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