ida-umb: visualizing with the al abacus march 2011

© Joan A. Cotter, Ph.D., 2011

Overcoming Obstacles Learning Arithmeticthrough Visualizing with the AL Abacus

VII

IDA-UMB ConferenceMarch 12, 2011

Saint Paul, Minnesota

by Joan A. Cotter, [email protected]

7

5 2


Children with MD (Math Difficulties)Often experience difficulties with:


Children with MD (Math Difficulties)

• Counting in its various forms.

Often experience difficulties with:




• Composing numbers.






• Memorizing the facts.






• Understanding and applying math symbols.







• Understanding and applying math symbols.

• Learning algorithms.




Children with MD (Math Difficulties)Often learn best when:



• They are taught visually, not orally.

Often learn best when:




• They use the “math way” of counting initially.






• They truly understand math concepts.






• They are given the “big picture” before details.








• They use part/whole circles for solving problems








• They use part/whole circles for solving problems

• They are provided with references as needed.




Learning Arithmetic Traditionally

Counting



Counting

Memorizing390 Facts



Counting

Memorizing390 Facts

LearningProcedures



Counting

Memorizing390 Facts

LearningProcedures

SolvingProblems



Counting

Memorizing390 Facts

LearningProcedures

SolvingProblems

PlaceValue


Learning Arithmetic Visually

Place Value

Place value is the single most important topic in arithmetic.




NamingQuantities

Place Value





NamingQuantities

Visualizing390 Facts

Place Value





NamingQuantities


LearningProcedures

Place Value





NamingQuantities


LearningProcedures

SolvingProblems

Place Value




Counting Based-ArithmeticArithmetic is deemed to be based on counting.


Counting Based-Arithmetic

• Rote counting to 100 in kindergarten.

Arithmetic is deemed to be based on counting.




• Calendars (mis)used to teach counting.







• Addition and subtraction taught with counting.





• Number lines, a counting artifact.







• Skip counting used for multiplication facts.









• Graphing primarily a counting activity.









• Graphing primarily a counting activity.




• Doesn’t work well for fractions or algebra.


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 ModelCounting:


Counting Model

• Is not natural; it takes years of practice.Counting:


Counting Model

• Is not natural; it takes years of practice.

• Provides poor concept of quantity.

Counting:


Counting Model



• Ignores place value.

Counting:


Counting Model




• Is very error prone.

Counting:


Counting Model





• Is tedious and time-consuming.

Counting:


Counting Model





• Is tedious and time-consuming.

Counting:

• 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

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



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




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.


Memorizing Math

Percentage Recall

Immediately After 1 day After 4 wks

Rote 32 23 8

Concept 69 69 58


Memorizing Math

Percentage Recall


Rote 32 23 8

Concept 69 69 58

Even worseif you haveMD.


Memorizing Math

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

Richard Skemp

Percentage Recall


Rote 32 23 8

Concept 69 69 58

Even worseif you haveMD.


Memorizing MathFlash Cards

9 + 7


• Are often used to teach rote.


9 + 7



• Are liked only by those who don’t need them.


9 + 7




• Give the false impression that math isn’t about thinking.


9 + 7





• Often produce stress – children under stress stop learning.

Memorizing MathEven worseif you haveMD.Flash Cards

9 + 7





• Often produce stress – children under stress stop learning.

• Are not concrete – use abstract symbols.

Memorizing MathEven worseif you haveMD.Flash Cards

9 + 7


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





• Children are discouraged from using counting for adding.

• They consistently group in 5s.

In Japanese schools:


Visualizing Mathematics

Visualizing is an alternative to copious counting and mind-numbing memorization.



“Think in pictures, because the

brain remembers images better

than it does anything else.”

Ben Pridmore, World Memory Champion, 2009



“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)



“The process of connecting symbols to

imagery is at the heart of mathematics

learning.”

Dienes



“Mathematics is the activity of

creating relationships, many of which

are based in visual imagery.”

Wheatley and Cobb



“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.


Naming QuantitiesUsing fingers



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 QuantitiesA typical worksheet

The child counts all the horsies and forgets the fact before turning the page.

The child counts all the horsies and forgets the fact before turning the page.


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


AL Abacus1000 10 1100

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

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.


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 facts that total 10:

1 + 92 + 83 + 74 + 65 + 5

Object of the game: To collect the most pairs that equal ten.




Go to the Dump Game




Go to the Dump Game

Starting

A game viewed from above.

A game viewed from above.


72795

7 42 6138 349

Go to the Dump Game

Starting

Each player takes 5 cards.Each player takes 5 cards.


72795

72 4 6138 349

Go to the Dump Game

Finding pairs

Does YellowCap have any pairs? [no]Does YellowCap have any pairs? [no]


72795

72 4 6138 349

Go to the Dump Game

Finding pairs

Does BlueCap have any pairs? [yes, 1]



72795

72138

Go to the Dump Game

Finding pairs

4 6349




4 6

72795

72138 349

Go to the Dump Game

Finding pairs




4 6

72795

72138 349

Go to the Dump Game

Finding pairs

Does PinkCap have any pairs? [yes, 2]



4 6

72795

349

Go to the Dump Game

Finding pairs

72138




4 6

72795

21 8 349

Go to the Dump Game

Finding pairs

7 3




4 6

72795

1 349

Go to the Dump Game

Finding pairs

7 32 8




2

4 6

72795

1 349

Go to the Dump Game

2 8

Playing

The player asks the player on her left.



2

4 6

72795

1 349

Go to the Dump GameBlueCap, do you

have a 3?BlueCap, do you

have an 3?

2 8

Playing




7

4 6

2795

1 49



have an 3?

2 8

Playing




7

4 6

2795

1 49



have an 3?

2 8

Playing

3




4 6

2795

1 49



have an 3?

2 8

Playing

7 3


4 6

2795

1 49



have an 8?

2 8

Playing

7 3

YellowCap gets another turn.YellowCap gets another turn.


4 6

2795

1 49



have an 8?

Go to the dump.

2 8

Playing

7 3

YellowCap gets another turn.YellowCap gets another turn.


2

4 6

7 3

2795

1 49



have an 8?

Go to the dump.

2 8

Playing


2 8 4 6

7 3

22795

1 49

Go to the Dump Game

Playing

1


2 8 4 6

7 3

22795

1 49

Go to the Dump Game

PinkCap, do youhave a 6?Playing

1


2 8 4 6

7 3

22795

1 49

Go to the Dump Game

PinkCap, do youhave a 6?Playing

1

Go to the dump.


5

2 8 4 6

7 3

22795

1 49

Go to the Dump Game

Playing

1


2 8

5

4 6

7 3

22795

1 49

Go to the Dump Game

Playing

1


1 92 8

5

4 6

7 3

22795

49

Go to the Dump Game

YellowCap, doyou have a 9? Playing

1


1 92 8

5

4 6

7 3

227 5

49

Go to the Dump Game


1


1 92 8

5

4 6

7 3

227 5

49

Go to the Dump Game


19


1 91 9

5

4 6

7 3

227 5

49

Go to the Dump Game

Playing


1 9

5

4 6

7 3

227 5

49

Go to the Dump Game

Playing

29177

PinkCap is not out of the game. Her turn ends, but she takes 5 more cards.

PinkCap is not out of the game. Her turn ends, but she takes 5 more cards.


Go to the Dump Game

6 5

1

Winner?

4 5

9

5


Go to the Dump Game

Winner?

4 5

9

6 5

1

No counting. Combine both stacks.No counting. Combine both stacks.


Go to the Dump Game

Winner?

46 55

91

Whose pile is the highest?Whose pile is the highest?


Go to the Dump Game

Next game

No shuffling needed for next game.

No shuffling needed for next game.


Part-Whole Circles

Part-whole circles help children see relationships and solve problems.



Part-Whole Circles

Whole




Part-Whole Circles

Whole

Part Part




Part-Whole Circles

10

If 10 is the whole


Part-Whole Circles

10

4

and 4 is one part,


Part-Whole Circles

10

4

What is the other part?


Part-Whole Circles

10

4 6

What is the other part?


Part-Whole Circles

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

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.


Part-Whole Circles


Is 3 a part or whole?


Part-Whole Circles



3


Part-Whole Circles

3




Part-Whole Circles


Is 5 a part or whole?5

3


Part-Whole Circles


5

3

What is the missing part?


Part-Whole Circles


What is the missing part?5

3 2


Part-Whole Circles

5

3 2


Write the equation.


Part-Whole Circles


2 + 3 = 55

3 2

Write the equation.


Part-Whole Circles


2 + 3 = 55

3 2

3 + 2 = 5

Write the equation.


Part-Whole Circles


2 + 3 = 55

3 2

3 + 2 = 55 – 3 = 2

Write the equation.

Is this an addition or subtraction problem?

Is this an addition or subtraction problem?


Part-Whole Circles

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


Part-Whole Circles

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


“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.

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]

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

10

20

30

40

50

60

70

80

90

100

4 5 6Ages (yrs.)




Chinese

U.S.

Ave

rage

Hig

hest

Num

ber

Cou

nted

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

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



0

10

20

30

40

50

60

70

80

90

100

4 5 6Ages (yrs.)




Chinese

U.S.

Ave

rage

Hig

hest

Num

ber

Cou

nted

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



0

10

20

30

40

50

60

70

80

90

100

4 5 6Ages (yrs.)

Ave

rage

Hig

hest

Num

ber

Cou

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

90

100

4 5 6Ages (yrs.)




Chinese

U.S.

Ave

rage

Hig

hest

Num

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:


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

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.

• Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way).

Compared to reading:



“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 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


Composing Numbers

7-ten

70 is visualizable—again because of the fives’ grouping.



Composing Numbers

7-ten

7 07 0




Composing Numbers

7-ten 8

7 07 0




Composing Numbers

7-ten 8

7 07 0 88

70 is visualizable—again because of the fives grouping.

70 is visualizable—again because of the fives grouping.


Composing Numbers

7-ten 8

7 87 888

Place the 8 on top of the 0 of the 70.Place the 8 on top of the 0 of the 70.


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



Composing Numbers

2 hundred


Composing Numbers

2 hundred

2 0 02 0 0

Just the edges of the abacuses are shown.Just the edges of the abacuses are shown.


Composing Numbers

6 hundred

6 0 06 0 0

Maintaining the fives’ grouping.

Maintaining the fives’ grouping.


1 0 0 01 0 0 0

Composing Numbers

10 hundred


Composing Numbers

10 hundred

1 0 0 01 0 0 0


Composing Numbers

1 thousand

1 0 0 01 0 0 0

Of course, we can also read it as one th-ou-sand.

Of course, we can also read it as one th-ou-sand.


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




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


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


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.


Some Important Conclusions


Some Important Conclusions• We need to use quantity, not counting words, and place value as the foundation of arithmetic.



• We need to introduce the thousands much sooner to give children the big picture.




• Fostering visualization reduces the heavy memory load, allowing our disadvantaged youngsters to succeed.




• Fostering visualization reduces the heavy memory load, allowing our disadvantaged youngsters to succeed.

• Games, not flash cards, not timed tests, are the best way to help our students understand, master, apply, and enjoy mathematics.



VII




7

5 2



VII




7

5 2

(PowerPoint is available on Alabacus.com under Resources.)

ida-umb: visualizing with the al abacus march 2011

Education

md math difficulties

math way

math concepts

learning arithmetic

math symbols

counting memorizing

place value place value

learning algorithms