Students’ perception of infinity Perception, obstacles, conception SOKRATIS THEODORIDIS SUPERVISOR John D. Monaghan This master’s thesis is carried out as a part of the education at the University of Agder and is therefore approved as a part of this education. However, this does not imply that the University answers for the methods that are used or the conclusions that are drawn. University of Agder, 2017 Faculty of Engineering and Science Department of Mathematical Sciences Summary Several studies in mathematics education have shown that students face significant difficulties for understanding the concept of infinity. The purpose of this study is to suggest an explanation of how students understand the concept of infinity and identify the difficulties for understanding it. Explanation will be given through an investigation on the following aspects of understanding and the implementation of the corresponding theoretical background: i) primary perception in terms of Singer and Voica’s(2008) categories and secondary perception in the context of comparison of infinite sets ii) obstacles in coming to understand the concept in terms of the Theory of Epistemological Obstacles and iii) the construction of understanding in terms of APOS theory. The empirical part of this study was conducted in Norway. The subjects are five 12th graders and data for this study were collected by means of a questionnaire and an interview. A precise overview of the aim and contents of the thesis is given in the introductory part. The thesis begins with a historical analysis of the concept of infinity, aiming to reveal its paradoxical and contradictory nature. Next, the theoretical background is presented along with the basis of each theory. There has been a qualitative analysis on the collected data, by methods found in the literature. The results have shown that most of the students perceived infinity as a process while specific difficulties related to the notion of actual infinity were noticed. The analysis of the structure of understanding, raised the importance of “encapsulation” as a way to understand its actual form. The thesis hopes to shed light on students’ understanding and thus make a small contribution to mathematics education research. Sammendrag Flere tidligere studier matematikkdidaktikk har vist at elever har store vanskeligheter når det kommer til å forstå konseptet uendelighet. Hensikten med denne studien er å foreslå en forklaring på hvordan elever forstår konseptet uendelighet og å identifisere vanskeligheter knyttet til forståelsen. Forklaringen vil bli gitt gjennom en undersøkelse av følgende aspekter for forståelse og implementeringen av tilhørende teoretisk rammeverk: i) primær persepsjon etter Singer og Voica (2008) sine kategorier og sekundær persepsjon i konteksten av å sammenlikne uendelige mengder, ii) hindringer i å nå en forståelse av konseptet ved bruk av teorien om epistemologiske hindringer, og iii) konstruksjonen av forståelse ved bruk av APOS teorien. Innsamling av empiri til denne oppgaven ble gjort i Norge og informantene er fem VG2 elever. Data for studien er samlet inn ved hjelp av spørreskjema og intervju. Et detaljert overblikk av mål og innhold i studien blir gitt i innledningen. Studien begynner med å ta for seg en historisk analyse av konseptet uendelighet, med mål om å avdekke dets paradoksale og motstridende natur. Deretter blir det teoretiske rammeverket presentert sammen med grunnlaget til hver teori. Det har blitt utført en kvalitativ analyse på innsamlet data ved hjelp av metoder fra funnet i litteraturen. Resultatene har vist at de fleste elevene oppfattet uendelighet som en prosess, samtidig som spesifikke vanskeligheter knyttet til forestillingen av faktisk uendelighet ble oppdaget. Analysen av strukturen av forståelse viser viktigheten av «innkapsling» som en måte å forstå forståelses faktiske form. Studien håper å kaste lys på elevenes forståelse og dermed gjøre et lite bidrag til matematikkdidaktisk i Preface Before you lies the thesis “Students’ understanding of infinity: perception, obstacles, conception”. It has been submitted in partial fulfillment of the requirements of the double degree of the Joint Nordic Master’s Programme in Didactics of Mathematics – NORDIMA. I was engaged in researching and writing this thesis from December 2016 to May 2017. The basis for this research stemmed from my passion in mathematical counterintuitive facts and problems that push the mind to new areas and challenge the cognitive versatility. One of the hardest mathematical concepts responsible for intellectual struggle and controversies, is infinity. As a future mathematics teacher, the passion has turned into curiosity about infinity in a student’s mind. How do students understand infinity? I wrote this thesis in an attempt to find out. The project was undertaken at the request of the Faculty of Engineering and Science, University of Agder. The research was not easy, but conducting extensive investigation has allowed me to formulate the research questions and to reflect on appropriate theory to answer them. The contribution of my advisor, John Monaghan, had been critical though all the process. I would like to thank him for his guidance and support during the academic year 2016-17. I would also like to thank professor Reinhard Siegmund-Schultze for his inspiration on the topic of the thesis as well as his comments on the historical part. I also wish to thank all the participants of the study, both students and teachers. A big thanks to my colleagues at UiA who contributed with their educational experience and their native Norwegian speaking skills. Last but not least, a note to friends and family: Thank you for being rational, thank you for being there. I wish you a pleasant reading. Sokratis Theodoridis Kristiansand, May 2017 ii Table of Contents Table of Contents .................................................................................................................................. iii CHAPTER 1: Introduction ....................................................................................................................... 1 CHAPTER 2: From mythology to Cantor ................................................................................................ 5 2.1 Mythology ......................................................................................................................................... 5 2.2 Philosophy and science ..................................................................................................................... 6 2.3 Hellenistic Period ............................................................................................................................ 13 2.4 The Decline of Greek Mathematics................................................................................................. 15 2.5 The Middle Ages ............................................................................................................................. 15 2.6 17th Century-An explosion of mathematical ideas .......................................................................... 16 2.7 18th and 19th Century, the appearance of “menge” ....................................................................... 21 2.8 Hilbert’s Paradox and Dedekind’s “cuts” ........................................................................................ 24 2.9 The triumph .................................................................................................................................... 26 2.10 End of the journey ........................................................................................................................ 28 CHAPTER 3: Theoretical Framework ................................................................................................... 29 3.1 On the nature of infinity ................................................................................................................. 29 3.1.1 What is infinity? ........................................................................................................................... 29 3.1.2 The contradictory nature of infinity ............................................................................................. 30 3.2 Identification of Students’ Perceptions ........................................................................................... 31 3.3 The Theory of Genetic Epistemology .............................................................................................. 32 3.3.1 APOS Theory ................................................................................................................................ 33 3.4 Psychological Recapitulation .......................................................................................................... 35 3.4.1 The Theory of Epistemological Obstacles .................................................................................... 35 3.4.1.1 The function of history in the Theory of Epistemological Obstacles ......................................... 37 3.5 Refining and implementing the theory ........................................................................................... 38 CHAPTER 4: Methodology ................................................................................................................... 41 4.1 Research Design ............................................................................................................................. 41 4.2 Pilot Study ......................................................................................................................................