This study focuses on curriculum integration for several reasons. Firstly, because there appeared to be no clear definition of integration nor a consensus on what constituted good integration. Secondly, there were few studies on integration and the type of learning involved. I believe that a study looking at an integrated unit in depth should help to shed light on what integration is and how it can be accommodated within the school system. Thirdly, an opportunity existed to observe such a unit within an established middle school. As integration is purported as being "the way" young adolescents should be taught, a middle school setting seemed ideal to me. I thought that this setting should be far enough removed from the content driven senior school to allow for its complete and uninhibited development, especially given the philosophy of this well developed middle school. Fourthly, I am interested in the potential of integration as a way of focusing on learning outcomes rather than curriculum inputs. My own theoretical perspective, with a heavy leaning toward constructivist ideas, caused me to lean towards qualitative rather than quantitative research methodologies and methods. I wanted to do justice to the study by clearly describing the social context of the school and the curriculum. Basil Bemstein's pedagogic code was seen as a way of providing the framework for the development of such a method of description. As this pedagogic code had seldom been used in a study such as this, a complete investigation of its descriptive and analytic power was seen as being of benefit to future curriculum research. The study involved two major tasks. The first task was to develop the framework to a point that it would provide a descriptive language for the recording and analysis of a school culture. / This was done by reconceptualising theories about the sociology of knowledge drawing on research by Bemstein (1971a; 1971b; 1977; 1990; 1996; 2000), Young (1971), Daniels (1987; 1989; 1995; 2001), Morais (1992) and Parker (1994) and modifying the resulting mapping tool developed to suit the complexity of the data gathered. The second task was to apply this framework to the observational data and to derive a description of the culture of the school and the micro-cultures of the two units of study observed within this school. From this description meaning was generated in the form of propositional statements about the development of an integrated unit of study within the culture of a school.

Student Number: 0111229X. PhD Thesis. Faculty of Science. School of Education / This study is part of a larger national study, the Learners’ Perspectives Study. The main focus of this particular study is to explore, describe and explain learners’ perspectives regarding the incorporation of the everyday in mathematics. Two Grade 8 mathematics classrooms in two different schools, Umhlanga and Settlers are used as sites for empirical data. Learners’ perspectives are sought through a series of post-lesson interviews with different groups of learners for lessons in which the everyday was summoned for a mathematics lesson. During these interviews, learners whether 1. They welcomed or appreciated the use of the everyday in class or not and 2. The everyday inhibited or enabled easy access to mathematics content. In order to understand the background against which these perspectives are held, mathematics lessons wherein the everyday was incorporated were observed, recorded and transcribed. In addition, teachers’ views about these lessons were explored through interviews and activities which incorporated the everyday were analysed. I used Bernstein’s notions of classification and framing as a theoretical lens through which to account for my observations. It became necessary though, to supplement these through Dowling’s domains of text analysis (esoteric, expressive, public and descriptive). I also introduced the notion of authentic/inauthentic and close/far descriptions. It is a combination of these three broad theoretical frameworks which assisted in the provision of a comprehensive theoretical account. The significance of mathematics-everyday aspect in mathematics education is highlighted by the number of studies, as discussed in the study, and different orientations from which this aspect is engaged. What can be teased out of these studies is that mathematics education debates are seldom informed by the learners’ perspectives. The study suggests that most of the learners who participated in the interviews welcomed and appreciated the use of the everyday in mathematics. However, most learners (particularly from Umhlanga) viewed mathematics as a platform to raise genuine concerns about the everyday used. In contrast, some learners (particularly from Settlers) viewed the everyday as vehicles or see-throughs towards the mathematics content. What this study viii suggests is that, firstly, the everyday is multifaceted and the nature of the context summoned tends to influence views learners hold about the role of the everyday in mathematics. Secondly, the study suggests that learners’ perspectives about the everyday cannot be divorced from the classroom context in which they encounter these everyday contexts.

Nous affinons le crible asymptotique de Bombieri afin d’obtenir un asymptotique en variables localisées. Comme conséquence, nous démontrons, sous la conjecture d’Elliott-Halberstam, qu’il existe une infinité de nombres presque premiers jumeaux, c’est à dire tels que pour tout ε > 0, p est premier et p−2 est soit premier, soit de la forme p1p2 où p1 < Xε, et nous en donnons un asymptotique. A ce travail s’ajoutent deux chapitres : d’un côté, une preuve montrant comment une méthode sans crible préliminaire donne un résultat plus faible en nécessitant une hypothèse plus forte, ce qui nous permettra de détailler plusieurs estimations et de souligner l’intérêt de notre approche. D’un autre côté une exposition pédagogique d’une méthode donnant un accès facile et explicite à plusieurs estimations de moyennes de fonctions multiplicatives. / We improve Bombieri’s asymptotic sieve to localise the variables. As a consequence, we prove, under a Elliott-Halberstam conjecture, that there exists an infinity of twins almost prime. Those are prime numbers p such that for all ε > 0, p −2 is either a prime number or can be written as p1p2 where p1 and p2 are prime and p1 < Xε, and we give the explicit asymptotic. In addition to this main work, there are two other chapters: the first one gives an asymptotic of prime numbers p such p−2is either a prime number or a product of three primes without using a preliminary sieve and so a stronger conjecture was needed. Hence this part shows the strength of the preliminary sieve and presents a few detailed sommations, most of them involving the Möbius fonction, that could be useful. The second one presents an easy and explicit method to calculate an average order of multiplicative functions.

In this Bachelor's thesis we present a few results about the absolute convergence of Fourier series, followed by an example of a differentiable function whose Fourier series does not converge absolutely. In the end we provide a suggestion for future work on generalizing the given example, and we briefly discuss an issue that has not been given much attention in the existing literature on the subject.

Soit P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ un polynôme de degré n et M:=\sup_{|z|=1}|P(z)|.$ Sans aucne restriction suplémentaire, on sait que $|P'(z)|\leq Mn$ pour $|z|\leq 1$ (inégalité de Bernstein). Si nous supposons maintenant que les zéros du polynôme $P$ sont à l'extérieur du cercle $|z|=k,$ quelle amélioration peut-on apporter à l'inégalité de Bernstein? Il est déjà connu [{\bf \ref{Mal1}}] que dans le cas où $k\geq 1$ on a $$(*) \qquad |P'(z)|\leq \frac{n}{1+k}M \qquad (|z|\leq 1),$$ qu'en est-il pour le cas où $k < 1$? Quelle est l'inégalité analogue à $(*)$ pour une fonction entière de type exponentiel $\tau ?$ D'autre part, si on suppose que $P$ a tous ses zéros dans $|z|\geq k \, \, (k\geq 1),$ quelle est l'estimation de $|P'(z)|$ sur le cercle unité, en terme des quatre premiers termes de son développement en série entière autour de l'origine. Cette thèse constitue une contribution à la théorie analytique des polynômes à la lumière de ces questions. / Let P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ a polynomial of degree n and M:=\sup_{|z|=1}|P(z)|$. Without any additional restriction, we know that $|P '(z) | \leq Mn$ for $| z | \leq 1$ (Bernstein's inequality). Now if we assume that the zeros of the polynomial $P$ are outside the circle $| z | = k$, which improvement could be made to the Bernstein inequality? It is already known [{\bf \ref{Mal1}}] that in the case where $k \geq 1$, one has$$ (*) \qquad | P '(z) | \leq \frac{n}{1 + k} M \qquad (| z | \leq 1),$$ what would it be in the case where $k < 1$? What is the analogous inequality for an entire function of exponential type $\tau$? On the other hand, if we assume that $P$ has all its zeros in $| z | \geq k \, \, (k \geq 1),$ which is the estimate of $| P '(z) |$ on the unit circle, in terms of the first four terms of its Maclaurin series expansion. This thesis comprises a contribution to the analytic theory of polynomials in the light of these problems.