ICTMT 5 Klagenfurt - 2001, August 6-9

Abstracts

Mary Abboud, Lebanon:

Animation, a Tool for Understanding Polar Coordinates

Students in undergraduate classes have a great deal of difficulty in plotting graphs of functions given in polar co-ordinates. In previous work done, animation was used as a tool to understand how a linear transformation affects the graph of a function, and here I am extending this work to enable students to better understand polar Coordinates and the relationship to Cartesian Coordinates. In our experience, the use of a Computer Algebra System such as Mathematica does not by itself guarantee that students will improve their visualisation skills or their understanding of mathematical concepts. It is necessary to design projects where students are encouraged to observe phenomena, make conjectures and then test whether these conjectures are really true. We present in this paper the work that we have done and which can be used with students of Calculus.

George Adie, Sweden:

Practical applications of CAS using sinusoidal functions

A lot of physics is involved with the study of sinusoidal variation. In this talk we will show how handheld technology with CAS changes our approach making the physics more accessible for students and allowing the physics course to become deeper and more meaningful. We will also discuss appropriate changes in the students´maths course.

George Adie, Sweden:

Differential Equations in maths and physics instead of analytical methods

Handheld technology with CAS makes it easier to study common scientific phenomena at undergraduate level directly using differential equations and numerical techniques instead of the conventional analytical methods. We will highlight areas of study where requirements are changing in physics using linear and non linear differential equations in one or more dimensions. This leads to changing requirements in mathematics. These changes will be discussed.

Bengt Ahlander, Sweden

How to Make Tests for Students Using CAS Tools (TI-89)

In my school, Ostrabogymnnasiet an upper secondary school in Sweden, I work with a math class where every student uses the TI-89. The age of the students is 17 year. My thoughts about how to examine students using this powerful tool and still testing the understanding of mathematics will be explained. Questions such as “What are the roots of the equation x^2-6x + 5 = 0?” are not testing the understanding if you use the TI-89. But if you give the students the answer (the roots of a quadratic equation are x = 5 and x = 1), you can ask the students to give examples of equations that will give this answers. This is a kind of jeopardy in maths and really tests if they have the understanding behind the solutions of quadratic equations. We can also give questions with some solutions and ask the students to control and explain the steps in the solution. That will also test if the students can explain in proper way mathematical thinking. I will give some more examples in my presentation from my classroom experience.

Mara Alagic, USA:

Mapping for Learning: Differentiating Mathematics Instruction for Personalised Learning

In the context of WHAT? - HOW? - WHO?, if the WHAT is a mathematics and/or technology standards-based curriculum and the WHO? are learners, could the HOW explain our way of thinking, our teaching/learning/reflecting philosophy, and/or our sense-making processes? Where is the place of technology in these processes? This paper attempts to give some answers/examples and pose more questions about the power of technology in the learning of mathematics: How technology can make a difference in the way we differentiate instruction for personalised learning in mathematics classroom?

G. Albano*, C. D'Apice, M. Desiderio, Italy:

Laplace transform and electric circuits: an interdisciplinary learning tool

The present work is addressed to high school students with scientific trend and it aims at supporting the pupils in learning two subjects: the solution of second order linear differential equations and the study of electric circuits. The two subjects are correlated because one of the presented methods to solve the differential equations uses the Laplace transform, and this is the best way to solve the integral-differential equations that are met in the study of the electric circuits. A package is created using a CAS as Mathematica. The package provides a theoretical framework and many exercises where the students are leaded step by step to solve the differential equations. Using this package equations describing electric circuits can be solved, and consequently physical quantities evolution (current intensity and voltage) can be obtained.

Burkhard Alpers, Germany:

Mathematical Application Projects for Mechanical Engineers - Concept, Guidelines and Examples

In the article, we present the concept of mathematical application projects as a means to enhance the capabilities of engineering students to use mathematics for solving problems in larger projects as well as to communicate and present mathematical content. As opposed to many case studies, we concentrate on stating criteria and project classes from which instructors can build instances (i.e. specific projects). The main goal of this paper is to facilitate the definition of new 'good' projects in a certain curricular setting.

Halil Ardahan, Turkey:

Issues on Integrating CAS in Teaching Mathematics: A Functional and Programming Approach to some Questions

In recent years we have attempted to study main issues and various research questions about integrating and implementing cognitive tools such as computer algebra systems (CAS) environments, in particular TI-92 calculator in both teaching and learning mathematics in Turkey. In this presentation, after overviewing the main issues and obstacles on the subject matter very briefly, we construct a new function, named digit spare function (dsf), a functional approach to two digit prime numbers and a programming approach to find the greatest common divisor (GCD) of integers. Finally, we present a few instructional materials, which were designed and developed in the viewpoint of new learning theories and models, namely constructive and discovery learning.

Deane Arganbright, USA:

Creative Spreadsheet Graphics in Mathematics Teaching and Modeling

The spreadsheet is an excellent and readily available tool for teaching and learning mathematics. Mathematical models, algorithms, and visualization techniques can be implemented in spreadsheets in an interactive format in a way that the creation process itself conveys the underlying mathematics. Examples show how mathematical modeling and teaching are enhanced through innovative and animated spreadsheet graphics. Mathematical illustrations include the investigation of functions, geometry constructions, computational algorithms, and mathematical visualization. Examples come from geometry, calculus, numerical methods, linear algebra, and operations research, as well as such applied fields as population modeling, heat flow, epidemics, genetics, business, and cultural and computer graphics.

Brigitta & Klaus Aspetsberger, Austria:

Cross curriculum teaching and experimenting in math & science courses using new technologies

Cross curriculum teaching and learning by experimenting are important objectives for future math & science courses. Various practical as well as mathematical skills of the students are trained by carrying out experiments, analysing the results and finally using functions for fitting data points obtained by the experiments. The students have to combine knowledge about different types of functions with knowledge about chemical and physical theorems. As an additional aspect, the students also have to take care of accuracy in experimenting for obtaining good results. The collection of large lists of experimental data is supported by the TI-CBL system. Mathematical experimenting, complicated computations and visualisation are supported by the graphical pocket calculator TI-92. We report about the experiences made with several groups of students at the age of 17 to 18 and about one group of students of high ability at the age of 14. Skills and abilities of the students for carrying out the experiments

Adnan Baki, Turkey:

Investigating teachers' perceptions on their preparation to use IT in classroom instruction

The researcher taught a two-term required course within mathematics teacher education program to train student teachers and to investigate perceptions on their preparation to use computers in their own teaching. This paper describes issues emerging from the analysis of the course. Data were gathered through questionnaires. Students who felt prepared made the link between computer-based activities and school mathematics, and had more experience on the instructional software during the course than others. The implications of these results for the designing and implementing of computer-based undergraduate courses and further research in this field are discussed.

Maria Bako; France:

Mathematical software in the educational process of the French and Hungarian teachers

The French and Hungarian education systems spend a lot of energy to keep up with the new developments in the field of technology. Informatics is taught through out high school all over but the computers had no enough role yet in the teaching process of various subjects. The poll's aim, presented in the article, is to show how much and how well the college professors and their students knows and uses mathematical programs. The subjects of this poll are the professors and the students at the Faculty of Mathematics of the University Paul Sabatier of Toulouse and the University of Debrecen of Hungary. The parallel study of this two, culturally and economically different countries brought our attention to some very interesting particular and general problems, which are presented in details in this paper. This and the ideas on the questionnaires can help to set new goals in the application of the computers in the teaching process of mathematics.

Yuriko Baldin, Brazil:

A study of conics with Maple V and Cabri-Géomètre II

The usual presentation of conics in elementary instruction is based on the plane geometry, starting from focal properties and then connecting geometry to algebra by means of quadratic expressions.With 3-dimensional approach, conics are presented as plane sections of a symmetric cone and the fundamental focal properties are usually hard to be understood by students. Nevertheless, the most beautiful and motivating applications of conics to real world problems demand the conics to be worked out in 3-dimensional settings. In this paper, we present a study with combined use of CAS(Maple V) and DGS(Cabri-Géomètre) which integrates both approaches in the classroom, stressing the capabilities of each program suited to specific situations. We include useful exercises on Dandelin constructions with Maple V and Cabri-Géomètre, which would help teachers to construct concrete teaching material on the subject.

Rafael Barbastefano, Brazil:

Tabulae and Mangaba: Dynamical Geometry with a Distance Twist

We report on the ongoing development of two complementary DGS, for plane and space geometry. The design briefs of both softwares were tailored bearing in mind the needs of distance teaching and Web communication. The current implementation is described in some detail, and we also discuss some of the issues that brought about the decision to engage in the project, as well as the implications for the technology driven teacher training program that provided the initial motivation for it.

Elizabeth Belfort*, Rafael Barbastefano, Luiz Carlos Guimaraes, Brazil:

Using Computers in Mathematics Teacher Training Programs: a Reflection upon an Experiment

As part of the requisites for an in service graduate course at our university, Secondary school teachers attend a discipline regarding the use of computers for teaching mathematics. Among other activities, they are asked to produce their own instructional materials, which should be supported by one of the educational computer packages made available to them during the course. The authors have designed this discipline and have also been ministering it for the past three years. Meanwhile, we have been investigating its consequences on teacher's opinions and practices. In this article, we analyse qualitatively the instructional materials produced during the course of the discipline by these teachers, as well as some medium term consequences of these activities for their subsequent classroom work.

Lyudmyla Belousova, Ukraine:

Using of spreadsheets for developing mathematic skills

The article is devoted to the questions of using spreadsheets with the aim of forming the educational curriculum. The main purpose is to developed mathematics skills and habit. The results of the research were probated while teaching several chapters from the mathematics course. The set of tasks showing aims and outcomes of the work with the students are given in the article. At the end of the research interdisciplinary connection were revealed.

Stephan Berchtold, Austria:

School Development - a Systems Perspective

During the last decade the need to do something about schools has increased significantly. The reasons are myriad. In this presentation the author will give a short abstract of how schools came to their current status. This builds the basis for an analysis of a school from a systems perspective. Questions such as "Is a school an organisation?" or "Can school be seen as a social system?" will highlight some of the major weaknesses. Based on 5 years work in School Development the presenter will also give a short case study using a systems tool, the so called Causal Loop Diagrams, to show how systems tools can be applied in Organisational Development.

Stephan Berchtold, Ernst Gebetsroither, Stefan Gueldenberg

Causal Loop Diagramming - a practical 'crash course'

In this presentation three systems modelling experts from the System Dynamics Group Austria with different professional backgrounds (educational, managerial and scientific) will offer a practical "crash course" for everyone who wants to learn the basics of Causal Loop Diagramming, a simple yet flexible tool for diagramming systemic situations. In mathematical terms causal loop diagrams are oriented graphs (of nodes and edges), with "+" and "-" signs attached to the edges. So causal loop diagramming can be considered as a kind of "applied graph theory", which has become prominent in many fields of systemic enterprise. The scope of this workshop will be:

Positive and negative causal effects

indirect causal effects

escalating and stabilising loops of causal effects

basic archetypes of causal loop structures in systems

Detlef Berntzen, Germany:

Movies from the TI-PLUS

Screenshots from the TI-92PLUS can be arranged to little movies (storage capacity of less than 30 KB) by using a GIF Construction tool. The technical details are easy to use and therefore of interest for pupils activities in math lessons. The lecture will be used to show the technic as well as to discuss the usage in math education.

Plenary: John Berry, UK:

The use of technology in developing mathematical modelling skills

An important part of teaching and learning mathematics at all levels of education is the development of the skills needed to solve "real problems". The process of solving real-world problems in mathematics is called mathematical modelling. It can be summarised by the following diagram (see page of the strand, hotkey for the plenary). Technology has an important role to play in this process. The use of software and calculators are natural in the solution phase. It is now well established that the formulation phase of mathematical modelling represents the ?bottleneck? stage of the modelling process. Helping students to develop good problem solving skills often involves much time and effort in this phase. Data logging equipment is a powerful means of collecting and analysing data as part of the Interpretation phase of the process. The aim of this plenary lecture is to reflect on ways that we can bring technology to the teaching, learning and assessing mathematical skills.

John Berry, Andy Smith, UK:

Observing student working styles using Graphic Calculators

When students are working with hand-held technology, such as a graphic calculator, we usually only see the outcomes of their activities in the form of a contribution to a written solution of a mathematical problem. It is more difficult to capture their process of thinking or actions as they use the technology to solve the problem. In this paper we describe an empirical investigation of student working styles with a graphic calculator using software that captures the keystrokes that are used. In this way the students were able to work naturally without the feeling of 'being observed'. After the student problem solving session we were able to playback the sequence of keystrokes to explore how the students actually used the technology, whether they used 'trial and error' mode and how their working related to the training they had received.

Piotr Bialas, USA:

ANOVA with the TI-83 Graphing Calculator

This presentation will demonstrate how the Catalog and cut-and paste utilities of the TI-83 graphing calculator can be used to complete a one-factor ANOVA table. The worksheet including two examples will be distributed. Extension may include Two-Factor ANOVA such as 2X2, 2X3, or 3X3 factorial designs. (Hands-on session with the participants using TI-83 graphing calculator suggested.) The presenter will share a handout for the TI-89 graphic calculator if needed.

Piotr Bialas, USA:

Linking Graphing Calculators to the Internet (LGCI)

LGCI increases access to the numerical data files, provides no need to type the data into the graphing calculator, and makes possible that the selected data files may be used for Excel, Minitab, SPSS, and many other statistical software products. The TI-83 example of the data transfer will be demonstrated. Participants will be provided with written materials about data transfer to the TI-83/TI-89 graphing calculator.

Piotr Bialas, USA:

Spreadsheet uses in elementary statistics course

Many important statistical concepts that seem too obscure for the beginning student can be readily understood through visualization and the ability to perform complex computations rapidly. Available commercially various spreadsheet files computer application programs allow the instructor/student to perform complicated calculations, draw graphs and animate these in real time. The presenter will share the results of his investigation about the effects of the spreadsheet on achievement in selected statistical topics of undergraduate students in an elementary statistics course.

Josef Böhm, Austria

From Pole to Pole, A numerical journey with an analytical destination

The TI-89/92 Data - Editor is an excellent tool to have a numerical approach to basics of calculus. We show how to combine numerical and graphical means to introduce discontinuities, differentiability and curvature. We find not only numerical, but also analytical solutions without using any calculus. Our starting point is a pole of a rational function and our destination is a pole of an evolute. This teaching unit can easily be presented with any other CAS.

Francisco Botana Ferreiro, Spain:

The Three and Four Bar Linkages Revisited: Graphs and Equations

This paper reviews the behavior of current dynamic geometry systems (The Geometer's Sketchpad, Cabri Géomètre, Cinderella, Geometry Expert and Locus) when dealing with two simple linkages: the three and four bar linkages. The different approaches to numerical generation of loci are discussed, highlighting their success and limitations. Dynamic linkage generation can be used in engineering education and real design, overcoming the needs of books for designers.

Denis Bouhineau*, Jean-François Nicaud, Xavier Pavard, Emmanuel Sander, France:

A Microworld For Helping Students To Learn Algebra

This paper describes the design principles of a microworld devoted to the manipulation of algebraic expressions. This microworld contains an advanced editor with classical actions and direct manipulation. Most of the actions are available in two or three modes; the three action modes are: a text mode that manipulates characters, a structure mode that takes care of the algebraic structure of the expressions, and an equivalence mode that takes into account the equivalence between the expressions. The microworld also allows to represent reasoning trees. The equivalence of the expressions built by the student is evaluated and the student is informed of the result. The paper also describes the current state of the implementation of the microworld. A first prototype has been realised at the beginning of February 2001.

Per Broman, Sweden:

Mathematical modelling with CABRI

I will show some examples how Cabri can be used in order to form functions out of geometrical constructions. For example: Properties and use of directrix lines and circles of the different conics. What if we inscribe a rectangle in an acute angled triangle? How can we use Cabri and Derive in combination? I also want to say a few words about TiM, a Nordic network and conference series on Technology in Mathematics education.

Douglas Butler, UK:

Adding a sparkle to classroom teaching - Using Word, Excel and the Internet

A live large-screen demonstration of the creative use of generic software tools both in the classroom and in the creation of worksheets. Surprisingly complex single line mathematical expressions can be created in Word as text using the Unicode font set and user-defined ALT-keys (in preference to the equation editor, though that is still required for multi-layer expressions). These expressions can be pasted into single-font environments such as an Excel cell or an email. Also the drawing toolbar can be used to create a wide variety of diagrams, though there are disappointing limitations. The finding and categorising of useful web resources will be discussed; and the associate pasting of text, graphics and data (often with difficulties to overcome) off the internet will also be covered, including a trawl through the amazing web resources linked from the Oundle School (UK) site

http://www.argonet.co.uk/oundlesch

There will also be a look at some of the pitfalls when using Excel, and an introduction to the concept of using dynamically linked objects to visualise mathematics.

Douglas Butler, UK:

Autograph: Dynamic Coordinate Geometry and Statistics

This presentation will demonstrate how dynamic and dependent objects can be used to enhance understanding in the teaching of mathematics at school and college level, and how they give the teacher an exciting new repertoire of moving images.

Tatyana Byelyavtseva, Ukraine:

Power Point computer support during mathematics lessons in secondary school

The article includes the analyses of computer support during mathematics lessons in secondary school. One of the main purposes of this is to analyse lessons dedicated to developing basic geometric concepts. The influence of new computer technologies on the process of stimulation the scientific research among pupils of secondary schools is also shown.

Jaime Carvalho e Silva*, José Carlos Balsa, Maria José Ramos, Portugal:

Internet as a tool in the preparation of future mathematics teachers

We describe a project that was developed with two groups of seven future mathematics teachers (7th-12th grades) that worked in different schools (30km apart). They sent messages with weekly reports of their activities, comments, and files to a mailing list. The participation was considered to be very fruitful, and these future mathematics teachers became more aware of activities outside their daily routine, developing at the same time their communication skills; they exchanged more than 90 messages and 50 files (mainly with activities and exams). All considered this project to be a very important part of their preparation as teachers of mathematics, showing how they can get new ideas and fight their isolation using the Internet. This project showed that the Internet is a very powerful tool for the preparation of teachers and should be used more frequently.

Neil Challis, UK:

The Role of Technology in Mathematical Diagnosis

In the UK and elsewhere, access to higher education is widening. Students arriving on the same course can have widely different mathematical backgrounds. The issue arises of identifying students' individual mathematical needs, and following up appropriately, as well as making courses appropriate to those students. We report on a project at Sheffield Hallam University addressing this issue, particularly examining the role that technology, for both learning and doing mathematics, can and cannot play.

Plenary: Alison Clark-Jeavons / Rosalyn Hyde, UK:

Developing a technologically rich scheme of work for 11 - 12 year olds in mathematics for electronic delivery

• Background

There is a major change happening in the English and Welsh education system in relation to the use of technology. Generally, in the last couple of years, schools have moved over from using a variety of computer platforms, including, commonly, Archimedes to using IBM-compatible networks of personal computers. There has been a huge increase over this period in the number of schools connected to the Internet, although the level of access in schools does vary. It is now common to find mathematics classrooms equipped with one or two PCs and there have been schemes to help teachers buy laptops for personal use. Schools are also beginning embrace other forms of technology. Some schools now have some access to electronic whiteboards and data projectors. The government is helping to fuel these developments in the use of ICT through its Department for Education and Employment who are implementing various initiatives, one of which is described here. The use of calculators, four function, scientific and graphics, at all levels of the curriculum has been a matter for great debate in England and Wales for some time. The associated issues of choosing software and training teachers to use this technology are also matters for consideration.

At the end of September 2000, the National Numeracy Strategy published a draft Framework for Teaching Mathematics for Key Stage 3 (11 - 14 year olds). This should have a significant impact on the use of technology in the teaching of mathematics as it contains exemplification of the use of PCs (principally spreadsheets and dynamic geometry) and graphics calculators.

• Project

In order to respond pro-actively to this climate of changing technology, the Department for Education and Employment has commissioned Research Machines plc to develop a year 7 (pupils aged 11 - 12) scheme of work for mathematics that makes extensive use of these technologies. The materials forming the scheme of work are all delivered to the 20 pilot schools electronically. Each of these pilot schools have been equipped with 3 classroom PCs, a laptop for the teacher, an electronic whiteboard, a data projector, and 15 graphics calculators. In terms of software, the schools have Microsoft Office, The Geometer's Sketchpad, MSW LOGO, Easiteach for using the electronic whiteboard, and some custom-written software. The project has been developed to motivate and engage students and is aimed at evaluating the contribution of ICT in raising standards in the teaching of mathematics.

Developing materials for using this level of technology in classrooms is a real challenge and is uncharted territory, certainly for a project of this scale and with this level of impact nationally. The opportunity exists to develop the pedagogy for the appropriate use of technology and result in a real impact on the teaching and learning of mathematics.

The paper will examine the background to this work and relate recent research as to the effects of different types of access to ICT on the learning process. It will develop a rationale for development of such materials and examine the implications and effects of such development.

The plenary lecture will present this work as well as showcase materials developed by the project and present some of the preliminary findings.

Alison Clark-Jeavons, UK:

Why DGS is such an effective tool in math education

Many school curricula are advocating the use of dynamic gemoetry software. This presentation will outline why DGS is such an effective tool in the maths classroom, relating current views on how we learn in an ICT environment. The presenter will suggest generic ways in which the software can be used to enhance learning for understanding.

Peter Cooper, USA:

Design of Content Independent Instructional Systems

In designing and implementing instructional systems for remote use, the more sophisticated development environment allow for the use of multimedia content in a packaged environment. In such systems, the container/interface is bound to the content at compile time and prior to distribution. As part of a joint project with the United State Corps of Engineers Research Laboratory, the researchers investigated methods of separating the interface and data container from the content in ways that support a more dynamic approach to maintaining currency of content and distributed storage of instructional materials. The presentation session will include demonstrations of the training application, data entry applications and a look at existing training developed through the system.

John Cosgrave, Ireland

Fermat's 'little' theorem

To mark the 400th anniversary (on 17th August 2001) of the birth of Pierre de Fermat I will present a survey paper - using Maple - on his renowned 'little' theorem. I will treat the theorem itself, and present ideas relating to its applications to periods of decimal expansions, solutions to congruences, primality testing, Pollard's p-1 factoring method, and public-key cryptography. I will also consider some open questions relating to Fermat's 'little' theorem. I will pitch my talk at a general, non-specialist audience.

Jean Jaques Dahan, France:

Cabri Java: A new communication and pedagogical tool

1. Presentation of the software "Cabriweb": I will show how to create a Cabrijava applet (internet file) starting from a Cabri file and what it is possible to do with this applet (automatic animations and manipulation of the figure on the Web). 2. Exemples of problems under cabrijava: like "inversed problems" that can be shared with different persons in different countries (black boxes are particular "inversed problems" but I will present others of different levels). 3. How to write an article using this tool in order to get a dynamic communication between us.

Hans Dirnboeck, Austria

The Evolvente-Curve of a Circle, Used for Gear-Wheels. You Need It Everyday

Gear-wheels are an important chapter of Kinematic Geometry. The terms to construct or to plot the evolvent curve of a circle are given. The fundamental law of gearing is explained. On two wheels we fix two evolvent curves; we proof that they can work as profiles of two gear-wheels. Special case: An evolvent curve fixed on a wheel and a straight line fixed on a rack are working as profiles. This gear-wheel mechanism You are using everyday in Your car, in the railway, the aircraft; in Your coffee-mill etc. You need it and You need Geometry. DERIVE, drawings, models to visualize it.

Alfred Dominik, Austria:

Taylor Series and Finding Zeros with DERIVE and MATHEMATICA

The meaningful use of the two Computer - Algebra - Sytems MATHEMATICA and DERIVE in the Calculus Curriculum for 16 to 18 year old students in Austrian Grammar Schools will be demonstrated with the help of Taylor - Polynomials, Bisection- and Newton's method. Specially prepared functions help the students to get better insights into basic ideas of Calculus such as approximation and limit. Additionally the influence of inital values to iteration - processes will be discussed.

Plenary: Tommy Dreyfus, Israel:

The construction of meaning for abstract algebraic concepts

The teaching and learning of algebra, whether elementary, linear or modern algebra, seems to virtually cry out for computer support, for several reasons: A large variety of multi-representational tools are available, the heavier calculations can easily be taken over by the computer, and most importantly, appropriate software can be used to bridge the existing gap between the concrete and the abstract (see, e.g., Schwarz & Dreyfus, 1995). Indeed, there are examples of success in using technology for students' construction of meaning for abstract algebraic concepts but there are also examples of failure. In the lecture, I will examine a number of possible reasons for failure, including inadequate task design (Sierpinska, Dreyfus & Hillel, 1999) and the ambiguity of representatives for mathematical objects (Dreyfus & Hillel, 1998; Schwarz & Hershkowitz, in press). I will conclude that there is no simple explanation. I will then make the point that in order to deepen our understanding of the relevant learning processes, a re-conceptualisation of abstraction is in order, as well as a research program that allows describing processes of abstraction. Such a re-conceptualisation will be proposed and a research program will be outlined (Hershkowitz, Schwarz & Dreyfus, in press).

Timo Ehmke, Germany:

Geometria: A Tool for the Production of Interactive Worksheets on the Web

With this contribution I will introduce the Java-Applet Geometria, a tool for interactive worksheets to be presented on web-pages. Worksheets generally contain a dynamic figure together with some kind of geometric learning content. This content is described by means of a script-language (GeoScript) which provides the possibility to construct a euclidian figure and also supports the analytical definition of points, vectors and curves. A special feature is the feedback given to the student, while he/she is interacting with the figure. A tutoring component enables Geometria to evaluating and commenting on the student's answer.

Hans-Jürgen Elschenbroich, Germany:

Teaching and Learning Geometry: dynamic and visual

"A generation has grown up that may be far more visual than verbal ... . The state of mind of young mathematicians is not what it was fifty or hundred years ago ..." (Davis)

Dynamic Geometry Software like Cabri II, Cinderella or Euklid-Dynageo offers new chances by using dragmode and loci to learn and to teach geometry in a visual and dynamic way. Classical ideas can be brought to life.

DGS is not seen as a substitute, but as a complement to and an extension of the classic tools compass and ruler. Electronic worksheets will give a safe basis, which avoids lengthy phases full of mistakes and will support experimental and heuristic activities of the students.

After some basic reflections about visual learning and teaching, well-tried examples of electronic worksheets and pre-formal, visual-dynamic proofs will be presented.

Joachim Engel*, Marcus Otto, Germany:

Simulation and Modelling with Lisp-Stat: A Flexible Software for Teaching Statistics

We illustrate how a simulation based use of computers supports conceptual learning in statistics. We focus on three areas of application: 1. simulation via bootstrap 2. modelling functional relationships between two variables that are corrupted by noise 3. demonstration of the central limit theorem. The basis is the programming environment Lisp-Stat.

Björn Felsager, Denmark:

Through the Looking Glass: A glimpse of the Minkowski Geometry

The Minkowski geometry offers the possibility of seeing well-known concepts from high school mathematics in a new perspective. The investigation of Minkowski Geometry requires the use of a Hyberbolic compass. This is introduced using Cabrii, which supports conic sections as a primitive geometric object. Thus the use of modern technology makes it feasible to investigate Minkowski Geometry in almost the same elementary way as Euclidean Geometry.

Roger Fentem, UK:

The impact of training for students on learning mathematics

Training for teachers in the use of graphing calculator technology is widely accepted. To what extent are the training needs of the users of the technology addressed i.e. the students? This paper introduces a research project designed to investigate the issues of technology training for both teacher and student in studying mathematics post 16. Attitude, relative achievement and practice are studied, recorded and analysed.

Roger Fentem, UK:

Investigation into Student Attitudes to using Calculators in Learning Mathematics

In many countries curriculum designers, educators and examiners receive mixed messages about the role that should be played by information technology: imposition of severe restrictions to active encouragement of CAS. We present an international study exploring student attitudes to the use of technology, their training needs, and their ability in mathematics when learning in a CAS intensive but assessment hostile environment.

Isabel Fevereiro*; Carmo Belchior, Portugal:

Changing the Classroom Practices. The use of Technology in Mathematics Teaching

Since 1997/98 the Department of Secondary Education, Ministry of Education of Portugal, has created a training teachers NET constituted by 80 mathematics teachers to improve meetings and promote training sessions and workshops with mathematics teachers in all secondary schools in Portugal. The aim of this project is to change the classroom practices according to the curriculum guidelines, which focus on experimental teaching/learning process, centred in the students themselves, in knowing how to do, and with a strong emphasis in the use of technology. Since then, after three years, the use of the graphic calculators in the classroom is generalised (the use of the graphic calculators is compulsory in the final national 12ª grade exams). We will take a look at the final exams. Since our truing teachers are working with many different activities in the classroom we will take a look in some of these activities.

Ruhal Floris, Switzerland:

Evolution of mathematical tasks in a CAS-classroom

We will propose a study on how technology modify the kind of work made by students and teacher in the lesson and the kind of mathematical objects discussed. We will analyse some new possible tasks such as studying equations as objects, or solving function interpolation problems; we will also study the consequences of the constant use of CAS and graphic technology, tending to modify some didactical situations, with the production of complex outputs and discuss the ways and conditions to convert these outputs in learning objects.

Plenary: Jean Flower, UK:

Interactive web-based resources and a new perspective on algebra and geometry

This paper will reflect upon the use of DaC (dynamic geometry and computer algebra software) in two contexts - two undergraduate Linear Algebra courses taught at different UK universities. The main questions of this strand will be considered in the light of this experience. It is hard to compare the two linear algebra modules and claim that one was "more successful" than the other. One covered more pure algebra topics, whereas the other included more applications of Linear Algebra. Both used DaC. One used Maple and JavaSketchpad, and the other used TI92's algebra and Sketchpad on the PC. The students on one module were mainly training to become teachers, whereas the students on the other were studying for a mix of maths degrees, heading for business.

Is it necessary to achieve widespread use of DaC throughout a course for best benefits? The students who had a wider exposure to Sketchpad in a range of modules over many semesters made better use of the Linear Algebra images than the students who were unfamiliar with DGS. How do the costs (time as well and money) of introducing DaC in a single module compare with the benefits?

Is it necessary to integrate DaC into assessment at the same time as its introduction to the teaching? The students whose assessment included a Maple test learned to use Maple mainly for the purposes of completing the test, whereas the students with TI92s used them more widely to shortcut rote algebra. Use of the handheld technology was not required for successful completion of the course, but the TI-92s were used more widely.

How can we tie in a DaC approach to a subject whose key texts take a more traditional approach? There is a mismatch between the students' experience of Linear Algebra in the classroom (and in the website) and the students' experience of Linear Algebra from books. Does this contribute to confusion? Can we make use of this contrast to deepen understanding of the different facets of a subject?

The use of DaC allows for revitalisation of some "tough" topics which were getting taught later on in a degree. Tasks which required intensive numerical calculation can now be completed quickly, allowing more space for understanding the results of the calculation. The use of technology itself can provide relevant applications for study (eg. computer graphics). Different approaches to proof and argument contrasts axiomatics (a traditional way in to Linear Algebra) with investigation (assisted by DaC).

What is the relationship between working on the computer and working with paper and pencil? This question is critical when introducing DaC into courses which maintain traditional assessment strategies like exams, where students may not have access to DaC.p>

Looking at the changing nature of algebra and geometry is like trying to gaze into a crystal ball. But we can have some fun looking there.

Greg Foley, USA:

Mathematics Teacher Development That Works

Several characteristics make for an effective professional development program for secondary mathematics teachers. The program needs a clearly focused purpose that is relevant to the participants. It must have expert and stimulating presenters with participant involvement and reflection. Participants should create and implement an action plan, with ongoing support. Good facilities and organisation are important. U.S. examples will illustrate these key features.

Ruth Forrester, UK:

Data Collection and Manipulation using Graphic Calculators with 10-14 year olds

A teacher researcher group at the Edinburgh Centre for Mathematical Education is currently investigating the use of graphic calculators in Mathematics classes for pupils aged 10 -14 years. One focus has been on the development of data handling skills. Activities have been devised where pupils use graphic calculators in the collection of data and its subsequent analysis. Classroom implementation has produced positive results. Evidence has been found of gains in understanding of statistical concepts attributable to the use of this technology. Positive motivational effects were also seen. The graphic calculators enabled the use of pupils' own data and allowed the teacher to pace and vary the learning experience appropriately.

Wolfgang Fraunholz*, Frank Postel, Germany:

A Computer Learning Environment in Linear Algebra using CAS MuPAD

The Computer Learning Environment in Linear Algebra offers an introduction to Linear Algebra (vector space, basis, dimension, matrices, determinants, systems of linear equations, linear operators, dot product, vector product). Representing the development and the examples, the solution of exercises step by step and the controlling of solutions is done by the Computer Algebra System MuPAD. Important is also a three-dimensional graphic tool, which visualises vectors, vector algebra, linear equations, mappings in three dimensions. The talk will give aspects of math education (Wolfgang Fraunholz) as well as those of programming and software (Frank Postel).

Nils Fruensgaard, Denmark:

Danish experiences with technology in mathematics teaching in upper secondary school

A co-operation between the Association of Mathematics Teachers and the Ministry of Education has resulted in building a new experimental curriculum, based on enhanced use of ICT, which teachers are encouraged to use in their classes in grade 10-12 in the Gymnasium. The old and new curricula are presented, with emphasise on the national written assignments and the intended use of technology in the experimenting classes. The general curriculum problems that ICT has created in teaching mathematics are discussed.

Karl Josef Fuchs*, Christian Kraler, Austria:

Programming in the Age of CAS and the Algorithm as Fundamental idea in mathematics education

The authors will concentrate on the basic question of the Special Group by taking How much Programming (knowledge / skills) must a Mathematics - teacher have in the Age of CAS as their theme. Reasons for the motivation and necessity of this question for the process of teaching mathematics with new technology will be given. Different accents in defining the term of Programming will show that fundamental ideas of mathematics such as algorithm, function or modelling are essential parts of these terms. Further discussions will mainly focus on the idea of the algorithm and its importance as a connecting piece between mathematics and computerscience.

Jenny Gage, UK:

The Role of the Graphic Calculator in Early Algebra Lessons

This is a study of first algebra lessons at secondary school using the lettered stores of a graphic calculator to form a model of a variable. The calculator provides a tool for thinking and for building up concepts. In this paper is a discussion of what happened in the classroom, and how the calculator helped in the remediation of a specific misconception without any need for teacher intervention. There is also discussion of what ideas the students bring to the work, and how these ideas change during the lessons.

Giuliano Gargiulo*, C. D'Apice, R. Manzo, Italy:

Mathematica and symbolic-numerical methods for solving first order ODEs

The use of information technology in addition to traditional lectures affords a means to develop student intuition and curiosity, reaching in the same time a deep knowledge of the subject of study. The aim of this work is to show the didactic use of a Computer Algebra System, as Mathematica 4.0/4.1, to illustrate and compare different symbolic-numerical methods for solving first order ordinary differential equations (ODEs). In particular, we apply, relate and compare the built-in functions of Mathematica, the method of integration by series, the Picard process and the linearization method in solving some first order ODEs. Moreover, numerical solutions are compared with symbolical ones at the various stages of computation. This includes use of numerical methods (internally adaptive) to look for and analyse singular points for maximal solutions.

Ernst Gebetsroither, Austria:

Modelling Carbon Dioxide Pollution in Austria

One of the major ecological issues world-wide is the increase on carbon dioxide in the atmosphere. At the international Climate Conference in Kyoto 1991 Austria has committed itself to reduce until 2010 the Carbon Dioxide emissions by 13%, compared to the emissions of 1990. To fulfil this goal it is necessary to understand the national carbon cycle of Austria, an interdisciplinary team lead by Austrian Research Centre Seibersdorf (ARCS) has developed a system dynamics model of the Austrian Carbon Balance Cycle. The findings of this modelling are being used by political decision-makers in Austria. Ernst Gebetsroither, the co-ordinator of this project, will report about the experiences of this project.

Luiz Carlos Guimarães, Brazil:

Tabulae and Mangaba: Dynamical Geometry with a Distance Twist

We report on the ongoing development of two complementary DGS, for plane and space geometry. The design briefs of both softwares were tailored bearing in mind the needs of distance teaching and Web communication. The current implementation is described in some detail, and we also discuss some of the issues that brought about the decision to engage in the project, as well as the implications for the technology driven teacher training program that provided the initial motivation for it.

Stefan Gueldenberg*, Werner H. Hoffmann, Austria:

Leadership, Management and Management Control - a System Dynamics Approach

The purpose of leadership is to create a viable organisation capable of development that is both internally guided and externally oriented. Normally leadership is understood as the capability of a person - the charismatic leader. In this manner leadership is given someone by birth and not teachable. In contrast to this personal and determined view we understand leadership as a capacity of an organisation, a human community, to create its own future and can be built by its members. Utilising a system dynamics interpretation of the term leadership, we aim to identify in our work the current challenges to companies from their environments, and toexplain the consequences of these challenges for company design and control. For a company to achieve sustained development, there must be a healthy proportion of growth and balance. The conclusion of our work is that system dynamics is a prerequisite for educating successful organisational leaders to help them to understand complex organisation and design viable structures.

Samer Habre, Lebanon:

The ODE Curriculum: Traditional vs. Non-Traditional - The Case of One Student

A Traditional course in ordinary differential equations consists of tricks to find formulas for solutions with very little emphasis on the geometry of the solutions or on an analysis of the outcomes. Since differential equations are important in many fields, educators have come to believe that this approach is obsolete. With the advancement of computer graphics, it is now possible to offer a course on differential equations using a qualitative approach. This paper examines the two approaches as offered by the same instructor at the Lebanese American University in Lebanon. In particular, the point of view of one student who took the course twice using a different approach each time is presented. Results show that the qualitative approach is more appreciated, and that technology plays an essential role in the understanding of the material.

Mary. S. Hall, USA

Creating and Teaching Online Mathematics Courses

As distance learning has expanded, so also has the use of the Internet. More and more we are seeing the expansion of course material to the Internet. What are the issues for teaching course material on the Internet?

What students will benefit from such opportunities? These are some of the issues addressed in creating an online developmental mathematics course and other mathematics courses. This presentation will provide both resources and methods for teaching a course on the Internet as well as an emphasis on the new technologies becoming available.

Several online mathematics courses will be used to demonstrate some basic forms of communication and evaluation that are necessary for a course to be successful.

Andre Heck, Netherlands:

Modelling Human Growth

Many a pupil at secondary school asks oneself questions like 'Am I too thick or too thin?', 'Am I short or tall in comparison with persons of my age?', and 'What adult length may I expect to reach?'. To answer such questions one needs real data. We have used the recent Dutch growth study to create learning material for pupils in upper general secondary education (age 15-16 yrs.) to carry out practical investigation tasks. A mathematical highlight is the ICP-model that models length for age within millimetres. It is used in the medical literature and yet consists of growth models that are studied at school, viz., exponential growth, quadratic growth, and logistic growth. We shall present the learning material and discuss the classroom experiences.

Andre Heck, Netherlands:

A Practical Investigation Task with the Computer at Secondary School: Bridges and Hanging Chains

Almost everywhere you can come across hanging chains and cables. Examples are necklaces, high-voltage cables, and cables that support a bridge surface. Do these cables all hang in the same mathematical shape? The first thought of many a pupil will be: this is a parabola, isn't it? In the computer learning environment Coach you can easily measure this on digital images. It will turn out that the parabolic shape quite often occurs with bridges, but that an ordinary chain does not hang as a parabola. Can you understand this? We shall show that a key idea for solving the problem can be discovered by measuring digital images and that it can be theoretically explained with basic physics afterwards. It also leads to a simple computer model of hanging chains. We shall discuss our learning material and classroom experiences, and in this way present an example of how ICT and context situations can contribute to the realisation of challenging mathematical investigation tasks.

Judith Hector, USA:

Teaching Probability and Statistics via the Internet

The author has taught a one-semester Probability and Statistics course via the Internet four times. The course is offered for university transfer credit at an American community college. The course is conducted totally online for students at a distance, but local students may meet for an orientation, midterm exam and final exam. From her experiences and research, the author discusses basic principles of teaching and learning mathematics on the Internet.

Judith Hector, USA:

Programming Principles for Mathematics/Engineering Students

The author has taught computer programming since 1970. She has developed an introductory programming course for mathematics/engineering students. Students develop structured programs on a computer using FORTRAN and the same programs for a TI-92 calculator. Students learn to program certainnumerical techniques such as Newton's method of root finding and Euler'smethods of solving a differential equation. Such techniques are available preprogrammed as black boxes in CAS.

Guido Herweyers, Belgium

Elimination of Parameters and Substitution with Computer Algebra

Elimination of parameters and substitution with computeralgebra. Starting with the geometrical concept of parametric equations of lines and planes, we illustrate the method of elimination to obtain a cartesian equation. This elimination can be done in a direct and simple way by using the procedures "solve" and "substitute" (the basic algebraic manipulations of formulas) of a CAS. Without a CAS this method is difficult to realize by hand (e.g. solution of a system of two equations in a context with different "letters"). Therefore it was necessary to introduce in advance more elegant (but also more sophisticated) algebraic techniques like determinants. The result was that, for a lot of pupils, the meaning of the elimination process disappeared behind these algebraic manipulations. Later on in the educational process, we have the opportunity to show the equivalence and strength of the new algebraic techniques. These ideas will be illustrated in a few (geometric) examples.

Iavor Hristov, Bulgaria:

Model of deformations of fluid particles due to electric field

A mathematical model of finite deformations of compound drop containing another drop due to electric field are obtained. The fluids are homogenous, incompressible and Newtonian. The cases of concentric and eccentric particles are investigated together.

Ros Hyde, UK:

Creating a Professional Development Network

Recent developments in T3 (Teachers Teaching with Technology) in England will be used as a case study to explore the setting up of formal and informal networks for professional development in the context of an increased emphasis from government on continuing professional development for teachers. The intention is to explore the creation of networks that are enabling and empowering for teachers and that provide teachers with the support and resources they need to take responsibility for their own professional development.

Nicholas Jackiw, USA:

Functions as First-Class Dynamic Geometry Objects

The Geometer's Sketchpad version 4.0, arriving Summer 2001, includes support for functions as first-class objects in Dynamic Geometry, allowing users to define, combine, and differentiate functions symbolically, evaluate them numerically, and plot them through a variety of coordinate projections. While in isolation, these capabilities have been long present in other mathematics technologies (e. g. graphing calculators and CAS packages), their meaning is altered by the rich possibilities of interaction and manipulation afforded by the dynamic geometry environment. In this talk, Sketchpad's designer will summarize the research leading to these new developments, demonstrate some models of their classroom use observed in software field tests, and outline possibilities for how representations of functions as first-class dynamic geometry objects engage various strands of a secondary-level mathematics curriculum.

Youngcook Jun, Austria

Theorema-based TI-92 Simulator for exploratory learning

One of the Theorema system's capabilities provides computing environment which can simulate the existing graphing calculator such as TI-92. Moreover, the deductive reasoning facility of Theorema allows the simulator to deal with propositional and predicate logic for pedagogical purposes. We present how to apply the use of such a simulator to help students explore mathematical ideas in terms of black box/white box principle. This experimental approach is demonstrated with our prototype by explicitly generating the sequences of calculator keystokes. Exploratory learning as a part of cretivity cycle is realized with algorithmic and logical empowerments built in the Theorema system.

Henryk Kakol, Poland:

Integrated Teaching Mathematics with Elements of Computer Science

At present nearly all Polish schools have computer rooms well equipped while teaching mathematics generally is traditional. During school lessons chalk and blackboard are still teaching instruments frequently applied. What are the reasons for such a situation? There are many of them. It will be list some of them. Special Programme of Teaching Mathematics with Elements of Computer Science in Gymnasium eliminates many of the above mentioned problems. It offers teaching mathematics and elements of computer science in the form of one thematic block.

Jan Kaspar, Czech Republic:

Programming as a tool for the precision

Using the TI-83 graphing calculator I would like to demonstrate how programming requires precision in step-by-step description of mathematics tasks.

Karl-Heinz Keunecke, Germany

Curvature of Functions as a Limit

A road sign "Dangerous Curve" can introduce to the problem. A car driving through a curve must not "cut" but osculate the road. For a short while, when the steering wheel is in a certain position the car moves on a arc of a circle. From this discussion all the expressions are available to define the curvature of a function by means of the radius r of the osculating circle as k = 1/r. We will realize the teaching unit using DERIVE 5´s new features to enable the students producing their own "notebooks" combining text, graphs and calculations.

Mark Klespis, USA:

An on-going program of professional development program in hand held technology for instructors of prospective teachers

The Mathematics Teacher Educator (MTE) program is an on-going professional development program of Teachers Teaching with Technology (T3) and is designed to assist US college faculty integrate technology into their mathematics content courses for prospective elementary teachers. The paper focuses on a collaboration of the MTE program with a similar NSF-funded program directed by the author. This collaboration began in 1998 and has provided a forum for 39 faculty interested in restructuring their mathematics courses for prospective elementary teachers. Data collected at the workshops indicate participant improvements in teaching and using technology. In May 2001, members of the original cohort and new faculty will participate in an updated workshop. Longitudinal data will be collected and included in the paper.

Heiko Knechtel, Germany

Mathematic with Graphic and Symbolic Calculators - Teacher Training in Lower Saxony, Germany

History - organisation - contents of teachertraining in Lower Saxony: In Lower Saxony a new concept of teacher-training was developed from the mathematics advisers: Every math-teacher at highschool have to take part in 4 math workshops within 3 years. They should learn, how to integrate the new technology of the handheld calculators and dynamic geometry in their own math lessons. Interested teachers were trained within 2 years for math-multipliers. The math-multiplier-groups were divided in teams of two persons. Each team is responsible for six schools in their region. Each team focussing on special interests for each school and go ahead for four times with the groups. They will visit the colleagues in their own school and give several workshops there. Items of the workshos are handling with graphic and symbolic calculators and dynamic geometry; developing units with the new technology basing on their traditional math lessons by their own. After testing their own units during half a year the last 2 workshops give them a view on new possibilities in math lessons, specially in advanced or real-world mathematics. Supplementary every year in each region there are Regional T³-Conferences with a main lecture and up to 15 workshops all over the day.

Mykola M. Kolodnytsky a.o., Ukraine

Teaching Elementary Number Theory with a Software System

In this paper we show how to teach and to solve some computational problems of elementary number theory including modular arithmetic using the software tool "DSR Open Lab 1.0" designed and developed by the authors. We consider such computational problems as follows: to run the prime number test, to determine all prime numbers in some range ("the sieve of Eratosthenes"), to factorise a number into primes, to compute the GCD for a pair (or more) of numbers, to solve the systems of linear or polynomial congruences, i.e. polynomials modula m, to compute residue classes, i.e. modulo m, as well as the Euler phi-function, quadratic and power residues, reciprocal number modulo m, primitive roots modulo m, modular exponential, indexes, discrete logarithm, etc. We also give the comparison of the user interface implemetation of our software with the following: Maple V release 5, Mathematica 4 and DERIVE. The shown examples convince that the process of elementary number theory problem solving and teaching became easier now due to the visual interface of the presented software.

Michael Kourkoulos, Marianne A. Keyling:

Self-correction in algebraic algorithms with the use of educational software: an experimental work

Our work points out that self-correction is a complex but fruitful activity concerning the learning of elementary algebraic algorithms. Pupils who have worked with an adequate software («Arithm»), both in Greece and in France, present a significant improvement of their strategies of localisation of errors, which are an essential element of the self-correction procedures. Furthermore, the work done led these pupils to a significant amelioration concerning the treatment of the examined algorithms.

The software allowed teachers to be alone in their class (or in a half-class in the case of weak pupils) but nevertheless to offer adequate individual support to the pupils in their self-correction work, which is very difficult to realise in usual teaching conditions.

Konrad Krainer, Austria:

Innovations in Mathematics, Science and Technology Teaching (IMST²) - First outcomes of a nation-wide initiative for upper secondary schools in Austria

The bad results of Austrian high school students with regard to the TIMSS achievement test led to a research project where the results were analysed and additional investigations into the situation of mathematics and science teaching were started. As a consequence, a pilot project called IMST² - Innovations in Mathematics, Science and Technology Teaching - was launched in the school year 2000-01. The project aims at supporting mathematics and science teachers' efforts for raising quality in learning and teaching. 126 Austrian schools participated in this project, about one quarter collaborated more intensively with the IMST²-team and documented one or more innovations at their school. The concept, experiences and findings of IMST² will be presented and discussed.

Krivsky Stefanie, Germany:

Didactic innovations of teaching by internet

While in the beginning the internet was designed by scientists for the purpose of exchanging information, it is nowadays more and more adopted by entertainment and commercial use. The internet project matheprisma (math prism) tries to combine these two objectives with the aim to simplify learning of complex mathematics using multimedia and entertainment aspects of internet. Matheprisma is a collection of modules addressing several mathematical questions on different educational levels. Technical and didactic possibilities of internet pages are presented by means of some examples of matheprisma-modules.

Ewa Lakoma, Poland:

On the impact of hand-held technology on mathematics learning - from the epistemological point of view

Recently in the most of countries, mathematics became to be treated as one of the most important components of general education and general culture. Thus it is extremely important to enable students to develop their own mathematics as a language for communication. Thus, it is necessary to consider a process of mathematics learning from the epistemological perspective and to recognise students' ways of mathematical thinking, especially when students use information technology. In this presentation I would like to show the main results of my educational research, concerning exploring and analysing a process of mathematics learning from epistemological point of view- at secondary and tertiary level - in which graphing calculators TI-83 and TI-92 are used as supporting tools.

Duncan Lawson*, J. Reed, and S. Tyrrell, UK:

Extending a Mathematics Support Centre via the Web

The Mathematics Support Centre at Coventry University offers support to any student in the University who wants help with any area of mathematics, statistics or quantitative methods. The support offered by the Centre is in addition to that routinely received in lectures, tutorials, seminars, problems classes, etc. The primary mechanism of support is one-to-one contact with students offered on a 'drop-in' basis. This support is staff intensive and in order to optimise the use of staff time alternative methods of supporting students are continually under review. A recent development has been the introduction of a web-site for the Centre. This paper describes the background to the Mathematics Support Centre, the development to-date of the web-site and an evaluation of its use.

Duncan Lawson, UK:

A Discrete Introduction to Modelling

In applications focused mathematics degree courses there is an understandable desire to introduce students to the ideas and practice of mathematical modelling at an early stage. However, many mathematical models depend on a level of mathematical sophistication, such as differential equations, which most undergraduates do not have on entry to university. Furthermore, it is often the case with such models that specialist mathematical software is required for the solution of the model equations. This combination of sophisticated mathematics and unknown software can be a considerable deterrent to new undergraduates. This paper describes a way of introducing a range of key ideas within modelling, initially without using any new mathematical concepts, and relying on software which is both familiar and not specifically mathematical, namely the spreadsheet. A short description is given of a number of models which are easily explored with spreadsheets.

Josef Lechner, Austria

Standardisierung der Normalverteilung - ein Anachronismus?

Während der numerische Taschenrechner alle anderen Funktionstabellen aus dem Schulunterricht vertrieben hat, hat bis zum heutigen Tag die Tabelle für PHI(z) mit den Parametern 0 und 1 für den Erwartungswert bzw. die Standardabweichung bei der Normalverteilung in den Lehrbüchern überlebt. Welche Ursachen hat dieser Anachronismus (traditionelle, technische oder andere)? Was würde es bedeuten, auf die mehr oder weniger aufwendige Skalentransformation im Unterricht zu verzichten?

Carl Leinbach, USA

Using a CAS to Teach Algebra - Going Beyond the Manipulations

In this paper I will examine two of the basic theorems from a first year algebra class, the Division Algorithm and its corollary, the Remainder Theorem for polynomials. These two theorems are the basis of much of the teaching and learning in a first course in algebra. Unfortunately, most of the students efforts are devoted to factoring polynomials and finding their roots with little gained in terms of insight as to why they are performing these tasks. In this paper we will show how we can use these theorems to write expansions of polynomials about x = a for a not equal to 0. Once this is done, students can learn about the idea of local linearity and tangent lines to the graphs of polynomials. I intend to develop two applications of these ideas. One is an application to pure mathematics, the other is to more real world settings.

Pavel Leischner, Czech Republic:

The collection of interactive solids figures and spatial situations in the Cabri - geometry

The article gives information on the collection of interactive solid figures and spatial situations in the program Cabri-geometry. These aids would facilitate the teaching of stereometry at high and elementary schools. It is intended for the spatial imagery evolving. It should make students pass from experimental manipulations with the spatial situation to mental ones.

Key Words: High school stereometry, spatial imagery, teaching with software, Cabri-geometry.

Gisèle Lemoyne, Canada:

Cognitive and didactic ideas in ICT environments for the learning and teaching of mathematics

Over the past few years, we have designed computer environments for the teaching of arithmetic, pre-algebra and algebra. We describe some of these to demonstrate how cognitive and didactic ideas are put into practice and how these environments engage both learners and teachers in non trivial problem-solving activities. The first environment is devoted to additive and multiplicative problems. Three different tasks were planned:

• construct an iconic representation of a problem, using the tools in the environment

• write a mathematical sentence that corresponds with an iconic representation of a problem

• write a problem that corresponds with a mathematical sentence.

In the second environment, teachers have access to a calculator and can create problems by specifying numbers and operations and then choosing on the key pad of the calculator which keys will be non functional. Each subgroup of students receives specific calculations. The third environment consists of a task of abstraction of properties and characteristics of numbers and operations

Auxencia Limjap, Philippines:

Current Educational Theories & New Tech: Development of a Training Programme for Math Teachers in the Philippines

Reform movements on mathematics education in different parts of the world point out to the need to adopt a cognitivist view of instruction that focuses on the nature and process of mathematics learning. Proponents advocate constructive learning and gear teaching towards the development of meaningful quantitative thinking. They adhere to the social origins of cognition and situate learning in realistic settings. They harness technology as a learning resource that provides both context and support for meaningful problem solving activities. Consequently, learner centred educational theories proliferated with the advances in educational technologies. These developments in pedagogy and didactics pose a big challenge to school mathematics teachers especially those who have neither experienced the constructive process of learning mathematics in the classroom, nor employed the current educational technologies.

Wolfgang Lindner, Germany:

The Digraph-CAS-Environment and Misconceptions around Matrixoperations

A longtime research at the University Duisburg, Germany, studies the impact of CAS on the belief structur of high school students and on the development of conceptions and skills of Elementary Linear Algebra with special consideration of animated visualisations and algorithmic semiautomations. The design of a Digraph-CAS-Environment (realized in MuPAD) is shown, which represents e.g. airline connections in an informal-visual way. The usual matrixoperations on the quadratic adjacency matrices are introduced and programmed to enhance understanding. Afterwards the extracted concepts and intuitions are transfered to rectangular matrices and the effect of this singular local perturbation of the individual knowlege net is studied. We compare the handling of misconceptions by the students with and without CAS.

Alex Lobregt, Netherlands

Introducing Fourier Series with DERIVE

In Electrical Engineering Courses functions such as the square wave Sq(t) and the sawtooth Saw(t) are frequently used. These periodic functions may well be approximated by a so-called Forier Series. In a workshop we will present some examples leading to an application, which can be shown by means of DERIVE as a first step in the filtering theory.

Marie-Thérèse Loeman; Belgium:

To learn from and make history of maths with the help of ICT

Results from the EEP Comenius Action 1 : "The history of some aspects of mathematics like: history of mathematical persons, symbols, algorithms..." Looking through different aspects of history of maths, in co-operation with people from other nationalities and cultures, convinced our students that maths, having its special common language and symbolic notations, has no boundaries. Digging in history of maths and working cross-subject ( English, religion, philosophy, chemistry, geography, physics...) revealed to them that as it comes to solve a problem, not only the solution is to be appreciated but certainly getting to know a nice, perhaps different and original way of reasoning can be a source of inspiration for the scientist being superior to the machine ! In addition they were encouraged to learn from the stronger elements in each partner country.

Victor Lysytsya, Ukraine:

University level Geometry Course and DG

Computer experiments within the course of "Analytical Geometry" are suggested. This course is taught at the Department of Mechanics and Mathematics of Kharkov National University. The most interesting are the tasks devoted to the geometrical sets of points on the plane. The experiments are constructed with the help of geometrical packet DG, which has been worked out at Kharkov State Pedagogical University.

Li Ma, Sweden:

Supervision of students projects

This paper concerns supervising students projects in information technology.

Li Ma, Sweden:

Maple and a unified approach

This paper will discuss the use of Maple in teaching Linear Algebra and Calculus as a unified approach.

Li Ma, Sweden:

Technology and History of Mathematics

This paper will discuss some aspects of using technology in teaching history of mathematics.

Eoghan MacAogain, Ireland:

A CAS-index applied to engineering mathematics Ps

A CAS-index is applied to a set of first year university engineering mathematics examination papers; the results are analysed. The CAS-index is an index of suitability; its purpose is to try to answer the following question: given a mathematics examination paper which was written for a CAS-free environment how suitable is that examination paper for use in a CAS-supported environment?

Tom G Macintyre, UK:

A CAS project carried out in Scotland with 16-17 year olds using TI-92s

This study explored the impact of using hand-held technology throughout a course of study in a year 12 mathematics course - leading towards the Scottish Higher Grade. Students in the study sample had dedicated access to Texas Instruments TI-92 calculators, utilising the built in Computer Algebra System (CAS) as they developed their knowledge of the various components of mathematics studied. Both quantitative and qualitative data was gathered from the study sample students and teachers, who were based in three secondary comprehensive schools. Additionally, data was gathered from the three paired-control groups, providing evidence of algebraic ability at the start and end of the period of intervention. Performance in algebraic skills was of particular interest in this study, ascertaining whether extended use of technology had a positive or negative impact on students' abilities. The quantitative findings, taken from the two assessments administered at the start and end of the one-year course, demonstrate a significantly better performance in the study sample compared with the control group. This affected performance in items that were common to both assessments, resulting in a 7% increase in the study sample compared to the control (p=0.004). A similar trend was noted in new items that assessed mathematics studied during the course of the year; taking the base level of performance into consideration there was a 5% increase in the study sample compared with the control (p=0.046). Some underlying reasons for these differences in algebraic ability are explored. The discussion includes consideration of: the teaching approaches promoted by the staff; the impact of mathematical rigour and syntax demanded by the technologies; the emphasis on equivalence when interpreting screen displays; and the general motivational effect that dedicated access to the technology has had on the students in the study. A number of questions remain, for current debate and future research into the use of a CAS in mathematics education.

Katherine Mackrell, UK:

The role of dynamic geometry packages in visualisation and animation

This session will comprise a report of discussions held at the CabriWorld conference in Montreal in June 2001 regarding the use of Cabri-Geometre to create interactive teaching materials using visual imagery and animation to introduce mathematics from a wide range of areas.

Giora Mann*, Nurit Zehavi, Israel:

Virtual Experiments and Probability

A good model in probability must agree with observations. It is not practical to perform the real experiment many times. In a CAS environment we can perform a virtual experiment many times with relative ease. This changes modelling in probability to be twofold - programming a virtual experiment which controls the traditional modelling.

Robert Mayes, USA:

Cinderella: Software Tool for Euclidean and Non-Euclidean Geometry

Although axiomatics account for a small part of the current boom in geometric research, the study of the axiomatic approach dominates the geometry taught in high school and college. The result is a curriculum where the geometry of plane figures is developed from a very narrow point of view. Students view geometry as an intellectual game of proof that has little or no relation to the "real world". In addition, many students do not see a connection between geometry and other areas of mathematics. If teachers present solely an axiomatic approach, they will propagate this approach among their students. The outcome is an isolated and outdated geometry course that serves to turn students off, rather than demonstrating the beauty and utility of geometry in our world. Breaking away form the current narrow curriculum provides for a variety of societal and mathematically desirable goals. Modern Geometry should aspire to attain some of the goals recommended by the NCTM in the Curriculum and Evaluation Standards for School Mathematics and the NCTM 1987 Yearbook: Learning and Teaching Geometry, and by COMAP in Geometry's Future.

Michael McCabe*, Ann Heal, Alison White, UK

Computer Assisted Assessment of Mathematical Proof = Proof of Computer Assisted Assessment : An Integrated Approach to Higher Level Learning using Group Response Systems and On-Line Assessment

In the School of Computer Science and Mathematics at the University of Portsmouth, computer assisted assessment (CAA) has been used successfully in support of maths teaching for almost 10 years. CAA is most commonly used for first year university modules, where the numbers of students are greatest and the topics covered are basic. This leads to the common conception that CAA is only appropriate for low-level learning.

Mathematical proof is a topic which students find difficult to grasp and involves a higher level of learning. Traditional exam questions on proof are time-consuming to mark, but CAA can provide an efficient and effective alternative. The speed and accuracy of marking objective questions and the ability to give immediate feedback are among its obvious benefits. It remains to demonstrate that CAA can generate results equivalent to those of a written, hand-marked examination. We will explain how this has been achieved:

• by carefully designing test questions and considering learning objectives

• by exploiting both on-line assessment and group response systems (also referred to as an audience (or class or personal response) system

• by integrating both public and private practice of CAA into learning

• by analysing the results of computer marked exams

Claus Meyer-Bothling, Germany:

Thinking the Unthinkable - Understanding 4 Dimensions

The existence of a fourth spatial dimension is confirmed by the Theory of General Relativity. Furthermore some simple properties of 4-dimensional objects, say of a 4-D-cube, can be deduced by analogy. The 3-D-projections of such objects can even be illustrated. Although we can state the properties of a 4-D-cube, we cannot picture the object itself. Our brain is not equipped to do that - following today's accepted wisdom anyway. My claim is that with the aid of modern resources we will probably be able to overcome this obstacle: With today's technology of illustration it should be possible to train our perception in such a way that we will be able to imagine 4-D-bodies.

Claus Meyer-Bothling, Germany:

More is more? More is less?

Does IT really improve the educational process, or does it merely get in the way of communication between teachers and students? Why do so many teachers persistently neglect or even refuse the use of IT in their day-to-day teaching practice - doesn't the great number of successful pilot projects prove that IT enhances the educational process? Drawing mostly on examples from Schools of Baden-Württemberg, and reporting on good as well as poor practice, I shall try to provoke a search for criteria that will be continued in our following sessions. Criteria that are sufficiently complex to be useful, but sufficiently simple to be practical, in order to distinguish between success and failure with IT in teaching.

Eva Milkova*, Milan Turcani ,Czech Rep.:

Integration ICT into teaching and learning the subject Discrete Mathematics

ICT enables new approach to the education of various subjects, also of mathematics. The education with help of visualisation is interesting and more understandable. Because our faculty disposes with good and modern equipment and there are several students who are able to prepare nice programs, we decided to improve lectures of the subject Discrete mathematics with help of students teaching packages. In our article three programs developed by students as part of their thesis will be briefly introduced.

Kent Neuerburg, USA:

Introductory statistics with spreadsheets

Spreadsheets are ideally suited for use in an introductory statistics course. These programs have the ability to handle large amounts of data and are easy to use. As an added benefit, a working knowledge of spreadsheets is a marketable skill for many students. We will focus on our experience in using spreadsheets to teach an introductory statistics course. In section one, we consider the pedagogical strengths and weaknesses of using spreadsheets in statistics. In section two, we discuss the computational strengths and limitations of spreadsheets. Finally, in section three, we provide some resources for real data and offer suggestions as to how to integrate these data into the course by demonstrating a few applications of spreadsheets to descriptive and inferential statistics.

Walther A. Neuper, Austria:

What teachers can request from CAS-designers

The basic functionality of computer algebra systems (CAS), increasingly introduced to math classes, is considered not yet optimal for education: CAS show up with the final result in one go, and do not show their built in knowledge. concept for re-engineering the interactive features of CAS is presented from the users point of view: An example session illustrates what a teacher (and a student!) can request w.r.t. the assistance in modelling and specifying a problem, and w.r.t. the user-guidance in stepwise solving a problem. Brief explanations point out, how the concept presented makes the example session work; and tasks for teachers are mentioned.

Erich Neuwirth, Austria:

The spreadsheet paradigm as a new mathematical notation

One of the fundamental properties of spreadsheets is creating formulas by relative an absolute references. These references represent spatial relationships, and therefore mathematical structures are represented visually and geometrically. Some exaples (e.g. from combinatorics and difference equations) will demonstrate how using these representations as conceptual tool can help in not only performing calculations in a very user friendly way, but also in gaining mathematical and structural insights.

Erich Neuwirth, Austria:

Let the spreadsheet throw the dice - Spreadsheets as Monte Carlo simulation engines

Monte Carlo simulation (using computer generated pseudo random numbers) is an extremely helpful tool for illustrating concepts in probability and statistics. It is surprisingly easy (and surprisingly unknown) that this kind of simulation can easily be done with spreadsheet programs. We will show some simple examples from probability and some moderately advanced examples from inductive statistics (testing and estimation) to demonstrate how simulation can help "getting the feeling" for randomness convergence of frequencies to probabilities.

Hitoshi Nishizawa, Japan:

Remedial Education of Quadratic Functions Using a WWW-based On-line Exercise System

The method and the effectiveness of remedial education using a WWW-based on-line exercise system are reported. The system displays a graph of a quadratic function and requests the student to express it in a symbolical expression. Six students were selected to attend the remedial course using the system. Although they used only one formula to express the graphs before the exercises, they have extended the variety of their expressions through the exercises.

Vladimir Nodelman, Israel:

Parametric nature of mathematics' objects and computer environment

Although the simplest mathematics' objects may be considered as based on parameters. Most of parameters are numeric. In computer software it is a regular task to implement numeric input. The problem is in:

• visually discrete nature of an "input box" entry opposite to continuity of most mathematics notions' parameters,

• not friendly interface with static changes in correspondence to entered values.

We present an approach which let the student DYNAMICALLY enter and change parameters in not pure numeric way, even prepare such input by himself in order to analyse parameters' rule and mathematics' objects "behaviour".

Plenary: Walter Oberschelp, Germany:

Chances and limits for teaching in the information age - human mind models and society demands

Successful IT-based teaching requires motivation, understanding, training and didactic sugar. The main problem is to adapt the problem structure to the intellectual structure of the learner and to his needs. Moreover there must be results which are useful for the society. We experience more and more, that the charm of having huge information resources e.g. via internet is only temporary: The present IT scratches only the surface of the human and social demands. The main need of man is not the consumption of news, but production of and interaction with personal signals on a reliable basis in order to be sure of ones own uniqueness. Surfing for information through open and heterogeneous nets will loose importance against new types of devices, which guarantee, e.g., security of transmission, legal control of transactions and semantic reliability of information. The task to keep the society in good order is incompatible with unrestricted informational liberalism, and the society needs more than a netiquette without obligations. New problems for jurisdiction arise: Information crimes cannot be judged by simply counting bits like peas.

Some epistemological problems which are connected with the concept of information are discussed. And the realisation of a global justice will have to be recognised as one fundamental basis for the global society. In particular, we investigate, how math-learning will have to develop: The special problems of math-teaching are the alienation by formalism, the lack of personal appeal and the somewhat metaphysical nature of mathematics, whereas its pragmatic value is often invisible. Since mathematical ideas are often very compact, the abundant information of the internet is hard to combine with mathematical thinking. And yet, mathematical teaching establishes useful tools for the complex existence in the global society. We exemplify problems in private and global economy and in our real physical world and discuss essential and obsolete material. We sketch, how methods for self-guided instruction may be improved. But we emphasise, that, due to the anthropological situation, personal instruction and care will become even more important in the future. The satisfaction of really understanding an argument from the scratch and the experience of responsibly solving problems without the assistance of non-transparent tools will become a source of creativity and a well accepted motive in the education of independent and mature citizens.

Regis Ockerman, Belgium:

Probability simulation with TI-83

Taking advantage of the possibilities of the TI-83, it's easy to do simulations, dealing with problems of probability. In this workshop, we will use programs for those simulations. This will be done in a way, that you can also use these things in class.

Tatyana Oleinik, Ukraine:

Project on critical thinking development using technology

This paper represents the results of special courses given to undergraduate teacher students of «mathematics-computer science» speciality. A general problem of its study is understanding the possibilities of technologies for realisation of ideas of Project on Critical Thinking development. The most interesting and significant aspect of this study was modification of views on the essence and kinds of teaching and learning activity. Obviously it is necessary to modify curricula and methodical frameworks which should focus to formation successful learners. So and why CAS like DERIVE and dynamic geometry software like DG are good medium for encouragement of pupils' interests and reflection. Besides new standards of the mathematics education require to understand how meaningful classroom dialog can stimulate collaboration of teacher, students and software.

Judy O'Neal, USA:

Technology as a Vehicle for Updating Middle Grades Content and Pedagogy

Technology has certain unique capabilities that support the learning, doing, teaching, and assessing of mathematics. Accepting that these capabilities are ever changing as new tools are developed, the design of innovative and effective professional development programs for motivating and inspiring the current and next generation of mathematics teachers is a continuously evolving and stimulating endeavour. A description of the guiding principles, planning and development phase, and initial implementation and evaluations efforts that support technology training from a slightly different perspective will be presented.

Guenther Ossimitz, Austria:

System Dynamics modelling: a new perspective for math classes? - An introduction for all who are interested in this field

In this presentation I will give an introduction to the topics of the working group "System Dynamics and Systems Thinking". I will address the following issues:

• What is systems thinking?

• What are the basic ideas of System Dynamics Modelling?

• Can Systems Thinking / Systems Dynamics be a topic for math classes?

• SD/ST: a section in Austrian Mathematics curriculum

• Results of empirical studies concerning SD / ST in math classes.

Guenther Ossimitz, Austria:

Practical Examples for Teaching System Dynamics in Mathematics Classes

In this presentation I will give an overview about some practical examples of teaching System Dynamics Modelling in Math Classes. Each example will include some application context. I plan to present some of the following examples:

• A variety of simple growth models: linear, exponential, logistic, "overshoot and collapse".

• A homeostatic feedback model and how simple time delays may cause even an elementary model to run into (deterministic) chaos.

• Population dynamics: development of the age-structure in Austria, problem of an over-aged society

• A model of balanced age structures of faculty staff: how to keep a healthy relation between assistant, associate and full professors over a longer period of time?

• The ecological balance between deer and mountain lions in the Kaibab Plateau (USA) and how human "protection" of the deer induced their doom.

Marcus Otto*, Joachim Engel, Germany:

Design and Use of a Computer Language for Teaching Mathematics - Some Examples from Statistics

During the last years, we designed a computer language and used it in mathematics education. Our aim was to establish a tool for learning and doing mathematics. The language can be shaped to meet the needs of a course. Besides using such a language for algorithmic purposes, one can create its own mathematical structures based on their features, relations and operations. Students can use this to investigate the concepts presented in a course. Taking concepts from probability and statistics as examples, we illustrate how to incorporate our language into mathematical teaching.

Bronislav Pabich, Poland:

Magic Polyhedrons

Close your eyes and imagine that you are connecting the midpoint of a cube with its vertices by line segments, creating in this way six congruent square pyramids, which will completely fill this cube. Now duplicate each of these pyramids by reflecting each of them on the plane given by its base. You get now 6 square pyramids positioned onto the faces of the cube outside. The cube together with these six pyramids perform a new polyhedron. Draw this polyhedron in that way you can imagine it. Then answer the following questions:

How many vertices, faces, edges does have this new polyhedron?

Which kind of polygonal shapes are its faces of?

Are its faces congruent?

Is this polyhedron a regular one?

What's its volume? (Compare the volume of this polyhedron with the volume of the cube in regard with the method you did create it.).......

John Pappas; Greece:

Integrating Mathematics, Physics and Interactive Digital Video

Previous research on Digital Interactive Video Technologies (DIVT) is limited to the domain of kinematics and graph interpretation in particular. This pilot study is part of a full-scale research that aims to extend the field of investigation using Digital Video Technologies as a connecting link for the Integration of Mathematics and Science. Five students participated in this study, which consisted of two parts, one without and one with DIVT support. The analysis of data gathered indicate that being able to manipulate the reference frame in the environment of the DIVT software and notice how it affects co-ordinates, graphs and equations improves the students' conceptual knowledge on this subject, in two levels:

• Students realise that there is a dynamic linking of the reference frame position and orientation and the way that graphs and the matrix of co-ordinates look.

• By bringing the reference frame to particular positions of 'special' interest, such as positioning one of the axes to be parallel to an inclined level, they can deal with their misconceptions and gain a better understanding and insight to the role of a co-ordinate system.

Pavel Pech, Czech Rep.:

Cubics and quartics on computer

In basic courses of geometry at universities are mainly linear and quadratic objects studied. Using computers enables us to include into this courses also objects, which are described by an algebraic equation of the order higher than two. With the co-operation with the students of the Pedagogical Faculty at the University of South Bohemia the software has been developed by means of which cubics and quartics (and conics as well) can be mapped in a high quality.

Valentyna Pikalova, Ukraine:

Learning Explorations and its DG Support in Geometry Course for Secondary School

The article includes the analyses of DG support in geometry course for secondary school. As a result the Dynamic Demonstrative library was developed. It includes sketches for learning explorations in geometry. This library is recommended to use in geometry course by the minister of science and education of Ukraine. The attention is also paid to the methodological questions of implementing learning explorations in secondary school curriculum.

Neil Pitcher, UK:

Evaluating the Effectiveness of Computer-Based Learning in Mathematics

This session will discuss effective ways of integrating computer-based learning environments into university Mathematics courses. The system 'Mathwise' will be used as an exemplar. Mathwise contains materials both for learning and for assessment. Such a system needs to be used carefully if it is to promote good study skills. Different teaching methods will be examined and some evaluation results presented.

Rein Prank*, Eno Tonisson, Estonia:

Computers in School Mathematics - a pilot training program for Estonian Mathematics teachers

Most of the software for the national schools' computerisation program called 'Tiigrihüpe' (Tiger Leap) has been acquired in such a way that the programs are available to all/most of the schools in Estonia. This will also simplify the training of teachers. Each county has a well-equipped pilot school, which shall organise local training and consultation for teachers of different subjects. This report describes the training cycle (9 sessions with 144 hours plus homework in the scope of more than 300 hours) conducted for 40 teachers in 2000. The cycle consisted of thematic modules based on special packages (StudyWorks, dynamic geometry, computer algebra systems, graphing functions, proofs in geometry, probability theory and statistics, spreadsheets, testing software, Internet and distance education tools) and the final integrative module on the use of computers.

Pavel Prazak, Czech Rep.:

Software Maple and Matlab in teaching of ordinary differential equations

Matlab and Maple are the powerful interactive numerical computation programmes. They have efficient built in routines enabling wide variety of computations. They have also easy to use graphical commands to make visualisation available. In our contribution we will focus on selected possibility of using symbolic calculations, numerical and graphical methods for support and illustration of the subject of ordinary differential equations and outline various possibilities of visualisation of the solutions of these equations and show the samples of application of above mentioned problems

Stefan Priselac, Nancy Priselac, USA:

Technologically Presented Learning Material: The Communiversity Project in Maryland, USA.

The presentation is multi-media in nature and can last from fifteen minutes to one hour depending on the allocation of time. The Communiversity at Garrett Community College provides diverse ways to deliver training, courses, programs and interaction across the globe as we redefine access from set time to anytime and from one place to many places as we create a new future in education.

Wolfgang Pröpper, Germany

The TI-89/92 as a Tool for Analytic Geometry

The CAS calculators by Texas Instruments seem to be primarily suited for algebra and calculus at a first glance. The home screen menus give special emphasis to operations like "factor" and "comDenom" or "limit" and "taylor" respectively. For problems that typically appear in Analytic Geometry assistance is scarcely found. Solving vec­torial equations can only be achieved after a large-scale (and by that faulty) rewriting into systems of equations or into matrices. Functions of vector algebra are not available in the home screen but must awkwardly be looked for in a catalog. Texas Instruments however took care for a way out of that dilemma when designing the operating system. The user can easily create customized menus and complete not available functions by pro­grams of his own. In the contribution a menu together with some desirable functions is presented and shown how it can be put into action for solving problems that usually occur in classical Analytic Geometry.

Chantal Randour, Belgium:

Cabri and anamorphoses

Des élèves de 17-18 ans ont traité le problème des anamorphoses, tant perspectives que celles utilisant des miroirs. La principale source utilisée est La Perspective Curieuse du Père Niceron (1652). La littérature peu abondante traite uniquement ce sujet sur le plan analytique. Nous avons préféré utiliser la géométrie descriptive pour concevoir des constructions simples pouvant être ensuite communiquées à Cabri. Les élèves ont ainsi réalisé des anamorphoses perspectives, coniques, cylindriques et pyramidales. Le travail mathématique s'est accompagné d'une recherche artistique en bibliothèque, dans les musées et sur internet. Un CD-rom (en power-point) montre quelques extraits de cette recherche. Une exposition des travaux a eu lieu dans l'école. Je me propose d'expliquer les différentes figures Cabri crées pour ce travail et de montrer le diaporama (+/- 20 min.) réalisé. Quelques modèles d'anamorphoses réalisées par les élèves seront visibles, ainsi qu'un pantographe (Scheiner-Parré) permettant de réaliser un type particulier d'anamorphoses coniques.

T adeusz Ratusinski, Poland:

The role of the computer in discovering mathematical theorems

Pedagogical University, where I work, prepares mathematicians for being mathematics teachers in the future. The pre-service teachers ought to be educated enough to work in a modern school. In this paper I would like to present my observation I made during my classes with Four Year mathematics students (approx. 22-year-old). The students were supposed to discover, using computer, some properties of the monotonic functions. I would like to show the results the students work and also a few characteristic errors they made formulating mathematical hypothesis.

Plenary: Eugenio Roanes-Lozano, Spain:

Co-operation Between Dynamic Geometry Systems and Computer Algebra Systems - Investigating, Guessing, Checking and Proving with the computer

Computer Algebra Systems (CASs), like Maple, Derive, Mathematica, Axiom, Macsyma, Reduce, MuPad..., are specialised in exact and algebraic calculations. They use Exact Arithmetic and can handle non-assigned variables (i.e. variables in the "mathematical" sense, not in the usual sense in Computer Science). Many extensions like symbolic differentiation and integration, linear and non-linear equation and polynomial systems solving, 2D and 3D plotting... are usually included too. II, Cinderella, Euklid, Dr. Geo, WinGeom..., are specialised in rule and compass Geometry. The adjective dynamic comes from the fact that, once a construction is finished, the first objects drawn (points) can be dragged and dropped with the mouse, subsequently changing the whole construction. They usually incorporate animation and tracing too.

Unfortunately CASs and DGSs have evolved independently. Some CASs like Maple include specific and powerful packages devoted to Euclidean Geometry, but no CAS has incorporated Dynamic Geometry capabilities. On the other hand, Dynamic Geometry Systems can't handle (at least from the point of view of the user) non-assigned variables. Therefore, what can be saved from a DGS is only live graphic (to be read by the DGS), a geometric algorithm (script or macro, to be interpreted by the DGS) or a dead (fixed) graphic in one of the standard graphic formats. More precisely, what is missing in the DGSs is the possibility to handle and export parametric data about the plot: co-ordinates of points (allowing parameters as co-ordinates), equations of objects (allowing parameters as coefficients), length of objects (depending on parameters)...

Some DGSs (like Cabri Geometry II or Cinderella) include theorem-checking capabilities. This theorem-checking is based in altering the initial data: they find counterexamples if the result is false and suppose that the result is true if they find no counterexample (i.e., they are not "proofs" from the mathematical point of view). This lack of co-operation is more surprising in cases like the TI-92, where both technologies are simultaneously available. A straightforward application of this co-operation would be to treat with the computer the whole mathematical process of discovery (or re-discovery):

Investigating - Guessing - Checking - Proving.

The talk will begin presenting an overview of the main capabilities of CASs and DGSs. A basic introduction to Automatic Theorem Proving in Geometry (Gröbner bases method and Wu's pseudoremainder method) will follow. The missing co-operation between CASs and DGSs will be detailed afterwards. Finally, the (ideal) whole mathematical process of discovery mentioned above will be presented. All steps will be illustrated with adequate examples.

Jarmila Robova, Czech Republic:

Graphic solutions of equations and their systems

The contribution deals with using graphing calculator TI-83 in teaching of algebra in secondary school. Several techniques of graphic solution are presented (geometric representation of problems, boolean function, graphic substitution).

Ana Rosendo*, Jaime Carvalho e Silva, Portugal:

Computers and calculators in the preparation of future mathematics teachers - an experience

We will describe how future mathematics teachers are being prepared to use technology in mathematics teaching (at the Mathematics Department of the University of Coimbra)

Anna Salvadori*, Primo Brandi, Italy:

A modern approach of limit process

A new approach to limit process is proposed. The aim is to drive students from perception to usual epsilon-delta definition gradually. This path involves the three fundamental aspects: geometric, numeric, algebraic. To supply the graphic support a software ad hoc is implemented.

Susanne Saminger, Austria:

IMMENSE - a tool for visualization and mathematical experiments

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Csaba Sárvári*, Mihály Klincsik, Hámori Ildikó Perjésiné, Hungary:

How can we combine the CAS with authoring system tools to create a flexible learning environment

Using Maple CAS as a powerful mathematical tool and the Toolbook Instructor object oriented authoring system we can create new learning environments.We illustrate with case studies the step by step learning methods within an example and from the easier examples towards the complicated ones. With these new methods the user can be focus, concentrate on the local and the global know-ledges together. Our applications particularly applicable via Internet and local network, too.

Ralf Schaper, Germany:

Mathematica graphics in the internet

An extended version of LiveGraphics3D will be presented.

Franz Schlöglhofer, Austria:

Teaching System Dynamics Modelling in Secondary Schools: The Teacher's perspective

In this presentation the following issues will be addressed:

• What are the basic ideas of the didactics of System Dynamics?

• What aspects of math teaching are involved when teaching system dynamics?

• What are the main ideas of the section "Investigation of interrelated Systems" ("Untersuchung vernetzter Systeme") of Austrians math curriculum at 11^th grade for a science-oriented subtype of high-school ("Realgymnasium")

What are the experiences with practical teaching SD in math classes?

Karsten Schmidt, Germany

The Use of CAS in the Thuringian School System: Present and Future

Based on a recent survey carried out in all 450 secondary schools in the state of Thuringia, Germany, the following questions will be investi­gated: Which level of computer equipment is available for classroom use? Which kinds (simple / scientific / graphical / symbolic) of pocket calculators are used in which grades? Does the school possess a license for a CAS? In a second part of the survey, the person filling in the questionnaire is asked to give some of his/her personal attitudes, which will also be analysed in the presentation: Which kinds (simple / scientific / graphical / symbolic) of pocket calculators should be used in which grades? Which knowledge does he/she have of symbolic calculators and CAS? What are the advantages and disadvantages associated with the use of symbolic calculators and CAS in the classroom?

Alfred Schreiber, Germany:

Project ZERO: Developing Online Material for Mathematics Teacher Education

This paper reports about a project dealing with the conception and production of supplementary learning material for mathematics teachers. It surveys the various types of courseware-modules presented herein online (e.g., dynamic geometry, computer-based-training-like frames, paper-and-pencil-exercises), and discusses their specific purpose and use. Emphasis is put on the problem of how to embody appropriate functions that provide the opportunity to evaluate user inputs - thus enabling an author to give "local" feedback to the student. Finally, some questions are raised concerning the form that should be used in the future to represent both data and logical structure of the underlying content.

Monika Schwarze, Germany:

Self directed learning in maths - szenarios, material from a german case study

Information about a german case study of self-directed learning in high schools supported in different ways by new media, e.g. interactive tools or learning environments: there will be an exemplarily presentation of szenarios of learning linear algebra, statistics, analysis or geometry and some results of evaluation of the first projects.

Angela Schwenk, Germany:

Mathematical Abilities of University Entrants

Looking the future of mathematical teaching should also include a view on the situation at the moment: University entrants to engineering courses have poor knowledge in mathematics. The presented results base on investigations in 1995 and 2000:

• Comparison of the results from 1995 and 2000

• Comparison of entrants with 12 (Fachabitur) and 13 (allgemeine Hochschulreife) years of high school education

• Influence of a mathematical bridging course

Peter Sedlmeier, Germany:

Improving statistical reasoning: a computer program for high-school students

New results in research on judgment under uncertainty show a way of how to improve the teaching of statistical reasoning (Sedlmeier, 1999). The implications of this research are that (i) successful learning needs doing, and (ii) that the format in which information is represented plays a decisive role. Statistical problems are, for instance, solved much better if the relevant pieces of information are presented as frequencies rather than probabilities. It also helps a lot if random processes can be observed rather than only read about. A computer program is presented that incorporates these implications from psychological research (Sedlmeier & Köhlers, 2001). The software accompanies an elementary text book on probability theory to be used in high school.

Mazen Shahin, USA:

Modelling with Difference equations using DERIVE

In this discussion we share the pedagogy and the methodology of modelling real life situations with difference equations using the computer algebra system Derive. This is a part of a reform finite mathematics course in which students explore and discover mathematical ideas on their own as they complete specially designed tasks whose emphasis on applications helps them see the relevance of the abstract concepts. We will emphasise the use of graphical and numerical techniques, rather than theoretical techniques, to investigate and analyse the behaviour of solutions of the difference equations. We will investigate interesting linear and non-linear models as well as systems of difference equations from such diverse disciplines as business, economics, life sciences and social sciences.

Mazen Shahin, USA:

Discrete Delayed Population Models with DERIVE

In this paper we show how Derive can be used efficiently in modelling and investigating discrete delayed population models. In particular we are interested in some population models represented by non-linear second order difference equations. We will explore the stability of the equilibrium values of the systems. We will apply an interesting method to control the chaos in a dynamical system represented by a first order non-linear difference equation. Some of the pedagogical issues related to the use of a CAS in modelling will be discussed.

Harry Silfverberg, Finland:

Using Voronoi diagrams produced by DGS as a tool in an educational study

The Voronoi diagram of a collection of points is a partition of space into cells, each of which consists of the points closer to one particular point than to any others. According to the prototype theoretical explanation students at the lowest van Hiele levels tend to classify geometrical figures on the basis of extent of the similarity of the figure and the visual prototypes. The poster will graphically show how well Voronoi diagrams and partitions based on the different selection of prototypes fit to the empirical data gathered in Silfverberg's research (1999) about the ways how students at the lowest van Hiele levels classified a given collection of triangles into acute, right, obtuse, equilateral and isosceles triangles.

Plenary: Branca Silveira (Portugal):

Teacher training: the role of technology

We can't have a change in our schools without teachers and teachers don't change if they are not convinced that the change is going to improve something. The world is changing, society is changing, pupils are changing, and the schools? How are schools coping with this? Technology is everywhere. No discussion about that. Everyday appears new software, new computers, new calculators, etc. Are the schools ready for this? Does technology play a significant role in the change of the curriculum? How do teachers face this? Are they prepared to use technology effectively? Which kind of difficulties do teachers face? Some teachers have been using technology; did they really changed their methodologies or are they using them in a inadequate environment? What about teacher training? Which kind of training is more effective? Initial training? In service training? But, what should we do for making teachers include technology in their practice? More computers? More training? A different schedule for the classroom? Making the use of technology compulsory? In Portugal the use of graphic calculators is compulsory in secondary schools, so, everybody has to use them. Should we do the same with computers and other technology? What about Internet? How should we train teachers for the use of Internet in the classroom? How can teachers develop the ability to analyse and integrate in an intelligent way, in their teaching the technological developments to come (software, hardware, communication...)?

Those are some of the questions we are going to discuss in this talk, based on the Portuguese experience, focusing my point of view as a teacher, as a teacher trainer and as a member of the board of directors of APM (the Portuguese "Association of Teachers of Mathematics").

Edgar Smith* and A. Waterson, Australia:

Online mathematics teaching:the development of student-instructor interaction

We discuss differences between teaching styles in online mathematics teaching and other less technical subjects. We discuss how to "lean over a student's shoulder" online. Techniques are both automatic and software mediated discussions with students. Discussions are extremely expensive in terms of staff time, so we consider automated responses. These are illustrated with sample problems in elementary fluid mechanics in a subject delivered via WebCT. We discuss how to evaluate and improve automated responses.

Robert Smith, USA:

Spreadsheets across the curriculum

Excel and other such electronic spreadsheet programs have found their way in to a variety of undergraduate mathematics courses. In this presentation we will demonstrate some spreadsheet uses in a variety of undergraduate courses from precalculus to abstract algebra.

Grosio Stanilov, Bulgaria:

Mittels Computergraphik zu mathematischen Entdeckungen

Wir untersuchen die Parallelogramm-und die Wuerfelschnitten nur mittels Schulmathematik.Um die Besonderkeiten der Laengenschnitten und die Flaecheninhalten zu entdecken,verwenden wir zunaechst die Computergraphik.Wir erreichen zu wichtigen Saetzen in der Analysis,zur besonderen Schnitten und zur neuen exotischen Flaechen in der Differentialgeometrie.Einiges ist auch in die Bildkunst zu verwenden.In der hyperbolischen Geometrie erreichen wir zu einer Konstante,die die Seiten des Morleys Dreiecks fuer jedes beliebigen Dreiecks von oben beschraenkt.

Steve Sudgen :

Teaching Discrete Mathematics With Excel

The modern spreadsheet as exemplified by Microsoft Excel offers almost unlimited opportunities for the illustration of fundamental mathematical concepts. Further, the same software allows the teacher to encourage an investigative or experimental approach to mathematics learning. This talk will present some examples of these ideas plus an overall framework for the use of Excel for the enhancement of laboratory work. It is claimed that the approach outlined is especially useful for tertiary IT students with a relatively modest background in mathematics. The discussion will focus on topics from fairly traditional courses in discrete mathematics.

Fred Szabo*, Miroslaw Majewski , Canada:

Integrating MuPAD into the Teaching of Mathematics

Computer Algebra Systems are becoming more and more popular in mathematics education. However, many teaching issues are still unresolved, and no one is able to give a simple recipe how to integrate computer algebra systems into the teaching process. In this paper, we discuss some proven strategies for using MuPAD in the teaching of mathematics.

Christian Thune Jacobsen, Denmark:

Experimental Mathematics. Someone invented the knife - everybody uses it

Computer algebra systems (CAS), such as Derive and Maple, will naturally be an integrated part of teaching mathematics in the future - just as the use of calculators has been for the last two decades. The question is only how to implement CAS.

Eno Tonisson, Estonia:

Expression Equivalence Checking in Computer Algebra Systems

This paper investigates the possible educational application of equivalence checking and the capability of expression equivalence checking in some common computer algebra systems. The applications of equivalence checking can be analysed from the viewpoint of three types of users: that of the teacher, that of the student, and that of an Intelligent Tutoring System.

This paper deals with the way a computer algebra system copes with the checking of the basic equivalencies of algebra and trigonometry. It appears that the tools are far from perfect and require improvements.

Yulian Tsankov, Bulgaria:

Cubic Section by moving plane

By Computer graphic and Schoolmathematic we investigate all cubic sections. They depend of three parameters. If we fix two of them, the interval (-infinity, +infinity) for the third parameter divided in six subintervals, where the sections are from different type. We visualize these sections and corresponding them area functions. The dividing - points arise some surfaces geometrically connected with the cube.

Nelson Urrego, Columbia:

Using DERIVE for beginner courses of recursion theory

In this Paper, the author gives a short introduction to the main concepts about Recursive Functions and some examples are programmed using DERIVE. These exercises can motivate students in the design of algorithms for solving rigorous arithmetic problems such as the implementation of a procedure for generate of a 1-1 Primitive Recursive correspondence between N2 and N.

Aynur Uysal, Turkey:

Importance of Mathematics in Engineering Education

Two different approaches have traditionally influenced mathematics teaching in engineering education. First one considers mathematics only as a tool for professional practice ,while the second one relates mathematics education with the development of the logical and critical thinking without which no tool will be efficient to the learners for their understanding and interpretation of the world. As well known , the second approach has been receiving a growing importance in the last years. In this paper , the second approach are described with detailed examples. A rich set of experience are also presented from the mathematics teaching in the Technical University of Istanbul.

Mithat Uysal, Turkey:

An Internet-Based Course Structure for Teaching Mathematics in an Engineering School

This study sets out to present a detailed and integrated approach for teaching mathematics using world wide web. Previous works and existing www-based teaching structures are first discussed. Then the concept of a course portal following the comprehensive and integrated approach are presented. Main modules of the portal, namely, the main page, multimedia page, courseware page, contact page and the search page are described. The ways to improve the portal are discussed. Some observations from the ITU model (Istanbul Technical University) are also presented.

José Luis Valcarce Gómez, Spain:

Bridging the Gap between Dynamic Geometry and Computer Algebra: The Case of Loci Discovery

A basic problem in elementary geometry consists of finding the equation of a locus, given some conditions defining it. This problem remains unsolved in the field of mathematics education from a technological point of view: no friendly tool exists that allows a student to specify the conditions of a locus in a diagram and it returns the equation of the locus. Numerical approaches to this problem have been tackled in cuurent dynamic geometry environments but they share an essential incompleteness: an object must be constrained to move along a predefined path in order to get the trace of some other object. This paper describes a symbolic-dynamic approach to this problem: a computer algebra system solves it within a dynamic geometry environment.

Piet van Blokland, Netherlands:

A sample of ideas in teaching statistics

Probability and statistics in secondary school should be presented in such a way that it demonstrates the importance of this subjects in society. Some realistic simulations will be shown. Polls are an often used tool in modern society to investigate opinions. In this lecture a huge dataset of 50000 students will be presented The effect of sampling will be shown. In order for the students to grasp the idea of central limit theorem, technology will help. Pictures which can be manipulated by students will help students to understand better the ideas behind hypothesis testing.

Carel van de Giessen, Netherlands:

The Visualisation of a parameter

Based on the ideas of David Tall we, Piet van Blokland and I, have developed a program to investigate graphs and formulas. Two aspects may be of special interest: variables and parameters. For the young students (12-14 years) it is easier to understand the concepts involved with graphs and formulas when using word-variables. The concept of 'parameter' in formulas is difficult to grasp, because the mathematical level needed to understand a parameter is high. We therefore introduced a so called 'sliding parameter'. In the programme this concept arises interactively using a scrollbar: the parameter value changes and so does the graph. This is a dynamic way to investigate a graph and the role of a parameter. One graph, one value of the parameter.

Henk van der Kooij, Netherlands:

Functional Algebra with the Use of the Graphing Calculator

Algebra is a very important topic in mathematical programs for upper secondary education, but a vast majority of students is weak in understanding and using formal algebraic tools. This paper discusses some ideas about using the graphing calculator to support the learning of algebra in the context of functions and to help students overcome algebra-anxiety. Accepting the graphing calculator as a supportive toolkit in the learning of algebra has far-going consequences for the way in which what kind of algebra should be learned and taught.

Peter van Wijk*, Hans Stam, Netherlands:

Mathematics and Internet

The Internet is primarily used as a source of information, as reference work and as a medium in which to look things up. There is, it is true, a lot to be found on the Internet, but for (arithmetic) education the Internet can be more than an encyclopaedia or library.

In order to organise the various ways in which the Internet can be used in education, we take the classification based on the idea that there are various sorts of places on the Internet.

Ödön Vancso, Hungary:

Classical and Bayes-statistics in the school supported by computer

In this presentation I would like to show such software which help to understand by visualisation, representation or counting some main ideas of the classical statistics for example: normal distribution and Laplace-condition, confidence-interval, testing hypothesis. On the other side I talk about working (following one idea of Dieter Wickmann) on a program which also can be used in the school and give a possibility to teach Bayes-statistics earlier than the Universities and Highschools. This software have been developed by mathematics and informatics students of Eötvös Lóránd University of Budapest leading by Éva Vásárhelyi, László Szabadi and me.

Eva Vasarhelyi*, Karl Josef Fuchs, Hungary/ Austria:

Problem - Analysis - Encoding - Testing = About Program- and Data-Structures

The two authors will show examples for the use of Hand-Held-CAS-Technology in computerscience. From the educational point of view the different problems of interpretation, stepwise refining and modification concentrate on the flexible, effective use of basic comments of an imperative programming tool in many different ways.

Herrmann Vogel, Germany:

Use of Cinderella in higher elementary geometry

I will presentate a paper created with Cinderella, which deals with the "Wallace line" of a triangle and a generalaziation of this line. It demonstrate the possibilities of Cinderella how one can - illustrate well known geometry facts by using the moving mode or the animation mode, - find new suppositons by doing exercises, - create the envelope of a set of straight lines, - construct conics with certain conditions, - create algebraic curves of higher order.

Rolf Wasen, Sweden

Computers in Engineering Education

I will present experiences from 1 ½ years at a mathematical Study Center and the use of computers and computer algebra in project works in the basic analysis courses. A model of how to use computer algebra in mathematical education was developed and will also be presented. It turned out that the computer was an indispensable tool for illustrating and testing mathematical ideas ­ this not at least for the teacher ­ and that the objections can be met with. There is an attractive possibility to continue these project works into research at different levels of ambition.

Wilhelm Weiskirch, Germany

Ortskurven - Loci

Kurven sind mehr als Graphen von Funktionen. Dass die verbreitete unterrichtliche Reduktion des Kurvenbegriffs auf das Bild einer Funktion dessen mathematische Bedeutung und das didaktische Potential nicht annähernd ausschöpft, ist unbestreitbar. Insbesondere geomtrische Zugänge zu nichttrivialen Kurven und deren analytische Betrachtung werden durch DGS und CAS ermöglicht und können dazu beitragen, die gegenwärtige Starrheit der Oberstufenmathematik zu durchbrechen. Am Beispiel nichttrivialer Kurven als Ortslinien abhängiger Punkte, bzw. Massenpunktbahnen sollen unter Ausnutzung der genetischen Methode deren Bedeutung und Potential für den Mathematikunterricht erörtert werden.

Otto Wurnig, Austria

Advantages and Dangers in the Teaching of Stochastics by using CAS

The use of CAS in the teaching of stochastics can be dangerous because the students like to use standard functions and functions which the teacher programmed as a tool without thinking. In student oriented thinking, however, CAS can well be used to gradually develop definitions and to help with the understanding of formulas and ways of solutions. The simulation of experiments by direct input of CAS commands makes it possible to put a stronger accent on the building of models.

Maria Zajac, Poland:

Internet materials in mathematics teaching

In the paper the idea of an Internet educational website will be presented. The learning materials are divided into three groups: Power Point presentations, web pages and lesson scenarios. All of them are intended to be a tool for computer assisted learning. The resources for Math lessons will be of special interest in the paper.

Zulkardi, Netherlands:

CASCADE-IMEI: Web site support for student teachers to learn realistic mathematics in Indonesia

CASCADE-IMEI is a learning environment in the form of a face-to-face course and a Web site (www.cascadeimei.com) which aims to support student teachers in Indonesia to learn Realistic Mathematics Education (RME). RME is an instructional theory in mathematics education that was originally developed in the Netherlands. So far, two prototypes have been developed and evaluated both by student teachers and several experts in the Netherlands. This paper presents the origins of the learning environment, with a more detailed on its Web site as well as the results of first two cycles of its prototyping process.