developing spatial mathematics

Developing Spatial Mathematics

Richard LehrerVanderbilt 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 EverydayBuilding & Designing---> Structuring SpaceCounting ---> Measuring & Structuring SpaceDrawing ---> Representing Space (Diagram, Net)Walking ---> Position and Direction in Space

Whats 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: Whats a triangle? A perfect solid? Magnify: Whats the same?

Designing Quilts

Investigating Symmetries

Art-Mathematics:Design Spaces

Gateways to Algebra

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

Mother

Movers

Mother

Movers

Form Represents Situation

Properties of Form Emerge From Modeling

The Fairest Form of All?Investigate Properties of Circle, Finding CenterDevelop Units of Length MeasureShape as Generalization

Whats a Triangle?Whats straight?Whats corner?Whats tip?

3 Sides, 3 Corners

Defining Properties (Rules)

Building and Defining in KKindergarten: Closed

Open vs. Closed in Kindergarten

Modeling 3-D StructurePhysical Unfolding--> Mathematical Representation

Investigating Surface and Edge

Solutions for Truncated Cones

Truncated Cone-2

Truncated Cone - 3,4

Truncated Cone-5

Shifting to Representing WorldHow can we be sure?

Is It Possible?

System of Systems

Circumference-Height of Cylinders

Student InvestigationsGood Forum for Density

Extensions to Modeling Nature

Dealing with Variation

Root vs. Shoot Growth

Mapping the Playground

Measuring Space Structuring Space Practical Activity

Childrens Theory of Measure Build Understanding of Measure as a Web of Components

Childrens 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: VolumeAppearance - 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 arent 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

5. Johnny like making buildings from cubes. He made bulding A by

putting 8 cubes like this together.

A

C

B

Brief Items

25. Susan likes to make buildings with cubes. She made building A by putting

8 cubes like this together.

A

B

Brief Items

18. The area of the base of the cylinder below is 5 square inches (5 in.2). The

height of the cylinder is 8 in. What is the volume of the cylinder?

_______________________

8 in.

5 in. 2

Comparative PerformanceGrade 2Hidden 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%

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

InterviewsRectangular 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 HeightWith Different Tools

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The Shape of Data

Shape of Data (2)

The Construction ZoneBuilding Mathematics from Experience of SpaceAs Moved InAs MeasuredAs SeenAs ImaginedVisual Support for Mathematical ReasoningDefining, Generalizing, Modeling, ProvingCONNECTING