<p>The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated. A group of mathematics students at undergraduate level at the National University of Lesotho (NUL) was used as the sample for the study. Empirical data were collected by using interviews and questionnaires. These data were analysed using both the APOS framework and a semiotic perspective.</p> <p><br /> Within the APOS framework, the pieces of knowledge that have to be constructed in coming to understand the limit concept are actions, processes and objects. Actions are interiorised into processes and processes are encapsulated into objects. The conceptual structure is called a schema. In investigating the idea of limit within the context of a function some main epistemological obstacles that were encountered when actions were interiorised into processes are over-generalising and taking the limit value as the function value. For example, in finding the limit value L for f(x) as x tends to 0, 46 subjects out of 251 subjects said that they would calculate f(0) as the limit value. This method is appropriate for calculating the limit values for continuous functions. However, in this case, the method is generalised to all the functions. When these subjects encounter situations in which the functional value is equal to the limit value, they take the two to be the same. However, the two are different entities conceptually.</p>

<p>Problems of understanding fundamental calculus concepts by students in tertiary education colleges and universities are evidenced by a body of research studies conducted in different parts of the world. The researchers have identified, classified and analysed these problems from historical, epistemological, and learning theory perspectives. History is important because mathematical concepts are a result of the developments of the past. The way knowledge is acquired is an epistemological issue and the major purpose of learning is to acquire knowledge. Hence, these three perspectives qualify to be used as lenses in understanding problems that students encounter in a learning situation. The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated, because communication in the mathematics classroom takes place by using language and symbols.</p>

This dissertation deals with the development of mathematical concepts from a historical and didactical perspective. In particular, the development of concepts in mathematical analysis during the 19th century is considered. The thesis consists of a summary and three papers. In the first paper we investigate the Swedish mathematician E.G. Björling's contribution to uniform convergence in connection with Cauchy's sum theorem from 1821. In connection to Björling's convergence theory we discuss some modern interpretations of Cauchy's expression x=1/n. We also consider Björling's convergence conditions in view of Grattan-Guinness distinction between history and heritage. In the second paper we study visualizations in mathematics from historical and didactical perspectives. We consider some historical debates regarding the role of intuition and visual thinking in mathematics. We also consider the problem of what a visualization in mathematics can achieve in learning situations. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. In the third paper we consider Cauchy's theorem on power series expansions of complex valued functions on the basis of a paper written by E.G. Björling in 1852. We discuss Björling's, Lamarle's and Cauchy's different conditions for expanding a complex valued function in a power seris. In the third paper we also discuss the problem of the ambiguites of fundamental concpets that existed during the mid-19th century. We argue that Cauchy's and Lamarle's proofs of Cauchy's theorem on power series expansions of complex valued functions are correct on the basis of their own definitions of the fundamental concepts involved.

Thesis (Ph. D.)--University of Adelaide, Dept. of Pure Mathematics, 1984. / Includes bibliographical references.

Thesis (Ph. D.)--Carleton University, 2001. / Includes bibliographical references (p. 141-144). Also available in electronic format on the Internet.

Thesis (M.S.)--Kutztown University of Pennsylvania, 1993. / Source: Masters Abstracts International, Volume: 45-06, page: 3184. Typescript. Includes bibliographical references (leaves 55-56).

Thesis (M.S.)--Oregon State University, 2004. / Printout. Includes bibliographical references (leaves 63-64). Also available on the World Wide Web.

Thesis (M.S.)--Oregon State University, 2004. / Typescript (photocopy). Includes bibliographical references (leaves 53-54). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 127-132). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 127-132). Also available on the Internet.