MAPSA: Spirit of Asian Math Oct 2010

Applying the Spiritof Asian Mathematics

MAPSA ConferenceNovember 2, 2010Detroit, Michigan

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

Handout and Presentation:

ALabacus.com

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.

• 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 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

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

G I J KHA FC D EB

Now memorize the facts!!

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

Place Value From a Child’s Perspective

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

(12)(one 10)

(two 1s).

(twelve)

Calendar Math

August

Sometimes calendars are used for counting.

Calendar Math

August

Sometimes calendars are used for counting.

Calendar Math

August

Calendar Math

August

Calendar Math

September123489101115161718222324252930

567121314192021262728

August

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

Calendar Math

September123489101115161718222324252930

567121314192021262728

August

1 2 3 4 5 6

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

Calendar Math

August

3 4 5 6 7

Always show the whole calendar. A child needs to see the whole before the parts. Children also need to learn to plan ahead.

Calendar Math 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.

• 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.

• 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.

• 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.

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.

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.

Counting Model

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

Contrast naming quantities with this early counting model.

Counting Model

What we see

Counting Model

What we see

Counting Model

What we see

Counting Model

What we see

Counting Model

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

What the young child sees

Counting Model

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.

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.

Tally Sticks

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

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.

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

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.

Abacus Cleared

Entering Quantities

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

Entering Quantities

Relate quantities to hands.Relate quantities to hands.

Entering Quantities

Stairs

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

4 + 3 =

Adding

4 + 3 =Adding

4 + 3 = 7Adding

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.

Go to the Dump Game

Starting

A game viewed from above.

7 42 61 38 349

Go to the Dump Game

Starting

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

72 4 61 38 349

Go to the Dump Game

Finding pairs

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

72 4 61 38 349

Go to the Dump Game

Finding pairs

Does BlueCap have any pairs? [yes, 1]

721 38 349

Go to the Dump Game

Finding pairs

Does PinkCap have any pairs? [yes, 2]

21 8 349

Go to the Dump Game

Finding pairs

7 32 8

Does PinkCap have any pairs? [yes, 2]

Go to the Dump GameBlueCap, do you

have a 3?BlueCap, do you

have an 8?

Go to the dump.

Playing

The player asks the player on his left.

Go to the Dump Game

PinkCap, do youhave a 6?Playing

Go to the dump.

1 92 8

Go to the Dump Game

YellowCap, doyou have a 9? Playing

Go to the Dump Game

Playing

291 77

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

Go to the Dump Game

Winner?

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

Go to the Dump Game

Winner?

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

Go to the Dump Game

Winner?

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

Part-Whole Circles

Part Part

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

Part-Whole Circles

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.

Part-Whole Circles

Is 3 a part or whole?

Part-Whole Circles

Is 5 a part or whole?5

Part-Whole Circles

What is the missing part?

Part-Whole Circles

What is the missing part?

Part-Whole Circles

Write the equation.

Is this an addition or subtraction problem?

Part-Whole Circles

2 + 3 = 53 + 2 = 5

5 – 3 = 2

Is this an addition or subtraction problem?

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.

Language Effect on Counting

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

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.

• 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.

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.

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

Paper Abacus

Paper Abacus4 + 3 =

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

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.

15 – 9 = ___Strategy: Going Down

Subtract 5, then 4.

15 – 9 = ___Strategy: Subtract from 10

Subtract 9 from the 10.

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.

1 + 3 =

1 + 3 = 4

Traditional Names

4-ten = forty

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.”

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.

Traditional Names

ten 4 teen 4 fourteen

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.

Traditional Names

two left twelve

nickel

quarter

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.

Mental Addition

24 + 38 =

+ 3024 + 8 =

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

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

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

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

Adding

1000 100 10 1

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.

1000 100 10 1

Do we need to trade?

Adding

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

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

1000 100 10 1

Bead Trading

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

1000 100 10 1

Bead Trading

Enter 7 beads.Enter 7 beads.

1000 100 10 1

Bead Trading

Turn over another card.

1000 100 10 1

Bead Trading

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

1000 100 10 1

Bead Trading

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

1000 100 10 1

Bead Trading

1000 100 10 1

Bead Trading

1000 100 10 1

Bead Trading

Turn over another card.Turn over another card.

1000 100 10 1

Bead Trading

Add 9 ones.Add 9 ones.

1000 100 10 1

Bead Trading

Add 9 ones.Add 9 ones.

1000 100 10 1

Bead Trading

1000 100 10 1

Bead Trading

1000 100 10 1

Bead Trading

1000 100 10 1

Bead Trading

1000 100 10 1

Bead 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

Addition

1000 100 10 1

Addition

1000 100 10 1

Addition

1000 100 10 1

Addition

1000 100 10 1

Addition

1000 100 10 1

Addition

1000 100 10 1

Addition

1000 100 10 1

Addition

1000 100 10 1

Addition

Critically important to write down what happened after each step.

1000 100 10 1

Addition

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

1000 100 10 1

Addition

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

1000 100 10 1

Addition

1000 100 10 1

Addition

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

1000 100 10 1

273896

Addition

1000 100 10 1

273896

Addition

1000 100 10 1

273896

Addition

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

1000 100 10 1

273896

Addition

1000 100 10 1

273896

Addition

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

273896

Addition

1000 100 10 1

2738396

Addition

1000 100 10 1

2738396

Addition

1000 100 10 1

2738396

Addition

1000 100 10 1

2738396

Addition

1000 100 10 1

27386396

Addition

1000 100 10 1

27386396

Addition

Skip Counting Patterns

2 4 6 8 10

12 14 16 18 20

Recognizing multiples necessary for simplifying fractions and doing algebra.

4 8 12 16 20

24 28 32 36 40

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

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.

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

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.

21 24 27

Threes

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

Threes

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

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]

Sevens

28 35 42

49 56 63

7 14 21

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?

16171819

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?

A hyperbola floating down.A hyperbola floating down.

Non-unit Fractions

or 2 ÷ 3.23 means two s1

16171819

Fraction Chart

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

Partial Chart

1 2 3 4 5 6

Especially useful for learning to read a ruler with inches.

Fraction War

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

Fraction War

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

Fraction War

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

Fraction War1

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

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

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

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

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

MAPSA: Spirit of Asian Math Oct 2010

Education

Transcript of MAPSA: Spirit of Asian Math Oct 2010

Food Security Challenges Faced by Developing Asian ......Resilience Food Sovereignty Healty, active and productive individual and community Spirit/foundation Performance measure Outcome

Success in Logistics – An Asian Perspective · 2020. 3. 6. · Success in Logistics – An Asian Perspective Page 4 In the spirit of open collaboration and innovation, the BVL Singapore

Tackling challenges in Asian market with strong spirit Toru Tokushige / Terra Motors CEO

Kevin Nash MaPSA-Light test system 1. The System 2.

24 March 2014 The ups and downs of the MaPSA project March 2014francois.vasey@cern.ch1.

Title Newsletter : Center for Southeast Asian Studies ...repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/180691/1/cseasnl64.pdfflowers and incense. Spirit communications are important

Richard King (Professor of Buddhist and Asian Studies ...€¦ · Richard King (Professor of Buddhist and Asian Studies ... spirit of Buddhism, but in fact it is the spirit of all

ASIAN JEWISH LIFEasianjewishlife.org/images/issues/Issue12-June2013/PDFs/AJL-Issue… · A JOURNAL OF SPIRIT, SOCIETY AND CULTURE ASIAN JEWISH LIFE ISSUE 12 (JUNE 2013) ISSN 2224-3011

Communing Spirit to Spirit

NORTH ASIAN INTERNATIONAL North Asian International … · 2017-02-02 · North Asian International Researc h Journal of Multidisciplinary North Asian Internation North Asian International

BECKY CARLTON · BECKY CARLTON. Director of Communications Strategy • Worked at MAPSA over 9 years • Social media and marketing expertise • Constant Contact email all-star award

Asian Americans. Symbol – Asian American Woman Symbol – Asian American Man.

Inmobiliaria Mapsa S.A.

AQUACLEAN I Kolekcja SPIRIT I tkanina ułatwiająca czyszczenie · 2018-09-12 · AQUACLEAN I Kolekcja SPIRIT I tkanina ułatwiająca czyszczenie SPIRIT 01 SPIRIT 02 SPIRIT 07 SPIRIT

The SPIRIT The SPIRIT

GETRÄNKEKARTE - werkstatt-rankweil.ws · Green Spot 4cl 8,60 ... Bio Asian Spirit Ingwer Lemongrass . Säfte von Rauch Mango, Johannisbeere, 0,20l 3,10 ...

Asian Summit. The Fastest Economic Growth To deal with sustainable issues To enhance the spirit of the 4th CECAR: Working for Asian Sustainability.

Fruits of the Holy Spirit. Love - the Spirit lives! Joy - the Spirit dances! Peace - the Spirit rests! Patience - the Spirit waits! Kindness - the Spirit.

Geometric Surface Processing via · Geometric Surface Processing via Normal Mapsa ... to the processing of 3D surfaces has become an important problem in computer graphics, visualization,

Inmobiliaria Mapsa S.A. Estados financieros e informe de ... · Inmobiliaria Mapsa S.A. 4 Notas 31.12.2016 31.12.2015 M$ M$ Patrimonio neto y pasivos Pasivos corrientes: Cuentas por