Although there have been calls for secondary mathematics education in the U.S. to incorporate problem-solving and creativity, the lion’s share of instruction is designed to train students to accurately use procedures or understand concepts made by mathematicians in the past (National Research Council, 1999; Watson, 2008). This disconnect highlights a need to know more about student mathematical creativity. The goal of the study was to examine the nature of student mathematical creativity and identify how it can be influenced by social and aesthetic factors. Therefore, I performed a qualitative analysis of video and audio recordings of student and teacher interactions from eight high school mathematics lessons taught in the Northeast in the United States. To demonstrate the range of creativity of which students are capable, I identified and categorized potentially creative actions. I also developed episodes of creative action, explaining how some created new mathematical possibilities, and others were blocked in doing so. From these episodes, I identified a set of key moments in their development: taking the action, the reception by others, advocacy for the action, and an additional creative action by other members of the student group or class. Finally, from comparing multiple episodes, I found that experiencing mystery or mathematical discomfort motivated students to take actions with creative potential, and that positive relationships and strong group participation contributed to interactive discussions between group members that enabled the actions’ creative potential to be realized. Findings from this process could support educators in giving more students the opportunity to create new ways of doing mathematics.

Knowing how to correctly count, is fundamental to the future mathematics success of young children. Earlier studies show that many South African primary school students underperform in mathematics even when evaluated with task below grade level. Reports suggest that this is a problem stemming from the poor pedagogic, and or content knowledge of classroom mathematics teachers. Shulman (1986; 1987) refers to this area of knowledge as Pedagogic Content Knowledge (PCK). In the field of mathematics teaching and learning, Ball, Thames and Phelps (2008) refer to it as Mathematics Knowledge for Teaching (MKfT). Teachers' mathematics PCK, comprises of three core knowledge domain: (i) Teacher's Knowledge of Content and Teaching (KCT); (ii) Teacher's Knowledge of Content and Student (KCS); and (iii) teacher's Knowledge of Content and Curriculum (KCC). Teachers' KCS was considered in this study as it concerns what teachers know about what learners know and how they learn. The general interest of this project was to study the construction of experience of mathematics (non-core domain knowledge) by genetic endowment on the basis of contextual data. More specifically, the particular interest of the study is on the construction of the experience of counting in the pedagogic situations of Grade 3 schooling. For that purpose, video records of mathematics teaching in two schools situated in working-class communities were analysed. The study adopted an Integrated Causal Model approach which drew on resources from different disciplines such as mathematics education, cognitive science, evolutionary psychology and mathematics. The study was partly framed by Bernstein's pedagogic device, particularly with respect to his notion of evaluation, as well as the inter-related constructs of PCK, MKfT and KCS. The theoretical resources used to describe computations were drawn largely from Davis (2001, 2010b, 2011a, 2012, 2013a, 2015, 2018) and related work on the use of morphisms as elaborated in Baker et al. (1971), Gallistel & King, (2010), Krause (1969) and Open University (1970). These resources were used to produce the analytic framework for the production of and analysis of data. The analysis describes the computational activities of teachers and learners during the recorded lessons, specifically the computational domains made available pedagogically. In so doing, I was able to provide more illumination on what is described as teacher's KCS for teaching counting at the Grade 3 level. From the generated data, the study finds that counting proper was restricted to the constitution and identification of very small ordered discrete aggregates which can be handled by human core domain object tracking system and approximate number system, and that an implicit reliance on numerical order derived from computations on aggregates was central to the teaching and learning of counting.

Bibliography: leaves 78-82. / Assessment practices are an integral part of schooling. The prominence of assessment within schooling in providing information to students and teachers about students' "ability" in learning school subjects, raises an important question: what sort of influence do assessment practices have on how school subjects are perceived by students and teachers? This dissertation focuses on two themes - the way in which assessment practices construct school mathematics, and the way in which these constructions of school mathematics work dynamically with assessment practices to produce descriptions of students.

Includes bibliographical references. / This study focused on Foundation Phase teachers' pedagogical and content knowledge. It investigated the impact that a geometry course (Shape and Space), had on the teachers levels of understanding of Shape and Space. The course was conducted over 5 days. A literature search revealed a few different tools in designing the course, the majority of which referred to either Van Hiele or Hoffer. Our course design however was instructed by the requirements of the Revised National Curriculum Statement (RNCS) and had to follow it closely.

Bibliography: leaves 149-155. / This study deals with a systematic description of student production of mathematics texts and focused on written texts that appeared to be legitimate yet could not be upheld by a principled verbal description. A search of the literature on the analysis of students texts revealed that semiotic analysis, was not only scarce, but ideally suited to examining the social organisation of school mathematics practice. This study examines how student texts produced in response to typical school mathematics problems can, via a systematic analysis of texts, index the construction of mathematics knowledge. It outlines Dowlings' model of Social Activity Theory (1998) to produce a textual analysis which focuses on textual strategies to distribute message. These strategies and the message underpin the analysis. Practices that establish the message distributed indexes mathematics knowledge and curriculum practices. The notion of a mathematising gaze informing school practice was explored and was related to the construction of existing and pre-existing mathematics knowledge. To locate the effects of a mathematics gaze that could produce texts that lacked discursive elaboration in verbal discriptions, a comprehensive list of ideal types were developed to act as an interface between the empirical text produced that acted as a reading for constructive description of the theoretical terrain.

Bibliography: p. 101-107. / This research project focuses on the "new primary mathematics" curriculum that has been implemented in the schools in the Western Cape over the past six years. The specific question I addressed was, 'What access do teachers have to the regulating principles underpinning the 'new primary mathematics' curriculum". The term "regulating principles" is drawn from the work of Paul Dowling (1993;98). In terms of this research, the regulating principles are the theoretical underpinnings to the new curriculum, which include substantially a theory of learning. I explore access to the regulating principles through semi-structured interviews with six teachers, who have implemented this new approach with different degrees of success, as measured in their own terms. I also investigate the official Teachers' Guide for Mathematics (Cape Education Department, 1993) for explicitness of theoretical underpinnings. An analysis of the teachers' guide indicated that the regulatory principles were not made explicit and the research indicates that the teachers in my sample have restricted access to these principles. I conclude that teachers who have little access to the regulating principles are constructed as a subordinate voice in relation to teacher educators, and must of necessity rely on procedure for their practice and be subject to external validation. This raises questions as to the successful implementation of the curriculum, in that it limits access by teachers to the educational debates surrounding theories of knowledge and theories of learning, and so inhibits teacher involvement in curriculum implementation. It also limits the ability of teachers to interrogate their own practice.

This dissertation is an investigation into the constitution of school mathematics within the context of the Revised National Curriculum Statement in a selection of Grade 8 mathematics lessons in five working-class schools in the Western Cape Province of South Africa. The study is located within the broad framework of the sociology of education, specifically drawing on Bernstein's (1996) sociological theory of education and his pedagogic device. This study focuses on the way in which the content of the evaluative rule of the pedagogic device is realised in the particular selection of schools. My theoretical framework relies on of the work of Davis (2010a, 2010b, 2010c, 2011a, 2011b, 2011c, 2012, 2013a & 2013b) and Bernstein (ibid.). These theoretical resources were drawn on to describe and analyse the mathematical activity in the five schools as well as serving as a means for generating analytical resources for describing the constitution of mathematics. In my analysis I present an account of the computational activity of teachers and their learners and the regulation of mathematical activity in fifteen Grade 8 mathematics lessons. I use these descriptions of computational activity to discuss the realisation of content against a general background of curriculum reform that has de-emphasised explicit use of formal definitions. I explore what mathematical content was recognised and constituted in relation to topics announced by teachers and use the mathematics encyclopaedia as a resource to ascertain the content that substitutes for formal mathematical definitions, axioms and propositions.

A small-scale study was conducted within a historic and traditional, independent, all-boys secondary school in the Western Cape, the focus of which is the exploration of subject positions potentially available to the black male students of the "Black Scholarship Programme" in their study of school mathematics. This includes an examination of the particular nature of the schooling ethos and culture, and its role in creating and maintaining boundaries, producing and reproducing forms of power and control which assist in holding these black students to positions of subordination. It is proposed that the hierarchical and differentiating rituals and codes within the school context provide the means by which the Black Scholarship students are constructed as disadvantaged. Particular emphasis is placed on the discourse of mathematics within the Academic Support Programme of the school, designed to assist these black students in "bridging the gap" in their academic knowledge and experience; and in the differentiated nature of the mathematics discourse available to the Black Scholarship students within the Mainstream Programme. There is an examination of the power relations between these two discourses and other discourses within the social domain which shape the way in which these students are positioned in terms of deficit and disadvantage. Four students of the Black Scholarship Programme were interviewed in their initial year at the secondary school (Standard Six) as were the two teachers of the Academic Support Programme. The discussions were taped and transcribed and formed the basis of the analysis. Field notes were taken of discussions with academic staff within the Mathematics Department and school documentation reflecting school policies and discussions within the school were used, where relevant, in relation to the Black Scholarship students and mathematics. The methodological framework was drawn, in the main, from the work of Basil Bernstein and Paul Dowling in focusing on context, discourse and subjectivity. The study was used to interrogate previous research work in the area of Social Inequality and Mathematics Education. It also raised questions about taken-for-granted assumptions, both within the school as well as the wider community, regarding race, social class, language and cultural difference. The study attempts to investigate and bring into focus how "difference" is created and maintained, produced and reproduced within the context of the school, providing boundaries rather than bridges, and how this difference is recontextualised into disadvantage in relation to the Black Scholarship students and mathematics.

Bibliography: leaves 88-91. / This study explores the notion of "spatial ability" from the perspective of mathematics education. A review of the literature on "spatial ability" is used to compile a preliminary model of the phenomenon. Certain questions related to interaction in space arising from the literature review are noted in this model. Three aspects of this interaction are the focus of the empirical study. The results of the research are used to shed light on the preliminary model of "spatial ability". The three themes of the empirical study can be described as follows: the visualisation of "objects" from different perspectives (in "small-scale" space); the visualisation of "objects" from different perspectives (in "large-scale" space): and the representation of a three-dimensional "object" in a two-dimensional net. The results of the study suggest that a range of strategies can be used on the same "spatial" activity, and that a learner can adopt a variety of different strategies on a set of activities. Of the ten strategies identified in the study, some appear to rely on the manipulation of visual imagery, while others suggest that the manipulation process has been generalised resulting in a more "abstract" strategy. Interesting features related to the use of physical manipulation in solving spatial tasks and the communication of visual processes in the form of drawings and verbal responses are discussed. These results are used to expand the preliminary model of "spatial ability". This updated model suggests that a learner who has mastered four "abilities" and has a working knowledge of visual conventions will be able to interact successfully in the visual world. The different strategies identified in the empirical study are required during this interaction in space. The researcher also identifies areas for further research in. the field of "spatial ability" and reflects on the methodology associated with the assessment of this phenomenon. The potential value of the results of the study for use in mathematics education is discussed.

Includes abstract.|Includes bibliographical references (leaves 94-99). / This study investigates students' perceptions of their learning experiences at the Cape Peninsula University of Technology (CPUT) as well as their perceptions of their previous high school learning experiences. Eight first time entering Black township-schooled foundation Civil Engineering students were interviewed. The students indicated that they had difficulties with the medium of instruction, English. It also appears that certain teaching and learning practices at school do not prepare students for study at a tertiary institution. Social factors, such as transport and residence issues, were also named as issues influencing students' learning. An important focus of the study was on students' perceptions of group work, since the study took place during a period in which a group work intervention was conducted in the class from which the eight participants were selected. Seven of the eight participating students in the study made use of some form of group work at high school. The students had a positive disposition towards group work at school and towards the group work intervention programme at CPUT. They also had particular views of what group work is. The study also claims that students benefited from group work and that group work had a positive effect on students' performances in Mathematics. This study advocates for and contributes to a theoretical perspective on student communities, an aspect of the community perspective (Allie et al., 2007) on student learning. Group work as a form of participation that was investigated in this study was beneficial in student learning. Thus the theoretical perspective for the study, student communities, is appropriate. The study makes a contribution to the existing theoretical perspective in that it provides some insight into the school communities from which students entering higher education come; it suggests what classroom communities at tertiary level might look like; and it argues for the importance of the development of student communities outside the classroom.