Enriching Montessori Math with Visualization

13
1 © Joan A. Cotter, 2010 Enriching Montessori Mathematics with Visualization VII 7 x 7 AMS Fall Conference October 22, 2010 San Diego, California by Joan A. Cotter, Ph.D. [email protected] Handout and Presentation: ALabacus.com 7 5 2 2 © 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 3 © Joan A. Cotter, 2010 National Math Crisis Ready, Willing, and Unable to Serve says that 75% of 17 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 the ACT test, only 42% are ready for college algebra. • A generation ago, the US produced 30% of the world’s college grads; today it’s 14%. (CSM 2006) • Two-thirds of 4-year degrees in Japan and China are in science and engineering; one-third in the U.S. • U.S. students, compared to the world, score high at 4th grade, average at 8th, and near bottom at 12th. 4 © 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 workplace needs analytical thinkers and problem solvers. • State exams require more than arithmetic: including geometry, algebra, probability, and statistics. • Brain research is providing clues on how to better facilitate learning, including math. • Increased emphasis on mathematical reasoning, less emphasis on rules and procedures.

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Transcript of Enriching Montessori Math with Visualization

Page 1: Enriching Montessori Math with Visualization

1

© 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

7

5 2

2

© 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

10

4 6

What is the other part?

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© Joan A. Cotter, 2010

Language Effect on Counting

0

10

20

30

40

50

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

14

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

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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

Page 10: Enriching Montessori Math with Visualization

© 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.

Page 11: Enriching Montessori Math with Visualization

© 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 ×

Page 12: Enriching Montessori Math with Visualization

© 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.

Page 13: Enriching Montessori Math with Visualization

© 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.”