Table Of ContentMEASURING AND VISUALIZING
SPACE IN ELEMENTARY
MATHEMATICS LEARNING
Measuring and Visualizing Space in Elementary Mathematics Learning explores
the development of elementary students’ understanding of the mathemat-
ics 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 math-
ematical operations, yet it 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 progres-
sion that describes benchmarks of student learning about measure in length,
angle, area, volume, and rational number, exploring related concepts, class-
room experiences, and instructional practices at each stage.
Offering a unique, research-driven resource for helping students develop a
deep understanding of measurement to enhance mathematical understanding,
as well as further learning in other STEM disciplines, the book will be rele-
vant 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 teach-
er-focused materials, helping to clarify how to bring concepts of measure and
rational number to life in classrooms.
Richard Lehrer is a Professor of Education in the Department of Teaching
and Learning at Vanderbilt University, Nashville, Tennessee, USA.
Leona Schauble is a Professor of Education in the Department of Teaching
and Learning at Vanderbilt University, Nashville, Tennessee, USA.
MEASURING AND
VISUALIZING SPACE
IN ELEMENTARY
MATHEMATICS
LEARNING
Richard Lehrer
Leona Schauble
Designed cover image: © Trisha Flores
First published 2023
by Routledge
605 Third Avenue, New York, NY 10158
and by Routledge
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Routledge is an imprint of the Taylor & Francis Group, an informa business
© 2023 Richard Lehrer and Leona Schauble
The right of Richard Lehrer and Leona Schauble to be identified as
authors of this work has been asserted in accordance with sections
77 and 78 of the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this book may be reprinted or
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or other means, now known or hereafter invented, including
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Trademark notice: Product or corporate names may be trademarks
or registered trademarks, and are used only for identification and
explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Names: Lehrer, Richard, author. | Schauble, Leona, author.
Title: Measuring and visualizing space in elementary mathematics learning /
Richard Lehrer, Leona Schauble.
Description: New York, NY : Routledge, 2023. | Includes bibliographical
references and index.
Identifiers: LCCN 2022039860 (print) | LCCN 2022039861 (ebook) | ISBN
9781032262734 (hardback) | ISBN 9781032262727 (paperback) | ISBN
9781003287476 (ebook)
Subjects: LCSH: Geometry--Study and teaching (Elementary)
Classification: LCC QA461 .L466 2023 (print) | LCC QA461 (ebook) | DDC
372.7/6--dc23/eng20230110
LC record available at https://lccn.loc.gov/2022039860
LC ebook record available at https://lccn.loc.gov/2022039861
ISBN: 9781032262734 (hbk)
ISBN: 9781032262727 (pbk)
ISBN: 9781003287476 (ebk)
DOI: 10.4324/9781003287476
Typeset in Bembo Std
by KnowledgeWorks Global Ltd.
Access the Support Material: www.routledge.com/9781032262727
CONTENTS
About the Authors x
Acknowledgments xi
1 Measure is Fundamental 1
Developing a Theory of Measure 2
Relation to Existing Scholarship 8
Organization of the Book 12
References 13
2 The Context, Goals, and Design of the Research 17
Research Sites and Participants 17
Design Research: Engineering Learning to Support Its Study 18
The Educational Design 19
Professional Development Collaboration with Participating
Teachers 20
Learning Constructs 22
Supporting Curriculum Units 24
Assessment System 26
Digital Tools for Collecting, Displaying, and Interpreting
Student Data 29
Studies of Student Learning 33
Exploratory Learning Studies 33
Yearly Interview Data 34
Formative Assessment Tasks and In Situ Observations 36
Studies of Teacher Learning and Practice 36
References 38
vi Contents
3 Origins of Quantitative Reasoning in the Measure
of Length 41
Overview of Children’s Understanding of Length 43
Benchmarks in Thinking About Length 44
Directly Comparing Magnitudes 44
Explaining How Properties of Units Affect Measure 48
Unit Iteration and Constructing a Measurement Scale 52
2-Splitting and Symbolizing 2-Split Units as Measures 55
3-Splitting and Symbolizing 3-Split Units 59
Generalizing Relationships Among Units and Measures 61
References 64
4 Creating New Quantities in the Dynamic Generation
of Area 67
Two Coordinated Perspectives on Measure 67
Integrating the Two Perspectives on Measure 68
Direct Comparison of Magnitudes 68
Comparing Magnitudes of Area Indirectly Through Dissection
and Unit Dissection 68
Properties of Units of Area Measure 72
Dynamic Generation of Area and Product 76
Guided Reinvention of Area Measure Formulas 83
References 86
5 Extending Motion to Three Dimensions: Volume
and Its Measure 88
Structuring and Dynamic Approaches to Volume Measure 88
Volume Conceived as Space Inside 90
Measuring Volume by Accumulating Units 91
Visualizing Volume as Composites of Layers 93
Finding Volumes of Prisms with Fractional Dimension 96
Generating Volume Dynamically 97
References 101
6 Integrating Figure and Motion in the Measure of Angle 104
Dynamic and Figural Perspectives 104
Benchmarks in Thinking About Angle 108
Noticing Canonical Examples 108
Representing Angles-as-Figures, Angles-as-Turns 108
Integrating Angle-as-Turn with Angle-as-Figure, Interior vs.
Turn Angles 112
Contents vii
Generating and Justifying Angle Measure Theorems 116
Developing New Understandings of Figures and Structures via
Angle Theorems 120
References 122
7 Measurement Models of Arithmetic Operations
and Rational Number 124
Initial Resources for Reasoning About Rational Number in
Measure 125
Unit Iteration 125
Length Measure as a Point Along a Path 126
Measure-Magnitude Distinction 126
Symbolizing Measure 126
Additive and Multiplicative Comparison 127
Rational Numbers as Measured Quantities 127
Two-Split of a Unit Length and Half-Unit Iteration 127
Four- and Eight-Splits of a Unit Length and Measures in
Fourth-Unit and Eighth-Unit 129
Three-Splits and Compositions of Two- and Three-Splits of a
Unit Length 131
Fractions as Operators on Measured Quantities 132
Initial Steps in Developing a Sense of Fraction as Operator 133
Extending the Reach of Fraction-as-Operator to Refine Measure 134
Extending Multiplication and Multiplicative Comparisons 135
References 138
8 Highlights of Student Learning Research 140
The Two Phases of Research 140
Phase I: Student Conceptions of Measure as Indicated by Yearly
Interviews 142
Early-Developing Conceptions of the Measure of Length 143
Direct comparison 144
Tiling, unit, and iteration 145
Unit iteration and symbolizations of unit on scale 147
Equipartitioning fractured units 148
Measurement arithmetic 148
Conceptions of Area Measure 149
Necessary conditions for area 150
Unit structuring of area 151
Differentiating area and length measure conceptually and
symbolically 152
viii Contents
Conceptions of Angle Measure 154
Embodied turns in walking paths in primary grades 154
Conceptions of angle and measure in later grades 155
Understandings of angle theorems 156
Conceptions of Volume Measure 158
Strategies employed to measure prisms constructed of cubic
units 158
Strategies employed to measure prisms with partial
structuring 159
Volume of pentagonal prism 159
Cavalieri’s principle 160
Phase II: Summative, Formative, and In-Situ Evidence of Student
Learning 161
Summative Assessment 161
Formative Assessment and In-Situ Evidence of Student
Learning 164
Guiding instruction by monitoring conceptual
development 164
Formative assessment and dialogic space 167
Reflections and Prospects 173
References 173
9 Teacher Learning 175
Initiating and Sustaining a Teacher-Researcher Partnership 176
Supporting Development of a Shared Professional Vision 177
Constructs 177
Curriculum 179
Digital observation tools 180
Activity Structures That Forge a Professional Learning
Community 182
Learning labs 183
Mathematical investigations 187
Communal critique 188
Investigations of Change in Professional Practice 188
Teacher Noticings During Video Episodes 189
Video episodes 189
Interview procedure 191
Interview analysis 191
Interview results 192
Teacher Perspectives on Professional Development 197
Teachers’ views of learning labs 198
Contents ix
Teachers’ views of mathematical investigations 199
Teachers’ views of communal critique 200
Changes in Teachers’ Construct-Centered Judgments About
Students’ Ways of Thinking 201
Professional Vision as a Fulcrum for Learning Progression 203
References 205
10 Measures and Models in Elementary Science 207
Characterizing Growth 210
Describing Change by Determining Differences in Quantity 210
Describing Change with Intensive Quantity: Differences in
Rates 212
Describing Change in Population with Measures of
Distribution 216
Cultivating Distributional Thinking 220
Initial Steps 220
Revisiting Cause and Chance in Generating Variability 225
Melding Chance and Cause: Investigating Precision of
Measure 228
Modeling Measurements as Signal and Noise 232
Expanding the reach of modeling chance 235
Modeling Broadens the Scope of Measures 236
References 237
Index 240
