Measuring and Visualizing Space in Elementary Mathematics Learning

Cover
Half Title
Title Page
Copyright Page
CONTENTS
About the Authors
Acknowledgments
1. Measure is Fundamental
Developing a Theory of Measure
Relation to Existing Scholarship
Organization of the Book
References
2. The Context, Goals, and Design of the Research
Research Sites and Participants
Design Research: Engineering Learning to Support Its Study
The Educational Design
Professional Development Collaboration with Participating Teachers
Learning Constructs
Supporting Curriculum Units
Assessment System
Digital Tools for Collecting, Displaying, and Interpreting Student Data
Studies of Student Learning
Exploratory Learning Studies
Yearly Interview Data
Formative Assessment Tasks and In Situ Observations
Studies of Teacher Learning and Practice
References
3. Origins of Quantitative Reasoning in the Measure of Length
Overview of Children's Understanding of Length
Benchmarks in Thinking About Length
Directly Comparing Magnitudes
Explaining How Properties of Units Affect Measure
Unit Iteration and Constructing a Measurement Scale
2-Splitting and Symbolizing 2-Split Units as Measures
3-Splitting and Symbolizing 3-Split Units
Generalizing Relationships Among Units and Measures
References
4. Creating New Quantities in the Dynamic Generation of Area
Two Coordinated Perspectives on Measure
Integrating the Two Perspectives on Measure
Direct Comparison of Magnitudes
Comparing Magnitudes of Area Indirectly Through Dissection and Unit Dissection
Properties of Units of Area Measure
Dynamic Generation of Area and Product
Guided Reinvention of Area Measure Formulas
References
5. Extending Motion to Three Dimensions: Volume and Its Measure
Structuring and Dynamic Approaches to Volume Measure
Volume Conceived as Space Inside
Measuring Volume by Accumulating Units
Visualizing Volume as Composites of Layers
Finding Volumes of Prisms with Fractional Dimension
Generating Volume Dynamically
References
6. Integrating Figure and Motion in the Measure of Angle
Dynamic and Figural Perspectives
Benchmarks in Thinking About Angle
Noticing Canonical Examples
Representing Angles-as-Figures, Angles-as-Turns
Integrating Angle-as-Turn with Angle-as-Figure, Interior vs. Turn Angles
Generating and Justifying Angle Measure Theorems
Developing New Understandings of Figures and Structures via Angle Theorems
References
7. Measurement Models of Arithmetic Operations and Rational Number
Initial Resources for Reasoning About Rational Number in Measure
Unit Iteration
Length Measure as a Point Along a Path
Measure-Magnitude Distinction
Symbolizing Measure
Additive and Multiplicative Comparison
Rational Numbers as Measured Quantities
Two-Split of a Unit Length and Half-Unit Iteration
Four- and Eight-Splits of a Unit Length and Measures in Fourth-Unit and Eighth-Unit
Three-Splits and Compositions of Two- and Three-Splits of a Unit Length
Fractions as Operators on Measured Quantities
Initial Steps in Developing a Sense of Fraction as Operator
Extending the Reach of Fraction-as-Operator to Refine Measure
Extending Multiplication and Multiplicative Comparisons
References
8. Highlights of Student Learning Research
The Two Phases of Research
Phase I: Student Conceptions of Measure as Indicated by Yearly Interviews
Early-Developing Conceptions of the Measure of Length
Direct comparison
Tiling, unit, and iteration
Unit iteration and symbolizations of unit on scale
Equipartitioning fractured units
Measurement arithmetic
Conceptions of Area Measure
Necessary conditions for area
Unit structuring of area
Differentiating area and length measure conceptually and symbolically
Conceptions of Angle Measure
Embodied turns in walking paths in primary grades
Conceptions of angle and measure in later grades
Understandings of angle theorems
Conceptions of Volume Measure
Strategies employed to measure prisms constructed of cubic units
Strategies employed to measure prisms with partial structuring
Volume of pentagonal prism
Cavalieri's principle
Phase II: Summative, Formative, and In-Situ Evidence of Student Learning
Summative Assessment
Formative Assessment and In-Situ Evidence of Student Learning
Guiding instruction by monitoring conceptual development
Formative assessment and dialogic space
Reflections and Prospects
References
9. Teacher Learning
Initiating and Sustaining a Teacher-Researcher Partnership
Supporting Development of a Shared Professional Vision
Constructs
Curriculum
Digital observation tools
Activity Structures That Forge a Professional Learning Community
Learning labs
Mathematical investigations
Communal critique
Investigations of Change in Professional Practice
Teacher Noticings During Video Episodes
Video episodes
Interview procedure
Interview analysis
Interview results
Teacher Perspectives on Professional Development
Teachers' views of learning labs
Teachers' views of mathematical investigations
Teachers' views of communal critique
Changes in Teachers' Construct-Centered Judgments About Students' Ways of Thinking
Professional Vision as a Fulcrum for Learning Progression
References
10. Measures and Models in Elementary Science
Characterizing Growth
Describing Change by Determining Differences in Quantity
Describing Change with Intensive Quantity: Differences in Rates
Describing Change in Population with Measures of Distribution
Cultivating Distributional Thinking
Initial Steps
Revisiting Cause and Chance in Generating Variability
Melding Chance and Cause: Investigating Precision of Measure
Modeling Measurements as Signal and Noise
Expanding the reach of modeling chance
Modeling Broadens the Scope of Measures
References
Index