developing spatial mathematics
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DESCRIPTIONDeveloping 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
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?
Gateways to Algebra
The Shape of Fairness Game of Tag-- Whats 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 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 - 3,4
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
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
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
5. Johnny like making buildings from cubes. He made bulding A by
putting 8 cubes like this together.
25. Susan likes to make buildings with cubes. She made building A by putting
8 cubes like this together.
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?
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
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