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![Page 1: Developing Spatial Mathematics Richard Lehrer Vanderbilt University Thanks to Nina Knapp for collaborative study of evolution of volume concepts.](https://reader031.fdocuments.us/reader031/viewer/2022032200/56649cf85503460f949c97f2/html5/thumbnails/1.jpg)
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.
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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.
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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
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What’s a Perfect Solid?What’s a Perfect Solid?
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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?
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Designing QuiltsDesigning Quilts
COPYCORE
SIDE-WAYSFLIP
UP-DOWNFLIP
TURNRIGHT1/ 2
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Investigating SymmetriesInvestigating SymmetriesInvestigating SymmetriesInvestigating Symmetries
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Art-Mathematics:Art-Mathematics:Design SpacesDesign Spaces
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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
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The Shape of FairnessThe Shape of Fairness
Game of Tag-- What’s fair? (Gr 1/2:Liz Penner)
• Mother
• • • • • • • • • • • • • • • •Movers
• Mother
• • • • • • • • • • • • • • • •Movers
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Form Represents SituationForm Represents Situation
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Properties of Form Emerge Properties of Form Emerge From ModelingFrom Modeling
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The Fairest Form of All?The Fairest Form of All?
Investigate Properties of Circle, Finding Center
Develop Units of Length Measure
Shape as Generalization
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What’s a Triangle?What’s a Triangle?
What’s “straight?”
What’s “corner?”
What’s “tip?”
3 Sides, 3 Corners
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Defining Properties (“Rules”)Defining Properties (“Rules”)
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Building and Defining in KBuilding and Defining in K
Kindergarten: “Closed”
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Open vs. Closed in Open vs. Closed in KindergartenKindergarten
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Modeling 3-D StructureModeling 3-D Structure
• Physical Unfolding--> Mathematical Representation
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Investigating Surface and EdgeInvestigating Surface and Edge
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Solutions for Truncated ConesSolutions for Truncated Cones
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Truncated Cone-2Truncated Cone-2
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Truncated Cone - 3,4Truncated Cone - 3,4
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Truncated Cone-5Truncated Cone-5
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Shifting to Representing WorldShifting to Representing World
“How can we be sure?”
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Is It Possible?Is It Possible?
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““System of Systems”System of Systems”
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Circumference-Height of CylindersCircumference-Height of Cylinders
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Student InvestigationsStudent Investigations
Good Forum for Density
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Extensions to Modeling NatureExtensions to Modeling Nature
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Dealing with VariationDealing with VariationDealing with VariationDealing with Variation
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Root vs. Shoot GrowthRoot vs. Shoot Growth
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Mapping the PlaygroundMapping the Playground
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Measuring SpaceMeasuring Space
• Structuring Space
• Practical Activity
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Children’s Theory of MeasureChildren’s Theory of Measure
• Build Understanding of Measure as a Web of Components
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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
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Inventing Units of AreaInventing Units of Area
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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
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Structuring Space: VolumeStructuring Space: Volume
Appearance - Reality Conflict
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Supporting Visualization Supporting Visualization
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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
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Move toward ContinuityMove toward Continuity
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Re-purposing for VolumeRe-purposing for Volume
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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?
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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
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Brief ItemsBrief Items5. Johnny like making buildings from cubes. He made bulding A by
putting 8 cubes like this together.
A
C
B
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Brief ItemsBrief Items25. Susan likes to make buildings with cubes. She made building A by putting
8 cubes like this together.
A
B
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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
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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%
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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.
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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]
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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
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The Shape of DataThe Shape of DataThe Shape of DataThe Shape of Data
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Shape of Data (2)Shape of Data (2)Shape of Data (2)Shape of Data (2)
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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