AMS: Counting-Necessary or Detrimental? March 2011

© Joan A. Cotter, Ph.D., 2011

VII

AMS ConferenceMarch 25, 2011Chicago, Illinois

by Joan A. Cotter, [email protected]

7

5 2

Counting: Necessary or Detrimental?

Presentation available: ALabacus.com7 x 7


National Math Crisis


National Math Crisis• 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.


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.




• U.S. students, compared to the world, score high at 4th grade, average at 8th, and near bottom at 12th.


Math Education is Changing


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.





• Brain research is providing clues on how to better facilitate learning, including math.

• Calculators and computers have made computation with many digits an unneeded skill.


Counting Model


Counting Model

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

In Montessori materials, counting is pervasive:


Counting ModelFrom the perspective of most teachers


Counting Model

• Memorize counting sequence.

From the perspective of most teachers


Counting Model


• One-to-one correspondence.



Counting Model



• Cardinality principal.



Counting Model




• Add by counting all.



Counting Model





• Add by counting on.



Counting Model






• Add by counting from the larger number.



Counting Model







• Subtract by counting backward.



Counting Model







• Subtract by counting backward.

• Multiply by skip counting.



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

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)


Counting ModelSummary


Counting Model

• Is not natural; it takes years of practice.

Summary


Counting Model


• 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 Math

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

Sometimes calendars are used for counting.

Sometimes calendars are used for counting.


Calendar Math

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


Calendar Math

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 3 doesn’t include the 1 and the 2.

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


Calendar Math

September123489101115161718222324252930

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.

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


Calendar Math

September123489101115161718222324252930

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

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


Calendar Math

August

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.

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.


Research on CountingKaren Wynn’s research

Show the baby two teddy bears.

Show the baby two teddy bears.


Research on Counting

Karen Wynn’s research

Then hide them with a screen.Then hide them with a screen.




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

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




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

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.

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.




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





• Children who can subitize perform better in mathematics.—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


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 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.

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



Who could read the music?

:

Music needs 10 lines, two groups of five.Music needs 10 lines, two groups of five.


Research on CountingTeach Counting



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


Naming QuantitiesUsing fingers



Naming quantities is a three-period lesson.



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

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


Naming Quantities

Yellow 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.

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.

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.

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.

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



Start a new row for every ten.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?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.

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.

Use two sets of finger cards and match them.


Naming Quantities














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Ordering Finger Cards

Putting the finger cards in order.

Putting the finger cards in order.


Naming Quantities





















10

5 1

Matching Numbers to Finger Cards

Match the number to the finger card.

Match the number to the finger card.


Naming Quantities



9 4

Matching Fingers to Number Cards

1 610

2 83 57















Match the finger card to the number.

Match the finger card to the number.


Naming Quantities









































Finger Card Memory game

Use two sets of finger cards and play Memory.

Use two sets of finger cards and play Memory.


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.This was the inspiration to group in 5s.


Naming QuantitiesNumber Rods


Naming QuantitiesNumber Rods

Using different colors.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. Could be in separate containers.



45 dark-colored and 10 light-colored spindles in two 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 QuantitiesBlack and White Bead Stairs

This was the inspiration to group in 5s.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


3

AL AbacusEntering quantities

Quantities are entered all at once, not counted.Quantities are entered all at once, not counted.


5


Relate quantities to hands.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.

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.

Answer is seen immediately, no counting needed.


Problem Solving


Problem Solving

• A mathematical problem is a problem only when the solution is not obvious. A good analogy is a puzzle.


Problem Solving


• Story problems color-coded by operation are no longer problems. Real life is not color-coded!


Problem Solving


• Story problems color-coded by operation are no longer problems. Real life is not color-coded!

• Do not try to help children solve story problems by teaching “key” words.


Problem SolvingPart-Whole Circles

Part-whole circles children see relationships and solve problems.



Problem Solving

Whole

Part-Whole Circles




Problem Solving

Whole

Part Part

Part-Whole Circles




Problem Solving

10

If 10 is the whole

Part-Whole Circles


Problem Solving

10

4

and 4 is one part,

Part-Whole Circles


Problem Solving

10

4

What is the other part?

Part-Whole Circles


Problem Solving

10

4 6

What is the other part?

Part-Whole Circles


Problem Solving

Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with?

Solving a problem

A missing addend problem, considered very difficult for first graders. They can do it with Part-Whole Circles.

A missing addend problem, considered very difficult for first graders. They can do it with Part-Whole Circles.


Problem Solving


Is 3 a part or whole?

Solving a problem


Problem Solving



3

Solving a problem


Problem Solving

3



Solving a problem


Problem Solving


Is 5 a part or whole?5

3

Solving a problem


Problem Solving


5

3

What is the missing part?Solving a problem


Problem Solving


What is the missing part?5

3 2

Solving a problem


Problem Solving

5

3 2


Write the equation.Solving a problem


Problem Solving


2 + 3 = 55

3 2



Problem Solving


2 + 3 = 55

3 2

3 + 2 = 5



Problem Solving


2 + 3 = 55

3 2

3 + 2 = 55 – 3 = 2


Is this an addition or subtraction problem?

Is this an addition or subtraction problem?


Problem Solving

Part-whole circles young children solve problems. Writing equations do not.


Go to the Dump GameObjective: 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 GameObjective: To learn the ways to partition 10:

1 + 92 + 83 + 74 + 65 + 5

Object of the game: To collect the most pairs that equal ten.It is similar to “Go Fish.”




Go to the Dump Game

The ways to partition 10.The ways to partition 10.


“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





0

10

20

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40

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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]

Chinese

U.S.

Ave

rage

Hig

hest

Num

ber

Cou

nted

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

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



0

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20

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4 5 6Ages (yrs.)




Chinese

U.S.

Ave

rage

Hig

hest

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ber

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nted

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

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



0

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4 5 6Ages (yrs.)




Chinese

U.S.

Ave

rage

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nted

Dark green is Korean “math” way.Dark green is Korean “math” way.



0

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4 5 6Ages (yrs.)

Ave

rage

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nted




Chinese

U.S.

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

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



0

10

20

30

40

50

60

70

80

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100

4 5 6Ages (yrs.)




Chinese

U.S.

Ave

rage

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ber

Cou

nted

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

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.




• 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:




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:


Children who understand tens remove a ten and 4 ones.


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.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 game

fireplace place-fire

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




A word game


paper-newsnewspaper





A word game


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 03 0


Composing Numbers

3-ten

3 03 0

Point to the 3 and say 3.Point to the 3 and say 3.


Composing Numbers

3-ten

3 03 0

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


Composing Numbers

3-ten 7

3 03 0


Composing Numbers

3-ten 7

3 03 0 77


3 03 0

Composing Numbers

3-ten 7

77

Place the 7 on top of the 0 of the 30.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 03 077


Composing Numbers

7-ten 8

7 87 888

Another example.

Another example.


Composing Numbers

10-ten


Composing Numbers

10-ten

1 0 01 0 0


Composing Numbers

1 hundred


Composing Numbers

1 hundred

1 0 01 0 0


Composing Numbers

1 hundred

1 0 01 0 0

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


Composing Numbers

1 hundred

11 001 01 0 01 0 0



Composing Numbers

1 hundred

1 0 01 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.


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.

• Powerful strategies are often visualizable representations.


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 bead simultaneously.

Use two hands and move the bead simultaneously.



9 + 5 =




9 + 5 = 14



Fact StrategiesTwo Fives

8 + 6 =



8 + 6 =

Two fives make 10. Two fives make 10.



8 + 6 =

Just add the “leftovers.”Just add the “leftovers.”



8 + 6 =

10 + 4 = 14

Just add the “leftovers.”Just add the “leftovers.”



7 + 5 =

Another example.

Another example.



7 + 5 =



7 + 5 = 12


Fact StrategiesGoing Down

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


Bead Frame

1

10

100

1000


Bead FrameMontessori’sPatent

QuickTime™ and a decompressor



Bead Frame



Montessori’sPatent


Bead FrameMontessori’sPatent




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.



AL Abacus1000 10 1100




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

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).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 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


6

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

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


1000 10 1100


6

Trade 10 ones for 1 ten.


1000 10 1100


6


1000 10 1100


9


1000 10 1100


9

Another trade.


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?. . . 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?

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]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]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.

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


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 =



Tens:


7 8 =



Tens: 20+ 30


7 8 =50 +



Tens: 20+ 30

50


7 8 =50 +



Tens: Ones:20+ 30

50


7 8 =50 +



Tens: Ones: 3 2

20+ 30

50


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 Board

1 2 3 4 5 6 7 8 9 10

6

6 4

7 x 7 on original multiplication board.



1 2 3 4 5 6 7 8 9 10

6


6 4

Using two colors.Using two colors.



1 2 3 4 5 6 7 8 9 10

7

7 7




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.

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



7 7

Less clutter.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.




Multiples PatternsFours

4 8 12 16 20

24 28 32 36 40


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

Again the ones repeat in the second row.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 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

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.


Multiples PatternsSevens

7 14 21

28 35 42

49 56 63

70

The 7s have the 1, 2, 3… pattern.


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

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

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

Giving the child the big picture, a Montessori principle.

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?


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 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

17

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.


“Pie” Model

Are we comparing angles, arcs, or area?


“Pie” Model

61

61

61

61

61

61

51

41

21

31

51

51

51

51

41

41

41

31

31

21

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.”







part of a whole.”

• Difficult for the child to see how fractions

relate to each other.







part of a whole.”

• Difficult for the child to see how fractions

relate to each other.

• Is the user comparing angles, arcs, or area?


112

12

14

14

14

14

18

18

18

18

18

18

18

18

Fraction War


Fraction War


Fraction War

1 2 3 4 5 6

Especially useful for learning to read a ruler with inches.

Especially useful for learning to read a ruler with inches.


Fraction War1

12

12

14

14

14

18

18

18

18

18

18

18

14

18


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


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


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

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


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


21212828

In what column would you put 21/28?In what column would you put 21/28?


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


21212828

45457272


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


12 12 1616


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


12 12 1616

6/8 needs further simplifying.6/8 needs further simplifying.


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


12 12 1616

12/16 could have put here originally.12/16 could have put here originally.


In Conclusion


In Conclusion

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


In Conclusion


• Subitizing needs to be encouraged.


In Conclusion



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


In Conclusion




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


In Conclusion




• 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.


VII

AMS ConferenceMarch 25, 2011Chicago, Illinois

by Joan A. Cotter, [email protected]

7

5 2

Counting: Necessary or Detrimental?

Presentation available: ALabacus.com7 x 7

AMS: Counting-Necessary or Detrimental? March 2011

Education

Transcript of AMS: Counting-Necessary or Detrimental? March 2011