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Doctor of Philosophy / Department of Physics / Eleanor C. Sayre / In this dissertation I will start with the broad research question of what does problem solving in upper division physics look like? My focus in this study is on students' problem solving in physics theory courses. Some mathematical formalisms are common across all physics core courses such as using the process of separation of variables, doing Taylor series, or using the orthogonality properties of mathematical functions to set terms equal to zero. However, there are slight differences in their use of these mathematical formalisms across different courses, possibly because of how students map different physical systems to these processes. Thus, my first main research question aims to answer how students perform these recurring processes across upper division physics courses. I break this broad question into three particular research questions: What knowledge pieces do students use to make connections between physics and procedural math? How do students use their knowledge pieces coherently to provide reasoning strategies in estimation problems? How do students look ahead into the problem to read the information out of the physical scenario to align their use of math in physics? Building on the previous body of the literature, I will use the theory family of Knowledge in Pieces and provide evidence to expand this theoretical foundation. I will compare my study with previous studies and provide suggestions on how to generalize these theory expansions for future use. My experimental data mostly come from video-based classroom data. Students in groups of 2-4 students solve in-class problems in quantum mechanics and electromagnetic fields 1 courses collaboratively. In addition, I will analyze clinical interviews to demonstrate how a single case study student plays an epistemic game to estimate the total energy in a hurricane. My second research question is more focused on a particular instructional context. How do students frame problem solving in quantum mechanics? I will lay out a new theoretical framework based in epistemic framing that separates the problem solving space into four frames divided along two axes. The first axis models students' framing in math and physics, expanded through the second axis of conceptual problem solving and algorithmic problem solving. I use this framework to show how students navigate problem solving. Lastly, I will use this developed framework to interpret existing difficulties in quantum mechanics.

This study investigated the deliberate use of learners‟ home languages in the teaching and learning of linear programming. The study involved a Grade 11 teacher and his Grade 11 multilingual learners in a township school in the East Rand. Data was collected through lesson observations for five consecutive days, reflective interview with teacher and clinical interview with two learners. Analysis of data revealed that the teacher used learners‟ home languages to probe learners‟ understanding of specific terms frequently used in linear programming concepts, for example terms such as, „at least‟ and „at most‟. Learners‟ responses suggest that they drew on their home languages for the meaning of these words. Learners explained the term „at least‟ in their home languages as „buncinci‟ in Isixhosa, „bonnyane‟ in Sesotho and Sepedi and „okungenani‟ in IsiZulu. Learners also used mathematical English term minimum to explain „at least‟ and maximum to explain „at most‟.

Dans cette thèse en didactique des mathématiques, nous étudions une possible transposition à la classe de pratiques de chercheurs utilisant la modélisation mathématique en nous intéressant à la dévolution aux élèves du secondaire (11 ans à 18 ans) du travail de mathématisation horizontale nécessaire pour envisager un traitement mathématique d’une situation ancrée dans le réel. Nous inscrivons notre travail dans la méthodologie de l'ingénierie didactique, en y intégrant, en outre, des problématiques liées aux pratiques enseignantes, présentant, en cela, certaines similitudes avec la démarche de l’ingénierie didactique de deuxième génération.Nous avons conduit une étude d’épistémologie contemporaine visant à identifier des éléments invariants dans les pratiques de chercheurs relevant de la mathématisation horizontale, en sciences du vivant. En appui sur ces résultats, nous avons caractérisé un énoncé de type fiction réaliste relevant d’une adaptation d’une problématique professionnelle de modélisation et avons élaboré un tel énoncé pour la classe autour de la prévision de la croissance dans un arbre. Nous avons mené des expérimentations dans des classes du secondaire, au sein d'un dispositif de formation continue de résolution collaborative de problèmes comportant une phase initiale de questions-réponses. Nous soutenons l’hypothèse que, les caractéristiques d’une fiction réaliste conçue comme une adaptation d’une problématique professionnelle de modélisation, et sa mise en œuvre dans les classes avec une phase de questions-réponses entre pairs pour débuter sa résolution, favorisent la dévolution de la mathématisation horizontale aux élèves. Les analyses didactiques des données recueillies ont été conduites en appui sur les résultats issus de l’étude épistémologique. Elles ont mis en évidence la dévolution aux élèves de la mathématisation horizontale et des traces de transposition des pratiques invariantes identifiées dans l'étude épistémologique. En appui sur le choix du cadre de la double approche didactique et ergonomique et sur une étude des obstacles à l’enseignement de la modélisation mathématique à travers la littérature en éducation mathématique, nous avons émis des hypothèses de travail sur les obstacles et les conditions à propos des pratiques enseignantes relevant de l’enseignement de la mathématisation horizontale. Nous avons utilisé l'effet de loupe potentiel offert par le dispositif de formation continue pour émettre et mettre à l’épreuve des hypothèses portant sur des leviers potentiels internes à la logique de ce dispositif, répondant, dans une certaine mesure, aux hypothèses concernant les obstacles et les conditions. L'étude de la portée générale de nos résultats à propos des pratiques ordinaires reste à faire. / In this thesis in didactics of mathematics, we study a possible transposition to the classroom of practices of researchers using mathematical modelling. We are focusing on the devolution to secondary pupils (11 years to 18 years) of the work of horizontal mathematization necessary to make a situation rooted in reality accessible to a mathematical treatment.We frame our work in the methodology of didactic engineering. We also include issues related to teaching practices and also we have some similarities with the approach of second-generation didactic engineering.Our epistemological analyses allowed us to identify invariant practices of researchers in life sciences concerning horizontal mathematization. We then relied on these analyses to characterize, develop and analyse a realistic fiction designed as an adaptation of a professional problem of modelling on the prediction of growth of a tree. We conducted our experiments in a training device of collaborative problem solving with an initial phase of questions and answers. We support the hypothesis that the characteristics of a realistic fiction designed as an adaptation of a professional problem of modelling with an initial phase of questions-answers between peers are likely to favour the devolution of horizontal mathematization to pupils.The didactic analyses of the collected data were conducted based on the results of the epistemological study. They highlighted the devolution to students of horizontal mathematization. In addition, some traces of transposition of the invariant practices identified in the epistemological study were attested. Relying on the framework of the didactic and ergonomic double approach and on a study of literature in mathematics education on possible obstacles to the teaching of mathematical modelling, we have made hypotheses on the obstacles and conditions about teaching practices related to the teaching of horizontal mathematization. We used the potential magnifying effect offered by the in-service teachers’ training device to emit and test hypotheses about potential internal levers within this device, in respect to a certain extent, to possible obstacles and conditions. The study of the general scope of our results concerning ordinary teaching practices is still to be done.

L’étude des ouvrages de physique publiés au 18e siècle montre que l’évolution depuis le 17e siècle est loin de se limiter à l’approfondissement des seules découvertes de Newton, comme on a souvent tendance à le présenter aujourd’hui. La physique mécaniste de Descartes, attaquée par Newton, va continuer de se développer avec l’aide de nombreux savants, en particulier de l’Académie des Sciences parisienne. Les débats entre cartésiens et newtoniens ne sont toujours pas éteints dans les années 1740. Ce véritable duel scientifique de plus d’un demi-siècle est au cœur d’une réflexion plus large sur la physique et s’exerce sur plusieurs plans : Mathématisation, Concepts, définitions, lois, rôle de l’expérience et des hypothéses, Problèmes philosophiques : les principes, la recherche des causes, les problèmes théologiques. L'objet de cette thèse est de montrer que la diversité des approches et des méthodes tout au long du premier 18e siècle va permettre l’émergence d’une nouvelle conception de la physique. Cette diversité se manifeste dans les écrits d’auteurs nombreux, les savants eux-mêmes, mais aussi ceux qu’on a appelés les transmetteurs, dont le rôle fut très important. / The study of books on physics published in the 18th century shows that the evolution since the 17th century is much more than just a furthering of the discoveries of Newton, as we often tend to present it these days. Descartes’s mechanistic physics, severely criticized by Newton, was to develop with help from many scientists, particularly from the Academy of Sciences in Paris. The discussions between Cartesians and Newtonians did not end in the 1740’s. This real scientific duel, which lasted over half a century, was the heart of a broader way of thinking about physics which operated on several levels: Mathématization, Concepts, définitions, laws, the role of experimentation and hypotheses, Philosophical problems: principles, the search of the causes, theological problems. The aim of this thesis is to show that the variety of the approaches and the methods throughout the early 18th century was to allow the creation of a new conception of physics. This variety appears in the works of many authors, who were either scientists themselves, or transmitters of science, who played a very important role, too.

Submitted by Mauricio Martinelli Silva Luperi (mluperi@gmail.com) on 2012-08-22T20:36:42Z No. of bitstreams: 1 TESE FINAL BIBLIOTECA.pdf: 2792504 bytes, checksum: d5e859c60db6e24376ba31b2f0a0d511 (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2012-08-23T17:41:32Z (GMT) No. of bitstreams: 1 TESE FINAL BIBLIOTECA.pdf: 2792504 bytes, checksum: d5e859c60db6e24376ba31b2f0a0d511 (MD5) / Made available in DSpace on 2012-08-23T18:01:30Z (GMT). No. of bitstreams: 1 TESE FINAL BIBLIOTECA.pdf: 2792504 bytes, checksum: d5e859c60db6e24376ba31b2f0a0d511 (MD5) Previous issue date: 2012-08-10 / The possibility of the economy becoming more mathematicized began with the marginalist revolution in the late nineteenth century. However, effectively, the process of mathematization of economic discourse would only become widespread, according to MIROWSKI (1991), from 1925 onwards. In an attempt to elucidate how this process took place and when it occurred in Brazil, we wrote three critical essays on the subject. The goal of the first essay is to make more accessible to Brazilian students and researchers a discussion that is held somewhat dispersedly in our country, and also to encourage further research on the subject. This subject is the discussion about the influence of the crisis in mathematics and physics of the early twentieth century on economic discourse. To see how this happened, we investigated the texts of some of the main authors who dealt with the subject. So we seek to elucidate the differences in rigor between the different models in Economics before and after the creation by mathematical physicists of quantum physics and non-Euclidean geometry, as well as its impact on general equilibrium theory. In the second essay, we begin by presenting the main benefits generated by the mathematization of economics, according to what is proclaimed by some of the advocates of the progress of the mathematical formalization of the economic discourse. After that, we point out the more traditional criticism to this mathematization process. Then we focus on the recent criticism of GILLIES (2005) about the the prevalence of operational numbers in economics. Later we analyze the criticism presented by BRESSER-PEREIRA (2008), who considers that the hypothetical-deductive method used by the "mainstream" is inadequate for economics. Finally, bearing in mind the definitions of BRESSER-PEREIRA (2008), we tentatively associate the reproduction of the hypothetical-deductive method to a metatheoretical process triggered by the theory of general equilibrium. In our third essay, we check how mathematical formalization in economics advanced in Brazil in the last three decades. To see this, we classified into several categories all the articles published in three major economic journals of the country (Revista Brasileira de Economia, Estudos Econômicos and Revista de Economia Política) and also the papers presented at the meetings of ANPEC (Associação Nacional dos Centros de Pós Graduação em Economia) from 1981 to 2010, according to the type of argument used. The total of articles analyzed was 5733. We try to notice if there was a turning point in the trajectory of economic discourse, making it more mathematical. Finally, in order to reinforce our analysis, we focus on the process of mathematization through the observation of a quantitative variable: equations per article. / A possibilidade da economia se tornar mais matematizada se iniciou com a revolução marginalista no final do século XIX. Entretanto, efetivamente, o processo de matematização do discurso econômico apenas teria se propagado, segundo MIROWSKI (1991), a partir de 1925. A fim de tentar elucidar como se deu esse processo e quando teria ocorrido no Brasil é que escrevemos três ensaios críticos sobre o tema. O objetivo do primeiro ensaio é o de tornar mais acessível aos estudantes e pesquisadores brasileiros uma questão que é tratada de maneira pouco orgânica em nosso país, e também incentivar novas pesquisas. Trata-se da discussão sobre as principais influências da crise da matemática e da física do final do século XX sobre o discurso econômico. Para verificar como isso se deu, investigamos os textos de alguns dos principais autores que tratam do tema. E daí buscamos elucidar as diferenças de rigor entre os diferentes modelos físicos matemáticos antes e depois da física quântica e da geometria não euclidiana, bem como seus impactos na teoria do equilíbrio geral. No segundo ensaio, iniciamos definindo os principais benefícios gerados pela matematização da economia, proclamados por alguns dos defensores do avanço do processo de formalização matemática sobre o discurso econômico. Em seguida, apontamos as críticas mais tradicionais a esse processo de matematização. Depois nos concentramos nas críticas mais recentes de GILLIES (2005) sobre a prevalência de números operacionais em economia. Para afinal, analisarmos a crítica de BRESSER-PEREIRA (2008) que considera o método hipotético-dedutivo utilizado pelo 'mainstream' inadequado à economia. Por último, de maneira tentativa, tendo em mente as definições de BRESSER-PEREIRA (2008), buscamos associar a reprodução do método hipotético-dedutivo a um processo metateórico deflagrado pela teoria do equilíbrio geral. No nosso terceiro ensaio, buscamos verificar como a formalização matemática avançou na ciência econômica brasileira nas três últimas décadas. Para observar isso, classificamos em diversas categorias todos os artigos publicados em três das principais revistas de economia do país (Revista Brasileira de Economia, Estudos Econômicos e Revista de Economia Política), bem como as publicações efetuadas nos encontros da ANPEC (Associação Nacional dos Centros de Pós-graduação em Economia) desde 1981 até 2010, de acordo com o tipo de argumentação utilizada. O total de artigos analisados soma 5.733. Procuramos observar quando houve um ponto de inflexão na trajetória do discurso econômico, tornando-o mais matemático. Por fim, para atestar nossas conclusões, focamos o processo de matematização na observação da variável quantitativa: equações por artigo.

Aux XVIIIe et XIXe siècles, les historiens des sciences associaient généralement Francis Bacon et Galilée pour leur rôle dans l’émergence de la science moderne, mais, à la fin du XIXe et au début du XXe siècle, la Révolution scientifique fut identifiée de manière stricte à la construction de la physique mathématique, ce qui eut souvent pour conséquence de rejeter Bacon hors de l’histoire des sciences. Nous reprenons l’étude conjointe de ces deux auteurs pour mesurer quelle est exactement la nature de leur divergence. Dans la première partie de notre travail, nous abordons les questions cosmologiques. Sur quels arguments Galilée fonde-t-il sa défense publique du copernicianisme entre 1610 et 1616, jusqu’à la première condamnation de l’opinion copernicienne par l’Eglise Catholique ? Pour quelles raisons Bacon, qui suit cette campagne copernicienne, rejette-t-il la plupart des découvertes astronomiques de Galilée ? Pourquoi Bacon, tout en réussissant à percevoir le caractère (trop peu) systématique du géocentrisme, refuse-t-il l’héliocentrisme ? Dans la deuxième partie de notre travail, nous abordons les questions relatives à la méthode, ainsi que les théories de la matière et du mouvement. Quel est le rôle de la perception sensible et la fonction des mathématiques dans les théories de Bacon ? Quelle est la signification de sa théorie du mouvement, qui multiplie les objets d’étude en proposant une typologie des différents mouvements concrets, alors que la physique mathématique tend à réduire tout déplacement au seul mouvement linéaire inertiel ? Quelle est la fonction de l’atomisme mathématique de Galilée ? Dans quelle mesure sa science du mouvement se distingue-t-elle de l’approche baconienne ? La différence fondamentale entre la science galiléenne et la démarche de Bacon consiste, selon nous, dans la nature des expériences et des observations qui sont convoquées, ainsi que dans le type d’abstraction que ces deux auteurs veulent conférer à la philosophie naturelle. / During the eighteenth and nineteenth centuries, historians of science usually considered that Francis Bacon and Galileo had respectively played their role in the merging of modern science, but, at the end of the nineteenth and the beginning of the twentieth century, Scientific Revolution has been strictly reduced to the elaboration of mathematical physics, which had for consequence to exclude Bacon from the history of science. Our aim is to underline the exact nature of the difference between those two authors. In the first part, we deal with the cosmological problems. What arguments did Galileo produce to sustain his public commitment for the Copernican system, from 1610 to 1616, until the first condemnation of copernicanism by the Roman Church ? For what reasons did Bacon reject most of Galileo’s astronomical discoveries ? Why Bacon, who clearly perceived the fact that the geocentric theory lacked systematic character, refused heliocentrism ? In the second part, we deal with the methodological questions, we analyse matter theories and the science of motion. What is the role of sense perception and what is the fonction of mathematics in Bacon’s theories ? What is the significance of his theory of motion, which multiplies the objects of study, proposing a typology of concrete movements, while mathematical physics aims at reducing any motion to the rectilinear inertial movement ? What is the fonction of the mathematical atomism proposed by Galileo ? In what measure does his science of motion distinguish from the baconian approach ? We think that the fondamental difference between the science of Galileo and the theories of Bacon consists in the nature of the experiments and observations used, and in the type of abstraction they are looking for in natural philosophy.

The study intends to investigate the nature and cognitive demands of contextual word-problems posed in the FET mathematics examinations of IEB and NSC. The analysis of the mathematization of real-life situations to form contextual word-problems is based on the theory of authentic task situations. The theoretical basis for analyzing mathematics teaching and learning is the Realistic Mathematics Education (RME) theory. Data was obtained using the schedule of mathematization of real-life situations and the schedule of total marks of contextual word-problems and national performance. All contextual word-problems included in the 2008-2013 question papers of IEB and NSC mathematics examinations were analysed. The research revealed that 509 marks out of 1800 marks were allocated to contextual word-problems in IEB examinations; whereas 473 marks out of 1800 marks were allocated to contextual word-problems in NSC examinations. / Mathematics Education / M. Sc. (Mathematical Education)

The study intends to investigate the nature and cognitive demands of contextual word-problems posed in the FET mathematics examinations of IEB and NSC. The analysis of the mathematization of real-life situations to form contextual word-problems is based on the theory of authentic task situations. The theoretical basis for analyzing mathematics teaching and learning is the Realistic Mathematics Education (RME) theory. Data was obtained using the schedule of mathematization of real-life situations and the schedule of total marks of contextual word-problems and national performance. All contextual word-problems included in the 2008-2013 question papers of IEB and NSC mathematics examinations were analysed. The research revealed that 509 marks out of 1800 marks were allocated to contextual word-problems in IEB examinations; whereas 473 marks out of 1800 marks were allocated to contextual word-problems in NSC examinations. / Mathematics Education / M. Sc. (Mathematical Education)

The integration of modern science and indigenous knowledge systems in the science education curriculum for South African schools represents social justice for the majority of South Africans as they determine the knowledge necessary for themselves and for future generations in the new South Africa. An exploratory research reveals tension and a dichotomous relationship between modern science and IKS, caused by false hierarchies that are influenced by factors such as colonialism, capitalism and modernisation to the exclusion of the core values held by indigenous people in their relationship with nature. The thesis demonstrates that the integration requires an epistemology that puts humanity first and a framework that accommodates both ways of knowing. This should allow for the best in the two systems of knowing to serve humanity in a dialogical manner. / Educational Studies / M. Ed. (Philosophy of Education)