NJMAC Visualization

© Joan A. Cotter, Ph.D., 2012

NJMAC ConferenceMarch 2, 2012

Edison, New Jersey

by Joan A. Cotter, [email protected]

Enriching Montessori Mathematics with Visualization

Presentations available: rightstartmath.com

7 x 71000

101

100 53

52


Verbal Counting Model


Verbal 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


Verbal Counting Model From a child's perspective

F + E



A

F + E



A B

F + E



A CB

F + E



A FC D EB

F + E



AA FC D EB

F + E



A BA FC D EB

F + E



A C D EBA FC D EB

F + E



A C D EBA FC D EB

F + E

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



K

G I J KHA FC D EB

F + E



Now memorize the facts!!

G + D




G + D

H + F




G + D

H + F

D + C




G + D

H + F

C + G

D + C



E + I


G + D

H + F

C + G

D + C



Try subtractingby “taking away”

H – E



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?



Lis written ABbecause it is A J and B A’s




huh?




(twelve)




(12)(twelve)




(12)(one 10)

(twelve)




(12)(one 10)

(two 1s).

(twelve)


Verbal Counting ModelSummary



• Is not natural; it takes years of practice.Summary



• Is not natural; it takes years of practice.

• Provides poor concept of quantity.

Summary





• Ignores place value.

Summary






• Is very error prone.

Summary







• Is tedious and time-consuming.

Summary







• Is tedious and time-consuming.

Summary

• Does not provide an efficient way to master the facts.


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.


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


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.


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.


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 a ruler. On a ruler the numbers are not in the spaces.


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.


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.




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.


• 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:


Learning ArithmeticCompared to reading:

Show the baby two teddy bears.

• A child learns to read.

• Later a child uses reading to learn.

• A child learns to do arithmetic.

• Later a child uses arithmetic to solve problems.


Research on CountingKaren Wynn’s research

Show the baby two teddy bears.


Research on Counting

Karen Wynn’s research

Then hide them with a screen.




Show the baby a third teddy bear and put it behind the screen.



Raise screen. Baby seeing 3 won’t look long because it is expected.




Researcher can change the number of teddy bears behind the screen.



A baby seeing 1 teddy bear will look much longer, because it’s unexpected.


Research on CountingOther research



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

Other research

These groups matched quantities without using counting words.




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

Other research






• Adults, ages 18-50, from Boston.Edward Gibson and Michael Frank, MIT, 2008.

Other research






• 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



Research on CountingIn Japanese schools:

• Children are discouraged from using counting for adding.


Research on CountingIn Japanese schools:

• Children are discouraged from using counting for adding.

• They consistently group in 5s.


Research on CountingSubitizing

• Subitizing is quick recognition of quantity without counting.



• Subitizing is quick recognition of quantity without counting.• Human babies and some animals can subitize small quantities at birth.



• Subitizing is quick recognition of quantity without counting.• Human babies and some animals can subitize small quantities at birth.• Children who can subitize perform better in mathematics long term.—Butterworth




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




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

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


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)



“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


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

Visualizing MathematicsJapanese criteria for manipulatives

Japanese Council ofMathematics Education



• Reading• Sports• Creativity• Geography• Engineering• Construction

Visualizing also needed in:



• Reading• Sports• Creativity• Geography• Engineering• Construction

• Architecture• Astronomy• Archeology• Chemistry• Physics• Surgery

Visualizing also needed in:


Visualizing MathematicsReady: How many?


Visualizing MathematicsTry again: How many?


Visualizing MathematicsReady: How many?


Visualizing MathematicsTry again: How many?


Visualizing MathematicsTry to visualize 8 identical apples without grouping.


Visualizing MathematicsNow try to visualize 5 as red and 3 as green.



I II III IIII V VIII

1 23458

Early Roman numerals

Romans grouped in fives. Notice 8 is 5 and 3.



Who could read the music?

:

Music needs 10 lines, two groups of five.


Very Early ComputationNumerals

In English there are two ways of writing numbers:

Numerals: 3578




Numerals:

Words: Three thousand five hundred seventy-eight

3578



In ancient Chinese there was only one way of writing numbers:

3 Th 5 H 7 T 8 U(8 characters)


Numerals:

Words: Three thousand five hundred seventy-eight

3578


Very Early ComputationCalculating rods

Because their characters are cumbersome to use for computing, the Chinese used calculating rods, beginning in the 4th century BC.



Numerals for Ones and Hundreds (Even Powers of Ten)



Numerals for Tens and Thousands (Odd Powers of Ten)




3578



3578

3578,3578They grouped, not in thousands, but ten-thousands!


Naming QuantitiesUsing fingers



Naming quantities is a three-period lesson.



Use left hand for 1-5 because we read from left to right.



Always show 7 as 5 and 2, not for example, as 4 and 3.


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.


Naming QuantitiesRecognizing 5


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.


Naming QuantitiesTally sticks

Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.



Stick is horizontal, because it won’t fit diagonally and young children have problems with diagonals.



Start a new row for every ten.


Naming Quantities

What is 4 apples plus 3 more apples?

Solving a problem without counting

How would you find the answer without counting?


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.


Naming Quantities

1

2

3

4

5

NumberChart


Naming Quantities

1

2

3

4

5

NumberChart

To help the child learn the symbols


Naming Quantities

61

72

83

94

105

NumberChart

To help the child learn the symbols


Naming Quantities

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Pairing Finger Cards

Use two sets of finger cards and match them.


Naming Quantities














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Ordering Finger Cards

Putting the finger cards in order.


Naming Quantities

















10

5 1

Matching Numbers to Finger Cards

Match the number to the finger card.


Naming Quantities



9 4Matching Fingers to Number Cards

1 610

2 83 57















Match the finger card to the number.


Naming Quantities







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Finger Card Memory game

Use two sets of finger cards and play Memory.


Naming QuantitiesNumber Rods



Using different colors.


Naming QuantitiesSpindle Box

45 dark-colored and 10 light-colored spindles. Could be in separate containers.



45 dark-colored and 10 light-colored spindles in two containers.



1 2 30 4

The child takes blue spindles with left hand and yellow with right.



6 7 85 9



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.


AL AbacusCleared


3

AL AbacusEntering quantities

Quantities are entered all at once, not counted.


5


Relate quantities to hands.


7



AL Abacus

10

Entering quantities


AL AbacusThe stairs

Can use to “count” 1 to 10. Also read quantities on the right side.


AL AbacusAdding


AL AbacusAdding

4 + 3 =


AL AbacusAdding

4 + 3 = 7

Answer is seen immediately, no counting needed.


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.


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.


Go to the Dump Game

The ways to partition 10.


Go to the Dump Game

A game viewed from above.

Starting


7 2 7 9 5

7 42 61 3 8 3 4 9

Go to the Dump Game

Starting

Each player takes 5 cards.


Go to the Dump Game

7 2 7 9 5

7 42 61 3 8 3 4 9

Does YellowCap have any pairs? [no]

Finding pairs


Go to the Dump Game

7 2 7 9 5

7 42 61 3 8 3 4 9

Does BlueCap have any pairs? [yes, 1]

Finding pairs


Go to the Dump Game

7 2 7 9 5

7 42 61 3 8 3 4 9


Finding pairs


Go to the Dump Game

7 2 7 9 5

7 2 1 3 8 3 4 9

4 6


Finding pairs


Go to the Dump Game

7 2 7 9 5

7 2 1 3 8 3 4 9

4 6

Does PinkCap have any pairs? [yes, 2]

Finding pairs


Go to the Dump Game

7 2 7 9 5

7 2 1 3 8 3 4 9

4 6


Finding pairs


Go to the Dump Game

7 2 7 9 5

2 1 8 3 4 9

4 67 3


Finding pairs


Go to the Dump Game

Finding pairs

7 2 7 9 5

1 3 4 9

4 62 82 8



Go to the Dump Game

7 2 7 9 5

1 3 4 9

4 62 82 8

The player asks the player on her left.

Playing


Go to the Dump GameBlueCap, do you

have a 3?BlueCap, do you

have an 3?

7 2 7 9 5

1 3 4 9

4 62 82 8


Playing




have an 3?

7 2 7 9 5

1

3

4 9

4 62 82 8


Playing




have an 3?

2 7 9 5

1 4 9

4 62 82 8

7 3

Playing




have an 8?

2 7 9 5

1 4 9

4 62 82 8

7 3

YellowCap gets another turn.

Playing




have an 8?

Go to the dump.

2 7 9 5

1 4 9

4 62 82 8

7 3

YellowCap gets another turn.

Playing


2



have an 8?

Go to the dump.

2 7 9 5

1 4 9

4 62 82 8

7 3

Playing


Go to the Dump Game

2 2 7 9 5

1 4 9

4 62 82 8

7 3

Playing


Go to the Dump Game

PinkCap, do youhave a 6?

2 2 7 9 5

1 4 9

4 62 82 8

7 3

Playing


Go to the Dump Game

PinkCap, do youhave a 6?Go to the dump.

2 2 7 9 5

1 4 9

4 62 82 8

7 3

Playing


5

Go to the Dump Game

2 2 7 9 5

1 4 9

4 62 82 8

7 3

Playing


Go to the Dump Game

5

2 2 7 9 5

1 4 9

4 62 82 8

7 3

Playing


Go to the Dump Game

YellowCap, doyou have a 9?

5

2 2 7 9 5

1 4 9

4 62 82 8

7 3

Playing


Go to the Dump Game


5

2 2 7 5

1 4 9

4 62 82 8

7 3

Playing


Go to the Dump Game


5

2 2 7 5

1 4 9

4 62 82 8

7 3

9

Playing


Go to the Dump Game

5

2 2 7 5

4 9

4 62 81 9

7 3

Playing


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.


Go to the Dump Game

Winner?

5 54 6

9 1


Go to the Dump Game

Winner?

5546

91

No counting. Combine both stacks.


Go to the Dump Game

Winner?

5546

91

Whose stack is the highest?


Go to the Dump Game

Next game

No shuffling needed for next game.


“Math” Way of Naming Numbers



11 = ten 1



11 = ten 112 = ten 2



11 = ten 112 = ten 213 = ten 3



11 = ten 112 = ten 213 = ten 314 = ten 4



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.




20 = 2-ten 21 = 2-ten 1





20 = 2-ten 21 = 2-ten 122 = 2-ten 2





20 = 2-ten 21 = 2-ten 122 = 2-ten 223 = 2-ten 3





20 = 2-ten 21 = 2-ten 122 = 2-ten 223 = 2-ten 3 . . . . . . . .99 = 9-ten 9



137 = 1 hundred 3-ten 7

Only numbers under 100 need to be said the “math” way.



137 = 1 hundred 3-ten 7or

137 = 1 hundred and 3-ten 7

Only numbers under 100 need to be said the “math” way.



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.

Shows how far children from 3 countries can count at ages 4, 5, and 6.

Ave

rage

Hig

hest

Num

ber C

ount

ed



0

10

20

30

40

50

60

70

80

90

100

4 5 6Ages (yrs.)



ChineseU.S.

Purple is Chinese. Note jump between ages 5 and 6.

Ave

rage

Hig

hest

Num

ber C

ount

ed



0

10

20

30

40

50

60

70

80

90

100

4 5 6Ages (yrs.)



ChineseU.S.

Dark green is Korean “math” way.

Ave

rage

Hig

hest

Num

ber C

ount

ed



0

10

20

30

40

50

60

70

80

90

100

4 5 6Ages (yrs.)



ChineseU.S.

Dotted green is everyday Korean; notice smaller jump between ages 5 and 6.

Ave

rage

Hig

hest

Num

ber C

ount

ed



0

10

20

30

40

50

60

70

80

90

100

4 5 6Ages (yrs.)



ChineseU.S.

Red is English speakers. They learn same amount between ages 4-5 and 5-6.

Ave

rage

Hig

hest

Num

ber C

ount

ed


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.


Math Way of Naming NumbersCompared to reading:


Math Way of Naming Numbers

• Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic.

Compared to reading:




• 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 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.




“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



Using 10s and 1s, ask the child to construct 48.

Research task:




Then ask the child to subtract 14.

Research task:





Children thinking of 14 as 14 ones count 14.

Research task:





Children thinking of 14 as 14 ones counted 14.

Research task:




Research task:







Children who understand tens remove a ten and 4 ones.

Research task:






Research task:


Math Way of Naming NumbersTraditional names

4-ten = forty

The “ty” means tens.



4-ten = forty


The traditional names for 40, 60, 70, 80, and 90 follow a pattern.



6-ten = sixty




3-ten = thirty

“Thir” also used in 1/3, 13 and 30.



5-ten = fifty

“Fif” also used in 1/5, 15 and 50.



2-ten = twenty

Two used to be pronounced “twoo.”



A word gamefireplace place-fire

Say the syllables backward. This is how we say the teen numbers.




paper-newsnewspaper





paper-news

box-mail mailbox

newspaper




ten 4

“Teen” also means ten.



ten 4 teen 4




ten 4 teen 4 fourteen




a one left



a one left a left-one



a one left a left-one eleven



two left

Two pronounced “twoo.”



two left twelve

Two pronounced “twoo.”


Composing Numbers

3-ten


Composing Numbers

3-ten

3 0


Composing Numbers

3-ten

3 0

Point to the 3 and say 3.


Composing Numbers

3-ten

3 0

Point to 0 and say 10. The 0 makes 3 a ten.


Composing Numbers

3-ten 7

3 0


Composing Numbers

3-ten 7

3 07


3 0

Composing Numbers

3-ten 7

7

Place the 7 on top of the 0 of the 30.


Composing Numbers

3-ten 7

Notice the way we say the number, represent the number, and write the number all correspond.

3 07


Composing Numbers

7-ten 6

7 86

Another example.


Composing Numbers

7-ten 6

7 86

Another example.

In the UK, pupils are expected to know the amount remaining: 24, that is 100 – 76.


Composing Numbers

10-ten


Composing Numbers

10-ten

1 0 0


Composing Numbers

1 hundred


Composing Numbers

1 hundred

1 0 0


Composing Numbers

1 hundred

1 0 0

Of course, we can also read it as one-hun-dred.


Composing Numbers

1 hundred

1 01 01 0 0



Composing Numbers

1 hundred

1 0 0



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


2584 58

Composing Numbers




2584258

Composing Numbers




2584258

Composing Numbers

4




2584258

Composing Numbers

4



The Decimal Cards encourage reading numbers in the normal order.


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


Fact Strategies


Fact Strategies

• A strategy is a way to learn a new fact or recall a forgotten fact.


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.


Fact StrategiesComplete the Ten

9 + 5 =



9 + 5 =

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



9 + 5 =


Use two hands and move the beads simultaneously.



9 + 5 =




9 + 5 = 14



Fact StrategiesTwo Fives

8 + 6 =



8 + 6 =

Two fives make 10.



8 + 6 =

Just add the “leftovers.”



8 + 6 =10 + 4 = 14

Just add the “leftovers.”



7 + 5 =

Another example.



7 + 5 =



7 + 5 = 12


Fact StrategiesGoing Down

15 – 9 =


Fact StrategiesDifference

7 – 4 =

Subtract 4 from 5; then add 2.



15 – 9 =



15 – 9 =

Subtract 5;then 4.



15 – 9 = 6

Subtract 5;then 4.


Fact StrategiesSubtract from 10

15 – 9 =



15 – 9 =

Subtract 9 from 10.



15 – 9 = 6

Subtract 9 from 10.


Fact StrategiesGoing Up

13 – 9 =



13 – 9 =

Start with 9; go up to 13.



13 – 9 =1 + 3 = 4

Start with 9; go up to 13.


MoneyPenny


MoneyNickel


MoneyDime


MoneyQuarter


Base-10 Picture Cards

One



Ten One



Hundred Ten One



Thousand Hundred Ten One



3658+2724

Add using the base-10 picture cards.



3 0 0 06 0 05 08



2 0 0 07 0 02 04



3 0 0 06 0 05 082 0 0 07 0 02 04

Add them together.



3 0 0 06 0 05 082 0 0 07 0 02 04



3 0 0 06 0 05 082 0 0 07 0 02 04

Trade 10 ones for 1 ten.




3 0 0 06 0 05 082 0 0 07 0 02 04



3 0 0 06 0 05 082 0 0 07 0 02 04

Trade 10 hundreds for 1 thousand.




3 0 0 06 0 05 082 0 0 07 0 02 04



6 0 0 03 0 08 02

3 0 0 06 0 05 082 0 0 07 0 02 04


Bead Frame

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


8+ 614

1

10

100

1000

Bead Frame


Bead FrameDifficulties for the child


• 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.







• Trading done before second number is completely added: Addends need to combined before trading.

• Answer is read going up: We read top to bottom.



Trading SideCleared

1000 10 1100


1000 10 1100

Trading SideThousands


1000 10 1100

Trading SideHundreds

The third wire from each end is not used.


1000 10 1100

Trading SideTens



1000 10 1100

Trading SideOnes



1000 10 1100

Trading SideAdding

8+ 6


1000 10 1100

Trading SideAdding

8+ 614


1000 10 1100

Trading SideAdding

8+ 614

Too many ones; trade 10 ones for 1 ten.

You can see the 10 ones (yellow).


1000 10 1100

Trading SideAdding

8+ 614

Too many ones; trade 10 ones for 1 ten.


1000 10 1100

Trading SideAdding

8+ 614

Same answer before and after trading.


1000 10 1100

Trading SideCleared


1000 10 1100

Trading SideBead Trading game

Object: To get a high score by adding numbers on the green cards.


1000 10 1100


Object: To get a high score by adding numbers on the green cards.

7


1000 10 1100


Turn over another card. Enter 6 beads. Do we need to trade?

6


1000 10 1100



6


1000 10 1100


6


1000 10 1100


9


1000 10 1100


Another trade.

9


1000 10 1100


3



• In the Bead Trading game 10 ones for 1 ten occurs frequently;



• In the Bead Trading game 10 ones for 1 ten occurs frequently;10 tens for 1 hundred, less often;



• 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.




• 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.)


1000 10 1100

Trading SideAdding 4-digit numbers

3658+ 2738


1000 10 1100


3658+ 2738

Enter the first number from left to right.


1000 10 1100


3658+ 2738

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


1000 10 1100


3658+ 2738

6


. . . 6 ones. Did anything else happen?


1000 10 1100


3658+ 2738

6


1

Is it okay to show the extra ten by writing a 1 above the tens column?


1000 10 1100


3658+ 2738

6


1


1000 10 1100


3658+ 2738

6


1

Do we need to trade? [no]


1000 10 1100


3658+ 2738

96


1


1000 10 1100


3658+ 2738

96


1

Do we need to trade? [yes]


1000 10 1100


3658+ 2738

96


1

Notice the number of yellow beads. [3] Notice the number of blue beads left. [3] Coincidence? No, because 13 – 10 = 3.


1000 10 1100


3658+ 2738

96


1


1000 10 1100


3658+ 2738

396


1


1000 10 1100


3658+ 2738

396


11


1000 10 1100


3658+ 2738

6396


11


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


Multiplication on the AL AbacusBasic facts

6 4 =(6 taken 4 times)



9 3 =



9 3 =30



9 3 =30 – 3 = 27



4 8 =



4 8 =20 + 12 = 32



7 7 =



7 7 =25 + 10 + 10 + 4 = 49


Multiplication on the AL AbacusCommutative property

5 6 =


Multiplication on the AL AbacusCommutative property

5 6 = 6 5


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


7 8 =




7 8 =



Tens:


7 8 =50 + 6



Tens: Ones: 3 26

20+ 30

50


7 8 =50 + 6 = 56



Tens: Ones: 3 26

20+ 30

50


9 7 =



9 7 =


Tens:


9 7 =


Tens: 40+ 20


9 7 =60 +


Tens: 40+ 20

60


9 7 =60 +


Tens: Ones:40+ 20

60


9 7 =60 +


Tens: Ones: 1 3

40+ 20

60


9 7 =60 + 3


Tens: Ones:40+ 20

60

1 3

3


9 7 =60 + 3 = 63


Tens: Ones: 1 3

3

40+ 20

60


The Multiplication Board1 2 3 4 5 6 7 8 9 10

6

6 4

6 x 4 on original multiplication board.


1 2 3 4 5 6 7 8 9 10

6

The Multiplication Board

6 4

Using two colors.


The Multiplication Board1 2 3 4 5 6 7 8 9 10

7

7 7

7 x 7 on original multiplication board.


1 2 3 4 5 6 7 8 9 10

7


7 7

Upper left square is 25, yellow rectangles are 10. So, 25, 35, 45, 49.



7 7

Less clutter.


Multiples PatternsTwos

2 4 6 8 10

12 14 16 18 20

Recognizing multiples needed for fractions and algebra.


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.


Multiples PatternsFours

4 8 12 16 20

24 28 32 36 40

The ones repeat in the second row.


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 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80



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.



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.



6 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80




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).



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).


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.


Multiples PatternsThrees

3 6 9

12 15 18

21 24 27

30

The 3s have several patterns: Observe the ones.



3 6 9

12 15 18

21 24 27

30




3 6 9

12 15 18

21 24 27

30

The 3s have several patterns: The tens are the same in each row.



3 6 9

12 15 18

21 24 27

30

The 3s have several patterns: Add the digits in the columns.



3 6 9

12 15 18

21 24 27

30




3 6 9

12 15 18

21 24 27

30

The 3s have several patterns: Add the “opposites.”



3 6 9

12 15 18

21 24 27

30



Multiples PatternsSevens

7 14 21

28 35 42

49 56 63

70

The 7s have the 1, 2, 3… pattern.



7 14 21

28 35 42

49 56 63

70




7 14 21

28 35 42

49 56 63

70

Look at the tens.


Multiples Memory

“Multiples” are sometimes referred to as “skip counting.”


Multiples Memory

Aim: To help the players learn the multiples patterns.

“Multiples” are sometimes referred to as “skip counting.”


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.


Multiples Memory

The 7s envelope contains 10 cards, each with one of the numbers listed.

7 14 2128 35 4249 56 63

70


Multiples Memory

The 8s envelope contains 10 cards, each with one of the numbers listed.

8 16 24 32 4048 56 64 72 80


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.

8 16 24 32 4048 56 64 72 80


Multiples Memory

7 14 2128 35 4249 56 6370

7 14 2128 35 4249 56 6370

Players may refer to their envelopes at all times.

8 16 24 32 4048 56 64 72 80


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.


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.


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.


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.


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.


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


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


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


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


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


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.


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.


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


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


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.


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?


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


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


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.


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.


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.


Multiplication Tables



A rectangle 3 6



4 7



4 7Grouping in fives makes counting over unnecessary.



Removing duplicates.



9 3

Removing duplicates.



6 6



4 7



7 9



7 9

9 7



squares


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

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17

17

18

18

18

18

19

19

19

19

19

19

110

110

110

110

110

110

110

110

110

Giving the student the big picture.


Fraction Chart1

12

12

13

14

15

16

17

18

19

110

13

13

14

15

16

17

18

19

14

15

16

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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.


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? Giving the student the big picture.


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?


Fraction Chart1

12

12

13

14

15

16

17

18

19

110

13

13

14

15

16

17

18

19

14

15

16

17

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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? Giving the child the big picture.


Fraction Chart

1

12

13

14

15

17

18

110

16

19

Stairs (Unit fractions)


Fraction Chart

1

12

13

14

15

17

18

110

16

19

A hyperbola.

Stairs (Unit fractions)


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

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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.


Circle Model

Are we comparing angles, arcs, or area?


Circle 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.


Circle 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


Circle 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.

www.statcan.ca


Circle ModelDifficulties


• Perpetuates cultural myth fractions are < 1.




• Does not give the child the “big picture.”




• Does not give the child the “big picture.” • Limits understanding of fractions: they are

more than “a part of a whole or part of a set.”




• Does not give the child the “big picture.” • Limits understanding of fractions: they are

more than “a part of a whole or part of a set.”• Makes it difficult for the child to see how

fractions relate to each other.



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


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


The fraction 4/8 can be reduced on the multiplication table as 1/2.


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


21212828

In what column would you put 21/28?


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



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


21212828

In what column would you put 21/28?


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


2121282845457272


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


12 12 1616


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


12 12 1616

6/8 needs further simplifying.


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


12 12 1616

12/16 could have put here originally.


NJMAC ConferenceMarch 2, 2012

Edison, New Jersey

by Joan A. Cotter, [email protected]

Enriching Montessori Mathematics with Visualization

7 x 71000

101

100 53

52

Presentations available: rightstartmath.com

NJMAC Visualization

Education

Transcript of NJMAC Visualization