9
Playing
© Joan A. Cotter, Ph.D., 2012
- 206. 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
© Joan A. Cotter, Ph.D., 2012
- 207. 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
© Joan A. Cotter, Ph.D., 2012
- 208. 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
© Joan A. Cotter, Ph.D., 2012
- 209. Go to the Dump Game
7 3
2 2 7 5
2
1 8
9 4 6
5 4 9
Playing
© Joan A. Cotter, Ph.D., 2012
- 210. 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. A. Cotter, Ph.D., 2012
© Joan
- 211. Go to the Dump Game
9 1
4 6 5 5
Winner?
© Joan A. Cotter, Ph.D., 2012
- 212. Go to the Dump Game
9
1
4
6 5
Winner?
No counting. Combine both stacks. © Joan A. Cotter, Ph.D., 2012
- 213. Go to the Dump Game
9
1
4
6 5
Winner?
Whose stack is the highest? © Joan A. Cotter, Ph.D., 2012
- 214. Go to the Dump Game
Next game
No shuffling needed for next game. © Joan A. Cotter, Ph.D., 2012
- 216. ―Math‖ Way of Naming Numbers
11 = ten 1
© Joan A. Cotter, Ph.D., 2012
- 217. ―Math‖ Way of Naming Numbers
11 = ten 1
12 = ten 2
© Joan A. Cotter, Ph.D., 2012
- 218. ―Math‖ Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
© Joan A. Cotter, Ph.D., 2012
- 219. ―Math‖ Way of Naming Numbers
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
© Joan A. Cotter, Ph.D., 2012
- 220. ―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
- 221. ―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
- 222. ―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
- 223. ―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
- 224. ―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
- 225. ―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
- 226. ―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
- 227. ―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
- 228. ―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
- 229. ―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
- 230. ―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
- 231. ―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 6. Cotter, Ph.D., 2012
© Joan A.
- 232. ―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 andJoan A. Cotter, Ph.D., 2012
©
5-6.
- 233. 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
- 234. 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
- 235. 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
- 236. 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
- 237. Math Way of Naming Numbers
Compared to reading:
© Joan A. Cotter, Ph.D., 2012
- 238. 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
- 239. 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
- 240. 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. © Joan A. Cotter, Ph.D., 2012
- 241. 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
- 242. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask
the child to construct
48.
© Joan A. Cotter, Ph.D., 2012
- 243. Math Way of Naming Numbers
Research task:
Using 10s and 1s, ask
the child to construct
48.
Then ask the child to
subtract 14.
© Joan A. Cotter, Ph.D., 2012
- 244. 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.
© Joan A. Cotter, Ph.D., 2012
- 245. 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.
© Joan A. Cotter, Ph.D., 2012
- 246. 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.
© Joan A. Cotter, Ph.D., 2012
- 247. 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.
© Joan A. Cotter, Ph.D., 2012
- 248. 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.
© Joan A. Cotter, Ph.D., 2012
- 249. 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.
© Joan A. Cotter, Ph.D., 2012
- 250. 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.
© Joan A. Cotter, Ph.D., 2012
- 251. 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.
© Joan A. Cotter, Ph.D., 2012
- 252. 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.
© Joan A. Cotter, Ph.D., 2012
- 253. 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.
© Joan A. Cotter, Ph.D., 2012
- 254. 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.
© Joan A. Cotter, Ph.D., 2012
- 255. Math Way of Naming Numbers
Traditional names
4-ten = forty
The ―ty‖
means
tens.
© Joan A. Cotter, Ph.D., 2012
- 256. 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
- 257. Math Way of Naming Numbers
Traditional names
6-ten = sixty
The ―ty‖
means
tens.
© Joan A. Cotter, Ph.D., 2012
- 258. 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
- 259. 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
- 260. Math Way of Naming Numbers
Traditional names
2-ten = twenty
Two used to
be
pronounced
―twoo.‖
© Joan A. Cotter, Ph.D., 2012
- 261. 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
- 262. 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
- 263. 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
- 264. Math Way of Naming Numbers
Traditional names
ten 4
―Teen‖ also
means ten.
© Joan A. Cotter, Ph.D., 2012
- 265. Math Way of Naming Numbers
Traditional names
ten 4 teen 4
―Teen‖ also
means ten.
© Joan A. Cotter, Ph.D., 2012
- 266. Math Way of Naming Numbers
Traditional names
ten 4 teen 4 fourteen
―Teen‖ also
means ten.
© Joan A. Cotter, Ph.D., 2012
- 267. Math Way of Naming Numbers
Traditional names
a one left
© Joan A. Cotter, Ph.D., 2012
- 268. Math Way of Naming Numbers
Traditional names
a one left a left-one
© Joan A. Cotter, Ph.D., 2012
- 269. Math Way of Naming Numbers
Traditional names
a one left a left-one eleven
© Joan A. Cotter, Ph.D., 2012
- 270. Math Way of Naming Numbers
Traditional names
two left
Two
pronounce
d ―twoo.‖
© Joan A. Cotter, Ph.D., 2012
- 271. Math Way of Naming Numbers
Traditional names
two left twelve
Two
pronounce
d ―twoo.‖
© Joan A. Cotter, Ph.D., 2012
- 275. Composing Numbers
3-ten
30
Point to the 3 and say 3. © Joan A. Cotter, Ph.D., 2012
- 276. Composing Numbers
3-ten
30
Point to 0 and say 10. The 0 makes 3 a ten. © Joan A. Cotter, Ph.D., 2012
- 280. Composing Numbers
3-ten
7
30
7
Place the 7 on top of the 0 of the 30. © Joan A. Cotter, Ph.D., 2012
- 281. 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
- 289. Composing Numbers
1
hundred
100
Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
- 290. Composing Numbers
1
hundred
100
Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
- 291. Composing Numbers
1
hundred
100
Of course, we can also read it as one-hun-dred. © Joan A. Cotter, Ph.D., 2012
- 292. 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
- 293. 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
- 294. 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
- 295. 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
- 296. 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.
© Joan A. Cotter, Ph.D., 2012
- 297. 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.)
© Joan A. Cotter, Ph.D., 2012
- 299. Fact Strategies
• A strategy is a way to learn a new fact
or recall a forgotten fact.
© Joan A. Cotter, Ph.D., 2012
- 300. Fact Strategies
• A strategy is a way to learn a new fact
or recall a forgotten fact.
• A visualizable representation is part
of a powerful strategy.
© Joan A. Cotter, Ph.D., 2012
- 304. Fact Strategies
Complete the
Ten
9+5=
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 305. Fact Strategies
Complete the
Ten
9+5=
Take 1 from
the 5 and give
it to the 9.
Use two hands and move the beads simultaneously. © Joan A. Cotter, Ph.D., 2012
- 306. Fact Strategies
Complete the
Ten
9+5=
Take 1 from
the 5 and give
it to the 9.
© Joan A. Cotter, Ph.D., 2012
- 307. 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
- 310. Fact Strategies
Two Fives
8+6=
Two fives make 10. © Joan A. Cotter, Ph.D., 2012
- 311. Fact Strategies
Two Fives
8+6=
Just add the ―leftovers.‖ © Joan A. Cotter, Ph.D., 2012
- 312. Fact Strategies
Two Fives
8+6=
10 + 4 = 14
Just add the ―leftovers.‖ © Joan A. Cotter, Ph.D., 2012
- 313. Fact Strategies
Two Fives
7+5=
Another example. © Joan A. Cotter, Ph.D., 2012
- 317. Fact Strategies
Difference
7–4=
Subtract 4
from 5; then
add 2.
© Joan A. Cotter, Ph.D., 2012
- 319. Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 320. Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 321. Fact Strategies
Going Down
15 – 9 =
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 322. Fact Strategies
Going Down
15 – 9 = 6
Subtract 5;
then 4.
© Joan A. Cotter, Ph.D., 2012
- 323. Fact Strategies
Subtract from 10
15 – 9 =
© Joan A. Cotter, Ph.D., 2012
- 324. Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 325. Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 326. Fact Strategies
Subtract from 10
15 – 9 =
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 327. Fact Strategies
Subtract from 10
15 – 9 = 6
Subtract 9
from 10.
© Joan A. Cotter, Ph.D., 2012
- 329. Fact Strategies
Going Up
13 – 9 =
Start with 9;
go up to 13.
© Joan A. Cotter, Ph.D., 2012
- 330. Fact Strategies
Going Up
13 – 9 =
Start with 9;
go up to 13.
© Joan A. Cotter, Ph.D., 2012
- 331. Fact Strategies
Going Up
13 – 9 =
Start with 9;
go up to 13.
© Joan A. Cotter, Ph.D., 2012
- 332. Fact Strategies
Going Up
13 – 9 =
Start with 9;
go up to 13.
© Joan A. Cotter, Ph.D., 2012
- 333. Fact Strategies
Going Up
13 – 9 =
1+3=4
Start with 9;
go up to 13.
© Joan A. Cotter, Ph.D., 2012
- 354. Base-10 Picture Cards
6 5 8
3 0 0 0
7 2 4
2 0 0 0
Trade 10 ones for 1
ten. © Joan A. Cotter, Ph.D., 2012
- 355. Base-10 Picture Cards
6 5 8
3 0 0 0
7 2 4
2 0 0 0
Trade 10 ones for 1
ten. © Joan A. Cotter, Ph.D., 2012
- 356. Base-10 Picture Cards
6 5 8
3 0 0 0
7 2 4
2 0 0 0
Trade 10 ones for 1
ten. © Joan A. Cotter, Ph.D., 2012
- 358. Base-10 Picture Cards
6 5 8
3 0 0 0
7 2 4
2 0 0 0
Trade 10 hundreds for 1
thousand. © Joan A. Cotter, Ph.D., 2012
- 359. Base-10 Picture Cards
6 5 8
3 0 0 0
7 2 4
2 0 0 0
Trade 10 hundreds for 1
thousand. © Joan A. Cotter, Ph.D., 2012
- 360. Base-10 Picture Cards
6 5 8
3 0 0 0
7 2 4
2 0 0 0
Trade 10 hundreds for 1
thousand. © Joan A. Cotter, Ph.D., 2012
- 361. Base-10 Picture Cards
6 5 8
3 0 0 0
7 2 4
2 0 0 0
Trade 10 hundreds for 1
thousand. © Joan A. Cotter, Ph.D., 2012
