MAPSA: Spirit of Asian Math Oct 2010

© Joan A. Cotter, 2010

Applying the Spiritof Asian Mathematics

VII

MAPSA ConferenceNovember 2, 2010Detroit, Michigan

by Joan A. Cotter, [email protected]

Handout and Presentation:

ALabacus.com

7

5 2


Some Features of Asian Math• Explicit number naming (math way of counting).



• Grouping in fives, as well as tens.




• A function of good instruction and hard work.





• Manipulatives used judiciously.






• Little time spent reviewing.






• Little time spent reviewing.

• Low SES and low-achievers also taught concepts.


Japanese Teaching Principles• The Intellectual Engagement Principle.

Students must be engaged with important math.




• The Goal Principle.Lesson explicitly addresses student motivation, performance, and understanding.





• The Flow Principle.The lesson builds on students’ previous knowledge and supports them in learning the lesson’s big math ideas.






• The Unit Principle.Teacher fits lesson with past and future lessons.







• The Adaptive Instruction Principle.All students do math at their current understanding.







• The Adaptive Instruction Principle.All students do math at their current understanding.

• The Preparation Principle.Coherent lesson plan must be well-thought-out and detailed.


Adding by Counting From 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



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



E

+ I

Now memorize the facts!!

G + D

H + F

C + G

D + C


Subtracting by Counting BackFrom a Child’s Perspective

Try subtractingby ‘taking away’

H – E


Skip CountingFrom a Child’s Perspective

Try skip counting by B’s to T: B, D, . . . T.


Place Value From a Child’s Perspective

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

huh?


Place Value From a Child’s Perspective

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

huh?

(12)(one 10)

(two 1s).

(twelve)


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.



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


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

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 Drawbacks• The calendar is not a number line.

• No quantity is involved.• Numbers are in spaces, not at lines like a ruler.




• Children need to see the whole month, not just part.• Purpose of calendar is to plan ahead.• Many ways to show the current date.




• 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.


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• 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.

• Increased emphasis on mathematical reasoning, less emphasis on rules and procedures.


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


Memorizing Math

Percentage Recall


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

Percentage Recall


Rote 32 23 8

Concept 69 69 58


Flash Cards• Are often used to teach rote.



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




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





• Often produce stress – children under stress stop learning.





• Often produce stress – children under stress stop learning.

• Are not concrete – use abstract symbols.


Visualizing Needed in:

• Reading

• Mathematics

• Botany

• Geography

• Engineering

• Construction

• Architecture

• Astronomy

• Archeology

• Chemistry

• Physics

• Surgery


Visualization

“Think in pictures, because the

brain remembers images better

than it does anything else.”

Ben Pridmore, World Memory Champion, 2009


5-Month Old Babies CanAdd and Subtract Up to 3

Show the baby two teddy bears. 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.

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



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.


Counting without Words

• Australian Aboriginal children from two tribes.

Brian Butterworth, University College London, 2008.

These groups matched quantities without using counting words.




• Australian Aboriginal children from two tribes.

Brian Butterworth, University College London, 2008.

• Adult Pirahã from Amazon region.

Edward Gibson and Michael Frank, MIT, 2008.





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

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

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





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

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

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

• Baby chicks from Italy.Lucia Regolin, University of Padova, 2009.




Quantities with 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.




Yellow is the SunYellow is the sun.Six is five and one.

Why is the sky so blue?Seven is five and two.

Salty is the sea.Eight is five and three.

Hear the thunder roar.Nine is five and four.

Ducks will swim and dive.Ten is five and five.

–Joan A. Cotter

Also set to music. Listen and download sheet music from Web site.

Also set to music. Listen and download sheet music from Web site.


Counting Model

How many?Contrast naming quantities with this early counting model.

Contrast naming quantities with this early counting model.


Counting Model

1

What we see


Counting Model

2

What we see


Counting Model

3

What we see


Counting Model

4

What we see


Counting Model

2

What the young child seesChildren think we’re naming the stick, not the quantity.Children think we’re naming the stick, not the quantity.


Counting Model

3

What the young child sees


Counting Model

4

What the young child sees


Counting Model DrawbacksCounting:


Counting Model Drawbacks

• Is not natural.Counting:



• Is not natural.

• Provides poor concept of quantity.

Counting:



• Is not natural.


• Ignores place value.

Counting:



• Is not natural.



• Is very error prone.

Counting:



• Is not natural.




• Is inefficient and time-consuming.

Counting:



• Is not natural.




• Is inefficient and time-consuming.

• Is a hard habit to break for mastering the facts.

Counting:


Counting in Japanese Schools

• Children are discouraged from counting to add.

• They group in 5s.


Recognizing 5

5 has a middle; 4 does not.

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.


Ready: How Many?


Ready: How Many?

Which is easier?Which is easier?


Visualizing 8

Try to visualize 8 apples without grouping.


Visualizing 8

Next try to visualize 5 as red and 3 as green.


Grouping by 5s

I II III IIII V VIII

1 23458

Early Roman numeralsRomans grouped in fives. Notice 8 is 5 and 3.

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


Grouping by 5s

Who could read the music?

:

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


Tally 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.


Tally Sticks


Tally Sticks


Tally Sticks

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.


Tally Sticks


Tally Sticks

Start a new row for every ten.Start a new row for every ten.


Tally Sticks

What is 4 apples plus 3 more apples?

How would you find the answer without counting?How would you find the answer without counting?


Tally Sticks

What is 4 apples plus 3 more apples?

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.


Materials for Visualizing

“In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.”

Mindy Holte (E I)



• Representative of structure of numbers.

• Easily manipulated by children.

• Imaginable mentally.

Japanese Council ofMathematics Education

Japanese criteria.Japanese criteria.



“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


Number Chart

61

72

83

94

105To help children learn the symbols.

To help children learn the symbols.


AL Abacus1000 100 10 1

Double-sided AL abacus. Side 1 is grouped in 5s.Side 2 allows both addends to be entered before trading.

Double-sided AL abacus. Side 1 is grouped in 5s.Side 2 allows both addends to be entered before trading.


Abacus Cleared


3

Entering Quantities

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


5

Entering Quantities

Relate quantities to hands.Relate quantities to hands.


7

Entering Quantities


10

Entering Quantities


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.


4 + 3 =

Adding


4 + 3 =Adding


4 + 3 = 7Adding


4 + 3 = 7Adding

Mentally, think take 1 from 3 and give it to 4, making 5 + 2.

Mentally, think take 1 from 3 and give it to 4, making 5 + 2.


Sums Adding to Ten

1 and 9; 2 and 8; 3 and 7; and so forth.1 and 9; 2 and 8; 3 and 7; and so forth.


Go to the Dump GameObjective: To 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.

Children use the abacus while playing this “Go Fish” type game.


Go to the Dump Game

Starting

A game viewed from above.

A game viewed from above.


72795

7 42 61 38 349

Go to the Dump Game

Starting

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


72795

72 4 61 38 349

Go to the Dump Game

Finding pairs

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


4 6

72795

72 4 61 38 349

Go to the Dump Game

Finding pairs

Does BlueCap have any pairs? [yes, 1]

Does BlueCap have any pairs? [yes, 1]


4 6

72795

721 38 349

Go to the Dump Game

Finding pairs

7 3

Does PinkCap have any pairs? [yes, 2]



4 6

72795

21 8 349

Go to the Dump Game

Finding pairs

7 32 8




2

4 6

7 3

72795

1 349

Go to the Dump GameBlueCap, do you

have a 3?BlueCap, do you

have an 8?

Go to the dump.

2 8

Playing

The player asks the player on his left.

The player asks the player on his left.


2 8

5

4 6

7 3

22795

1 49

Go to the Dump Game

PinkCap, do youhave a 6?Playing

1

Go to the dump.


1 92 8

5

4 6

7 3

22795

49

Go to the Dump Game

YellowCap, doyou have a 9? Playing

1


1 9

5

4 6

7 3

22795

49

Go to the Dump Game

Playing

291 77

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

No counting. Combine both stacks. (Shuffling not necessary for next game.)



Go to the Dump Game

Winner?

4 5

9

6 5

1




Go to the Dump Game

Winner?

46 55

91

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


Part-Whole Circles

Whole

Part Part

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

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


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 a Part-Whole Circles.

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


Part-Whole Circles


Is 3 a part or whole?


Part-Whole Circles



3


Part-Whole Circles

3




Part-Whole Circles

3


Is 5 a part or whole?5


Part-Whole Circles

5

3


What is the missing part?


Part-Whole Circles

5

3 2


What is the missing part?


Part-Whole Circles

5

3 2


Write the equation.

Is this an addition or subtraction problem?



Part-Whole Circles

5

3 2


2 + 3 = 53 + 2 = 5

5 – 3 = 2




Part-Whole Circles

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


“Math” Way of Counting

11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9

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

Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.

Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.


Language Effect on Counting

0

10

20

30

40

50

60

70

80

90

100

4 5 6Ages (yrs.)

Ave

rage

Hig

hest

Num

ber

Cou

nted

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.

Purple is Chinese. Note jump during school year. Dark green is Korean “math” way. Dotted green is everyday Korean; notice jump during school year.Red is English speakers. They learn same amount between ages 4-5 and 5-6.

Purple is Chinese. Note jump during school year. Dark green is Korean “math” way. Dotted green is everyday Korean; notice jump during school year.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.


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

Math Way of CountingCompared to Reading


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

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

Math Way of CountingCompared to Reading


Research Quote

“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, 2005


Subtracting 14 From 48

Using 10s and 1s, ask the childto construct 48.Then ask the child to subtract 14.

Children thinking of 14 as 14 ones will count 14. Those understanding place value will remove a ten and 4 ones.

Children thinking of 14 as 14 ones will count 14. Those understanding place value will remove a ten and 4 ones.


3-ten 33 003 0

Place-value card for 3-ten. Point to the 3, saying three and point to 0, saying ten. The 0 makes 3 a ten.

Place-value card for 3-ten. Point to the 3, saying three and point to 0, saying ten. The 0 makes 3 a ten.


3-ten 7 33 00 7700


3-ten 7 33 000077


10-ten 11 00 001 0 0

Now enter 10-ten.Now enter 10-ten.


1 hundred 11 00 001 0 0

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


2584 8

Column Method for Reading Numbers

To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:


2584 58




2584258




2584258



4


Paper Abacus


Paper Abacus4 + 3 =


Paper Abacus4 + 3 =


Paper Abacus4 + 3 =


Paper Abacus4 + 3 =


Paper Abacus4 + 3 =


Paper Abacus4 + 3 =


Paper Abacus3-ten 7


Paper Abacus3-ten 7


Paper Abacus3-ten 7


Paper Abacus3-ten 7


Strategies

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

• Powerful strategies are often visualizable representations.


9 + 5 =Strategy: Complete the Ten

14

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


8 + 7 = 10 + 5 = 15Strategy: Two Fives

Two fives make 10. Just add the “leftovers.”Two fives make 10. Just add the “leftovers.”


7 + 5 = 12Strategy: Two Fives

Another example.

Another example.


15 – 9 = ___Strategy: Going Down





Subtract 5, then 4.



Subtract 5, then 4.



Subtract 5, then 4.



Subtract 5, then 4.

6


15 – 9 = ___Strategy: Subtract from 10

Subtract 9 from the 10.






6


Then add 1 and 5.Then add 1 and 5.


13 – 9 =Strategy: Going Up



Start at 9; go up to 13.




To go up, start with 9.To go up, start with 9.




Then complete the 10 and 3 more.Then complete the 10 and 3 more.




Then complete the 10 and 3 more.Then complete the 10 and 3 more.




1 + 3 =




1 + 3 = 4


Traditional Names

4-ten = forty

4-ten has another name: “forty.” The “ty” means ten.

4-ten has another name: “forty.” The “ty” means ten.


Traditional Names

6-ten = sixty

The same is true for 60, 70, 80, and 90.The same is true for 60, 70, 80, and 90.


Traditional Names

3-ten = thirty

The “thir” is more common than “three,” 3rd in line, 1/3, 13, and 30.The “thir” is more common than “three,” 3rd in line, 1/3, 13, and 30.


Traditional Names

5-ten = fifty

The same is true for “fif.”The same is true for “fif.”


Traditional Names

2-ten = twenty

Twenty is twice ten or twin ten. Note “two” is spelled with a “w.”

Twenty is twice ten or twin ten. Note “two” is spelled with a “w.”


Traditional Names

A word game

fireplace place-fire

paper-news

box-mail mailbox

newspaper

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

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


Traditional Names

ten 4


Traditional Names

ten 4 teen 4 fourteen

Ten 4 becomes teen 4 (teen = ten) and then fourteen. Similar for other teens.

Ten 4 becomes teen 4 (teen = ten) and then fourteen. Similar for other teens.


Traditional Names

a one left a left-one eleven

1000 yrs ago, people thought a good name for this number would be “a one left.” They said it backward: a left-one, which became: eleven.

1000 yrs ago, people thought a good name for this number would be “a one left.” They said it backward: a left-one, which became: eleven.


Traditional Names

two left twelve


Money

penny


Money

nickel


Money

dime


Money

quarter


Counting by Fives


Counting by Fives


Counting by Fives


Counting by Fives


Mental Addition

You need to find twenty-four plus thirty-eight.How do you do it?

You are sitting at your desk with a calculator, paper and pencil, and a box of teddy bears.

Research shows a majority of people do it mentally. “How would you do it mentally?” Discuss methods.

Research shows a majority of people do it mentally. “How would you do it mentally?” Discuss methods.


Mental Addition

24 + 38 =

+ 3024 + 8 =

A very efficient way, taught to Dutch children, especially oral.

A very efficient way, taught to Dutch children, especially oral.


Evens

To experience “evens”, touch each row with two fingers, (e-ven).

To experience “evens”, touch each row with two fingers, (e-ven).


Odds

To experience “odd”, touch each row with two fingers. Last row will feel odd.

To experience “odd”, touch each row with two fingers. Last row will feel odd.


1000 100 10 1

Cleared

Side 2


1000 100 10 1

Thousands1000

Side 2


1000 100 10 1

Hundreds100

Side 2


1000 100 10 1

Tens10

Side 2


1000 100 10 1

Ones1

Side 2

The third wire from each end is not used. Red wires indicate ones.The third wire from each end is not used. Red wires indicate ones.


1000 100 10 1

8+ 6

Adding


1000 100 10 1

8+ 6

Adding


1000 100 10 1

8+ 614

Adding

You can see the ten (yellow).You can see the ten (yellow).


1000 100 10 1

8+ 614

Adding

Trading ten ones for one ten. Trade, not rename or regroup.Trading ten ones for one ten. Trade, not rename or regroup.


1000 100 10 1

8+ 614

Adding


1000 100 10 1

8+ 614

Adding

Same answer, ten-4, or fourteen.

Same answer, ten-4, or fourteen.


1000 100 10 1

Do we need to trade?

Adding

If the columns are even or nearly even, trading is much easier.

If the columns are even or nearly even, trading is much easier.


Paper Abacus

1000 100 10 1 8+ 614


Paper Abacus

1000 100 10 1 8+ 614


Paper Abacus

1000 100 10 1 8+ 614


Paper Abacus

1000 100 10 1 8+ 614


Paper Abacus

1000 100 10 1 8+ 614


Paper Abacus

1000 100 10 1 8+ 614


Paper Abacus

1000 100 10 1 8+ 614


Paper Abacus

1000 100 10 1 8+ 614


Paper Abacus

1000 100 10 1 8+ 614


1000 100 10 1

Bead Trading

99

In this activity, children add numbers to get as high a score as possible.

In this activity, children add numbers to get as high a score as possible.


1000 100 10 1

Bead Trading

997

Turn over the top card.Turn over the top card.


1000 100 10 1

Bead Trading

997

Enter 7 beads.Enter 7 beads.


1000 100 10 1

Bead Trading

996

Turn over another card.

Turn over another card.


1000 100 10 1

Bead Trading

996

Enter 6 beads. Do we need to trade?Enter 6 beads. Do we need to trade?


1000 100 10 1

Bead Trading

996

Trading 10 ones for 1 ten.Trading 10 ones for 1 ten.


1000 100 10 1

Bead Trading

996



1000 100 10 1

Bead Trading

996



1000 100 10 1

Bead Trading

999

Turn over another card.Turn over another card.


1000 100 10 1

Bead Trading

999

Add 9 ones.Add 9 ones.


1000 100 10 1

Bead Trading

999

Add 9 ones.Add 9 ones.


1000 100 10 1

Bead Trading

999



1000 100 10 1

Bead Trading

999



1000 100 10 1

Bead Trading

999



1000 100 10 1

Bead Trading

993


1000 100 10 1

Bead Trading

993

No trading.

No trading.


Bead Trading

• Trading 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.

• To appreciate a pattern, there must be at least three examples in the sequence.


1000 100 10 1

3658+

2738

Addition


1000 100 10 1

3658+

2738

Addition


1000 100 10 1

3658+

2738

Addition


1000 100 10 1

3658+

2738

Addition


1000 100 10 1

3658+

2738

Addition


1000 100 10 1

3658+

2738

Addition


1000 100 10 1

3658+

2738

Addition


1000 100 10 1

3658+

2738

Addition


1000 100 10 1

3658+

2738

Addition

Critically important to write down what happened after each step.

Critically important to write down what happened after each step.


1000 100 10 1

3658+

27386

Addition

. . . 6 ones. Did anything else happen?. . . 6 ones. Did anything else happen?


1000 100 10 1

3658+

27386

1

Addition

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 100 10 1

3658+

27386

1

Addition


1000 100 10 1

3658+

27386

1

Addition

Do we need to trade? [no]Do we need to trade? [no]


1000 100 10 1

3658+

273896

1

Addition


1000 100 10 1

3658+

273896

1

Addition


1000 100 10 1

3658+

273896

1

Addition

Do we need to trade? [yes]Do we need to trade? [yes]


1000 100 10 1

3658+

273896

1

Addition


1000 100 10 1

3658+

273896

1

Addition

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

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


1000 100 10 1

3658+

273896

1

Addition


1000 100 10 1

3658+

2738396

1

Addition


1000 100 10 1

3658+

2738396

1 1

Addition


1000 100 10 1

3658+

2738396

1 1

Addition


1000 100 10 1

3658+

2738396

1 1

Addition


1000 100 10 1

3658+

27386396

1 1

Addition


1000 100 10 1

3658+

27386396

1 1

Addition

6


Skip Counting Patterns

2 4 6 8 10

12 14 16 18 20

Twos

2

2

4

4

6

6

8

8

0

0

Recognizing multiples necessary for simplifying fractions and doing algebra.

Recognizing multiples necessary for simplifying fractions and doing algebra.



4 8 12 16 20

24 28 32 36 40

Fours

4

4

8

8

2

2

6

6

0

0

Notice the ones repeat in the second row.Notice the ones repeat in the second row.



Sixes and Eights

6 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80

6

6

2

2

8

8

4

4

0

0

8

8

6

6

4

4

2

2

0

0

8 16 24 32 40

Also with the 6s and 8s, the ones repeat in the second row.

Also, the ones in the eights are counting by 2s backward, 8, 6, 4, 2, 0.

Also with the 6s and 8s, the ones repeat in the second row.

Also, the ones in the eights are counting by 2s backward, 8, 6, 4, 2, 0.


6x4

Skip Counting PatternsSixes and Eights

6 12 18 24 30

36 42 48 54 60

8 16 24 32 40

48 56 64 72 80

8x7

6 x 4 is the fourth number (multiple).6 x 4 is the fourth number (multiple).


Skip Counting PatternsNines

9 18 27 36 45

90 81 72 63 54

Second row done backward to see digits reversing. Also the digits in each number add to 9.

Second row done backward to see digits reversing. Also the digits in each number add to 9.


15 5


12 18

21 24 27

3 6 9

30

Threes

2 8

1 4 7

3 6 9

0

Threes have several patterns. First see 0, 1, 2, 3, . . . 9.Threes have several patterns. First see 0, 1, 2, 3, . . . 9.



12 15 18

21 24 27

3 6 9

30

Threes

The tens in each column are 0, 1, 2.The tens in each column are 0, 1, 2.


6

15

24

6

18

27

9

12

21

3

30

18

27

9

12

21

3

30

15

24

6


Threes

The second column. [6]The second column. [6] And the third column – the 9s.And the third column – the 9s.

Now add the digits in each number in the first column. [3]

Now add the digits in each number in the first column. [3]



Sevens

28 35 42

49 56 63

7 14 21

70

8

9

7

0

5

6

4

2

3

1

Start in the upper right to see the 1, 2, 3 pattern.Start in the upper right to see the 1, 2, 3 pattern.


6 4 (6 taken 4 times)

Multiplying on the Abacus


5 7 (30 + 5)


Groups of 5s to make 10s.Groups of 5s to make 10s.


7 7 = Multiplying on the Abacus

25 + 10 + 10 + 4


9 3 (30 – 3)



9 3 3 9

Commutative property



Fraction Chart

How many fourths make a whole? How many sixths?

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

Giving the child the big picture, a Montessori principle.Giving the child the big picture, a Montessori principle.


Fraction Stairs

Are the fraction stairs similar to the pink tower?

1

12

13

14

15

17

18

110

16

19

A hyperbola floating down.A hyperbola floating down.


113

13

13

Non-unit Fractions

or 2 ÷ 3.23 means two s1

3


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

Showing 9/8 is 1 plus 1/8.Showing 9/8 is 1 plus 1/8.


“Pie” Model

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

Specialists also suggest refraining from using more than one pie chart for comparison.

www.statcan.ca


• Perpetuates cultural myth that fractions < 1.

• It does not give child the “big picture.”

• A fraction is much more than “a part of a set of part of a whole.”

• Difficult for the child to see how fractions relate to each other.

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

“Pie” ModelDifficulties


112

12

14

14

14

14

18

18

18

18

18

18

18

18

Partial Chart

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 War

14

18

112

12

14

14

14

18

18

18

18

18

18

18

14

18

Which is more, 1/8 or 1/4?Which is more, 1/8 or 1/4?


112

12

14

14

14

18

18

18

18

18

18

18

14

18

Fraction War

34

58

Which is more, 5/8 or 3/4?Which is more, 5/8 or 3/4?


Fraction War

34

34

38

14

112

12

14

14

14

18

18

18

18

18

18

18

14

18

When cards are equal, a “war,” players put 1 card face down and 1 face up.

When cards are equal, a “war,” players put 1 card face down and 1 face up.


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

21212828

45457272

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.The fraction 21/28 can be reduced on the multiplication table as 3/4.The fraction 21/28 can be reduced on the multiplication table as 3/4.


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

12 12 1616

6/8 needs further simplifying.6/8 needs further simplifying.12/16 could have put here originally.12/16 could have put here originally.


Research Highlights

TASK EXPER CTRL

TEENS 10 + 3 94% 47%6 + 10 88% 33%

CIRCLE TENS 78 75% 67%

3924 44% 7%

14 as 10 & 4 48 – 14 81% 33%


Research Highlights

TASK EXPER CTRL

26-TASK (tens) 6 (ones) 94% 100%2 (tens) 63% 13%

MENTAL COMP: 85 – 70 31% 0%2nd Graders in U.S. (Reys): 9%

38 + 24 = 512 or 0% 40%

57 + 35 = 812

Other research questions asked.

Other research questions asked.


Some Important Conclusions

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

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

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

• When we reduce the heavy memory load for our disadvantaged youngsters, more of them will succeed.


Current Early Math• Counting words.

• Child must memorize 100 words in order.• One-to-one correspondence.

• Child must coordinate words with hand.• Cardinality principle.

• No model exists in child’s everyday life.• Written numbers.

• Why is twelve written with a 1 and a 2?• Place value.

• Quantity is taught as a collection of ones.


References

• Cotter, Joan. “Using Language and Visualization to Teach Place Value.” Teaching Children Mathematics 7 (October, 2000): 108-114.

• Also reprinted in NCTM (National Council of Teachers of Mathematics) On-Math Journal and in Growing Professionally: Readings from NCTM Publications for Grades K-8, in 2008.


Some Features of Asian Math

• Explicit number naming (math way of counting).



• Manipulatives: representative of math concept, for children’s use, and imaginable mentally.

• Less time spent reviewing.


Applying the Spiritof Asian Mathematics

VII

MAPSA ConferenceNovember 2, 2010Detroit, Michigan

by Joan A. Cotter, [email protected]

Handout and Presentation:

ALabacus.com

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