Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Humanities, School of Education, 2011 / This study is a located in a foundational undergraduate mathematics course designed to facilitate the transition from school mathematics to advanced mathematics. The focus of the study is on two innovations in the course; the use of practical problems that make links to non-mathematical practices and a learner-centred pedagogy. While these innovations are part of the discourse of the mathematics education community in terms of access to school mathematics, this study investigates the relationship between these innovations and access to advanced mathematics. The texts of three practical problems from the course and texts representing the verbal and non-verbal action of 17 students as they worked collaboratively in small groups on these problems were analyzed. The analysis of these texts is used to describe and explain, firstly, how the practical problems in the foundational course represent the practice of foundational undergraduate mathematics and its relationship to other practices in the educational space (for example, school mathematics, calculus reform, advanced mathematics, and non-mathematical practices). Secondly, the students‟ enabling and constraining mathematical action on the practical problems is described and explained. Answering the empirical questions in this study has required theoretical work to develop a socio-political perspective of mathematical practice. This theoretical perspective is based on Fairclough‟s social practice perspective from critical linguistics, but has been supplemented with recontextualized theoretical constructs used by Morgan, Moschkovich and Sfard in mathematics education. These constructs are used to conceptualize the notion of mathematical discourse and action on mathematical objects in this discourse. The methodological work of this study has involved supplementing Fairclough‟s method of critical discourse analysis with Sfard‟s method of focal analysis to analyze mathematical, discursive, social and political action in a socio-political mathematical practice. The central finding of this thesis is that foundational mathematical practice represents both continuities and disruptions in its relationship to other practices in the space. As a result, participation in the foundational practice is complex, requiring control over the how and when of boundary crossings across practices, social events and texts. On the basis of this complexity, innovative foundational practice is positioned paradoxically in the higher education space. On the one hand, it represents an alternative to the dominant representation of mathematical practice and positioning of the foundational student in higher education. On the other hand, the complexity of foundational practice makes access to advanced mathematics problematic and foundational practice thus reproduces the dominant ordering.

This study provides a framework to guide educators and researchers within departments of mathematics at institutions of higher education involved in reform efforts for undergraduate mainstream calculus. It does so by using motivational constructs from Self-Determination Theory to define and measure student-centeredness within both traditional and reformed calculus learning environments within a large-scale, quasi-experimental study. Motivational inventories assessing students’ perceived satisfaction of basic psychological needs and self-determined motivation were analyzed together with demographic variables, course outcomes, and prior math knowledge within traditional and reformed conditions in both Calculus I and Calculus II courses. Results include 1) positive correlates among students’ perceptions of satisfaction of basic psychological needs, intrinsic motivation, and achievement; 2) overall increased student perception of BPN-satisfaction in the reformed condition; and 3) directional variation in achievement and perception of BPN-satisfaction between conditions across subpopulations. The results demonstrate how student-centered calculus learning environments operate through motivational processes to improve academic outcomes and how learning environments may differentially affect demographic categories at institutions of higher education. Specifically, learning environments are not culturally or socially neutral and may, despite good intentions, be centered about privileged populations to the detriment of historically disenfranchised groups.