Supporting Independent Thinking Through Mathematical Education

1 Supporting Independent Thinking Through Mathematical Education Editors Bożena Maj The University of Rzeszów Rzeszów, Poland Marta Pytlak The University of Rzeszów Rzeszów, Poland Ewa Swoboda The University of Rzeszów Rzeszów, Poland Wydawnictwo Uniwersytetu Rzeszowskiego 2008 2 Reviewers Andras Ambrus Peter Appelbaum Jenni Back Stanis³aw Domoradzki Jan Gunèaga Carlo Marchini Tatyana Oleinik Jana Slezakova Lambrecht Spijkerboer Sophie Stathopoulou Ewa Swoboda Cover Design Dorota Domaga³a Layout Design Iwona Ga³uszka, Jaros³aw Naus Manufacturing Nowa Era Sp. z o.o. ISBN 978-83-7338-420-0 © Wydawnictwo Uniwersytetu Rzeszowskiego Rzeszów 2008 No part of the material protected by this copyright notice may be reproduced or utilized in any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in Poland 3 TABLE OF CONTENTS Introduction ......................................................................................................................... 7 Part 1. Open questions in mathematical education Sense and representation in elementary mathematics Peter Appelbaum ................................................................................................................. 10 Factors hampering independent mathematical thinking Zbigniew Semadeni ............................................................................................................. 18 Various manipulation functions in solving geometrical tasks Ewa Swoboda, Joanna Syno, Danuta Pluta ..................................................................... 26 Is it possible to teach our pupils to think independently? Agata Hoffmann ................................................................................................................... 34 Problems of formation learning environment Tatyana Oleinik, Nikolay Ivashenko, Andrey Prokopenko ............................................... 36 Part 2. Suggestions regarding the support of independent thinking in mathematics I. Changes and trends in arithmetics and early algebra Scheme oriented educational strategy in mathematic Milan Hejný ......................................................................................................................... 40 Perceptions of numbers of 5 to 6 year old children Gabriela Pavlovièová, Valéria Vasková ............................................................................. 49 Primary school pupils' misconceptions in number Graham H. Littler, Darina Jirotková .................................................................................. 55 Focusing on children's early ideas of fractions Ema Mamede ....................................................................................................................... 61 Connections as a fundamental element to constructing mathematical knowledge (exemplify of one pre-algebra task) Marta Pytlak ....................................................................................................................... 68 Preparing of pupils for notion of limits Ján Gunèaga ....................................................................................................................... 76 II. Changes and trends in geometry How to reveal geometrical independent thinking in the lower primary years Carlo Marchini, Ewa Swoboda, Paola Vighi ..................................................................... 82 4 Table of contents Activities in the preparation for and establishment of the concept of length and perimeter Margit Tarcsi ....................................................................................................................... 89 Developing problem solving strategies via estimating the area of irregular shapes Ioannis Papadopoulos ........................................................................................................ 95 Geometrical transformations and the concept of cyclic ordering Eszter Herendiné-Kónya ..................................................................................................... 102 Building the concept of line symmetry Edyta Jagoda ...................................................................................................................... 109 Part 3. Teachers' professional development Investigating mathematics: A challenge for students, teachers, and mathematics education researchers João Pedro Da Ponte .......................................................................................................... 122 Developing creative mathematical activities during lessons of mathematics Bo¿ena Maj .......................................................................................................................... 138 Teaching practices revealed through arithmetic misconceptions Carlo Marchini, Anne Cockburn ........................................................................................ 145 Students discussing their mathematical ideas: Group-tests and mind-maps Monique Pijls, Dain De Kramer .......................................................................................... 152 Developing teachers' subject didactic competence: Case of problem posing. Marie Tichá ......................................................................................................................... 154 Thinking to the future: Prospective teachers encouraging children's mathematical thinking. Example of a workshop Anne Cockburn, Eleanor Cockerton, Ralph Manning, Paul Parslow-Williams ............... 156 Part 4. Language and communication in mathematics Analysing interaction processes with jigsaw during mathematics lessons in elementary school Birgit Brandt, Konstantinos Tatsis ..................................................................................... 160 Discussing on the fairness of probabilistic games: The creation of a discursive community with kindergarten children Konstantinos Tatsis, Sonia Kafoussi, Chrysanthi Skoumpourdi ....................................... 167 Change: What change? Considering whether a national initiative has had an impact on the quality of classroom talk in primary mathematics Jenni Back, Tony Beauchamp ............................................................................................. 174 The pupils' interpretation of mathematical writings Filip Roubíèek .................................................................................................................... 181 Table of contents 5 Part 5. Problem posing, problem solving Approaches to solving processes of one fuzzy problem by primary school children Ivana Kovárová ................................................................................................................... 188 Problem stories in the education for numeracy and literacy Maria de Fátima Sardinha, Pedro Palhares, Fernando Azevedo...................................... 196 The differences in mathematical thinking of children in the in-school and out-of-school situations Daria Muzyczka .................................................................................................................. 202 Designing mathematics walks Lambrecht Spijkerboer........................................................................................................ 208 Mathematics through investigations and problem solving Silva Kmetiè ........................................................................................................................ 210 Part 6. Information technology as an educational environment Development of abstract thinking in mathematics in computer environment Ivona Grzegorczyk .............................................................................................................. 214 Teaching computer-assisted mathematics to primary students Renata Raszka (Kozie³) ...................................................................................................... 219 Procedural and paving way of building-up of geometrical object in concepts and strategies of primary school pupils Jiøí Vaníèek ......................................................................................................................... 226 Promoting thinking through a modern approach for problem solving Ronit Hoffmann, Ronith Klein ............................................................................................. 233 Part 7. Brief texts on independent thinking in mathematical education From a caterpillar to a butterfly: a learning project for kindergarten Vighi Paola.......................................................................................................................... 236 Children's models of number 3 at the age of 67 Valéria Vasková, Gabriela Pavlovièová ............................................................................. 237 On the numerical competence of six-year-old children Aleksandra Urbañska ......................................................................................................... 238 On mathematical education of 6 year old children in Ukraine Iryna Zarichna .................................................................................................................... 239 Percentages are not another name for fractions Malka Sheffet, Ronit Bassan-Cincinatus ............................................................................ 240 The method of incomplete pictures and pupils´ ideas Jana Cachova ....................................................................................................................

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