Innovative Practices That Increase Mathematics Achievement

1
Innovative Practices That Increase Mathematics
Achievement
by Joan A. Cotter, Ph.D.JoanCotter_at_ALabacus.com
Slides/handouts ALabacus.com
Cotter Tens Fractal
FCSC Orlando, FL November 17, 2009 1230 - 130
p.m. Cape Canaveral Volusia
How many little black triangles do you see?
2
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 worlds college grads today its 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.

3
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.

4
Finland

• Teachers from top 10 of undergraduate class.
  Need masters to teach. Held in high esteem.
• Teachers work together on lessons and visit each
  others classrooms. Half day/week for PD.
• Work with students as soon as they fall behind.

5
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.

6
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.

7
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.

8
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.

9
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.

10
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.
11
Memorizing Math
Math needs to be taught so 95 is understood and
only 5 memorized. Richard Skemp
12
Flash Cards

• Often used to teach rote.
• Liked only by are those who dont need them.
• Give the false impression that math isnt about
  thinking.
• Often produce stress children under stress
  stop learning.
• Not concrete use abstract symbols.

13
Rigorous Mathematics

• To develop deep understanding.
• To justify reasoning.
• To connect ideas to prior knowledge.
• To explore concepts.

14
Adding by CountingFrom a Childs Perspective
15
Adding by CountingFrom a Childs Perspective
F E
16
Adding by CountingFrom a Childs Perspective
F E
What is the sum? (It must be a letter.)
17
Adding by CountingFrom a Childs Perspective
F E
K
G
I
J
K
H
A
F
C
D
E
B
18
Adding by CountingFrom a Childs Perspective
Now memorize the facts!!
19
Place ValueFrom a Childs Perspective
L is written AB because it is A J and B As
huh?
20
Place ValueFrom a Childs Perspective
(twelve)
L is written AB because it is A J and B As
(12)
(one 10)
(two 1s).
huh?
21
Subtracting by Counting BackFrom a Childs
Perspective
H E
Try subtracting by taking away
22
Skip CountingFrom a Childs Perspective
Try skip counting by Bs to T B, D, . . . T.
23
Adding on a Number Line
A B C D E F G H I J K L M
D C
Start at D and count C more. Also were counting
spaces, not lines.
24
Calendars
A calendar is NOT a number line day 4 does not
include days 1 to 4.
25
Calendars
September
1
2
3
4
5
6
7
8
9
10
Always show the whole calendar. A child wants to
see the whole before the parts. Children also
need to learn to plan ahead.
26
Calendars
27
Counting Model Drawbacks

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

28
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.
29
5-Month Old Babies CanAdd and Subtract up to 3
Raise screen. Baby seeing 3 wont look long
because it is expected.
30
5-Month Old Babies CanAdd and Subtract up to 3
A baby seeing 1 teddy bear will look much longer,
because its unexpected.
31
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.
32
Ready How Many?
33
Ready How Many?
Which is easier?
34
Visualizing 8
Try to visualize 8 apples without grouping.
35
Visualizing 8
Next try to visualize 5 as red and 3 as green.
36
Grouping by 5s
I II III IIII V VIII
1 2 3 4 5 8
Early Roman numerals
Romans grouped in fives. Notice 8 is 5 and 3.
37
Grouping by 5s

Who could read the music?
Music needs 10 lines, two groups of five.
38
Materials for Visualizing
Japanese Council of Mathematics Education
Japanese criteria.
39
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)
40
Manipulatives
41
Visualizing Needed in

• Mathematics
• Botany
• Geography
• Engineering
• Construction
• Spelling

• Architecture
• Astronomy
• Archeology
• Chemistry
• Physics
• Surgery

42
Manipulatives
A manipulative must not only be visual, but also
visualizable.
Can you visualize this rod?
Most countries stopped using these by early 1990s.
43
Colored Rod Drawbacks

• Young children think each rod is one.

• Adding rods doesnt instantly give the sum
  still need to count or compare.

44
Manipulatives
The 4-rod plus the 2-rod does not give the
immediate answer.
You must count or compare.
45
Colored Rod Drawbacks

• Young children often think each rod is one.

• Adding rods doesnt 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.

46
Quantities With Fingers
Use left hand for 1-5 because we read from left
to right.
47
Quantities With Fingers
48
Quantities With Fingers
49
Quantities With Fingers
Always show 7 as 5 and 2, not for example, as 4
and 3.
50
Quantities With Fingers
51
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.
52
Tally Sticks
Lay the sticks flat on a surface, about 1 inch
(2.5 cm) apart.
53
Tally Sticks
54
Tally Sticks
55
Tally Sticks
Stick is horizontal, because it wont fit
diagonally and young children have problems with
diagonals.
56
Tally Sticks
57
Tally Sticks
Start a new row for every ten.
58
Tally Sticks
What is 4 apples plus 3 more apples?
How would you find the answer without counting?
59
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.
60
AL Abacus
Many types of abacuses. AL abacus shown is
designed to help children learn math.
61
Abacus Cleared
62
3
Entering Quantities
Quantities are entered all at once, not counted.
63
5
Entering Quantities
Relate quantities to hands.
64
7
Entering Quantities
65
10
Entering Quantities
66
Stairs
Stairs. Can use to count 1-10.
67
4 3
Adding
68
4 3
Adding
69
4 3
Adding
7
70
4 3
Adding
7
Mentally, think take 1 from 3 and give to 4,
making 5 2.
71
Typical Worksheet
72
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.
73
Go to the Dump Game
A game viewed from above.
74
Go to the Dump Game
Each player takes 5 cards.
75
Go to the Dump Game
Does YellowCap have any pairs? no
76
Go to the Dump Game
Does BlueCap have any pairs? yes, 1
77
Go to the Dump Game
Does PinkCap have any pairs? yes, 2
78
Go to the Dump Game
Does PinkCap have any pairs? yes, 2
79
Go to the Dump Game
BlueCap, do you have a 3?
BlueCap, do you have an 8?
Go to the dump.
The player asks the player on his left.
80
Go to the Dump Game
PinkCap, do you have a 6?
Go to the dump.
81
Go to the Dump Game
YellowCap, do you have a 9?
82
Go to the Dump Game
PinkCap is not out of the game. Her turn ends,
but she takes 5 more cards.
83
Go to the Dump Game
No counting. Combine both stacks. (Shuffling not
necessary for next game.)
84
Go to the Dump Game
No counting. Combine both stacks. (Shuffling not
necessary for next game.)
85
Go to the Dump Game
Whose pile is the highest?
86
Part-Whole Circles
Whole
Part-whole circles help children see
relationships and solve problems.
87
Part-Whole Circles
10
4
6
What is the other part?
88
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.
89
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?
90
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?
91
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?
92
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?
93
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?
94
Part-Whole Circles
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?
95
Part-Whole Circles
5
Write the equation.
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?
Is this an addition or subtraction problem?
96
Part-Whole Circles
Part-whole circles help young children solve
problems. Writing equations do not.
97
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.
98
Language Effect on Counting
100
Chinese
U.S.
90
Korean formal math way
Korean informal not explicit
80
70
60
50
Average Highest Number Counted
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
children's counting A natural experiment in
numerical bilingualism. International Journal of
Psychology, 23, 319-332.
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.
99
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.

100
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).

101
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.
102
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.
103
3-ten
3
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.
104
3-ten 7
3
0
7
0
105
10-ten
1
0
0
0
Now enter 10-ten.
106
1 hundred
1
0
0
1
0
0
Of course, we can also read it as one-hun-dred.
107
2
0
0
2 hundred
How could you make 200?
108
1
0
0
0
0
0
10 hundred
109
1
0
0
1
0
0
0
0
1 thousand
Point to the digits and say, one-th-ou-sand.
Sorry for the extra syllable in thousand, but
its the best we can do.
110
Place-Value Cards
3- ten
3
0
0
3 hun-dred
3
0
0
0
3 th- ou-sand
111
Place-Value Cards
8
3
0
0
0
3
0
0
0
6
0
0
5
0
8
8
112
Place-Value Cards
3
0
0
0
8
3
0
0
0
3
0
0
0
8
8
No problem when some denominations are missing.
113
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
2
5
8
4
2
5
8
4
114
Traditional Names
4-ten forty
4-ten has another name forty. The ty means
ten.
115
Traditional Names
6-ten sixty
The same is true for 60, 70, 80, and 90.
116
Traditional Names
3-ten thirty
The thir is more common than three, 3rd in
line, 1/3, 13, and 30.
117
Traditional Names
5-ten fifty
The same is true for fif.
118
Traditional Names
2-ten twenty
Twenty is twice ten or twin ten. Note two is
spelled with a w.
119
Traditional Names
A word game
fireplace
place-fire
paper-news
newspaper
box-mail
mailbox
Say the syllables backward. This is how we say
the teen numbers.
120
Traditional Names
ten 4
121
Traditional Names
ten 4
Ten 4 becomes teen 4 (teen ten) and then
fourteen. Similar for other teens.
122
Traditional Names
a one left
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.
123
Traditional Names
two left
Two used to be pronounced (twoo).
124
Money
penny
125
Money
nickel
126
Money
dime
127
Money
quarter
128
9 5
Strategy Complete the Ten
14
Take 1 from the 5 and give it to the 9.
129
8 6
Strategy Two Fives
10 4 14
Two fives make 10. Just add the leftovers.
130
7 5
Strategy Two Fives
10 2 12
Another example.
131
15 9
Strategy Going Down
6
Subtract 5, then 4
Subtract the 9 from the 10. Then add 1 and 5.
132
15 9
Strategy Going Down
6
Subtract 9 from the 10
Subtract the 9 from the 10. Then add 1 and 6.
133
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.
134
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.
135
Mental Addition
24 38
30
24
8
A very efficient way, especially for oral
problems, taught to Dutch children.
136
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
137
Cleared
Side 2
138
Thousands
Side 2
1000
139
Hundreds
Side 2
100
140
Tens
Side 2
10
141
Ones
Side 2
1
The third wire from each end is not used. Red
wires indicate ones.
142
8 6
Adding
143
8 6
Adding
144
8 6
Adding
14
You can see the ten (yellow) and 4 (purple).
145
8 614
Adding
Trading ten ones for one ten. Trade, not rename
or regroup.
146
8 614
Adding
147
8 614
Adding
Same answer, ten-4, or fourteen.
148
Do we need to trade?
Adding
If the columns are even or nearly even, trading
is much easier.
149
Bead Trading
7
In this activity, children add numbers to get as
high a score as possible. Turn over the top card.
Enter 7 beads.
150
Bead Trading
6
Turn over another card. Enter 6 beads. Do we need
to trade?
151
Bead Trading
6
Trading 10 ones for 1 ten.
152
Bead Trading
9
Turn over another card. Enter 9 beads. Do we need
to trade?
153
Bead Trading
9
Trading 10 ones for 1 ten.
154
Bead Trading
3
No trading.
155
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.

156
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Critically important to write down what happened
after each step.
165
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Addition
. . . 6 ones. Did anything else happen?
166
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1
Is it okay to show the extra ten by writing a 1
above the tens column?
167
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1
168
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1
Do we need to trade? no
169
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1
170
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1
171
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1
Do we need to trade? yes
172
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1
173
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1
Notice the number of yellow beads. 3 Notice the
number of purple beads left. 3 Coincidence? No,
because 13 10 3.
174
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1
175
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Addition
1
176
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1 1
177
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1 1
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1 1
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Addition
1 1
180
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1 1
6
181
3658 2738396
Addition
1 1
Most children who learn to add on the AL abacus
transition to the paper and pencil algorithm
without further instruction.
6
182
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.
183
Multiplying on the Abacus
6 x 4 (6 taken 4 times)
184
Multiplying on the Abacus
5 x 7
(30 5)
Groups of 5s to make 10s.
185
Multiplying on the Abacus
7 x 7
25 10 10 4
186
Multiplying on the Abacus
9 x 3 (30 3)
187
Multiplying on the Abacus
9 x 3
3 x 9
Commutative property
188
Research Highlights
TASK
EXPER
CTRL
189
Research Highlights
TASK
EXPER
CTRL
6 (ones)
26-TASK (tens)
94
100
Other research questions asked.
190
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.

191
Innovative Practices That Increase Mathematics
Achievement
by Joan A. Cotter, Ph.D.JoanCotter_at_ALabacus.com
Slides/handouts ALabacus.com
Cotter Tens Fractal
FCSC Orlando, FL November 17, 2009 1230 - 130
p.m. Cape Canaveral Volusia
How many little black triangles do you see?