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Developing Spatial Mathematics

Developing Spatial Mathematics. Richard Lehrer Vanderbilt University. Thanks to Nina Knapp for collaborative study of evolution of volume concepts. Why a Spatial Mathematics?. HABITS OF MIND. Generalization (This Square --> All Squares). Definition. Making Mathematical Objects.

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Developing Spatial Mathematics

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  1. Developing Spatial Mathematics Richard Lehrer Vanderbilt University Thanks to Nina Knapp for collaborative study of evolution of volume concepts.

  2. Why a Spatial Mathematics? HABITS OF MIND • Generalization (This Square --> All Squares) • Definition. Making Mathematical Objects • System. Relating Mathematical Objects • Relation Between Particular and General (Proof) • Writing Mathematics. Representation.

  3. Capitalizing on the Everyday • Building & Designing---> Structuring Space • Drawing ---> Representing Space (Diagram, Net) • Counting ---> Measuring & Structuring Space • Walking ---> Position and Direction in Space

  4. What’s a Perfect Solid?

  5. Pathways to Shape and Form • Design: Quilting, City Planning (Whoville) • Modeling: The Shape of Fairness • Build: 3-D Forms from 2-D Nets • Classify: What’s a triangle? A perfect solid? • Magnify: What’s the same?

  6. Designing Quilts

  7. Investigating Symmetries

  8. Art-Mathematics:Design Spaces

  9. Gateways to Algebra

  10. The Shape of Fairness Game of Tag-- What’s fair? (Gr 1/2:Liz Penner)

  11. Form Represents Situation

  12. Properties of Form Emerge From Modeling

  13. The Fairest Form of All? Investigate Properties of Circle, Finding Center Develop Units of Length Measure Shape as Generalization

  14. What’s a Triangle? What’s “straight?” What’s “corner?” What’s “tip?” 3 Sides, 3 Corners

  15. Defining Properties (“Rules”)

  16. Building and Defining in K Kindergarten: “Closed”

  17. Open vs. Closed in Kindergarten

  18. Modeling 3-D Structure • Physical Unfolding--> Mathematical Representation

  19. Investigating Surface and Edge

  20. Solutions for Truncated Cones

  21. Truncated Cone-2

  22. Truncated Cone - 3,4

  23. Truncated Cone-5

  24. Shifting to Representing World “How can we be sure?”

  25. Is It Possible?

  26. “System of Systems”

  27. Circumference-Height of Cylinders

  28. Student Investigations Good Forum for Density

  29. Extensions to Modeling Nature

  30. Dealing with Variation

  31. Root vs. Shoot Growth

  32. Mapping the Playground

  33. Measuring Space • Structuring Space • Practical Activity

  34. Children’s Theory of Measure • Build Understanding of Measure as a Web of Components

  35. Children’s Investigations

  36. Inventing Units of Area

  37. Constructing Arrays Grade 2: 5 x 8 Rectangle as 5 rows of 8 or as 8 columns of 5 (given a ruler) L x W = W x L, rotational invariance of area

  38. Structuring Space: Volume Appearance - Reality Conflict

  39. Supporting Visualization

  40. Making Counts More Efficient • Introducing Hidden Cubes Via Rectangular Prisms (Shoeboxes) • Column or row structure as a way of accounting for hidden cubes • Layers as a way of summing row or column structures • Partial units (e.g., 4 x 3 x 3 1/2) to promote view of layers as slices

  41. Move toward Continuity

  42. Re-purposing for Volume

  43. Extensions to Modeling Nature Cylinder as Model Given “Width,” What is the Circumference? Why aren’t the volumes (ordered in time) similar?

  44. Yes, But Did They Learn Anything? • Brief Problems (A Test) - Survey of Learning • Clinical Interview - Strategies and Patterns of Reasoning

  45. Brief Items

  46. Brief Items

  47. Brief Items

  48. Comparative Performance Grade 2 Hidden Cube 23% ---> 64% Larger Lattice 27% ---> 68% Grade 3 (Comparison Group, Target Classroom) Hidden Cube 44% vs. 86% Larger Lattice 48% vs. 82% Cylinder 16% vs. 91% Multiple Hidden Units: 68%

  49. Interviews • Wooden Cube Tower, no hidden units (2 x 2 x 9) • - Strategies: Layers, Dimensions, Count-all • Wooden Cube Tower, hidden units (3 x 3 x 4) • - Strategies: Dimensions, Layers, Count-all • Rectangular Prism, integer dimensions, ruler, some cubes, grid paper • -Strategies: Dimension (including A x H), Layer, Count-All • NO CHILD ATTEMPTS TO ONLY COUNT FACES AND ONLY A FEW (2-3/22) Count-all.

  50. Interviews • Rectangular Prism, non-integer dimensions • -Strategies: Dimension (more A x H), Layer, Only 1 Counts but “not enough cubes.” • Hexagonal Prism • - Strategy A x H (68%) [including some who switched from layers to A x H]

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