mctm future primary math

© Joan A. Cotter, Ph.D., 2012

The Future of Primary Math: More Understanding/Less Counting

MCTMSaturday, May 5, 2012

Duluth, Minnesota

by Joan A. Cotter, [email protected]

3 03 077

3 03 0

77

1000 10 1100

PowerPoint PresentationRightStartMath.com >Resources


Verbal Counting Model


Verbal Counting ModelFrom 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


Verbal Counting Model From a child's perspective

F + E



A

F + E



A B

F + E



A CB

F + E



A FC D EB

F + E



AA FC D EB

F + E



A BA FC D EB

F + E



A C D EBA FC D EB

F + E



A C D EBA FC D EB

F + E

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



K

G I J KHA FC D EB

F + E



Now memorize the facts!!

G + D




G + D

H + F




G + D

H + F

D + C




G + D

H + F

C + G

D + C



E

+ I


G + D

H + F

C + G

D + C



H – E

Subtract with your fingers by counting backward.



J – F

Subtract without using your fingers.



Try skip counting by B’s to T: B, D, . . . T.



Try skip counting by B’s to T: B, D, . . . T.

What is D E?



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




huh?




(twelve)




(12)(twelve)




(12)(one 10)

(twelve)




(12)(one 10)

(two 1s).

(twelve)


Calendar Math

August

29

22

15

8

1

30

23

16

9

2

24

17

10

3

25

18

11

4

26

19

12

5

27

20

13

6

28

21

14

7

31


Calendar Math

August

29

22

15

8

1

30

23

16

9

2

24

17

10

3

25

18

11

4

26

19

12

5

27

20

13

6

28

21

14

7

31

Calendar Counting


Calendar Math

September123489101115161718222324252930

567121314192021262728

August

29

22

15

8

1

30

23

16

9

2

24

17

10

3

25

18

11

4

26

19

12

5

27

20

13

6

28

21

14

7

31

Calendar Counting


Calendar Math

September123489101115161718222324252930

567121314192021262728

August

29

22

15

8

1

30

23

16

9

2

24

17

10

3

25

18

11

4

26

19

12

5

27

20

13

6

28

21

14

7

31

This is ordinal counting, not cardinal counting.

Calendar Counting


Calendar Math

August

8

1

9

2

10

3 4 5 6 7

Partial Calendar


Calendar Math

August

8

1

9

2

10

3 4 5 6 7

Partial Calendar

Children need the whole month to plan ahead.


Calendar Math

September123489101115161718222324252930

567121314192021262728

August

29

22

15

8

1

30

23

16

9

2

24

17

10

3

25

18

11

4

26

19

12

5

27

20

13

6

28

21

14

7

31

Patterns are rarely based on 7s or proceed row by row.Patterns go on forever; they don’t stop at 31.

Calendar Patterning


Minnesota Standards

K: Represent quantities using whole numbers and understand relationships among whole numbers.

1–2: Understand place value and relationships among whole numbers.

Number Sense

With the counting model, how difficult are the associated benchmarks for children to master?


Minnesota Standards

Represent quantities using whole numbers and understand relationships among whole numbers.

Kindergarten


Minnesota Standards


• Count forward to 31, backward from 10.

Kindergarten


Minnesota Standards



• Count number of objects and identify the quantity.

Kindergarten


Minnesota Standards




• Compare the number of objects in two or more sets.

Kindergarten


Minnesota Standards





• Given a number, identify one more or one less.

Kindergarten


Minnesota Standards






• Recognize number of objects up to 6, without counting.

Kindergarten


Minnesota Standards







• Add and subtract whole numbers up to 6, using objects.

Kindergarten


Minnesota Standards

Understand place value and relationships among whole numbers.

Grade 1


Minnesota Standards


• Read, write, compare and order numbers to 120.

Grade 1


Minnesota Standards



• Count by 2s to 30 and by 5s to 120.

Grade 1


Minnesota Standards




• Count backwards from 30.

Grade 1


Minnesota Standards





• Demonstrate understanding of odd and even to 12.

Grade 1


Minnesota Standards






• Represent whole numbers up to 20 in various ways.

Grade 1


Minnesota Standards


Grade 2


Minnesota Standards



Grade 2


Minnesota Standards



• Count by 2s, 5s, 10s from any given whole number.

Grade 2


Minnesota Standards




• Understand the significance of groups of ten.

Grade 2


Minnesota Standards





• Demonstrate understanding of odd and even up to 12.

Grade 2


Minnesota Standards







Grade 2


Research on CountingKaren Wynn’s research


Research on Counting

Karen Wynn’s research


Research on CountingOther research



• Australian Aboriginal children from two tribes.Brian Butterworth, University College London, 2008.

Other research




• Adult Pirahã from Amazon region.Edward Gibson and Michael Frank, MIT, 2008.

Other research





• Adults, ages 18-50, from Boston.Edward Gibson and Michael Frank, MIT, 2008.

Other research





• Adults, ages 18-50, from Boston.Edward Gibson and Michael Frank, MIT, 2008.

• Baby chicks from Italy.Lucia Regolin, University of Padova, 2009.

Other research


Research on CountingIn Japanese schools:

• Children are discouraged from using counting for adding.


Research on CountingIn Japanese schools:

• Children are discouraged from using counting for adding.

• They consistently group in 5s.


Research on CountingSubitizing

• Subitizing is quick recognition of quantity without counting.




• Human babies and some animals can subitize small quantities at birth.





• Children who can subitize perform better in mathematics long term.—Butterworth






• Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit






• Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit

• Subitizing seems to be a necessary skill for understanding what the counting process means.—Glasersfeld


Visualizing Quantities



“Think in pictures, because the

brain remembers images better

than it does anything else.”

Ben Pridmore, World Memory Champion, 2009



“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


• Representative of structure of numbers.• Easily manipulated by children.• Imaginable mentally.

Visualizing QuantitiesJapanese criteria for manipulatives

Japanese Council ofMathematics Education



• Reading

• Sports

• Creativity

• Geography

• Engineering

• Construction

Visualizing also needed in:



• Reading

• Sports

• Creativity

• Geography

• Engineering

• Construction

• Architecture

• Astronomy

• Archeology

• Chemistry

• Physics

• Surgery

Visualizing also needed in:


Visualizing QuantitiesReady: How many?


Visualizing QuantitiesTry again: How many?


Visualizing QuantitiesTry to visualize 8 identical apples without grouping.


Visualizing QuantitiesNow try to visualize 5 as red and 3 as green.



I II III IIII V VIII

1 23458

Early Roman numerals



Who could read the music?

:


Subitizing (groups of five)

Math Way (of number naming)

Place Value Cards

Trading (with 4-digit numbers)

AN ALTERNATIVE to learning place value:


Grouping in FivesUsing fingers


Grouping in Fives

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

Yellow is the Sun


Grouping in FivesRecognizing 5


Grouping in Fives

5 has a middle; 4 does not.

Recognizing 5


Grouping in FivesTally sticks


Grouping in FivesEntering quantities


3



5



7



Grouping in Fives

10

Entering quantities


Grouping in FivesThe stairs


Grouping in FivesAdding


Grouping in FivesAdding

4 + 3 =


Grouping in Fives

4 + 3 = Adding


Grouping in Fives

4 + 3 = 7 Adding


Grouping in Fives

4 + 3 = Adding


Go to the Dump GameObjective: To learn the facts that total 10:

1 + 92 + 83 + 74 + 65 + 5


Go to the Dump GameObjective: 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.


Go to the Dump Game

6 + = 10


“Math” Way of Naming Numbers



11 = ten 1



11 = ten 112 = ten 2



11 = ten 112 = ten 213 = ten 3



11 = ten 112 = ten 213 = ten 314 = ten 4



11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9




20 = 2-ten




20 = 2-ten 21 = 2-ten 1




20 = 2-ten 21 = 2-ten 122 = 2-ten 2




20 = 2-ten 21 = 2-ten 122 = 2-ten 223 = 2-ten 3




20 = 2-ten 21 = 2-ten 122 = 2-ten 223 = 2-ten 3 . . . . . . . .99 = 9-ten 9



137 = 1 hundred 3-ten 7



137 = 1 hundred 3-ten 7or

137 = 1 hundred and 3-ten 7



0

10

20

30

40

50

60

70

80

90

100

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

U.S.

Ave

rage

Hig

hest

Num

ber

Cou

nted



0

10

20

30

40

50

60

70

80

90

100

4 5 6Ages (yrs.)




Chinese

U.S.

Ave

rage

Hig

hest

Num

ber

Cou

nted


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.




• 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 Naming NumbersCompared to reading:


Math Way of Naming Numbers

• Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic.

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 must first teach the name of the quantity (math way).

Compared to reading:



“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, 2005Researchers


Math Way of Naming NumbersTraditional names

4-ten = forty

The “ty” means tens.



6-ten = sixty

The “ty” means tens.



3-ten = thirty

“Thir” also used in 1/3, 13 and 30.



5-ten = fifty

“Fif” also used in 1/5, 15 and 50.



2-ten = twenty

Two used to be pronounced “twoo.”



A word game

fireplace place-fire



A word game


paper-newsnewspaper



A word game


paper-news

box-mail mailbox

newspaper



ten 4

“Teen” also means ten.



ten 4 teen 4




ten 4 teen 4 fourteen




a one left



a one left a left-one



a one left a left-one eleven



two left

Two pronounced “twoo.”



two left twelve

Two pronounced “twoo.”


Composing Numbers

3-ten


Composing Numbers

3-ten

3 03 0


Composing Numbers

3-ten 7

3 03 0


Composing Numbers

3-ten 7

3 03 0

77


3 03 0

Composing Numbers

3-ten 7

77


Composing Numbers

3-ten 7

Notice the way we say the number, represent the number, and write the number all correspond.

3 03 077


Composing Numbers

7-ten

7 07 0

Another example.


Composing Numbers

7-ten 8

7 07 0


Composing Numbers

7-ten 8

7 07 0

88


Composing Numbers

7-ten 8

7 87 888


Composing Numbers

10-ten


Composing Numbers

10-ten

1 0 01 0 0


Composing Numbers

1 hundred


Composing Numbers

1 hundred

1 0 01 0 0


Composing Numbers

1 hundred

11 001 01 0 01 0 0


Composing Numbers

1 hundred

1 0 01 0 0


Composing Numbers

2 hundred


Composing Numbers

2 hundred

2 0 02 0 0


Counting by 2s and 5s


Counting by 2s and 5sCounting by 2s



2



2 4



2 4 6



2 4 6 8



2 4 6 8 10



2 4 6 8 10

12



2 4 6 8 10

12 14



2 4 6 8 10

12 14 16



2 4 6 8 10

12 14 16 18



2 4 6 8 10

12 14 16 18 20



5

Counting by 5s



5 10

Counting by 5s



5 10

15

Counting by 5s



5 10

15 20

Counting by 5s



5 10

15 20

25

Counting by 5s



5 10

15 20

25 30

Counting by 5s


Evens and OddsEvens


Evens and OddsEvens

Use two fingers and touch each pair in succession.


Evens and OddsEvens


EVEN!


Evens and OddsOdds



Evens and OddsOdds


ODD!


Fact Strategies


Fact StrategiesComplete the Ten

9 + 5 =



9 + 5 =

Take 1 from the 5 and give it to the 9.



9 + 5 = 14

Take 1 from the 5 and give it to the 9.


Fact StrategiesTwo Fives

8 + 6 =


Fact StrategiesTwo Fives

8 + 6 =

10 + 4 = 14


Fact StrategiesGoing Down

15 – 9 =



15 – 9 =

Subtract 5;then 4.



15 – 9 = 6

Subtract 5;then 4.


Fact StrategiesSubtract from 10

15 – 9 =



15 – 9 =

Subtract 9 from 10.



15 – 9 = 6

Subtract 9 from 10.


Fact StrategiesGoing Up

15 – 9 =



15 – 9 =

Start with 9; go up to 15.



15 – 9 =

1 + 5 = 6

Start with 9; go up to 15.


Fact Strategies

6 4 =(6 taken 4 times)

Multiplication


Place ValueTwo aspects



Static



Static • Value of a digit is determined by position



Static • Value of a digit is determined by position.• No position may have more than nine.



Static • Value of a digit is determined by position.• No position may have more than nine.• As you progress to the left, value at each position is ten times greater than previous position.



Static • Value of a digit is determined by position.• No position may have more than nine.• As you progress to the left, value at each position is ten times greater than previous position.• Place value cards show this aspect.




Dynamic




Dynamic • Ten ones = 1 ten; ten tens = 1 hundred; ten hundreds = 1 thousand, ….


Trading

1000 10 1100


TradingThousands

1000 10 1100


TradingHundreds

1000 10 1100


TradingTens

1000 10 1100


TradingOnes

1000 10 1100


1000 10 1100

TradingAdding

8+ 6


TradingAdding

8+ 614

1000 10 1100


TradingAdding

8+ 614

Too many ones; trade 10 ones for 1 ten.

1000 10 1100


1000 10 1100

TradingAdding

8+ 614

Too many ones; trade 10 ones for 1 ten.


1000 10 1100

TradingAdding

8+ 614

Same answer before and after trading.


TradingBead Trading Activity

1000 10 1100



Object: To get a high score by adding numbers on the green cards.

1000 10 1100




71000 10 1100


1000 10 1100



7


1000 10 1100


6


1000 10 1100


Trade 10 ones for 1 ten.

6


1000 10 1100


6


1000 10 1100


9


1000 10 1100


Another trade.

9


1000 10 1100


3


Trading

• In the Bead Trading activity trading10 ones for 1 ten occurs frequently;

Bead Trading Activity



• In the Bead Trading activity trading10 ones for 1 ten occurs frequently;10 tens for 1 hundred, less often;



• In the Bead Trading activity trading10 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 from left to right.


Trading


• Bead trading helps the child experience the greater value of each column from left to right.

• To detect a pattern, there must be at least three examples in the sequence. Place value is a pattern.

Bead Trading Activity


1000 10 1100

TradingAdding 4-digit numbers

3658+ 2738


1000 10 1100


3658+ 2738

Enter the first number from left to right.


1000 10 1100


3658+ 2738

Add starting at the right. Write results after each step.


1000 10 1100


3658+ 2738

6



1000 10 1100


3658+ 2738

6


1


1000 10 1100


3658+ 2738

96


1


1000 10 1100


3658+ 2738

396


1


1000 10 1100


3658+ 2738

396


11


1000 10 1100


3658+ 2738

6396


11


Minnesota Standards

K: Represent quantities using whole numbers and understand relationships among whole numbers.

1–2: Understand place value and relationships among whole numbers.

Number Sense

With this alternate model, how difficult are the associated benchmarks for children to master?


Minnesota Standards



Kindergarten


Minnesota Standards




Kindergarten


Minnesota Standards





Kindergarten


Minnesota Standards






Kindergarten


Minnesota Standards







Kindergarten


Minnesota Standards







• Add and subtract whole numbers up to 6, using objects.

Kindergarten


Minnesota Standards



Grade 1


Minnesota Standards




Grade 1


Minnesota Standards





Grade 1


Minnesota Standards






Grade 1


Minnesota Standards







Grade 1


Minnesota Standards



Grade 2


Minnesota Standards




Grade 2


Minnesota Standards





Grade 2


Minnesota Standards






Grade 2


Minnesota Standards







Grade 2


Research Highlights


Research Highlights

Using 10s and 1s, ask the child to construct 48.

Research task:


Research Highlights


Research task:


Research Highlights


Research task:

Then ask the child to subtract 14.


Research Highlights


Research task:


Children thinking of 14 as 14 ones counted 14.


Research Highlights


Research task:




Research Highlights


Research task:


Children who understand tens remove a ten and 4 ones.


Research Highlights


Research task:




Research HighlightsTASK EXPER CTRL

14 as 10 & 4 48 – 14 81% 33%



Teens 10 + 3 94% 47%14 as 10 & 4 48 – 14 81% 33%



Teens 10 + 3 94% 47%6 + 10 88% 33%

14 as 10 & 4 48 – 14 81% 33%



Teens 10 + 3 94% 47%6 + 10 88% 33%

Circle TensPlace

78 75% 67%

14 as 10 & 4 48 – 14 81% 33%



Teens 10 + 3 94% 47%6 + 10 88% 33%

Circle TensPlace

78 75% 67%

14 as 10 & 4 48 – 14 81% 33%

3924 44% 7%



Teens 10 + 3 94% 47%6 + 10 88% 33%

Circle TensPlace

78 75% 67%

3924 44% 7%

14 as 10 & 4 48 – 14 81% 33%

Mental Computation

85 – 70 31% 0%



Teens 10 + 3 94% 47%6 + 10 88% 33%

Circle TensPlace

78 75% 67%

3924 44% 7%

14 as 10 & 4 48 – 14 81% 33%

Mental Computation

85 – 70 31% 0% 2nd Graders in US (Reys): 9%



Teens 10 + 3 94% 47%6 + 10 88% 33%

Circle TensPlace

78 75% 67%

3924 44% 7%

14 as 10 & 4 48 – 14 81% 33%

Mental Computation

85 – 70 31% 0% 2nd Graders in US (Reys): 9%

38 + 24 = 512 or 0% 40%

57 + 35 = 812


Framing the Future of Mathematics in Minnesota

Place value, not counting, is the key to under-standing numbers.

Place value is best taught by:

• Subitizing (with groups of fives),• Initially using Math Way of number naming,• Incorporating Place Value Cards,• Patterning (trading with 4-digit numbers).


The Future of Primary Math: More Understanding/Less Counting

MCTMSaturday, May 5, 2012

Duluth, Minnesota

by Joan A. Cotter, [email protected]

3 03 077

3 03 0

77

1000 10 1100

PowerPoint PresentationRightStartMath.com >Resources

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