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 - System. Relating Mathematical Objects - Relation Between Particular and General (Proof) - Writing Mathematics. Representation.

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

What’s a Perfect Solid?

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?

Designing Quilts

Investigating Symmetries

Art-Mathematics: Design Spaces

Gateways to Algebra UDRLRDLD UD RL RD LD

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

Form Represents Situation

Properties of Form Emerge From Modeling

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

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

Defining Properties (“Rules”)

Building and Defining in K Kindergarten: “Closed”

Open vs. Closed in Kindergarten

Modeling 3-D Structure Physical Unfolding--> Mathematical Representation

Investigating Surface and Edge

Solutions for Truncated Cones

Truncated Cone-2

Truncated Cone - 3,4

Truncated Cone-5

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

Is It Possible?

“System of Systems”

Circumference-Height of Cylinders

Student Investigations Good Forum for Density

Extensions to Modeling Nature

Dealing with Variation

Root vs. Shoot Growth

Mapping the Playground

Measuring Space Structuring Space Practical Activity

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

Children’s Investigations

Inventing Units of Area

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

Structuring Space: Volume Appearance - Reality Conflict

Supporting Visualization

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

Move toward Continuity

Re-purposing for Volume

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

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

Brief Items

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%

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.

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]

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

The Shape of Data

Shape of Data (2)

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