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 Counting ---> Measuring & Structuring Space Drawing ---> Representing Space (Diagram, Net) 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 90180270360UDRLRDLD 90 180 270 360 UD RL RD LD

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

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. InterviewsInterviews 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 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]

51. Do differences in measures have a structure? Repeated Measure of Height With Different Tools

52. The Shape of Data

53. Shape of Data (2)

54. The Construction Zone Building Mathematics from Experience of Space –As Moved In –As Measured –As Seen –As Imagined Visual Support for Mathematical Reasoning –Defining, Generalizing, Modeling, Proving CONNECTING