enriching montessori math with visualization
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© Joan A. Cotter, 2010
Enriching Montessori
Mathematics with Visualization
VII 7 x 7
AMS Fall ConferenceOctober 22, 2010
San Diego, California
by Joan A. Cotter, [email protected]
Handout and
Presentation:
ALabacus.com
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© Joan A. Cotter, 2010
Key Decisions of a First-year‘Turnaround’ Principal
1) Elimination of an ineffective instructional program.
2) Creation of a culture of teacher accountability.
3) Development of an effective reading program.
D. Duke and M. Salmonowicz
Educational Administration Management & Leadership, 2010
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© Joan A. Cotter, 2010
National Math Crisis
• Ready, Willing, and Unable to Serve says that 75% of17 to 24 year-olds are unfit for military service. (2010)
• 25% of college freshmen take remedial math.
• In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra.
• A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006)
• Two-thirds of 4-year degrees in Japan and Chinaare in science and engineering; one-third in the U.S.
• U.S. students, compared to the world, score high at4th grade, average at 8th, and near bottom at 12th.
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© Joan A. Cotter, 2010
Math Education is Changing• The field of mathematics is doubling every 7 years.
• Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers.
• State exams require more than arithmetic: includinggeometry, algebra, probability, and statistics.
• Brain research is providing clues on how to betterfacilitate learning, including math.
• Increased emphasis on mathematical reasoning,less emphasis on rules and procedures.
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© Joan A. Cotter, 2010
Calendar Math Drawbacks• 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.
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© Joan A. Cotter, 2010
Memorizing Math
Math needs to be taught so 95% isunderstood and only 5% memorized.
Richard Skemp
586969 Concept
82332 Rote
After 4 wksAfter 1 dayImmediately
Percentage Recall
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© Joan A. Cotter, 2010
Yellow is the SunYellow 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
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© Joan A. Cotter, 2010
Counting Model Drawbacks
• Is not natural.
• Provides poor concept of quantity.
• Ignores place value.
• Is very error prone.
• Is inefficient and time-consuming.
• Is a hard habit to break for masteringthe facts.
Counting:
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© Joan A. Cotter, 2010
Recognizing 5
5 has a middle; 4 does not.
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© Joan A. Cotter, 2010
Materials for Visualizing
“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)
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© Joan A. Cotter, 2010
Materials for Visualizing
• Representative of structure of numbers.
• Easily manipulated by children.
• Imaginable mentally.
Japanese Council of
Mathematics Education
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© Joan A. Cotter, 2010
Materials for Visualizing
“Mathematics is the activity of
creating relationships, many of which
are based in visual imagery.”
Wheatley and Cobb
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© Joan A. Cotter, 2010
Materials for Visualizing
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
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© Joan A. Cotter, 2010
Number Rods
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© Joan A. Cotter, 2010
Spindle Box
1 2 30 4
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© Joan A. Cotter, 2010
Spindle Box
6 7 85 9
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© Joan A. Cotter, 2010
• Distracting: Room is visible through the frame.
• Not visual: Beads need to be grouped in fives.
• Inconsistent with equation order when beads aremoved right: Beads need to be moved left.
• Hierarchies represented sideways: They need to bein vertical columns.
• Trading done before second number is completelyadded: Addends need to combined before trading.
• Answer is read going up: We read top to bottom.
Bead Frame Challenges
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© Joan A. Cotter, 2010
AL Abacus
1000 100 10 1
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© Joan A. Cotter, 2010
4 + 3 = 7
Adding
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© Joan A. Cotter, 2010
Sums Adding to Ten
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© Joan A. Cotter, 2010
Part-Whole Circles
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4 6
What is the other part?
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Language Effect on Counting
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10
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60
70
80
90
100
4 5 6Ages (yrs.)
Avera
ge H
ighest N
um
ber
Counte
d
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 Journalof Psychology, 23, 319-332.
Korean formal [math way]
Korean informal [not explicit]
Chinese
U.S.
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© Joan A. Cotter, 2010
Math Way of Naming Numbers
• Only 11 words are needed to count to 100 themath way, 28 in English. (All Indo-Europeanlanguages are non-standard in number naming.)
• Asian children learn mathematics using themath way of counting.
• They understand place value in first grade;only half of U.S. children understand placevalue at the end of fourth grade.
• Mathematics is the science of patterns. Thepatterned math way of counting greatly helpschildren learn number sense.
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© Joan A. Cotter, 2010
• 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
first teach the name of the quantity (math way).
Math Way of CountingCompared to Reading
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© Joan A. Cotter, 2010
3-ten 7 3 03 0 77
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© Joan A. Cotter, 2010
8 + 7 = 10 + 5 = 15
Strategy: Two Fives
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© Joan A. Cotter, 2010
1000 100 10 1
8
+ 6
Adding
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© Joan A. Cotter, 2010
1000 100 10 1
8
+ 6
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Adding
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© Joan A. Cotter, 2010
7 x 7
The Multiplication Board
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© Joan A. Cotter, 2010
Fraction Chart
How many fourths make a whole? How many sixths?
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© Joan A. Cotter, 2010
• 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?
“Pie” Model Difficulties
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© Joan A. Cotter, 2010
1 2 3 4 5 6 7 8 9
2 4 6 8 10 12 14 16 18
3 6 9 12 15 18 21 24 27
4 8 12 16 20 24 28 32 36
10
20
30
40
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 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
2121
2828
4545
7272
This page may be duplicated for a single teacher or a single fam
ily’s use.©
Activities for Learning, Inc. 2010
!in
ger (
ar*s
AP
PE
ND
I' 1
© 2010 Joan A
. Cotter, Ph.D. • JoanCotter@
ALabacus.com
• alabacus.com
5
GO
TO TH
E DU
MP
(From M
ath Card Gam
es: 300 Gam
es for Learning and Enjoying Math. Fifth edition by
Joan A. C
otter (2010); published by Activities for Learning, Inc.: H
azelton, ND
.)O
bjectiveTo learn the com
binations that total 10Num
ber of players2 to 4
Cards4 or 6 of each basic num
ber card 1 to 9Deal
Each player takes five cards; the remaining cards face dow
n form the
dump, or stack.
Object of the gam
eTo collect the m
ost pairs that equal 10M
aterialsBeginners need an abacus or at least a list of the facts.
PreparationBefore starting, the players check over their hands for pairs that total 10.To do this, they look at each card in turn, determ
ine what is needed to
make 10 and look for that num
ber among their other cards. (Som
echildren m
ay need to spread the cards out on the playing surface.)Store paired cards face up on tw
o piles. (This allows verification and
keeps the cards shuffled for the next game.)
6 is need
ed w
ith 4 to
make 10.
4
4
6
6
8
8
2
2
1 + 9
2 + 8
3 + 7
4 + 6
5 + 5
PlayW
hen all are ready, the first player asks the player on her left for anum
ber needed to complete a pair. If he has it, he m
ust give it to her,w
hereupon she receives another turn. If he does not have it, he says, “Go
to the Dum
p,” 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.Meanw
hile, the firstplayer 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 neededcard. H
owever, she m
ay put a new pair on top of her other pairs.
A player running out of cards takes five m
ore cards, but the turn isended. W
hen the dump is exhausted, players m
ay ask any player (notonly the players on their left) for a card.A
t the end of the game, players com
bine their two stacks and com
parethe heights. (C
ounting the cards is too time consum
ing.) No shuffling is
necessary for subsequent games.
Player 1.
Player 2.
© 2010 Joan Cotter • JoanCotter@
ALabacus.com
• More G
ames at: alabacus.com
> Resources > Presentations
10
10
12
12
5
5
2
2
4
4
6
6
A g
ame in
pro
gress: T
he
player o
n th
e left collects
the 2s w
hile th
e player o
nth
e righ
t collects th
e 5s.
SKIP C
OU
NTIN
G M
EMO
RY
Objective
To learn the skip counting patterns on previous page.Preparation
To prepare the envelopes, see page 13. The players use the envelopes forreference during the gam
e to mem
orize the patterns.Num
ber of players2 or 2 team
sCards
Each player or team chooses an envelope and rem
oves the cards. Mix the
cards together and shuffle lightly. Lay the cards out face down in a 5 by 4
array.O
bject of the game
To be the first player to collect in order the complete set of cards
PlayThe first player turns over one card so both players can see it. If it is theneeded card, the player collects the card and receives another turn. If it isnot the needed card, the card is returned. N
ext the second player takes aturn. Turns alternate until one player has picked up all ten cards.Stress the im
portance of returning the cards to the correct envelopes follow
ing a game.
246810
1214161820
510
1520
2530
3540
4550
MU
LTIPLICA
TION
MEM
OR
YO
bjectiveTo help the players m
aster the multiplication facts.
Cards10 basic num
ber cards with num
bers 1 to 10 and one set of product cards.A
lso a sticky note with the set num
ber and “×” and another note with “=.”
Number of players
Two. Beginners should sit on the sam
e side of the cards.O
bject of the game
To collect the most cards by m
atching the multiplier w
ith the product.Layout
Lay the basic number cards face dow
n in two row
s. To the right in separaterow
s 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 tim
es is 12.” He
then decides where it could be am
ong the product cards. If he is correct, hecollects both cards and takes another turn. If it is not a m
atch, both cardsare returned face dow
n in their original places, and the other player takes aturn.
12
12
4
4
=3 ×
© 2010 Joan Cotter • JoanCotter@
ALabacus.com
• More G
ames at: alabacus.com
> Resources > Presentations
38
58 CO
NC
ENTR
ATIN
G O
N O
NE
(From M
ath Card Gam
es: 300 Gam
es for Learning and Enjoying Math. Fifth edition by Joan
A. C
otter (2010); published by Activities for Learning, Inc.: H
azelton, 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.
BackgroundExplain that – m
eans two –s. Then lay dow
n various fraction cards and askthe children to find the equivalent fraction pieces.N
ow, ask a child to lay the fraction pieces for – under the 1. Then ask her
how m
any more fifths are needed to m
ake 1. [Two –s] Repeat this for other
fractions, such as – and —. C
hildren often have a problem w
ith –.Som
e children find the fraction chart to be veryhelpful. W
ith it they can see what they have and
count how m
any more are needed. W
ith the leftindex finger, the child counts w
hat she has. With
the left finger still in place, she counts with her
right index finger how m
any more she needs.
Explain that these are the pairs for this game.
CardsTw
enty fraction cards are needed: two 1⁄2s and
one of each of the following: 1⁄3, 2⁄3, 1⁄4, 3⁄4, 1⁄5,
2⁄5, 3⁄5, 4⁄5, 1⁄6, 5⁄6, 1⁄8, 3⁄8, 5⁄8, 7⁄8, 1⁄10, 3⁄10, 7⁄10,and 9⁄10.
Number of players
Two or tw
o teams.
LayoutLay the fraction cards out on the table face dow
n in rows as show
n.O
bject of the game
To collect the most pairs of fractions totaling one.
PlayThe first player turns over a card and decides how
many m
ore are neededto m
ake 1. She then chooses a probable card. If she is correct, she collectsboth cards and takes another turn. If they do not m
atch, both cards arereturned face dow
n. The second player then takes his turn. Turns continueuntil all the cards are collected.
1
1515
1515
15
A b
egin
nin
g g
ame sh
ow
ing
two
fraction
s that eq
ual 1.
23
3512
1516
7 10 13
113
15161718191010
1010
1010
1010
1010
1919
1919
1919
1919
1818
1818
1818
18
1616
1616
16
1515
1515
1313
12
1717
1717
1717 1
1 1
1 1
1 1
1 1
1
1414
1414
12
Sh
ow
ing
that five –s eq
ual 1.
15
Th
e fraction
chart.
© 2010 Joan A
. Cotter, Ph.D. • JoanCotter@
ALabacus.com
• alabacus.com
14
1212
1
1818
1818
1818
1818
1414
14
FRA
CTIO
N W
AR
(From M
ath Card Gam
es: 300 Gam
es for Learning and Enjoying Math. Fifth edition by Joan
A. C
otter (2010); published by Activities for Learning, Inc.: H
azelton, ND
.)
Objective
To provide practice in comparing tw
o fractions between the 1s, halves,
fourths, and eighths, the fractions needed for reading a ruler.M
aterialsThe 1, halves, fourths, and eighths of the fraction pieces, arranged asshow
n below.
CardsThe fraction cards w
ith 1s, halves, fourths, and eighths.Num
ber of playersTw
o only.Deal
With the cards face dow
n, divide the stack in half by comparing heights.
Object of the gam
eTo capture all the cards.
PlayEach player takes the top card from
his stack and lays it down in the
middle of the table face up. The player w
hose card is greater takes bothcards. Players should alternate deciding w
hose card is higher.Players continue com
paring cards until they put down cards of equal
value, which constitutes a “w
ar.” To resolve a war, both players play tw
ocards face dow
n and then play a third face up to be compared. The player
who has the high card in the last com
parison takes all eight cards.
Th
e fraction
pieces fo
rmin
g a “ru
ler.”