Dynamic Software in the Teaching and Learning of Geometry: Selected Papers from ICTMT-7

Guest editors:
Keith Jones, University of Southampton, UK
Federica Olivero, University of Bristol, UK

Vectors in use in a 3D juggling game simulation
Chronis Kynigos and Maria Latsi
Educational Technology Laboratory, School of Philosophy, University of Athens, Greece
kynigos@ppp.uoa.gr , mlatsi@ppp.uoa.gr

The new representations enabled by the educational computer game the ‘Juggler’ can place vectors in a central role both for controlling and measuring the behaviours of objects in a virtual environment simulating motion in three-dimensional spaces. The mathematical meanings constructed by 13 year-old students in relation to vectors as objects, as a set of properties and as representations of vectorial entities are reported and discussed in this document. It is argued that playing in the ‘Juggler’ game could make a considerable contribution to the development of intuitions and to the generation of meanings related to vectors through the use of vectorial properties.

Using 3D diagrams for teaching geometry
Giuseppe Accascina¹ and Enrico Rogora^2
1 Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza” accascina@dmmm.uniroma¹.it
²Dipartimento di Matematica, Università di Roma “La Sapienza” rogora@mat.uniroma.it

Cabri3D is a potentially very useful software for learning and teaching 3D geometry. The dynamic nature of the digital diagrams produced with it provides a useful aid for helping students to better develop concept images of geometric concepts. However, since any Cabri3D diagram represents three-dimensional objects on the two dimensional screen of a computer, some care is needed in order to avoid serious misconceptions which can arise from its use, in particular those due to the fact that projections do not preserve, in general, angles and distances. In this paper, after comparing digital diagrams (i.e. diagrams on a computer screen) with the more usual diagrams and models, we illustrate an experience around the use of Cabri3D with prospective high school teachers, aimed at clarifying which misconceptions may arise while interpreting a Cabri3D diagram.

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Encouraging the use of technology in problem-solving: some examples from an initial teacher education programme
Francis Lopez-Real and Arthur Lee
Faculty of Education, The University of Hong Kong, Hong Kong S.A.R., CHINA
lopezfj@hkucc.hku.hk , amslee@hkucc.hku.hk

As part of a module on mathematical problem solving in an Initial Teacher Education programme, the student teachers are encouraged to produce alternative solutions to the problems they tackle and, in particular, to consider whether ICT can help. In this paper we discuss a number of unusual solutions produced for some of the problems, specifically using Dynamic Geometry Software (DGS). These examples illustrate distinct functions of DGS in problem solving which we describe as follows: (i) DGS opening up new perspectives, (ii) DGS as explanation and insight, (iii) DGS highlighting the nature of exact and approximate methods, (iv) DGS as illustration and generalisation. However, underlying these different functions there is an important common feature, namely the re-formulation of problems in an ICT environment.

Designing tasks with Interactive Geometry Applets for use in research: some methodological issues
Margaret Sinclair
Faculty of Education, York University, Canada
msinclair@edu.yorku.ca

This paper discusses some of the results of a study carried out with two classes of grade 7 students (11-12 years old); the aim of the project was to design, develop, and test interactive geometry tasks for use in future research into how (or whether) interactive applets help students learn mathematics. The study tasks were developed around the topic of transformations. This paper focuses on two of the tasks; it draws attention to the complexity of the design process and the need to explicitly address methodological issues in applet design research.

Exploring Necessary and Sufficient Conditions in a Dynamic Geometry Environment
Allen Leung and Yip-Cheung Chan
Faculty of Education, The University of Hong Kong, Hong Kong SAR;
aylleung@hkucc.hku.hk , mathchan@graduate.hku.hk

This paper describes a possible dragging experiment in a dynamic geometry environment (DGE) that explores a necessary and sufficient condition for cyclic quadrilateral. The dragging modalities identified by Arzarello, Olivero, Paola, and Robutti (2002) and the functions of variation realized in DGE discussed by Leung (2003) will be used as tools to interpret and analyse the conjecture forming process. We will describe how different dragging strategies in DGE could serve different functions of variation aimed to bring about speculation of mathematical statements (in this case, a necessary and sufficient condition). It is our hope that this paper may initiate further discussion on how functions of variation could instrumentalise (Vérillon and Rabardel, 1995) dragging in DGE, within the process of geometric exploration.

Ideas for Teaching and Learning
Researching With Software: CAS, DGS and Cabri3D
Adrian Oldknow
University Of Chichester, UK
a_oldknow@compuserve.com
www.adrianoldknow.org.uk

Software tools are often designed with a particular purpose, and user community, in mind. Tools designed for one community may be seen to have potential value in educational contexts. Their use may make accessible aspects of mathematical content that would otherwise be out of reach, but may also require the acquisition of new techniques, terminology, notation etc. The associated benefits and costs need to be assessed before such tools are introduced to education; but how and by whom? This paper exemplifies approaches to mathematical research using such tools designed to contribute to that process, including work in number and algebra with computer algebra systems (CAS), 2D geometry with dynamic geometry software (DGS) and, as yet unpublished, discoveries in 3D geometry with Cabri 3D.

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