MATHEMATICS TEACHING FOR THE FUTURE MATHEMATICS Editors: Margita Pavleković, Zdenka Kolar-Begović, Ružica Kolar-Šuper MATHEMATICS TEACHING FOR THE FUTURE Editors: Margita Pavleković Zdenka Kolar-Begović Ružica Kolar-Šuper ISBN 978-953-197-583-4 Josip Juraj Strossmayer University of Osijek Faculty of Teacher Education and Department of Mathematics Sveuciliˇ steˇ Josipa Jurja Strossmayera u Osijeku Uciteljskiˇ fakultet i Odjel za matematiku MATHEMATICS TEACHING FOR THE FUTURE POUCAVANJEˇ MATEMATIKE ZA BUDUCNOST´ monography / monografija Editors / Urednice: Margita Pavlekovic´ Zdenka Kolar-Begovic´ Ruzicaˇ Kolar-Superˇ Zagreb, 2013 Josip Juraj Strossmayer University of Osijek / Sveuciliˇ steˇ Josipa Jurja Strossmayera u Osijeku Mirta Bensiˇ c´ Radoslav Galic´ Zdenka Kolar-Begovic´ Ruzicaˇ Kolar-Superˇ Ivan Matic´ Margita Pavlekovic´ Rudolf Scitovski Nenad Suvakˇ Simeˇ Ungar Marijana Zekic-Su´ sacˇ University of Zagreb / Sveuciliˇ steˇ u Zagrebu Blazenkaˇ Divjak Sibe Mardesiˇ c,´ academician Zeljkaˇ Milin Sipuˇ sˇ Mirko Polonijo Vladimir Volenec Foreign universities / Sveuciliˇ staˇ izvan Hrvatske Sefketˇ Arslanagic´ (Bosnia and Herzegovina / Bosna i Hercegovina) Mara Coticˇ (Slovenia / Slovenija) Darjo Felda (Slovenia/Slovenija) Eva´ Kopasz (Hungary/Madarska) Emil Molnar´ (Hungary/Madarska) Marta Takacˇ (Serbia/Srbija) Monograph referees / Recenzenti monografije: Sibe Mardesiˇ c,´ academician (Croatia / Hrvatska) Simeˇ Ungar (Croatia / Hrvatska) Eva´ Kopasz (Hungary / Madarska) Mara Coticˇ (Slovenia / Slovenija) A Word from the Editors The overall social turmoil of the present times, especially in the area of econ- omy and finance, is reflected in science, education and culture. The leadership of many countries is seeking the way out of the global crisis by stimulating the development of science and education. It is expected of scientists and teams of educationists to create new solutions by means of which to tackle the problems of shaken existence within the layers of contemporary social reality. Accordingly, the area of mathematics occupies an important position in the national curricula. Therefore, upon entering the third millenium we considered the scope of mathe- matical competences of fifteen-year-olds at the international level (PISA1 research in 1999 and TIMSS2 research in 1995), as well as those of pupils in the fourth grade of primary school (TIMSS research in 1995). The PISA studies are conducted every three years and are sponsored by OECD3 in cooperation with participating countries. The research was founded with the goal of assessing the levels of education of fifteen-year-olds in developed industrial coun- tries. In 1999 the PISA research was conducted in 43 countries, whereas in 2005 it included sixty countries. Consequently, the number of participating countries keeps continually increasing. The research sample in each country involves 4500 to 10 000 pupils. Each pupil is required to take the written exam in the duration of two hours. One part of the exam is comprised of multiple choice questions, whereas the remaining part of the examination is composed of open-ended questions which demand independently constructed answers to specific questions. Furthermore, pupils are also requested to fill out a questionnaire on their learning habits, their motivation and their family. The most recent PISA study was conducted in 2011. The TIMSS project devised for mathematics and science subjects has been implemented worldwide since 1995 in 4-year cycles for pupils of the fourth grade and those of the final grades of primary school. The project additionally monitors the application of curricula and identifies the best examples of teaching practice around the world. The project leader is the International Association for the Evalua- tion of Educational Achievement IEA4. Accordingly, the Association encompasses institutions and agencies around the world who do research in evaluating pupils’ achievements. The headquarters of this institution are in Amsterdam. Moreover, the design and application od the TIMSS research also includes the participation of the TIMSS and PIRLS5 International Study Center at the Lynch School of Educa- tion at Boston College, with the samples from the participating countries structured, gathered and analysed by the IEA Data Processing Center, situated in Hamburg, as well as Statistics Canada: Canada’s national statistical agency, situated in Ottawa. Due to the fact that the TIMSS project is realised in 4-year cycles, the first study was conducted in 1995, followed by those in 1999, 2003, 2007 and 2011. 1 PISA – Programme for International Student Assessment 2 TIMSS – Trends in International Mathematics and Science Study 3 OECD – Organisation for Economic Co-operation and Development 4 IEA – International Association for the Evaluation of Educational Achievement 5 PIRLS – Progress in International Reading Literacy Study Consequently, TIMSS 2011 included approximately 600 000 pupils aged 9–11 in 52 countries, and this has been so far the widest and very ambitious international study of measuring pupils’ achievements in mathematics and science in the domains of content and cognition with the purpose of improving learning and teaching of mathematics and natural sciences. The last study included, in addition to the pupils of the fourth grade of primary school, the pupils of the final (eighth and ninth) grade of primary school (aged 14–16) with reference to various content domains (Num- bers, Algebra, Geometric Objects and Measurements, and Data Presentation) and cognitive domains (Factual Knowledge, Application and Deduction). The research TIMSS 2011 was based on the mathematics and natural science curriculum, as well as the manner in which the mathematical and natural science contents are taught. The tested areas in natural science were biology, chemistry, physics and geography. The questions and tasks included in the tests varied from multiple choice questions to short answer or open-ended questions which required a pupil to solve a problem or explain their answer. TIMSS 2011 also included pupil, teacher and headmaster surveys on the con- ditions in which mathematics and science instruction is carried out. The obtained results included the information on the systems of education, curricula, teach- ing practice, characteristics of pupils, teachers and schools, which enable us to thoroughly observe the teaching and learning process within the mathematics and science instruction. All segments of the research were implemented in accord with the detailed instructions provided with the aim of achieving the highest possible level of standardized conditions in which the research is conducted. The universities and faculties which educate teachers of mathematics for teach- ing pupils/students of any age group from pre-school age to higher education care- fully monitor and compare valuable results of this research, detect the areas in which the mathematical achievements of pupils should be improved at the national level and propose the ways to improve the quality of education and achieve better learning outcomes. We deem particularly valuable contributions of the researchers regarding the samples of poorer mathematical achievements at the “local level” in neighbouring countries, as well as the presentation of examples of good practice by means of which we pave the path towards better achievements of pupils and, con- sequently, the improvement in the quality of teaching mathematics at the “local”, as well as the global level. This book contains the results of the research on teaching mathematics and examples of good practice provided by the scholars from the neighbouring coun- tries: Bosnia and Herzegovina, Croatia, Hungary, Germany, Romania, Slovakia and Slovenia. Mathematics teachers and parents can use this book to find answers to specific questions connected to the understanding of contemporary trends and procedures in teaching mathematics in our sorroundings. In this manner we are contributing to the global efforts to improve the quality of teaching mathematics. Thus, we are also affecting better learning outcomes in the area of mathematics, and consequently upward moving trends in economy. Osijek, March 25, 2013 Margita Pavlekovic´ Zdenka Kolar-Begovic´ Ruzicaˇ Kolar-Superˇ Rijecˇ urednica Sveukupna drustvenaˇ previranja s negativnim predznakom, osobito u podrucjuˇ gospodarstva i financija, odrazavajuˇ se i na znanost, obrazovanje te kulturu. Vodstva mnogih zemalja izlaz iz globalne krize trazeˇ u poticanju razvoja znanosti i obra- zovanja. Ocekujeˇ se da znanstvenici i obrazovni timovi kreiraju nova rjesenjaˇ u poljuljanim egzistencijama sveukupne drustveneˇ stvarnosti. U nacionalnim kuriku- lumima pridaje se vaznoˇ mjesto podrucjuˇ matematike. Ulaskom u treci´ milenij, sagledavaju se na medunarodnoj razini dosezi matematickihˇ kompetencija pet- naestogodisnjakaˇ (PISA1 istrazivanjemˇ od 1999. godine te TIMSS2 istrazivanjemˇ od 1995. godine) kao i ucenikaˇ cetvrtogaˇ razreda osnovne skoleˇ (TIMSS istrazivan-ˇ jem od 1995. godine). PISA istrazivanjaˇ se provode svake trece´ godine pod pokroviteljstvom OECD- a3 u suradnji zemalja sudionica. Osnovana su s ciljem ocjenjivanja razine obrazo- vanja petnaestogodisnjakaˇ u razvijenim industrijskim zemljama. PISA istrazivanjeˇ je 1999. provedeno u 43 zemlje, a primjerice 2005. u sezdesetˇ zemalja. Broj ze- malja sudionica se stalno povecava.´ Uzorak ispitivanja u svakoj drzaviˇ krece´ se izmedu 4500 i 10 000 ucenika.ˇ Svaki ucenikˇ rjesavaˇ pisani ispit u trajanju