# developing spatial mathematics

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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. - PowerPoint PPT Presentation

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• 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

90180270360UDRLRDLD9090180180270270360360UDUDRLRLRDRDLDLD

• 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

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

• 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

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