Research in Mathematics Education Series Editors: Jinfa Cai · James A. Middleton Kelly S. S. Mix · Michael T. Battista Editors Visualizing Mathematics The Role of Spatial Reasoning in Mathematical Thought Research in Mathematics Education Series editors 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 mathematics 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 Kelly S. S. Mix • Michael T. Battista Editors Visualizing Mathematics The Role of Spatial Reasoning in Mathematical Thought Editors Kelly S. S. Mix Michael T. Battista Department of Human Development Department of Teaching and Learning and Quantitative Methodology The Ohio State University University of Maryland Columbus, OH, USA College Park, MD, USA ISSN 2570-4729 ISSN 2570-4737 (electronic) Research in Mathematics Education ISBN 978-3-319-98766-8 ISBN 978-3-319-98767-5 (eBook) https://doi.org/10.1007/978-3-319-98767-5 Library of Congress Control Number: 2018958712 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Foreword In 2016, we were approached by series editor, Dr. Jinfa Cai, with a novel idea— invite authors from the fields of developmental psychology and mathematics educa- tion to write about their work on spatial visualization and mathematics, and then ask them to write commentaries on one another’s chapters. The goal was to provide a unique view of research on this topic that encompassed both disciplines, as well as foster cross-field communication and intellectual synergy. We eagerly took up the challenge and invited scholars whose work we knew to be at the forefront of our respective fields. The chapters and commentaries contained in this volume are the products of this esteemed group. They reflect the state of the art in research on spa- tial visualization and mathematics from at least two perspectives. They highlight important new contributions, but they also reveal the fault lines between our respec- tive disciplines. The commentaries insightfully point out some of these fault lines, as well as the immense common ground and the potential for deeper collaboration in the future. The basic question of how spatial skill relates to mathematics has received steady attention over the years. In psychology, most of this work has focused on long-term outcomes in STEM fields for individuals with more advanced spatial skill (e.g., Wai, Lubinski, & Benbow, 2009), the possibility that spatial deficits contribute to poor mathematics outcomes in children (e.g., Geary, Hoard, Byrd-Craven, Nugent & Numtee, 2007), and the use of materials that physically embody (via spatial rela- tions) abstract mathematics concepts (see Mix, 2010, for a review). Running through these disparate research programs is the shared notion that spatial thinking plays a major role in understanding mathematics, but it has not been addressed head on in psychology until recent years. In mathematics education, Clements and Battista, in their 1992 research review, address just this issue. They wrote that both Hadamard and Einstein (renown math- ematicians) claimed that much of the thinking required in higher mathematics is spatial, and they cited positive correlations between spatial ability and mathematics achievement at all grade levels. However, even in that time period, the relations between spatial thinking and learning nongeometric concepts did not seem straight- forward, and there were conflicting findings. For some tasks, having high-spatial v vi Foreword skill seemed to improve performance, whereas in other tasks, processing mathemat- ical information using verbal-logical reasoning enhanced performance compared to students who processed the information visually. Other mathematics education researchers countered that the understanding of some low-spatial students who did well in mathematics was instrumental, whereas high-spatial students’ understand- ing was more relational, a difference often not captured by classroom or standard- ized assessments. Clements and Battista concluded that even though there was reason to believe that spatial reasoning is important in students’ learning and use of mathematical concepts—including nongeometric concepts—the role that such rea- soning plays in this learning remained elusive. Possibly because of this elusiveness, interest in the topic waned in mathematics education. However, currently there is intense interest in this general topic in both psychology and mathematics education due to its potential educational benefits (Newcombe, 2010) and the insights into the relations found through extensive and detailed student interviews (Bruce et al., 2017; Davis et al., 2015). The chapters contributed to this volume represent various approaches to advancing this work in education or moving the work in both fields toward educational application. The developmental psychology chapters tended to focus on the underlying men- tal representations used to understand mathematics, and the extent to which these representations already involve, or could be improved by spatial processing. Cipora, Schroeder, Soltanlou, and Nuerk provide a detailed analysis of the link between spatial and numerical processing purportedly demonstrated by spatial-numerical association (SNA) or mental number line effects. They conclude that spatial skills provide a crucial tool for understanding mathematics, but this relation may not be realized in the form of a fixed mental number line. Congdon, Vasileyva, Mix, and Levine examine a deep psychological structure that may underlie a range of math- ematics topics—namely, the structure involved in identifying and enumerating spa- tial units of measurement. They argue that mastery of this structure has the potential to support mathematics learning throughout the elementary grades and perhaps head off misconceptions related to fractions, proportions, and conventional later on. Similarly, Jirout and Newcombe focus on another spatial relation with strong ties to mathematics—namely, relative magnitude—outline its potential role in improving instruction on whole number ordering, fractions, and proportions. Casey and Fell discuss the difference between general spatial skill and spatial skill instantiated in specific mathematics problems, concluding that the most effective way to leverage spatial training to improve mathematics outcomes is likely the latter. They highlight a number of instructional techniques from existing curricula that successfully use spatial representations. Finally, Young, Levine, and Mix considered the multidimen- sional nature of spatial processing and mathematics processing and the inherent complexity involved in identifying possible instructional levers. Following a cri- tique of the existing literature, including recent factor analytic approaches, they conclude with a set of recommendations for improving these approaches and apply- ing what is already known in educational settings. The mathematics education chapters discuss the spatial processes involved in specific topics in mathematics. Sinclair, Moss, Hawes, and Stephenson examine Foreword vii how children can learn “through and from drawing,” focusing on spatial processes and concepts in primary school geometry. They argue that drawing is not innate but can be improved, and they illustrate through fine-grained analysis how the potential benefits of geometric drawing can be realized in classrooms. Gutiérrez, Ramírez, Benedicto, Beltrán-Meneu, and Jaime analyze the spatial reasoning of mathemati- cally gifted secondary school students as they worked on a collaborative, communication- intensive, task in which they were shown orthogonal projections of cube buildings along with related verbal information. The authors related the objec- tives of students’ actions and their visualization processes and students’ solution strategies and cognitive demand. Herbst and Boileau argue that high school geom- etry instruction can do more than provide names for 3D shapes and formulas for finding surface area and volume. They illustrate, and invite reflection on their design of, a 3D geometry modeling activity in which students write and interpret instruc- tions for how to move pieces of furniture up an L staircase. Lowrie and Logan dis- cuss how the frequency of encountering, and interacting with, information in visual/ graphic format, including on the web, has increased our need for research on the role of spatial reasoning in students’ encoding and decoding of information in math- ematics. To this end, they analyze the representational reasoning of students engaged in tasks that permit different types of representations, from diagrams to equations. Battista, Frazee, and Winer describe the spatial processes involved in reasoning about the geometric topics of measurement, shapes, and isometries. They introduce, and use in their analysis, the construct of spatial-numerical linked structuring as the coordinated process in which numerical operations on measurement numbers are linked to spatial structuring of, and operation on, the measured objects in a way that is consistent with properties of numbers and measurement. As the chapters and commentaries illustrate, there are still fundamental differences between how researchers in psychology and mathematics education view and investi- gate the fundamental relations between spatial and mathematical reasoning. However, these differences provide fertile ground for exciting new investigations as each field respectively has advanced knowledge in some areas while leaving gaps in others. The commentaries are a starting point for identifying these points of contact and com- plementarity. We encourage readers to reflect on how the research in the two fields might be further integrated and how to build productive collaborations between the two sets of researchers. We thank all of our authors for taking a first step in this direction. Michael T. Battista Department of Teaching and Learning The Ohio State University, Columbus, OH, USA Kelly S. S. Mix Department of Human Development and Quantitative Methodology University of Maryland, College Park, MD, USA viii Foreword References Bruce, C., Davis, B., Sinclair, N, McGarvey, L., Hallowell, D., …, Woolcott, G. (2017). Understanding gaps in research networks: using “spatial reasoning” as a window into the importance of networked educational research. Educational Studies in Mathematics, 95(2), 143–161. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). New York: Macmillan. Davis, B., & the Spatial Reasoning Study Group. (2015). Spatial Reasoning in the Early Years: Principles, assertions, and speculations. New York: Routledge. Geary, D. C., Hoard, M. K., Byrd-Craven, J., Nugent, L., & Numtee, C. (2007). Cognitive mecha- nisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78(4), 1343–1359. Mix, K. S. (2010). Spatial tools for mathematical thought. In K.S. Mix, L. B. Smith, & M. Gasser (Eds.), The spatial foundations of language and thought (pp. 41–66). New York: Oxford University Press. Newcombe, N. S. (2010). Picture this: Increasing math and science learning by improving spatial thinking. American Educator, 34(2), 29–43. Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of Educational Psychology, 101(4), 817–835. Contents Part I Psychological Perspectives 1 How Much as Compared to What: Relative Magnitude as a Key Idea in Mathematics Cognition . . . . . . . . . . . . . . . . . . . . . . . 3 Jamie Jirout and Nora S. Newcombe 2 From Intuitive Spatial Measurement to Understanding of Units . . . . 25 Eliza L. Congdon, Marina Vasilyeva, Kelly S. S. Mix, and Susan C. Levine 3 Spatial Reasoning: A Critical Problem-Solving Tool in Children’s Mathematics Strategy Tool-Kit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Beth M. Casey and Harriet Fell 4 More Space, Better Mathematics: Is Space a Powerful Tool or a Cornerstone for Understanding Arithmetic? . . . . . . . . . . . . . . . . . . . . 77 Krzysztof Cipora, Philipp Alexander Schroeder, Mojtaba Soltanlou, and Hans-Christoph Nuerk 5 What Processes Underlie the Relation Between Spatial Skill and Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Christopher Young, Susan C. Levine, and Kelly S. S. Mix Part II Commentaries 6 Part I Commentary 1: Deepening the Analysis of Students’ Reasoning About Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Michael T. Battista, Leah M. Frazee, and Michael L. Winer 7 Part I Commentary 2: Visualization in School Mathematics Analyzed from Two Points of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 A. Gutiérrez ix