1. The Construction of Number A Constance Kamii Perspective Paterson Public Schools Dept. of Early Childhood Education Craig Wachsman

2. Program Philosophy Children are constructivists. Children are constructivists. Children are active learners. Children are active learners. Children are intrinsically motivated and self directed. Children are intrinsically motivated and self directed.

3. Program Goals The construction of “Number.” The construction of “Number.” Cognitive Process: Order Cognitive Process: Order Cognitive Process: Hierarchical Inclusion Cognitive Process: Hierarchical Inclusion

4. 3 Stages of Quantification Global Global One-to-One Correspondence One-to-One Correspondence Counting Counting

5. Global Stage of Quantification Young children are guided initially by their perceptions when quantifying. Young children are guided initially by their perceptions when quantifying. At this stage, children make a visual approximation of the quantity they are attempting to match. At this stage, children make a visual approximation of the quantity they are attempting to match. When asked to take as many objects as are in another group, they may create a pile or row that looks about the same as the model set. When asked to take as many objects as are in another group, they may create a pile or row that looks about the same as the model set. (Andre-Anna-Christian-Nicole) (Andre-Anna-Christian-Nicole)

6. One-to-One Correspondence Stage Children attempt to make an equivalent set by taking one object for each item in the original set. Children attempt to make an equivalent set by taking one object for each item in the original set. Children may point to an item in the original set each time they select a new object for their own set. Children may point to an item in the original set each time they select a new object for their own set. They may also line up the materials in rows so that each object they take is opposite one item in the original set. They may also line up the materials in rows so that each object they take is opposite one item in the original set.(Omerlyn-David)

7. Counting Stage: Conservers Children understand that the last item they count represents the total. Children understand that the last item they count represents the total. Children realize that there is a particular order to saying the counting words, and each object can be counted one, and only one time. Children realize that there is a particular order to saying the counting words, and each object can be counted one, and only one time.

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

9. Conservers Rationale The Identity Argument: The Identity Argument: “There’s just as many blue ones as red ones because you didn’t add anything or take anything away.” “There’s just as many blue ones as red ones because you didn’t add anything or take anything away.”

10. Conservers Rationale The Reversibility Argument: The Reversibility Argument: “We could put all the red ones back to the way they were before, and you’ll see that there’s the same number.” “We could put all the red ones back to the way they were before, and you’ll see that there’s the same number.”

11. Conservers Rationale The Compensation Argument: The Compensation Argument: “The red row is longer, but there’s more space in between. So the number is still the same.” “The red row is longer, but there’s more space in between. So the number is still the same.”

12. Teacher’s Role Entering the Child’s World Imitating the child as a means of entering their world of play Imitating the child as a means of entering their world of play When playing math games with children, teachers or parents can carefully observe a child’s level of quantification and model the next stage. When playing math games with children, teachers or parents can carefully observe a child’s level of quantification and model the next stage.

13. Math Games Aiming Games Aiming Games Bowling & Ball Games Bowling & Ball Games Hiding Games Hiding Games Hide and Seek Hide and Seek 6 Tongue Depressors 6 Tongue Depressors Races & Chasing Games Races & Chasing Games Musical Chairs Musical Chairs Duck, Duck, Goose Duck, Duck, Goose

14. Materials Math Games Strawberry Game Strawberry Game Children roll a die to determine how many cloth strawberries they can put in their baskets. At the end of the game, children can compare quantities of strawberries if they chose. Children roll a die to determine how many cloth strawberries they can put in their baskets. At the end of the game, children can compare quantities of strawberries if they chose.

15. Materials Math Games Mail a Letter Game Mail a Letter Game Each play has a game board with a short path leading from a child sticker to a mailbox. Children use a die or spinner to move along the path. (A Letter To Amy by Keats) Each play has a game board with a short path leading from a child sticker to a mailbox. Children use a die or spinner to move along the path. (A Letter To Amy by Keats) Children quantify and create equilvent sets as they attempt to move the same number of spaces as indicated on the die or spinner. Children quantify and create equilvent sets as they attempt to move the same number of spaces as indicated on the die or spinner.

16. Materials Math Games Balloon (Long Path) Game Balloon (Long Path) Game Children move small people along a path made from balloon stickers and try to reach the party at the end of the game. Younger children should use one die, while older children could use two dice and add them together. Children move small people along a path made from balloon stickers and try to reach the party at the end of the game. Younger children should use one die, while older children could use two dice and add them together.

17. Mathematical Opportunities Throughout the Day

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

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

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

21. “Do you want the story now or would you rather finish your collage.” Order

22. Nicole looks at the schedule to see what comes after group time. Sequence

23. David breaks the play dough in two pieces and gives a part to Maria. Parts of the Whole

24. Stephanie puts cube blocks on her construction: red, yellow, blue, red, yellow, blue Pattern

25. “Can we have just five more minutes?” Time

26. “I need four more bears.” Number

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

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

29. Assessment/Evaluation Observation Observation Conservation Tasks Conservation Tasks