IMF: Visualizing and Montessori Math PART 1

How Visualization Enhances
Montessori Mathematics PART 1
by Joan A. Cotter, Ph.D.
JoanCotter@RightStartMath.com

Montessori Foundation 30
30
Conference 77
Friday, Nov 2, 2012
Sarasota, Florida 30
370
7
1000 100 10 1

7
7
7 3
3
3

PowerPoint Presentation
RightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012

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

Verbal Counting Model
From a child's perspective



Because we’re so familiar with 1, 2, 3, we’ll use letters.

A=1
B=2
C=3
D=4
E = 5, and so forth



F
+E



F
+E

A



F
+E

A B



F
+E

A B C



F
+E

A B C D E F



F
+E

A B C D E F A



F
+E

A B C D E F A B



F
+E

A B C D E F A B C D E



F
+E

A B C D E F A B C D E

What is the sum?
(It must be a letter.)


F
+E
K

A B C D E F G H I J K



Now memorize the facts!!

G
+D



H
+
G F
+D



H
+
G F
+D
D
+C


H
+
G F
+D
D C
+C +G


H
+
E

G F
I
+

+D
D C
+C +G


H
–E

Subtract with your fingers by counting backward.



J
–F

Subtract without using your fingers.



Try skip counting by B’s to T:
B, D, . . . T.



Try skip counting by B’s to T:
B, D, . . . T.

What is D × E?



L
is written AB
because it is A J
and B A’s



L
is written AB
because it is A J
and B A’s
huh?



L (twelve)
is written AB
because it is A J
and B A’s



L (twelve)
is written AB (12)
because it is A J
and B A’s



L (twelve)
is written AB (12)
because it is A J (one 10)
and B A’s



L (twelve)
is written AB (12)
because it is A J (one 10)
and B A’s (two 1s).


Calendar Math


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


Calendar Math
Calendar Counting
August
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31


Calendar Math
Septemb
Calendar Counting

1234567
August

89101214
1 2
113
11921
15112628
8
122820
67527
9
3 4 5 6
10 11 12 13 14
7

2234
20
15 16 17 18 19 20 21

29
3
22 23 24 25 26 27 28
29 30 31


Calendar Math
Septemb
Calendar Counting

1234567
August

89101214
1
113
11921
2

15112628
122820
8
67527
9
3 4 5 6
10 11 12 13 14
7

2234
20
15 16 17 18 19 20 21

29
3
22 23 24 25 26 27 28
29 30 31

This is ordinal counting, not cardinal counting.


Calendar Math
Partial Calendar
August
1 2 3 4 5 6 7
8 9 10


Calendar Math
Partial Calendar
August
1 2 3 4 5 6 7
8 9 10

Children need the whole month to plan ahead.


Calendar Math
Septemb
Calendar patterning

1234567
August

89101214
1 2
113
11921
15112628
8
122820
67527
9
3 4 5 6
10 11 12 13 14
7

2234
20
15 16 17 18 19 20 21

29
3
22 23 24 25 26 27 28
29 30 31

Patterns are rarely based on 7s or proceed row by row.
Patterns go on forever; they don’t stop at 31.

Research on Counting
Karen Wynn’s research


Other research


Other research
• Australian Aboriginal children from two tribes.
Brian Butterworth, University College London, 2008.


Other research

• Adult Pirahã from Amazon region.
Edward Gibson and Michael Frank, MIT, 2008.


Other research


• Adults, ages 18-50, from Boston.


Other research


• Adults, ages 18-50, from Boston.

• Baby chicks from Italy.
Lucia Regolin, University of Padova, 2009.


In Japanese schools:

• Children are discouraged from using
counting for adding.


In Japanese schools:

• Children are discouraged from using
counting for adding.
• They consistently group in 5s.


Subitizing Quantities
(Identifying without counting)



• Five-month-old infants can subitize to 3.




• Three-year-olds can subitize to 5.




• Four-year-olds can subitize 6 to 10 by
using five as a subbase.




• Four-year-olds can subitize 6 to 10 by
using five as a subbase.
• Counting is like sounding out each letter;
subitizing is recognizing the quantity.


Subitizing
• Subitizing “allows the child to grasp the whole
and the elements at the same time.”—Benoit


Subitizing
• Subitizing seems to be a necessary skill for
understanding what the counting process means.
—Glasersfeld


Subitizing
—Glasersfeld
• Children who can subitize perform better in
mathematics long term.—Butterworth


Subitizing
—Glasersfeld
• Counting-on is a difficult skill for many children.
—Journal for Res. in Math Ed. Nov. 2011


Subitizing
—Glasersfeld
• Counting-on is a difficult skill for many children.
—Journal for Res. in Math Ed. Nov. 2011
• Math anxiety affects counting ability, but
not subitizing ability.

Visualizing Quantities


“Think in pictures, because the
brain remembers images better
than it does anything else.”
Ben Pridmore, World Memory Champion, 2009


“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


Japanese criteria for manipulatives

• Representative of structure of numbers.
• Easily manipulated by children.
• Imaginable mentally.

Japanese Council of
Mathematics Education


Visualizing also needed in:
• Reading
• Sports
• Creativity
• Geography
• Engineering
• Construction

Visualizing also needed in:
• Reading • Architecture
• Sports • Astronomy
• Creativity • Archeology
• Geography • Chemistry
• Engineering • Physics
• Construction • Surgery

Ready: How many?


Try again: How many?


Try to visualize 8 identical apples without grouping.


Now try to visualize 5 as red and 3 as green.


Early Roman numerals

1 I
2 II
3 III
4 IIII
5 V
8 VIII


:

Who could read the music?


Grouping in Fives


Grouping in Fives

• Grouping in fives extends subitizing.


Grouping in Fives
Using fingers

Grouping in Fives is a three-period lesson.

Grouping in Fives
Yellow is the Sun
Yellow is the sun.
Six is five and one.
Why is the sky so blue?
Seven is five and two.
Salty is the sea.
Eight is five and three.
Hear the thunder roar.
Nine is five and four.
Ducks will swim and dive.
Ten is five and five.
–Joan A. Cotter


Grouping in Fives
Recognizing 5


Grouping in Fives
Recognizing 5

5 has a middle; 4 does not.


Grouping in Fives
Tally sticks


Grouping in Fives
Pairing Finger Cards
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Grouping in Fives
Ordering Finger Cards

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are needed to see this picture. QuickTime™ and a

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Grouping in Fives
Matching Number Cards to Finger Cards

5 1

10

Grouping in Fives
Matching Finger Cards to Number Cards

9 1 10 4 6
QuickTime™ and a

2 3 7 8 5
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Grouping in Fives
Finger Card Memory game
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Grouping in Fives
Number Rods


Grouping in Fives
Spindle Box


Grouping in Fives
Spindle Box

0 1 2 3 4


Grouping in Fives
Spindle Box

5 6 7 8 9


Grouping in Fives
1000 100 10 1

1000 100 10 1

1000 100 10 1

1000 100 10 1

100 10 1

100 10 1

100 10 1

100 1
Stamp Game

Grouping in Fives
1000 1000 100 100 10 10 1 1

1000 1000 100 100 10 10 1 1

100 100 10 10 1 1

100 100 10 10 1 1

100 100 10 10 1 1

100 100 10

100 100

100 100
Stamp Game

Grouping in Fives
1000 1000 100 100 10 1 1

1000 1000 100 100 1 1
10 10

100 100 1 1
10 10

100 100 1 1
10 10

100 100 1 1
10 10

100 100
10 10

100 100

100 100
Stamp Game

Grouping in Fives
1000 1000 100 100 10 1 1

1000 1000 100 100 1 1
10 10

100 100 1 1
10 10

1 1
100 100 10 10

1 1
100 100 10 10

100 100 10 10

100 100

Stamp Game 100 100

Grouping in Fives
Black and White Bead Stairs

“Grouped in fives so the child does not
need to count.” A. M. Joosten


Grouping in Fives
Entering quantities


Grouping in Fives
Entering quantities

3


Grouping in Fives
Entering quantities

5


Grouping in Fives
Entering quantities

7


Grouping in Fives
Entering quantities

10


Grouping in Fives
The stairs


Grouping in Fives
Adding


Grouping in Fives
Adding
4+3=


Grouping in Fives
Adding
4+3=7


Math Card Games


Math Card Games
• Provide repetition for learning the facts.


Math Card Games
• Encourage autonomy.


Math Card Games
• Promote social interaction.


Math Card Games
• Promote social interaction.
• Are enjoyed by the children.


Go to the Dump Game
Objective:
To learn the facts that total 10:
1+9
2+8
3+7
4+6
5+5


Go to the Dump Game
Objective:
To learn the facts that total 10:
1+9
2+8
3+7
4+6
5+5
Object of the game:
To collect the most pairs that equal ten.


“Math” Way of Naming Numbers


11 = ten 1


11 = ten 1
12 = ten 2


11 = ten 1
12 = ten 2
13 = ten 3


11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4


11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9


11 = ten 1 20 = 2-ten
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9


11 = ten 1 20 = 2-ten
12 = ten 2 21 = 2-ten 1
13 = ten 3
14 = ten 4
....
19 = ten 9


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


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


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


137 = 1 hundred 3-ten 7



137 = 1 hundred 3-ten 7
or
137 = 1 hundred and 3-ten 7


100 Chinese
U.S.

Average Highest Number Counted
90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
50
40
30
20
10
0
4 5 6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
children's counting: A natural experiment in numerical bilingualism. International Journal
of Psychology, 23, 319-332.


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.


• 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.

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).




• Just as we first teach the sound of the letters, we must
first teach the name of the quantity (math way).

• Montessorians need to use the math way of naming
numbers for a longer period of time.



“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


Traditional names

4-ten =
forty

The “ty”
means tens.


Traditional names

6-ten = sixty

The “ty”
means tens.


Traditional names

3-ten = thirty

“Thir” also
used in 1/3,
13 and 30.


Traditional names

5-ten = fifty

“Fif” also
used in 1/5,
15 and 50.


Traditional names

2-ten = twenty

Two used to be
pronounced
“twoo.”


Traditional names

A word game
fireplace place-fire


Traditional names

A word game
newspaper paper-news


Traditional names

A word game
newspaper paper-news
box-mail mailbox


Traditional names
ten 4

“Teen” also
means ten.


Traditional names
ten 4 teen 4

“Teen” also
means ten.


Traditional names
ten 4 teen 4 fourtee
n

“Teen” also
means ten.


Traditional names
a one left


Traditional names
a one left a left-one


Traditional names
a one left a left-one eleven


Traditional names
two left

Two said
as “twoo.”


Traditional names
two left twelve

Two said
as “twoo.”


Composing Numbers
3-ten


Composing Numbers
3-ten
30
30


Composing Numbers
3-ten 7
30
30


Composing Numbers
3-ten 7
30
30
7
7


Composing Numbers
3-ten 7
30
37
0
7


Composing Numbers
3-ten 7
30
37
0
7

Note the congruence in how we say the number,
represent the number, and write the number.

Composing Numbers
1-ten
10
10

Another example.


Composing Numbers
1-ten 8
10
10


Composing Numbers
1-ten 8
10
10
8
8


Composing Numbers
1-ten 8
18
18


Composing Numbers
10-ten


Composing Numbers
10-ten
100
100


Composing Numbers
1 hundred


Composing Numbers
1 hundred
100
100


Composing Numbers
2 hundred


Composing Numbers
2 hundred
200
200


Evens and Odds


Evens and Odds
Evens


Evens and Odds
Evens

Use two fingers
and touch each
pair in succession.


Evens and Odds
Evens

Use two fingers
and touch each
pair in succession.

EVEN!


Evens and Odds
Odds

Use two fingers
and touch each
pair in succession.


Evens and Odds
Odds

Use two fingers
and touch each
pair in succession.

ODD!


Learning the Facts


Learning the Facts
Limited success when:
• Based on counting.
Whether dots, fingers, number lines, or
counting words.


Learning the Facts
counting words.

• Based on rote memory.
Whether by flash cards or timed tests.


Learning the Facts
counting words.

• Based on rote memory.
Whether by flash cards or timed tests.

• Based on skip counting for multiplication facts.


Fact Strategies


Fact Strategies
Complete the Ten

9+5=


Fact Strategies
Complete the Ten

9+5=

Take 1 from
the 5 and give
it to the 9.


Fact Strategies
Complete the Ten

9 + 5 = 14

Take 1 from
the 5 and give
it to the 9.


Fact Strategies
Two Fives

8+6=


Fact Strategies
Two Fives

8+6=
10 + 4 = 14


Fact Strategies
Going Down

15 – 9 =


Fact Strategies
Going Down

15 – 9 =

Subtract 5;
then 4.


Fact Strategies
Going Down

15 – 9 = 6

Subtract 5;
then 4.


Fact Strategies
Subtract from 10

15 – 9 =


Fact Strategies
Subtract from 10

15 – 9 =

Subtract 9
from 10.


Fact Strategies
Subtract from 10

15 – 9 = 6

Subtract 9
from 10.


Fact Strategies
Going Up

15 – 9 =


Fact Strategies
Going Up

15 – 9 =

Start with 9;
go up to 15.


Fact Strategies
Going Up

15 – 9 =
1+5=6

Start with 9;
go up to 15.


Rows and Columns Game
Objective:
To find a total of 15 by adding 2, 3, or 4
cards in a row or in a column.


Objective:
To find a total of 15 by adding 2, 3, or 4
cards in a row or in a column.

Object of the game:
To collect the most cards.


8 7 1 9

6 4 3 3

2 2 5 6

6 3 8 8


1 9

6 4 3 3

6 3 8 8


7 6 1 9

6 4 3 3

2 1 5 1

6 3 8 8


1

6 4 3 3

1 5 1

3 8 8


Money
Penny


Money
Nickel


Money
Dime


Money
Quarter


Place Value
Two aspects


Place Value
Two aspects
Static


Place Value
Two aspects
Static
• Value of a digit is determined by position


Place Value
Two aspects
Static
• Value of a digit is determined by position.
• No position may have more than nine.


Place Value
Two aspects
Static
• As you progress to the left, value at each position
is ten times greater than previous position.


Place Value
Two aspects
Static
(Shown by the Decimal Cards.)


Place Value
Two aspects
Static
Dynamic


Place Value
Two aspects
Static
Dynamic
• 10 ones = 1 ten; 10 tens = 1 hundred;
10 hundreds = 1 thousand, ….


Place Value
Two aspects
Static
Dynamic
• 10 ones = 1 ten; 10 tens = 1 hundred;
10 hundreds = 1 thousand, ….
(Represented on the Abacus and other materials.)

Exchanging
1000 100 10 1


Exchanging
Thousands
1000 100 10 1


Exchanging
Hundreds
1000 100 10 1


Exchanging
Tens
1000 100 10 1


Exchanging
Ones
1000 100 10 1


Exchanging
Adding
1000 100 10 1

8
+6


Exchanging
Adding
1000 100 10 1

8
+6
14


Exchanging
Adding
1000 100 10 1

8
+6
14

Too many ones;
trade 10 ones for
1 ten.


Exchanging
Adding
1000 100 10 1

8
+6
14

Same answer
before and after
exchanging.


Bead Frame

1

10

100

1000


Bead Frame

1 8
10 +6
100

1000


Bead Frame

1 8
10 +6
100 14
1000


1

Bead Frame
10

100

1000
Difficulties for the child


1

Bead Frame
10

100

1000
• Not visualizable: Beads need to be grouped in fives.


1

Bead Frame
10

100

1000
• When beads are moved right, inconsistent with
equation order: Beads need to be moved left.


1

Bead Frame
10

100

1000
• Hierarchies of numbers represented sideways:
They need to be in vertical columns.


1

Bead Frame
10

100

1000
• Exchanging done before second number is
completely added: Addends need to be combined before
exchanging.


1

Bead Frame
10

100

1000
• Exchanging done before second number is
completely added: Addends need to be combined before
exchanging.
• Answer is read going up: We read top to bottom.


1

Bead Frame
10

100

1000
• Exchanging before second number is completely
added: Addends need to be combined before exchanging.
• Answer is read going up: We read top to bottom.
• Distracting: Room is visible through the frame.

Exchanging
Adding 4-digit numbers
1000 100 10 1

3658
+ 2738


Exchanging
1000 100 10 1

3658
+ 2738

Enter the first
number from left
to right.


Exchanging
1000 100 10 1

3658
+ 2738

Add starting at
the right. Write
results after each
step.

Exchanging
1000 100 10 1

3658
+ 2738
6

Add starting at
the right. Write
results after each
step.

Exchanging
1000 100 10 1 1
3658
+ 2738
6

Add starting at
the right. Write
results after each
step.

Exchanging
1000 100 10 1 1
3658
+ 2738
96

Add starting at
the right. Write
results after each
step.

Exchanging
1000 100 10 1 1
3658
+ 2738
396

Add starting at
the right. Write
results after each
step.

Exchanging
1000 100 10 1 1 1
3658
+ 2738
396

Add starting at
the right. Write
results after each
step.

IMF: Visualizing and Montessori Math PART 1

IMF: Visualizing and Montessori Math PART 1

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IMF: Visualizing and Montessori Math PART 1

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