656 resultados para Mathematics -- Philosophy
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Introduction Fundamental to the philosophy of Buddhism, is the insight that there is "unsatisfactohness" (dukkha) in the world and that it can be eliminated through the practice of the Noble Eight Fold Path. Buddhism also maintains that the world as we experience and entities that exist are bereft of any substantiality. Instead existence is manifest through dependent origination. All things are conditional; nothing is permanent. However, inherent in this dependent existence is the interconnectedness of all beings and their subjection to the cosmic law of karma. Part of cultivating the Eight Fold path includes a deep compassion for all other living things, 'trapped' within this cycle of dependent origination. This compassion or empathy (karuna) is crucial to the Buddhist path to enlightenment. It is this emphasis on karuna that shows itself in Mahayana Buddhism with respect to the theory of the boddhisatva (or Buddha-to-be) since the boddhisatva willingly postpones his/her own enlightenment to help others on the same path. One of the ramifications of the theory of dependent origination is that there is no anthropocentric bias placed on humans over the natural world. Paradoxically the doctrine of non-self becomes an ontology within Buddhism, culminating in the Mayahana realization that a common boundary exists between samsara and nirvana. Essential to this ontology is the life of dharma or a moral life. Ethics is not separated from ontology. As my thesis will show, this basic outlook of Buddhism has implications toward our understanding of the Buddhist world-view with respect to the current human predicament concerning the environment. While humans are the only ones who can 4 attain "Buddhahood", it is because of our ability to understand what it means to follow the Eight fold path and act accordingly. Because of the interconnectedness of all entities {dharmas), there is an ontological necessity to eliminate suffering and 'save the earth' because if we allow the earth to suffer, we ALL suffer. This can be understood as an ethical outlook which can be applied to our interaction with and treatment of the natural environment or environment in the broadest sense, not just trees plants rocks etc. It is an approach to samsara and all within it. It has been argued that there is no ontology in Buddhism due to its doctrine of "non-self". However, it is a goal of this thesis to argue that there does exist an original ontology in Buddhism; that according to it, the nature of Being is essentially neither "Being nor non-being nor not non-being" as illustrated by Nagarjuna. Within this ontology is engrained an ethic or 'right path' (samma marga) that is fundamental to our being and this includes a compassionate relationship to our environment. In this dissertation I endeavour to trace the implications that the Buddhist worldview has for the environmental issues that assail us in our age of technology. I will explore questions such as: can the Buddhist way of thinking help us comprehend and possibly resolve the environmental problems of our day and age? Are there any current environmental theories which are comparable to or share common ground with the classical Buddhist doctrines? I will elucidate some fundamental doctrines of early Buddhism from an environmental perspective as well as identify some comparable modern environmental theories such as deep ecology and general systems theory, that seem to share in the wisdom of classical Buddhism and have much to gain from a deeper appreciation of Buddhism.
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This research studioo the effect of integrated instruction in mathematics and~ science on student achievement in and attitude towards both mathematics and science. A group of grade 9 academic students received instruction in both science and mathematics in an integrated program specifically developed for the purposes of the research. This group was compared to a control group that had received science and mathematics instruction in a traditional, nonintegrated program. The findings showed that in all measures of attitude, there was no significant difference between the students who participated in the integrated science and mathematics program and those who participated in a traditional science and mathematics program. The findings also revealed that integration did improve achievement on some of the measures used. The performance on mathematics open-ended problem-solving tasks improved after participation in the integrated program, suggesting that the integrated students were better able to apply their understanding of mathematics in a real-life context. The performance on the final science exam was also improved for the integrated group. Improvement was not noted on the other measures, which included EQAO scores and laboratory practical tasks. These results raise the issue of the suitability of the instruments used to gauge both achievement and attitude. The accuracy and suitability of traditional measures of achievement are considered. It is argued that they should not necessarily be used as the measure of the value of integrated instruction in a science and mathematics classroom.
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Ontario bansho is an emergent mathematics instructional strategy used by teachers working within communities of practice that has been deemed to have a transformational effect on teachers' professional learning of mathematics. This study sought to answer the following question: How does teachers' implementation of Ontario bansho within their communities of practice inform their professional learning process concerning mathematics-for-teaching? Two other key questions also guided the study: What processes support teachers' professional learning of content-for-teaching? What conditions support teachers' professional learning of content-for-teaching? The study followed an interpretive phenomenological approach to collect data using a purposive sampling of teachers as participants. The researcher conducted interviews and followed an interpretive approach to data analysis to investigate how teachers construct meaning and create interpretations through their social interactions. The study developed a model of professional learning made up of 3 processes, informing with resources, engaging with students, and visualizing and schematizing in which the participants engaged and 2 conditions, ownership and community that supported the 3 processes. The 3 processes occur in ways that are complex, recursive, nonpredictable, and contextual. This model provides a framework for facilitators and leaders to plan for effective, content-relevant professional learning by placing teachers, students, and their learning at the heart of professional learning.
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This thesis research was a qualitative case study of a single class of Interdisciplinary Studies: Introduction to Engineering taught in a secondary school. The study endeavoured to explore students' experiences in and perceptions of the course, and to investigate the viability of engineering as an interdisciplinary theme at the secondary school level. Data were collected in the form of student questionnaires, the researcher's observations and reflections, and artefacts representative of students' work. Data analysis was performed by coding textual data and classifying text segments into common themes. The themes that emerged from the data were aligned with facets of interdisciplinary study, including making connections, project-based learning, and student engagement and affective outcomes. The findings of the study showed that students were positive about their experiences in the course, and enjoyed its project-driven nature. Content from mathematics, physics, and technological design was easily integrated under the umbrella of engineering. Students felt that the opportunity to develop problem solving and teamwork skills were two of the most important aspects of the course and could be relevant not only for engineering, but for other disciplines or their day-to-day lives after secondary school. The study concluded that engineering education in secondary school can be a worthwhile experience for a variety of students and not just those intending postsecondary study in engineering. This has implications for the inclusion of engineering in the secondary school curriculum and can inform the practice of curriculum planners at the school, school board, and provincial levels. Suggested directions for further research include classroom-based action research in the areas of technological education, engineering education in secondary school, and interdisciplinary education.
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This thesis considers that the purport of the Bhagavadgita is to prioritize the philosophy of loving devotion to God (bhakti), not the propagation of color-coded-caste (varna system). The distinction between bhakti and caste becomes clear when one sees their effect on human life and on the society. Jnana and karma, two of the other polarities with which the Gita contends, finally support bhakti towards betterment, not deterioration, if done selflessly and with balance. Caste, however, is a totally different tension, which is always detrimental to the well-being of the person and the society. In the Gita, the devotees' mystical or emotional love of, God apprehends their ~ oneness with the Supreme God and with all beings, and transcends the pitiless segregation of the caste system, and opens the path of salvation to all irrespective of race, color, caste, class or gender in life. In spite of much opposition from orthodoxy, the bhakti movement spread allover India, and bhakti itself rose to the level of orthodoxy and has become the faith of millions of people especially of the south, and surprisingly, of even of those of the so called highest caste. And yet, caste still remains as an indelible mark of every Hindu, even after they change their religion. Although caste is less venomous now, it is still openly present in all walks of Indian life and shows up its ugly head at important moments such as marriage, elections for public office, admission to school or employment. True, bhakti is the antidote for. caste; but only real bhakti can remove caste completely, not mere lip-service to it. This thesis claims that bhakti is the deliberate major thrust of the teaching of the Gita while caste seems to be a contradiction of this thrust.
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This project addressed the need for more insightful, current, and applicable resources for intermediate math teachers in Canadian classrooms. A need for a handbook in this division seemed warranted by a lack of government resource support. Throughout an extensive review of the literature, themes and topics for the handbook emerged. The handbook was designed to not only provide educators with examples of effective teaching strategies within the mathematics classroom but to also inform them about the ways in which their personal characteristics and personality type could affect their students and their own pedagogical practices. Three teaching professionals who had each taught in an intermediate math class within the past year evaluated the handbook. The feedback received from these educators was directly applied to the first draft of the handbook in order to make it more accessible and applicable to other math teachers. Although the handbook was written with teachers in mind, the language and format used throughout the manual also make it accessible to parents, tutors, preservice education students, and educational administrators. Essentially, any individual who is hoping to inspire and educate intermediate math students could make use of the content within the handbook.
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This is a study of the implementation and impact of formative assessment strategies on the motivation and self-efficacy of secondary school mathematics students. An explanatory sequential mixed methods design was implemented where quantitative and qualitative data were collected and analyzed sequentially in 2 different phases. The first phase involved quantitative data from student questionnaires and the second phase involved qualitative data from individual student and teacher interviews. The findings of the study suggest that formative assessment is implemented in practice in diverse ways and is a process where the strategies are interconnected. Teachers experience difficulty in incorporating peer and self-assessment and perceive a need for exemplars. Key factors described as influencing implementation include teaching philosophies, interpretation of ministry documents, teachers’ experiences, leadership in administration and department, teacher collaboration, misconceptions of teachers, and student understanding of formative assessment. Findings suggest that overall, formative assessment positively impacts student motivation and self-efficacy, because feedback is provided which offers encouragement and recognition by highlighting the progress that has been made and what steps need to be taken to improve. However, students are impacted differently with some considerations including how students perceive mistakes and if they fear judgement. Additionally, the impact of formative assessment is influenced by the connection between self-efficacy and motivation, namely how well a student is doing is a source of both concepts.
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La thèse présente une analyse conceptuelle de l'évolution du concept d'espace topologique. En particulier, elle se concentre sur la transition des espaces topologiques hérités de Hausdorff aux topos de Grothendieck. Il en ressort que, par rapport aux espaces topologiques traditionnels, les topos transforment radicalement la conceptualisation topologique de l'espace. Alors qu'un espace topologique est un ensemble de points muni d'une structure induite par certains sous-ensembles appelés ouverts, un topos est plutôt une catégorie satisfaisant certaines propriétés d'exactitude. L'aspect le plus important de cette transformation tient à un renversement de la relation dialectique unissant un espace à ses points. Un espace topologique est entièrement déterminé par ses points, ceux-ci étant compris comme des unités indivisibles et sans structure. L'identité de l'espace est donc celle que lui insufflent ses points. À l'opposé, les points et les ouverts d'un topos sont déterminés par la structure de celui-ci. Qui plus est, la nature des points change: ils ne sont plus premiers et indivisibles. En effet, les points d'un topos disposent eux-mêmes d'une structure. L'analyse met également en évidence que le concept d'espace topologique évolua selon une dynamique de rupture et de continuité. Entre 1945 et 1957, la topologie algébrique et, dans une certaine mesure, la géométrie algébrique furent l'objet de changements fondamentaux. Les livres Foundations of Algebraic Topology de Eilenberg et Steenrod et Homological Algebra de Cartan et Eilenberg de même que la théorie des faisceaux modifièrent profondément l'étude des espaces topologiques. En contrepartie, ces ruptures ne furent pas assez profondes pour altérer la conceptualisation topologique de l'espace elle-même. Ces ruptures doivent donc être considérées comme des microfractures dans la perspective de l'évolution du concept d'espace topologique. La rupture définitive ne survint qu'au début des années 1960 avec l'avènement des topos dans le cadre de la vaste refonte de la géométrie algébrique entreprise par Grothendieck. La clé fut l'utilisation novatrice que fit Grothendieck de la théorie des catégories. Alors que ses prédécesseurs n'y voyaient qu'un langage utile pour exprimer certaines idées mathématiques, Grothendieck l'emploie comme un outil de clarification conceptuelle. Ce faisant, il se trouve à mettre de l'avant une approche axiomatico-catégorielle des mathématiques. Or, cette rupture était tributaire des innovations associées à Foundations of Algebraic Topology, Homological Algebra et la théorie des faisceaux. La théorie des catégories permit à Grothendieck d'exploiter le plein potentiel des idées introduites par ces ruptures partielles. D'un point de vue épistémologique, la transition des espaces topologiques aux topos doit alors être vue comme s'inscrivant dans un changement de position normative en mathématiques, soit celui des mathématiques modernes vers les mathématiques contemporaines.
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Partant des travaux séminaux de Boole, Frege et Russell, le mémoire cherche à clarifier l‟enjeu du pluralisme logique à l‟ère de la prolifération des logiques non-classiques et des développements en informatique théorique et en théorie des preuves. Deux chapitres plus « historiques » sont à l‟ordre du jour : (1) le premier chapitre articule l‟absolutisme de Frege et Russell en prenant soin de montrer comment il exclut la possibilité d‟envisager des structures et des logiques alternatives; (2) le quatrième chapitre expose le chemin qui mena Carnap à l‟adoption de la méthode syntaxique et du principe de tolérance, pour ensuite dégager l‟instrumentalisme carnapien en philosophie de la Logique et des mathématiques. Passant par l‟analyse d‟une interprétation intuitive de la logique linéaire, le deuxième chapitre se tourne ensuite vers l‟établissement d‟une forme logico-mathématique de pluralisme logique à l‟aide de la théorie des relations d‟ordre et la théorie des catégories. Le troisième chapitre délimite le terrain de jeu des positions entourant le débat entre monisme et pluralisme puis offre un argument contre la thèse qui veut que le conflit entre logiques rivales soit apparent, le tout grâce à l‟utilisation du point de vue des logiques sous-structurelles. Enfin, le cinquième chapitre démontre que chacune des trois grandes approches au concept de conséquence logique (modèle-théorétique, preuve-théorétique et dialogique) forme un cadre suffisamment général pour établir un pluralisme. Bref, le mémoire est une défense du pluralisme logique.
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Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal