NCSM April 2013

Counting: Necessary or Detrimental?

Teaching Primary Mathematics with More Understanding and Less CountingNational Council of Supervisors of MathematicsMonday, April 16, 2013Denver, ColoradoJoan A. Cotter, [email protected] Mittleider, [email protected]

1

Joan A. Cotter, Ph.D., 20132ObjectivesReview the traditional counting trajectory.

Joan A. Cotter, Ph.D., 2013In her work on strategies for learning the number combinations, Steinberg (1985) states that it appears that the counting-on procedure is not a necessary one for the learning of strategies. She also noted that the use of strategies was accompanied by a decrease in counting.

2

3ObjectivesReview the traditional counting trajectory.Experience traditional counting like a child.


3

4ObjectivesReview the traditional counting trajectory.Experience traditional counting like a child.Group in 5s and 10s: an alternative to counting.


4

5ObjectivesReview the traditional counting trajectory.Experience traditional counting like a child.Group in 5s and 10s: an alternative to counting.Meet CCSS without counting.


5

6Traditional Counting ModelMemorizing counting sequence.

Joan A. Cotter, Ph.D., 20137Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.

Joan A. Cotter, Ph.D., 20138Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal.

Joan A. Cotter, Ph.D., 20139Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.

Joan A. Cotter, Ph.D., 201310Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on.

Joan A. Cotter, Ph.D., 201311Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on.Adding by counting from the larger number.

Joan A. Cotter, Ph.D., 201312Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on.Adding by counting from larger number.Subtracting by counting backward.

Joan A. Cotter, Ph.D., 201313Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on.Adding by counting from larger number.Subtracting by counting backward.Multiplying by skip counting.

Joan A. Cotter, Ph.D., 201314Traditional Counting ModelMemorizing counting sequence. String level Unbreakable list Breakable chain Numerable chain Bidirectional chain

Joan A. Cotter, Ph.D., 2013Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence. Requires stable order for counting words Common errors: double counting and missed count15

Joan A. Cotter, Ph.D., 201316Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Unlike anything else in childs experience (e.g. in naming family, baby all others).

Joan A. Cotter, Ph.D., 201317Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Unlike anything else in childs experience (e.g. in naming family, baby all others). How many not a good test; take n is better.

Joan A. Cotter, Ph.D., 201318Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all. Focuses more on counting than adding.

Joan A. Cotter, Ph.D., 201319Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on. Leads to counting words.

Joan A. Cotter, Ph.D., 201320Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on. Leads to counting words. No need to learn strategies.

Joan A. Cotter, Ph.D., 201321Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on. Leads to counting words. No need to learn strategies. Very difficult. (article in Nov. 2011, JRME)

Joan A. Cotter, Ph.D., 201322Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on.Adding by counting from larger number. First need to determine larger number.

Joan A. Cotter, Ph.D., 201323Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on.Adding by counting from the larger number.Subtracting by counting backward. Extremely difficult. (Easier to go forward.)

Joan A. Cotter, Ph.D., 201324Traditional Counting ModelMemorizing counting sequence.One-to-one correspondence.Cardinality principal. Adding by counting all.Adding by counting on.Adding by counting from larger number.Subtracting by counting backward.Multiplying by skip counting. Tedious for finding multiplication facts.

Joan A. Cotter, Ph.D., 201325Traditional CountingFrom a child's perspective

Joan A. Cotter, Ph.D., 201326Traditional CountingFrom a child's perspectiveBecause we're so familiar with 1, 2, 3, well use letters.A = 1B = 2C = 3D = 4E = 5, and so forth

Joan A. Cotter, Ph.D., 201327Traditional Counting From a child's perspective

F + E =

Joan A. Cotter, Ph.D., 201327

28Traditional Counting From a child's perspective

AF + E =



A

BF + E =



A

C

BF + E =



A

F

C

D

E

BF + E =



A

A

F

C

D

E

BF + E =



A

B

A

F

C

D

E

BF + E =



A

C

D

E

B

A

F

C

D

E

BF + E =



A

C

D

E

B

A

F

C

D

E

BWhat is the sum?(It must be a letter.)F + E =


GIJKHAFCDEBF + E = K

Joan A. Cotter, Ph.D., 201337Traditional Counting From a child's perspectiveE + D =Find the sum without counters.

Joan A. Cotter, Ph.D., 201338Traditional Counting From a child's perspectiveG + E =Find the sum without fingers.

Joan A. Cotter, Ph.D., 201339Traditional Counting From a child's perspectiveNow memorize the facts!!

G + D


G + D

H + F


G + D

H + F

D + C


G + D

H + F

C + G

D + C


E + INow memorize the facts!!

G + D

H + F

C + G

D + C

Joan A. Cotter, Ph.D., 201344Traditional Counting From a child's perspectiveSubtract counting backward by using your fingers.H C =

Joan A. Cotter, Ph.D., 201345Traditional Counting From a child's perspectiveSubtract by counting backward without fingers.J F =

Joan A. Cotter, Ph.D., 201346Traditional Counting From a child's perspectiveTry skip counting by B's to T: B, D, . . . , T.

Joan A. Cotter, Ph.D., 201347Traditional Counting From a child's perspectiveTry skip counting by B's to T: B, D, . . . , T.What is D x E?

Joan A. Cotter, Ph.D., 201348Traditional Counting Special cases of place value (1.NBT.2)Lis a bundle of J Asand B A's.

Joan A. Cotter, Ph.D., 201349Traditional Counting Special cases of place value (1.NBT.2)Lis a bundle of J Asand B A's. huh?

Joan A. Cotter, Ph.D., 201350Traditional Counting Special cases of place value (1.NBT.2)Lis a bundle of J Asand B A's. (12)

Joan A. Cotter, Ph.D., 201351Traditional Counting Special cases of place value (1.NBT.2)Lis a bundle of J Asand B A's. (ten ones)(12)

Joan A. Cotter, Ph.D., 201352Traditional Counting Special cases of place value (1.NBT.2)Lis a bundle of J Asand B A's. (ten ones)(two ones)(12)

Joan A. Cotter, Ph.D., 2013Grouping in Fives


Chinese abacus

Joan A. Cotter, Ph.D., 2013Grouping in Fives I II III IIII V VIII 1 23458

Early Roman numerals


Grouping in Fives

Musical staff

Joan A. Cotter, Ph.D., 2013Clocks and nickelsGrouping in Fives


Clocks and nickels

Joan A. Cotter, Ph.D., 2013Grouping in FivesTally marks

Joan A. Cotter, Ph.D., 2013Grouping in FivesSubitizing Instant recognition of quantity is called subitizing.

Joan A. Cotter, Ph.D., 2013Grouping in FivesSubitizing Instant recognition of quantity is called subitizing. Grouping in fives extends subitizing beyond five.

Joan A. Cotter, Ph.D., 2013SubitizingFive-month-old infants can subitize to 13.

Joan A. Cotter, Ph.D., 2013SubitizingThree-year-olds can subitize to 15.Five-month-old infants can subitize to 13.

Joan A. Cotter, Ph.D., 2013SubitizingThree-year-olds can subitize to 15.Four-year-olds can subitize 110 by grouping with five.Five-month-old infants can subitize to 13.

Joan A. Cotter, Ph.D., 2013SubitizingThree-year-olds can subitize to 15.Four-year-olds can subitize 110 by grouping with five.Five-month-old infants can subitize to 13.Counting is analogous to sounding out a word; subitizing, recognizing the word.

Joan A. Cotter, Ph.D., 201366Research on Subitizing

Joan A. Cotter, Ph.D., 2013Research on SubitizingKaren Wynn's research

Joan A. Cotter, Ph.D., 201367Show the baby 2 bears.

Research on Subitizing

Karen Wynn's research


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70




71













Other research


Subitizing allows the child to grasp the whole and the elements at the same time.BenoitOther research


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 Other research


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 seems to be a necessary skill for understanding what the counting process means. Glasersfeld Other research

Joan A. Cotter, Ph.D., 2013Other researchResearch on SubitizingAustralian Aboriginal children from two tribes.Brian Butterworth, University College London, 2008.79

Joan A. Cotter, Ph.D., 2013Other researchResearch on SubitizingAustralian Aboriginal children from two tribes.Brian Butterworth, University College London, 2008.Adult Pirah from Amazon region.Edward Gibson and Michael Frank, MIT, 2008.

80


Adults, ages 18-50, from Boston.Edward Gibson and Michael Frank, MIT, 2008.81


Adults, ages 18-50, from Boston.Edward Gibson and Michael Frank, MIT, 2008.Baby chicks from Italy.Lucia Regolin, University of Padova, 2009.82

Joan A. Cotter, Ph.D., 2013Research on SubitizingIn Japanese schoolsChildren are discouraged from using counting for adding.83

Joan A. Cotter, Ph.D., 2013Research on SubitizingIn Japanese schoolsChildren are discouraged from using counting for adding.They consistently group in 5s.84

Joan A. Cotter, Ph.D., 2013Research on SubitizingFinger gnosiaFinger gnosia is the ability to know which fingers can been lightly touched without looking.85

Joan A. Cotter, Ph.D., 2013Research on SubitizingFinger gnosiaFinger gnosia is the ability to know which fingers can been lightly touched without looking.Part of the brain controlling fingers is adjacent to math part of the brain.86

Joan A. Cotter, Ph.D., 2013Research on SubitizingFinger gnosiaFinger gnosia is the ability to know which fingers can been lightly touched without looking.Part of the brain controlling fingers is adjacent to math part of the brain.Children who use their fingers as representational tools perform better in mathematics.Butterworth87

Joan A. Cotter, Ph.D., 2013Research on SubitizingFinger gnosiaFinger gnosia is the ability to know which fingers can been lightly touched without looking.Part of the brain controlling fingers is adjacent to math part of the brain.Children who use their fingers as representational tools perform better in mathematics.Butterworth88Children learn subitizing up to 5 before counting.Starkey & Cooper

Joan A. Cotter, Ph.D., 2013Learning 110Using fingers

Joan A. Cotter, Ph.D., 2013Learning 110

Using fingers


Using fingers


Using fingers


Using fingers


Using fingers


Subitizing 5


Subitizing 5

Joan A. Cotter, Ph.D., 2013Learning 1105 has a middle; 4 does not.

Subitizing 5


Learning 110Tally sticks


Learning 110Tally sticks


Learning 110

Tally sticks

Joan A. Cotter, Ph.D., 2013101Learning 110Tally sticks

Five as a group.


Tally sticks


Tally sticks


Learning 110

Entering quantities


3Learning 110

Entering quantities


5Learning 110

Entering quantities


7Learning 110

Entering quantities


Learning 11010

Entering quantities


Learning 110

The stairs

Joan A. Cotter, Ph.D., 2013109Stairs

Learning 110

Adding


Learning 110

4 + 3 =

Adding


Learning 110

4 + 3 =

Adding


Learning 110

4 + 3 =

Adding


Learning 110

4 + 3 =

Adding


Learning 110

4 + 3 = 7

Adding


Learning 110

4 + 3 = 7

VisualizingJapanese children learn to do this mentally.

Joan A. Cotter, Ph.D., 2013117VisualizingVisual is related to seeing.Visualize is to form a mental image.

Joan A. Cotter, Ph.D., 2013118Visualizing

Think in pictures, because the brain remembers images better than it does anything else.

Ben Pridmore, World Memory Champion, 2009

Joan A. Cotter, Ph.D., 2013119VisualizingThe 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

Joan A. Cotter, Ph.D., 2013VisualizingJapanese criteria for manipulatives

Joan A. Cotter, Ph.D., 2013 Representative of structure of numbers.VisualizingJapanese criteria for manipulatives

Joan A. Cotter, Ph.D., 2013 Representative of structure of numbers. Easily manipulated by children.VisualizingJapanese criteria for manipulatives

Joan A. Cotter, Ph.D., 2013 Representative of structure of numbers. Easily manipulated by children. Imaginable mentally.VisualizingJapanese criteria for manipulativesJapanese Council ofMathematics Education

Joan A. Cotter, Ph.D., 2013Visualizing Reading Sports Creativity Geography Engineering ConstructionNecessary in:

Joan A. Cotter, Ph.D., 2013Visualizing Reading Sports Creativity Geography Engineering Construction Architecture Astronomy Archeology Chemistry Physics SurgeryNecessary in:

Joan A. Cotter, Ph.D., 2013VisualizingTry to visualize 8 identical apples without grouping.

Joan A. Cotter, Ph.D., 2013Visualizing

Try to visualize 8 identical apples without grouping.

Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green.

Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green.

Joan A. Cotter, Ph.D., 2013Learning 110Partitioning


Learning 110

5 = +

Partitioning


Learning 110

5 = 4 + 1

Partitioning


Learning 110

5 = 3 + 2

Partitioning


Learning 110

5 = 2 + 3

Partitioning


Learning 110

5 = 1 + 4

Partitioning


Learning 110

5 = 5 + 0

Partitioning


Learning 110

5 = 0 + 5

Partitioning

Joan A. Cotter, Ph.D., 2013Learning 110Place valuePlace value is the foundation of modern arithmetic.

Joan A. Cotter, Ph.D., 2013Learning 110Place valuePlace value is the foundation of modern arithmetic.Critical for understanding algorithms.

Joan A. Cotter, Ph.D., 2013Learning 110Place valuePlace value is the foundation of modern arithmetic.Critical for understanding algorithms.Must be taught, not left for discovery.

Joan A. Cotter, Ph.D., 2013Learning 110Place valuePlace value is the foundation of modern arithmetic.Critical for understanding algorithms.Children need the big picture, not tiny snapshots.Must be taught, not left for discovery.

Joan A. Cotter, Ph.D., 2013Place ValueCCSS (K.NBT.1, 1.NBT.2)Does it make sense that students should:Work with numbers 1119 to gain foundations for place value. (They are the most difficult numbers we have in English.)

Joan A. Cotter, Ph.D., 2013Place ValueCCSS (K.NBT.1, 1.NBT.2)Does it make sense that students should:Work with numbers 1119 to gain foundations for place value. (They are the most difficult numbers we have in English.)

Are these really special cases?10 can be thought of as a bundle of ten ones called a ten.

100 can be thought of as a bundle of ten tens called a hundred.

Joan A. Cotter, Ph.D., 2013Place ValueTwo aspects 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 place-value cards.) Dynamic (Trading) 10 ones = 1 ten; 10 tens = 1 hundred; 10 hundreds = 1 thousand, . (Represented on the abacus and other materials.)

Joan A. Cotter, Ph.D., 2013Place ValueAsian number-naming(Math way of number naming) Asian children do not struggle with the teens.

Joan A. Cotter, Ph.D., 2013Place ValueAsian number-naming(Math way of number naming) Their languages are completely ten-based.Asian children do not struggle with the teens.

Joan A. Cotter, Ph.D., 2013Place ValueAsian number-naming(Math way of number naming) Their languages are completely ten-based.Asian children do not struggle with the teens.Asian countries use the ten-based metric system.

Joan A. Cotter, Ph.D., 2013148Math Way of Number Naming

Joan A. Cotter, Ph.D., 2013149Math Way of Number Naming11 = ten 1

Joan A. Cotter, Ph.D., 2013150Math Way of Number Naming11 = ten 112 = ten 2

Joan A. Cotter, Ph.D., 2013151Math Way of Number Naming11 = ten 112 = ten 213 = ten 3

Joan A. Cotter, Ph.D., 2013152Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4

Joan A. Cotter, Ph.D., 2013153Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 9

Joan A. Cotter, Ph.D., 2013154Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 920 = 2-ten

Joan A. Cotter, Ph.D., 2013155Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 920 = 2-ten 21 = 2-ten 1

Joan A. Cotter, Ph.D., 2013156Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 920 = 2-ten 21 = 2-ten 122 = 2-ten 2

Joan A. Cotter, Ph.D., 2013157Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 920 = 2-ten 21 = 2-ten 122 = 2-ten 223 = 2-ten 3

Joan A. Cotter, Ph.D., 2013158Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4 . . . .19 = ten 920 = 2-ten 21 = 2-ten 122 = 2-ten 223 = 2-ten 3 . . . . . . . .99 = 9-ten 9

Joan A. Cotter, Ph.D., 2013159Math Way of Number Naming137 = 1 hundred 3-ten 7

Joan A. Cotter, Ph.D., 2013160Math Way of Number Naming137 = 1 hundred 3-ten 7or137 = 1 hundred and 3-ten 7


160

161Math Way of Number Naming

0102030405060708090100456Age (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





















Joan A. Cotter, Ph.D., 2013166Math Way of Number NamingOnly 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.)

Joan A. Cotter, Ph.D., 2013167Math Way of Number NamingOnly 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.

Joan A. Cotter, Ph.D., 2013168Math Way of Number NamingOnly 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.

Joan A. Cotter, Ph.D., 2013169Math Way of Number NamingOnly 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.

Joan A. Cotter, Ph.D., 2013170Math Way of Number NamingCompared to reading

Joan A. Cotter, Ph.D., 2013171Math Way of Number NamingJust as reciting the alphabet doesnt teach reading, counting doesnt teach arithmetic.Compared to reading

Joan A. Cotter, Ph.D., 2013172Math Way of Number NamingJust as reciting the alphabet doesnt teach reading, counting doesnt 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


Math Way of Number NamingRegular names

4-ten = fortyThe ty means tens.



4-ten = fortyThe ty means tens.



6-ten = sixtyThe ty means tens.



3-ten = thirtyThir also used in 1/3, 13 and 30.



5-ten = fiftyFif also used in 1/5, 15 and 50.



2-ten = twentyTwo used to be pronounced twoo.

Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplaceplace-fire

Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplaceplace-firepaper-news

newspaper

Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplaceplace-firepaper-news

box-mailmailbox

newspaper



ten 4

Prefix -teen means ten.



ten 4

teen 4

Prefix -teen means ten.



ten 4

teen 4

fourteenPrefix -teen means ten.



a one left



a one left

a left-one



a one left

a left-one

eleven



two left

Two said as twoo.



two left

twelveTwo said as twoo.


Composing Numbers

3-ten


Composing Numbers

3-ten


Composing Numbers

3-ten3 0


Composing Numbers

3-ten3 0


Composing Numbers

3-ten3 0


Composing Numbers3-ten 7

3 0



3 0



3 0

7


3 0Composing Numbers3-ten 7

7



Note the congruence in how we say the number, represent the number, and write the number.3 07


Composing Numbers

1-ten1 0Another example.


Composing Numbers

1-ten 81 0


Composing Numbers

1-ten 81 0


Composing Numbers

1-ten 81 0

8



1 88


Composing Numbers10-ten



1 0 0



1 0 0



1 0 0


Composing Numbers1 hundred



1 0 0



1 0 0



10

10

1 0 0



1 0 0

Joan A. Cotter, Ph.D., 2013Composing Numbers2 hundred

Joan A. Cotter, Ph.D., 2013Composing Numbers2 hundred

Joan A. Cotter, Ph.D., 2013Composing Numbers2 hundred2 0 0

Joan A. Cotter, Ph.D., 2013217Learning the Facts

Joan A. Cotter, Ph.D., 2013218Learning the FactsLimited success, especially for struggling children, when learning is:

Joan A. Cotter, Ph.D., 2013219Learning the FactsBased on counting: whether dots, fingers, number lines, or counting words.Limited success, especially for struggling children, when learning is:

Joan A. Cotter, Ph.D., 2013220Learning the FactsBased on counting: whether dots, fingers, number lines, or counting words.Limited success, especially for struggling children, when learning is:Based on rote memory: whether flash cards, timed tests, or computer games.

Joan A. Cotter, Ph.D., 2013221Learning the FactsBased on counting: whether dots, fingers, number lines, or counting words.Limited success, especially for struggling children, when learning is:Based on rote memory: whether flash cards, timed tests, or computer games. Based on skip counting: whether fingers or songs.

Joan A. Cotter, Ph.D., 2013222Fact Strategies


Fact StrategiesComplete the Ten9 + 5 =

Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =

Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =

Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 from the 5 and give it to the 9.



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





Fact StrategiesComplete the Ten9 + 5 = 14



229

Fact StrategiesTwo Fives8 + 6 =








Fact StrategiesTwo Fives8 + 6 =10 + 4 = 14


Fact StrategiesGoing Down15 9 =





Subtract 5;then 4.



Subtract 5;then 4.



Subtract 5;then 4.


Fact StrategiesGoing Down15 9 = 6

Subtract 5;then 4.


Fact StrategiesSubtract from 1015 9 =



Subtract 9 from 10.



Subtract 9 from 10.



Subtract 9 from 10.


Fact StrategiesSubtract from 1015 9 = 6

Subtract 9 from 10.


Fact StrategiesGoing Up15 9 =



Start with 9; go up to 15.











Fact StrategiesGoing Up15 9 =1 + 5 = 6


Joan A. Cotter, Ph.D., 2013MoneyPenny


MoneyNickel


MoneyDime


MoneyQuarter


MoneyQuarter


MoneyQuarter


MoneyQuarter

Joan A. Cotter, Ph.D., 2013Trading

1000101100

Joan A. Cotter, Ph.D., 2013TradingThousands

1000101100

Joan A. Cotter, Ph.D., 2013TradingHundreds

1000101100

Joan A. Cotter, Ph.D., 2013TradingTens

1000101100

Joan A. Cotter, Ph.D., 2013TradingOnes

1000101100


1000101100

TradingAdding8+ 6


1000101100

TradingAdding8+ 6


1000101100

TradingAdding8+ 6


1000101100

TradingAdding8+ 6

Joan A. Cotter, Ph.D., 2013TradingAdding8+ 614

1000101100

Joan A. Cotter, Ph.D., 2013TradingAdding8+ 614

Too many ones; trade 10 ones for 1 ten.

1000101100


1000101100

TradingAdding8+ 614



1000101100

TradingAdding8+ 614



1000101100

TradingAdding8+ 614

Same answer before and after trading.


1000101100

TradingAdding 4-digit numbers

3658+ 2738


1000101100

TradingAdding 4-digit numbers3658+ 2738

Enter the first number from left to right.


1000101100

TradingAdding 4-digit numbers3658+ 2738

Enter numbers from left to right.


1000101100


3658+ 2738



1000101100


3658+ 2738



1000101100


3658+ 2738



1000101100


3658+ 2738



1000101100


3658+ 2738

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


1000101100


3658+ 2738



1000101100


3658+ 2738



1000101100


3658+ 2738



1000101100


3658+ 27386



1000101100


3658+ 27386


1


1000101100


3658+ 27386


1


1000101100


3658+ 27386


1


1000101100


3658+ 273896


1


1000101100


3658+ 273896


1


1000101100


3658+ 273896


1


1000101100


3658+ 273896


1


1000101100


3658+ 273896


1


1000101100


3658+ 2738396


1


1000101100


3658+ 2738396


1

1


1000101100


3658+ 2738396


11


1000101100


3658+ 2738396


11


1000101100


3658+ 27386396


11


1000101100


3658+ 27386396


11

Joan A. Cotter, Ph.D., 2013299Meeting the Standards

Joan A. Cotter, Ph.D., 2013300Meeting the StandardsPage 5These 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. 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. 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. CCSS

Joan A. Cotter, Ph.D., 2013301Meeting the StandardsPage 5 summaryStandards do not dictate curriculum or teaching methods.

Joan A. Cotter, Ph.D., 2013302Meeting the StandardsPage 5 summaryStandards do not dictate curriculum or teaching methods.

Within a grade, topics may be taught in any order or taught indirectly.

Joan A. Cotter, Ph.D., 2013303Meeting the StandardsKindergarten (K.NBT)Know number names and the count sequence.Count to 100 by ones and by tens.Count forward beginning from a given number within the known sequence (instead of having to begin at 1).Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

Joan A. Cotter, Ph.D., 2013304Meeting the StandardsKindergarten (K.CC)Count to 100 by ones and by tens.Count forward beginning from a given number.

Joan A. Cotter, Ph.D., 2013305Meeting the StandardsKindergarten (K.CC)Count to 100 by ones and by tens.Count forward beginning from a given number.

Joan A. Cotter, Ph.D., 2013306Meeting the StandardsKindergarten (K.CC)

Count to 100 by ones and by tens.Count forward beginning from a given number.

Joan A. Cotter, Ph.D., 2013Meeting the Standards

6

1

7

2

8

3

9

4

10

5Kindergarten (K.CC)Write numbers from 0 to 20. Number Chart

Joan A. Cotter, Ph.D., 2013308Meeting the StandardsKindergarten (K.NBT)Work with numbers 1119.Compose and partition numbers from 11 to 19 into ten ones and some further ones.



1 861 06

Joan A. Cotter, Ph.D., 2013311Meeting the StandardsKindergarten (K.OA)Understand addition and subtraction.Represent addition and subtraction with objects, fingers, . . . equations.Solve addition and subtraction word problems, and add and subtract within 10.Partition numbers less than or equal to 10 into pairs.For any number from 1 to 9, find the number that makes 10.Fluently add and subtract within 5.

Joan A. Cotter, Ph.D., 2013312Meeting the StandardsKindergarten (K.OA)Solve addition and subtraction word problems, and add and subtract within 10.

WholePartPartPart-whole circles

Joan A. Cotter, Ph.D., 2013313Meeting the StandardsUsing part-whole circles to solve problems

Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with?


Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with?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?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?What is the missing part?35


Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with?What is the missing part?352

Joan A. Cotter, Ph.D., 2013320Meeting the StandardsKindergarten (K.OA)For any number from 1 to 9, find the number that makes 10.

1073

Joan A. Cotter, Ph.D., 2013321Meeting the StandardsGrade 1 (1.OA)Understand and apply properties of operations and the relationship between addition and subtraction.Apply properties of operations as strategies to add and subtract, commutative property and associative property of addition.Understand subtraction as an unknown-addend problem. [Subtract by going up.]

Joan A. Cotter, Ph.D., 2013322Meeting the StandardsGrade 1 (1.OA)Apply properties of operations as strategies to add and subtract, commutative property and associative property of addition.

6 + 3 = 93 + 6 = 9

Joan A. Cotter, Ph.D., 2013323Meeting the StandardsGrade 1 (1.OA)Work with addition and subtraction equations.Understand the meaning of the equal sign.Determine the unknown whole number in an addition or subtraction equation.

Joan A. Cotter, Ph.D., 2013324Meeting the StandardsGrade 1 (1.OA)Understand the meaning of the equal sign.

1098765432112345678910

Math balance


1098765432112345678910

7 = 7

Joan A. Cotter, Ph.D., 2013326Meeting the StandardsGrade 1 (1.OA)Understand the meaning of the equal sign.10 = 3 + 7

1098765432112345678910


10987654321123456789108 + 2 = 10


109865432112345678910

7 + 7 = 147

Joan A. Cotter, Ph.D., 2013329Meeting the StandardsGrade 1 (1.OA)

1098654321123456789107

8 + _ = 11

Determine the unknown whole number in an addition or subtraction equation.

Joan A. Cotter, Ph.D., 2013330Meeting the StandardsGrade 1 (1.OA)

10987654321123456789108 + 3 = 11Determine the unknown whole number in an addition or subtraction equation.

Joan A. Cotter, Ph.D., 2013331Meeting the StandardsGrade 1 (1.OA)Extend the counting sequence.Count to 120, starting at any number less than 120.

Joan A. Cotter, Ph.D., 2013332Meeting the StandardsGrade 1 (1.OA)Extend the counting sequence.Count to 120, starting at any number less than 120.1 0 0

1 091 0 01 09

Joan A. Cotter, Ph.D., 2013333Meeting the StandardsGrade 1 (1.NBT)Understanding place value.Compare two two-digit numbers, recording the results of comparisons with symbols >, =, , =, , =, , =, , =, , =, , =, , =, , =, and , =, and 670

Joan A. Cotter, Ph.D., 2013363ObjectivesReview the traditional counting trajectory.Experience traditional counting like a child.Group in 5s and 10s: an alternative to counting.Meet CCSS without counting.


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Teaching Primary Mathematics with More Understanding and Less CountingNational Council of Supervisors of MathematicsMonday, April 16, 2013Denver, ColoradoJoan A. Cotter, [email protected] Mittleider, [email protected]


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NCSM April 2013