Innovative Practices That Increase
Mathematics Achievement
by Joan A. Cotter, Ph.D.
JoanCotter@ALabacus.com
Cotter Tens Fractal
How many little black
triangles do you see?
Slides/handouts:
ALabacus.com
FCSC
Orlando, FL
November 17, 2009
12:30 - 1:30 p.m.
Cape Canaveral
Volusia
Math Crisis
• 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.
• Close to 60% of those in jail under the age of 30 have no
high school diploma and math is often the reason.
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
According to a study with college students, it took them:
• 93 minutes to learn 200 nonsense syllables.
• 24 minutes to learn 200 words of prose.
• 10 minutes to learn 200 words of poetry.
This shows the importance of meaning before memorizing.
Memorizing Math
Percentage Recall
Immediately After 1 day After 4 wks
Rote
Concept
32
69
23
69
8
58
Math needs to be taught so 95% is
understood and only 5% memorized.
Richard Skemp
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 Counting
From a Child’s Perspective
Because we’re so familiar with 1, 2, 3, we’ll use letters.
A=1
B=2
C=3
D=4
E = 5, and so forth
Adding by Counting
From a Child’s Perspective
F
+E
A
B
C
D
E
F
A
B
C
D
E
Adding by Counting
From a Child’s Perspective
F
+E
A
B
C
D
E
F
A
B
What is the sum?
(It must be a letter.)
C
D
E
Adding by Counting
From a Child’s Perspective
F
+E
K
A
B
C
D
E
F
G
H
I
J
K
Adding by Counting
From a Child’s Perspective
Now memorize the facts!!
G
+D
D
+C
C
+G
Place Value
From a Child’s Perspective
L
is written AB
because it is A J
and B A’s
huh?
Place Value
From a Child’s Perspective
L (twelve)
is written AB (12)
because it is A J (one 10)
and B A’s (two 1s).
huh?
Subtracting by Counting Back
From a Child’s Perspective
Try subtracting
by ‘taking away’
H
–E
Skip Counting
From 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
September
1
2
3
8
9
10
4
5
6
7
Always show the whole calendar. A child wants to see the whole
before the parts. Children also need to learn to plan ahead.
Calendars
first, second, third, fourth
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 Can
Add 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.
5-Month Old Babies Can
Add and Subtract up to 3
Raise screen. Baby seeing 3 won’t look long because it is expected.
5-Month Old Babies Can
Add and Subtract up to 3
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?
Ready: How Many?
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
1
2
3
4
5
8
I
II
III
IIII
V
VIII
Early Roman numerals
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.
Materials for Visualizing
• Representative of structure of numbers.
• Easily
manipulated by children.
• Imaginable
mentally.
Japanese Council of
Mathematics Education
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
• Architecture
• Botany
• Astronomy
• Geography
• Archeology
• Engineering
• Chemistry
• Construction
• Physics
• Spelling
• 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.
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.
Quantities With Fingers
Quantities With Fingers
Quantities With Fingers
Always show 7 as 5 and 2, not for example, as 4 and 3.
Quantities With Fingers
Yellow is the Sun
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
Also set to music.
AL Abacus
1000
Many types of abacuses. AL abacus shown is designed to
help children learn math.
100
10
1
Abacus Cleared
Entering Quantities
3
Quantities are entered all at once, not counted.
Entering Quantities
5
Relate quantities to hands.
Entering Quantities
7
Entering Quantities
10
Stairs
Stairs. Can use to count 1-10.
Adding
4+3=
Adding
4+3=
Adding
4+3= 7
Adding
4+3= 7
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&9
2&8
3&7
4&6
5&5
Children use the abacus while playing this game.
Go to the Dump Game
Starting
A game viewed from above.
Go to the Dump Game
72 7 9 5
4 6 34 9
72 1 3 8
Starting
Each player takes 5 cards.
Go to the Dump Game
72 7 9 5
72 1 3 8
4 6 34 9
Finding pairs
Does YellowCap have any pairs? [no]
Go to the Dump Game
72 7 9 5
4 6
72 1 3 8
4 6 34 9
Finding pairs
Does BlueCap have any pairs? [yes, 1]
Go to the Dump Game
72 7 9 5
7 3
4 6
72 1 3 8
34 9
Finding pairs
Does PinkCap have any pairs? [yes, 2]
Go to the Dump Game
72 7 9 5
7
2 3
8
21
4 6
8
34 9
Finding pairs
Does PinkCap have any pairs? [yes, 2]
Go to the Dump Game
7
3
BlueCap, do you
have
havean
a 3?
8?
2 795
72
2 8
4 6
34 9
1
Playing
The player asks the player on his left.
Go to the dump.
Go to the Dump Game
7
3
22 7 9 5
2 8
4 6
549
1
Go to the dump.
Playing
PinkCap, do you
have a 6?
Go to the Dump Game
7
3
22 7 9 5
1 8
2
9
4 6
549
1
YellowCap, do
you have a 9?
Playing
Go to the Dump Game
7
3
22 7 9 5
1 9
4 6
549
29 1 7 7
Playing
PinkCap is not out of the game. Her turn ends, but she takes 5 more cards.
Go to the Dump Game
9
4 6
1
5 5
Winner?
No counting. Combine both stacks. (Shuffling not necessary for next game.)
Go to the Dump Game
91
4
6
5
Winner?
No counting. Combine both stacks. (Shuffling not necessary for next game.)
Go to the Dump Game
91
4
6 5
Winner?
Whose pile is the highest?
Part-Whole Circles
Whole
Part
Part
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.
Part-Whole Circles
Is 3 a part or whole?
Lee received 3 goldfish as a gift. Now Lee
has 5. How many did Lee have to start with?
Part-Whole Circles
Is 3 a part or whole?
3
Lee received 3 goldfish as a gift. Now Lee
has 5. How many did Lee have to start with?
Part-Whole Circles
Is 5 a part or whole?
3
Lee received 3 goldfish as a gift. Now Lee
has 5. How many did Lee have to start with?
Part-Whole Circles
5
Is 5 a part or whole?
3
Lee received 3 goldfish as a gift. Now Lee
has 5. How many did Lee have to start with?
Part-Whole Circles
5
What is the missing part?
3
Lee received 3 goldfish as a gift. Now Lee
has 5. How many did Lee have to start with?
Part-Whole Circles
5
3
What is the missing part?
2
Lee received 3 goldfish as a gift. Now Lee
has 5. How many did Lee have to start with?
Part-Whole Circles
5
3
Write the equation.
2
2+3=5
3+2=5
5–3=2
Lee received 3 goldfish as a gift. Now Lee
has 5. How many did Lee have to start with?
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 1
12 = ten 2
13 = ten 3
14 = ten 4
....
19 = ten 9
20 = 2-ten
21 = 2-ten 1
22 = 2-ten 2
23 = 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
Average Highest Number Counted
100
90
80
Chinese
U.S.
Korean formal [math way]
Korean informal [not explicit]
70
60
50
40
30
20
10
0
4
5
6
Ages (yrs.)
Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young
Purple is Chinese.
Note
jump
during
school
year. Dark
green
is Korean
children's
counting:
A natural
experiment
in numerical
bilingualism.
International
Journal“math” way.
of Psychology,
319-332. notice jump during school year.
Dotted green is
everyday23,Korean;
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.
Math Way of Counting
Compared 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).
Subtracting 14 From 48
Using 10s and 1s,
ask the child
to construct 48.
Then ask the child
to subtract 14.
Children thinking of 14 as 14 ones will count 14.
Subtracting 14 From 48
Using 10s and 1s,
ask the child
to construct 48.
Then ask the child
to subtract 14.
Those understanding place value will remove a ten and 4 ones.
3-ten
3 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
3 0
7
7
10-ten
Now enter 10-ten.
1 0 0
1 hundred 1 0 0
Of course, we can also read it as one-hun-dred.
2 hundred 2 0 0
How could you make 200?
10 hundred 1 0 0 0
1 thousand 1 0 0 0
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
3 0
3- ten
3 0 0
3 hun-dred
3 0 0 0
3 th- ou-sand
Place-Value Cards
3 0 0 0
6 0 0
5 0
8
3 0 0 0
6 0 0
5 0
8
8
3 0
60
50
Place-Value Cards
3 0 0 0
3 0 0 0
8
8
No problem when some denominations are missing.
3 0 0 0
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:
4258
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.
Traditional Names
3-ten = thirty
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.”
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
newspaper
box-mail
place-fire
paper-news
mailbox
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.
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
“Two” used to be pronounced (twoo).
twelve
Money
penny
Money
nickel
Money
dime
Money
quarter
Strategy: Complete the Ten
9 + 5 = 14
Take 1 from the 5 and give it to the 9.
Strategy: Two Fives
8 + 6 = 10 + 4 = 14
Two fives make 10. Just add the “leftovers.”
Strategy: Two Fives
7 + 5 = 10 + 2 = 12
Another example.
Strategy: Going Down
15 – 9 = 6
Subtract 5,
then 4
Subtract the 9 from the 10. Then add 1 and 5.
Strategy: Going Down
15 – 9 = 6
Subtract 9
from the 10
Subtract the 9 from the 10. Then add 1 and 6.
Strategy: Going Up
13 – 9 = 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 are sitting at your desk with a calculator,
paper and pencil, and a box of teddy bears.
You need to find twenty-four plus thirty-eight.
How do you do it?
Research shows a majority of people do it mentally. “How
would you do it mentally?” Discuss methods.
Mental Addition
24 + 38 =
24 + 30 + 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
Side 2
1000
100
10
1
Cleared
Side 2
1000
100
10
1
Thousands
Side 2
1000
100
10
1
Hundreds
Side 2
1000
100
10
1
Tens
Side 2
1000
100
10
1
Ones
The third wire from each end is not used. Red wires indicate ones.
Adding
1000
100
10
1
8
+6
Adding
1000
100
10
1
8
+6
Adding
1000
100
10
1
8
+6
14
You can see the ten (yellow) and 4 (purple).
Adding
1000
100
10
1
8
+6
14
Trading ten ones for one ten. Trade, not rename or regroup.
Adding
1000
100
10
1
8
+6
14
Adding
1000
100
10
1
8
+6
14
Same answer, ten-4, or fourteen.
Adding
1000
100
10
1
Do we need
to trade?
If the columns are even or nearly even, trading is much easier.
Bead Trading
1000
100
10
1
97
In this activity, children add numbers to get as high a score as possible.
Turn over the top card. Enter 7 beads.
Bead Trading
1000
100
10
1
96
Turn over another card. Enter 6 beads. Do we need to trade?
Bead Trading
1000
100
10
1
96
Trading 10 ones for 1 ten.
Bead Trading
1000
100
10
1
99
Turn over another card. Enter 9 beads. Do we need to trade?
Bead Trading
1000
100
10
1
99
Trading 10 ones for 1 ten.
Bead Trading
1000
100
10
1
93
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.
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
1000
100
10
1
3658
+
2738
Critically important to write down what happened after each step.
Addition
1000
100
10
1
3658
+
2738
6
. . . 6 ones. Did anything else happen?
Addition
1000
100
10
1
1
3658
+
2738
6
Is it okay to show the extra ten by writing a 1 above the tens column?
Addition
1000
100
10
1
1
3658
+
2738
6
Addition
1000
100
10
1
1
3658
+
2738
6
Do we need to trade? [no]
Addition
1000
100
10
1
1
3658
+
2738
96
Addition
1000
100
10
1
1
3658
+
2738
96
Addition
1000
100
10
1
1
3658
+
2738
96
Do we need to trade? [yes]
Addition
1000
100
10
1
1
3658
+
2738
96
Addition
1000
100
10
1
1
3658
+
2738
96
Notice the number of yellow beads. [3] Notice the number of purple beads left. [3]
Coincidence? No, because 13 – 10 = 3.
Addition
1000
100
10
1
1
3658
+
2738
96
Addition
1000
100
10
1
1
3658
+
2738
396
Addition
1000
100
10
1
1
1
3658
+
2738
396
Addition
1000
100
10
1
1 1
3658
+
2738
396
Addition
1000
100
10
1
1
1
3658
+
2738
396
Addition
1000
100
10
1
1
1
3658
+
2738
6396
Addition
1000
100
10
1
1
1
3658
+
6
2738
396
Addition
1
Most children who
learn to add on the
AL abacus transition
to the paper and
pencil algorithm
without further
instruction.
1
3658
+
6
2738
396
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.
Multiplying on the Abacus
6 x 4 (6 taken 4 times)
Multiplying on the Abacus
5 x 7 (30 + 5)
Groups of 5s to make 10s.
Multiplying on the Abacus
7 x 7 = 25 + 10 + 10 + 4
Multiplying on the Abacus
9 x 3 (30 – 3)
Multiplying on the Abacus
9x3
3x9
Commutative property
Research Highlights
TASK
TEENS
CIRCLE TENS
14 as 10 & 4
EXPER
94%
88%
CTRL
47%
33%
78
3924
75%
44%
67%
7%
48 – 14
81%
33%
10 + 3
6 + 10
Research Highlights
TASK
26-TASK (tens)
6 (ones)
2 (tens)
EXPER
94%
63%
MENTAL COMP: 85 – 70
31%
2nd Graders in U.S. (Reys): 9%
38 + 24 = 512 or
57 + 35 = 812
Other research questions asked.
0%
CTRL
100%
13%
0%
40%
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.
Innovative Practices That Increase
Mathematics Achievement
by Joan A. Cotter, Ph.D.
JoanCotter@ALabacus.com
Cotter Tens Fractal
How many little black
triangles do you see?
Slides/handouts:
ALabacus.com
FCSC
Orlando, FL
November 17, 2009
12:30 - 1:30 p.m.
Cape Canaveral
Volusia