imf: visualization october 2011
DESCRIPTION
TRANSCRIPT
© Joan A. Cotter, Ph.D., 2011
VII
IMF ConferenceOctober 21, 2011Sarasota, Florida
by Joan A. Cotter, [email protected]
Adding Visualization toMontessori Mathematics
Presentation available: ALabacus.com
7 x 7
100010
1
100
7 3
7 3
© Joan A. Cotter, Ph.D., 2011
Counting Model
• Number Rods• Spindle Boxes• Golden Bead materials• Snake Game• Dot Game • Stamp Game• Multiplication Board• Bead Frame
In Montessori, counting is pervasive:
© Joan A. Cotter, Ph.D., 2011
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., 2011
Counting Model From a child's perspective
F + E
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
A
F + E
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
A B
F + E
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
A CB
F + E
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
A FC D EB
F + E
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
AA FC D EB
F + E
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
A BA FC D EB
F + E
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
A C D EBA FC D EB
F + E
© Joan A. Cotter, Ph.D., 2011
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., 2011
Counting Model From a child's perspective
K
G I J KHA FC D EB
F + E
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
Now memorize the facts!!
G + D
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
Now memorize the facts!!
G + D
H + F
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
Now memorize the facts!!
G + D
H + F
D + C
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
Now memorize the facts!!
G + D
H + F
C + G
D + C
© Joan A. Cotter, Ph.D., 2011
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., 2011
Counting Model From a child's perspective
Try subtractingby “taking away”
H – E
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
Try skip counting by B’s to T: B, D, . . . T.
© Joan A. Cotter, Ph.D., 2011
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., 2011
Counting Model From a child's perspective
Lis written ABbecause it is A J and B A’s
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
Lis written ABbecause it is A J and B A’s
huh?
© Joan A. Cotter, Ph.D., 2011
Counting Model From a child's perspective
Lis written ABbecause it is A J and B A’s
(twelve)
© Joan A. Cotter, Ph.D., 2011
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., 2011
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., 2011
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., 2011
Counting ModelSummary
© Joan A. Cotter, Ph.D., 2011
Counting Model
• Is not natural; it takes years of practice.Summary
© Joan A. Cotter, Ph.D., 2011
Counting Model
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
Summary
© Joan A. Cotter, Ph.D., 2011
Counting Model
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
Summary
© Joan A. Cotter, Ph.D., 2011
Counting Model
• Is not natural; it takes years of practice.
• Provides poor concept of quantity.
• Ignores place value.
• Is very error prone.
Summary
© Joan A. Cotter, Ph.D., 2011
Counting Model
• 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.
Summary
© Joan A. Cotter, Ph.D., 2011
Counting Model
• 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.
Summary
• Does not provide an efficient way to master the facts.
© Joan A. Cotter, Ph.D., 2011
Calendar MathAugust
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
Sometimes calendars are used for counting.
© Joan A. Cotter, Ph.D., 2011
Calendar MathAugust
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
Sometimes calendars are used for counting.
© Joan A. Cotter, Ph.D., 2011
Calendar MathAugust
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., 2011
Calendar MathAugust
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, not cardinal counting. The 3 doesn’t include the 1 and the 2.
© Joan A. Cotter, Ph.D., 2011
Calendar MathSeptember123489101115161718222324252930
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, not cardinal counting. The 4 doesn’t include 1, 2 and 3.
© Joan A. Cotter, Ph.D., 2011
Calendar MathSeptember123489101115161718222324252930
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
1 2 3 4 5 6
A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.
© Joan A. Cotter, Ph.D., 2011
Calendar MathAugust
8
1
9
2
10
3 4 5 6 7
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., 2011
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., 2011
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., 2011
Calendar Math The calendar is not a number line.
• No quantity is involved.• Numbers are in spaces, not at lines like a ruler.
Children need to see the whole month, not just part.• Purpose of calendar is to plan ahead.• Many ways to show the current date.
Calendars give a narrow view of patterning.• Patterns do not necessarily involve numbers.• Patterns rarely proceed row by row.• Patterns go on forever; they don’t stop at 31.
© Joan A. Cotter, Ph.D., 2011
Memorizing Math
Percentage RecallImmediately After 1 day After 4 wks
Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2011
Memorizing Math
Percentage RecallImmediately After 1 day After 4 wks
Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2011
Memorizing Math
Percentage RecallImmediately After 1 day After 4 wks
Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2011
Memorizing Math
Percentage RecallImmediately After 1 day After 4 wks
Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2011
Memorizing Math
Percentage RecallImmediately After 1 day After 4 wks
Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2011
Memorizing Math
Percentage RecallImmediately After 1 day After 4 wks
Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2011
Memorizing Math
Math needs to be taught so 95% is understood and only 5% memorized.
Richard Skemp
Percentage RecallImmediately After 1 day After 4 wks
Rote 32 23 8 Concept 69 69 58
© Joan A. Cotter, Ph.D., 2011
Memorizing MathFlash cards:
9 + 7
© Joan A. Cotter, Ph.D., 2011
• Are often used to teach rote.
Memorizing MathFlash cards:
9 + 7
© Joan A. Cotter, Ph.D., 2011
• Are often used to teach rote.
• Are liked only by those who don’t need them.
Memorizing MathFlash cards:
9 + 7
© Joan A. Cotter, Ph.D., 2011
• Are often used to teach rote.
• Are liked only by those who don’t need them.
• Don’t work for those with learning disabilities.
Memorizing MathFlash cards:
9 + 7
© Joan A. Cotter, Ph.D., 2011
• 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.
Memorizing MathFlash cards:
9 + 7
© Joan A. Cotter, Ph.D., 2011
• 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.
Memorizing Math 9 + 7Flash cards:
© Joan A. Cotter, Ph.D., 2011
• Are often used to teach rote.
• Are liked only by those who don’t need them.
• Don’t work for those with learning disabilities.
• Give the false impression that math isn’t about thinking.
• Often produce stress – children under stress stop learning.
• Are not concrete – use abstract symbols.
Memorizing Math 9 + 7Flash cards:
© Joan A. Cotter, Ph.D., 2011
Research on CountingKaren Wynn’s research
Show the baby two teddy bears.
© Joan A. Cotter, Ph.D., 2011
Research on Counting
Karen Wynn’s research
Then hide them with a screen.
© Joan A. Cotter, Ph.D., 2011
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., 2011
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., 2011
Research on CountingKaren Wynn’s research
Raise screen. Baby seeing 3 won’t look long because it is expected.
© Joan A. Cotter, Ph.D., 2011
Research on Counting
Karen Wynn’s research
Researcher can change the number of teddy bears behind the screen.
© Joan A. Cotter, Ph.D., 2011
Research on CountingKaren Wynn’s research
A baby seeing 1 teddy bear will look much longer, because it’s unexpected.
© Joan A. Cotter, Ph.D., 2011
Research on CountingOther research
© Joan A. Cotter, Ph.D., 2011
Research on Counting
• Australian Aboriginal children from two tribes.Brian Butterworth, University College London, 2008.
Other research
These groups matched quantities without using counting words.
© Joan A. Cotter, Ph.D., 2011
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
These groups matched quantities without using counting words.
© Joan A. Cotter, Ph.D., 2011
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
These groups matched quantities without using counting words.
© Joan A. Cotter, Ph.D., 2011
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
These groups matched quantities without using counting words.
© Joan A. Cotter, Ph.D., 2011
Research on CountingIn Japanese schools:
• Children are discouraged from using counting for adding.
© Joan A. Cotter, Ph.D., 2011
Research on CountingIn Japanese schools:
• Children are discouraged from using counting for adding.
• They consistently group in 5s.
© Joan A. Cotter, Ph.D., 2011
Research on CountingSubitizing
• Subitizing is recognizing a quantity without counting.
© Joan A. Cotter, Ph.D., 2011
Research on CountingSubitizing
• Subitizing is recognizing a quantity without counting.• Human babies and some animals can subitize small quantities at birth.
© Joan A. Cotter, Ph.D., 2011
Research on CountingSubitizing
• Subitizing is recognizing a quantity without counting.
• “One could have grasped the idea of a square without being able to count to four, that is, without learning the number of sides and corners.”—Montessori
• Human babies and some animals can subitize small quantities at birth.
Said to contrast to teaching the names of a triangle and square by counting.
© Joan A. Cotter, Ph.D., 2011
Research on CountingSubitizing
• Subitizing is recognizing a quantity without counting.
• “One could have grasped the idea of a square without being able to count to four, that is, without learning the number of sides and corners.”—Montessori
• Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit
• Human babies and some animals can subitize small quantities at birth.
© Joan A. Cotter, Ph.D., 2011
Research on CountingSubitizing
• Subitizing is recognizing a quantity without counting.
• “One could have grasped the idea of a square without being able to count to four, that is, without learning the number of sides and corners.”—Montessori
• Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit
• Subitizing seems to be a necessary skill for under-standing what the counting process means.—Glasersfeld
• Human babies and some animals can subitize small quantities at birth.
© Joan A. Cotter, Ph.D., 2011
Research on CountingSubitizing
• Subitizing is recognizing a quantity without counting.
• “One could have grasped the idea of a square without being able to count to four, that is, without learning the number of sides and corners.”—Montessori
• 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 under-standing what the counting process means.—Glasersfeld
• Human babies and some animals can subitize small quantities at birth.
© Joan A. Cotter, Ph.D., 2011
Research on CountingFinger gnosia
• Finger gnosia is the ability to know which fingers can been lightly touched without looking.
© Joan A. Cotter, Ph.D., 2011
Research on CountingFinger 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., 2011
Research on CountingFinger 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., 2011
Visualizing Mathematics
© Joan A. Cotter, Ph.D., 2011
Visualizing Mathematics
“In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.”
Mindy Holte (E I)
© Joan A. Cotter, Ph.D., 2011
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., 2011
Visualizing Mathematics
“Mathematics is the activity of creating relationships, many of which are based in visual imagery.”
Wheatley and Cobb
© Joan A. Cotter, Ph.D., 2011
Visualizing Mathematics
“The process of connecting symbols to imagery is at the heart of mathematics learning.”
Dienes
© Joan A. Cotter, Ph.D., 2011
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., 2011
• Representative of structure of numbers.• Easily manipulated by children.• Imaginable mentally.
Visualizing MathematicsJapanese criteria for manipulatives
Japanese Council ofMathematics Education
© Joan A. Cotter, Ph.D., 2011
Visualizing Mathematics
• Reading• Sports• Creativity• Geography• Engineering• Construction
Visualizing also needed in:
© Joan A. Cotter, Ph.D., 2011
Visualizing Mathematics
• Reading• Sports• Creativity• Geography• Engineering• Construction
• Architecture• Astronomy• Archeology• Chemistry• Physics• Surgery
Visualizing also needed in:
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsReady: How many?
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsReady: How many?
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsTry again: How many?
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsTry again: How many?
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsTry again: How many?
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsReady: How many?
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsTry again: How many?
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsTry to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsTry to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsNow try to visualize 5 as red and 3 as green.
© Joan A. Cotter, Ph.D., 2011
Visualizing MathematicsNow try to visualize 5 as red and 3 as green.
© Joan A. Cotter, Ph.D., 2011
Visualizing Mathematics
I II III IIII V VIII
1 23458
Early Roman numerals
Romans grouped in fives. Notice 8 is 5 and 3.
© Joan A. Cotter, Ph.D., 2011
Visualizing Mathematics
Who could read the music?
:
Music needs 10 lines, two groups of five.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesUsing fingers
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesUsing fingers
Naming quantities is a three-period lesson.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesUsing fingers
Use left hand for 1-5 because we read from left to right.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesUsing fingers
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesUsing fingers
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesUsing fingers
Always show 7 as 5 and 2, not for example, as 4 and 3.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesUsing fingers
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesYellow 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
Also set to music. Listen and download sheet music from Web site.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesRecognizing 5
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesRecognizing 5
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
5 has a middle; 4 does not.
Recognizing 5
Look at your hand; your middle finger is longer to remind you 5 has a middle.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesTally sticks
Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesTally sticks
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesTally sticks
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesTally sticks
Stick is horizontal, because it won’t fit diagonally and young children have problems with diagonals.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesTally sticks
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesTally sticks
Start a new row for every ten.
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
What is 4 apples plus 3 more apples?
Solving a problem without counting
How would you find the answer without counting?
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
What is 4 apples plus 3 more apples?
Solving a problem without counting
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., 2011
Naming Quantities
1
2
3
4
5
NumberControlChart
To help the child learn the symbols
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
61
72
83
94
105
To help the child learn the symbols
NumberControlChart
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Pairing Finger Cards
Use two sets of finger cards and match them.
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Ordering Finger Cards
Putting the finger cards in order.
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.QuickTime™ and a
TIFF (LZW) decompressorare needed to see this picture.
10
5 1
Matching Numbers to Finger Cards
Match the number to the finger card.
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
9 4Matching Fingers to Number Cards
1 610
2 83 57
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Match the finger card to the number.
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
Finger Card Memory game
Use two sets of finger cards and play Memory.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesNumber Rods
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesNumber Rods
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesNumber Rods
Using different colors.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesSpindle Box
45 dark-colored and 10 light-colored spindles. Could be in separate containers.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesSpindle Box
45 dark-colored and 10 light-colored spindles in two containers.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesSpindle Box
1 2 30 4
The child takes blue spindles with left hand and yellow with right.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesSpindle Box
6 7 85 9
The child takes blue spindles with left hand and yellow with right.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesSpindle Box
6 7 85 9
The child takes blue spindles with left hand and yellow with right.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesSpindle Box
6 7 85 9
The child takes blue spindles with left hand and yellow with right.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesSpindle Box
6 7 85 9
The child takes blue spindles with left hand and yellow with right.
© Joan A. Cotter, Ph.D., 2011
Naming QuantitiesSpindle Box
6 7 85 9
The child takes blue spindles with left hand and yellow with right.
© Joan A. Cotter, Ph.D., 2011
6 7 85 9
Naming QuantitiesSpindle Box
The child takes blue spindles with left hand and yellow with right.
© Joan A. Cotter, Ph.D., 2011
Naming Quantities
“Grouped in fives so the child does not need to count.”
Black and White Bead Stairs
A. M. Joosten
This was the inspiration to group in 5s.
© Joan A. Cotter, Ph.D., 2011
AL Abacus1000 10 1100
Double-sided AL abacus. Side 1 is grouped in 5s.Trading Side introduces algorithms with trading.
© Joan A. Cotter, Ph.D., 2011
AL AbacusCleared
© Joan A. Cotter, Ph.D., 2011
3
AL AbacusEntering quantities
Quantities are entered all at once, not counted.
© Joan A. Cotter, Ph.D., 2011
5
AL AbacusEntering quantities
Relate quantities to hands.
© Joan A. Cotter, Ph.D., 2011
7
AL AbacusEntering quantities
© Joan A. Cotter, Ph.D., 2011
AL Abacus
10
Entering quantities
© Joan A. Cotter, Ph.D., 2011
AL AbacusThe stairs
Can use to “count” 1 to 10. Also read quantities on the right side.
© Joan A. Cotter, Ph.D., 2011
AL AbacusAdding
© Joan A. Cotter, Ph.D., 2011
AL AbacusAdding
4 + 3 =
© Joan A. Cotter, Ph.D., 2011
AL AbacusAdding
4 + 3 =
© Joan A. Cotter, Ph.D., 2011
AL AbacusAdding
4 + 3 =
© Joan A. Cotter, Ph.D., 2011
AL AbacusAdding
4 + 3 =
© Joan A. Cotter, Ph.D., 2011
AL AbacusAdding
4 + 3 = 7
Answer is seen immediately, no counting needed.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump GameAim: To learn the facts that total 10:
1 + 92 + 83 + 74 + 65 + 5
Children use the abacus while playing this “Go Fish” type game.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump GameAim: 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.
Children use the abacus while playing this “Go Fish” type game.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Starting
A game viewed from above.
© Joan A. Cotter, Ph.D., 2011
7 2 7 9 5
7 42 61 3 8 3 4 9
Go to the Dump Game
Starting
Each player takes 5 cards.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 42 61 3 8 3 4 9
Does YellowCap have any pairs? [no]
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 42 61 3 8 3 4 9
Does BlueCap have any pairs? [yes, 1]
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 42 61 3 8 3 4 9
Does BlueCap have any pairs? [yes, 1]
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 2 1 3 8 3 4 9
4 6
Does BlueCap have any pairs? [yes, 1]
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 2 1 3 8 3 4 9
4 6
Does PinkCap have any pairs? [yes, 2]
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Finding pairs
7 2 7 9 5
7 2 1 3 8 3 4 9
4 6
Does PinkCap have any pairs? [yes, 2]
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Finding pairs
7 2 7 9 5
2 1 8 3 4 9
4 67 3
Does PinkCap have any pairs? [yes, 2]
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Finding pairs
7 2 7 9 5
1 3 4 9
4 62 82 8
Does PinkCap have any pairs? [yes, 2]
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Playing
7 2 7 9 5
1 3 4 9
4 62 82 8
The player asks the player on her left.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 3?
Playing
7 2 7 9 5
1 3 4 9
4 62 82 8
The player asks the player on her left.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 3?
Playing
7 2 7 9 5
1
3
4 9
4 62 82 8
The player asks the player on her left.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 3?
Playing
2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2011
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 8?
Playing
2 7 9 5
1 4 9
4 62 82 8
7 3
YellowCap gets another turn.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 8?
Go to the dump.Playing
2 7 9 5
1 4 9
4 62 82 8
7 3
YellowCap gets another turn.
© Joan A. Cotter, Ph.D., 2011
2
Go to the Dump GameBlueCap, do you
have a 3?BlueCap, do you
have an 8?
Go to the dump.Playing
2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Playing
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
PinkCap, do youhave a 6?Playing
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
PinkCap, do youhave a 6?PlayingGo to the dump.
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2011
5
Go to the Dump Game
Playing
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Playing
5
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
YellowCap, doyou have a 9? Playing
5
2 2 7 9 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
YellowCap, doyou have a 9? Playing
5
2 2 7 5
1 4 9
4 62 82 8
7 3
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
YellowCap, doyou have a 9? Playing
5
2 2 7 5
1 4 9
4 62 82 8
7 3
9
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Playing
5
2 2 7 5
4 9
4 62 81 9
7 3
© Joan A. Cotter, Ph.D., 2011
2 9 1 7 7
Go to the Dump Game
Playing
5
2 2 7 5
4 9
4 62 81 9
7 3
PinkCap is not out of the game. Her turn ends, but she takes 5 more cards.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Winner?
5 54 6
9 1
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Winner?
5546
91
No counting. Combine both stacks.
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Winner?
5546
91
Whose stack is the highest?
© Joan A. Cotter, Ph.D., 2011
Go to the Dump Game
Next game
No shuffling needed for next game.
© Joan A. Cotter, Ph.D., 2011
“Math” Way of Naming Numbers
© Joan A. Cotter, Ph.D., 2011
“Math” Way of Naming Numbers
11 = ten 1
© Joan A. Cotter, Ph.D., 2011
“Math” Way of Naming Numbers
11 = ten 112 = ten 2
© Joan A. Cotter, Ph.D., 2011
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 3
© Joan A. Cotter, Ph.D., 2011
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4
© Joan A. Cotter, Ph.D., 2011
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9
© Joan A. Cotter, Ph.D., 2011
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9
20 = 2-ten
Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.
© Joan A. Cotter, Ph.D., 2011
“Math” Way of Naming Numbers
11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9
20 = 2-ten 21 = 2-ten 1
Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.
© Joan A. Cotter, Ph.D., 2011
“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
Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.
© Joan A. Cotter, Ph.D., 2011
“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
Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.
© Joan A. Cotter, Ph.D., 2011
“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., 2011
“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., 2011
“Math” Way of Naming Numbers
137 = 1 hundred 3-ten 7or
137 = 1 hundred and 3-ten 7
Only numbers under 100 need to be said the “math” way.
© Joan A. Cotter, Ph.D., 2011
“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]
ChineseU.S.
Ave
rage
Hig
hest
Num
ber C
ount
ed
Shows how far children from 3 countries can count at ages 4, 5, and 6.
© Joan A. Cotter, Ph.D., 2011
“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]
ChineseU.S.
Ave
rage
Hig
hest
Num
ber C
ount
ed
Purple is Chinese. Note jump between ages 5 and 6.
© Joan A. Cotter, Ph.D., 2011
“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]
ChineseU.S.
Ave
rage
Hig
hest
Num
ber C
ount
ed
Dark green is Korean “math” way.
© Joan A. Cotter, Ph.D., 2011
“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]
ChineseU.S.
Ave
rage
Hig
hest
Num
ber C
ount
ed
Dotted green is everyday Korean; notice smaller jump between ages 5 and 6.
© Joan A. Cotter, Ph.D., 2011
“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]
ChineseU.S.
Ave
rage
Hig
hest
Num
ber C
ount
ed
Red is English speakers. They learn same amount between ages 4-5 and 5-6.
© Joan A. Cotter, Ph.D., 2011
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., 2011
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., 2011
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., 2011
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., 2011
Math Way of Naming NumbersCompared to reading:
© Joan A. Cotter, Ph.D., 2011
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., 2011
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., 2011
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).
• 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.
Compared to reading:
© Joan A. Cotter, Ph.D., 2011
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., 2011
Math Way of Naming NumbersTraditional names
4-ten = forty
The “ty” means tens.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional 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., 2011
Math Way of Naming NumbersTraditional names
6-ten = sixty
The “ty” means tens.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
3-ten = thirty
“Thir” also used in 1/3, 13 and 30.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
5-ten = fifty
“Fif” also used in 1/5, 15 and 50.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
2-ten = twenty
Two used to be pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
A word gamefireplace place-fire
Say the syllables backward. This is how we say the teen numbers.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
A word gamefireplace place-fire
paper-newsnewspaper
Say the syllables backward. This is how we say the teen numbers.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
A word gamefireplace place-fire
paper-news
box-mail mailbox
newspaper
Say the syllables backward. This is how we say the teen numbers.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
ten 4
“Teen” also means ten.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
ten 4 teen 4
“Teen” also means ten.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
ten 4 teen 4 fourteen
“Teen” also means ten.
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
a one left
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
a one left a left-one
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
a one left a left-one eleven
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
two left
Two pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2011
Math Way of Naming NumbersTraditional names
two left twelve
Two pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
3-ten
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
3-ten
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
3-ten
3 0
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
3-ten
3 0
Point to the 3 and say 3.
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
3-ten
3 0
Point to 0 and say 10. The 0 makes 3 a ten.
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
3-ten 7
3 0
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
3-ten 7
3 0
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
3-ten 7
3 0 7
© Joan A. Cotter, Ph.D., 2011
3 0
Composing Numbers
3-ten 7
7
Place the 7 on top of the 0 of the 30.
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
3-ten 7
Notice the way we say the number, represent the number, and write the number all correspond.
3 07
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
10-ten
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
10-ten
1 0 0
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
10-ten
1 0 0
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
10-ten
1 0 0
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
1 hundred
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
1 hundred
1 0 0
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
1 hundred
1 0 0
Of course, we can also read it as one-hun-dred.
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
1 hundred
1 01 01 0 0
Of course, we can also read it as one-hun-dred.
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
1 hundred
1 0 0
Of course, we can also read it as one-hun-dred.
© Joan A. Cotter, Ph.D., 2011
2584 8
Composing Numbers
To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
Reading numbers backward
© Joan A. Cotter, Ph.D., 2011
2584 58
Composing Numbers
To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
Reading numbers backward
© Joan A. Cotter, Ph.D., 2011
2584258
Composing Numbers
To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
Reading numbers backward
© Joan A. Cotter, Ph.D., 2011
2584258
Composing Numbers
4
To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
Reading numbers backward
© Joan A. Cotter, Ph.D., 2011
2584258
Composing Numbers
4
To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
Reading numbers backward
The Decimal Cards encourage reading numbers in the normal order.
© Joan A. Cotter, Ph.D., 2011
Composing Numbers
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.)
Scientific Notation
4000 = 4 x 103
© Joan A. Cotter, Ph.D., 2011
Fact Strategies
© Joan A. Cotter, Ph.D., 2011
Fact Strategies
• A strategy is a way to learn a new fact or recall a forgotten fact.
© Joan A. Cotter, Ph.D., 2011
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., 2011
Fact StrategiesComplete the Ten
9 + 5 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesComplete the Ten
9 + 5 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesComplete the Ten
9 + 5 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesComplete the Ten
9 + 5 =
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesComplete 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., 2011
Fact StrategiesComplete the Ten
9 + 5 =
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesComplete the Ten
9 + 5 = 14
Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesTwo Fives
8 + 6 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesTwo Fives
8 + 6 =
Two fives make 10.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesTwo Fives
8 + 6 =
Just add the “leftovers.”
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesTwo Fives
8 + 6 =10 + 4 = 14
Just add the “leftovers.”
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesTwo Fives
7 + 5 =
Another example.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesTwo Fives
7 + 5 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesTwo Fives
7 + 5 = 12
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Down
15 – 9 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Down
15 – 9 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Down
15 – 9 =
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Down
15 – 9 =
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Down
15 – 9 =
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Down
15 – 9 = 6
Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesSubtract from 10
15 – 9 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesSubtract from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesSubtract from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesSubtract from 10
15 – 9 =
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesSubtract from 10
15 – 9 = 6
Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Up
13 – 9 =
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Up
13 – 9 =
Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Up
13 – 9 =
Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Up
13 – 9 =
Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Up
13 – 9 =
Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2011
Fact StrategiesGoing Up
13 – 9 =1 + 3 = 4
Start with 9; go up to 13.
© Joan A. Cotter, Ph.D., 2011
MoneyPenny
© Joan A. Cotter, Ph.D., 2011
MoneyNickel
© Joan A. Cotter, Ph.D., 2011
MoneyDime
© Joan A. Cotter, Ph.D., 2011
MoneyQuarter
© Joan A. Cotter, Ph.D., 2011
MoneyQuarter
© Joan A. Cotter, Ph.D., 2011
MoneyQuarter
© Joan A. Cotter, Ph.D., 2011
MoneyQuarter
© Joan A. Cotter, Ph.D., 2011
Bead Frame
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
Bead Frame
8+ 6
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
Bead Frame
8+ 6
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
Bead Frame
8+ 6
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
Bead Frame
8+ 6
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
Bead Frame
8+ 6
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
Bead Frame
8+ 6
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
Bead Frame
8+ 6
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
Bead Frame
8+ 6
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
Bead Frame
8+ 6
1
10
100
1000
© Joan A. Cotter, Ph.D., 2011
8+ 614
1
10
100
1000
Bead Frame
© Joan A. Cotter, Ph.D., 2011
Bead FrameDifficulties for the child
© Joan A. Cotter, Ph.D., 2011
• Distracting: Room is visible through the frame.
Bead FrameDifficulties for the child
© Joan A. Cotter, Ph.D., 2011
• Distracting: Room is visible through the frame.
• Not visualizable: Beads need to be grouped in fives.
Bead FrameDifficulties for the child
© Joan A. Cotter, Ph.D., 2011
• Distracting: Room is visible through the frame.
• Not visualizable: Beads need to be grouped in fives.
• Inconsistent with equation order when beads are moved right: Beads need to be moved left.
Bead FrameDifficulties for the child
© Joan A. Cotter, Ph.D., 2011
• Distracting: Room is visible through the frame.
• Not visualizable: Beads need to be grouped in fives.
• Inconsistent with equation order when beads are moved right: Beads need to be moved left.
• Hierarchies of numbers represented sideways: They need to be in vertical columns.
Bead FrameDifficulties for the child
© Joan A. Cotter, Ph.D., 2011
• Distracting: Room is visible through the frame.
• Not visualizable: Beads need to be grouped in fives.
• Inconsistent with equation order when beads are moved right: Beads need to be moved left.
• Hierarchies of numbers represented sideways: They need to be in vertical columns.
• Trading done before second number is completely added: Addends need to combined before trading.
Bead FrameDifficulties for the child
© Joan A. Cotter, Ph.D., 2011
• Distracting: Room is visible through the frame.
• Not visualizable: Beads need to be grouped in fives.
• Inconsistent with equation order when beads are moved right: Beads need to be moved left.
• Hierarchies of numbers represented sideways: They need to be in vertical columns.
• Trading done before second number is completely added: Addends need to combined before trading.
• Answer is read going up: We read top to bottom.
Bead FrameDifficulties for the child
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideCleared
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideThousands
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideHundreds
The third wire from each end is not used.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideTens
The third wire from each end is not used.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideOnes
The third wire from each end is not used.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding
8+ 6
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding
8+ 6
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding
8+ 6
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding
8+ 6
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding
8+ 614
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding
8+ 614
Too many ones; trade 10 ones for 1 ten.
You can see the 10 ones (yellow).
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding
8+ 614
Too many ones; trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding
8+ 614
Too many ones; trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding
8+ 614
Same answer before and after trading.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
Object: To get a high score by adding numbers on the green cards.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
Object: To get a high score by adding numbers on the green cards.
7
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
Object: To get a high score by adding numbers on the green cards.
7
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
6
Turn over another card. Enter 6 beads. Do we need to trade?
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
6
Turn over another card. Enter 6 beads. Do we need to trade?
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
6
Turn over another card. Enter 6 beads. Do we need to trade?
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
6
Trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
6
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
6
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
9
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
9
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
9
Another trade.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
9
Another trade.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
3
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideBead Trading game
3
© Joan A. Cotter, Ph.D., 2011
Trading SideBead Trading game
• In the Bead Trading game 10 ones for 1 ten occurs frequently;
© Joan A. Cotter, Ph.D., 2011
Trading SideBead Trading game
• In the Bead Trading game 10 ones for 1 ten occurs frequently;10 tens for 1 hundred, less often;
© Joan A. Cotter, Ph.D., 2011
Trading SideBead Trading game
• In the Bead Trading game 10 ones for 1 ten occurs frequently;10 tens for 1 hundred, less often;10 hundreds for 1 thousand, rarely.
© Joan A. Cotter, Ph.D., 2011
Trading SideBead Trading game
• In the Bead Trading game 10 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., 2011
Trading SideBead Trading game
• In the Bead Trading game 10 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.)
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
6
Add starting at the right. Write results after each step.
. . . 6 ones. Did anything else happen?
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
6
Add starting at the right. Write results after each step.
1
Is it okay to show the extra ten by writing a 1 above the tens column?
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
6
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
6
Add starting at the right. Write results after each step.
1
Do we need to trade? [no]
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
Do we need to trade? [yes]
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
Notice the number of yellow beads. [3] Notice the number of blue beads left. [3] Coincidence? No, because 13 – 10 = 3.
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
96
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
396
Add starting at the right. Write results after each step.
1
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
6396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2011
1000 10 1100
Trading SideAdding 4-digit numbers
3658+ 2738
6396
Add starting at the right. Write results after each step.
11
© Joan A. Cotter, Ph.D., 2011
The Stamp Game
1000 100 10 1
1000 100 10 1
100 10 1
100 10 1
100 10 1
100 1
1000 100 10 1
1000 100 10 1
© Joan A. Cotter, Ph.D., 2011
The Stamp Game
1000 100 10 1
1000 100 10 1
100 10 1
100 10 1
100 10 1
100 1
1000 100 10 1
1000 100 10 1
© Joan A. Cotter, Ph.D., 2011
The Stamp Game
100 10 1100 10 1
100 10 1100 10 1
10 1 1
1000 100 10 11000 100 10 1
1000 100 10 11000 100 10 1
10
10
100 100
100 100
100 100
100 100
© Joan A. Cotter, Ph.D., 2011
The Stamp Game
100 10 1100 10 1
100 10 1100 10 1
10 1 1
1000 100 10 11000 100 1
1000 100 10 11000 100 10 1
10
10
100 100
100 100
100 100
100 100
10
© Joan A. Cotter, Ph.D., 2011
The Stamp Game
100 10 1100 10 1
100 10 1100 10 1
10 1 1
1000 100 10 11000 100 1
1000 100 10 11000 100 10 1
10
10
100 100
100 100
100 100
100 100
10
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
6 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
9 3 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
9 3 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
9 3 =30
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
9 3 =30 – 3 = 27
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
4 8 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
4 8 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
4 8 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
4 8 =20 + 12 = 32
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
7 7 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
7 7 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusBasic facts
7 7 =25 + 10 + 10 + 4 = 49
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusCommutative property
5 6 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusCommutative property
5 6 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusCommutative property
5 6 =
© Joan A. Cotter, Ph.D., 2011
Multiplication on the AL AbacusCommutative property
5 6 = 6 5
© Joan A. Cotter, Ph.D., 2011
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
© Joan A. Cotter, Ph.D., 2011
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
© Joan A. Cotter, Ph.D., 2011
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens:
© Joan A. Cotter, Ph.D., 2011
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens:
© Joan A. Cotter, Ph.D., 2011
7 8 =
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: 20+ 30
© Joan A. Cotter, Ph.D., 2011
7 8 =50 +
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: 20+ 30
50
© Joan A. Cotter, Ph.D., 2011
7 8 =50 +
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:20+ 30
50
© Joan A. Cotter, Ph.D., 2011
7 8 =50 +
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:20+ 30
50
© Joan A. Cotter, Ph.D., 2011
7 8 =50 +
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 3 2
20+ 30
50
© Joan A. Cotter, Ph.D., 2011
7 8 =50 + 6
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 3 26
20+ 30
50
© Joan A. Cotter, Ph.D., 2011
7 8 =50 + 6 = 56
This method was used in the Middle Ages, rather than memorize the facts > 5 5.
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 3 26
20+ 30
50
© Joan A. Cotter, Ph.D., 2011
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
© Joan A. Cotter, Ph.D., 2011
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
© Joan A. Cotter, Ph.D., 2011
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
Tens:
© Joan A. Cotter, Ph.D., 2011
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
Tens:
© Joan A. Cotter, Ph.D., 2011
9 7 =
Multiplication on the AL AbacusFor facts > 5 5
Tens: 40+ 20
© Joan A. Cotter, Ph.D., 2011
9 7 =60 +
Multiplication on the AL AbacusFor facts > 5 5
Tens: 40+ 20
60
© Joan A. Cotter, Ph.D., 2011
9 7 =60 +
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:40+ 20
60
© Joan A. Cotter, Ph.D., 2011
9 7 =60 +
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:40+ 20
60
© Joan A. Cotter, Ph.D., 2011
9 7 =60 +
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 1 3
40+ 20
60
© Joan A. Cotter, Ph.D., 2011
9 7 =60 + 3
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones:40+ 20
60
1 3
3
© Joan A. Cotter, Ph.D., 2011
9 7 =60 + 3 = 63
Multiplication on the AL AbacusFor facts > 5 5
Tens: Ones: 1 3
3
40+ 20
60
© Joan A. Cotter, Ph.D., 2011
The Multiplication Board1 2 3 4 5 6 7 8 9 10
6
6 4
6 x 4 on original multiplication board.
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 9 10
6
The Multiplication Board
6 4
Using two colors.
© Joan A. Cotter, Ph.D., 2011
The Multiplication Board1 2 3 4 5 6 7 8 9 10
7
7 7
7 x 7 on original multiplication board.
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 9 10
7
The Multiplication Board
7 7
Upper left square is 25, yellow rectangles are 10. So, 25, 35, 45, 49.
© Joan A. Cotter, Ph.D., 2011
The Multiplication Board
7 7
Less clutter.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
The ones repeat in the second row.
Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
The ones repeat in the second row.
Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
The ones repeat in the second row.
Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
The ones repeat in the second row.
Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsTwos
2 4 6 8 10
12 14 16 18 20
The ones repeat in the second row.
Recognizing multiples needed for fractions and algebra.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsFours
4 8 12 16 20
24 28 32 36 40
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsFours
4 8 12 16 20
24 28 32 36 40
The ones repeat in the second row.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
Again the ones repeat in the second row.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
The ones in the 8s show the multiples of 2.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80
6 4
6 4 is the fourth number (multiple).
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSixes and Eights
6 12 18 24 30
36 42 48 54 60
8 16 24 32 40
48 56 64 72 80 8 7
8 7 is the seventh number (multiple).
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsNines
9 18 27 36 45
90 81 72 63 54
The second row is written in reverse order.Also the digits in each number add to 9.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Observe the ones.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: The tens are the same in each row.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the digits in the columns.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the digits in the columns.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsThrees
3 6 9
12 15 18
21 24 27
30
The 3s have several patterns: Add the digits in the columns.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
The 7s have the 1, 2, 3… pattern.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
© Joan A. Cotter, Ph.D., 2011
Multiples PatternsSevens
7 14 21
28 35 42
49 56 63
70
Look at the tens.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
Aim: To help the players learn the multiples patterns.
“Multiples” are sometimes referred to as “skip counting.”
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
Object of the game: To be the first player to collect all ten cards of a multiple in order.
Aim: To help the players learn the multiples patterns.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
The 7s envelope contains 10 cards, each with one of the numbers listed.
7 14 2128 35 4249 56 63
70
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
The 8s envelope contains 10 cards, each with one of the numbers listed.
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
Players may refer to their envelopes at all times.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
Players may refer to their envelopes at all times.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
Players may refer to their envelopes at all times.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
14
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
The 7s player is looking for a 7.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
Wrong card, so it is turned face down in its original space.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
The 8s player takes a turn.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
40
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
Cannot use this card yet.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
Card returned.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
The needed card is collected. Receives another turn.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
8856
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
Who needs 56? [both 7s and 8s] At least one card per game is a duplicate.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
The needed card.
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
Where is that 14?
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7
14
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14
7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
© Joan A. Cotter, Ph.D., 2011
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14
24 7 14 2128 35 4249 56 6370
7 14 2128 35 4249 56 6370
A another turn.
© Joan A. Cotter, Ph.D., 2011
7 14 2128 35 4249 56 6370
Multiples Memory
8 16 24 32 4048 56 64 72 80
88
7 14
7 14 2128 35 4249 56 6370
8 16 24 32 4048 56 64 72 80
We’ll never know who won.
© Joan A. Cotter, Ph.D., 2011
7 14 2128 35 4249 56 6370
Multiples Memory
8 16 24 32 4048 56 64 72 80
8 16 24 32 4048 56 64 72 80
7 14 2128 35 4249 56 6370
We’ll never know who won.
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Giving the child the big picture, a Montessori principle.
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole? Giving the child the big picture, a Montessori principle.
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole? Giving the child the big picture, a Montessori principle.
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole? Giving the child the big picture, a Montessori principle.
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole? Giving the child the big picture, a Montessori principle.
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many fourths in a whole? Giving the child the big picture, a Montessori principle.
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
How many eighths in a whole?
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Which is more, 3/4 or 4/5?
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Which is more, 3/4 or 4/5?
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Which is more, 3/4 or 4/5?
© Joan A. Cotter, Ph.D., 2011
Fraction Chart1
12
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Which is more, 3/4 or 4/5?
© Joan A. Cotter, Ph.D., 2011
Fraction Chart
1
12
13
14
15
17
18
110
16
19
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2011
Fraction Chart
1
12
13
14
15
17
18
110
16
19
A hyperbola.
Stairs (Unit fractions)
© Joan A. Cotter, Ph.D., 2011
112
12
13
14
15
16
17
18
19
110
13
13
14
15
16
17
18
19
14
15
16
17
18
14
15
16171819
15
16
16
17
17
17
18
18
18
18
19
19
19
19
19
19
110
110
110
110
110
110
110
110
110
Fraction Chart
18
9/8 is 1 and 1/8.
© Joan A. Cotter, Ph.D., 2011
“Pie” Model
Are we comparing angles, arcs, or area?
© Joan A. Cotter, Ph.D., 2011
“Pie” Model
61
61
61
61
61
61
51
41
21 3
1
51
51
51
51
41
41
41
31
31
21
Try to compare 4/5 and 5/6 with this model.
© Joan A. Cotter, Ph.D., 2011
“Pie” Model
Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com
© Joan A. Cotter, Ph.D., 2011
“Pie” Model
Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com
Specialists also suggest refraining from using more than one pie chart for comparison.
statcan.ca
© Joan A. Cotter, Ph.D., 2011
“Pie” ModelDifficulties
© Joan A. Cotter, Ph.D., 2011
“Pie” ModelDifficulties
• Perpetuates cultural myth fractions always < 1.
© Joan A. Cotter, Ph.D., 2011
“Pie” ModelDifficulties
• Perpetuates cultural myth fractions always < 1.
• Requires counting pieces.
© Joan A. Cotter, Ph.D., 2011
“Pie” ModelDifficulties
• Perpetuates cultural myth fractions always < 1.
• Requires counting pieces.
• It does not give child the “big picture.”
© Joan A. Cotter, Ph.D., 2011
“Pie” ModelDifficulties
• Perpetuates cultural myth fractions always < 1.
• Requires counting pieces.
• It does not give child the “big picture.”
• A fraction is much more than “a part of a set of part of a whole.”
© Joan A. Cotter, Ph.D., 2011
“Pie” ModelDifficulties
• Perpetuates cultural myth fractions always < 1.
• Requires counting pieces.
• It does not give child the “big picture.”
• A fraction is much more than “a part of a set of part of a whole.”
• Difficult for the child to see how fractions relate to each other.
© Joan A. Cotter, Ph.D., 2011
“Pie” ModelDifficulties
• Perpetuates cultural myth fractions always < 1.
• Requires counting pieces.
• It does not give child the “big picture.”
• A fraction is much more than “a part of a set of part of a whole.”
• Difficult for the child to see how fractions relate to each other.
• Is the user comparing angles, arcs, or area?
© Joan A. Cotter, Ph.D., 2011
112
12
14
14
14
14
18
18
18
18
18
18
18
18
Fraction War
© Joan A. Cotter, Ph.D., 2011
112
12
14
14
14
14
18
18
18
18
18
18
18
18
Fraction War
© Joan A. Cotter, Ph.D., 2011
Fraction War
© Joan A. Cotter, Ph.D., 2011
Fraction War
1 2 3 4 5 6
Especially useful for learning to read a ruler with inches.
© Joan A. Cotter, Ph.D., 2011
Fraction War1
12
12
14
14
14
18
18
18
18
18
18
18
14
18
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
The fraction 4/8 can be reduced on the multiplication table as 1/2.
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
The fraction 4/8 can be reduced on the multiplication table as 1/2.
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
21212828
In what column would you put 21/28?
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
21212828
In what column would you put 21/28?
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
21212828
In what column would you put 21/28?
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
2121282845457272
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
2121282845457272
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
2121282845457272
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
12 12 1616
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
12 12 1616
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
12 12 1616
6/8 needs further simplifying.
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
12 12 1616
6/8 needs further simplifying.
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
12 12 1616
6/8 needs further simplifying.
© Joan A. Cotter, Ph.D., 2011
1 2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 36
10203040
6 12 18 24 30 36 42 48 54 607 14 21 28 35 42 49 56 63 708 16 24 32 40 48 56 64 72 809 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 1
5 10 15 20 25 30 35 40 45 50
00
Simplifying Fractions
12 12 1616
12/16 could have put here originally.
© Joan A. Cotter, Ph.D., 2011
In Conclusion
© Joan A. Cotter, Ph.D., 2011
In Conclusion• We need to use quantity, not counting words, as the basis of arithmetic.
© Joan A. Cotter, Ph.D., 2011
In Conclusion• We need to use quantity, not counting words, as the basis of arithmetic.
• Subitizing needs to be encouraged.
© Joan A. Cotter, Ph.D., 2011
In Conclusion• We need to use quantity, not counting words, as the basis of arithmetic.
• Subitizing needs to be encouraged.
• Children need to have visual images based on fives to remember the facts.
© Joan A. Cotter, Ph.D., 2011
In Conclusion• We need to use quantity, not counting words, as the basis of arithmetic.
• Subitizing needs to be encouraged.
• Children need to have visual images based on fives to remember the facts.
• Visualizing helps our disadvantaged children because it reduces the heavy memory load.
© Joan A. Cotter, Ph.D., 2011
In Conclusion• We need to use quantity, not counting words, as the basis of arithmetic.
• Subitizing needs to be encouraged.
• Children need to have visual images based on fives to remember the facts.
• Visualizing helps our disadvantaged children because it reduces the heavy memory load.
• We need to use the math way of number naming for a longer period of time.
© Joan A. Cotter, Ph.D., 2011
VII
IMF ConferenceOctober 21, 2011Sarasota, Florida
by Joan A. Cotter, [email protected]
Adding Visualization toMontessori Mathematics
Presentation available: ALabacus.com
7 x 7
100010
1
100
7 3
7 3