IMF: Visualizing and Montessori Math PART 1

© Joan A. Cotter, Ph.D., 2012

How Visualization Enhances Montessori Mathematics PART 1

Montessori FoundationConference

Friday, Nov 2, 2012Sarasota, Florida

by Joan A. Cotter, [email protected]

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PowerPoint PresentationRightStartMath.com >Resources

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Counting Model

• Number Rods• Spindle Boxes• Decimal materials• Snake Game• Dot Game • Stamp Game• Multiplication Board• Bead Frame

In Montessori, counting is pervasive:


Verbal Counting ModelFrom a child's perspective


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


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

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

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


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.


Subitizing Quantities(Identifying without counting)



• Five-month-old infants can subitize to 3.



• Three-year-olds can subitize to 5.





• Four-year-olds can subitize 6 to 10 by using five as a subbase.





• Four-year-olds can subitize 6 to 10 by using five as a subbase.


• Counting is like sounding out each letter; subitizing is recognizing the quantity.


Research on CountingSubitizing

• 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



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




• Counting-on is a difficult skill for many children. —Journal for Res. in Math Ed. Nov. 2011





• Counting-on is a difficult skill for many children. —Journal for Res. in Math Ed. Nov. 2011



• Math anxiety affects counting ability, but not subitizing ability.


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 QuantitiesReady: How many?


Visualizing QuantitiesTry again: How many?


Visualizing QuantitiesTry again: How many?


Visualizing QuantitiesTry to visualize 8 identical apples without grouping.


Visualizing QuantitiesTry to visualize 8 identical apples without grouping.


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


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?

:


Grouping in Fives


Grouping in Fives

• Grouping in fives extends subitizing.


Grouping in FivesUsing fingers

Grouping in Fives is a three-period lesson.














Grouping in FivesYellow 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 FivesRecognizing 5


Grouping in Fives

5 has a middle; 4 does not.

Recognizing 5


Grouping in FivesTally sticks












Grouping in Fives

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Pairing Finger Cards


Grouping in Fives














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Ordering Finger Cards


Grouping in Fives

















10

5 1

Matching Number Cards to Finger Cards


Grouping in Fives



9 4Matching Finger Cards to Number Cards

1 610

2 83 57
















Grouping in Fives







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Finger Card Memory game


Grouping in FivesNumber Rods






Grouping in FivesSpindle Box





1 2 30 4



6 7 85 9



6 7 85 9



6 7 85 9



6 7 85 9



6 7 85 9


6 7 85 9



Grouping in Fives

1000 100 10 1

1000 100 10 1

100 10 1

100 10 1

100 10 1

100 1

1000 100 10 1

1000 100 10 1

Stamp Game


Grouping in Fives

1000 100 10 1

1000 100 10 1

100 10 1

100 10 1

100 10 1

100 1

1000 100 10 1

1000 100 10 1

Stamp Game


Grouping in Fives

100 10 1100 10 1

100 10 1100 10 1

10 1 1

1000 100 10 11000 100 10 1

1000 100 10 11000 100 10 1

10

10

100 100

100 100

100 100

100 100Stamp Game


Grouping in Fives

100 10 1100 10 1

100 10 1100 10 1

10 1 1

1000 100 10 11000 100 1

1000 100 10 11000 100 10 1

10

10

100 100

100 100

100 100

100 100

10

Stamp Game


Grouping in Fives

100 10 1100 10 1

100 10 1100 10 1

10 1 1

1000 100 10 11000 100 1

1000 100 10 11000 100 10 1

10

10

100 100

100 100

100 100

100 100

10

Stamp Game


Grouping in Fives

“Grouped in fives so the child does not need to count.”

Black and White Bead Stairs

A. M. Joosten


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 = Adding


Grouping in Fives

4 + 3 = Adding


Grouping in Fives

4 + 3 = 7 Adding


Math Card Games


Math Card Games

• Provide repetition for learning the facts.


Math Card Games


• Encourage autonomy.


Math Card Games



• Promote social interaction.


Math Card Games



• Promote social interaction.

• Are enjoyed by the children.


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.


“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



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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]

ChineseU.S.

Average Highest Number

Counted



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ChineseU.S.


Counted



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ChineseU.S.


Counted



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ChineseU.S.


Counted



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ChineseU.S.


Counted


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 we first teach the sound of the letters, we must first teach the name of the quantity (math way).





• Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way).

• Montessorians need to use the math way of naming numbers for a longer period of time.




“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.



4-ten = forty




6-ten = sixty




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 gamefireplace place-fire




paper-newsnewspaper




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 said as “twoo.”



two left twelve

Two said as “twoo.”


Composing Numbers

3-ten


Composing Numbers

3-ten


Composing Numbers

3-ten

3 0


Composing Numbers

3-ten

3 0


Composing Numbers

3-ten

3 0


Composing Numbers

3-ten 7

3 0


Composing Numbers

3-ten 7

3 0


Composing Numbers

3-ten 7

3 07


3 0

Composing Numbers

3-ten 7

7


Composing Numbers

3-ten 7

Note the congruence in how we say the number, represent the number, and write the number.

3 07


Composing Numbers

1-ten

1 0

Another example.


Composing Numbers

1-ten 8

1 0


Composing Numbers

1-ten 8

1 0


Composing Numbers

1-ten 8

1 08


Composing Numbers

1-ten 8

1 88


Composing Numbers

10-ten


Composing Numbers

10-ten

1 0 0


Composing Numbers

10-ten

1 0 0


Composing Numbers

10-ten

1 0 0


Composing Numbers

1 hundred


Composing Numbers

1 hundred

1 0 0


Composing Numbers

1 hundred

1 0 0


Composing Numbers

1 hundred

1 01 01 0 0


Composing Numbers

1 hundred

1 0 0


Composing Numbers

2 hundred


Composing Numbers

2 hundred


Composing Numbers

2 hundred

2 0 0


Evens and Odds


Evens and OddsEvens


Evens and OddsEvens

Use two fingers and touch each pair in succession.


Evens and OddsEvens



Evens and OddsEvens



Evens and OddsEvens


EVEN!


Evens and OddsOdds



Evens and OddsOdds



Evens and OddsOdds



Evens and OddsOdds



Evens and OddsOdds


ODD!


Learning the Facts


Learning the Facts

• Based on counting.

Limited success when:

Whether dots, fingers, number lines, or counting words.


Learning the Facts



• Based on rote memory.


Whether by flash cards or timed tests.


Learning the Facts


• Based on skip counting for multiplication facts.


• Based on rote memory.


Whether by flash cards or timed tests.


Fact Strategies


Fact StrategiesComplete the Ten

9 + 5 =



9 + 5 =



9 + 5 =



9 + 5 =

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



9 + 5 =




9 + 5 =




9 + 5 = 14



Fact StrategiesTwo Fives

8 + 6 =



8 + 6 =



8 + 6 =



8 + 6 =



8 + 6 =10 + 4 = 14


Fact StrategiesGoing Down

15 – 9 =



15 – 9 =



15 – 9 =

Subtract 5;then 4.



15 – 9 =

Subtract 5;then 4.



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 =

Subtract 9 from 10.



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 =




15 – 9 =




15 – 9 =




15 – 9 =1 + 5 = 6



Rows and Columns GameObjective: To find a total of 15 by adding 2, 3, or 4 cards in a row or in a column.


Rows and Columns GameObjective: To find a total of 15 by adding 2, 3, or 4 cards in a row or in a column.

Object of the game: To collect the most cards.


Rows and Columns Game8 7 1 9

6 4 3 3

2 2 5 6

6 3 8 8


Rows and Columns Game

6 3 8 8

8 7 1 9

6 4 3 3

2 2 5 6


Rows and Columns Game8 7 1 9

6 4 3 3

2 2 5 6

6 3 8 8


Rows and Columns Game1 9

6 4 3 3

6 3 8 8



6 4 3 3

6 3 8 8

7 6

2 1 5 1



6 4 3 3

6 3 8 8

7 6

2 1 5 1



6 4 3 3

6 3 8 8

7 6

2 1 5 1


Rows and Columns Game1

6 4 3 3

3 8 8

1 5 1


Rows and Columns Game


MoneyPenny


MoneyNickel


MoneyDime


MoneyQuarter


MoneyQuarter


MoneyQuarter


MoneyQuarter


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.




(Shown by the Decimal Cards.)





Dynamic





Dynamic • 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand, ….





Dynamic • 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand, ….

(Represented on the Abacus and other materials.)


Exchanging

1000 10 1100


ExchangingThousands

1000 10 1100


ExchangingHundreds

1000 10 1100


ExchangingTens

1000 10 1100


ExchangingOnes

1000 10 1100


1000 10 1100

ExchangingAdding

8+ 6


1000 10 1100

ExchangingAdding

8+ 6


1000 10 1100

ExchangingAdding

8+ 6


1000 10 1100

ExchangingAdding

8+ 6


ExchangingAdding

8+ 614

1000 10 1100


ExchangingAdding

8+ 614

Too many ones; trade 10 ones for 1 ten.

1000 10 1100


1000 10 1100

ExchangingAdding

8+ 614



1000 10 1100

ExchangingAdding

8+ 614



1000 10 1100

ExchangingAdding

8+ 614

Same answer before and after exchanging.


Bead Frame

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


Bead Frame

8+ 6

1

10

100

1000


8+ 614

1

10

100

1000

Bead Frame


Bead FrameDifficulties for the child

1

10

100

1000


• Not visualizable: Beads need to be grouped in fives.


1

10

100

1000



• When beads are moved right, inconsistent with equation order: Beads need to be moved left.


1

10

100

1000




• Hierarchies of numbers represented sideways: They need to be in vertical columns.


1

10

100

1000





• Exchanging done before second number is completely added: Addends need to be combined before exchanging.


1

10

100

1000





• Exchanging done before second number is completely added: Addends need to be combined before exchanging.

• Answer is read going up: We read top to bottom.


1

10

100

1000





• Exchanging before second number is completely added: Addends need to be combined before exchanging.

• Answer is read going up: We read top to bottom.

• Distracting: Room is visible through the frame.


1

10

100

1000


1000 10 1100

ExchangingAdding 4-digit numbers

3658+ 2738


1000 10 1100


3658+ 2738

Enter the first number from left to right.


1000 10 1100


3658+ 2738



1000 10 1100


3658+ 2738



1000 10 1100


3658+ 2738



1000 10 1100


3658+ 2738



1000 10 1100


3658+ 2738



1000 10 1100


3658+ 2738

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


1000 10 1100


3658+ 2738



1000 10 1100


3658+ 2738



1000 10 1100


3658+ 2738



1000 10 1100


3658+ 2738

6



1000 10 1100


3658+ 2738

6


1


1000 10 1100


3658+ 2738

6


1


1000 10 1100


3658+ 2738

6


1


1000 10 1100


3658+ 2738

96


1


1000 10 1100


3658+ 2738

96


1


1000 10 1100


3658+ 2738

96


1


1000 10 1100


3658+ 2738

96


1


1000 10 1100


3658+ 2738

96


1


1000 10 1100


3658+ 2738

396


1


1000 10 1100


3658+ 2738

396


1 1


1000 10 1100


3658+ 2738

396


1 1


1000 10 1100


3658+ 2738

396


1 1


1000 10 1100


3658+ 2738

6396


1 1


1000 10 1100


3658+ 2738

6396


1 1


Common Core State Standards

These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B.

Page 5


Common Core State StandardsPage 5

A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time.


Common Core State StandardsPage 5

Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.


How Visualization Enhances Montessori Mathematics PART 1

Montessori FoundationConference

Friday, Nov 2, 2012Sarasota, Florida

by Joan A. Cotter, [email protected]

3 07

3 0

7

1000 10 1100

PowerPoint PresentationRightStartMath.com >Resources

7 37 37 3


Memorizing Math


Memorizing Math

Percentage RecallImmediately After 1 day After 4 wks

Rote 32 23 8 Concept 69 69 58

Some research


Memorizing Math


Rote 32 23 8 Concept 69 69 58

Some research


Memorizing Math


Rote 32 23 8 Concept 69 69 58

Some research


Memorizing Math


Rote 32 23 8 Concept 69 69 58

Some research


Memorizing Math


Rote 32 23 8 Concept 69 69 58

Some research


Memorizing Math


Rote 32 23 8 Concept 69 69 58

Some research


Memorizing Math


Rote 32 23 8 Concept 69 69 58

Some research


Memorizing Math 9 + 7Flash cards


• Are often used to teach rote.




• Liked by those who don’t need them.





• Don’t work for those with learning disabilities.






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







• Often produce stress – children under stress stop learning.




• Liked by those who don’t need them.• Don’t work for those with learning disabilities.


• Often produce stress – children under stress stop learning.

• Are not concrete – they use abstract symbols.

Memorizing MathFlash cards

9 + 7