Innovative Practices That Increase Mathematics Achievement

How many little How many little blackblacktrianglestriangles do you see? do you see?

Cotter Tens FractalCotter Tens FractalFCSC

Orlando, FLNovember 17, 200912:30 - 1:30 p.m.Cape Canaveral

Volusia

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

Slides/handouts:ALabacus.com

Math Crisis

• Close to 60% of those in jail under the age of 30 have no high school diploma and math is often the reason.

• 25% of college freshmen take remedial math; 38%, in California.

• 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 percent of the world’s college grads; today it’s 14 percent. 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.

What Makes Little Difference• Class size: engagement rises, but achievement gap remains. (40 in Japan, 50 in China, 26 in Singapore)

• Amount of homework.

• Counting ability.

• Poverty makes greater difference in US than in other countries.

Finland• Teachers from top 10% of undergraduate class. Need master’s to teach. Held in high esteem.

• Teachers work together on lessons and visit each other’s classrooms. Half day/week for PD.

• Work with students as soon as they fall behind.

Singapore• Although highest scorer in recent TIMSS, Singapore scored 16/26 in science in 1983-84.

• In 1990 curriculum changed to emphasize math concepts and problem solving, rather than rote learning.

• Stress visualization, patterning, number sense. (Not so much in US versions.)

• National curriculum.

China• Math specialists starting at grade 1.

• Teach 2 classes/day with 50 students/class.

• Teachers’ desks are near other math teachers in workroom to encourage collaboration.

• Half day every week for PD.

• Standard national curriculum.

Japan• Teacher stays with the same class for 3-4 years.

• Teachers’ desks in a huge room with references.

• Goal for math lesson: the class understands a new concept, not done something (worksheet).

• Teachers emphasize visualization; discourage counting for computation.

• Groups quantities into 5s as well as 10s.

• Uses part/whole model for problem solving.

What Does Matter• Knowing that learning math depends upon hard work and good instruction, not genes or talent.

• Having teachers who understand and like mathematics.

• Teaching for understanding.

• Supporting children who fall behind.

Innovative Math• Teach for understanding, not rote.

• Minimize counting; group in fives and tens.

• Practice facts with games; avoid flash cards.

• Use part/whole circles.

• Use math way of number naming initially.

• Teach visualizable strategies.

• Teach algorithms with four-digit numbers.

Time Needed to Memorize

• 93 minutes to learn 200 nonsense syllables.

• 24 minutes to learn 200 words of prose.

• 10 minutes to learn 200 words of poetry.

According to a study with college students, it took them:

This shows the importance of meaning before memorizing.

Memorizing Math

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

Richard Skemp

Percentage Recall

Immediately After 1 day After 4 wks

Rote 32 23 8

Concept 69 69 58

Flash Cards• Often used to teach rote.

• Liked only by are those who don’t need them.

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

• Often produce stress – children under stress stop learning.

• Not concrete – use abstract symbols.

Rigorous Mathematics

• To develop deep understanding.

• To justify reasoning.

• To connect ideas to prior knowledge.

• To explore concepts.

Adding by CountingFrom 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!!

Place ValueFrom a Child’s Perspective

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

Place ValueFrom a Child’s Perspective

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

(12)(one 10)

(two 1s).

(twelve)

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.

Calendars

A calendar is NOT a number line: day 4 does not include days 1 to 4.

Calendars

1 2 3 4

8 9 10

September

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

Calendars

fourthsecond, third,first,

Counting Model Drawbacks

• Poor concept of quantity.

• Ignores place value.

• Very error prone.

• Inefficient and time-consuming.

• Hard habit to break for the facts.

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.

Recognizing 5

5 has a middle; 4 does not.

Look at your hand; your middle finger is longer as a reminder 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.

Materials for Visualizing

• Representative of structure of numbers.

• Easily manipulated by children.

• Imaginable mentally.

Japanese Council ofMathematics Education

Japanese criteria.Japanese criteria.

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 (Montessori Elementary Teacher)

Manipulatives

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

Visualizing Needed in:

• Mathematics

• Botany

• Geography

• Engineering

• Construction

• Spelling

• Architecture

• Astronomy

• Archeology

• Chemistry

• Physics

• Surgery

Manipulatives

A manipulative must not only be visual, but also visualizable.

Can you visualize this rod?

Most countries stopped using these by early 1990s.Most countries stopped using these by early 1990s.

Colored Rod Drawbacks

• Young children think each rod is “one.”

• Adding rods doesn’t instantly give the sum; still need to count or compare.

Manipulatives

The 4-rod plus the 2-rod does not give the immediate answer.

You must count or compare.

Colored Rod Drawbacks

• Young children often think each rod is “one.”

• Adding rods doesn’t instantly give the sum; still need to count or compare.

• 8% of children have a color-deficiency; they cannot see 10 distinct colors.

• Many small pieces hard to manage.

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.Also set to music.

AL Abacus1000 100 10 1

Many types of abacuses. AL abacus shown is designed to help children learn math.

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

Stairs. Can use to count 1-10.Stairs. Can use to count 1-10.

4 + 3 = Adding

4 + 3 =Adding

4 + 3 = 7Adding

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

Typical Worksheet

Go to the Dump Game

A “Go Fish” type of game where the pairs are:

1 & 92 & 83 & 74 & 65 & 5

Children use the abacus while playing this 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.

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.

• 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

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. Children thinking of 14 as 14 ones will count 14.

Subtracting 14 From 48

Using 10s and 1s, ask the childto construct 48.

Then ask the child to subtract 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 770077

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.

2 hundred 22 00 00

How could you make 200?How could you make 200?

10 hundred 11 00 001 0 0 000

1 thousand 11 00 001 0 0 000

Point to the digits and say, one-th-ou-sand. Sorry for the extra syllable in thousand, but it’s the best we can do.

Place-Value Cards

33 00 00 003 th- ou-sand

3 hun-dred

3- ten

33 00 00

Place-Value Cards

33 00 00 00 33 00 00 0066 00 00

33 00 00 0066 00 0055 0088

55 008866 00 00

Place-Value Cards

33 00 00 00 33 00 00 33 00 0000 0088

No problem when some denominations are missing.No problem when some denominations are missing.

2584258

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:

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

“Two” used to be pronounced (twoo).“Two” used to be pronounced (twoo).

nickel

quarter

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 + 6 = 10 + 4 = 14Strategy: Two Fives

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

7 + 5 = 10 + 2 = 12Strategy: Two Fives

Another example.

15 – 9 =Strategy: Going Down

Subtract 5, then 4

Subtract the 9 from the 10. Then add 1 and 5.Subtract the 9 from the 10. Then add 1 and 5.

15 – 9 =Strategy: Going Down

Subtract 9 from the 10

Subtract the 9 from the 10. Then add 1 and 6.Subtract the 9 from the 10. Then add 1 and 6.

13 – 9 =Strategy: Going Up

1 + 3 = 4

Start at 9; go up to 13

To go up, start with 9; then complete the 10; then 3 more.

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, especially for oral problems, taught to Dutch children.

Mental Addition

“…The now well established fact that

those who are mathematically effective

in daily life seldom make use in their

heads of the standard written methods

which are taught in the classroom.”

W. H. Cockroft, 1982

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) and 4 (purple).You can see the ten (yellow) and 4 (purple).

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.

1000 100 10 1

Bead Trading

In this activity, children add numbers to get as high a score as possible.Turn over the top card. Enter 7 beads.

1000 100 10 1

Bead Trading

Turn over another card. 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

Turn over another card. Enter 9 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

No trading.

Bead Trading

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

• Bead trading helps the child experience the greater value of each column.

• Trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely.

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

2738396

Addition

2738396

Addition

Most children who learn to add on the AL abacus transition to the paper and pencil algorithm without further instruction.

Why Thousands So Early

To appreciate a pattern, at least three samples must be presented.

Therefore, to understand the never-ending pattern of trading, the child must trade 10 ones for 1 ten, 10 tens for 1 hundred, and 10 hundreds for 1 thousand.

6 x 4 (6 taken 4 times)

Multiplying on the Abacus

5 x 7 (30 + 5)

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

7 x 7 = Multiplying on the Abacus

25 + 10 + 10 + 4

9 x 3 (30 – 3)

9 x 3 3 x 9

Commutative property

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

Innovative Practices That Increase Mathematics Achievement

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