2006:08 DOCTORAL T H E SI S Limits of Functions University Students’ Concept Development Kristina Juter Luleå University of Technology Department of Mathematics 2006:08|: 02-544|: - -- 06 ⁄08 -- Limits of Functions- University Students’ Concept Development by Kristina Juter Department of Mathematics Luleå University of Technology SE-971 87 Luleå February 2006 2 Abstract This PhD thesis consists of six articles and a concise synopsis that generally frame these articles. The theories and methods used are outlined and discussed in the synopsis to show their relation. The articles are published or accepted for publication (see the list of publications at page 9) with the following abstracts: I: Limits of functions – how do students handle them? This paper aims at formulating and analysing the development of 15 students in their creations of mental representations of limits of functions during a basic mathematics course at a Swedish university. Their concept images are sought via questionnaires and interviews. How do students respond to limits of functions? The data indicate that some students have incoherent representations, but they do not recognise it themselves. II: Limits of functions: Traces of students’ concept images Students at a Swedish university were subjects in a study about learning limits of functions. The students’ perceptions were investigated in terms of traces of concept images through interviews and problem solving. The results imply that most students’ foundations were not sufficiently strong for them to understand the concept of limit well enough to be able to form coherent concept images. The traces of the students’ concept images reveal confusion about different features of the limit concept. III: Students’ attitudes to mathematics and performance in limits of functions The main aim of this article is to discuss the attitudes to mathematics of students taking a basic mathematics course at a Swedish university, and to explore possible links between how well such students manage to solve tasks about limits of functions and their attitudes. Two groups, each of about a hundred students, were investigated using questionnaires, field notes and interviews. From the results presented a connection can be inferred between students’ attitudes to mathematics and their ability to solve limit tasks. Students with positive attitudes perform better in solving limit problems. The educational implications of these findings are also discussed. IV: Limits of functions as they developed through time and as students learn them today In this article, the ways first year university students’ develop the concept of limits are compared to the historical development of the concept. The 3 aim is to find out if the students perceive the notion as mathematicians of the past did as understandings of the concept evolved. The results imply that there are some similarities, for example, the struggle with rigour and attainability. Knowledge of such critical areas can be used to improve the students’ opportunities of learning limits of functions. Some teaching aspects are also discussed. V: Limits of functions - students solving tasks Students solve problems to learn mathematics. How do they cope with the problems and explain their solutions? This study is about students at a Swedish university, solving tasks about limits of functions. The students are enrolled in their first university course in mathematics. They study algebra and calculus for 20 weeks fulltime. There are many possible problems for the students to handle including the notion of function, the formal definition and limits as unattainable values for functions. Many students do not possess enough knowledge about limits to solve and explain solutions to non-standard problems. Some students arrive at correct answers through erroneous reasoning. VI: Limits and infinity: A study of university students’ performance Students’ work with limits of functions includes the concept of infinity. These concepts are complex but necessary for mathematics studies. A study of student solutions to limit tasks was conducted to reveal how students handled limits of functions and infinity at a Swedish university. As a background to the students’ and teachers’ situations we have also analysed textbooks and curricula. One result is that many students were unable to handle limits correctly. Some students gave correct answers with incorrect explanations to tasks. Textbooks used at upper secondary schools do not provide much theory or many tasks about limits and infinity so most students new to university do not have a developed image of these concepts. 4 Acknowledgements My supervisors, Professor Barbro Grevholm, Professor Lars-Erik Persson and Professor Gilah Leder, have all been there for me throughout the period of my research education and I wish to thank each of them for that. Barbro, I am very happy to have been able to discuss all sorts of issues with you and for your encouragement when I needed it. Thank you for tirelessly reading and commenting on my texts at all times. Lars-Erik, thank you for your support and your very positive attitude to me and my work. You really strengthened my confidence with your enthusiasm. Gilah, thank you for your constructive advice and all your efforts to help me, in particular during my visit at La Trobe University. It was really an adventure, which you made possible. Especially thanks for the kangaroo expedition at the golf course. I am very grateful for the hospitality of the Mathematics department at Lund University for letting me take time and space to carry out my two studies. Especially thanks to Per-Anders Ivert, Bengt Gamstedt, Tomas Claesson, Kjell Elfström, Catarina Petersson and Agneta Ohlzon. I want to thank ‘Luleå Tekniska Universitet’ for the generous team work together with ‘Högskolan Kristianstad’ that has given me the opportunity to do my doctoral studies in Kristianstad near my home. I also want to thank ‘Vetenskapsrådet’ and ‘Riksbankens Jubileumsfond’ for funding my research studies and the board of ‘Forskarskolan i matematik med ämnesdidaktisk inriktning’ for making the project possible. I would also like to thank ‘Högskolan Kristianstad’ for supporting me. There have been two important seminars during my research education where my thesis has been read and commented by an opponent. The first was a licentiate seminar in January 2004 and the second was a 90% seminar in January 2006. I would like to thank Professor Johan Lithner and Professor Mogens Niss respectively for their valuable remarks and suggestions at those seminars. Last, but perhaps most, I want to thank every single one of the students who participated in the studies. Their endless willingness to answer my questions was inspiring. Kristina 5 6 Contents Abstract 3 Acknowledgements 5 Contents 7 List of Publications 9 Introduction 11 Background and Questions 11 Theoretical Overview 12 Limits of Functions 12 Learning 14 Terminology 20 The Empirical Study 21 Rationale of the Empirical Study 22 The First Study 22 The Second Study 24 Methodological Remarks 27 Analysis of Data 29 Results 30 Ethical Issues and Scientific Quality 33 Discussion 34 New Results for the Field 34 Implications for Teaching and Learning 35 Further Research 37 Conclusions 38 References 40 Appendix 47 Article I Article II Article III Article IV Article V Article VI 7 8 List of Publications I Juter, K. (2005a). Limits of functions – how do students handle them? Pythagoras 61, 11-20. II Juter, K. (2005c). Limits of functions: Traces of students’ concept images, Nordic Studies in Mathematics Education 10(3-4), 65-82. Parts of the content of this article was presented at a conference and published as: II* Juter, K. (2004b). University students’ perceptions of limits of functions. In B. Tadich, S. Tobias, C. Brew, B. Beatty & P. Sullivan (Eds.), MAV (The Mathematics Association of Victoria) annual conference 2004, Towards excellence in mathematics (pp. 228-236). Melbourne, Australia: The Mathematics Association of Victoria. III Juter, K. (2005b). Students’ attitudes to mathematics and performance in limits of functions, Mathematics Education Research Journal 17(2), 91- 110. IV Juter, K. (2006a). Limits of functions as they developed through time and as students learn them today, to appear in Mathematical Thinking and Learning. V Juter, K. (2006b). Limits of functions - students solving tasks, to appear in Australian Senior Mathematics Journal 20(1). Parts of the content of this article was presented at a conference and published as: V* Juter, K. (2004a). Limits of functions – How students solve tasks. In C. Bergsten & B. Grevholm (Eds.), Mathematics and language, Proceedings of MADIF 4, the 4th Swedish Mathematics Education Research Seminar (pp. 146-156). Linköping, Sweden: Svensk Förening för Matematikdidaktisk Forskning. VI Juter, K., & Grevholm, B. (2006). Limits and infinity: A study of university students' performance. To appear in C. Bergsten, B. Grevholm, H. Måsøval, & F. Rønning (Eds.), Relating practice and research in mathematics education. Fourth Nordic Conference on Mathematics Education, Trondheim, 2nd-6th of September 2005. Trondheim: Sør- Trøndelag University College. 9 10 Introduction Background and Questions Learning mathematics is an endeavour requiring a number of abilities, which vary for different mathematical topics. How individuals learn mathematics can also vary. Some go through hard work followed by a period of enlightenment (Hadamard, 1949), whereas others simply endure hard work. Mathematics