Exploiting Mental Imagery with Computers in Mathematics Education

337 Pages·1995·9.751 MB·English

by  Tommy Dreyfus (auth.), Rosamund Sutherland, John Mason (eds.)

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Series F: Computer and Systems Sciences, Vol. 138
The ASI Series F Books Published as a Result of
Activities of the Special Programme on
ADVANCED EDUCATIONAL TECHNOLOGY
This book contains the proceedings of a NATO Advanced Research Workshop held
within the NATO Special Programme on Advanced Educational Technology, running
under the auspices of the NATO Science Committee.The volumes published so far in
the Special Programme are as follows (further details are given at the end of this
volume):
67: Designing Hypermedia for Learning. 1990
76: Multimedia Interface Design in Education. 1992, 2nd corr. print 1994
78: Integrating Advanced Technology into Technology Education. 1991
80: Intelligent Tutoring Systems for Foreign Language Learning. 1992
81: Cognitive Tools for Learning. 1992
84: Computer-Based Learning Environments and Problem Solving. 1992
85: Adaptive Learning Environments: Foundations and Frontiers. 1992
86: Intelligent Learning Environments and Knowledge Acquisition in Physics. 1992
87: Cognitive Modelling and Interactive Environments in Language Learning. 1992
89: Mathematical Problem Solving and New Information Technologies. 1992
90: Collaborative Learning Through Computer Conferencing. 1992
91: New Directions for Intelligent Tutoring Systems. 1992
92: Hypermedia Courseware: Structures of Communication and Intelligent Help. 1992
93: Interactive Multimedia Learning Environments. 1992
95: Comprehensive System Design: A New Educational Technology. 1993
96: New Directions in Educational Technology. 1992
97: Advanced Models of Cognition for Medical Training and Practice. 1992
104: Instructional Models in Computer-Based Learning Environments. 1992
105: Designing Environments for Constructive Learning. 1993
107: Advanced Educational Technology for Mathematics and Science. 1993
109: Advanced Educational Technology in Technology Education. 1993
111: Cognitive Models and Intelligent Environments for Learning Programming. 1993
112: Item Banking: Interactive Testing and Self-Assessment. 1993
113: Interactive Learning Technology for the Deaf. 1993
115: Learning Electricity and Electronics with Advanced Educational Technology. 1993
116: Control Technology in Elementary Education. 1993
119: Automating Instructional Design, Development, and Delivery. 1993
121: Learning from Computers: Mathematics Education and Technology. 1993
122: Simulation-Based Experiential Learning. 1993
125: Student Modelling: The Key to Individualized Knowledge-Based Instruction. 1994
128: Computer Supported Collaborative Learning. 1995
129: Human-Machine Communication for Educational Systems Design. 1994
132: Design of Mathematical Modelling Courses for Engineering Education. 1994
133: Collaborative Dialogue Technologies in Distance Learning. 1994
135: Technology Education in School and Industry. 1994
137: Technology-Based Learning Environments. 1994
138: Exploiting Mental Imagery with Computers in Mathematics Education. 1995
140: Automating Instructional Design. 1995
141: Organizational Learning and Technological Change. 1995
142: Dialogue and Instruction. 1995
146: Computers and Exploratory Learning. 1995
Exploiting Mental Imagery
with Computers in
Mathematics Education
Edited by
Rosamund Sutherland
Department of Mathematics, Statistics and Computing
Institute of Education, University of London
London WC1H0AL, UK
John Mason
Mathematics Faculty, The Open University
Milton Keynes MK7 6AA, UK
Springer
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Research Workshop on Exploiting Mental
Imagery with Computers in Mathematics Education, held at Eynsham Hall,
Oxford, UK, May 20-25, 1993
CR Subject Classification (1991): K.3. I.3, J.2
ISBN 978-3-642-63350-8 ISBN 978-3-642-57771-0 (eBook)
DOI 10.1007/978-3-642-57771-0
CIP data applied for
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Preface
This book results from the NATO Advanced Research Workshop, Exploiting
Mental Imagery with Computers in Mathematics Education, held at Eynsham Hall
Oxford in May 1993. The advent of fast and sophisticated computer graphics has
brought dynamic and interactive images under the control of professional mathe
maticians and mathematics teachers at school and university. The workshop
brought together researchers from a variety of disciplines, computer science,
mathematics, mathematics education, psychology and design, in order to explore
the ways in which images generally and computer generated images in particular
can be exploited and developed. Throughout the workshop participants actively
engaged in discussion of their collective research and also worked on their own
use of mental imagery. Participants were involved in peer group reviewing of the
conference papers and the chapters in this book represent the lively debate which
was provoked by the conference.
A number of central issues for further research emerged from the conference.
One of these was the role of sketching in mathematical activity. There was general
agreement that the sparseness of a sketch facilitates the creation, manipulation and
transformation of visual images, but debate as to whether or not computer
environments would be able to support this sketching process. The workshop drew
attention to the need for research to investigate the characteristics and role of
mathematical sketching on paper. It seems that competent mathematical students
make use of a range of representations within this sketching process, but further
research should be carried out on how student's use of multiple representations
relates to their presentation within computer environments.
The workshop was concerned with the educational implications of computer
based imagery. There was agreement that students actively construct mental
images and that the student's constructions will not necessarily be the same as
those presented by the teacher or the computer. We became aware throughout the
conference that some participants were more able than others to construct and
manipulate mental images of mathematical objects, and that this ability cannot be
separated from mathematical knowledge and previous experience.
A number of presentations centred around the idea that geometry offers a
privileged domain for an analysis of the dialectic process of conceptualisation and
visualisation. But attention was also drawn to the fact that geometrical figures are
VI Preface
a special kind of image taking on a dual discursive and semiotic role. Imagery
plays a role within all mathematical domains (for example algebra) and we must
be cautious about generalising research results from one domain to another. We
agreed about the potential of computer-based dynamic imagery which can be
directly manipulated by students, because movement provides the opportunity for
students to discover mathematical properties by attending to what is variable and
what is invariant. The workshop highlighted the need for more research on how
students appropriate and make use of these dynamic images. Throughout the
conference there was debate about the relationship between mental imagery and
spoken and written discourse. Some participants believed that it is not sufficient to
build up, in the imagination, representations of mathematical objects and concepts,
because mathematical objects (for example the natural numbers) are essentially
discursive.
We have organised the book into four parts. The authors in Part I are concerned
with the role of the external imagery. In Part 2 the authors focus on the role of
mental imagery in the teaching and learning of geometry. Authors in Part 3
develop the links between screen and mental imagery. Finally in Part 4 the authors
focus on the ways in which students read and use imagery.
We would like to thank the members of the organising committee, Nicolas
Balacheff, Sandy Dawson and Maria Alessandra Mariotti for all their support in
the planning of the conference. We are also very grateful to Lulu Healy and
Stefano Pozzi for their energetic support throughout the conference. Finally the
production of this book could not have been completed without the systematic
endeavour of Magdalen Meade.
January 1995 Rosamund Sutherland
John Mason
Contents
Part 1. Emphasizing the External
Imagery for diagrams 3
Tommy Dreyfus
External representations in arithmetic problem solving 20
Giuliana Dettori and Enrica Lemut
Visualisation in mathematics and graphical mediators:
an experience with 11-12 year old pupils ................................................... 34
Angela Pesci
Visual organisers for formal mathematics 52
David Tall
Mediating mathematical action 71
Rosamund Sutherland
Mathematical objects, representations, and imagery 82
Willibald D6rjler
Part 2. Imagery in Support of Geometry 95
Images and concepts in geometrical reasoning 97
M. Alessandra Mariotti
Between drawing and figure 117
Reinhard Holzl
The functions of visualisation in learning geometry 125
Eric Love
Geometrical pictures: kinds of representation and specific processings 142
Raymond Duval
VIII Contents
Part 3. Linking Screen and Mental Imagery 159
Overcoming physicality and the external present: cybernetic manipulatives 161
James J. Kaput
On visual and symbolic representations ...................................................... 178
Luis E. Moreno A. and Ana Isabel Sacristan R.
The dark side of the Moon .. ........ ........ ...... ........ .......... ...... ............ ............... 190
Richard Noss and Celia Hoyles
Ruminations about dynamic imagery (and a strong plea for research) 202
E. Paul Goldenberg
On designing screen images to generate mental images 225
Richard J. Phillips, John Gillespie, and Daniel Pead
Learning as embodied action ........ ............ .......... ........ .......... ........ ............... 233
Stephen Campbell and A. J. (Sandy) Dawson
Part 4. Employing Imagery 251
The importance of mental perception when creating research pictures 252
Monique Sicard and Jean-Alain Marek
Random images on mental images 263
Mario Barra
Imagery as a tool to assist the teaching of algebra 277
Dave Hewitt
Mathematical screen metaphors 291
John Mason and Benedict Heal
Exploiting mental imaging: reflections of an artist on a mathematical
excursion ............ ................ .............................................. ....... ....... .............. 309
Stephen A.R. Scrivener
Index 323
Part 1
Emphasising the External
This part focuses on the ways in which external images and discourse interact
with students' approaches to solving problems in mathematics. In the first chapter
Tommy Dreyfus suggests that visual images contain strongly interpreted
variations, arguing against the classical view of imagery as internalised
perception. In contrast to Eric Love (Part 2) he maintains that the ability to
transform mentally visual images can be enhanced by external visual support. He
stresses that students have to learn to read mathematical diagrams, and suggests
that there could be advantages in students constructing these diagrams for
themselves and disadvantages in diagrams being automatically constructed by a
computer. In school mathematics students are not encouraged to communicate
with diagrams and a diagram does not usually count as an answer to a problem,
and so, if we want students to use diagrams, their communciation function needs
more emphasis. Dreyfus ends this chapter by discussing the potential importance
of thinking with 'vague' images, a notion for which the term knodeling was
coined during the conference.
Giuliana Dettori and Enrica Lemut are concerned with the role of external
representations in arithmetic problem solving. They stress that external represen
tations become tools for dialogue both inter-personally and intra-personally and
emphasise the link between external and internal representations. They agree with
Tommy Dreyfus that students have to be taught how to represent mathematicsl
problems, and discuss the differences between representing a resolution strategy
and representing a computation strategy. They suggest that good problem solvers
use representations which contain the germs of their resolution strategies. They
also suggest that pupils can use external representations to support and unblock
arithmetic problem solving processes. Angela Pesci also focuses on the use of
external representations in arithmetic problem solving, in particular the use of
arrow schemes in inverse problems. She maintains that these graphical images
mediate, in the sense of Vygotsky, the cognitive activities needed to solve inverse
arithmetic problems. Finally in this chapter Angela Pesci discusses some of the
methodological issues involved in carrying out research in school settings.
David Tall discusses his research with students who were training to be
mathematics teachers. These students were presented with computer-based visual
images in order to encourage them to visualise and verhalise advanced mathe-
2 Part 1
matical ideas. He discusses ways of displaying meaningful images of functions,
which discriminate between rational and irrational numbers, and the relevance of
these visual displays in developing a sense of continuity and discontinuity,
differentiability and non-differentiability. He also discusses the weakness of using
a finite computer screen to display images of objects for which infinity is an
essential component.
In the fifth chapter, Rosamund Sutherland considers the importance of external
representations as mediators of mathematical problem solving and maintains that
it is the "person acting with mediational means" which should be the focus of
study. This allows for students to solve problems with a range of external visual
and symbolic tools, and places value on these external productions as opposed to
internal mental processing.
All the authors of the preceding chapters, whilst focusing on external imagery,
do not deny the role of mental imagery. Willibald Dorfler, on the other hand,
maintains that there can be no mental imagery of abstract mathematical objects,
for example the mathematical idea of even-ness. He stresses that mathematical
objects are created by mathematical discourse and it is discourse which allows us
to talk about these abstract objects and which constitutes the rules of how to talk
about them. From this perspective a system of representations provides meaning
for the mathematical discourse, but is qualitatively different from this discourse.
Students have to become convinced about the sense of the mathematical
discourse, and Dorfier suggests that representation systems can support this sense
making.