The Construction of The Construction of NumberNumber

A Constance Kamii PerspectiveA Constance Kamii Perspective

Paterson Public SchoolsPaterson Public Schools

Dept. of Early Childhood EducationDept. of Early Childhood Education

Craig WachsmanCraig Wachsman

Program PhilosophyProgram Philosophy

Children are constructivists.Children are constructivists. Children are active learners.Children are active learners. Children are intrinsically motivated Children are intrinsically motivated

and self directed.and self directed.

Program GoalsProgram Goals

The construction of The construction of “Number.”“Number.” Cognitive Process: OrderCognitive Process: Order Cognitive Process: Hierarchical InclusionCognitive Process: Hierarchical Inclusion

3 Stages of Quantification3 Stages of Quantification

GlobalGlobal One-to-One One-to-One

CorrespondenceCorrespondence CountingCounting

Global Stage of Global Stage of QuantificationQuantification

Young children are guided initially by their Young children are guided initially by their perceptionsperceptions when quantifying. when quantifying.

At this stage, children make a visual At this stage, children make a visual approximation of the quantity they are approximation of the quantity they are attempting to match.attempting to match.

When asked to take as many objects as are When asked to take as many objects as are in another group, they may create a pile or in another group, they may create a pile or row that looks about the same as the row that looks about the same as the model set.model set.

((Andre-Anna-ChristianAndre-Anna-Christian--Nicole) Nicole)

One-to-One Correspondence One-to-One Correspondence StageStage

Children attempt to make an equivalent Children attempt to make an equivalent set by taking one object for each item in set by taking one object for each item in the original set.the original set.

Children may point to an item in the Children may point to an item in the original set each time they select a new original set each time they select a new object for their own set.object for their own set.

They may also line up the materials in They may also line up the materials in rows so that each object they take is rows so that each object they take is opposite one item in the original set.opposite one item in the original set.

(Omerlyn-David)(Omerlyn-David)

Counting Stage: ConserversCounting Stage: Conservers

Children understand that the last Children understand that the last item they count represents the total. item they count represents the total.

Children realize that there is a Children realize that there is a particular order to saying the particular order to saying the counting words, and each object can counting words, and each object can be counted one, and only one time.be counted one, and only one time.

Conservation TaskConservation Task

To conserve number means to think To conserve number means to think that the quantity remains the same that the quantity remains the same when the spatial arrangement of the when the spatial arrangement of the objects has been changed.objects has been changed.

Conservers RationaleConservers Rationale

The Identity Argument:The Identity Argument: ““There’s just as many blue ones as red There’s just as many blue ones as red

ones because you didn’t add anything or ones because you didn’t add anything or take anything away.”take anything away.”

Conservers RationaleConservers Rationale

The Reversibility Argument:The Reversibility Argument: ““We could put all the red ones back to We could put all the red ones back to

the way they were before, and you’ll see the way they were before, and you’ll see that there’s the same number.”that there’s the same number.”

Conservers RationaleConservers Rationale

The Compensation Argument:The Compensation Argument: ““The red row is longer, but there’s more The red row is longer, but there’s more

space in between. So the number is still space in between. So the number is still the same.”the same.”

Teacher’s RoleTeacher’s RoleEntering the Child’s WorldEntering the Child’s World

Imitating the child as a means of Imitating the child as a means of entering their world of play entering their world of play

When playing math games with When playing math games with children, teachers or parents can children, teachers or parents can carefully observe a child’s level of carefully observe a child’s level of quantification and model the next quantification and model the next stage. stage.

Math GamesMath Games Aiming GamesAiming Games

Bowling & Ball GamesBowling & Ball Games Hiding GamesHiding Games

Hide and SeekHide and Seek 6 Tongue Depressors6 Tongue Depressors

Races & Chasing GamesRaces & Chasing Games Musical ChairsMusical Chairs Duck, Duck, GooseDuck, Duck, Goose

MaterialsMaterialsMath GamesMath Games

Strawberry GameStrawberry Game Children roll a die to determine how Children roll a die to determine how

many cloth strawberries they can put in many cloth strawberries they can put in their baskets. At the end of the game, their baskets. At the end of the game, children can compare quantities of children can compare quantities of strawberries if they chose.strawberries if they chose.

MaterialsMaterialsMath GamesMath Games

Mail a Letter GameMail a Letter Game Each play has a game board with a short Each play has a game board with a short

path leading from a child sticker to a path leading from a child sticker to a mailbox. Children use a die or spinner to mailbox. Children use a die or spinner to move along the path. (A Letter To Amy by move along the path. (A Letter To Amy by Keats)Keats)

Children quantify and create equilvent sets Children quantify and create equilvent sets as they attempt to move the same number as they attempt to move the same number of spaces as indicated on the die or spinner.of spaces as indicated on the die or spinner.

MaterialsMaterialsMath GamesMath Games

Balloon (Long Path) GameBalloon (Long Path) Game Children move small people along a Children move small people along a

path made from balloon stickers and try path made from balloon stickers and try to reach the party at the end of the to reach the party at the end of the game. Younger children should use one game. Younger children should use one die, while older children could use two die, while older children could use two dice and add them together.dice and add them together.

Mathematical Mathematical Opportunities Throughout Opportunities Throughout

the Daythe Day

““I found all the rough I found all the rough paper.”paper.”

TextureTexture

Andre uses all the triangle Andre uses all the triangle blocks and looks for more.blocks and looks for more.

ShapeShape

Michele lines up the blocks Michele lines up the blocks from shortest to longest.from shortest to longest.

SeriationSeriation

““Do you want the story now Do you want the story now or would you rather finish or would you rather finish

your collage.”your collage.”

OrderOrder

Nicole looks at the schedule Nicole looks at the schedule to see what comes after to see what comes after

group time.group time.

SequenceSequence

David breaks the play dough David breaks the play dough in two pieces and gives a in two pieces and gives a

part to Maria.part to Maria.

Parts of the WholeParts of the Whole

Stephanie puts cube blocks Stephanie puts cube blocks on her construction:on her construction:

red, yellow, blue, red, yellow, red, yellow, blue, red, yellow, blueblue

PatternPattern

““Can we have just five Can we have just five more minutes?”more minutes?”

TimeTime

““I need four more bears.”I need four more bears.”

NumberNumber

““These cars are bigger These cars are bigger than those.”than those.”

SizeSize

““I have a new striped I have a new striped shirt.”shirt.”

PatternPattern

Assessment/EvaluationAssessment/Evaluation

ObservationObservation Conservation TasksConservation Tasks