Essener Beiträge zur Mathematikdidaktik Angel Mizzi The Relationship between Language and Spatial Ability An Analysis of Spatial Language for Reconstructing the Solving of Spatial Tasks Essener Beiträge zur Mathematikdidaktik Reihe herausgegeben von B. Barzel, Essen, Deutschland A. Büchter, Essen, Deutschland B. Rott, Köln, Deutschland F. Schacht, Essen, Deutschland P. Scherer, Essen, Deutschland In der Reihe werden ausgewählte exzellente Forschungsarbeiten publiziert, die das breite Spektrum der mathematikdidaktischen Forschung am Hochschulstand ort Essen repräsentieren. Dieses umfasst qualitative und quantitative empirische Studien zum Lehren und Lernen von Mathematik vom Elementarbereich über die verschiedenen Schulstufen bis zur Hochschule sowie zur Lehrerbildung. Die publizierten Arbeiten sind Beiträge zur mathematikdidaktischen Grundlagen und Entwicklungsforschung und zum Teil interdisziplinär angelegt. In der Reihe erscheinen neben Qualifikationsarbeiten auch Publikationen aus weiteren Essener Forschungsprojekten. Weitere Bände in der Reihe http://www.springer.com/series/13887 Angel Mizzi The Relationship between Language and Spatial Ability An Analysis of Spatial Language for Reconstructing the Solving of Spatial Tasks With a Preface by Prof. Dr. Andreas Büchter Angel Mizzi Essen, Germany Dissertation der Universität DuisburgEssen, 2017 Von der Fakultät für Mathematik der Universität DuisburgEssen genehmigte Dissertation zur Erlangung des Doktorgrades „Dr. rer. nat.“ Datum der mündlichen Prüfung: 27.07.2017 Gutachter: Prof. Dr. Andreas Büchter, Prof. Dr. Benjamin Rott ISSN 25093169 ISSN 25093177 (electronic) Essener Beiträge zur Mathematikdidaktik ISBN 9783658206314 ISBN 9783658206321 (eBook) https://doi.org/10.1007/9783658206321 Library of Congress Control Number: 2017963266 Springer Spektrum © Springer Fachmedien Wiesbaden GmbH 2017 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. Printed on acidfree paper This Springer Spektrum imprint is published by Springer Nature The registered company is Springer Fachmedien Wiesbaden GmbH The registered company address is: AbrahamLincolnStr. 46, 65189 Wiesbaden, Germany Preface Spatial-geometrical competencies are required in mathematics continuously, starting from preschool education, then during primary and secondary education until mathematics at university level. Besides arithmetic (which is based on the concept of time), geometry (based on the concept of space) is one of the two ini- tial ways of operating mathematically. The crucial role of geometrical thinking in general and the solving of spatial-geometrical tasks specificially lead to the long research tradition in spatial ability in both mathematics education and psy- chology. In traditional psychometric-based intelligence research, spatial-geometrical com- petencies form a fundamental factor of intelligence in all models. Ordinarily, this factor is considered as largely independent from verbal competencies; especially in popular science discussions, whereby assumptions about different intelligence profiles (e.g., with local strengths in "figural intelligence" or in "verbal intelli- gence") are sometimes based on such hypotheses. The interdisciplinary research about the relationship between language and con- tent learning has been researched intensively in Germany in recent years; in mathematics education in particular by Susanne Prediger's research group in TU Dortmund. This research strand emphasises the dependence of other cognitive competencies from the verbal competencies, i.e. the interdependency between verbal and other cognitive competencies. Previous findings in language and con- tent learning research can be categorised either as general results about the inte- grated verbal and content learning or as results which are specific for particular content in mathematics education. Relevant works are present for mathematical content beyond geometry, e.g. fractions, percentages, descriptive statistics, func- tions, and elementary algebra. In contrast, there is still a substantial need for re- search regarding the integrated language and content learning in geometry, in particular in the area of spatial geometry. A fundamental question arising in the context of mathematics education and mathematics classroom is: Which role do verbal competencies or language use play during the solving of spatial-geometrical tasks? The dissertation of Angel Mizzi addresses this issue - according to the interdisciplinary character of the question - by successfully synthesising theoretical principles from mathematics (cid:2) V education, from linguistics and language teaching, and from psychology. The design of the reconstruction method in his work generates a data collection situation which enables a methodologically-controlled empirical study of his underlying research questions. Hereby, a student is required to rebuild a pre- designed object made up of building cubes based on oral instructions from another student. The foci of this study include the strategies developed by students and potential obstacles arising during the solving process, which can be observed in the communication setting. These goals were achieved by an analysis of spatial language used by students in the solving processes. In view of this theoretical and methodological background, the author succeeds in developing important results which show that language plays an important role during the solving of spatial-geometrical tasks. The analysis of spatial language used by students leads to a differentiation of previous models about strategies used for solving spatial-geometrical tasks. By his own way of use and enrichment of the construct of spatial language, Angel Mizzi succeeds to create an important foundation work for a deeper investigation of the relationship between language and spatial ability. Due to the importance of the underlying issue focussed on in this work further future research projects are expected to emerge following this dissertation. Prof. Dr. Andreas Büchter University of Duisburg-Essen VI (cid:2) Acknowledgements I would like to express my sincere gratitude to my supervisor Prof. Dr. Andreas Büchter for his continuous support of my Ph.D study and for his patience and immense knowledge attributing to the intense and fruitful discussions which led to the development of this work. I would also like to thank my second supervisor, Prof. Dr. Benjamin Rott from the University of Cologne, whose guidance helped me at the time of this research and who was always present for answering any questions concerning this research project and mentored my work. My sincere thanks also goes to the mathematics teachers, especially Daniel Jung, Okan Kaplan and Christina Hohenstein, who helped me conduct the test instruments and select students from their classrooms or schools for this research study. I would like to thank my colleagues from the Buechter working group at University of Duisburg-Essen, especially Sabine Schlager and Dr. Christina Krause, and overseas colleagues, especially Prof. Boon Liang Chua and Prof. Ban Heng Choy from the National Institute of Education in Singapore and Prof. Wes Maciejewski from San José State University, for supporting me throughout the writing of this thesis. Last but not least, I would like to thank my family which always supported me throughout my life for achieving a good education. (cid:2) VII Table of Contents Preface ................................................................................................................. V Acknowledgements .......................................................................................... VII 1. Introduction ...................................................................................................... 1 1.1 Overture ...................................................................................................... 1 1.2 Motivation ................................................................................................... 1 1.3 Background and purpose of study ............................................................... 3 1.4 Aims of research ......................................................................................... 6 2. Theoretical framework .................................................................................... 9 2.1 Spatial ability .............................................................................................. 9 2.1.1 Definitions of spatial ability and approaches to its research ............. 10 2.1.2 Cognitive processes in solving spatial tasks ..................................... 12 2.1.3 Selected spatial ability models .......................................................... 17 2.1.4 Spatial abilities in German mathematics curriculum and classroom 22 2.1.5 Strategies for solving spatial tasks .................................................... 23 2.1.6 Sex differences in spatial ability performance .................................. 28 2.1.7 Summary: Spatial ability ................................................................... 30 2.2 Language in mathematics classroom ........................................................ 31 2.2.1 Definition of the notion of language ................................................. 32 2.2.2 Language and thinking ...................................................................... 34 2.2.3 The functions of language ................................................................. 36 2.2.4 Levels of language acquisition .......................................................... 37 2.2.5 Metaphors in mathematics classroom ............................................... 40 2.2.6 Roles of language in German mathematics curriculum and classroom ........................................................................................... 42 2.2.7 Summary: Language in mathematics classroom ............................... 44 2.3 Interplay of spatial ability and language ................................................... 45 2.3.1 Spatial language ................................................................................ 46 2.3.2 Spatial task strategies and language .................................................. 54 2.3.3 Summary: Interplay of language and spatial ability .......................... 56 2.4 Representations of mathematical knowledge ............................................ 56 2.4.1 Sfard’s dual nature of mathematical conceptions ............................. 57 2.4.2 Bruner’s modes of representation ..................................................... 58 2.4.3 Summary: Representations of mathematical knowledge .................. 60 (cid:2) IX 3. Methodology ................................................................................................... 61 3.1 Research questions .................................................................................... 61 3.2 Research paradigm .................................................................................... 63 3.3 Research design process ............................................................................ 66 3.4 Research method ....................................................................................... 67 3.4.1 Reconstruction method ...................................................................... 68 3.4.2 Design principles of the reconstruction method ................................ 70 3.4.3 Limitations of the reconstruction method ......................................... 76 4. Design and Implementation .......................................................................... 77 4.1 Pilot study ................................................................................................. 77 4.1.1 Aims of the pilot study ...................................................................... 77 4.1.2 Design of the pilot study ................................................................... 78 4.1.3 Sampling and implementation ........................................................... 81 4.1.4 Results and consequences of pilot study ........................................... 81 4.2 Design of main study ................................................................................ 85 4.2.1 Reconstruction method in the main study ......................................... 86 4.2.2 Task design ........................................................................................ 87 4.2.3 Sampling ............................................................................................ 97 4.2.4 Quality criteria ................................................................................. 108 4.2.5 Implementation of main study ......................................................... 113 4.2.6 Data analysis ................................................................................... 116 5. Results and discussion from the inductive data analyses ......................... 125 5.1 Description of the identified strategies ................................................... 125 5.1.1 Spatial metaphors ............................................................................ 125 5.1.2 Object break-down strategy ............................................................ 142 5.1.3 Assembling strategy ........................................................................ 145 5.1.4 Rotation strategy ............................................................................. 147 5.1.5 Cubes controlling strategy ............................................................... 149 5.1.6 Structure controlling strategy .......................................................... 150 5.1.7 Discussion of the identified strategies and review of literature ...... 151 5.2 Description of identified obstacles .......................................................... 156 5.2.1 Spatial metaphors as obstacles ........................................................ 158 5.2.2 Describing spatial relations ............................................................. 160 5.2.3 Verbalising rotation ......................................................................... 164 5.2.4 Dimension reduction ....................................................................... 169 X (cid:2)
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