Measuring and Visualizing Space in Elementary Mathematics Learning

Description

Measuring and Visualizing Space in Elementary Mathematics Learning explores the development of elementary students’ understanding of the mathematics of measure, and demonstrates how measurement can serve as an anchor for supporting a deeper understanding of number operations and rational numbers. The concept of measurement is centrally implicated in a number of mathematical operations, yet is not often given the placement it deserves in the elementary mathematics curriculum. By drawing on K-5 classroom research, authors Lehrer and Schauble have been able to articulate a learning progression that describes benchmarks of student learning about measure in length, angle, area, volume, and rational number, exploring related concepts, classroom experiences, and instructional practices at each stage. Offering a unique, research driven resource for helping students develop a deep understanding of measurement to further enhance mathematical understanding, as well as further learning in other STEM disciplines; the book will be relevant for scholars, teacher educators, and specialists in math education. The book is accompanied by online resources developed for practitioners, including instructional guides, examples of student thinking, and other teacher-focused materials, helping clarify how to bring concepts of measure and rational number to life in classrooms. 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