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226159-CP-1-2005-1-AT-COMENIUS-C21 527269-LLP-1-2012-1-AT-COMENIUS-CAM Pavel Boytchev Hannes Hohenwarter Evgenia Sendova Neli Dimitrova Emil Kostadinov Andreas Ulovec Vladimir Georgiev Arne Mogensen Henning Westphael Oleg Mushkarov MEETING IN MATHEMATICS 2nd edition • Universität Wien • Dipartimento di Matematica, Universita' di Pisà • VIA University College – Læreruddannelsen i Århus • Институт по математика и информатика , Българска академия на науките Authors Pavel Boytchev, Neli Dimitrova, Vladimir Georgiev, Hannes Hohenwarter, Emil Kostadinov, Arne Mogensen, Oleg Mushkarov, Evgenia Sendova, Andreas Ulovec, Henning Westphael Editors Evgenia Sendova Andreas Ulovec Project Evaluator Jarmila Novotná, Charles University, Prague, Czech Republic Reviewers Jarmila Novotná, Charles University, Prague, Czech Republic Nicholas Mousoulides, University of Nicosia, Cyprus Cover design by Pavel Boytchev Cartoons by Yovko Kolarov All Rights Reserved © 2013 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the authors. For educational purposes only (i.e. for use in schools, teaching, teacher training etc.), you may use this work or parts of it under the “Attribution Non-Commercial Share Alike” license according to Creative Commons, as detailed in http://creativecommons.org/licenses/by-nc-sa/3.0/legalcode. This project has been funded with support from the European Commission. This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein. Published by Demetra Publishing House, Sofia, Bulgaria ISBN 978-954-9526-49-3 iii Contents PREFACE................................................................................................................... vii CHAPTER 1 Meeting the gifted – while not forgetting the others Author in charge: Arne Mogensen Contributor: Henning Westphae .............................................................................. 1 CHAPTER 2 Mathematical competences Authors in charge: Henning Westphael, Arne Mogensen...................................... 21 CHAPTER 3 Mathematics competitions Author in charge: Vladimir Georgiev Contributors: Arne Mogensen, Oleg Mushkarov, Henning Westphael ......................................................................... 41 Appendix to Chapter 3 The Unsung Hero – some selected problems Vladimir Georgiev................................................................................................. 55 CHAPTER 4 Mathematics clubs Author in charge: Vladimir Georgiev Contributor: Arne Mogensen ................................................................................. 65 CHAPTER 5 Mathematics explorations and interdisciplinary work enhanced by IT Author in charge: Andreas Ulovec Contributor: Evgenia Sendova............................................................................... 79 Appendix to Chapter 5 A dynamic approach to a classical geometric problem Andreas Ulovec ..................................................................................................... 95 Unity via diversity Pavel Boytchev ...................................................................................................... 97 iv Contents “The more it changes the samer it gets” Andreas Ulovec ................................................................................................... 110 Fractal geometry Hannes Hohenwarter, Andreas Ulovec, Pavel Boytchev .................................... 115 CHAPTER 6 Math research at school age Authors in charge : Oleg Mushkarov, Evgenia Sendova Contributor: Neli Dimitrova ................................................................................ 137 Appendix to Chapter 6 How to cut two circles Oleg Mushkarov .................................................................................................. 173 Inequalities for sums of powers Vladimir Georgiev , Oleg Mushkarov .................................................................. 179 On a metric property of the equilateral triangles Emil Kostadinov .................................................................................................. 189 Malfatti’s problems Emil Kostadinov .................................................................................................. 199 v Authors Dr. Pavel Boytchev Faculty of Mathematics and Informatics, Sofia University James Baurchier 5, Sofia 1126, Bulgaria [email protected] Dr. Neli Dimitrova Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev, bl. 8, Sofia 1113, Bulgaria [email protected] DSc. Vladimir Georgiev Department of Mathematics, University of Pisa Largo Bruno Pontecorvo, 5, Pisa 56127, Italy [email protected] MSc. Hannes Hohenwarter High School Villach St. Martin St. Martiner Strasse 7, Villach 9500, Austria [email protected] Emil Kostadinov PhD student of Economics at the University of Essex, United Kingdom [email protected] Dr. Arne Mogensen Dept. of Education, VIA University College Aarhus, Denmark [email protected] DSc. Oleg Mushkarov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev, bl. 8, Sofia 1113, Bulgaria [email protected] vi Dr. Evgenia Sendova Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad. G. Bonchev, bl. 8, Sofia 1113, Bulgaria [email protected] Dr. Andreas Ulovec Faculty of Mathematics, University of Vienna Nordbergstrasse 15, Vienna 1090, Austria [email protected] MSc. Henning Westphael Department of Teacher Education at Aarhus VIA University College Trøjborgvej 82, 8200 Aarhus N, Denmark [email protected] Preface Dear readers, to encourage many more young people to appreciate the real nature and spirit of mathematics and possibly to be enrolled in mathematics study it is important to involve them in doing mathematics (not just learning about mathematics) as early as possible. This goal could be achieved if mathematics teachers are prepared to identify and work with mathematically gifted students (without losing the rest), to integrate the informatics and IT tools with mathematics explorations, to introduce some topics and activities (not present in the traditional mathematics curriculum) that would be more appealing and relevant to the students and teachers alike. With the idea of addressing some of these problems the project Meetings in Mathematics : Mathematics Enrolment and Effectiveness of Teaching in Mathematics was carried out in the years 2005–2008 in the frames of Comenius 2.1 program of the European Commission. The goal of the participants (University of Pisa, Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences, University of Vienna, and VIA University College – Department of Teacher Education, Aarhus) was to share their experience in identifying mathematically gifted students, in offering mathematical challenges for those who love competing and for the ones who prefer to attack an open problem not being pressed by the time limitations. Furthermore, the spectrum of “mathematical giftedness” was not determined so as to address just mathematical elite but rather to help teachers in revealing the potential of young people to be creative in a mathematical context. One of the products of this project was the book Meeting in Mathematics, oriented to the needs of mathematics teachers in the Project countries (Austria, Bulgaria, Denmark and Italy). After publishing the first edition of the book we distributed it among math educators and teachers and got various valuable remarks and suggestions by the readers. The most encouraging feedback was “More, please!”, so here it is – what you are holding in your hands is the second edition of Meeting in mathematics . The preparation of the new edition was brought to life thanks to a new Comenius project, MeetMe – 527629-LLP-1-2012-1-AT-COMENIUS-CAM. The authors’ team (enriched by a new member from VIA University College – Department of Teacher Education) took the chance to turn back to the original materials developed several years ago and to make some improvements based on the accumulated experience in the context of teacher training courses and the users’ suggestions for refinements and enrichments. Here is the updated content in a nut shell. viii Preface In Chapter 1 we present some characteristics of gifted students , as they are experienced in research and practice. The chapter also deals with some of the challenges that teachers might meet. In Chapter 2 we present and use the concept of mathematical competence for description of mathematics teaching and learning in accordance with the most recent theoretical developments. In Chapter 3 we present various types of math competitions, both individual and team ones. The specifics of the individual math competitions of IMO type are compared with those of the team competitions and ways to combine the advantages of each kind are discussed. Special emphasis is put on the team competitions since in general they: • involve larger groups of pupils; • have attractive and dynamic form; • integrate knowledge from other fields, e.g. arts, modeling, physics; • could be considered as natural elements of some traditional school events. Ideas related to the organization of math team competitions are presented based on a concrete example – the international math competition The
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