This research presents a case study at a South African University, involving students who had studied mathematics in a pre-undergraduate Foundation Programme (FP) and who were currently in their first year of study in Information Technology (IT) at the same institution. The study investigated a possible relationship between the teaching approach used in the FP mathematics classroom and the extent of students’ abilities to use important mathematical practices, such as using procedures flexibly; using representation; understanding/explaining concepts; questioning; justifying claims; disagreeing; strategising; and generalising, in an undergraduate IT context. Focus group interviews and task-based interviews were used to answer three related questions: “To what extent are students aware of differences in teaching approaches between FP mathematics and undergraduate study?”; “To what extent do students believe that their experiences of the teaching approaches in the Foundation Programme mathematics class have helped them in undergraduate study in other courses?”; and “In what ways are the mathematical practices taught in the Foundation Programme used in undergraduate study in IT?” A bricolage of learning theories was used as a framework for understanding the possible relationships between teaching approach, development of mathematical practices and learning transfer. The students in the focus groups described the teaching approach used in the FP mathematics classes as studentcentred, whereas many of the undergraduate IT lectures and tutorials were described as teachercentred. The students felt that the approach used in the FP mathematics classroom was beneficial to further study, in that it taught them how to become responsible for their own learning and brought about deep understanding of the mathematical concepts learned in the FP. The task-based interviews showed that all students used mathematical practices to solve IT problems to a greater or lesser extent. The use of these mathematical practices was best understood as being influenced by all past cognitive, social and cultural experiences, and was therefore not a case of “transfer” in the traditional sense of the word. Instead, the use of mathematical practices could be described as an extreme case of “cognitive accommodation” from a cognitive constructivist perspective, or a case of “generality” from a situative perspective. Furthermore, an inter-relationship emerged between student-centred teaching, students’ productive disposition towards mathematics, and the extent of “transfer” of mathematical practices to the IT domain. This interesting relationship warrants further investigation.

Graphing is a fundamental topic in algebra that is notoriously difficult for students. Much of the past research has focused on conceptions and misconceptions. This study extends past research by looking at the mathematical practices of a practitioner, specifically one instructor of a function-based covariation-focused algebra class in the linear functions unit. Considering practices in addition to conception adds dramatically to our understanding of mathematical activity because it leads to explicit descriptions of normative purposes that are connected to particular situations or problems and also specifies how tools and symbols are coordinated to achieve these purposes. The results of this study are three levels of empirically proven practices associated with the conception of one advanced level of covariational reasoning, chunky continuous covariation. This study not only describes how practices may be described at different levels of complexity, but also demonstrates how smaller practices may be combined to form larger, more complex practices. These practices can be used to guide instruction of those who want to participate in and become practitioners in the community of teachers of function-based covariation-focused algebra curricula.

A research report submitted to the WITS SCHOOL OF EDUCATION, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Master of Science (Science Education). Johannesburg 2015 / Using qualitative methods, this study reports on the selection and use examples in Probability by a teacher in a multilingual mathematics classroom where learners learn in a language which is not their first or home language. The study involved one teacher together with his Grade 11 multilingual class in a township school in Ekurhuleni South Johannesburg. Data was collected through audio-visual recording of four lessons. In addition two one-on-one semi-structured interviews were conducted with the teacher. Data was analysed using Rowland‘s (2008) categories of exemplification alongside Staples' (2007) conceptual model of collaborative inquiry mathematics practices. In the study it emerged that it is important for teachers to select examples by considering the context, ability of the example to be generalised, consistency in the use of symbols, syllabus requirements and accessibility. It also emerged that the selection of examples together with the accompanying mathematical practices has the potential to support or impede the learning of mathematics. In particular the findings revealed that the practice of ‗guiding the learners with the map‘ declines the cognitive level of examples and hence impedes learning. Code- switching and re-voicing were most frequently used practices seen in the findings with the use of code-switching encouraging full participation of the learners. The study recommends that methodology courses offered at tertiary institutions to pre-service teachers should include the selection, how to select or design and use examples in multilingual classrooms e.g. what constitutes a good example and how to maintain the cognitive level of an example. The study also recommends that more research needs to be done on effective mathematical practices that may be used to implement worked-out examples in multilingual classrooms.

Students do not always evaluate explanations based on the mathematics despite their teacher's effort to be the guide-on-the-side and delegate evaluation to the students. This case study examined how the use of three features of the Discourse—authority, sociomathematical norms, and classroom mathematical practices—impacted students' evaluation and contributed to students' failure to evaluate. By studying three pre-service elementary school students' evaluation methods, it was found that the students applied different types of each of the features of the Discourse and employed them at different times. The way that the features of the Discourse were used contributed to some of the difficulties that the participants experienced in their evaluation of explanations. The results suggest that researchers in the field must come to believe that resistance to teaching methods is not the only reason for student failure to evaluate mathematical explanations and that authority is operating in the classroom even when the teacher is acting as the guide on the side. The framework developed for the study will be valuable for researchers who continue to use for their investigation of individual student's participation in mathematical activity.

Une étude récente auprès de 302 mathématiciens canadiens révèle un écart intriguant : tandis que 43% des sondés utilisent la programmation informatique dans leur recherche, seulement 18% indiquent qu'ils emploient cette technologie dans leur enseignement (Buteau et coll., 2014). La première donnée reflète le potentiel énorme qu'a la programmation pour faire et apprendre des mathématiques. La deuxième donnée a inspiré ce mémoire : pourquoi existe-t-il un tel écart ? Pour répondre à cette question, nous avons mené une étude exploratoire qui cherche à mieux comprendre la place de la programmation dans la recherche et la formation en mathématiques au niveau universitaire. Des entrevues semi-dirigées ont été conduites avec 14 mathématiciens travaillant dans des domaines variés et à différentes universités à travers le pays. Notre analyse qualitative nous permet de décrire les façons dont ces mathématiciens construisent des programmes informatiques afin d'accomplir plusieurs tâches (p.e., simuler des phénomènes réels, faire des mathématiques « expérimentales », développer de nouveaux outils puissants). Elle nous permet également d'identifier des moments où les mathématiciens exposent leurs étudiants à certains éléments de ces pratiques en recherche. Nous notons toutefois que les étudiants sont rarement invités à concevoir et à écrire leurs propres programmes. Enfin, nos participants évoquent plusieurs contraintes institutionnelles : le curriculum, la culture départementale, les ressources humaines, les traditions en mathématiques, etc. Quelques-unes de ces contraintes, qui semblent limiter l'expérience mathématique des étudiants de premier cycle, pourraient être revues. / A recent survey of 302 Canadian mathematicians points to an intriguing gap: while 43% of the participants use computer programming in their research, only 18% indicate that they use such technology in their teaching (Buteau et al., 2014). The first statistic reflects the enormous potential that programming has for doing and learning mathematics. The second served as the inspiration for our research: why would such a gap exist? In response to this question, we put forth an exploratory study aimed at better understanding the place of programming in mathematical research and university mathematics education. Semi-directed interviews were conducted with 14 mathematicians working within various mathematical subfields at different universities across Canada. Our qualitative analysis allows us to describe the ways in which these mathematicians construct computer programs in order to accomplish several tasks (e.g., simulating real-world phenomena, doing "experimental" mathematics, developing new powerful tools). It also allows us to identify some moments where the mathematicians expose their students to certain elements of these research practices. We notice, however, that the students are rarely invited to conceptualize and write their own programs. In the end, our participants highlight several institutional constraints: the curriculum, departmental culture, human resources, the traditions in mathematics, etc. Some of these constraints, which seem to be limiting the mathematical experience of some undergraduate students, could warrant re-examination.

In this thesis, focused attention is given to the idea of task solvers as active participants in mathematical practices. The theoretical assumptions of the study, reported in this thesis, are inspired by socio-political concerns. The aim of the study is to investigate the underlying view of participation in mathematical practices, as understood in a nationwide teacher professional development programme, the Boost for Mathematics, in Sweden. To be more precise, the study is arranged to problematise ways of encouraging students as active participants. This aim is approached by means of the following research questions: (1) What messages do mathematical tasks in the Boost for Mathematics send about people as participants in mathematical practices? and (2) What is the role of multiple representations in these messages? An empirical study is reported. The data of the study, i.e., three collections of problems, are drawn from the Boost for Mathematics. Data processing is conducted by using a modified version of a pre-existing data processing framework, focusing on mathematical practices as socio-political practices. The empirical study uncovers an implicit view of task solvers in mathematical practices and especially a detachment between students, as potential task solvers, and the social contexts where mathematical ideas and concepts are embedded. This implicit view is challenged from the assumption that it is motivating for a student to conceive him/herself as someone who is ‘qualified’ to take part in mathematical practices.

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Made available in DSpace on 2016-06-02T19:39:46Z (GMT). No. of bitstreams: 1 6037.pdf: 968767 bytes, checksum: 8fb7f515768cae960495efe96022b1da (MD5) Previous issue date: 2014-02-12 / This research has the purpose to analyze the influence of situations in order to deal with mathematics in different social practices. As a specific goal, it investigates the meanings in various school and non-school mathematical practices. Among these purposes there is an analysis about the switch of meanings between one and another practice. The development of this work is based on a review of studies about the exploitation of Mathematics on the day to day problems and in other areas of knowledge that could contribute to the learning of mathematics as a school subject. Considering the idea of mathematics as a social practice, the theoretical framework of the present research has centered on the design of means of structuring and situated learning by Jean Lave. The research follows a naturalistic perspective with ethnographicins piration and uses as a methodological resource the participant observation with a group of elementary school students. The data collected were recorded by the researcher in field activities, through interviews, diaries and field recordings. The activities observed occur inside and outside the school. For the analysis it is considered the resource association between the object and the theoretical framework consisting drawn from Lave s studies. Concerning the results obtained, it is possible to realize the strength of the situation and sometimes, how crucial it is in order to practice math. It stands out especially the prevalence of different meanings in different practices. The present study also promotes questions about the proposal to take the students everyday situations to inside the classroom and therefore, it intermediates my work as a mathematics teacher at elementary schools. / Esta pesquisa tem por objetivo geral analisar a influência das situações no modo de lidar com a matemática em diferentes práticas sociais. Como propósito específico, busca investigar os significados em diferentes práticas matemáticas escolares e não escolares. Destes propósitos decorre uma análise da transferência de significados entre uma prática e outra. O desenvolvimento deste trabalho se apoia em uma revisão bibliográfica de estudos sobre como a exploração da matemática nos problemas do dia a dia e nas demais áreas do conhecimento poderiam contribuir para o aprendizado da matemática escolar. Partindo da ideia da matemática como prática social, a referência teórica da pesquisa tem como eixo central a concepção de meios de estruturação e aprendizagem situada de Jean Lave. A pesquisa segue uma perspectiva naturalística com inspiração etnográfica e usa como recurso metodológico a observação participante com um grupo de estudantes do ensino fundamental. Os dados foram constituídos pela pesquisadora em atividades de campo, por meio de entrevistas, diários de campo e gravações. As atividades observadas ocorrem dentro e fora da escola. Para a análise é considerado o recurso de associação entre o objeto constituído e o referencial teórico elaborado a partir dos estudos de Lave. Dos resultados obtidos, é possível perceber a força da situação e, por vezes, como ela é determinante no modo de se praticar matemática. Destaca-se, sobretudo a prevalência de diferentes significados em práticas distintas. O presente estudo também promove questionamentos acerca da proposta de se levar as situações do cotidiano do aluno para a sala de aula e com isso, intervém na minha atuação como professora de matemática do ensino fundamental.

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