Table Of Content

Real-World Problems for Secondary School Mathematics Students Real-World Problems for Secondary School Mathematics Students Case Studies Edited by Juergen Maasz University of Linz, Austria John O’Donoghue University of Limerick, Ireland SENSE PUBLISHERS ROTTERDAM/BOSTON/TAIPEI A C.I.P. record for this book is available from the Library of Congress. ISBN: 978-94-6091-541-3 (paperback) ISBN: 978-94-6091-542-0 (hardback) ISBN: 978-94-6091-543-7 (e-book) Published by: Sense Publishers, P.O. Box 21858, 3001 AW Rotterdam, The Netherlands www.sensepublishers.com Printed on acid-free paper Image – The Living Bridge, University of Limerick. © Patrick Johnson, 2008. “The Living Bridge – An Droichead Beo” The Living Bridge is the longest pedestrian bridge in Ireland and links both sides of the University of Limerick’s campus across the river Shannon. The bridge is constructed of 6 equal spans and follows a 350 metre long curved alignment on a 300 metre radius. All Rights Reserved © 2011 Sense Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. TABLE OF CONTENTS Preface....................................................................................................................vii 1. Modelling in Probability and Statistics: Key Ideas and Innovative Examples.........................................................................................1 Manfred Borovcnik and Ramesh Kapadia 2. Problems for the Secondary Mathematics Classrooms on the Topic of Future Energy Issues...................................................................................45 Astrid Brinkmann and Klaus Brinkmann 3. Coding Theory.................................................................................................67 Tim Brophy 4. Travelling to Mars: A Very Long Journey: Mathematical Modelling in Space Travelling..........................................................................................87 Jean Charpin 5. Modelling the Storage Capacity of 2D Pixel Mosaics.....................................99 Simone Göttlich and Thorsten Sickenberger 6. Mathematics for Problems in the Everyday World.......................................113 Günter Graumann 7. Political Polls and Surveys: The Statistics Behind the Headlines.................123 Ailish Hannigan 8. Correlations between Reality and Modelling: “Dirk Nowitzki Playing for Dallas in the NBA (U.S.A.)”...................................................................137 Herbert Henning and Benjamin John 9. Exploring the Final Frontier: Using Space Related Problems to Assist in the Teaching of Mathematics....................................................................155 Patrick Johnson 10. What are the Odds?.......................................................................................173 Patrick Johnson and John O’Donoghue 11. Models for Logistic Growth Processes (e.g. Fish Population in a Pond, Number of Mobile Phones within a Given Population)................................187 Astrid Kubicek 12. Teaching Aspects of School Geometry Using the Popular Games Rugby and Snooker...................................................................................................209 Jim Leahy v TABLE OF CONTENTS 13. Increasing Turnover? Streamlining Working Conditions? A Possible Way to Optimize Production Processes as a Topic in Mathematics Lessons........221 Juergen Maasz 14. Mathematics and Eggs: Does this Topic Make Sense in Education?............239 Juergen Maasz and Hans-Stefan Siller 15. Digital Images: Filters and Edge Detection...................................................257 Thomas Schiller 16. Modelling and Technology: Modelling in Mathematics Education Meets New Challenges..................................................................................273 Hans-Stefan Siller List of Contributors..............................................................................................281 vi PREFACE We should start by pointing out that this is not a mathematics text book – this is an ideas book. This is a book full of ideas for teaching real world problems to older students (15 years and older, Upper Secondary level). These contributions by no means exhaust all the possibilities for working with real world problems in mathematics classrooms but taken as a whole they do provide a rich resource for mathematics teachers that is readily available in a single volume. While many papers offer specific well worked out lesson type ideas, others concentrate on the teacher knowledge needed to introduce real world applications of mathematics into the classroom. We are confident that mathematics teachers who read the book will find a myriad of ways to introduce the material into their classrooms whether in ways suggested by the contributing authors or in their own ways, perhaps through mini- projects or extended projects or practical sessions or enquiry based learning. We are happy if they do! Why did we collect and edit them for you, the mathematics teachers? In fact we did not collect them for you but rather for your students! They will enjoy working with them at school. Having fun learning mathematics is a novel idea for many students. Since many students do not enjoy mathematics at school, students often ask: “Why should we learn mathematics?” Solving real world problems is one (and not the only one!) good answer to this question. If your students enjoy learning mathematics by solving real world problems you will enjoy your job as a mathematics teacher more. So in a real sense the collection of examples in this book is for you too. Using real world problems in mathematics classrooms places extra demands on teachers and students that need to be addressed. We need to consider at least two dimensions related to classroom teaching when we teach real world problems. One is the complexity (intensity or grade) of reality teachers think is appropriate to import into the classroom and the other is about the methods used to learn and work with real problems. Papers in this collection offer a practical perspective on each dimension, and more. Solving real world problems often leads to a typical decision situation where you (we hope together with your students) will ask: Should we stop working on our problem now? Do we have enough information to solve the real world problem? These are not typical questions asked in mathematics lessons. What students should learn when they solve real world problems is that an exact calculation is not enough for a good solution. They should learn the whole process of modelling from the first step abstracting important information from the complex real world situation, to the next steps of the mathematical modelling process. For example, they should learn to write down equations to describe the situation; do calculations; interpret the results of calculation; improve the quality of the model; calculate again (several times if needed); and discuss the results with others. Last but not least, they should reflect on the solution process in order to learn for the future. vii PREFACE How real should real world problems be? More realistic problems are generally more complex and more complex problems demand more time to work them out. On the other hand a very simplified reality will not motivate students intrinsically to work for a solution (which is much better for a sustaining learning). Experience suggests starting with simple problems and simple open questions and moving to more complex problems. We think it is an impossible task for students without any experience of solving complex real problems to start by solving difficult real problems. It is better if you start with a simpler question and add complexity step by step. The second dimension of classroom teaching is concerned with methods of teaching real world problems. We are convinced that learning and teaching is more successful if you use open methods like group work, project planning, enquiry learning, practical work, and reflection. A lot of real world problems have more than one correct solution, and may in fact have several that are good from different points of view. The different solutions need to be discussed and con- sidered carefully and this is good for achieving general education aims like “Students should become critical citizens”. Students are better prepared for life if they learn how to decide which solution is better in relation to the question and the people who are concerned. Finally we would like to counter a typical “No, thank you” argument against teaching real world problems. Yes, you will need more time for this kind of teaching than you need for a typical lesson training students in mathematical skills and operations. Yes, you will need to prepare more intensively for these lessons and be prepared for lot of activity in your classroom. You will need to change your role from a typical teacher in the centre of the classroom knowing and telling everything to that of manager of the learning process who knows how to solve the problem. But you need help to get started! We hope you will use this book as your starter pack We don’t expect you to teach like this every day but only on occasions during the year. It should be one of your teaching approaches but not the only one. Try it and you will be happy because the results will be great for the students and for you! ACKNOWLEDGEMENTS We would like to thank all those who made this book possible especially the many authors who so generously contributed papers. This collaboration, sharing of in- sights, expertise and resources benefits all who engage in an enterprise such as this and offers potential benefits to many others who may have access to this volume. We are especially pleased to bring a wealth of material and expertise to an English speaking audience which might otherwise have remained unseen and untapped. The editors would like also to record their thanks to their respective organizations who have supported this endeavour viz. the Institut fur Didaktik der Mathematik, viii PREFACE Johannes Kepler University, Linz, and the National Centre for Excellence in Mathematics and Science Teaching and Learning (NCE-MSTL), at the University of Limerick. Juergen Maasz University of Linz, Austria John O’Donoghue NCE-MSTL University of Limerick and Linz Autumn 2010 ix