IMF: Visualization October 2011

IMF ConferenceOctober 21, 2011Sarasota, Florida

by Joan A. Cotter, Ph.D.JoanCotter@ALabacus.com

Adding Visualization toMontessori Mathematics

Presentation available: ALabacus.com

100010

Counting Model

• Number Rods• Spindle Boxes• Golden Bead materials• Snake Game• Dot Game • Stamp Game• Multiplication Board• Bead Frame

In Montessori, counting is pervasive:

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

Counting Model From a child's perspective

A FC D EB

AA FC D EB

A BA FC D EB

A C D EBA FC D EB

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

G I J KHA FC D EB

Now memorize the facts!!

Try subtractingby “taking away”

H – E

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

(twelve)

(12)(twelve)

(12)(one 10)

(twelve)

(12)(one 10)

(two 1s).

(twelve)

Counting ModelSummary

Counting Model

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

Counting Model

• Is not natural; it takes years of practice.

• Provides poor concept of quantity.

Summary

Counting Model

• Ignores place value.

Summary

Counting Model

• Is very error prone.

Summary

Counting Model

• Is tedious and time-consuming.

Summary

Counting Model

• Is tedious and time-consuming.

Summary

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

Calendar MathAugust

Sometimes calendars are used for counting.

Calendar MathAugust

Sometimes calendars are used for counting.

Calendar MathAugust

This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.

Calendar MathSeptember123489101115161718222324252930

567121314192021262728

August

This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and 3.

Calendar MathSeptember123489101115161718222324252930

567121314192021262728

August

1 2 3 4 5 6

A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.

Calendar MathAugust

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.

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.

Memorizing Math

Percentage RecallImmediately After 1 day After 4 wks

Rote 32 23 8 Concept 69 69 58

Memorizing Math

Rote 32 23 8 Concept 69 69 58

Memorizing Math

Rote 32 23 8 Concept 69 69 58

Memorizing Math

Rote 32 23 8 Concept 69 69 58

Memorizing Math

Rote 32 23 8 Concept 69 69 58

Memorizing Math

Rote 32 23 8 Concept 69 69 58

Memorizing Math

Math needs to be taught so 95% is understood and only 5% memorized.

Richard Skemp

Rote 32 23 8 Concept 69 69 58

Memorizing MathFlash cards:

• 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.

Memorizing Math 9 + 7Flash cards:

• Often produce stress – children under stress stop learning.

• Are not concrete – use abstract symbols.

Memorizing Math 9 + 7Flash cards:

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 recognizing a quantity without counting.

• Subitizing is recognizing a quantity without counting.• Human babies and some animals can subitize small quantities at birth.

• “One could have grasped the idea of a square without being able to count to four, that is, without learning the number of sides and corners.”—Montessori

• Human babies and some animals can subitize small quantities at birth.

Said to contrast to teaching the names of a triangle and square by counting.

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

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

• Children who can subitize perform better in mathematics.—Butterworth

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

Research on CountingFinger gnosia

• Finger gnosia is the ability to know which fingers can been lightly touched without looking.

• Part of the brain controlling fingers is adjacent to math part of the brain.

• Children who use their fingers as representational tools perform better in mathematics—Butterworth

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

“Mathematics is the activity of creating relationships, many of which are based in visual imagery.”

Wheatley and Cobb

“The process of connecting symbols to imagery is at the heart of mathematics learning.”

Dienes

“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 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.

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

NumberControlChart

To help the child learn the symbols

Naming Quantities

To help the child learn the symbols

NumberControlChart

Naming Quantities

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Pairing Finger Cards

Use two sets of finger cards and match them.

Naming Quantities

are needed to see this picture.QuickTime™ and aTIFF (LZW) decompressor

Ordering Finger Cards

Putting the finger cards in order.

Naming Quantities

Matching Numbers to Finger Cards

Match the number to the finger card.

Naming Quantities

9 4Matching Fingers to Number Cards

2 83 57

Match the finger card to the number.

Naming Quantities

QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.

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 Abacus1000 10 1100

Double-sided AL abacus. Side 1 is grouped in 5s.Trading Side introduces algorithms with trading.

AL AbacusCleared

AL AbacusEntering quantities

Quantities are entered all at once, not counted.

Relate quantities to hands.

AL Abacus

Entering quantities

AL AbacusThe stairs

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

AL AbacusAdding

4 + 3 =

AL AbacusAdding

4 + 3 =

AL AbacusAdding

4 + 3 =

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

Starting

A game viewed from above.

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

Finding pairs

7 2 7 9 5

7 42 61 3 8 3 4 9

Does YellowCap have any pairs? [no]

Go to the Dump Game

Finding pairs

7 2 7 9 5

7 42 61 3 8 3 4 9

Does BlueCap have any pairs? [yes, 1]

Go to the Dump Game

Finding pairs

7 2 7 9 5

7 42 61 3 8 3 4 9

Go to the Dump Game

Finding pairs

7 2 7 9 5

7 2 1 3 8 3 4 9

Go to the Dump Game

Finding pairs

7 2 7 9 5

7 2 1 3 8 3 4 9

Does PinkCap have any pairs? [yes, 2]

Go to the Dump Game

Finding pairs

7 2 7 9 5

7 2 1 3 8 3 4 9

Go to the Dump Game

Finding pairs

7 2 7 9 5

2 1 8 3 4 9

4 67 3

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

Playing

7 2 7 9 5

1 3 4 9

4 62 82 8

The player asks the player on her left.

Go to the Dump GameBlueCap, do you

have a 3?BlueCap, do you

have an 3?

Playing

7 2 7 9 5

1 3 4 9

4 62 82 8

have an 3?

Playing

7 2 7 9 5

4 62 82 8

have an 3?

Playing

2 7 9 5

4 62 82 8

have an 8?

Playing

2 7 9 5

4 62 82 8

YellowCap gets another turn.

have an 8?

Go to the dump.Playing

2 7 9 5

4 62 82 8

YellowCap gets another turn.

have an 8?

Go to the dump.Playing

2 7 9 5

4 62 82 8

Go to the Dump Game

Playing

2 2 7 9 5

4 62 82 8

Go to the Dump Game

PinkCap, do youhave a 6?Playing

2 2 7 9 5

4 62 82 8

Go to the Dump Game

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

2 2 7 9 5

4 62 82 8

Go to the Dump Game

Playing

2 2 7 9 5

4 62 82 8

Go to the Dump Game

Playing

2 2 7 9 5

4 62 82 8

Go to the Dump Game

YellowCap, doyou have a 9? Playing

2 2 7 9 5

4 62 82 8

Go to the Dump Game

2 2 7 5

4 62 82 8

Go to the Dump Game

2 2 7 5

4 62 82 8

Go to the Dump Game

Playing

2 2 7 5

4 62 81 9

2 9 1 7 7

Go to the Dump Game

Playing

2 2 7 5

4 62 81 9

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

Go to the Dump Game

Winner?

No counting. Combine both stacks.

Go to the Dump Game

Winner?

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.

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.

4 5 6Ages (yrs.)

ChineseU.S.

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

4 5 6Ages (yrs.)

ChineseU.S.

Dark green is Korean “math” way.

4 5 6Ages (yrs.)

ChineseU.S.

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

4 5 6Ages (yrs.)

ChineseU.S.

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

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.

• 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).

• 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

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

“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

Point to the 3 and say 3.

Composing Numbers

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

Composing Numbers

3-ten 7

Composing Numbers

3-ten 7

Composing Numbers

3-ten 7

Composing Numbers

3-ten 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.

Composing Numbers

10-ten

Composing Numbers

10-ten

Composing Numbers

10-ten

Composing Numbers

10-ten

Composing Numbers

1 hundred

Composing Numbers

1 hundred

Composing Numbers

1 hundred

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

Composing Numbers

1 hundred

1 01 01 0 0

Composing Numbers

1 hundred

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

2584258

Composing Numbers

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

• 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 =

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 =

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 =

Subtract 5;then 4.

15 – 9 =

Subtract 5;then 4.

15 – 9 =

Subtract 5;then 4.

15 – 9 = 6

Subtract 5;then 4.

Fact StrategiesSubtract from 10

15 – 9 =

Subtract 9 from 10.

15 – 9 =

Subtract 9 from 10.

15 – 9 =

Subtract 9 from 10.

15 – 9 = 6

Subtract 9 from 10.

Fact StrategiesGoing Up

13 – 9 =

Start with 9; go up to 13.

13 – 9 =

13 – 9 =1 + 3 = 4

MoneyPenny

MoneyNickel

MoneyDime

MoneyQuarter

Bead Frame

8+ 614

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.

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

1000 10 1100

Trading SideCleared

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

1000 10 1100

Trading SideAdding

1000 10 1100

Trading SideAdding

1000 10 1100

Trading SideAdding

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

1000 10 1100

Trading SideAdding

8+ 614

1000 10 1100

Trading SideAdding

8+ 614

Same answer before and after trading.

1000 10 1100

Trading SideBead Trading game

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

1000 10 1100

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

1000 10 1100

Trade 10 ones for 1 ten.

1000 10 1100

Another trade.

1000 10 1100

Another trade.

1000 10 1100

• 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.

• 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

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

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

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

. . . 6 ones. Did anything else happen?

1000 10 1100

3658+ 2738

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

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

Do we need to trade? [no]

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

Do we need to trade? [yes]

1000 10 1100

3658+ 2738

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

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

1000 10 1100

3658+ 2738

The Stamp Game

1000 100 10 1

100 10 1

1000 100 10 1

The Stamp Game

1000 100 10 1

100 10 1

1000 100 10 1

The Stamp Game

100 10 1100 10 1

10 1 1

1000 100 10 11000 100 10 1

100 100

The Stamp Game

100 10 1100 10 1

10 1 1

1000 100 10 11000 100 1

1000 100 10 11000 100 10 1

100 100

The Stamp Game

100 10 1100 10 1

10 1 1

1000 100 10 11000 100 1

1000 100 10 11000 100 10 1

100 100

Multiplication on the AL AbacusBasic facts

6 4 =(6 taken 4 times)

9 3 =30

9 3 =30 – 3 = 27

4 8 =20 + 12 = 32

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

Multiplication on the AL AbacusCommutative property

5 6 = 6 5

This method was used in the Middle Ages, rather than memorize the facts > 5 5.

Multiplication on the AL AbacusFor facts > 5 5

Tens: 20+ 30

7 8 =50 +

Tens: 20+ 30

7 8 =50 +

Tens: Ones:20+ 30

7 8 =50 +

Tens: Ones:20+ 30

7 8 =50 +

Tens: Ones: 3 2

20+ 30

7 8 =50 + 6

Tens: Ones: 3 26

20+ 30

7 8 =50 + 6 = 56

Tens: Ones: 3 26

20+ 30

Tens: 40+ 20

9 7 =60 +

Tens: 40+ 20

9 7 =60 +

Tens: Ones:40+ 20

9 7 =60 +

Tens: Ones:40+ 20

9 7 =60 +

Tens: Ones: 1 3

40+ 20

9 7 =60 + 3

Tens: Ones:40+ 20

9 7 =60 + 3 = 63

Tens: Ones: 1 3

40+ 20

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

6 x 4 on original multiplication board.

1 2 3 4 5 6 7 8 9 10

The Multiplication Board

Using two colors.

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

7 x 7 on original multiplication board.

1 2 3 4 5 6 7 8 9 10

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

Less clutter.

Multiples PatternsTwos

2 4 6 8 10

12 14 16 18 20

Recognizing multiples needed for fractions and algebra.

2 4 6 8 10

12 14 16 18 20

The ones repeat in the second row.

2 4 6 8 10

12 14 16 18 20

2 4 6 8 10

12 14 16 18 20

2 4 6 8 10

12 14 16 18 20

2 4 6 8 10

12 14 16 18 20

Multiples PatternsFours

4 8 12 16 20

24 28 32 36 40

Multiples PatternsFours

4 8 12 16 20

24 28 32 36 40

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 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

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

12 15 18

21 24 27

The 3s have several patterns: Observe the ones.

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

12 15 18

21 24 27

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

12 15 18

21 24 27

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

12 15 18

21 24 27

12 15 18

21 24 27

Multiples PatternsSevens

7 14 21

28 35 42

49 56 63

The 7s have the 1, 2, 3… pattern.

7 14 21

28 35 42

49 56 63

7 14 21

28 35 42

49 56 63

7 14 21

28 35 42

49 56 63

7 14 21

28 35 42

49 56 63

Look at the tens.

7 14 21

28 35 42

49 56 63

Look at the tens.

7 14 21

28 35 42

49 56 63

Look at the tens.

7 14 21

28 35 42

49 56 63

Look at the tens.

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

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

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

Players may refer to their envelopes at all times.

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

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

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

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

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

Multiples Memory

8 16 24 32 4048 56 64 72 80

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

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 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

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

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

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

7 14 2128 35 4249 56 6370

Multiples Memory

8 16 24 32 4048 56 64 72 80

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

7 14 2128 35 4249 56 6370

Where is that 14?

Multiples Memory

8 16 24 32 4048 56 64 72 80

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

Multiples Memory

8 16 24 32 4048 56 64 72 80

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

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

7 14 2128 35 4249 56 6370

We’ll never know who won.

Fraction Chart1

16171819

Giving the child the big picture, a Montessori principle.

Fraction Chart1

16171819

How many fourths in a whole? Giving the child the big picture, a Montessori principle.

Fraction Chart1

16171819

Fraction Chart1

16171819

Fraction Chart1

16171819

Fraction Chart1

16171819

Fraction Chart1

16171819

How many eighths in a whole?

Fraction Chart1

16171819

Which is more, 3/4 or 4/5?

Fraction Chart1

16171819

Fraction Chart1

16171819

Fraction Chart1

16171819

Fraction Chart

Stairs (Unit fractions)

Fraction Chart

A hyperbola.

Stairs (Unit fractions)

16171819

Fraction Chart

9/8 is 1 and 1/8.

“Pie” Model

Are we comparing angles, arcs, or area?

“Pie” Model

Try to compare 4/5 and 5/6 with this model.

“Pie” 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

“Pie” 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.

statcan.ca

“Pie” ModelDifficulties

• Perpetuates cultural myth fractions always < 1.

• Requires counting pieces.

• 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?

Fraction War

1 2 3 4 5 6

Especially useful for learning to read a ruler with inches.

Fraction War1

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

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

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

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

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

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

21212828

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

21212828

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

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

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

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

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

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

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

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

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

12 12 1616

12/16 could have put here originally.

In Conclusion

In Conclusion• We need to use quantity, not counting words, as the basis of arithmetic.

• Subitizing needs to be encouraged.

• Children need to have visual images based on fives to remember the facts.

• Visualizing helps our disadvantaged children because it reduces the heavy memory load.

• We need to use the math way of number naming for a longer period of time.

IMF ConferenceOctober 21, 2011Sarasota, Florida

by Joan A. Cotter, Ph.D.JoanCotter@ALabacus.com

Adding Visualization toMontessori Mathematics

Presentation available: ALabacus.com

100010

IMF: Visualization October 2011

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