The Relationship between Language and Spatial Ability: An Analysis of Spatial Language for Reconstructing the Solving of Spatial Tasks

251 Pages·2017·3.33 MB·English

by  Angel Mizzi (auth.)

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
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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
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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)