Developing Spatial Developing Spatial MathematicsMathematics

Developing Spatial Developing Spatial MathematicsMathematics

Richard Lehrer

Vanderbilt University

Thanks to Nina Knapp for collaborative study of evolution of volume concepts.

Why a Spatial Mathematics?Why a Spatial Mathematics?Why a Spatial Mathematics?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 EverydayCapitalizing 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?What’s a Perfect Solid?

Pathways to Shape and FormPathways 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 QuiltsDesigning Quilts

COPYCORE

SIDE-WAYSFLIP

UP-DOWNFLIP

TURNRIGHT1/ 2

Investigating SymmetriesInvestigating SymmetriesInvestigating SymmetriesInvestigating Symmetries

Art-Mathematics:Art-Mathematics:Design SpacesDesign Spaces

Gateways to AlgebraGateways to AlgebraGateways to AlgebraGateways to Algebra

90 180 270 360 UD RL RD LD

90 90

180 180

270 270

360 360

UD UD

RL RL

RD RD

LD LD

The Shape of FairnessThe Shape of Fairness

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

• Mother

• • • • • • • • • • • • • • • •Movers

• Mother

• • • • • • • • • • • • • • • •Movers

Form Represents SituationForm Represents Situation

Properties of Form Emerge Properties of Form Emerge From ModelingFrom Modeling

The Fairest Form of All?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 a Triangle?

What’s “straight?”

What’s “corner?”

What’s “tip?”

3 Sides, 3 Corners

Defining Properties (“Rules”)Defining Properties (“Rules”)

Building and Defining in KBuilding and Defining in K

Kindergarten: “Closed”

Open vs. Closed in Open vs. Closed in KindergartenKindergarten

Modeling 3-D StructureModeling 3-D Structure

• Physical Unfolding--> Mathematical Representation

Investigating Surface and EdgeInvestigating Surface and Edge

Solutions for Truncated ConesSolutions for Truncated Cones

Truncated Cone-2Truncated Cone-2

Truncated Cone - 3,4Truncated Cone - 3,4

Truncated Cone-5Truncated Cone-5

Shifting to Representing WorldShifting to Representing World

“How can we be sure?”

Is It Possible?Is It Possible?

““System of Systems”System of Systems”

Circumference-Height of CylindersCircumference-Height of Cylinders

Student InvestigationsStudent Investigations

Good Forum for Density

Extensions to Modeling NatureExtensions to Modeling Nature

Dealing with VariationDealing with VariationDealing with VariationDealing with Variation

Root vs. Shoot GrowthRoot vs. Shoot Growth

Mapping the PlaygroundMapping the Playground

Measuring SpaceMeasuring Space

• Structuring Space

• Practical Activity

Children’s Theory of MeasureChildren’s Theory of Measure

• Build Understanding of Measure as a Web of Components

Children’s InvestigationsChildren’s Investigations

1 unit and 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Desk

1 unit and 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Desk

Inventing Units of AreaInventing Units of Area

Constructing Arrays 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: VolumeStructuring Space: Volume

Appearance - Reality Conflict

Supporting Visualization Supporting Visualization

Making Counts More EfficientMaking 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 ContinuityMove toward Continuity

Re-purposing for VolumeRe-purposing for Volume

Extensions to Modeling NatureExtensions 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 Yes, But Did They Learn Anything?Anything?

• Brief Problems (A Test) - Survey of Learning

• Clinical Interview - Strategies and Patterns of Reasoning

Brief ItemsBrief Items5. Johnny like making buildings from cubes. He made bulding A by

putting 8 cubes like this together.

A

C

B

Brief ItemsBrief Items25. Susan likes to make buildings with cubes. She made building A by putting

8 cubes like this together.

A

B

Brief ItemsBrief Items18. 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 PerformanceComparative 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.

InterviewsInterviews

•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 Do differences in measures have a structure?have a structure?

Do differences in measures Do differences in measures have a structure?have a structure?

6 7 8 9 10 11 12 13 14 15

6 .2 7 .8 8 .7 9 .1 1 0 1 1 1 2. 3 1 3 1 5. 5

6 .4 8 .8 9 .2 1 0. 2

6 .8 9 .3 1 0. 3

9 .7 1 0. 6

9 .8

Repeated Measure of Height

With Different Tools

The Shape of DataThe Shape of DataThe Shape of DataThe Shape of Data

Shape of Data (2)Shape of Data (2)Shape of Data (2)Shape of Data (2)

The Construction ZoneThe 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