This thesis consists of a theoretical building of a cognitive approach to the calculus and an empirical testing of the theory in the classroom. A cognitive approach to the teaching of a knowledge domain is defined to be one that aims to make the material potentially meaningful at every stage (in the sense of Ausubel). As a resource in such an approach, the notion of a generic organiser is introduced (after Dienes), which is an environment enabling the learner to explore examples of mathematical processes and concepts, providing cognitive experience to assist in the abstraction of higher order concepts embodied by the organiser. This allows the learner to build and test concepts in a mode 1 environment (in the sense of Skemp) rather than the more abstract modes of thinking typical in higher mathematics. The major hypothesis of the thesis is that appropriately designed generic organisers, supported by an appropriate learning environment, are able to provide students with global gestalts for mathematical processes and concepts at an earlier stage than occurs with current teaching methods. The building of the theory involves an in-depth study of cognitive development, of the cultural growth and theoretical content of the mathematics, followed by the design and programming of appropriate organisers for the teaching of the calculus. Generic organisers were designed for differentiation (gradient of a graph), integration (area), and differential equations, to be coherent ends in themselves as well as laying foundations for the formal theories of both standard and non-standard analysis. The testing is concerned with the program GRADIENT, which is designed to give a global gestalt of the dynamic concept of the gradient of a graph. Three experimental classes (one taught by the researcher in conjunction with the regular class teacher) used the software as an adjunct to the normal study of the calculus and five other classes acted as controls. Matched pairs were selected on a pre-test for the purpose of statistical comparison of performance on the post-test. Data was also collected from a third school where the organisers functioned less well, and from university mathematics students who had not used a computer. The generic organiser GRADIENT, supported by appropriate teaching, enabled the experimental students to gain a global gestalt of the gradient concept. They were able to sketch derivatives. for given graphs significantly better than the controls on the post-test, at a level comparable with more able students reading mathematics at university. Their conceptualizations of gradient and tangent transferred to a new situation involving functions given by different formulae on either side of the point in question, performing significantly better than the control students and at least as well, or better, than those at university.

In this study, the reading experience of six young successful boy readers is examined with a view to identifying ways in which the reading achievements of all boys might be raised. Initially, the experiences and behaviours associated with young successful readers are identified, including aspects from the home and the school, and those characteristics from within children themselves. Next, literature on boys' reading is examined, and this shows that there are many negative influences on the reading lives of boys generally. The reading experience of the six young successful boy readers is then investigated through empirical work. The central approach adopted is multiple case study using ethnographic tools. The six boys were reading fluently and for pleasure by the end of their Reception year (aged 5 years) and were studied for a two year period. Observation and research conversation were the main data collection methods adopted; the boys' experience as young successful readers was examined by observing them in their homes and schools, and by talking to them, their parents and their teachers. The results illustrate that the six young boys who are successful readers have a masculine identity in which reading has a secure and positive place. They have overcome the negative influences which frequently impact on the reading experience of boys and have successfully integrated ways of being a boy and being a reader. The boys' reading is highly developed at home by living in a 'reading family'. The boys use their advanced achievement in reading to gain a high status position in the classroom; their reading behaviour makes them popular and powerful with their peers. Hence these boys make reading work for them and, subsequently, it is a desirable feature of their developing masculine identity. These results are reflected upon to identify ways in which the reading achievements of all boys might be raised. I have suggested that schools might be encouraged to develop their reading curriculum in a number of ways, including spending more time reading extended texts for pleasure and using high status texts from boys' vernacular reading in the formal reading curriculum. In addition, I argue that all boys might benefit from examining the gender assumptions on which texts and their own reading preferences are based.

Attention to how groups of students at university influence an individual learner's motivation within the group is the focus of this research. The uniqueness of this research lies in shifting the focus from an analysis of the individual's experience seen as being apart from the group to considering the individual's experience in relation to the social interactions within the group. This thesis begins with the examinations of the theoretical framework, including major issues in learning motivation and group dynamics, an area that has been gaining more and more attention in second language research. Then, it discusses the selection of a mixed methods approach, the employment of three research instruments (the classroom observation, the questionnaire, and the interview), and the research procedure. After presenting the findings from each research instrument, this study will integrate all the data and present key findings from the integration. Questionnaires were administered to 127 Taiwan university students from the Applied English Department of National Kaohsiung First University of Science and Technology (NKFUST). The results from the questionnaires show that there is a slight to moderate correlation between group processes (group cohesiveness and group norms) and students' level of motivation (self-efficacy and level of autonomy). A dozen students who participated in this study were asked to give further information during semi-structured in-depth interviews. During those interviews, several students commented that their classmates are indeed important to their learning, as being around more motivated classmates positively influences their own motivation and autonomy. Other relevant findings, such as what is a 'good' and 'bad' group, the importance of a mixed methods approach, and the role of culture aspects, will also be discussed.

This research study investigates the concept of function developed by a sample of secondary and university students in England and Malaysia studying mathematics as one of their subjects. It shows that whilst students may be able to do the 'mechanical' parts of this concept, their grasp of the 'theoretical' nature of the function concept may be tenuous and inconsistent. The hypothesis is that students develop 'prototypes' for the function concept in much the same way as they develop prototypes for concepts in everyday life. The definition of the function concept, though given in the curriculum, proves to be inoperative, with their understanding of the concept reliant on properties of familiar prototype examples: those having regular shaped graphs, such as x2 or sinx, those often encountered (possibly erroneously), such as a circle, those in which y is defined as an explicit formula in x, and so on. The results of the study in England revealed that even when the function concept was taught through the formal definition, the experiences which followed led to various prototypical conceptions. Investigations also show significant misconceptions. For example, three-quarters of a sample of students starting a university mathematics course considered that a constant function was not a function in either its graphical or algebraic forms, and three quarters thought that a circle is a function. The extension of the study in Malaysia was made with the hypothesis that there is a significant difference between the concept as perceived to be taught and as actually learned by the students. Although the intended curriculum emphasises conceptual understanding, in the perceived curriculum (curriculum as understood by the teachers), only 45% of the teachers follow this approach. The tested curriculum as reflected in the public examination questions, only emphasises the procedural skills and the results of the learned curriculum show that learning of functions is more consistent with the theory of prototypical learning. Students in Malaysia develop their own idiosyncratic mental prototypes for the function concept in much the same way as those students in the UK.

The fragmentation of strategies that distinguishes the more successful elementary grade students from those least successful has been documented previously. This study investigated whether this phenomenon of divergence and fragmentation of strategies would occur among undergraduate students enrolled in a remedial algebra course. Twenty-six undergraduate students enrolled in a remedial algebra course used a reform curriculum, with the concept of function as an organizing lens and graphing calculators during the 1997 fall semester. These students could be characterized as "victims of the proceptual divide," constrained by inflexible strategies and by prior procedural learning and/or teaching. In addition to investigating whether divergence and fragmentation of strategies would occur among a population assumed to be relatively homogeneous, the other major focus of this study was to investigate whether students who are more successful construct, organize, and restructure knowledge in ways that are qualitatively different from the processes utilized by those who are least successful. It was assumed that, though these cognitive structures are not directly knowable, it would be possible to document the ways in which students construct knowledge and reorganize their existing cognitive structures. Data reported in this study were interpreted within a multi-dimensional framework based on cognitive, sociocultural, and biological theories of conceptual development, using selected insights representative of the overall results of the broad data collection. In an effort to minimize the extent of researcher inferences concerning cognitive processes and to support the validity of the findings, several types of triangulation were used, including data, method, and theoretical triangulation. Profiles of the students characterized as most successful and least successful were developed.Analyses of the triangulated data revealed a divergence in performance and qualitatively different strategies used by students who were most successful compared with students who were least successful. The most successful students demonstrated significant improvement and growth in their ability to think flexibly to interpret ambiguous notation, switch their train of thought from a direct process to the reverse process, and to translate among various representations. They also curtailed their reasoning in a relatively short Period of time. Students who were least successful showed little, if any, improvement during the semester. They demonstrated less flexible strategies, few changes in attitudes, and almost no difference in their choice of tools. Despite many opportunities for additional practice, the least successful were unable to reconstruct previously learned inappropriate schemas. Students' concept maps and schematic diagrams of those maps revealed that most successful students organized the bits and pieces of new knowledge into a basic cognitive structure that remained relatively stable over time. New knowledge was assimilated into or added onto this basic structure, which gradually increased in complexity and richness. Students who are least successful constructed cognitive structures which were subsequently replaced by new, differently organized structures which lacked complexity and essential linkages to other related concepts and procedures. The bits and pieces of knowledge previously assembled were generally discarded and replaced with new bits and pieces in a new, differently organized structure.

This thesis is concerned with students' understanding of the core concept of function which cannot be represented by what is commonly called the multiple representations of functions. The function topic is taught to be the central idea of the whole of mathematics. In that sense, it is a model of mathematical simplicity. At the same time it has a richness and has mathematical complexity. Because of this nature, for students it is so difficult to grasp. The complexity of the function concept reveals itself as cognitive complications for weak students. This thesis investigates why the function concept is so difficult for students. In the Turkish context, students in high school are introduced to a colloquial definition and are presented with four different aspects of functions, set-correspondence diagrams, sets of ordered pairs, graphs and expressions. The coherency in recognizing these different aspects of functions by focusing on the definitional properties is considered as an indication of an understanding of the core concept of function. Focusing on a sample of a hundred and fourteen students, their responses in the questionnaires are considered to select nine students for individual interviews. The responses from these nine students in the interviews are categorized as they deal with different aspects of functions. The data indicates that there is a spectrum of performance of students. In this spectrum, responses range from the responses which handle the flexibility of the mathematical simplicity and complexity to the responses which are cognitively complicated. Successful students could focus on the definitional properties by using the colloquial definition for all different aspects of functions. Less successful students could use the colloquial definition for only set-correspondence diagrams and sets of ordered pairs and gave complicated responses for the graphs and expressions. Weaker students could not focus on the definitional properties for any aspect of functions.

AIM To investigate mathematically able adolescents' conceptions of the basic notions behind the Calculus: infinity (including the infinitely large, the infinitely small and infinite aggregates); limits (of sequences, series and functions); and real numbers. To observe the effect, if any, on these conceptions, of a one year calculus course. EXPERIMENTS Pilot interviews and questionnaires helped identify areas on which to focus the study. A questionnaire was administered to Lower Sixth Form students with 0-level mathematics passes. The questionnaire was administered twice, once in September and again the following May. The A-level mathematicians had received instruction in most of the techniques of the Calculus by May. Interviews, to clarify ambiguities, elicit reasoning behind the responses and probe typicality and atypicality, were conducted in the month following each administration. A second questionnaire, an amended version of the first, was administered to a larger but similar audience. The responses were analysed in the light of hypotheses formulated in the analysis of data from the first 5ample. PRINCIPAL FINDINGS Subjects have a concept of infinity. It exists mainly as a process, anything that goes on and on. It may exist as an object, as a large number or the cardinality of a set, but in these forms it is a vague and indeterminate form. The concept of infinity is inherently contradictory and labile. Recurring decimals are perceived as dynamic, not static, entities and are not proper numbers. Similar attitudes exist towards infinitesimals when they are seen to exist. Subjects' conception of the continuum do not conform to classical or nonstandard paradigms. Convergence / divergence properties are generally noted with infinite sequences and functions. With infinite series, however, convergence / divergence properties, when observed, are seen as secondary to the fact that any infinite series goes on indefinitely and is thus similar to any other infinite series. The concept that the hut is the saue type of entitiy as the finite tens is strong in subjects' thoughts. We coin the term generic hiuit for this phenomenon. The generic limit of 0.9, 0.99, is 0.9, not 1. Similarly the reasoning scheme that whatever holds for the finite holds for the infinite has widespread application. We coin the term generic law for this scheme. Many of the phrases used in calculus courses (in particular hut, tends to, approaches and converges) have everyday meanings that conflict with their mathematical definitions. Numeric/geometric, counting/measuring and static/dynamic contextual influences were observed in some areas. The first year of a calculus course has a negligible effect on students conceptions of limits, infinity and real numbers. IMPLICATIONS FOR TEACHING On introducing limits teachers should encourage full class discussion to ensure that potential cognitive obstacles are brought out into the open. Teachers should take great care that their use of language is understood. A-level courses should devote more of their time to studying the continuum. Nonstandard analysis is an unsuitable tool for introducing elementary calculus.

This thesis has two integrated components, one theoretical and one investigative. The theoretical component considers human reason about categories of objects. First, it proposes that the standards of argumentation in everyday life are variable, with emphasis on direct generalisation, whereas standards in mathematics are more fixed and require abstraction of properties. Second, it accounts for the difficulty of the transition to university mathematics by considering the impact of choosing formal definitions upon the nature of categories and argumentation. Through this it unifies established theories and observations regarding student behaviours at this level. Finally, it addresses the question of why Analysis seems particularly difficult, by considering the relative accessibility of its visual representations and its formal definitions. The investigative component is centred on a qualitative study, the main element of which is a series of interviews with students attending two different first courses in Real Analysis. One of these courses is a standard lecture course, the other involves a classroom-based, problem-solving approach. Grounded theory data analysis methods are used to interpret the data, identifying behaviours exhibited when students reason about specific objects and whole categories. These behaviours are linked to types of understanding as distinguished in the mathematics education literature. The student's visual or nonvisual reasoning style and their sense of authority, whether "internal" or "external" are identified as causal factors in the types of understanding a student develops. The course attended appears as an intervening factor. A substantive theory is developed to explain the contributions of these factors. This leads to improvement of the theory developed in the theoretical component. Finally, the study is reviewed and the implications of its findings for the teaching and learning of mathematics at this level are considered.

This study is an investigation of the potential of teacher-initiated action research for EFL/ESP teacher development and classroom innovation. The Collaborative Academic Writing Research Project (CAWRP), on which it is based, was carried out at the ESP Centre, Damascus University, in 1996-1997. It was in two phases, Baseline and Main. The researcher, a teacher in the context, assumed a participatory and facilitating role. The pedagogic problem was the teaching of research paper writing to postgraduate students. The CAWRP was proposed to ease this problem and introduce classroom innovation through teacher-initiated action research, the long-term aim of which was continuous professional development. The baseline research aimed at articulating a picture of teacher and context needs and assessing project viability. The proposal was refined in the light of the findings, and a programme of teacher development activities was agreed with the participants. This was implemented in the Main Phase, which had three stages: Orientation, Research and Reporting, and Summative Evaluation and Follow-up. The role of the researcher was to facilitate the teachers to self-direct their professional learning and introduce needed pedagogic innovations. The thesis is in eight chapters and 32 appendices. Chapter One sets the scene and introduces the study. Chapter Two focuses on the baseline investigation: its methodology, findings, and their implications for the Main Phase study. Chapter Three is a review of the relevant literature in the fields of teacher development and classroom innovation. Chapter Four focuses on project design and methodology and gives more details on the principles, values, strategies, and procedures that guided project implementation and how they worked out in action. Chapter Five reports the findings, focusing on the contribution of the Orientation Stage activities to the development of the teacher group as a whole (a total of 20 out of 23 Centre teachers). Its main sources of data are recordings, feedback questionnaires, and participant observation. Chapter Six focuses on the teachers who carried out action research and reported on it (8 out of the 20 Orientation Stage participants). It presents two case studies of frill participants, starting with their entry points and showing how they developed in the Research and Reporting Stage. One case exemplifies the experienced teachers and those who did research individually, and the other the novices and those who worked in collaboration. Chapter Seven reports on the participants' sununative evaluation of the project and the effect of this evaluation on project continuity. Chapter Eight summarises the main findings and evaluates them with reference to the literature, on the one hand, and design principles and methodology, on the other. In this chapter, I have looked critically at the lessons learnt from the study, discussed its significance and limitations, and put forward some recommendations. The appendices include some of the materials and documentary evidence used in the research.

The purpose of this research was threefold: 1) to study peer interactions at the computer; 2) to study the interactions of the teachers with the children and analyse the impact of those interactions on the children’s social behaviour; 3) to explore teacher’s beliefs, views and knowledge about computers use in four Malaysian pre-schools.  Qualitative research was employed using a multiple case study approach based on constructivism.  Data were obtained primarily through non-participant observations, interviews with principals, teachers and children and documentations.  The results indicated that children exhibited a considerable amount of positive social interaction behaviour while playing with the computer.  For the purpose of the study the researcher only concentrated on a specific set of behaviour using the Children Social Interactions Behaviour checklist. This study indicates that computer activities with teacher facilitation helped promote social interaction between children.  Teachers who observe carefully are likely to see that children who may seem to be minimally and only superficially involved with their peers at the computer are actually learning by carefully observing their peers’ success navigating developmentally appropriate software.  Computer experts are valuable assets to the classroom.  Computer centres that encourage collaboration and cooperation make important contributions to the development of a community of learners and children growing cognitive, fine motor, and social competencies.  However, this will not necessarily happen just because the computer is present in the early childhood education setting.  Teachers must be conscious of the kinds of learning interactions they would like to occur in the context of computer use (including between adults and children, or between children), and adopt pedagogical strategies to support these situations. From the case studies, it is also clear that the role of adults is vital in the computer centre and two types of teacher behaviour occurred.  One was a passive role in which the teacher was only involved in computer use when she was asked by the children.  The other type of teacher behaviour was in an active role that included being a technical helper, a conflict mediator and a tutor. This study also carries several serious implications for teachers and parents.  First and most important, it highlights the rich social environment offered by computer usage.  Second, gender variation was seen in the frequency of computer use, with boys using the computer more than girls.  Third, given the right directions, children are capable of resolving their own conflicts. Teachers, therefore, need to be careful about when they intervene and how much help they offer.