Technology in Mathematics Teaching Manfred Borovcnik and Hermann Kautschitsch Proceedings of the Fifth International Conference on Technology in Mathematics Teaching (ICTMT 5) in Klagenfurt 2001

Manfred Borovcnik and Hermann Kautschitsch (Eds.) (2002). Technology in Mathematics Teaching. Schriftenreihe Didaktik der Mathematik, Vol. 26 and 27. Vienna: öbv & hpt. For extended versions of the contributions, see the internet platform of Technology in Mathematics Teaching: http://wwwg.uni-klu.ac.at/stochastik.schule/ICTMT_5/ICTMT_5_CD/index.htm

Technology in Mathematics Teaching Contents of the first volume

Plenary Lectures and Strands Strand 1: Integration of IC technologies into learning processes Chair: Jean-Baptiste Lagrange Plenary: Tommy Dreyfus Mara Alagic Rebecca Langrall Mária Bakó John Berry Andy Smith Neil Challis, H. Gretton M. Robinson, St. Wan Roger Fentem Jenny Sharp Ruth Forrester Jenny Gage Samer Habre Christian Thune Jacobsen Gisèle Lemoyne François Brouillet Sophie René de Cotret Marie-Thérèse Loeman

Computer-rich learning environments and the construction of abstract algebraic concepts Differentiating mathematics instruction through technology: Deliberations about mapping personalized learning Mathematical software in the educational process of the French and Hungarian teachers Observing student working styles when using graphic calculators Diagnosing mathematical needs and following them up The impact of training for students on their learning of mathematics with a graphical calculator Data collection and manipulation using graphic calculators with 10-14 year olds The role of the graphic calculator in early algebra lessons The ODE curriculum: traditional vs. non-traditional. The case of one student Experimental mathematics

Cognitive and didactic ideas designed in TIC environments for the learning and teaching of arithmetic and pre-algebra knowledge and concepts To learn from and make history of maths with the help of ICT Claus Meyer-Bothling Thinking the unthinkable — Understanding 4 dimensions Hitoshi Nishizawa Remedial education of quadratic functions Y. Kajiwara T. Yoshioka using a web-based on-line exercise system John Pappas, E. Koleza Integrating mathematics, physics J. Rizos, C. Skordoulis and Interactive Digital Video Neil Pitcher How to use computer-based learning effectively in mathematics Carel van de Giessen The visualisation of parameters Henk van der Kooij Functional algebra with the use of the graphing calculator Manfred Borovcnik and Hermann Kautschitsch (Eds.) (2002). Technology in Mathematics Teaching. Schriftenreihe Didaktik der Mathematik, Vol. 26 and 27. Vienna: öbv & hpt. For extended versions of the contributions, see the internet platform of Technology in Mathematics Teaching: http://wwwg.uni-klu.ac.at/stochastik.schule/ICTMT_5/ICTMT_5_CD/index.htm

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Contents of Volume 1

Strand 2: Technologically presented learning material Chair: Bernard Winkelmann Plenary: Alison Clark-Jeavons Rosalyn Hyde May C. Abboud Douglas Butler Peter Cooper B. Magan, K. M. Dilks Timo Ehmke Mary Susan Hall Judith H. Hector Duncan A. Lawson J. Reed, S. Tyrrell Pavel Leischner Karel Kabelka Michael McCabe Ann Heal Alison White Vladimir Nodelman Nancy J. Priselac Stephen M. Priselac Alfred Schreiber Peter van Wijk Hans Stam A. Waterson E.R. Smith

Developing a technologically rich scheme of work for 11 – 12 year olds in mathematics for electronic delivery Animation — A tool for understanding polar coordinates Adding a sparkle to classroom teaching — Using Word, the Internet, and object-oriented software Design of content independent instructional systems Geometria: A tool for the production of interactive worksheets on the Web Creating and teaching online mathematics courses Teaching probability and statistics via the Internet A Web-site for a mathematics support centre The collection of interactive solids figures and spatial situations in the Cabri - geometry Computer assisted assessment of proof = Proof of CAA — New approaches to computer assisted assessment for higher level learning Parametric nature of mathematics’ objects and computer environment The Communiversity Project delivers a restructured pre-calculus distance learning course Project Zero: Developing online material for mathematics teacher education Mathematics and the Internet Online mathematics teaching: The development of student instructor interaction

Plenary Lectures and Strands

Strand 3: Technology in teacher education Chair: Jaime Carvalho e Silva Plenary: Branca Silveira George Adie Bogdan Zoltowski Adnan Baki Elizabeth Belfort Luiz C. Guimarães Rafael Barbastefano Primo Brandi Anna Salvadori Jaime Carvalho e Silva José Carlos Balsa Maria José Ramos Isabel Fevereiro Maria C. Belchior Henryk Kakol Konrad Krainer

Auxencia A. Limjap

Eva Milková Milan Turčáni Walther A. Neuper Rein Prank Eno Tonisson Ana Isabel Rosendo Jaime Carvalho e Silva Nelson Urrego P. Maria Zajac Zulkardi Nienke Nieveen

Teacher training: The role of technology Practical aspects of CAS using sinusoidal functions Investigating teachers’ perceptions on their preparation to use IT in classroom instruction Using computers in mathematics teacher training programs: A reflection upon an experiment A modern approach to limit processes Internet as a tool in the preparation of future mathematics teachers Changing the classroom practices — The use of technology in mathematics teaching Integrated teaching mathematics with elements of computer science Innovations in mathematics, science and technology teaching — IMST² — Initial outcome of a nation-wide initiative for upper secondary schools in Austria Current educational theories and New Technologies: Development of a training program for mathematics teachers in the Philippines Integrating ICT into the teaching and learning of discrete mathematics What teachers can request from CAS-designers Computers in school mathematics — A pilot course for school teachers of mathematics in Estonia Computers in mathematics education — An ongoing experience Using Derive for beginner courses of recursion theory Internet materials in mathematics teaching CASCADE-IMEI: Web site support for student teachers learning — Realistic mathematics education in Indonesia

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Strand 4: Changes in geometry and algebra via DGS and CAS Chair: Hans-Georg Weigand Plenary: Jean Flower Denis Bouhineau J. –F. Nicaud, X. Pavard E. Sander Hans-Jürgen Elschenbroich Thomas Gawlick

Fitting from function families with CAS and DGS A microworld for helping students to learn algebra

Michalis Kourkoulos M.-A. Keyling

Miroslaw L. Majewski M. E. Fred Szabo Robert Mayes

Self-correction in algebraic algorithms with the use of educational software: An experimental work with 13-15 years old pupils A CAS-index applied to engineering mathematics papers Improving maths skills with CAS technology: A CAS project carried out in Scotland with 16-17 year olds using TI-92s Integrating MuPAD into the teaching of mathematics Absolute geometry: Discovering common truths

Bronisław Pabich

Magic polyhedrons

Pavel Pech Jaroslav Hora Eno Tonisson

Cubics and quartics on computer

Eoghan MacAogáin Tom Macintyre

Teaching and learning geometry: Dynamic and visual Dynamic notions for Dynamic Geometry

Expression equivalence checking in Computer Algebra Systems

Plenary Lectures and Strands Strand 5: Cooperation between DGS and CAS Chair: Martín Garbayo Moreno Plenary: Eugenio Roanes-Lozano Yuriko Yamamoto Baldin Yolanda K. S. Furuya Francisco Botana José L. Valcarce Wolfgang Fraunholz José L. Valcarce Francisco Botana

Boosting the geometrical possibilities of Dynamic Geometry Systems and Computer Algebra Systems through cooperation A study of conics with Maple V and Cabri-Géomètre II The three and four bar linkages revisited: Graphs and equations A computer aided learning environment of linear algebra using the computer algebra system MuPAD Bridging the gap between dynamic geometry and computer algebra: The case of loci discovery

Strand 6: Mathematical modelling with technology Chair: Jenny Sharp Plenary: John Berry George Adie Bengt Löfstrand Bogdan Zoltowski G. Albano C. D’Apice M. Desiderio Burkhard Alpers Brigitta Aspetsberger Klaus Aspetsberger Per Broman André Heck André Holleman André Heck André Holleman Iavor V. Hristov Duncan A. Lawson J. H. Tabor Pavel Prazak Antonin Slaby Mazen Shahin

The use of technology in developing mathematical modelling skills Differential equations instead of analytical methods Laplace Transform and electrical circuits: An interdisciplinary learning tool Mathematical application projects for mechanical engineers — Concept, guidelines and examples Cross curriculum teaching and experimenting in math & science courses using New Technology Mathematical modelling with use of Cabri Modelling human growth Investigating bridges and hanging chains Model of deformations of fluid particles due to electric field Introducing models and modelling through spreadsheets Software Maple in the teaching of ODE’s Discrete delayed population models with Derive

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Strand 7: The global perspective of Information Technology Chair: Peter Bender Plenary: Walter Oberschelp John Berry Roger Fentem Stefanie Krivsky Ewa Lakoma Tatyana Oleinik Tadeusz Ratusinski Monika Schwarze Angela Schwenk Manfred Berger John Searl

Chances and limits for teaching in the information age — Human mind models and society demands Investigation into student attitudes to using calculators with CAS in learning mathematics The potential of the Internet for innovations in didactics of mathematics On the impact of hand-held technology on mathematics learning — From the epistemological point of view A project on the development of critical thinking by using technology The role of the computer in discovering mathematical theorems Self-guided learning — Scenarios and materials from a German pilot project Mathematical abilities of university entrants and the adapted use of computers in engineering education Of Babies and Bath Water

Technology in Mathematics Teaching Contents of the second volume

Special groups and working groups Special group 1: Derive, TI-89/92 and other CAS Organisers: Josef Böhm, Bernhard Kutzler, Marlene Torres-Skoumal Bengt Åhlander

How to make tests for students that are using a CAS tool (TI-89)

Halil Ardahan Yaşar Ersoy

Issues on integrating CAS in teaching mathematics: A functional and programming approach

Detlef Berntzen

Animiertes Grafiken-Zusammenspiel von PC und TC in der Mathematik

Josef Böhm

From pole to pole – A numerical journey to an analytical destination

John Cosgrave

Fermat’s Little Theorem: A thing of beauty is a joy for ever

Guido Herweyers Dirk Janssens

Elimination of parameters and substitution with computer algebra

Youngcook Jun

Theorema-based TI-92 simulator for exploratory learning

Karl-Heinz Keunecke Heiko Knechtel

Krümmung als Grenzwert – Curvature as limit Mathematics with graphic and symbolic calculators — Teacher training in Lower Saxony

Josef Lechner

Standardizing the normal probability distribution – An anachronism?!

Carl Leinbach

Using a CAS to teach algebra – Going beyond the manipulations

Alex J. Lobregt

Introducing Fourier Series with Derive

Wolfgang Pröpper

The TI-89/92 as a tool for analytic geometry

Karsten Schmidt

The use of CAS in the Thuringian school system: Present and future

Rolf Wasén

Computers and Computer Algebra Systems in engineering education

Wilhelm Weiskirch

Ortskurven – Loci

Otto Wurnig

Advantages and dangers in the teaching of stochastics by using CAS

Manfred Borovcnik and Hermann Kautschitsch (Eds.) (2002). Technology in Mathematics Teaching. Schriftenreihe Didaktik der Mathematik, Vol. 26 and 27. Vienna: öbv & hpt. For extended versions of the contributions, see the internet platform of Technology in Mathematics Teaching: http://wwwg.uni-klu.ac.at/stochastik.schule/ICTMT_5/ICTMT_5_CD/index.htm

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Special group 2: DGS — Dynamic Geometry Software Organiser: Adrian Oldknow Alison Clark-Jeavons

Why dynamic geometry software is such an effective tool in mathematics education

Björn Felsager

Through the looking glass: Euclid’s twin — The Minkowski Geometry

Chantal Gabriel-Randour Jean Drabbe Luiz Carlos Guimarães Rafael Barbastefano Elizabeth Belfort

Cabri and anamorphoses

Victor Lysytsya

Computer experiments in the lecture of analytical geometry

Valentyna Pikalova

Learning explorations and its DG support in the geometry course for secondary schools

Harry Silfverberg

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

Herrmann Vogel

Use of Cinderella in higher elementary geometry

Tabulæ and Mangaba: Dynamical geometry with a distance twist

Special group 3: Hand-held technology Organisers: Jan Kaspar and Alison Clark-Jeavons Piotr Bialas

Anova with the TI-83 graphing calculator

Piotr Bialas

Linking graphing calculators to the Internet

Jan Kaspar

Programming as a tool for the precision

Regis Ockerman

Probability simulations with TI 83(p)

Jarmila Robová

Graphic solutions of equations and their systems

Special group 4: Spreadsheets Organiser: Erich Neuwirth Deane Arganbright

Creative spreadsheet graphics in mathematics teaching and modeling

Piotr Bialas

Spreadsheet uses in elementary statistics course

Douglas Butler

Why are spreadsheets so unfriendly?

Kent M. Neuerburg

Elementary statistics with spreadsheets

Erich Neuwirth

The spreadsheet paradigm as a new mathematical notation

Robert S. Smith

Spreadsheets across the curriculum

Plenary Lectures and Strands – Special groups and Working groups Special group 5: Traditional programming — In the age of CAS Organiser: Karl Josef Fuchs Alfred Dominik

Taylor Series and finding zeros with Mathematica and Derive

Karl Josef Fuchs

Programming in the age of CAS

Karl Josef Fuchs Eva Vasarhélyi

Problem—Analysis—Encoding—Testing About program and data structures

Judith H. Hector

Programming principles for mathematics and engineering students

Wolfgang Lindner

The digraph-CAS-environment and corresponding elementary programming concepts

Csaba Sárvári M. Klincsik, I. Hámori

Combining CAS with authoring systems to create flexible learning environments

Working group 1: Computer animation, visualization and experimental mathematics Organiser: Gert Kadunz Douglas Butler

Adding a sparkle to classroom teaching — Introducing Autograph

Kate Mackrell

The role of dynamic geometry packages in visualization and animation

Susanne Saminger

MeetMATH — Visualizations and animations in a didactic framework

Ralf Schaper

Mathematica graphics in the Internet: Additional lighting and clipping in LiveGraphics3D

Grosio Stanilov Lidia Stanilova

Mittels Computer zu mathematischen Entdeckungen

Yulian Tsankov

Cubic section by moving plane

Working group 2: System dynamics and systems thinking Organiser: Günther Ossimitz Ernst Gebetsroither

Modelling carbon dioxide pollution — The Austrian carbon balance model

Stefan Gueldenberg Werner H. Hoffmann

Leadership, management and management control — A system dynamics approach

Guenther Ossimitz

Systems thinking and system dynamics: A new perspective for math classes?

Franz Schlöglhofer

Teaching system dynamics modelling in secondary schools

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Working group 3: Continued professional development Organiser: Edward Laughbaum Gregory D. Foley

Mathematics teacher development that works

Rosalyn Hyde

Creating a professional development network

Mark L. Klespis

An on-going program of professional development in hand-held technology for instructors of prospective teachers Technology as a vehicle for updating middle grades content and pedagogy

Judy O'Neal

Working group 4: Probability simulators and data analysis programs Organiser: Manfred Borovcnik Joachim Engel Marcus Otto

Simulation and modelling with Lisp-Stat

Giora Mann Nurit Zehavi

Virtual experiments and probability

Erich Neuwirth

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

Marcus Otto Joachim Engel

Design and use of a computer language for teaching mathematics — Some examples from statistics

Peter Sedlmeier

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

Piet van Blokland

A sample of ideas in teaching statistics

Working group 5: Computer technology in mathematics teaching: Dangers and limitations Organiser: Hartmut Köhler Working group 6: Curricular questions Organiser: Rolf Neveling Nils Fruensgaard

Addresses of authors Delegates of ICTMT 5 Contents of volume 1

Danish experiences with technology in mathematics teaching in upper secondary schools

Plenary Lectures and Strands

Technology in Mathematics Teaching Manfred Borovcnik and Hermann Kautschitsch Proceedings of the Fifth International Conference on (ICTMT 5) in Klagenfurt 2001

Preface In August 2001, the University of Klagenfurt hosted the Fifth International Conference on Technology in Mathematics Teaching – the ICTMT 5. Situated at the junction of three grand European cultures, the German, the Slavic, and the Roman culture, Carinthia has always been a transit region for Europe, important paths through the Alps are crossing the country. This has welcome symbolic implications for a conference like the ICTMT 5 to serve also as junction; as junction of – at least – three grand cultures of mathematics and mathematics teaching, i.e. Anschauliche Mathematik, experimental mathematics, and computer aided mathematics. The University of Klagenfurt is the youngest institution of tertiary education in Austria, established in 1972, an offspring of a personal idea and vision of the former vice mayor of Klagenfurt, Hofrat Hans Romauch which was later supported by the central government, by the federal state of Carinthia, and by the city of Klagenfurt. „Bildungswissenschaften“ in the Humboldtean sense was the principal idea behind the foundation of the Klagenfurt University – a radical new approach to science focussing on „Didaktik“ and teaching – a teaching research institute of European rank was the desired goal. Whilst developments took a different turn, the concept and educational philosophy of „Bildungswissenschaften“ are still much alive at our university. Consistent with this idea, the university hosted an institute for teaching technology and several professors for didactics at the mathematics department right from the beginning. In 1981, we organised a workshop on visualisation in mathematics teaching in co-operation with the visualisation groups at Kassel and Koblenz. And 15 more workshops were to follow over the years. Our equipment in those times looks outdated from today’s perspective, we did not even have a suitable computer for teaching. We worked with a simple video camera and a trick table. Our ideas in those times, however, are still relevant, as a whole series of films prove. The Fifth ICTMT, thus, coincides with our 20th anniversary of this first workshop. Anschauung and Experimental mathematics, our mottos from the beginning, once meant a cut compared to the then prevailing New Math, which was heavily theory, loaded. In general, such cuts lead to crises, which in the Greek origin means „danger and potential at the same time“. A parallel cut is forced by the use of technology in teaching that leads to great changes. History tells us that it is of no use to defend stubbornly the old; see e.g. the Manfred Borovcnik and Hermann Kautschitsch (Eds.) (2002). Technology in Mathematics Teaching. Schriftenreihe Didaktik der Mathematik, Vol. 26 and 27. Vienna: öbv & hpt. For extended versions of the contributions, see the internet platform of Technology in Mathematics Teaching: http://wwwg.uni-klu.ac.at/stochastik.schule/ICTMT_5/ICTMT_5_CD/index.htm

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controversy abacus – Adam Riese in the 16th century. Thus, we have to take up the challenge of the new; the challenge of the New Technologies. We should focus on maximising the potential, minimising the dangers – these proceedings should contribute to that process. The single sessions of the conference were attributed to strands, special groups, and working groups. It was the plan of the organisers and an international board preparing the conference to structure the presentations into strands each devoted to an important topic. Each strand should have a renowned plenary speaker as a leading figure. Moreover, to offer a plenty of discussion and group work, there are a number of special groups and working groups organised. Working groups were intended to focus around a common theme, while special groups should be working groups signified by a common tool. Presentations in special and working groups were much shorter and allowed for intense discussions and group work. The strands (Vol. 1) were: Integration of IC technologies into learning processes – chaired by J.-B. Lagrange, with the focus on the change of cognitive notions by new contents and new tools. Technologically presented learning material - chaired by Bernard Winkelmann; devoted to questions like: Criteria for the use of technologically presented material for the teaching of mathematics and how to implement these. Technology in teacher education – chaired by J. Carvalho e Silva; with the accent on teachers and how they are going to change their practice and how they may be influenced by teacher in-service. Changes in Geometry and Algebra via DGS and CAS – chaired by H.-G. Weigand; covering questions like the change of mathematics, change of curricula, change of presentation, change of concepts to acquire, and change of assessment. Co-operation between DGS and CAS - chaired by M. Garbayo Moreno; devoted to an aspect as far highly neglected; how to link these tools boosting thus the benefit of technologies. Mathematical modelling with technology – chaired by J. Sharp; devoted to new opportunities of working with concrete situations, simplifying the calculations, and giving insight with simulations. The global perspective of information technology – chaired by P. Bender; informing about the potential but also about dangers and limitations of the New Technologies. Volume 1 is devoted to the plenary sessions and presentations of all these strands. The results of the efforts in the special and working groups are published in Volume 2. Moreover, in accordance with the interactive character of many presentations, there is a CD-ROM with all the contributions in a much longer version with many valuable links to sources all over the Internet.

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The five special groups (part of Vol. 2) comprised: Derive, TI-89/92 and other CAS – organised by Josef Böhm, Bernhard Kutzler, Marlene Torres-Skoumal; contributions were based on teaching experience of presenters focusing on Computer Algebra Systems exploiting their advantages. DGS – Dynamic Geometry Software – organised by Adrian Oldknow; with the accent on illustrating the potential of well-known software of this type but also introducing some new, nationally created packages. Hand-held technology – organised by Jan Kaspar and Alison Clark-Jeavons; the presentations centred on the graphing calculator TI-83 illustrating its capacity and reflecting the overlap and borderline to PCs. Spreadsheets – organised by Erich Neuwirth; showing the wide range of possibilities of this type of software focusing also on questions of efficiency of input and output in availability and easiness of usage in comparison to CAS. Traditional programming – In the age of CAS – organised by Karl Josef Fuchs; pleading for traditional programming as a basis to create a special type of thinking, which is highly valuable even if we are in the age of CAS. The six working groups (part of Vol. 2) comprised: Computer animation, visualization and experimental mathematics – organised by Gert Kadunz; covering issues from mathematics education including questions like how to establish learning processes by discovery activities on the computer or in specially designed learning environments. System dynamics and systems thinking – organised by Günther Ossimitz; focusing on the special kind of thinking in systems and the open modelling approach from system dynamics ranging from a crash course in that field up to presenting teaching experience.

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Continued professional development – organised by Edward Laughbaum; reporting about various programs to integrate technology use into teacher in-service training putting the accent on establishing teacher networks to exchange their ideas and problems. Probability simulators and data analysis programs – organised by Manfred Borovcnik; dealing with difficult curricular topics by special educational software, forming intuitive ideas by training programs, supporting understanding by the simulation technique, or backing the whole course by a suitable programming language. Computer technology in mathematics teaching: Dangers and limitations – organised by Hartmut Köhler; facing problems that may arise from an all-too naïve approach, e.g. with the formation of adequate concepts and the danger of technologically caused new misconceptions; or the wide-spread overrating of executing actions as opposed to understanding actions. Curricular questions – organised by Rolf Neveling; discussing new aspects of learning with technologies and the necessary curricular change including a revision of the teacher’s role. Volume 2 contains the keynote papers and results of discussions of the special and working groups. The reader might also be interested in the plenary sessions and contributed papers of the conference. These are contained in the first volume of the proceedings. We summarise the strands and their goals briefly in what follows. Moreover, in accordance with the interactive character of many presentations, there is a CD-ROM with all the contributions in a much longer version with many valuable links to sources all over the Internet. The contributions and the discussions at the conference clearly showed that these New Technologies offer new possibilities with reference to elementary mathematics and school mathematics on the one hand and to applications of mathematics on the other hand. With respect to school mathematics, we get new approaches based on technology for understanding elementary concepts including new ways to establish understanding numbers and understanding geometric space for our students. Furthermore, an algorithmic understanding of mathematical concepts opens up completely new ways of thinking. With the focus on applications of mathematics, we come up with more flexible concepts and modelling techniques by the help of special software – be it realized on the PC or on hand-held technology. The experimental face of the concepts extends mathematics, its understanding, and its applicability. This helps to solve problems closer to reality than without technology. Moreover, this leads to new answers to problems in the form of algorithmic descriptions instead of mere numbers or formulae. The potential of the New Technologies will not be realized per se but only by specific efforts. This necessitates more and more qualified mathematicians and, one level below, or better, before, more qualified teachers of mathematics. There is an increasing demand for persons who are really capable to exploit the New Technologies, which gives a challenge and a chance to our youth.

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The contributions of the delegates of ICTMT 5 are very close to teaching in class, to applications of mathematics, or to establish new networks of continued professional development of teachers. There is a wide range of new ideas and activities on new contents, new teaching approaches, and a new role for teachers in class, and new efforts to cope with problems on the side of teachers in the two volumes of the proceedings. This conference could only be carried through by the combined effort of many. We had an excellent technical support from our computer centre, especially from Peter König there who also guided us through the cliffs of the proceedings. Our Internet specialist was Heinz Pozewaunig. We express our gratitude and thanks to the delegates who contributed so many challenging ideas so that we will have a long time to digest them. A great thank you also to the persons chairing the various sessions; they, too, gave a lot of effort in preparing the conference and the proceedings. Last but not least, we thank Prof. A. Oldknow, honorary president of this conference. The local organisers tried to offer a platform for the exchange of ideas. We hope you can use it to improve the benefit of the New Technologies in teaching for those whom we teach. We wish you challenging days in reading this and the following book with the results of ICTMT 5 in the interest of our science. Finally, two photos as a memory to our landscape and nature we missed now for more than one year in preparing the conference and the proceedings.

Klagenfurt, January/May 2002 Manfred Borovcnik Hermann Kautschitsch

For extended versions of the contributions, see the internet platform of Technology in Mathematics Teaching: http://wwwg.uni-klu.ac.at/stochastik.schule/ICTMT_5/ICTMT_5_CD/index.htm

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