Kontinuitetsbegrepet : En lærebokanalyse om behandlingen av kontinuitet i videregående skole

Abstract

The theory on complex numbers is often considered too complicated for upper secondary students.Because of that, students are told that it is not possible to calculate the square root of negative num-bers. For a similar reason, the formal definition of limits and continuity are not introduced in uppersecondary school. I believe it is not beneficial to withhold knowledge that may be concidered to be “toocomplicated”. On the contrary, I believe that students can become curious and motivated, if given anuncensored view into the world of mathematics.This thesis deals with the notion of continuity and will attempt to answer the following research ques-tion:•In what way does upper secondary textbooks help students obtain a correct mathematical under-standing on the notion of continuity?I have formulated the following research aims in order to address the research question:◦How does the content in textbooks strengthen students’ mathematical understanding of the notionof continuity?◦What significance does the presentations of the notion of continuity have for the development ofstudents’ concept image?◦To what extent does mathematical textbooks provide students with experience in mathematicalreasoning?In the study, I have examined how two Norwegian textbooks present the notion of continuity. Theanalysis is based on a framework designed by Charalambous et al. (2010) and uses a combination ofquantitative and qualitative research methods. I have used different frameworks to encode the variouselements in the textbooks. Among these, a research framework for imitative and creative reasoning isused in order to encode mathematical tasks (Lithner, 2008). Furthermore, to investigate how propertiesand definitions are presented to the students, I have adopted a framework by Thompson et al. (2012) tobe able to code how properties are justified in the narrative.The theory that deals with development of students’ concept image and concept definition is based onTall and Vinner’s (1981) framework, as well as findings from former studies completed by Juter (2006,2011, 2017). I discuss how two different conceptualizations of continuity (“natural continuity” and“Cauchy-Weierstrass continuity”) influence students understanding of the notion (N ́u ̃nez et al., 1999).In the study, I find that the textbooks define continuity purely based on an informal definition of thenotion of limits. This might limit the students’ opportunities to fully learn the notion of continuity andrestrict their basis for conducting mathematical reasoning. Many of the tasks provided are given rightafter an example, which does not contribute to help the students abilities for developing problem-solvingstrategies. The thesis discusses why the use of the word ‘connected’, since it is not properly defined,could conflict with the students future understanding of the mathematical definition. This paper alsopresents how an introduction of theε-δdefinition of limits and continuity can be accomplished anddiscuss how it can help the students to obtain a better understanding of the notions.vii