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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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