Table Of ContentResearch in Mathematics Education
Series Editors: Jinfa Cai · James A. Middleton
Anderson Norton
Martha W. Alibali Editors
Constructing
Number
Merging Perspectives from Psychology
and Mathematics Education
Research in Mathematics Education
Series Editor:
Jinfa Cai
Newark, DE, USA
James A. Middleton
Tempe, AZ, USA
This series is designed to produce thematic volumes, allowing researchers to access
numerous studies on a theme in a single, peer-reviewed source. Our intent for this
series is to publish the latest research in the field in a timely fashion. This design is
particularly geared toward highlighting the work of promising graduate students
and junior faculty working in conjunction with senior scholars. The audience for
this monograph series consists of those in the intersection between researchers and
mathematics education leaders—people who need the highest quality research,
methodological rigor, and potentially transformative implications ready at hand to
help them make decisions regarding the improvement of teaching, learning, policy,
and practice. With this vision, our mission of this book series is:
1. To support the sharing of critical research findings among members of the math-
ematics education community;
2. To support graduate students and junior faculty and induct them into the research
community by pairing them with senior faculty in the production of the highest
quality peer-reviewed research papers; and
3. To support the usefulness and widespread adoption of research-based
innovation.
More information about this series at http://www.springer.com/series/13030
Anderson Norton • Martha W. Alibali
Editors
Constructing Number
Merging Perspectives from Psychology and
Mathematics Education
Editors
Anderson Norton Martha W. Alibali
Department of Mathematics Department of Psychology
Virginia Tech University of Wisconsin–Madison
Blacksburg, VA, USA Madison, WI, USA
ISSN 2570-4729 ISSN 2570-4737 (electronic)
Research in Mathematics Education
ISBN 978-3-030-00490-3 ISBN 978-3-030-00491-0 (eBook)
https://doi.org/10.1007/978-3-030-00491-0
Library of Congress Control Number: 2018961408
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Foreword
As the editors of this series, we both consider ourselves mathematics education
researchers, but we have had significant training in cognitive psychology. Jinfa Cai
majored in Cognitive Studies in Mathematics Education and served as a research
assistant at the University of Pittsburgh’s Learning Research and Development
Center for 5 years. Jim Middleton is an educational psychologist whose doctoral
work at the University of Wisconsin was focused at the interface between applied
cognitive science and mathematics learning. For the past 25 years, our research has
focused on cognitive studies in mathematics education: On learning mathematics
and on designing curriculum, pedagogical strategies, and technology that supports
learning mathematics. In our experiences over the years, we find it interesting (and
a bit disappointing) that mathematics educators and cognitive psychologists—two
groups of researchers interested in many of the same issues related to mathematics
learning and teaching—have collaborated and interacted very little on the grand
scale. It is true that their research on mathematics learning and teaching is typically
conducted from different angles—each representing a different perspective on a
common problem—but our experiences have shown us that these perspectives are
complementary, not conflicting.
Because we both benefitted greatly from our interdisciplinary training, we have
long worked to facilitate common dialogue in our respective research circles, at the
National Science Foundation, and in our roles as leaders in mathematics education
research, where such dialogue has been deep and genuine, mathematics education
research has advanced both theoretically and pragmatically in ways that reflect the
strengths of the two perspectives. Moreover, new theory and new approaches to
teaching and learning have resulted from working in the interstices of our communi-
ties. This book is a product of such an effort. It critically examines research on the
learning of number that combines cognitive, developmental psychology and math-
ematics education approaches. This is a sister book to a previous volume in this
series on spatial visualization in mathematics (edited by Mix and Battista, 2019).
Both volumes use the device of scholars reporting their own work, followed by criti-
cal commentary written by colleagues with complementary expertise.
v
vi Foreword
In doing so, the editors of this book are developing mathematical epistemology,
asking us, “what does it mean to learn, know, and understand mathematics?” The
argument begins with mathematics itself—what it is as a field of knowledge and
practice—and flows from neuropsychological foundations, through perception,
through construction of number to the broadening of understanding that addresses
the fields of rational and negative numbers. Thus, the book grounds learners’ con-
struction and conceptual development in fundamental understanding of learning
processes, yet also reflects the important content which has puzzled students and
researchers alike for centuries.
Finally, as series editors, we wish to thank the editors for this volume on numbers
(Norton and Alibali) and the sister volume on spatial visualization (Mix and
Battista), as well as authors for the quality of the chapters and commentaries they
have provided.
Jinfa Cai
University of Delaware
Newark, DE, USA
James A. Middleton
Arizona State University
Tempe, AZ, USA
Preface
This book synergizes research across two disciplines—mathematics education and
psychology—to address how children construct number. The opening chapter
frames the problem in terms of children’s activity, including mental and physical
actions. Subsequent chapters are organized into sections that address specific
domains of number: natural numbers, fractions, and integers. Chapters within each
section address ways that children build upon biologically based foundational abili-
ties (e.g., subitizing, the approximate number system) and prior constructs (e.g.,
counting sequences) to construct number. The chapters address a range of change
mechanisms (e.g., reflective abstraction, analogy), a range of social contexts (e.g.,
informal interactions, formal educational settings), and a range of tools (e.g., cur-
ricular materials, technological tools). The book relies on co-authored chapters and
commentaries at the end of each section to create dialogue among scholars from
different disciplines. The final chapter brings this collective work together around
the theme of children’s activity and also considers additional themes that arise
within the chapters.
We hope that this book will foster additional dialogue between psychologists and
mathematics educators. As the chapters in the book demonstrate, mathematics edu-
cators can benefit from a better understanding of psychological constructs that they
might leverage to support students’ mathematical development. Conversely, psy-
chologists can benefit from a better understanding of ways that students’ activity
supports that development.
Blacksburg, VA, USA Anderson Norton
Madison, WI, USA Martha W. Alibali
vii
Contents
1 Mathematics in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Anderson Norton and Martha W. Alibali
Part I Natural Numbers and Operations on Natural Numbers
2 Subitizing: The Neglected Quantifier . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Douglas H. Clements, Julie Sarama, and Beth L. MacDonald
3 Discerning a Progression in Conceptions of Magnitude During
Children’s Construction of Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Catherine Ulrich and Anderson Norton
4 Spontaneous Mathematical Focusing Tendencies
in Mathematical Development and Education . . . . . . . . . . . . . . . . . . . . 69
Jake McMullen, Jenny Yun-Chen Chan, Michèle M. M. Mazzocco,
and Minna M. Hannula-Sormunen
5 Leveraging Relational Learning Mechanisms to Improve
Place Value Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Kelly S. Mix, Linda B. Smith, and Sandra Crespo
6 The Complexity of Basic Number Processing:
A Commentary from a Neurocognitive Perspective . . . . . . . . . . . . . . . 123
Bert De Smedt
Part II Fractions and Operations on Fractions
7 U nderstanding Fractions: Integrating Results from Mathematics
Education, Cognitive Psychology, and Neuroscience . . . . . . . . . . . . . . . 135
Andreas Obersteiner, Thomas Dresler, Silke M. Bieck, and
Korbinian Moeller
8 D eveloping Fractions as Multiplicative Relations:
A Model of Cognitive Reorganization . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Ron Tzur
ix
x Contents
9 Developing a Concept of Multiplication of Fractions:
Building on Constructivist and Sociocultural Theory . . . . . . . . . . . . . . 193
Martin A. Simon
10 What’s Perception Got To Do with It? Re-framing
Foundations for Rational Number Concepts . . . . . . . . . . . . . . . . . . . . . 213
Percival G. Matthews and Ryan Ziols
11 Commentary on Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Sybilla Beckmann
Part III Integers and Operations on Integers
12 Understanding Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Laura Bofferding
13 Integers as Directed Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Nicole M. Wessman-Enzinger
14 Cognitive Science Foundations of Integer Understanding
and Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Sashank Varma, Kristen P. Blair, and Daniel L. Schwartz
15 Commentary on Negative Numbers: Aspects of Epistemology,
Cognition, and Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Guershon Harel
16 Synergizing Research on Constructing Number:
Themes and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Martha W. Alibali and Anderson Norton
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
