The Mathematics
Education into the 21^{st} Century Project
The Decidable and the Undecidable in Mathematics Education
Editor
Alan Rogerson
September 19-25 2003
Major Sponsors
CASIO & AUTOGRAPH
All Rights Reserved.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright holder.
The Mathematics Education into the 21^{st} Century Project wishes to thank
Brno University of Technology
Statutory City of Brno
Masaryk University, Brno
The Union of Czech Mathematicians and Physicists
The Hong Kong Institute of Education
The Third World Forum
CASIO
Autograph
Greygum Software
ALMA (Teacher Training Centre, Poland)
SNM (Polish Association of Teachers of Mathematics)
VSG Enrichment
Wholemovement
Starobrno a.s.
Znovín Znojmo a.s
ISBN Number 83-919465-1-7
International Program Committee
Chairman Dr. Alan Rogerson, International Coordinator of the Mathematics in Society Project (Poland).
Prof Khaled Abuloum, University of Jordan (Jordan).
Professor Roberto Baldino, UNESP (Brazil).
Dr. Andy Begg , Senior Lecturer in Mathematics Education, University of Waikato (OU/New Zealand).
Dr. Donna F. Berlin, Mathematics, Science, and Technology Education, The Ohio State University (USA)
Prof. Dr. Werner Blum, University of Kassel (Germany).
Professor Ubiratan D'Ambrosio, Emeritus Professor of Mathematics, Campinas/UNICAMP, (Brazil).
Professor Bruno D'Amore, University of Bolgna (Italy).
Professor Dr.
Tilak de Alwis, Southeastern Louisiana
University (USA)
Prof Dr William
Ebeid, Emeritus Professor, Faculty of Education,
Ain Shams University (Egypt).
Prof. Omar Hassan El Sheikh, University of Jordan (Jordan).
Professor Paul Ernest, Professor of the Philosophy of Mathematics Education, University of Exeter (UK).
Prof. Dr. Gabriele Kaiser, Hamburg University (Germany).
Dr. Madeleine J. Long, Program Director, American Association for the Advancement of Science (USA).
Prof Dr Fayez Mina, Dept. of Curriculum & Instruction, Faculty of Education, Ain Shams University (Egypt).
Prof Nicolina Malara, University of Modena (Italy)
Prof Dr M. Ali M. Nassar, Director of Planning Center, Institute of National Planning (Egypt).
Professor Lionel Pereira Mendoza, Associate Professor, National Institute of Education (Singapore).
Professor Angela Pesci, University of Pavia (Italy)
Prof. Medhat Rahim, Lakehead University, Faculty of Education (Canada).
Professor Filippo Spagnolo, University of Palermo, Sicily, (Italy).
Professor Dr. Alicia Villar Icasuriaga, Instituto de Profesores "Artigas" (IPA), Montevideo, (Uruguay)
Dr. Arthur L. White, Mathematics, Science, and Technology Education, The Ohio State University ( USA)
Professor. Wacek Zawadowski, Warsaw University (Poland).
Local Organizing Committee
Chairman :
Prof.Dr. Ivan Mezník, Pro Dean, Faculty of Business and Management, Brno University of Technology
Local Conference Secretary:
Dr. Zuzana Chvátalová, Faculty of Business and Management, Brno University of Technology
Members :
Prof. Dr. Jan Chvalina, Head, Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology
Prof. Ing. Jiří Dvořák, Faculty of Business and Management, Brno University of Technology
Prof.dr.Jiří Hřebíček, Faculty of Informatics, Masaryk University Brno
Assoc.Prof. Ing. Miloš Koch, Dean, Faculty of Business and Management, Brno University of Technology
Assoc.Prof.Dr.Václav Viktora, Head, Department of Mathematics, Faculty of Education, Masaryk University Brno
Assoc.Prof. Dr. Jaromír Vosmanský, Faculty of Science, Masaryk University Brno
This volume contains the papers presented at the International Conference on The Decidable and the Undecidable in Mathematics Education held from September 19-25, 2003 at the Hotel Continental, Brno, Czech Republic. This Conference was organized by The Mathematics Education into the 21^{st} Century Project - a non-commercial international educational project founded in 1986. Our Project is dedicated to the improvement of mathematics education worldwide through the publication and dissemination of innovative materials and ideas. Previous successful Conferences have been held in Cairo, Egypt (1999), Amman, Jordan (2000), Zajaczkowo Poland, Palm Cove, Australia (2001) and Sicily, Italy (2002).
It has been my pleasure to read all of the papers and to edit them for these Proceedings. Not all papers are about research in mathematics education, a number of them report on innovative experiences in the classroom and on new technology. We believe that “mathematics education” is fundamentally a “practicum” and so in order to be “successful” all new materials, new ideas and new research must be tested and implemented in the classroom, the real “chalk face” of our discipline, and of our profession as mathematics educators .
These Proceedings begin with the Plenary Papers and otherwise contain contributions mostly in alphabetical order of the Principal Presenters. I sincerely thank all of the contributors for their time and creative effort. It is clear from the variety and quality of the papers that the conference has attracted many innovative mathematics educators from around the world. These Proceedings will therefore be useful in reviewing past work and looking ahead to the future. They will also serve as a stepping stone for the new and exciting International Initiative, SuperCourse, which will feature strongly throughout the conference programme. For further information about this initiative and the work of our Project, please email arogerson@vsg.edu.au
Finally a word about our conference theme - The Decidable and the Undecidable - a tribute of course to Kurt Godel, who was born and educated in Brno before moving to Vienna and the United States. The importance of his work, in freeing mathematics from the alluring but simplistic idea that all mathematics can be deduced from a few logical axioms, cannot be overestimated. Mathematics, like art, like music and literature, and like life itself, has the power to make us happy by means of our personal creativity and openness, and through our own personal resolution of what is decidable, and what is undecidable, not only in mathematics and not only in mathematics education, but in life itself.
Dr. Alan Rogerson
Chaiman of the Programme Committee
Presented Papers
Mathematical principles of graphic systems, Dalibor Martisek, Miloslav Druckmuller 1
Promoting Teachers’ Changes: Examples from an Educative Process in Early Algebra, Nicolina A. Malara 5
Fuzzy Logic, Fuzzy Thinking and the teaching/learning of mathematics in multicultural situations. Filippo Spagnolo 17
Implementing the intended mathematics curriculum: Teachers’ beliefs about the meaning and relevance of problem solving, Noor Azlan Ahmad Zanzali 34
WHAT ROLES DO MATHEMATICS TEACHERS PLAY WHEN THEIR STUDENTS SOLVE PROBEMS?, Othman N. Alsawaie 38
INTERVENTION RESEARCH: A TOOL FOR BRIDGING THE THEORY-PRACTICE GAP IN MATHEMATICS EDUCATION?, Erik DE CORTE 45
PRINCIPIA PROGRAM: TEACHING MATHEMATICS TO ENGINEERS WITH INTEGRATED CURRICULUM, TEAMWORK ENVIRONMENT AND USE OF TECHNOLOGY., Francisco Delgado, Rubén D. Santiago 56
SOLVING OF CONTEXT TYPE MATHEMATICAL PROBLEMS, Veronica Diaz Q. & Alvaro Poblete L. 61
Investigating the Relationship between Mental Imaging and Mathematical Problem Solving, Patricia Douville and David K. Pugalee 62
Does a proof render the truth of a claim decidable?, Nella Yollesh and Tommy Dreyfus 68
MENDEL: BIOLOGY, MATHEMATICS AND HISTORY OF SCIENCE, Soledad ESTEBAN, María P. GONZÁLEZ and Luis TEJERO 72
Deciding Optimal Experiences in Coordinate Geometry for Pre-Service Elementary School Teachers, Jack A. Carter, Beverly J. Ferrucci 75
MOM, TUT, SENSE MAKING and RICH LEARNING TASKS, Gary Flewelling 79
Development of a framework to assess primary students’ number sense in Malaysia, Munirah Ghazali, Shafia Abdul Rahman, Zurida Ismail Sharifah Norhaidah Idros, & Fatimah Salleh 83
AN INVESTIGATION INTO THE MATHEMATICAL KNOWLEDGE OF PRE-SERVICE PRIMARY TEACHERS., Maria Goulding 87
Assessment of General Aims like Ordering, Structuring and Analogising – a decidable task?, Günter Graumann, 91
Assessment of a Comprehensive, School-wide Intervention: The CAMMP Experience, Michael Green & John A. Piel 94
Linking procedural and conceptual mathematical knowledge in technology-based learning, Lenni Haapasalo 98
The Experience of the Arab Supercourse, Jean Michel Hanna 102
PRACTICAL APPLICATION OF THE ETHICS IN GEOMETRY, Bradford Hansen-Smith 105
Intertwining Research and Teacher Development in Lebanese Elementary Mathematics Classrooms: The MARAL Project, Marjorie Henningsen 110
The Internet as Instructional Tool: A Consideration of Its Potential Role in the Elementary Mathematics Classroom, Terese A. Herrera 114
Using Symbolic Computation Systems in Applications, Jiří Hřebíček, Jan Pešl, Jaroslav Ráček 116
Aspects of Critical Thinking in Classroom Instruction of Secondary School Mathematics Teachers in Jordan, Hanan Innabi 124
TEACHING STOCHASTIC CONCEPTS AT THE BUT FACULTY OF MECHANICAL ENGINEERING, Zdenek Karpisek – Pavel Popela – Josef Bednar 135
Empowerment of Teachers as Learners: Active Learning in the College Mathematics Classroom, Ada Katsap 140
Towards convergence of mathematics and computer science education, Alexander Khait 149
A model for professional development for graphics calculator use, Barry Kissane 153
Heron triangles and Heron´s Formula, Mgr. Daniela Křížová 158
Mathematical Problems and Mathematical Structures, George Malaty 166
A didactical proposal: perpendicular and vertical straight lines, Gianna Manno – Palmina Cutugno 172
Banach Fixed Point Theorem and the Stability of the Market, Ivan Mezník 177
The impact of interactive whiteboards on classroom practice: examples drawn from the teaching of mathematics in secondary schools in England., Derek Glover, David Miller and Doug Averis 181
The Decidable and the Undecidable in Mathematics Education “Going Beyond The Paradox”, Fayez M. Mina 186
A “Big Problem” as a Motivator and Backbone For Mathematics Learning, Dan Fendel, Nitsa Movshovitz-Hadar, Diane Resek, 193
Concept mapping and context in mathematics education, Dr. Willy Mwakapenda 198
Embedding, reorganization and construction of mathematical and didactical contents as an objective in teachers education, Engelbert Niehaus 204
Effects of Some Properties 5. Grade Students on the Performance of Mathematical Problem Solving, Nevin ORHUN 209
Simulation and Statistical Exploration of Data (Let’s Make a Deal – The Monty Hall Problem), using the new ClassPad300-technology, Ludwig Paditz 217
Could metaphorical discourse be useful for analysing and transforming individuals’ relationship with mathematics?, Angela Pesci 224
EVALUATION OF THE PROFESSIONAL COMPETENCES OF THE MATHEMATICS TEACHER, Alvaro Poblete & Verónica Díaz 231
Exercises of division theory leading to brand new results, Zdeněk Polický 232
Improving Statistical Education: Developing a CDRom orientation and training program for casual tutors., Dr Anne Porter and Dr Carolyn McPhail 235
NOVICES’ CHOICE OF EXAMPLES IN THE TEACHING OF ELEMENTARY MATHEMATICS, Tim Rowland, Anne Thwaites and Peter Huckstep 242
Applying the Graphic Calculus for Differentiation in Secondary School., Anna Rybak 246
STATISTICS TEACHING WITH AN EXAMPLE, BIZHAN SHABANKHANI, M.A.QAJAR KOOHESTANI 250
Perception of infinity: does it really help in problem solving?, Mihaela Singer, Cristian Voica 252
The comic of Clamat’: the use of a comic as a linguistic mediator, Claudia Sortino 262
The introduction of the symbolic language in secondary school: experimental analysis of a-didactic situation by Vigotskij semiotic tools, Claudia Sortino - Filippo Spagnolo 267
ON-LINE LEARNING FOR BRIDGING MATHEMATICS STUDENTS, GEERT MARIUS SPYKER 284
Attempts to Improve the Problem Solving Abilities of Practicing Teachers, Agnes Tuska 288
Application of mathematical concepts across disciplines: an example, Stamatis Voulgaris & Anastasia Evangelidou 294
Using PC and TI-92 in teaching Binomial and Normal Distribution in the final Grades of Austrian Grammar Schools, Otto WURNIG 298
Two-Way Tables: Introducing Probability Using Real Data, Gail Burrill 304
Special Casio New Technology Workshops
The Special Teacher Czech Day: