Changing views and practices? A study of the KappAbel mathematics competition Tine Wedege and Jeppe Skott Changing views and practices? A study of the KappAbel mathematics competition Tine Wedege and Jeppe Skott with Kjersti Wæge and Inge Henningsen Research report Changing views and practices? A Nordic study on the KappAbel mathematics competition - Research report Tine Wedege and Jeppe Skott 2006 Norwegian Center for Mathematics Education Norwegian University of Science and Technology (NTNU) Trondheim, Norway Copyright: The authors Illustrations: Tor Nielsen © Print: NTNU-tryk, Trondheim ISBN 82-471-6040-4 Preface The research project reported in this book is initiated by the Nordic Contact Committee (NCC) of the 10th International Congress on Mathematics Education (ICME-10). In the spring of 2004, they called for research projects on the Nordic mathematics competition KappAbel. The projects were to address significant questions about results of the competitions, in particular “possible changes in student attitudes towards mathematics, as well as possible changes in the practice of involved teachers”. The empirical focus of our study was the KappAbel mathematics competition in Norway in the academic year 2004-2005. We followed the competition from September 2004, when the theme “Mathematics and the Human Body” was announced, over the first and second round in November and January 2005, until the national final in April 2005. We did the first survey in December 2004 and the last observations and interviews in November 2005. The research project “Changing views and practices” is partly financed by a grant from NCC, and partly by the Norwegian Center for Mathematics Education where the project is situated. The members of the research team are Tine Wedege (research leader), the Norwegian Center for Mathematics Education and Malmö University, Jeppe Skott, the Danish University of Education, Inge Henningsen, the University of Copenhagen, and Kjersti Wæge, the Norwegian Centre for Mathematics Education (research assistant). Between us we have nine months of work financed within the project budget. We want to thank, first of all, the teachers and the students who gave their voices to this study and who spent their valuable time participating in surveys and interviews. Besides we thank Roald Buvig, Frolands Verk, project leader of KappAbel, and Knut H. Hassel Nielsen, NTNU, responsible for the web in the competition, for statistical information and contact to the teachers in KappAbel. And thank you to Svein Torkildsen, Samfundets Skole, Kristiansand, who constructively commented on the questionnaire and gave an interview, and to Anna Kristjánsdóttir, Agder University College, who also gave an interview. Many thanks to Tor Nielsen, who illustrated KappAbel with interest and nerve, and to Inger Sandnes, Trude Holum, Bjørg Riibe Ramskjell, Astrid Liestøl Henningsen, Anja Angelsen and Anna Folke Larsen who did transcriptions, translations and proofreading. Finally we want to thank Ingvill M. Stedøy and her colleagues at the Norwegian Center for Mathematics Education who always showed vital interest and support, not the least Merete Lysberg, who organised the sending out of the first survey, effectively supported by Margit K. Jensen and Ingunn Seem, who also coded the questionnaires; and Randi Håpnes, who has kept a check on the expenses. Tine Wedege Malmö University Content Chapter 1. The KappAbel study 7 1.1 Mathematical instruction in the Norwegian school 7 1.2 Introduction to KappAbel 9 Three basic ideas 10 1.3 Practice of KappAbel 10 The role of the teacher 11 Phase 1 – round 1 and 2 12 Phase 2 – project work and national finals 13 Phase 3 – Nordic final 18 1.4 Visions of change 18 Changing views among the students 20 Changing school mathematical practices 22 1.5 Problem field, research interest and research questions 25 International research on mathematics competitions 25 KappAbel is in line with international reform efforts 28 KappAbel has the role of an external source of influence on teaching/learning processes in the mathematics classrooms 29 Belief research in mathematics education (introduction to the problem) 30 Research questions 30 Chapter 2. Change in ”views and practices” 32 2.1 Beliefs, attitudes, emotions 32 2.2 Practices 37 2.3 “Didactical contract” as a metaphor in the study 40 2.4 Changing views and practices 44 Chapter 3. Research design 46 3.1 Methodological difficulties 48 Symbolic violence 53 3.2 The quantitative study – Teacher survey 1 54 Design of the questionnaire 55 3.3 The qualitative study 56 The questionnaire 57 Interviews, observations and documents 58 Ethics 60 Conclusions 61 Chapter 4. The teacher survey 64 4.1 The quantitative study 64 The teachers’ background, experience and participation in KappAbel 64 The teachers’ school mathematical priorities 72 The factor analysis of the teachers’ school mathematical priorities 74 Representativity 78 4.2 Taking the qualitative remarks into account 80 The time factor 80 The tasks, contents and approach of KappAbel 86 Lack of knowledge and information 89 Other comments related to participation or non-participation in KappAbel 91 Conclusions 94 4.3 Results from TS1 – a summary 98 Chapter 5. Meeting teachers and students in KappAbel 99 5.1 Getting in contact 100 5.2 Who responded? 101 Factual information 101 School mathematical priorities 103 5.3 KappAbel and three basic ideas 109 Co-operation 112 Project work 114 Gender equity 116 5.4 First meeting with six teachers 122 Arne – Ola – Peder – Randi – Steinar – Øystein 5.5 Second meeting with six of the teachers and their students 135 Arne – Ola – Peder – Randi – Steinar – Øystein Chapter 6. Mathematics teaching at Bjerkåsen lower secondary 168 school: the case of Kristin and her students. 6.1 Data and methods 169 6.2 Kristin and the classroom practices 171 Setting the stage: whole class introductions 172 Responding to students’ questions and comments 175 Relying on rules 183 Sequencing instructional tasks and questions 185 6.3 Kristin’s view of school mathematics and the role of KappAbel in it 188 6.4 The students’ views of school mathematics and of the role of KappAbel in it 194 6.5 Discussion – change in views and practices 198 Chapter 7. Conclusions 204 Who participate? 204 Why do they (not) participate? 206 What is the potential impact of participation? 208 Conclusions, challenges and proposals for consideration 211 References 215 Appendices 1-52 List of appendices A. Statistical analysis of Teacher Survey 1 (TS1) – Inge Henningsen B. Technical report on data collection and handling in TS1 Chapter 1 The KappAbel Study The study addresses the issue of potential and perceived influence of the KappAbel competition on the mathematical attitudes and practices of the participating teachers and students. In chapter 1, we will present the KappAbel competition: its organisation in three phases (two qualifying rounds; project work and semi-finals; final) and some examples of mathematical problems and project work. We will present the visions of KappAbel as stated at the official web-site and by three mathematics teachers and researchers who are deeply involved in the competition. Furthermore we will present and discuss the problem complex around the KappAbel competition, our research interest and our research questions. But firstly we will give a short presentation of mathematical instruction in the Norwegian primary and lower secondary school. 1.1 Mathematical instruction in the Norwegian school A reform of the Norwegian ten-year basic school was initiated in 1997, also introducing new syllabi for all school subjects. According to the mathematics syllabus, instruction is to create interest and insight in the subject on the part of the students, so as to develop their proficiency in using it in daily life and in other school subjects (L97,1996). The teachers are to introduce practical connections, examples and working methods in order to provide their students with opportunities to develop positive attitudes towards the subject, and instruction is to assist the students in developing their proficiency in using their mathematical qualifications for communicative purposes in a modern society. Also, the students are to experience mathematical learning as an investigative and reflective process. These intentions are outlined in slightly more detail for different age groups in a section of the syllabus describing the types of classroom activities envisaged for school mathematics. In the first phase of primary school (students from 6 to 9 years of age), play is a key term, and instruction is to take its point of departure 7 in the students’ everyday experiences. The syllabus also encourages cross- curricular activities, integrating mathematics with other subjects. In middle school (students from 10-12), mathematics education is to be linked to or embedded in practical activities. Theory and practice are to be linked by using play and games, and by working with nature and the students’ local environment. From such starting points, each student is to be challenged in a variety of ways and at her own level of expertise, while engaging with the more abstract aspects of mathematics. At the lower secondary level (students from 13-15), there is greater emphasis on the more formal and abstract sides of mathematics and on the applications of mathematics in society. Also at this level, instruction is to take its point of departure in the students’ own experiences. At all levels, the students’ involvement in genuine mathematical activities is emphasised. (L97, 1996). From 1998-2003 the Norwegian Research Council conducted an evaluation of the 1997-reform on behalf of the Ministry of Education and Research. Alseth et al. (2003) did the study on mathematics in primary and lower secondary school. In their interpretation, there are five main expectations concerning mathematics instruction in L97. These are that the school subject is to 1) be practical; 2) emphasise conceptual development; 3) be investigative; 4) be communicative and cooperative; 5) adopt also historical and cultural perspectives on the subject.