Enriching Montessori Math with Visualization

1 3

Enriching Montessori National Math Crisis
Mathematics with Visualization • 25% of college freshmen take remedial math.
• In 2009, of the 1.5 million students who took the
Handout and by Joan A. Cotter, Ph.D. ACT test, only 42% are ready for college algebra.
Presentation: JoanCotter@ALabacus.com
• A generation ago, the US produced 30% of the
ALabacus.com world’s college grads; today it’s 14%. (CSM 2006)
AMS Fall Conference
October 22, 2010
San Diego, California • Two-thirds of 4-year degrees in Japan and China
are in science and engineering; one-third in the U.S.
7 • U.S. students, compared to the world, score high at
4th grade, average at 8th, and near bottom at 12th.
5 2
• Ready, Willing, and Unable to Serve says that 75% of
7x7
VII 17 to 24 year-olds are unfit for military service. (2010)
© Joan A. Cotter, 2010 © Joan A. Cotter, 2010

2 4

Key Decisions of a First-year Math Education is Changing
‘Turnaround’ Principal • The field of mathematics is doubling every 7 years.
D. Duke and M. Salmonowicz • Math is used in many new ways. The workplace
needs analytical thinkers and problem solvers.
Educational Administration Management & Leadership, 2010
• State exams require more than arithmetic: including
1) Elimination of an ineffective instructional program. geometry, algebra, probability, and statistics.
2) Creation of a culture of teacher accountability. • Brain research is providing clues on how to better
facilitate learning, including math.
3) Development of an effective reading program.
• Increased emphasis on mathematical reasoning,
less emphasis on rules and procedures.


5 7

Calendar Math Drawbacks Yellow is the Sun
• The calendar is not a number line. Yellow is the sun.
Six is five and one.
• No quantity is involved.
Why is the sky so blue?
• Numbers are in spaces, not at lines like a ruler.
Seven is five and two.
• Children need to see the whole month, not just part. Salty is the sea.
• Purpose of calendar is to plan ahead. Eight is five and three.
• Many ways to show the current date. Hear the thunder roar.
Nine is five and four.
• Calendars give a narrow view of patterning.
Ducks will swim and dive.
• Patterns do not necessarily involve numbers.
Ten is five and five.
• Patterns rarely proceed row by row. –Joan A. Cotter
• Patterns go on forever; they don’t stop at 31.

6 8

Memorizing Math Counting Model Drawbacks
Counting:
Percentage Recall • Is not natural.
Immediately After 1 day After 4 wks
• Provides poor concept of quantity.
Rote 32 23 8
Concept 69 69 58 • Ignores place value.
• Is very error prone.
Math needs to be taught so 95% is • Is inefficient and time-consuming.
understood and only 5% memorized.
• Is a hard habit to break for mastering
Richard Skemp the facts.


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Recognizing 5 Materials for Visualizing
• Representative of structure of numbers.

• Easily manipulated by children.

• Imaginable mentally.

Japanese Council of
Mathematics Education
5 has a middle; 4 does not.


10 12

Materials for Visualizing Materials for Visualizing
“In our concern about the memorization of math
“Mathematics is the activity of
facts or solving problems, we must not forget creating relationships, many of which
that the root of mathematical study is the
creation of mental pictures in the imagination
are based in visual imagery.”
and manipulating those images and relationships
Wheatley and Cobb
using the power of reason and logic.”

Mindy Holte (E I)


13 15

Materials for Visualizing Spindle Box
The role of physical manipulatives
was to help the child form those
0 1 2 3 4
visual images and thus to eliminate
the need for the physical
manipulatives.
Ginsberg and others


14 16

Number Rods Spindle Box

5 6 7 8 9


17 19

Bead Frame Challenges Adding
4+3= 7
• Distracting: Room is visible through the frame.
• Not visual: Beads need to be grouped in fives.
• Inconsistent with equation order when beads are
moved right: Beads need to be moved left.
• Hierarchies 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.


18 20

AL Abacus Sums Adding to Ten
1000 100 10 1


21 23

Math Way of Naming Numbers
Part-Whole Circles
• 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.)
10
• Asian children learn mathematics using the
math way of counting.

4 6 • They understand place value in first grade;
only half of U.S. children understand place
value at the end of fourth grade.
What is the other part?
• Mathematics is the science of patterns. The
patterned math way of counting greatly helps
children learn number sense.

22 24

Language Effect on Counting Math Way of Counting
Compared to Reading
100 Chinese
U.S.
Average Highest Number Counted

90 Korean formal [math way]
Korean informal [not explicit]
80
70
60
• Just as reciting the alphabet doesn’t teach reading,
50 counting doesn’t teach arithmetic.
40
30 • Just as we first teach the sound of the letters, we
20 first teach the name of the quantity (math way).
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.


25 27
Adding
7
1000 100 10 1
3-ten 7 30 7 8
+6


26 28

Strategy: Two Fives Adding
1000 100 10 1
8 + 7 = 10 + 5 = 15 8
+6
14


29 31

The Multiplication Board
“Pie” Model Difficulties
7x7 • Perpetuates cultural myth that fractions < 1.
• 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?


30 32

Fraction Chart Simplifying Fractions
1 1 2 3 4 5 6 7 8 9 10
1 1
2 2 2 4 6 8 10 12 14 16 18 20
1 1 1
3 3 3 3 6 9 12 15 18 21 24 27 30 21
1 1 1 1
4 4 4 4 4 8 12 16 20 24 28 32 36 40 28
1 1 1 1 1
5 5 5 5 5 5 10 15 20 25 30 35 40 45 50
1 1 1 1 1 1
6 12 18 24 30 36 42 48 54 60
45
1
6
1
6
1
6
1
6
1
6
1
6
1 72
7 7 7 7 7 7 7 7 14 21 28 35 42 49 56 63 70
1 1 1 1 1 1 1 1
8 8 8 8 8 8 8 8 8 16 24 32 40 48 56 64 72 80
1 1 1 1 1 1 1 1 1
9 9 9 9 9 9 9 9 9 9 18 27 36 45 54 63 72 81 90
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10 10 20 30 40 50 60 70 80 90 100
How many fourths make a whole? How many sixths?

!inger (ar*s APPENDI' 1
© Activities for Learning, Inc. 2010 This page may be duplicated for a single teacher or a single family’s use.

5
GO TO THE DUMP
(From Math Card Games: 300 Games for Learning and Enjoying Math. Fifth edition by
Joan A. Cotter (2010); published by Activities for Learning, Inc.: Hazelton, ND.)
Objective To learn the combinations that total 10
Number of players 2 to 4
Cards 4 or 6 of each basic number card 1 to 9
Deal Each player takes five cards; the remaining cards face down form the
dump, or stack.
Object of the game To collect the most pairs that equal 10
Materials Beginners need an abacus or at least a list of the facts.
1+9
2+8
3+7
4+6
6 is needed with 4 to make 10. 5+5
Preparation Before starting, the players check over their hands for pairs that total 10.
To do this, they look at each card in turn, determine what is needed to
make 10 and look for that number among their other cards. (Some
children may need to spread the cards out on the playing surface.)
Store paired cards face up on two piles. (This allows verification and
keeps the cards shuffled for the next game.)
4 6 8 2
4 6 8 2
Player 1. Player 2.
Play When all are ready, the first player asks the player on her left for a
number needed to complete a pair. If he has it, he must give it to her,
whereupon she receives another turn. If he does not have it, he says, “Go
to the Dump,” which is also the signal for him to begin his turn. He takes
a turn by asking the player on his left and so forth.Meanwhile, the first
player concludes her turn by picking up the top card from the dump.
She does not receive an additional turn even if she picks up a needed
card. However, she may put a new pair on top of her other pairs.
A player running out of cards takes five more cards, but the turn is
ended. When the dump is exhausted, players may ask any player (not
only the players on their left) for a card.
At the end of the game, players combine their two stacks and compare
the heights. (Counting the cards is too time consuming.) No shuffling is
necessary for subsequent games.
© 2010 Joan A. Cotter, Ph.D. • JoanCotter@ALabacus.com • alabacus.com

SKIP COUNTING MEMORY
Objective To learn the skip counting patterns on previous page.
Preparation To prepare the envelopes, see page 13. The players use the envelopes for
reference during the game to memorize the patterns.
Number of players 2 or 2 teams
Cards Each player or team chooses an envelope and removes the cards. Mix the
cards together and shuffle lightly. Lay the cards out face down in a 5 by 4
array.
Object of the game To be the first player to collect in order the complete set of cards
Play The first player turns over one card so both players can see it. If it is the
needed card, the player collects the card and receives another turn. If it is
not the needed card, the card is returned. Next the second player takes a
turn. Turns alternate until one player has picked up all ten cards.
Stress the importance of returning the cards to the correct 5 10
envelopes following a game. 15 20
25 30
2 4 6 8 10 35 40
12 14 16 18 20 45 50
2 4 6 5 10
2 4 6 5 10
A game in progress: The
player on the left collects
the 2s while the player on 12
the right collects the 5s. 12
MULTIPLICATION MEMORY
Objective To help the players master the multiplication facts.
Cards 10 basic number cards with numbers 1 to 10 and one set of product cards.
Also a sticky note with the set number and “×” and another note with “=.”
Number of players Two. Beginners should sit on the same side of the cards.
Object of the game To collect the most cards by matching the multiplier with the product.
Layout Lay the basic number cards face down in two rows. To the right in separate
rows lay the product cards.
Play The first player turns over a basic number card and states the fact. For
example, if the card is 4, the player says, “Three taken four times is 12.” He
then decides where it could be among the product cards. If he is correct, he
collects both cards and takes another turn. If it is not a match, both cards
are returned face down in their original places, and the other player takes a
turn.
4
4
3× = 12
12
© 2010 Joan Cotter • JoanCotter@ALabacus.com • More Games at: alabacus.com > Resources > Presentations

CONCENTRATING ON ONE
(From Math Card Games: 300 Games for Learning and Enjoying Math. Fifth edition by Joan
A. Cotter (2010); published by Activities for Learning, Inc.: Hazelton, ND.)
Objective To help the children realize that two halves, three thirds, and so forth,
equal one. Being told this fact does not necessarily mean understanding it.
2 1
Background Explain that – means two –s. Then lay down various fraction cards and ask
3 3
the children to find the equivalent fraction pieces.
3
Now, ask a child to lay the fraction pieces for – under the 1. Then ask her
5 1
how many more fifths are needed to make 1. [Two 5 Repeat this for other
–s]
1 7 1
fractions, such as 6 and —. Children often have a problem with 2
– 10 –.
Some children find the fraction chart to be very
1
helpful. With it they can see what they have and 1 1
2 2
count how many more are needed. With the left 1
3
1
3
1
3
index finger, the child counts what she has. With 1
4
1
4
1
4
1
4
the left finger still in place, she counts with her 1
5
1
5
1
5
1
5
1
5
right index finger how many more she needs. 1
6
1
6
1
6
1
6
1
6
1
6
Explain that these are the pairs for this game. 1
7
1
7
1
7
1
7
1
7
1
7
1
7
1 1 1 1 1 1 1 1
Cards Twenty fraction cards are needed: two 1⁄2s and 8 8 8
1 1 1 1 1 1 1 1 1
8 8 8 8 8
one of each of the following: 1⁄3, 2⁄3, 1⁄4, 3⁄4, 1⁄5, 9 9 9 9 9 9 9 9 9
1 1 1 1 1 1 1 1 1 1
2⁄5, 3⁄5, 4⁄5, 1⁄6, 5⁄6, 1⁄8, 3⁄8, 5⁄8, 7⁄8, 1⁄10, 3⁄10, 7⁄10, 10 10 10 10 10 10 10 10 10 10
and 9⁄10. The fraction chart.
Number of players Two or two teams.
Layout Lay the fraction cards out on the table face down in rows as shown.
Object of the game To collect the most pairs of fractions totaling one.
Play The first player turns over a card and decides how many more are needed
to make 1. She then chooses a probable card. If she is correct, she collects
both cards and takes another turn. If they do not match, both cards are
returned face down. The second player then takes his turn. Turns continue
until all the cards are collected.
1 Showing that five 1 equal 1.
–s
5
1 1 1 1 1
5 5 5 5 5
5
8
A beginning game showing
3 two fractions that equal 1.
8
© 2010 Joan Cotter • JoanCotter@ALabacus.com • More Games at: alabacus.com > Resources > Presentations

FRACTION WAR
(From Math Card Games: 300 Games for Learning and Enjoying Math. Fifth edition by Joan
A. Cotter (2010); published by Activities for Learning, Inc.: Hazelton, ND.)
Objective To provide practice in comparing two fractions between the 1s, halves,
fourths, and eighths, the fractions needed for reading a ruler.
Materials The 1, halves, fourths, and eighths of the fraction pieces, arranged as
shown below.
1
1 1
2 2
1 1 1 1
4 4 4 4
1 1 1 1 1 1 1 1
8 8 8 8 8 8 8 8
The fraction pieces forming a “ruler.”
Cards The fraction cards with 1s, halves, fourths, and eighths.
Number of players Two only.
Deal With the cards face down, divide the stack in half by comparing heights.
Object of the game To capture all the cards.
Play Each player takes the top card from his stack and lays it down in the
middle of the table face up. The player whose card is greater takes both
cards. Players should alternate deciding whose card is higher.
Players continue comparing cards until they put down cards of equal
value, which constitutes a “war.” To resolve a war, both players play two
cards face down and then play a third face up to be compared. The player
who has the high card in the last comparison takes all eight cards.
© 2010 Joan A. Cotter, Ph.D. • JoanCotter@ALabacus.com • alabacus.com

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