Does it Matter How We Teach Math

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Does it Matter How We Teach Math?
Presented by Kathleen Cotter Lawler
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Math Crisis
u 25 of college students take remedial math.
u In 2004, of the 1.2 million students who took
the ACT test, only 40 are ready for college
algebra.
u Only 51 of worlds patents awarded to U.S.
citizens, down from 90 a few years ago.
u Bachelor degrees granted in mathematics fell
19 between 1990 and 2000, although enrollment
rose 9.
u A generation ago, the US produced 30 percent
of the worlds college grads today its 14
percent. CSM 2006
u U.S. students, compared to the world, score
high at 4th grade, average at 8th, and near
bottom at 12th.
u Two-thirds of careers require advanced math.
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Math Education is Changing

• The amount of known mathematics has doubled in
  the last 30 years.
• Math is used in many new ways. We need
  analytical thinkers and problem solvers.
• Brain research is providing clues on how to
  better facilitate learning, including math.
• Increased emphasis on mathematical reasoning,
  less on rules and procedures.
• Educators are learning that rote memorizing is
  not effective.

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Adding by CountingFrom a Childs Perspective
Because were so familiar with 1, 2, 3, well use
letters.
A 1 B 2 C 3 D 4 E 5, and so forth
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Adding by CountingFrom a Childs Perspective
F E
Use counters to find the sum, red for F and
yellow for E.
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Adding by CountingFrom a Childs Perspective
F E
K
G
I
J
K
H
A
F
C
D
E
B
Now count all to find the sum.
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Adding by CountingFrom a Childs Perspective
Now Memorize the Facts!!
H F
E I
G D
D C
C G
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Place ValueFrom a Childs Perspective
L is written AB because it is A J and B As
huh?
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Place ValueFrom a Childs Perspective
L is written AB because it is A J and B As
(twelve)
(12) (one 10)
(two 1s).
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Flash CardsFrom a Childs Perspective
2 3

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Flash CardsFrom a Childs Perspective
2 3 5

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Flash CardsFrom a Childs Perspective
2 x 3

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Flash CardsFrom a Childs Perspective
2 x 3 6

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Flash Cards

• Often used to teach rote.
• The only students who like flash cards are those
  who dont need them.
• Flash cards give students the false impression
  that math isnt about thinking.
• Flash cards often produce stresschildren under
  stress stop learning.

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Typical Worksheet
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Counting Model Drawbacks

• Poor concept of quantity.
• Ignores place value.
• Very error prone.
• Inefficient and time-consuming.
• Hard habit to break for the facts.

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How Math is Taught
Workbook Saxon, A-Beka, Horizon Show Tell
MUS Concept RS.
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Memorizing Math
Math needs to be taught so 95 is understood and
only 5 memorized.
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Time Needed to Memorize
According to a study of 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
We must tell our children that they need to
understand math.
This shows the importance of meaning before
memorizing.
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Math is about patterns and relationships.
How many LITTLE white triangles do you see?
Cotter Tens Fractal
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Math is about patterns and relationships, not
just memorizing facts and practicing skills.
Cotter Tens Fractal
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Skip Counting Patterns
Threes
3
6
9
15
12
18
21
24
27
30
Threes have several patterns. First see 0, 1, 2,
3, . . . 9.
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Skip Counting Patterns
Threes
3
6
9
3
6
9
15
5
12
18
2
8
21
24
27
1
4
7
30
0
Threes have several patterns. First see 0, 1, 2,
3, . . . 9.
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Skip Counting Patterns
Threes
3
6
9
12
15
18
21
24
27
30
The tens in each column are 0, 1, 2.
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Skip Counting Patterns
Threes
6
6
3
15
12
24
21
30
Now add the digits in each number in the first
row. 3
The second row. 6
And the third rowthe nines.
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Skip Counting Patterns
Sevens
7
14
21
28
35
42
49
56
63
70
Start in the upper right to see the 1, 2, 3
pattern.
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Fractions
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Fraction Chart
How many fourths in a whole?
How many sixths?
We use ordinal numbers, except for one-half, to
name fractions.
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Fraction Chart
What is more, 1/4 or 1/3?
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Fraction Chart
What is more, 1/4 or 1/3? What is more, 1/9 or
1/10?
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Fraction Chart
Which is more, 3/4 or 4/5?
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Fraction Chart
Which is more, 3/4 or 4/5? Which is more, 7/8 or
8/9?
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Fraction Chart
The pattern of 1/2, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9,
9/10.
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Fraction Chart
How many fourths equal a half?
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Fraction Chart
How many fourths equal a half? Eighths?
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Fraction Chart
How many fourths equal a half? Eighths? Sevenths?
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Reduce rote learning by incorporating your
childs natural ability to visualize.
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What is Visualizing?
To form a mental image or vision to imagine.
Japanese criteria.
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Visualizing Needed in

• Physics
• Chemistry
• Biology
• Engineering
• Surgery
• Construction

• Architecture
• Astronomy
• Geography
• Archeology
• Visual arts
• . . .

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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.
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5-Month Old Babies CanAdd and Subtract up to 3
Raise screen. Baby seeing 3 wont look long
because it is expected.
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5-Month Old Babies CanAdd and Subtract up to 3
Show the two bears, hide behind the screen, show
the third bear and hide.
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5-Month Old Babies CanAdd and Subtract up to 3
A baby seeing 1 teddy bear will look much longer,
because its unexpected.
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Quantities With Fingers
Use left hand for 1-5 because we read from left
to right.
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Quantities With Fingers
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Quantities With Fingers
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Quantities With Fingers
Always show 7 as 5 and 2, not for example, as 4
and 3.
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Quantities With Fingers
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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.
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Counting Model
What we see
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Counting Model
What we see
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Counting Model
3
What we see
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Counting Model
What we see
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Counting Model
What the young child sees
Children think were naming the stick, not the
quantity.
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Counting Model
What the young child sees
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Counting Model
What the young child sees
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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.
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Visualizing 8
Try to visualize 8 apples without grouping.
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Visualizing
Now try with grouping.
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Guided Discovery
To encourage and guide the child to discovery.
Ask questions, encouraging the child to find the
trick or secret pattern.
Japanese criteria.
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Guided Discovery
It is vitally important that children think about
what they are doing and not be satisfied with
memorizing a rule.
Japanese criteria.
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Abacus Cleared
Many types of abacuses. AL abacus shown is
designed to help children learn math.
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3
Entering Quantities
Quantities are entered all at once, not counted.
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5
Entering Quantities
Relate quantities to hands.
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7
Entering Quantities
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10
Entering Quantities
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Stairs
Stairs can be used to count 1-10.
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4 3
Adding
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4 3
Adding
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4 3
Adding
7
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4 3
Adding
7
Mentally, think take 1 from 3 and give to 4,
making 5 2.
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Part-Whole Circle Set
10
4
10 is the whole and 4 is a part.What is the
other part?
These part-whole circle sets help children solve
problems.
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Part-Whole Circle Set
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 circle set.
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Part-Whole Circle Set
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?
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Part-Whole Circle Set
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?
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Part-Whole Circle Set
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?
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Part-Whole Circle Set
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?
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Part-Whole Circle Set
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?
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Part-Whole Circle Set
5
What is the missing part?
3
2
Lee received 3 goldfish as a gift. Now Lee has 5.
How many did Lee have to start with?
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Part-Whole Circle Set
5
Write the equation.
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?
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Part-Whole Circle Set
5
2 3 5 3 2 5 5 3 2
3
2
Lee received 3 goldfish as a gift. Now Lee has 5.
How many did Lee have to start with?
2 3 5 is an equation, not a number sentence.
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Part-Whole Circle Set
Part-whole circles help young children solve
problems. Writing equations do not.
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Games to Learn Facts
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Why Games

Games provide interesting repetition needed for
automaticity.
More importantly, games provide a meaningful use
for math.
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Games and the Family

• Games are a good way for the family to be
  involved, in comparison to flash cards or
  worksheets.
• Different ability levels can play together.

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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.
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Strategies
A strategy is a way to learn a new fact or
recall a forgotten fact. A visual
representation is a powerful strategy.
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9 5
Strategy Complete the Ten
Take 1 from the 5 and give it to the 9.
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9 5
Strategy Complete the Ten
Take 1 from the 5 and give it to the 9.
90
9 5
Strategy Complete the Ten
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Take 1 from the 5 and give it to the 9.
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8 7
Strategy Two Fives
10 5 15
Two fives make 10. Just add the leftovers.
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7 5
Strategy Two Fives
10 2 12
Another example.
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Math Way of Counting
Dont say 2-tens. We dont say 3 hundreds
eleven for 311.
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Math Way of CountingCompared to Reading
Just as reciting the alphabet doesnt teach
reading, counting doesnt teach arithmetic.
Just as we first teach the sound of the letters,
we first teach the name of the quantity (math
way).
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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
Dont say 2-tens. We dont say 3 hundreds
eleven for 311.
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Math Way of Counting
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.
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Place-Value Cards
3- ten
3
0
0
3 hun-dred
3
0
0
0
3 th- ou-sand
98
Place-Value Cards
4
0
0
0
8
99
Place-Value Cards
4
0
0
0
4
0
0
0
2
0
0
5
0
8
8
100
Place-Value Cards
8
4
0
0
0
4
0
0
0
2
0
0
5
0
8
8
101
3-ten
3
0
0
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.
102
3-ten 7
3
0
0
7
103
10-ten
1
0
0
0
Now enter 10-ten.
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1 hundred
1
0
0
1
0
0
Of course, we can also read it as one-hun-dred.
105
15 9
Strategy Going Down
6
Subtract 5, then 4
Subtract the 9 from the 10. Then add 1 and 6.
106
15 9
Strategy Going Down
6
Subtract 9 from the 10
Subtract the 9 from the 10. Then add 1 and 6.
107
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.
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Cleared
Side 2
109
Thousands
Side 2
1000
110
Hundreds
Side 2
100
111
Tens
Side 2
10
112
Ones
Side 2
1
The third wire from each end is not used. Red
wires indicate ones.
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8 6
Adding
114
8 6
Adding

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8 6
Adding
14
You can see the ten (yellow).
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8 614
Adding
Trading ten ones for one ten. Trade, not rename
or regroup.
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8 614
Adding
118
8 614
Adding
Same answer, ten-4, or fourteen.
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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.
Through bead trading the child experiences the
magnitude of place value.
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Critically important to write down what happened
after each step.
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. . . 6 ones. Did anything else happen?
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1
Is it okay to show the extra ten by writing a 1
above the tens column?
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1
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Do we need to trade? no
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1
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1
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Do we need to trade? yes
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Notice the number of yellow beads. 3 Notice the
number of purple beads left. 3 Coincidence? No,
because 13 10 3.
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1 1
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Teaching Math

• u Talk positively about math.
• u Teach understanding use mental models.
• u Encourage your child to ask questions. I
  dont get it is not a question.
• u Use correct vocabulary.
• u Ask a question only once allow 3 seconds for a
  response. Resist giving hints.
• u Give a child 2 seconds to say a fact.

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Math is about patterns and relationships, not
just memorizing facts and practicing skills.
147
Does it Matter How We Teach Math?
Presented by Kathleen Cotter Lawler
148
Multiplying on the Abacus
6 x 4 (6 taken 4 times)
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Multiplying on the Abacus
5 x 7
(30 5)
Groups of 5s to make 10s.
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Multiplying on the Abacus
9 x 8 80 8
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Multiplying on the Abacus
7 x 7
25 10 10 4
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Multiplying on the Abacus
9 x 7
7 x 9
Commutative property
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Summary
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Dr. Joan A. Cotter
Math needs to be taught so 95 percent is
understood and only 5 percent memorized.