The appropriation of mathematical objects by undergraduate mathematics students: a study
Date
2014-06-24
Authors
Berger, Margot
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Abstract
In this thesis I consider how mathematics students in a traditional firstyear Calculus course at a South African university appropriate mathematical objects which are new to them but which are already part of the official mathematics discourse. Although several researchers have explained mathematical object appropriation in process-object terms (for example, Sfard, 1994; Dubinsky, 1991, 1997; Tall, 1991, 1995, 1999), my focus is largely on what happens prior to the object-process stage. In line with Vygotsky (1986), I posit that the appropriation of a new mathematical object by a student takes place in phases and that an examination of these phases gives a language of description for understanding this process. This theory, which I call “appropriation theory”, is an elaboration and application of Vygotsky’s (1986) theory of concept formation to the mathematical domain.
I also use Vygotsky’s (1986) notion of the functional use of a word to postulate that the mechanism for moving through these phases, that is, for appropriating the mathematical object, is a functional use of the mathematical sign. Specifically, I argue that the student uses new mathematical signs both as objects with which to communicate (like words are used) and as objects on which to focus and to organise his mathematical ideas (again as words are used) even before he fully comprehends the meaning of these signs. Through this sign usage the mathematical concept evolves for that student so that it eventually has personal meaning (like the meaning of a new word does for a child); furthermore, because the usage is socially regulated, the concept evolves so that its usage is concomitant with its usage in the mathematical community.
I further explicate appropriation theory by elaborating a link between the theoretical concept variables and their empirical indicators, illustrating these links with data obtained from seven clinical interviews. In these interviews, seven purposefully chosen students engage in a set of speciallydesigned tasks around the definition of an improper integral. I utilise the empirical indicators to analyse two of these interviews in great detail. These analyses further inform the development of appropriation theory and also demonstrate how the theory illuminates the process of mathematical object appropriation by a particular student.
Description
Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 2002.