This study In an attempt to promote equal access to education, we in South Africa, have adopted an instrumentalist approach to the debate of the right to education. In other words, we have provided an enabling legal framework and we simply assume that access to education has been granted to every one. We continue to pretend that we understand what exactly the concept of “access to education” means. We also assume that we all have a common understanding of what the Constitution means by the right to education. On 26 June 1955 the historic Freedom Charter of the African National Congress (ANC) was adopted. This charter declared “the doors of learning and culture shall be opened.” Over the next four decades, the demand for open and equal access to education became central platform in the anti-apartheid struggles that brought an ANC-led government to power in 1994. Yet, ten years later (2004) the problem of access continues to preoccupy education planners and activists against the backdrop of some of the most progressive policy positions including a Constitution that recognizes education as a basic right. The intellectual puzzle that motivates this study is to explain, therefore, why despite its prominence, it continues to be regarded as an intractable problem. The research strategy adopted in pursuit of this puzzle is to trace the changing meanings of the concept of “access to education” under and after apartheid, and its expression in the practices of two case study schools (comparative case studies). Data was collected from different sources to trace the concept of access to education in education from the apartheid era to the policies and practices that affirm access to education as a basic human right today. This study hopes to contribute significantly to the dialogue of “access” as a realisation of the basic right to education. For the conceptual framework Morrow’s distinction between epistemological access and physical access was used. Formal access to education refers to enrolment or registration at an education institution, in this case, a school, whereas epistemological access refers to access to knowledge and information that these education institutions hold. I expanded the conceptual framework to include dimensions of epistemological access such as how the topic is taught, who selected the topic, the value and political basis. I undertook documentary analysis and a series of interviews with individuals who were involved in the struggle either through intellectual contributions in the NEC and NEPI processes or in the National Education Co-ordinating Committee. I also conducted two case studies of schools located in vastly different social and political contexts. At these schools, I collected data through classroom and school observations, semi-structured interviews with principals, history teachers and learners. Findings: The first finding of this study is that the ways in which students experience access to knowledge (epistemological access) is strongly dependent on the history and politics of the school context and the institutional culture, rather than the formal prescriptions laid down in the school curriculum The second finding of this study is that even when students enjoy physical access to schools, they have highly uneven, even unequal, access to knowledge within those schools. The third finding is that despite the awareness and understanding of what good education entails, without physical access, it is difficult for individuals to entertain discussions about epistemological access. The fourth finding is that despite claims that the policy promotes increased access to education, it was not possible to find reports that refer to any significant degree of quality outcomes as a result of the implementation of the principle of “equality of access” to education. Increased access to education has not resulted in quality output. This dissertation contributes to knowledge by its nuanced exploration of the complexities of access to education as a human right. Most importantly it pushes the boundaries of knowledge pertaining to both physical and epistemological access at the time when each of these are crucial points in the education development agenda. / Thesis (PhD (Education Management and Policy Studies))--University of Pretoria, 2006. / Education Management and Policy Studies / unrestricted

The purpose of this study is to compare and contrast the motivations for reading in the subject syllabi for English in England and Sweden in the hopes of discovering how the English steering documents can inspire Swedish teachers to adopt new perspectives and approaches to reading. The aspects explored are epistemological approaches, reading content, general aims and aims within the subject, skills, requirements, discourses, and language usage. The study analyzes each of the subject syllabi from both countries and the final phase involves a comparison, which demonstrates the profitable and non-profitable similarities and dissimilarities. The study provides valuable perspectives on how teachers can broaden their interpretation of Swedish steering documents for the subject of English by departing from critical reading concepts and using a more encouraging discourse than that found in steering documents.

A tese propõe-se a analisar e a interpretar a representação social do usuário sobre a assistência social. Para tanto, foi utilizada a metodologia dos círculos epistemológicos, inspirados na teoria pedagógica dos círculos de cultura de Paulo Freire, que procura superar a dicotomia entre pesquisador-pesquisando. Como nos círculos de cultura, os pesquisandos participam ativamente do processo de pesquisa. Foram realizados nove círculos epistemológicos com os usuários, em três Centros de Referência de Assistência Social – CRAS de Porto Alegre RS, três em cada CRAS. Após a realização dos nove círculos epistemológicos, todas as informações foram transcritas, lidas, categorizadas a partir de sua dimensão semântica e analisadas à luz da Teoria das Representações Sociais. Da análise dos dados, foram construídos mapas representacionais com as categorias e subcategorias que revelaram os elementos centrais que constituem tal representação. Os resultados foram interpretados a partir de aportes teóricos de autores que trabalharam temáticas ligadas à assistência social. As contribuições dos usuários pesquisandos foram levadas em consideração na produção final da tese. A tese compõe-se de quatro capítulos organizados em forma de artigo. O primeiro apresenta o referencial teórico das representações sociais e discute a metodologia dos círculos epistemológicos mostrando as conexões entre eles. O segundo artigo, também teórico, trata da formação do Estado brasileiro e como sua estruturação influenciou as políticas de assistência social. O terceiro apresenta os resultados empíricos da representação social da assistência social na perspectiva dos usuários. Tal representação apresenta-se como contraditória revelando duas dimensões centrais, uma ideológica e outra crítica. No quarto e último artigo, também empírico, é enfatizada uma dimensão específica da representação social que se destacou das informações colhidas e que se julgou oportuno aprofundá-la: a representação social que os usuários revelaram sobre si mesmos. Tal representação revelou duas dimensões contraditórias, mas inter-relacionadas: por um lado os usuários discriminam-se como “bons” e “maus”; por outro, se reconhecem como iguais pertencentes a um mesmo grupo social e deixam entrever sinais de solidariedade, que oferece condições para a emergência de uma cidadania para além dos direitos. / The dissertation proposes to analyze and interpret the social representation of the user about social assistance. The methodology of epistemological circles inspired by the educational theory of cultural circles of Paulo Freire was used, which seeks to overcome the dichotomy between researcher and research subject. As in the circles of culture, the research subjects actively participate in the research process. Nine epistemological circles were conducted with users in three reference centers of Social Assistance-CRAS of Porto Alegre RS, three in each CRAS. After the completion of the nine epistemological circles , all the information was transcribed, read, categorized from its semantic dimension and analyzed in the light of the theory of social representations. From the data analysis, representational maps were constructed with the categories and subcategories which revealed the central elements that constitute such representation. The results were interpreted from theoretical contributions of authors who worked on welfare-related themes. Research subject users ' contributions were taken into consideration in the production end of the dissertation. The dissertation consists of four chapters arranged in article form. The first presents the theoretical framework of social representations and discusses the methodology of the epistemological circles demonstrating the connections between them. The second article, also theoretical, deals with the formation of the Brazilian State and how its structure influenced the policies of social assistance. The third presents the empirical results of social representation of social assistance from the perspective of the users. Such representation is presented as contradictory revealing two central dimensions, an ideological one and another critical one. In the fourth and last article, also empirical, emphasized a specific dimension of social representation that stood out from the information collected which was deemed appropriate to deepen into: social representation that users reveal about themselves. Such representation revealed two contradictory but interrelated dimensions: on the one hand users discriminate as "good" and "bad"; on the other, recognize each other as equals belonging to the same social group and they see signs of solidarity, which provides conditions for the emergence of a citizenship beyond rights.

Vilken historisk kunskap räknas som värdefull kunskap när det kommer till skolsammanhang och läroplansteori? Hur tolkar och använder skolans personal dessa kunskapsformuleringar i läroplanen i sin undervisning? Förra hösten presenterade Skolverket sina revideringsförslag av den nuvarande läroplanen: Läroplan för grundskolan, förskoleklassen och fritidshemmet 2011, reviderad 2019. Det första syftet med denna studie var att undersöka vad personalen i skolan ansåg om dessa revideringar och det tilltänkta implementeringsarbete de skulle innebära, med huvudfokus på historieämnet. Det andra syftet var att jämföra revideringsförslagen med den nuvarande läroplanen ur ett läroplansteoretiskt perspektiv, för att markera eventuella förskjutningar gällande kunskapssynen dem emellan. I denna studie användes kvalitativa intervjuer med skolpersonal samt kvalitativ textanalys av kursplanen i historia som metoder för att besvara studiens forskningsfrågor. Resultatet visade att skolpersonalen var väldigt positiva och trodde att det avsedda syftet med revideringen, att den skulle fungera som ett ännu bättre stöd och verktyg för dem i deras arbete, skulle uppnås. Resultatet visade också att för att utvecklas framåt har läroplanens formuleringar tagit ett steg tillbaka till att återigen betona begreppen kunskap om och förståelse för i flera skolämnen som exempelvis historia. I vilken utsträckning Skolverkets revideringsförslag når sin målsättning återstår dock att se. / What kind of knowledge about history counts as valuable knowledge when it comes to school context and curriculum theory? How does school staff interpret and transform these writings of knowledge and the curriculum framework into their teaching? Last fall, The Swedish National Agency for Education presented their revised version of the current curriculum: Curriculum for the compulsory school, preschool class and the leisure-time centre (2011). The first purpose of this study was to investigate what school staff thought about this revised version and the implementation work that would follow, in particularly when it came to the syllabus about the history subject. The second purpose was to compare the revised curriculum with the current one from a curriculum theoretic perspective, in order to highlight potential differences between how they visualize their epistemic beliefs. For this study, I used qualitative interviews with school staff as well as a qualitative analyse of the writings in the syllabus for history as methods to answer the both issues of this study. The results show that the school staff were very positive and believed that the intended purpose for the revised version, to work as an even better tool and support for them in their work, would be achieved. The results also show that in order to move forward, the writings in the curriculum has taken a step back in development, to once again enhancing the terms knowledge- and understanding about in several subjects such as history. In what extent this revision will reach its aim is not yet evaluated.

Philosophiae Doctor - PhD / This study was about the provision of epistemological access to Grade 10 to 12 learners in the business-related subjects of Accounting, Business Studies and Economics at two public high schools in the Western Cape Province. The aim of the study was twofold. Firstly, it investigated whether the Grade 10 to 12 learners had adequate subject content knowledge and skills in Accounting, Business Studies and Economics. Secondly, it wanted to identify the challenges that the learners at the two high schools were experiencing in these subjects and the reasons why.

The purpose of this study was to identify essential features of Epistemic Orientation toward Teaching Science (EOTS) and to explore the relationships between EOTS and instructional practices. This study proposes a new concept, EOTS: defined as a teacher's set of interrelated beliefs that are developed and used when teaching science, and are shaped by the Nature of Knowing in General, the Nature of Knowing in Science, the Nature of Learning, and the Nature of Teaching. The essential elements of EOTS were identified through a comprehensive literature review and refined through a multiple-case study. The participants of the study were three exemplary fifth grade teachers who had been implementing an Argument-based Inquiry (ABI) approach, called Science Writing Heuristic (SWH), for more than three years and were highly devoted to encouraging their students to engage in science practices addressed in Next Generation Science Standard. Data were collected from multiple sources including semi-structured interviews, Video-Stimulated Recall interviews, classroom observations, researchers' field notes, and classroom artifacts. Data was systematically coded, and each belief and practice analyzed in-depth. The results identified eleven interconnected beliefs held in common by all three teachers. Among the eleven elements, How to Learn was the core belief that was most connected to the others and also aligned well with the Source of Knowing, How to Learn, Evidence-based Argument, and How to Teach; this idea established a strong structural foundation for the EOTS. In addition, some elements were explicitly presented when the teachers made instructional decisions, while others were only presented implicitly. In addition, prominent patterns of instructional practice were evident across the three cases. The teachers did not plan how to teach in advance, rather they made instructional decisions based on their epistemic orientations. In particular, they emphasized a conceptual understanding of the big ideas in science by making connections between students' ideas and the big ideas in science. Constant negotiation (construction and critique) was another pattern observed throughout the lessons. In creating effective learning conditions for conceptual understanding and constant negotiation, teachers used language practices and social, group-work as epistemic tools to help students construct and critique knowledge. Moreover, physical resources, such as physical materials and time, were used in a way that encouraged students to engage in science practice. More importantly, the way in which classroom practices and dialogue were managed relied heavily on the essential elements of ETOS. Specifically, How to Learn and Control of Learning influenced the student-centeredness of their instructional practices. This study provides several implications for teacher education and research. Teacher-education programs should focus energy on shaping teacher ideas about learning, and address the epistemic foundations of science practices. Further investigation into the essential elements of EOTS, and the relationship between these elements and instructional practices must be pursued with diverse subjects, contexts, and methodologies, to develop a fuller understanding of how these elements work as a whole.

Philosophiae Doctor - PhD / This thesis is concerned with the definition of an inclusive democratic educational theory and practice which mediates the assumed tension between individualism and collectivism, difference and equality, and liberty and equality. In Part 1, I set out the elements of an inclusive theory of democracy and then proceed in Part 2 to examine various aspects of educational practice in the light of this. My main claim is that these assumed tensions can be mediated through the conceptualising of our composite identity in terms of the notion of dual social ontology. This refers to our two-fold identities as universal, common human beings and our multiple subjective positions as particular, different individuating beings. Together, these two aspects of our identity constitute the basis for conceptualising our simultaneous commonality and difference and for an inclusive notion of democracy. I argue further that the key to understanding the intersection of commonality and difference in social relations and institutional practices is the concept of the spheres of social relations and their constitutive meanings. The latter provide the criterion by which we can judge the appropriateness of difference or equality in that sphere or in practices which relate to it. In the light of these concepts, I trace the ideological contestation at the heart of democratic theory between liberalism and socialism. My claim is that the mutual limitations of these theories preclude constructing an inclusive theory of democracy which incorporates collective equality and individual liberty in a non-polarised way. I argue that the tension between individualism and collectivism can be mediated by analysing these cluster concepts into non-polarised simpler elements. My main contention is that only self-interested individualism, which assumes individuals as atomistic self-seekers, is necessarily in conceptual conflict with collectivism. The other two elements of individualism which I identify, namely, individuality, our universal common identity as bearers of rights, and individuation, the process of self-development through the expression of the unique difference, are shown to be compatible with collective concerns and the social view of human identity. Together, I suggest, individuality and individuation constitute our dual social ontology and the foundation for moral regard and an inclusive theory of democracy which accommodates difference and commonality. During the discussion, I draw from several theorists who provide inclusive frameworks in terms of the social, dialogical view of human nature and identity formation and who combine contemporary concerns for pluralism and critical social transformation. I examine the conceptual link between education and democracy through the educative notion of democracy and education for democracy. Critical educational theory is explored as an exemplar of an inclusive democratic educational practice incorporating individual and collective dimensions. The dynamics of commonality and difference are traced in key aspects of the educational process, namely, moral development, learning and the relationship between authority and freedom, and with regard to the democratisation of schooling, the appropriate boundary between the spheres of education and of politics, distributive justice in education and the curriculum. I argue throughout that the discursive tool of dual social ontology, along with the concept of the spheres of social relations and their constitutive meanings, provides the conceptual framework by which these tensions can be mediated and incorporated in an inclusive democratic educational theory and practice.

Doctor Educationis / This dissertation is a study of a primary school teacher researching her classroom practice within the broader specific aim of the project was to investigate how a primary school teacher could go about her work in more for democratic ways. The research method used in the study was that of action research, and the empirical basis for argument in the dissertation lies in the use of data created during the process of researching that classroom practice. A fundamental assumption of the study is that teachers are the central driving force in any meaningful development of a critical pedagogy. Teachers, acting as trans formative . intellectuals, can work towards socialist transformation, because viewing teachers as intellectuals redefines their work and the political nature of schooling. The study, therefore, takes the problematic relationship between authority and emancipation as central to its concerns and tries to develop a rationale for making an emancipatory view of authority, and thus a rationale for a particular notion of professionalism, a central category in the development of a critical theory of schooling. In taking the position that teachers are central to learning in the classroom, and are the nexus of the authority/emancipation problematic, critical reflection on the classroom practices was approached both reflexively and dialectically so that uncertainties and contradictions could surface and be explored. The epistemological radicalism inherent in action research made this form of reflection possible. The study views primary school teachers as important mediators of change. This meant examining the process of instruction more carefully. The task was to understand how mediation generates higher mental functioning. To this end, Vygotsky's notion of a zone of proximal development, as the zone in which mediation can take place, is explored. The problem of how to investigate the substance of the zone of proximal development is met by the use of rationally reconstructed mediational operators. Drawing from the data in the study, three mediational operators are fashioned to study the one-to-many interaction in the process of instruction. These mediational operators serve as explanatory constructs to explicate the interrelation between the teacher's (and other significant others') instructional process and the learners' existing levels of development. • The preoccupations of this reflective dissertation writing take a multi-disciplinary approach, for to consider critical pedagogy means also to consider the psychological functioning of human beings within society - the mind-in society dialectic. The study reveals that 'democratic practice' is something to be negotiated and contested continually, for the authoritative position of teachers has to be questioned endlessly to locate the contradictions within that position. The experience of this study suggests that action research is a powerful means whereby teachers can reflect both reflexively and dialectically on their practice, that action research is intrinsically educational. A significant realization in the study, therefore, is the ways in which the educators themselves may become educated to take responsibility for their agency in transformation. • Bringing power relations into question is a permanent political task inherent in all social existence, thus a local specific inquiry such as this one can take on a general significance at the level of that regime of truth which is essential to the structure and functioning of our society. • Chapters One to Three provide the social and theoretical context for the study, dealing as they do with the crisis in schooling, the role of the teacher in change and the ways in which a teacher can research her practice for a transformed pedagogy. Chapters Four to Six consider the teacher-researcher at work in her classroom and indicate how she goes about trying to understand her role as mediator in the instruction process her process of reflection and action while she was teaching. • Chapters Seven to Nine represent the reflection which took place after the initial investigation of actual day-to-day practice. These chapters arise historically out of the research process and include argument for developing a more critical pedagogical discourse, a deeper understanding of the instruction process (the teaching/learning dynamic), and discussion of the validity of an action research paradigm for emancipatory pedagogical practices .

Philosophiae Doctor - PhD / The purpose of this study was to investigate the epistemological obstacles that mathematics students at undergraduate level encounter in coming to understand the limit concept. The role played by language and symbolism in understanding the limit concept was also investigated. A group of mathematics students at undergraduate level at the National University of Lesotho (NUL) was used as the sample for the study. Empirical data were collected by using interviews and questionnaires. These data were analysed using both the APOS framework and a semiotic perspective. Within the APOS framework, the pieces of knowledge that have to be constructed in coming to understand the limit concept are actions, processes and objects. Actions are interiorised into processes and processes are encapsulated into objects. The conceptual structure is called a schema. In investigating the idea of limit within the context of a function some main epistemological obstacles that were encountered when actions were interiorised into processes are over-generalising and taking the limit value as the function value. For example, in finding the limit value L for./{x) as x tends to 0, 46 subjects out of 251 subjects said that they would calculate ./{O) as the limit value. This method is Within the context of a sequence everyday language acted as an epistemological obstacle in interiorising actions into processes. For example, in finding lim (_1)n ,the majority of x~oo n the subjects obtained the correct answer O. It was however revealed that such an answer was obtained by using an inappropriate method. The subjects substituted one big value for n in the formula. The result obtained was the number close to O. Then 0 was taken as the limit value because the subjects interpreted the word 'approaches' as meaning 'nearer to'. Other subjects rounded off the result. In everyday life when one object approaches another, we might say that they are nearer to each other. It seems that in this case the appropriate for calculating the limit values for continuous functions. However, in this case, the method is generalised to all the functions. When these subjects encounter situations in which the functional value is equal to the limit value, they take the two to be the same. However, the two are different entities conceptually. Within the context of a sequence everyday language acted as an epistemological obstacle in interiorising actions into processes. For example, in finding lim (_1)n ,the majority of x~oo n the subjects obtained the correct answer O. It was however revealed that such an answer was obtained by using an inappropriate method. The subjects substituted one big value for n in the formula. The result obtained was the number close to O. Then 0 was taken as the limit value because the subjects interpreted the word 'approaches' as meaning 'nearer to'. Other subjects rounded off the result. In everyday life when one object approaches another, we might say that they are nearer to each other. It seems that in this case the subjects used this meaning to get 0 as the limit value. We also round off numbers to the nearest unit, tenth, etc. The limit value is however a unique value that is found by using the limiting process of 'tending to' or 'approaching' which requires infinite values to be considered. Some are computed and others are contemplated. In constructing the coordinated process schema, f(x) ~ L as x ~ a, over-generalisation and everyday language were still epistemological obstacles. Subjects still perceived the limit value to exist where the function is defined. The limit was also taken as a bound, lower or upper bound. In a case where the function was represented in a tabular form, the first and the seemingly last functional value that appeared in the table of values were chosen as the limit values. Limit values were also approximated. In constructing the coordinated process an ~ L as n ~ 00, representation, generalisation and everyday language also acted as epistemological obstacles. An alternating sequence was perceived as not one but two sequences. Since the subjects will have met situations where convergence means meeting at a point, as in the case of rays of light, a sequence was said to converge to a number that did not change in the given decimal digits. For example, the limit of the sequence {3.1, 3.14, 3.141, 3.1415, ... } was taken to be 3 or 3.1 as these are the digits that are the same in all the terms. In encapsulating processes into objects, everyday language also acted as an epistemological obstacle. When subjects were asked what they understood the limit to be, they said that the limit is a boundary, an endpoint, an interval, or a restriction. Though these interpretations are correct they are however, inappropriate if used in the technical context such as the mathematical context. While some subjects referred to the limit as a noun to show that they refer to it as an object, other subjects described the limit in terms of the processes that give rise to it. That is, it was described in terms of either the domain process or the range process. This is an indication that full encapsulation of processes into objects was not achieved by the subjects. The role of language and symbolism has been identified in making different connections in building the concept of limit as: representation of mathematical objects, translation between modes of representation, communication of mathematical ideas, manipulation of surface or syntactic structures and the overcoming of epistemological obstacles. In representation some subjects were aware of what idea some symbolism signified while other subjects were not. For example, in the context of limit of a sequence, most subjects took the symbolism that represented an alternating sequence, an = (-lr, to represent two sequences. The first sequence was seen as {I, I, 1, 1,... } and the second as {-I, -1, - 1, -1, ... }.This occurred in all modes of representation. In translating from one mode of representation to another, the obscurity of the symbol lim/ex) = L was problematic to the students. This symbol could not be related to its X~a equivalent form lex) ~ L as x ~ a. The equal sign, '=', joining the part lex) and L does not reflect the process ofj{x) tending to L, rather it appears as if it is the functional value that is equal to L. Hence, instead of looking for the value that is approached the subjects chose one of the given functional values. The part of the symbol lim was a x~a source of difficulty in translating the algebraic form to the verbal or descriptive. The subjects saw this part to mean "the limit of x tends to a" rather than seeing the whole symbolism as the limit of j{x) as x tends to a. Some subjects actually wrote some formulae in the place of L because of this structure, e.g., lim/ex) = 2x. These subjects x~a seemed to have concentrated on the part lex) = .... This is probably because they are used to situations where this symbolism is used in representing functions algebraically. In communicating mathematical ideas the same word carried different meanings for the researcher and for the subjects in some cases. For example, when the subjects were asked what it means to say a sequence diverges, one of the interpretations given was that divergence means tending to infinity. So, over-generalisation here acted as an epistemological obstacle. Though a sequence that tends to infinity diverges, this is not the only case of divergence that exists and therefore cannot be generalised in that way. The manipulation of the surface structures was done instrumentally by some subjects. For 1 . ti di 1·.J x 2 examp e, m in mg im +29 - 3 ,urdmugri the mam.pu 1ati.on some subjiects 0 bttaaime d x.... o x 2 part of the expressions such as ~ by rationalising or .:;- by using L'Hospital's rule 2x x which needed to be simplified. Instead of simplifying the expressions further at this stage, the substitution of 0 was done. So, .o2. = 0 was obtained as the answer. This shows that neither the reasons for performing the manipulations, nor the process of rationalising for example was understood. The result was still an indeterminate form of limit. The numerator was also not yet in a rational form. In using language to overcome epistemological obstacles, subjects were exposed to a piece of knowledge that falsified the knowledge they had so that they could rethink replacing the old with the new. In some cases, this was successful but in others, the subjects did not surrender these old pieces of knowledge. For example, when asked what they understood the 'rate of change' to mean, the majority of the subjects associated the rate of change with time only. However, when referred to a situation that required them to find the rate of change of an area with respect to radius, some subjects changed their minds but others did not. Those who did not change their minds probably did not make any connections between ideas under discussion. The implications for practice of the findings include: In teaching one should discuss explicitly how answers to tasks concerning limits are obtained. The idea of the limit value as a unique value can only be recognized if the process by which it is obtained is discussed. It should not be taken for granted that students who respond correctly understand the answers. It is evident from the study that even when correct answers are given, improper methods may have been used. Hence, in investigating epistemological obstacles attention should also be paid to correct answers. Also beyond this, students should be exposed to different kinds of representation of the limit concept using simple functions and using a variety of examples of sequences. Words with dual or multiple meaning should also be discussed in mathematics classrooms so that students may be aware of the meanings they carry in the mathematical context. Different forms of indeterminate states of limit should be given attention. Relations should also be made between the surface structures and the deeper structures.

Doctor Educationis / Three questions are central to this thesis: First, can the practice of teaching be made safe from luck through the controlling power of knowledge and reason? Second, even if it can be made safe from luck, should it be? Third, if it is neither possible nor desirable to exclude luck from teaching, what knowledge and personal qualities will put practitioners in the strongest position to face the contingencies of luck and, more especially, to face those conflicts which arise as a consequence of circumstances beyond the practitioner's control? Martha Nussbaum's account of luck and ethics in Greek philosophy and tragedy prompts the questions and provides, with Aristotle, many of the conceptual tools for answering them; Thomas Nagel's work on moral luck provides the categories for a more refined account of luck and its place in teaching. With respect to the first two questions, I argue that as a human practice teaching is open to the vicissitudes of fortune and cannot be made safe from luck, except at the expense of its vitality. Like other human practices, teaching is mutable, indeterminate and particular. Both its primary and secondary agents (teachers and pupils) and the practice itself are vulnerable to luck in four categories: constitutive, circumstantial, causal and consequential. But teaching is not just a matter of luck; it is a public practice in which some people are put into the hands of others for specific purposes, usually at public expense. If we have no way of holding practitioners accountable for their actions, the practice loses credibility. Any money or trust put into it is simply a gamble. For these and other reasons, the drive to exclude luck from practice is strong. Yet strong luck-diminishment projects are themselves a threat to the vitality of the practice. During the twentieth century two strong luck-diminishment projects have been especially detrimental to teaching: one rooted in the science of management, the other in the empirical sciences. Both have resulted in a proliferation of unfruitful and often trivial research projects, to misconceived programmes of teacher education, to distorted notions of knowledge and excellence in teaching, and to self-defeating and impoverished practice. Luck-diminishment projects rooted in logic are more or less threatening to vital practice, depending on how far they are committed to instrumental reasoning and a science of measurement. These are blunt and controversial claims. A central task of the thesis is to refine and defend them. The refinement proceeds by way of a contrastive analysis of strong luck-diminishment projects and others which are more responsive to the indeterminacy of practice. With respect to the final question, I argue that there are at least three sets of necessary conditions for a flourishing practice in the face of luck. One concerns what Aristotle calls the virtues of intellect and character. Central among these are practical rationality (conceived non-instrumentally), situational appreciation, and the knowledge required for an intelligent pursuit of the definitive ends of teaching. A second set concerns enabling institutions. A third concerns the kind of community best able to nurture those qualities necessary for vital and excellent practice. All three sets are themselves vulnerable to reversal. Keeping the practice of teaching alive and ensuring that it remains true to its definitive ends is thus a matter of sustained struggle.