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Lima, Marcella Luanna da Silva. "Sobre pensamento geomátrico, provas e demonstrações matemáticas de alunos do 2º ano do Ensino Médio nos ambientes Lápis e Papel e Geogebra." Universidade Estadual da Paraíba, 2015. http://tede.bc.uepb.edu.br/tede/jspui/handle/tede/2336.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
Our research work aimed to investigate what type of proof, mathematical demonstration and level of geometrical thinking can occur from a didactic proposal within pencil, paper and GeoGebra environments. As qualitative research and study case, we used as instruments essays with Mathematical Proof and Demonstration themes, a didactic proposal developed by a team of five people who inserted worked collaboratively in the CAPES/OBEDUC/UFMS/UEPB/UFAL Project, field notes, participant observation, audios and photos. We elaborated a didactic proposal with eighteen activities, divided into four parts, which encouraged the students to reflect, justify, prove and demonstrate. The proposal application was carried out in July 2015 with High School 2nd year students of a public school in the town of Areia, Paraíba. For such, the students organized themselves in couples and one trio and the data collection happened in three moments. In the first moment we applied the essay, revised angles, triangles and theorems with the students and worked GeoGebra application with them. In the second moment we applied Parts I and II of the proposal with eight activities on Pythagoras Theorem and three activities on Sum of the Internal Angles of a Triangle Theorem, respectively. In the third moment we applied Part III, with two questions on External Angle Theorem and Part IV, with five question to be worked with the GeoGebra application on Pythagoras Theorem and Sum of the Internal Angles of a Triangle Theorem. In our research work we analyzed the work developed by the trio of students, once they were great in responding all the questions/activities. We analyzed Activity 8 of Part I, Activity 1 and 2 of Part II and all Activities of Part IV, totalizing in eight questions. We used the triangulation method for our study case and, firstly, we traced the profiles of the trio in relation to Mathematical Proof and Demonstration. Then we investigated the geometric thinking and the mathematical proof and demonstration used by the trio of students in the pencil and paper and GeoGebra environments. For such, we used discussions around the level of geometrical thinking proposed by Parzysz (2006) and the type of proofs proposed by Balacheff (2000) and Nasser and Tinoco (2003). From our research results we could conclude that the trio of students could not develop the justifications or proofs, once they did not understand what are mathematical proof and demonstration are, in their essays they understand mathematical proofs as bimestrial evaluations applied by the mathematics teacher. Moreover, the mathematical proofs performed by these students were in accordance with naive empiricism, pragmatic proof (Balacheff, 2000) and graphic justification (Nassar and Tinoco, 2003). In this way, when we observed the students geometrical thinking (Parzysz, 2006) we noted that it fits into two levels of the non-axiomatic Geometry: the Concrete Geomety (G0) and the Spatio-Graphique Geometry (G1), once these students used drawings to justify their affirmations, as the validation of the affirmation was done by the trio. We believe that if in Mathematic classes the teachers contemplate mathematical proof and demonstration, respecting the level of education, the degree of knowledge and maturity of the students, they could strongly contribute to the process of teaching and learning Mathematics and geometrical thinking, once the students would be led to reflect, justify, prove and demonstrate their ideas.
Nossa pesquisa investigou que tipo de provas, demonstrações matemáticas e nível de pensamento geométrico de alunos do 2º Ano do Ensino Médio podem ocorrer a partir de uma proposta didática nos ambientes lápis e papel e GeoGebra. Como pesquisa qualitativa, e estudo de caso, utilizamos como instrumentos redação com o tema Provas e Demonstrações Matemáticas, proposta didática desenvolvida por uma equipe de cinco pessoas que trabalhou de forma colaborativa inserida no Projeto CAPES/OBEDUC/UFMS/UEPB/UFAL Edital 2012, notas de campo, observação participante, gravações em áudio e fotos. Elaboramos uma proposta didática com 18 atividades, dividida em quatro partes, que incentivam alunos a refletirem, justificarem, provarem e demonstrarem. A aplicação dessa proposta se deu em julho de 2015 aos alunos do 2º Ano do Ensino Médio de uma escola pública na cidade de Areia, Paraíba. Para isso, os alunos se agruparam em duplas e um trio e a coleta dos dados se deu em três momentos. No primeiro momento, aplicamos a redação, revisamos com os alunos ângulos, triângulos e teoremas e trabalhamos com eles o aplicativo GeoGebra. No segundo momento, aplicamos as Partes I e II da proposta com 8 atividades sobre Teorema de Pitágoras e 3 atividades sobre Teorema da Soma dos Ângulos Internos de um Triângulo, respectivamente. No terceiro momento, aplicamos a Parte III, com 2 questões sobre o Teorema do Ângulo Externo e a Parte IV, com 5 questões à serem trabalhadas no aplicativo GeoGebra sobre o Teorema de Pitágoras e Teorema da Soma dos Ângulos Internos de um Triângulo. Em nossa pesquisa analisamos o trabalho desenvolvido pelo trio de alunos, uma vez que foram ricos na tentativa de r esponder a todas as perguntas/atividades. Analisamos a Atividade 8 da Parte I, as Atividades 1 e 2 da Parte II e todas as Atividades da Parte IV, totalizando em 8 questões. Utilizamos o método de triangulação de dados para nosso estudo de caso e, primeiramente, traçamos o perfil do trio de alunos com relação às Provas e Demonstrações Matemáticas. Em seguida, investigamos o pensamento geométrico e as provas e demonstrações matemáticas utilizadas pelo trio de alunos nos ambientes lápis e papel e GeoGebra. Para isso, utilizamos as discussões sobre os níveis do pensamento geométrico propostos por Parzysz (2006) e tipos de provas propostos por Balacheff (2000) e Nasser e Tinoco (2003). A partir de nossos resultados pudemos concluir que o trio de alunos não conseguiu desenvolver suas justificativas nem provas, uma vez que não entendem o que vem a ser provas e demonstrações matemáticas, e em suas redações percebemos que estes alunos tratam provas matemáticas como as avaliações aplicadas bimestralmente pelo professor de Matemática. Além disso, as provas matemáticas realizadas por estes alunos se enquadram no empirismo ingênuo, prova pragmática (Balacheff, 2000) e justificativa gráfica (Nasser e Tinoco, 2003). Dessa forma, quando observamos o pensamento geométrico (Parzysz, 2006) destes alunos, notamos que se enquadra nos dois níveis da Geometria não axiomática: a Geometria Concreta (G0) e a Geometria Spatio-Graphique (G1), uma vez que estes alunos se utilizam de desenhos para justificar suas afirmações, como também a validação das afirmações foi feita pela percepção do trio. Acreditamos que se nas aulas de Matemática os professores contemplassem provas e demonstrações matemáticas, respeitando o nível de escolaridade, o grau de conhecimento e a maturidade dos alunos, contribuiriam fortemente para o processo de ensino e aprendizagem da Matemática e do pensamento geométrico, uma vez que os alunos seriam levados a refletir, justificar, provar e demonstrar suas ideias.
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Swanzy, Michael John. "Analysis and demonstration: a proof-of-concept compass star tracker." Texas A&M University, 2005. http://hdl.handle.net/1969.1/4853.
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This research analyzes and demonstrates the local position determination problem
on Earth using a novel instrument, the Compass Star Tracker. Special focus is
given to the theoretical development of the mathematics of local position determination,
the design and fabrication of a proof-of-concept instrument, an error source
analysis, and the experimental tests used to validate the position determination concepts.
Star sensors are typically used as attitude determination instruments on spacecraft
orbiting Earth. In this capacity, the star sensor determines the orientation of
the spacecraft using digital images of the stars. This research utilizes the basic functionality
of the star sensor in a new way; the orientation information from the star
image is used to determine a user's latitude and longitude coordinates on Earth. This
concept is valuable because it allows users to determine their position autonomously.
The fundamental concepts that enable local position determination were originally
published in Drs. Samaan, Mortari, and Junkins (AAS 04-007). This research
improves upon that work by eliminating the zenith-orientation constraint and providing
several crucial theoretical corrections. In addition to the position determination
mathematics, this research provides analysis of the theoretical and practical error
sources associated with the position determination problem. This research also details
the design, fabrication, and experimental test program of a proof-of-concept Compass Star Tracker. Together, the theoretical development, error analysis, instrument
design, and test program serve as validation of the the position determination
concept. This work is intended as the first of many steps toward eventual deployment
of autonomous position determination sensors on the Moon and Mars.
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Pasini, Mirtes Fátima. "Argumentação e prova: explorações a partir da análise de uma coleção didática." Pontifícia Universidade Católica de São Paulo, 2007. https://tede2.pucsp.br/handle/handle/11282.
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Previous issue date: 2007-10-16Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
This work is inserted the research project Argumentation and Proof in School Mathematics (AProvaME), which aims to study the teaching and learning of mathematical proofs during compulsory schooling. The main research question of this contribution to the project relates to how proof is treated in particular geometry topics in one collection of mathematics textbooks for secondary school students. More specifically, the study aims to identify how the passage from empiricism to deduction is contemplated in the textbook activities as well as to document the interventions and strategies necessary on the part of the mathematics teacher in order to manage this transition. The types of proofs in the classification of Balacheff (1988) and the functions of proof identified by de Villiers (2001) serve as the principle theoretical tools for these analyses. Following a survey of the activities related to proof and proving in topics related to the theorem of Pythagoras and properties of straight lines and triangles, teaching sequences based on these activities were developed with students from the 8th Grade of a secondary school within the public school system of the municipal of Jacupiranga in the State of São Paulo. The main findings of the study indicate that the teacher has at his or her disposal material that permit a broad approach to proof and proving, although the passage from exercises involving reliance on empirical manipulations for validation to the construction of proofs based on mathematical properties is not very explicitly addressed, with the result that intense teacher intervention is necessary at this point. A particular difficulty faced by the teacher is knowing how to intervene without assuming responsibility for the resolution of the task in question. Finally, a dynamic geometry activity is presented, as an attempt to provide a learning situation which might enable students to engage more spontaneously in the transition from evidence-based arguments to valid mathematical proofs
Nosso trabalho está inserido no Projeto Argumentação e Prova na Matemática Escolar (AProvaME), que tem como objetivo estudar o ensino e aprendizagem de provas matemáticas na Educação Básica. A questão principal da pesquisa consiste em analisar o tratamento deste tema em determinados conteúdos geométricos de uma coleção de livros didáticos do Ensino Fundamental. Mais especificamente, o estudo busca identificar como a passagem do empirismo à dedução é contemplada nas atividades dos livros e quais as intervenções e estratégias necessárias por parte do professor para gerenciar essa passagem. Os tipos de prova na classificação de Balacheff (1988) e as funções de prova identificadas por De Villiers (2001) foram as principais ferramentas teóricas utilizadas para estas análises. Após um levantamento das atividades relacionadas à prova nos conteúdos Teorema de Pitágoras, Retas Paralelas e as propriedades dos Triângulos, seqüências baseadas nessas atividades foram desenvolvidas com alunos de 8.ª Série do Ensino Fundamental de uma escola pública no Município de Jacupiranga, do Estado da São Paulo. Concluímos que o professor tem à sua disposição material consistente para trabalhar com seus alunos, embora exista o problema na passagem brusca de exercícios empíricos em diversos níveis de verificação para as demonstrações formais, sendo necessária intervenção do professor por meio de revisões pertinentes, proporcionando ao aluno esclarecimentos para desenvolver uma atividade. A principal dificuldade para o professor foi interferir sem assumir a responsabilidade de resolver a situação em questão. Por fim, apresenta-se uma atividade no ambiente de geometria dinâmica, visando proporcionar uma transição mais espontânea entre argumentos baseados em evidência e argumentos baseados em propriedades matemáticas
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Macmillan, Emily. "Argumentation and Proof : Investigating the Effect of Teaching Mathematical Proof on Students' Argumentation Skills." Thesis, University of Oxford, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.517230.
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Hemmi, Kirsti. "Approaching Proof in a Community of Mathematical Practice." Doctoral thesis, Stockholm : Department of Mathematics, Stockholm University, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-1217.
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Owen, Stephen G. "Finding and using analogies to guide mathematical proof." Thesis, University of Edinburgh, 1988. http://hdl.handle.net/1842/27156.
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This thesis is concerned with reasoning by analogy within the context of auto-mated problem solving. In particular, we consider the provision of an analogical reasoning component to a resolution theorem proving system. The framework for reasoning by analogy which we use (called Basic APS) contains three major components -the finding of analogies (analogy matching), the construction of analogical plans, and the application of the plans to guide the search of a theorem prover. We first discuss the relationship of analogy to other machine learning techniques. We then develop programs for each of the component processes of Basic APS. First we consider analogy matching. We reconstruct, analyse and crticise two previous analogy matchers. We introduce the notion of analogy heuristics in order to understand the matchers. We find that we can explain the short-comings of the matchers in terms of analogy heuristics. We then develop a new analogy matching algorithm, based on flexible application of analogy heuristics, and demonstrate its superiority to the previous matchers. We go on to consider analogical plan construction. We describe procedures for constructing a plan for the solution of a problem, given the solution of a different problem and an analogy match between the two problems. Again, we compare our procedures with corresponding ones from previous systems. We then describe procedures for the execution of analogical plans. We demon-strate the procedures on a number of example analogies. The analogies involved are straightforward for a human, but the problems themselves involve.huge search spaees, if tackled directly using resolution. By comparison with unguided search, we demonstrate the dramatic reductfon in search entaile_d by the use of an ana-logical plan. We then consider some directions for development of our analogy systems, which have not yet been implemented. Firstly, towards more flexible and power-ful execution of analogical plans. Secondly, towards an analogy system which can improve its own ability to find and apply analogies over the course of experience.
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Almeida, Julio Cesar Porfirio de. "Argumentação e prova na matemática escolar do ensino básico: a soma das medidas dos ângulos internos de um triângulo." Pontifícia Universidade Católica de São Paulo, 2007. https://tede2.pucsp.br/handle/handle/11502.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
This study is about the demonstration of amount of measure the internal angles of triangles made by 8th grade from Fundamental School and the First year of High School, from of resolution of two specified questions. This work intends to contribute with the Argumentation and Proof in School Mathematics project (AprovaME), that has as one of objectives the mapping of conceptions about teenager s argumentation and proofs in public and private schools of São Paulo (state) For this was made a questionnaire in two books, five questions of Algebra and with five questions of Geometry. They were given to 1998 pupils aged between 14 and 16 years. The two analyzed questions are in the Geometry notebook. After checking the given information, took out 50 pupils as sample, that answers were classified in four progressive levels according their form of argument used in evolution of the Pragmatic proof (first principles methods of verification) to the Intellectual proof (elaborations of reasoning from logical-deduction nature and the production of explanation characterized as mathematics demonstration). In the following phase these pupils were put in groups according with the types of answers presented, to do the individual interviews aiming explanations about their choose. Finish the work a conclusive survey based in the results of the analysis, where are suggested forms of approach of subject Proofs and Demonstrations in the classroom, contemplating the execution of dynamic activities that give privilege the construction of mathematically consistent argument based in the expression of generalized reasoning
Este estudo trata da demonstração da soma da medida dos ângulos internos de um triângulo por alunos da oitava série do Ensino Fundamental e da primeira série do Ensino Médio, a partir da resolução de duas questões específicas. Procura contribuir com o Projeto Argumentação e Prova na Matemática Escolar (AprovaME), que tem como um de seus objetivos o mapeamento das concepções sobre argumentação e prova de alunos adolescentes em escolas públicas e particulares do Estado de São Paulo. Para esse levantamento foi elaborado um questionário contendo, em dois cadernos, cinco questões de Álgebra e cinco de Geometria, aplicados a 1998 alunos na faixa etária entre 14 e 16 anos. As duas questões analisadas estão inseridas no caderno de Geometria. Após a tabulação das informações coletadas, extraiu-se dessa população uma amostra de 50 alunos, cujas respostas foram classificadas em quatro níveis progressivos quanto às formas de validação dos argumentos empregados numa evolução da categoria Prova Pragmática (métodos rudimentares de verificação) à Prova Intelectual (elaboração de raciocínios de natureza lógico-dedutiva e produção de explicações caracterizadas como demonstrações matemáticas). Na etapa seguinte, esses alunos foram agrupados de acordo com os tipos de resposta apresentados para a realização de entrevistas individuais visando à obtenção de esclarecimentos adicionais sobre suas escolhas. Encerra o trabalho um panorama conclusivo baseado no resultado da análise em que são sugeridas formas de abordagem do tema Provas e Demonstrações em sala de aula, contemplando a realização de atividades dinâmicas que privilegiem a construção de argumentos matematicamente consistentes, fundamentados na expressão de raciocínios generalizadores
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Van, de Merwe Chelsey Lynn. "Student Use of Mathematical Content Knowledge During Proof Production." BYU ScholarsArchive, 2020. https://scholarsarchive.byu.edu/etd/8474.
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Proof is an important component of advanced mathematical activity. Nevertheless, undergraduates struggle to write valid proofs. Research identifies many of the struggles students experience with the logical nature and structure of proofs. Little research examines the role mathematical content knowledge plays in proof production. This study begins to fill this gap in the research by analyzing what role mathematical content knowledge plays in the success of a proof and how undergraduates use mathematical content knowledge during proofs. Four undergraduates participated in a series of task-based interviews wherein they completed several proofs. The interviews were analyzed to determine how the students used mathematical content knowledge and how mathematical content knowledge affected a proof’s validity. The results show that using mathematical content knowledge during a proof is nontrivial for students. Several of the proofs attempted by the students were unsuccessful due to issues with mathematical content knowledge. The data also show that students use mathematical content knowledge in a variety of ways. Some student use of mathematical content is productive and efficient, while other student practices are less efficient in formal proofs.
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Cramer, Marcos [Verfasser]. "Proof-checking mathematical texts in controlled natural language / Marcos Cramer." Bonn : Universitäts- und Landesbibliothek Bonn, 2013. http://d-nb.info/1045276626/34.
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Tanswell, Fenner Stanley. "Proof, rigour and informality : a virtue account of mathematical knowledge." Thesis, University of St Andrews, 2017. http://hdl.handle.net/10023/10249.
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This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour.
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Olivero, Federica. "The proving process within a dynamic geometry environment." Thesis, University of Bristol, 2003. http://hdl.handle.net/1983/ed52d690-e35f-4bd8-8a3a-74a8b7de5f7c.
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Wallen, Lincoln A. "Automated proof search in non-classical logics : efficient matrix proof methods for modal and intuitionistic logics." Thesis, University of Edinburgh, 1987. http://hdl.handle.net/1842/6600.
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In this thesis we develop efficient methods for automated proof search within an important class of mathematical logics. The logics considered are the varying, cumulative and constant domain versions of the first-order modal logics K, K4, D, D4, T, S4 and S5, and first-order intuitionistic logic. The use of these non-classical logics is commonplace within Computing Science and Artificial Intelligence in applications in which efficient machine assisted proof search is essential. Traditional techniques for the design of efficient proof methods for classical logic prove to be of limited use in this context due to their dependence on properties of classical logic not shared by most of the logics under consideration. One major contribution of this thesis is to reformulate and abstract some of these classical techniques to facilitate their application to a wider class of mathematical logics. We begin with Bibel's Connection Calculus: a matrix proof method for classical logic comparable in efficiency with most machine orientated proof methods for that logic. We reformulate this method to support its decomposition into a collection of individual techniques for improving the efficiency of proof search within a standard cut-free sequent calculus for classical logic. Each technique is presented as a means of alleviating a particular form of redundancy manifest within sequent-based proof search. One important result that arises from this anaylsis is an appreciation of the role of unification as a tool for removing certain proof-theoretic complexities of specific sequent rules; in the case of classical logic: the interaction of the quantifier rules. All of the non-classical logics under consideration admit complete sequent calculi. We anaylse the search spaces induced by these sequent proof systems and apply the techniques identified previously to remove specific redundancies found therein. Significantly, our proof-theoretic analysis of the role of unification renders it useful even within the propositional fragments of modal and intuitionistic logic.
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Nascimento, Anderson de Araújo. "Análise dos tipos de provas matemáticas e pensamento geométrico de alunos do 1º ano do Ensino Médio." Universidade Estadual da Paraíba, 2017. http://tede.bc.uepb.edu.br/jspui/handle/tede/2907.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
The present research work investigated the level of geometric thinking and the types of mathematical proofs by 1st year high school students from the application of a Didactic Proposal. This research was constituted as a qualitative one, and as case study, having instruments of the application an essay with the theme Proofs and Mathematical Demonstrations, Didactic Proposal developed by a team of five members who worked collaboratively, inserted in the Project CAPES/OBEDUC/UFMS/UEPB/UFAL Edital 2012, participant observation and audio recording. We developed the didactic proposal with 18 activities, divided into four parts, which stimulated students to reflect, justify, prove and demonstrate. The application of this proposal occurred in June 2015 for 1st year high school students in a public school in the city of Areia, Paraíba. Our research took place in three moments. In the first moment, we apply the essay on the subject mathematical proofs and demonstrations. In the second moment we did a didactic intervention approaching definitions, theorems, proofs and mathematical demonstrations with the objective of taking to the students this knowledge. In the third moment, Part I and II of the Didactic Proposal were applied, involving activities to conjecture and demonstrate the Pythagorean Theorem, Internal Angle Sum Theorem and External Angle Theorem. This proposal helped in the investigation of the mathematical knowledge of the 1st year high school students, divided into 8 pairs and one trio, chosen freely. The two pairs of students who achieved the best performance in our Didactic Proposal were chosen for our case study and the one of better performance had its dialogue recorded and transcribed as a source of evidence of our case study. In our research we analyzed the answers given by the two pairs on Activities 1 and 3 (Part II) and Activity 2 (Part III), totaling in 3 questions. We used the data triangulation method for our case study. Firstly, we draw the profile of the two pairs of students in relation to Proofs and Mathematical Demonstrations. Next, we investigate the types of mathematical proofs used by them and their geometric thinking. To do so, we use discussions about the levels of geometric thinking proposed by Van Hiele and the types of evidence. From our results we can conclude that the pairs of students were able to develop informal justifications, that is, informal proofs. Thus, the pairs presented pragmatic evidence and the types of evidence Pragmatic Justification and Crucial Example. Regarding the geometric thinking proposed by Van Hiele, only one pair could be classified in one of the levels of development of geometric thinking, Level 3, informal deduction. Therefore, we come to the end of this research convinced that it is necessary to start working mathematical proofs and demonstrations in the basic education level, adapting its teaching to the degree of maturity and to the mathematical knowledge of the students, since our results point out that this subject is not approached properly in the classroom.
A presente pesquisa investigou o nível do pensamento geométrico e os tipos de provas matemáticas de alunos do 1º ano do Ensino Médio a partir da aplicação de uma Proposta Didática. Esta pesquisa se constituiu como qualitativa, e estudo de caso, tendo como instrumentos a aplicação de uma redação com o tema Provas e Demonstrações Matemáticas, Proposta Didática desenvolvida por uma equipe de cinco membros que trabalhou de forma colaborativa, inserida no Projeto CAPES/OBEDUC/UFMS/UEPB/ UFAL Edital 2012, observação participante e gravação em audio do diálgo de umas das duplas participantes da pesquisa. Elaboramos uma proposta didática com 18 atividades, dividida em quatro partes, que estimulavam aos alunos refletirem, justificarem, provarem e demonstrarem. A aplicação dessa proposta se deu em junho de 2015 para alunos do 1º ano do Ensino Médio de uma escola pública da cidade de Areia, Paraíba. Nossa pesquisa se deu em três momentos. No primeiro momento, aplicamos a redação sobre o tema provas e demonstrações matemáticas. No segundo momento realizamos uma intervenção didática abordando definições, teoremas, provas e demonstrações matemáticas com o objetivo de levar aos alunos esses conhecimentos. No terceiro momento foi aplicado a Parte I e II da Proposta Didática, envolvendo atividades de conjecturar e demonstrar o Teorema de Pitágoras, Teorema da Soma dos Ângulos Internos e Teorema dos Ângulo Externo. Essa proposta auxiliou na investigação do conhecimento matemático dos alunos do 1º ano do Ensino Médio, divididos em 8 duplas e um trio, escolhidos livremente. As duas duplas de alunos que obteveram melhores desempenhos em nossa Proposta Didática foram escolhidas para o nosso estudo de caso e a de melhor desenpenho teve seu diálogo gravado e transcrito como fonte de evidência de nosso estudo de caso. Em nossa pesquisa analisamos as respostas dadas pelas duas duplas sobre Atividades 1 e 3 (Parte II) e Atividade 2 (Parte III), totalizando em 3 questões. Utilizamos o método de triângulação de dados para nosso estudo de caso. Primeiramente, traçamos o perfil das duas duplas de alunas com relação às Provas e Demonstrações Matemáticas. Em seguida, investigamos os tipos de provas matemáticas utilizadas por elas e o seu pensamento geométrico. Para tanto, utilizamos as discussões sobre os níveis do pensamento geométrico proposto por Van Hiele e os tipos de provas. A partir de nossos resultados pudemos concluir que as duplas de alunas conseguiram desenvolver justificativas informais, ou seja, provas informais. Assim, as duplas apresentaram provas pragmáticas e os tipos de provas Justificativa Pragmática e Exemplo Crucial. Com relação ao pensamento geométrico proposto por Van Hiele, apenas uma dupla pôde ser classificada em um dos níveis de desenvolvimento do pensamento geométrico, o Nível 3, dedução informal. Portanto, chegamos ao final desta pesquisa convictos de que é preciso iniciar o trabalho das provas e demonstrações matemáticas na Educação Básica, adequando seu ensino ao grau de maturidade e aos conhecimentos matemáticos dos alunos, visto que nossos resultados apontam que esse tema não é abordado adequadamente em sala de aula.
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Evans, Denis J., Debra J. Searles, and Stephen R. Williams. "A simple mathematical proof of boltzmann's equal a priori probability hypothesis." Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-190362.
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Using the Fluctuation Theorem (FT), we give a first-principles derivation of Boltzmann’s postulate of equal a priori probability
in phase space for the microcanonical ensemble. Using a corollary of the Fluctuation Theorem, namely the Second Law Inequality, we show that if the initial distribution differs from the uniform distribution over the energy hypersurface, then under very wide and commonly satisfied conditions, the initial distribution will relax to that uniform distribution. This result is somewhat analogous to the Boltzmann H-theorem but unlike that theorem, applies to dense fluids as well as dilute gases and also permits a nonmonotonic relaxation to equilibrium. We also prove that in ergodic systems the uniform (microcanonical) distribution is the only stationary, dissipationless distribution for the constant energy ensemble.
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Dickerson, David S. "High school mathematics teachers' understandings of the purposes of mathematical proof." Related electronic resource: Current Research at SU : database of SU dissertations, recent titles available full text, 2008. http://wwwlib.umi.com/cr/syr/main.
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Womack, Catherine A. "The crucial role of proof--a classical defense against mathematical empiricism." Thesis, Massachusetts Institute of Technology, 1993. http://hdl.handle.net/1721.1/12678.
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Evans, Denis J., Debra J. Searles, and Stephen R. Williams. "A simple mathematical proof of boltzmann's equal a priori probability hypothesis." Diffusion fundamentals 11 (2009) 57, S. 1-8, 2009. https://ul.qucosa.de/id/qucosa%3A14022.
Full textAbstract:
Using the Fluctuation Theorem (FT), we give a first-principles derivation of Boltzmann’s postulate of equal a priori probability
in phase space for the microcanonical ensemble. Using a corollary of the Fluctuation Theorem, namely the Second Law Inequality, we show that if the initial distribution differs from the uniform distribution over the energy hypersurface, then under very wide and commonly satisfied conditions, the initial distribution will relax to that uniform distribution. This result is somewhat analogous to the Boltzmann H-theorem but unlike that theorem, applies to dense fluids as well as dilute gases and also permits a nonmonotonic relaxation to equilibrium. We also prove that in ergodic systems the uniform (microcanonical) distribution is the only stationary, dissipationless distribution for the constant energy ensemble.
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Meikle, Laura Isabel. "Intuition in formal proof : a novel framework for combining mathematical tools." Thesis, University of Edinburgh, 2014. http://hdl.handle.net/1842/9663.
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This doctoral thesis addresses one major difficulty in formal proof: removing obstructions to intuition which hamper the proof endeavour. We investigate this in the context of formally verifying geometric algorithms using the theorem prover Isabelle, by first proving the Graham’s Scan algorithm for finding convex hulls, then using the challenges we encountered as motivations for the design of a general, modular framework for combining mathematical tools. We introduce our integration framework — the Prover’s Palette, describing in detail the guiding principles from software engineering and the key differentiator of our approach — emphasising the role of the user. Two integrations are described, using the framework to extend Eclipse Proof General so that the computer algebra systems QEPCAD and Maple are directly available in an Isabelle proof context, capable of running either fully automated or with user customisation. The versatility of the approach is illustrated by showing a variety of ways that these tools can be used to streamline the theorem proving process, enriching the user’s intuition rather than disrupting it. The usefulness of our approach is then demonstrated through the formal verification of an algorithm for computing Delaunay triangulations in the Prover’s Palette.
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Gawthorpe, Kateřina. "COMPETING CURRENCIES AS AN ALTERNATIVE SCENARIO TO LEGAL TENDER CLAUSE: MATHEMATICAL PROOF." Master's thesis, Vysoká škola ekonomická v Praze, 2013. http://www.nusl.cz/ntk/nusl-197885.
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Previous literature examining the scenario without the constraint of legal tender law is a rather theoretical analysis of the subject matter. Aside from the theoretical examination of the competition of money this paper offers dynamic structural macroeconomic model based on the money in the utility function. This model compares the current monetary conditions with the potential situation permitting more currencies circulating alongside. The main assumption about individuals' preferences over stable currencies underlines the whole paper with emphasis on the mathematical model. The uniqueness of this model lies in the incorporation of variables affecting respective money demand functions into the utility function of the DSGE model and in the purpose of its use as well as its variables, where representative agent is a household owning a bank rather than a firm. Overall the results of this paper favor the idea of exclusion of the legal tender law in a developed country without severe turmoil. Particularly, the ascent of competition among currencies leads to lower inflation than present scenario. However, final simulations of the model in Matlab supplements such so far "unambiguous" view with skepticism due to possible difficulties during discovery process in such scenario.
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Duff, Karen Malina. "What Are Some of the Common Traits in the Thought Processes of Undergraduate Students Capable of Creating Proof?" Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1856.pdf.
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Gruver, John David. "Growth in Students' Conceptions of Mathematical Induction." BYU ScholarsArchive, 2010. https://scholarsarchive.byu.edu/etd/2166.
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While proof and reasoning lie at the core of mathematical practice, how students learn to reason formally and build convincing proofs continues to invite reflection and discussion. To add to this discussion I investigated how three students grew in their conceptions of mathematical induction. While each of the students in the study had different experiences and grew in different ways, the grounded axes (triggering events, personal questions about mathematics, and personal questions about a particular solution) highlighted patterns in the narratives and from these patterns a theoretical perspective emerged. Reflection, both on mathematics in general and about specific problems, was central to students' growth. The personal reflections of students and triggering events influenced each other in the following way. The questions students wondered about impacted which trigger stimulated growth, while triggers caused students to rethink assumptions and reflect on mathematics or specific problems. The reflections that allowed triggers to stimulate growth along with the reflections that were results of triggering events constitute an "investigative orientation." Each narrative reflects a different investigative orientation motivated by different personal needs. These investigative orientations affected what type of knowledge was constructed.
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Liu, Yating. "Aspects of Mathematical Arguments that Influence Eighth Grade Students’ Judgment of Their Validity." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1373894064.
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Grudić, Gregory Z. "Iterative inverse kinematics with manipulator configuration control and proof of convergence." Thesis, University of British Columbia, 1990. http://hdl.handle.net/2429/42018.
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A complete solution to the inverse kinematics problem for a large class of practical manipulators,
which includes manipulators with no closed form inverse kinematics equations, is presented in this
thesis. A complete solution to the inverse kinematics problem of a manipulator is defined as a method
for obtaining the required joint variable values to establish the desired endpoint position, endpoint
orientation, and manipulator configuration; the only requirement being that the desired solution
exists. For all manipulator geometries that satisfy a set of conditions (THEOREM I), an algorithm
is presented that is theoretically guaranteed to always converge to the desired solution (if it exists).
The algorithm is extensively tested on two complex 6 degree of freedom manipulators which have no
known closed form inverse kinematics equations. It is shown that the algorithm can be used in real
time manipulator control. Applications of the method to other 6 DOF manipulator geometries and to
redundant manipulators are discussed.Applied Science, Faculty of
Electrical and Computer Engineering, Department of
Graduate
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Lai, Lan-chee Nancy. "A study of secondary three students' proof writing in geometry." Hong Kong : University of Hong Kong, 1995. http://sunzi.lib.hku.hk/hkuto/record.jsp?B17092292.
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Vincent, Jill. "Mechanical linkages, dynamic geometry software, and argumentation : supporting a classroom culture of mathematical proof /." Connect to thesis, 2002. http://eprints.unimelb.edu.au/archive/00001399.
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Plaxco, David Bryant. "Relating Understanding of Inverse and Identity to Engagement in Proof in Abstract Algebra." Diss., Virginia Tech, 2015. http://hdl.handle.net/10919/56587.
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In this research, I set out to elucidate the relationships that might exist between students' conceptual understanding upon which they draw in their proof activity. I explore these relationships using data from individual interviews with three students from a junior-level Modern Algebra course. Each phase of analysis was iterative, consisting of iterative coding drawing on grounded theory methodology (Charmaz, 2000, 2006; Glaser and Strauss, 1967). In the first phase, I analyzed the participants' interview responses to model their conceptual understanding by drawing on the form/function framework (Saxe, et al., 1998). I then analyzed the participants proof activity using Aberdein's (2006a, 2006b) extension of Toulmin's (1969) model of argumentation. Finally, I analyzed across participants' proofs to analyze emerging patterns of relationships between the models of participants' understanding of identity and inverse and the participants' proof activity. These analyses contributed to the development of three emerging constructs: form shifts in service of sense-making, re-claiming, and lemma generation. These three constructs provide insight into how conceptual understanding relates to proof activity.Ph. D.
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Kanellos, Ioannis. "Secondary students' proof schemes during the first encounters with formal mathematical reasoning : appreciation, fluency and readiness." Thesis, University of East Anglia, 2014. https://ueaeprints.uea.ac.uk/49759/.
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The topic of the thesis is proof. At Year 9 Greek students encounter proof for the first time in Algebra and Geometry. Thus the principal research question of the thesis is: How do students’ perceive proof when they first encounter it? The analysis tool in order to obtain an image of students’ perception of proof, the Harel and Sowder’s taxonomy, is itself a research question in what concerns its applicability under Greek conditions. Its applicability, of which there is strong evidence, provides the space to shape an image of students’ proof fluency, proof appreciation, proof readiness etc. In order to collect data with regard to answering the research questions in collaboration principally with the class teacher I constructed the two tests on proof that are presented in this thesis. The first test was administered to the students of Year 9 at the beginning of the school year 2010-2011 before the teaching of proof. The second was administered after the teaching of proof of the same school year. Students’ answers were analyzed and provided strong evidence that the Harel and Sowder’s taxonomy is applicable on them. Thus every answer was characterized in terms of the taxonomy. As a result every individual student but also the whole sample is depicted by proof schemes. The major findings of the analysis are the two following: • Students’ proof fluency is higher in simple proof issues. Although they face difficulties when the issues are more demanding, they show high proof appreciation. • There is strong evidence of the applicability of the Harel and Sowder’s taxonomy in a completely different socio-cultural and educational environment in comparison to that of its original invention and application. In the same vein the research proposes the mixture of proof schemes within one proof as theoretical and methodological contribution. Finally from the findings emerge new research questions as e.g. • How teaching and curriculum affect students’ proof schemes? • What is the origin of mixed proof schemes?
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Vasquez, Jose Eduardo. "Wiener-Lévy Theorem : Simple proof of Wiener's lemma and Wiener-Lévy theorem." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-104868.
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The purpose of this thesis is to formulate and proof some theorems about convergences of Fourier series. In essence, we shall formulate and proof Wiener's lemma and Wiener-Lévy theorem which give us weaker conditions for absolute convergence of Fourier series. This thesis follows the classical Fourier analysis approach in a straightforward and detailed way suitable for undergraduate science students.
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Yee, Sean P. "Students' Metaphors for Mathematical Problem Solving." Kent State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=kent1340197978.
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Braendlein, Marcel. "Lithographic fabrication, electrical characterization and proof-of-concept demonstration of sensor circuits comprising organic electrochemical transistors for in vitro and in vivo diagnostics." Thesis, Lyon, 2017. http://www.theses.fr/2017LYSEM007/document.
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Grâce à leurs excellentes propriétés mécaniques, électriques et chimiques, les dispositifs organiques électroniques à base de polymères conducteurs peuvent résoudre l’incompatibilité entre les modules électroniques rigides en silicone et les exigences des tissus mous qui constituent l’environnement biologique. Les avancées en matière de semiconducteurs organiques et en microélectronique ont donné naissance à la bioélectronique. Cette discipline emploie des capteurs à des fins diagnostiques, telles que la détection des métabolites ou la mesure d’un potentiel d’action neuronal, et des actionneurs à des fins thérapeutiques, comme l’application locale d’un traitement à l’intérieur même du corps, ou la stimulation cérébrale profonde afin de guérir un trouble neurologique. En bioélectronique, l’utilisation de matériaux organiques, tels que le polymère conducteur poly(3,4-éthylènedioxythiophène) polystyrène sulfonate de sodium (PEDOT:PSS) a permis de développer des composants électroniques biomédicaux de qualité exceptionnelle, comme par exemple le transistor organique électrochimique (OECT), qui ont été testés in vitro et in vivo. Ce manuscrit explique en détail la fabrication, la fonctionnalisation et la caractérisation du OECT à base de PEDOT:PSS. Afin de pouvoir intégrer ce capteur à des systèmes de mesure biomédicaux déjà établis, l’OECT est intégré à des circuits simples, tels qu’un amplificateur de tension ou un pont de Wheatstone. Ces circuits sont mis à l’épreuve de la pratique clinique, dans le cas de mesures électrocardiographiques, ou de détection de métabolites dans des cellules cancéreuses. Cela permet d’apprécier à la fois leur applicabilité, et leurs limitesDue to their outstanding mechanical, electrical and chemical properties, organic electronic devices based on conducting polymers can bridge the gap between the rigid silicon based read-out electronics and the soft biological environment and will have a huge impact on the medical healthcare sector. The recent advances in the field of organic semiconductors and microelectronics gave rise to a new discipline termed bioelectronics. This discipline deals with sensors for diagnostic purposes, ranging from metabolite detection and DNA recognition all the way to single neuronal firing events, and actuators for therapeutic purposes, through for example active local drug delivery inside the body or deep brain stimulation to cure neurological disorder. The use of organic materials such as the conducting polymer poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS) in the field of bioelectronics has brought about a variety of outstanding electronic biomedical devices, such as the organic electrochemical transistor (OECT), that have been implemented for both in vitro and in vivo applications. The present manuscript gives a detailed explanation of the fabrication, functionalization and characterization of OECTs based on PEDOT:PSS. To be able to intercept this sensor element with traditional biomedical recording systems, the OECT is implemented into simple circuit layouts such as a voltage amplifier or a Wheatstone bridge. These sensor circuits are then applied to real-life biomedical challenges, such as electrocardiographic recordings or metabolite detection in tumor cell cultures, to demonstrate their applicability as well as their limitations
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McClure, Bruce Alexander Carleton University Dissertation Engineering Mechanical and Aerospace. "The TEther LABoratory demonstration system (TE-LAB); design, operation and mathematical model validation using an eigenvalue approach." Ottawa, 1995.
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Tsiatouras, Vasilis. "Mathematics and the USSR : organising a discipline." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/19547.
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This thesis aspires to establish a new research direction in STS. In the first chapter a literature review is conducted and the research questions are being formulated. The second chapter is devoted to presenting research findings from the archaeological, biological and brain sciences in a unified form. The various stone tool technologies are analysed, and a brief introduction follows into human evolution and the effects that artefacts had on it; then recent neurobiological research on the deeper relationships between consciousness, artefacts and the brain is presented. In the third chapter, after an introduction in the deeper neurological relationships between language and gestures, a gestural analysis of mathematical speech follows, based on visual data generated from an interview with a working mathematician; the last section examines recent research on gesture and mathematics as special cases of Roman Ingarden’s aesthetic theory. In the fourth chapter, four approaches to the social history of mathematics in the USSR are presented, based on data generated from interviews with former professional Soviet mathematicians. Following a Maussian approach, the Soviet mathematical community is presented as a gift economy of scientific articles. Then, in line with a Marxian approach, the Soviet university mathematical school is presented as a factory with its own mode of self-production. In the following section, based on a Parsonian systemic approach, the Soviet mathematical community is presented as a banking system, with the scientific journals as the banking institutions. In the next section of the fourth chapter, following a Weberian approach, the mathematical community in the USSR is presented as a social estate, as separate and distinct from other Soviet social estates. The final section integrates the previous approaches and presents the Soviet mathematics research community as a modern version of an ancient city-state. In the fifth chapter Hilbert spaces are briefly presented, as an example of the fictional universe of modern mathematics, along with some conjectured differences between Soviet and Western mathematics research. In the final chapter, the conclusions of this research project are summarised, and this thesis is presented as an instance of a proposed revised version of David Bloor’s Strong Programme.
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Lai, Lan-chee Nancy, and 黎蘭芝. "A study of secondary three students' proof writing in geometry." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1995. http://hub.hku.hk/bib/B31957936.
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Alves, Thiago de Oliveira. "Lógica formal e sua aplicação na argumentação matemática." Universidade Federal de Juiz de Fora (UFJF), 2016. https://repositorio.ufjf.br/jspui/handle/ufjf/3248.
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O uso da Lógica é de fundamental importância no desenvolvimento de teorias matemáticas modernas, que buscam deduzir de axiomas e conceitos primitivos todo seu corpo de teoremas e consequências. O objetivo desta dissertação é descrever as ferramentas da Lógica Formal que possam ter aplicações imediatas nas demonstrações de conjecturas e teoremas, trazendo justificativa e significado para as técnicas dedutivas e argumentos normalmente utilizados na Matemática. Além de temas introdutórios sobre argumentação e âmbito da lógica, o trabalho todo é apresentado por método sistemático em busca de um critério formal que possa separar os argumentos válidos dos inválidos. Conclui-se que com uma boa preparação inicial no campo da Lógica Formal, o matemático iniciante possa ter uma referência sobre como proceder estrategicamente nos processos de provas de conjecturas e um conhecimento mais profundo ao entender os motivos da validade dos teoremas que encontrará ao se dedicar a sua área de formação.
TheuseofLogicisoffundamentalimportanceinthedevelopmentofmodernmathematical theories that seek deduce from axioms and primitive concepts all your body of theorems and consequences. The aim of this work is to describe the tools of Formal Logic that may have immediate applications in the statements of theorems and conjectures, bringing justification and meaning to the deductive techniques and arguments commonly used in Mathematics. In addition to introductory topics on argumentation and scope of Logic, all the work is presented by systematic method in search of a formal criterion that can separate the valid arguments of the invalids. It follows that with a good initial preparation in the field of Formal Logic, the novice mathematician could have a reference on how to strategically proceed in conjectures evidence processes and a deeper knowledge to understand the reasons for the validity of theorems found on their training area.
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Sommerhoff, Daniel [Verfasser], and Stefan [Akademischer Betreuer] Ufer. "The individual cognitive resources underlying students' mathematical argumentation and proof skills : from theory to intervention / Daniel Sommerhoff ; Betreuer: Stefan Ufer." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2017. http://d-nb.info/1163949361/34.
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Sommerhoff, Daniel Verfasser], and Stefan [Akademischer Betreuer] [Ufer. "The individual cognitive resources underlying students' mathematical argumentation and proof skills : from theory to intervention / Daniel Sommerhoff ; Betreuer: Stefan Ufer." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2017. http://nbn-resolving.de/urn:nbn:de:bvb:19-226879.
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Lelay, Catherine. "Repenser la bibliothèque réelle de Coq : vers une formalisation de l'analyse classique mieux adaptée." Thesis, Paris 11, 2015. http://www.theses.fr/2015PA112096/document.
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L'analyse réelle a de nombreuses applications car c'est un outil approprié pour modéliser de nombreux phénomènes physiques et socio-économiques. En tant que tel, sa formalisation dans des systèmes de preuve formelle est justifié pour permettre aux utilisateurs de vérifier formellement des théorèmes mathématiques et l'exactitude de systèmes critiques. La bibliothèque standard de Coq dispose d'une axiomatisation des nombres réels et d'une bibliothèque de théorèmes d'analyse réelle. Malheureusement, cette bibliothèque souffre de nombreuses lacunes. Par exemple, les définitions des intégrales et des dérivées sont basées sur les types dépendants, ce qui les rend difficiles à utiliser dans la pratique. Cette thèse décrit d'abord l'état de l'art des différentes bibliothèques d'analyse réelle disponibles dans les assistants de preuve. Pour pallier les insuffisances de la bibliothèque standard de Coq, nous avons conçu une bibliothèque facile à utiliser : Coquelicot. Une façon plus facile d'écrire les formules et les théorèmes a été mise en place en utilisant des fonctions totales à la place des types dépendants pour écrire les limites, dérivées, intégrales et séries entières. Pour faciliter l'utilisation, la bibliothèque dispose d'un ensemble complet de théorèmes couvrant ces notions, mais aussi quelques extensions comme les intégrales à paramètres et les comportements asymptotiques. En plus, une hiérarchie algébrique permet d'appliquer certains théorèmes dans un cadre plus générique comme les nombres complexes pour les matrices. Coquelicot est une extension conservative de l'analyse classique de la bibliothèque standard de Coq et nous avons démontré les théorèmes de correspondance entre les deux formalisations. Nous avons testé la bibliothèque sur plusieurs cas d'utilisation : sur une épreuve du Baccalauréat, pour les définitions et les propriétés des fonctions de Bessel ainsi que pour la solution de l'équation des ondes en dimension 1Real analysis is pervasive to many applications, if only because it is a suitable tool for modeling physical or socio-economical systems. As such, its support is warranted in proof assistants, so that the users have a way to formally verify mathematical theorems and correctness of critical systems. The Coq system comes with an axiomatization of standard real numbers and a library of theorems on real analysis. Unfortunately, this standard library is lacking some widely used results. For instance, the definitions of integrals and derivatives are based on dependent types, which make them cumbersome to use in practice. This thesis first describes various state-of-the-art libraries available in proof assistants. To palliate the inadequacies of the Coq standard library, we have designed a user-friendly formalization of real analysis: Coquelicot. An easier way of writing formulas and theorem statements is achieved by relying on total functions in place of dependent types for limits, derivatives, integrals, power series, and so on. To help with the proof process, the library comes with a comprehensive set of theorems that cover not only these notions, but also some extensions such as parametric integrals and asymptotic behaviors. Moreover, an algebraic hierarchy makes it possible to apply some of the theorems in a more generic setting, such as complex numbers or matrices. Coquelicot is a conservative extension of the classical analysis of Coq's standard library and we provide correspondence theorems between the two formalizations. We have exercised the library on several use cases: in an exam at university entry level, for the definitions and properties of Bessel functions, and for the solution of the one-dimensional wave equation
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Boudjani, Nadira. "Aide à la construction et l'évaluation des preuves mathématiques déductives par les systèmes d'argumentation." Thesis, Montpellier, 2018. http://www.theses.fr/2018MONTS060/document.
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L'apprentissage des preuves mathématiques déductives est fondamental dans l'enseignement des mathématiques. Pourtant, la dernière enquête TIMSS (Trends in International Mathematics and Science Study) menée par l'IEA ("International Association for the Evaluation of Educational Achievement") en mars 2015, le niveau général des étudiants en mathématiques est en baisse et les étudiants éprouvent de plus en plus de difficultés pour comprendre et écrire les preuves mathématiques déductives.Pour aborder ce problème, plusieurs travaux en didactique des mathématiques utilisent l’apprentissage collaboratif en classe.L'apprentissage collaboratif consiste à regrouper des étudiants pour travailler ensemble dans le but d'atteindre un objectif commun. Il repose sur le débat et l'argumentation. Les étudiants s'engagent dans des discussions pour exprimer leurs points de vue sous forme d'arguments et de contre-arguments dans le but de résoudre un problème posé.L’argumentation utilisée dans ces approches est basée sur des discussions informelles qui permettent aux étudiants d'exprimer publiquement leurs déclarations et de les justifier pour construire des preuves déductives. Ces travaux ont montré que l’argumentation est une méthode efficace pour l’apprentissage des preuves mathématiques : (i) elle améliore la pensée critique et les compétences métacognitives telles que l'auto-surveillance et l'auto-évaluation (ii) augmente la motivation des étudiants par les interactions sociales et (iii) favorise l'apprentissage entre les étudiants. Du point de vuedes enseignants, certaines difficultés surgissent avec ces approches pour l'évaluation des preuves déductives. En particulier, l'évaluation des résultats, qui comprend non seulement la preuve finale mais aussi les étapes intermédiaires, les discussions, les conflits qui peuvent exister entre les étudiants durant le débat. En effet, cette évaluation introduit une charge de travail importante pour les enseignants.Dans cette thèse, nous proposons un système pour la construction et l'évaluation des preuves mathématiques déductives. Ce système a un double objectif : (i) permettre aux étudiants de construire des preuves mathématiques déductives à partir un débat argumentatif structuré (ii) aider les enseignants à évaluer ces preuves et toutes les étapes intermédiaires afin d'identifier les erreurs et les lacunes et de fournir un retour constructif aux étudiants.Le système offre aux étudiants un cadre structuré pour débattre durant la construction de la preuve en utilisant les cadres d'argumentation proposés en intelligente artificielle. Ces cadres d’argumentation sont utilisés aussi dans l’analyse du débat qui servira pour représenter le résultat sous différentes formes afin de faciliter l’évaluation aux enseignants. Dans un second temps, nous avons implanté et validé le système par une étude expérimentale pour évaluer son acceptabilité dans la construction collaborative des preuves déductives par les étudiants et dans l’évaluation de ces preuves par les enseignantsLearning deductive proofs is fundamental for mathematics education. Yet, many students have difficulties to understand and write deductive mathematical proofs which has severe consequences for problem solving as highlighted by several studies. According to the recent study of TIMSS (Trends in International Mathematics and Science Study), the level of students in mathematics is falling. students have difficulties to understand mathematics and more precisely to build and structure mathematical proofs.To tackle this problem, several approaches in mathematical didactics have used a social approach in classrooms where students are engaged in a debate and use argumentation in order to build proofs.The term "argumentation" in this context refers to the use of informal discussions in classrooms to allow students to publicly express claims and justify them to build proofs for a given problem. The underlying hypotheses are that argumentation: (i) enhances critical thinking and meta-cognitive skills such as self monitoring and self assessment; (ii) increases student's motivation by social interactions; and (iii) allows learning among students. From instructors' point of view, some difficulties arise with these approaches for assessment. In fact, the evaluation of outcomes -- that includes not only the final proof but also all intermediary steps and aborted attempts -- introduces an important work overhead.In this thesis, we propose a system for constructing and evaluating deductive mathematical proofs. The system has a twofold objective: (i) allow students to build deductive mathematical proofs using structured argumentative debate; (ii) help the instructors to evaluate these proofs and assess all intermediary steps in order to identify misconceptions and provide a constructive feedback to students. The system provides students with a structured framework to debate during construction of proofs using the proposed argumentation frameworks in artificial intelligence. These argumentation frameworks are also used in the analysis of the debate which will be used to represent the result in different forms in order to facilitate the evaluation to the instructors. The system has been implemented and evaluated experimentally by students in the construction of deductive proofs and instructors in the evaluation of these proofs
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Dhaher, Yaser Yousef. "The Effect of a Modified Moore Method on Conceptualization of Proof Among College Students." Kent State University / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=kent1197511801.
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Bubp, Kelly M. "To Prove or Disprove: The Use of Intuition and Analysis by Undergraduate Students to Decide on the Truth Value of Mathematical Statements and Construct Proofs and Counterexamples." Ohio University / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1417178872.
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Myers, Jeremy. "Computational Fluid Dynamics in a Terminal Alveolated Bronchiole Duct with Expanding Walls: Proof-of-Concept in OpenFOAM." VCU Scholars Compass, 2017. http://scholarscompass.vcu.edu/etd/5011.
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Mathematical Biology has found recent success applying Computational Fluid Dynamics (CFD) to model airflow in the human lung. Detailed modeling of flow patterns in the alveoli, where the oxygen-carbon dioxide gas exchange occurs, has provided data that is useful in treating illnesses and designing drug-delivery systems. Unfortunately, many CFD software packages have high licensing fees that are out of reach for independent researchers. This thesis uses three open-source software packages, Gmsh, OpenFOAM, and ParaView, to design a mesh, create a simulation, and visualize the results of an idealized terminal alveolar sac model. This model successfully demonstrates that OpenFOAM can be used to model airflow in the acinar region of the lung under biologically relevant conditions.
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Varella, Márcia. "Prova e demonstração na geometria analítica: uma análise das organizações didática e matemática em materiais didáticos." Pontifícia Universidade Católica de São Paulo, 2010. https://tede2.pucsp.br/handle/handle/10844.
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Previous issue date: 2010-11-25Secretaria da Educação do Estado de São Paulo
This study aims to analyze how the authors of teaching materials of high school have organized tasks proposals with proofs and demonstrations on content proposed of Analytic Geometry on 3rd series of high school. With a view to proposing some thoughts in this respect, we analyze the collections of textbooks approved by the National Program of textbook for high school (PNLEM/2009) and the notebooks bimonthly adopted by the Education Secretary of the State of São Paulo (SEESP/2009), made available to students and teachers, distinctly. We judge the relevance in analyzing these materials because we act in the public network of State of São Paulo aiming contributions that may occur. The analysis of these materials was carried out considering the tasks proposed on the content Analytic Geometry, limited to studying the equation of a line. The theoretical contribution that substantiate our analyses followed the assumptions of Anthropological Theory of didactic Yves Chevallard (1999) that focuses the study of praxeological organization mathematics and didactics designed for teaching and learning of Mathematics and the work of Nicolas Balacheff (1988), which aims to study the typology of proofs produced by students. Supported by this theoretical, we realize our analyses with the purpose of responding to our question: Which mathematics and didactics organizations involving proofs and demonstration are proposed for didactics materials of high school, on content Analytic Geometry? Answering this question, we developed a qualitative research with approach documentary and from survey bibliographic we might have idea of problems involved in the teaching and learning of proofs and demonstrations on mathematical content, both in elementary and high school. The analysis of these materials confirmed two of our research hypotheses and showed that the work with proofs and demonstrations in didactics materials was not abandoned, but the clarity of the terms belonging to deductive system is unsatisfactory for understanding of what is demonstration in Mathematics
O presente estudo tem como objetivo analisar como os autores de materiais didáticos do Ensino Médio organizaram as tarefas propostas envolvendo provas e demonstrações no conteúdo Geometria Analítica para a 3ª. série do Ensino Médio. Com o intuito de propor algumas reflexões a esse respeito, decidimos analisar as coleções de livros didáticos aprovadas pelo Programa Nacional do Livro Didático para o Ensino Médio (PNLEM/2009) e os cadernos bimestrais adotados pela Secretaria da Educação do Estado de São Paulo (SEESP/2009), disponibilizados para alunos e professores, distintamente. Julgamos a pertinência de analisar conjuntamente esses materiais por atuarmos na rede pública estadual paulista, visando às contribuições que vierem a ocorrer. A análise desses materiais foi realizada considerando as tarefas propostas sobre o conteúdo Geometria Analítica, limitado ao estudo da equação de uma reta. O aporte teórico que fundamentou nossas análises seguiu os pressupostos da Teoria Antropológica do Didático de Yves Chevallard (1999), que focaliza o estudo das organizações praxeológicas Matemática e didática pensadas para o ensino e aprendizagem da Matemática, e o trabalho de Nicolas Balacheff (1988), que visa ao estudo da tipologia de provas produzidas por alunos. Apoiado por esse referencial teórico efetivamos nossas análises com o intuito de responder à nossa questão de pesquisa: Quais organizações Matemáticas e didáticas envolvendo prova e demonstração são propostas por materiais didáticos do Ensino Médio, no conteúdo Geometria Analítica? Visando a responder a esta questão, desenvolvemos uma pesquisa qualitativa com enfoque documental, e a partir do levantamento bibliográfico pudemos ter ideia da problemática envolvida no ensino e na aprendizagem de provas e demonstrações em conteúdos matemáticos, tanto no Ensino Fundamental quanto no Ensino Médio. A análise desses materiais confirmou duas de nossas hipóteses de pesquisa e nos revelou que o trabalho com provas e demonstrações em materiais didáticos não foi abandonado, porém a clareza dos termos pertencentes ao sistema dedutivo é insatisfatória no que diz respeito à compreensão do que seja passível de demonstração em Matemática
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Veng, Mengkoung. "Self-mixing interferometry for absolute distance measurement : modelling and experimental demonstration of intrinsic limitations." Thesis, Toulouse, INPT, 2020. http://www.theses.fr/2020INPT0077.
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La Self-Mixing Interférométrie (SMI) a été étudiée de manière approfondie au cours des cinq dernières décennies dans diverses applications. Les capteurs selon la technique SMI ont la diode laser comme la source de lumière, l'interféromètre et le détecteur. La lumière de la diode laser se propage vers une cible éloignée où elle est partiellement réfléchie ou rétrodiffusée avant d'être réinjectée dans la cavité active du laser. Lorsque la diode laser subit le retour optique externe, la lumière réfléchie imprimée avec des informations provenant de la cible éloignée ou du milieu de cavité externe induit une perturbation des paramètres de fonctionnement du laser. Pour les capteurs de mesure SMI tels que les applications de mouvement harmonique et de distance absolue, la méthode de comptage des franges est essentiellement utilisée pour déterminer respectivement le déplacement et la distance de la cible. La modélisation du phénomène SMI a été développée. L'équation unique qui décrit la condition de phase imposée par la rétro-injection optique est généralement appelée équation de phase. L'un des paramètres les plus importants cet équation est le paramètre C. Quand le C 1, le comportement du laser est stable. En revanche, quand le C > 1, des phénomènes plus complexes sont observés tels que l'effet d'hystérésis, la présence de multiples fréquences d'émission, la séparation de la ligne d'émission de fréquence causer la saut de mode et le phénomène de disparition des franges. Une approche bien acceptée dans la communauté décrit les régimes de SMI en fonction de la valeur du C de sorte que : le régime faible (0.1 < C < 1), le régime modéré (1 < C < 4.6) et le régime fort (C > 4,6). Le paramètre de rétro-injection C est directement impliqué dans le phénomène de disparition des franges. Bernal et al. a décrit que ce phénomène dépend de la régime qui décrits ci-dessus, c'està-dire que les franges commencent à disparaître uniquement dans le régime forte de la rétroinjection d’optique, tandis que Yu et al. a démontré que le nombre des franges est divisée par 2 dans la région 2 (7,8 < C < 14,0), 3 dans la région 3 (14,0 < C < 20,3) et ainsi de suite. Autres publications a proposé que deux paires de franges interférométriques pour une période complète de modulation disparaissent quand il y a une variation de C de 2. Cependant, à notre connaissance, aucune explication ou théorie précise sur le mécanisme de ce phénomène n'a été publiée jusqu'à présent. Dans cette thèse, nous rapportons l'observation de la disparition des franges dans le schéma de mesure de distance absolue. Par rapport au schéma de détection des vibrations, l'approche de la distance absolue garantit un paramètre de rétro-injection C stable permettant ainsi des conditions expérimentales plus répétables. Comme la cible est fixée à une certaine distance, l'amplitude de la lumière rétrodiffusée est facile à contrôler en utilisant des atténuateurs d’optiques variables. Les résultats expérimentaux montrent que le nombre de franges interférométriques continue de diminuer dans le signal SMI lorsque l'amplitude du coefficient de réflexion de la cible augmenteSelf-mixing Interferometry has been studied extensively in the last five decades in various sensing applications. Sensors under the SMI technique have the laser diode as the light source, the interferometer, and the detector. The light from the laser diode propagates towards a distant target where it is partially reflected or back-scattered before being re-injected into the active cavity of the laser. When the laser diode experiences the external optical feedback, the reflected light imprinted with information from the distant target or from the external cavity medium induces perturbation to the operating parameters of the laser. For SMI measurement sensors such as harmonic motion and absolute distance applications, the fringe counting method is basically used to determine the target's displacement and distance respectively. Two different approaches to modelling the SMI phenomenon have been developed: the three-mirror cavity and the perturbation of the rate equation. The single equation that describes the phase condition imposed by the optical feedback is usually referred to as the excess phase equation. One of the most important and most useful parameters in the excess phase equation is the feedback parameter C as it can be used to qualitatively categorize the regime of the laser under optical feedback. When the feedback level C < 1, the laser behaviour is stable. On the other hand, when the feedback level C > 1, more complex phenomena are observed such as hysteresis effect, presence of multiple emission frequencies (including the unstable frequencies), apparent splitting of the emission line due to mode hopping and fringe disappearance phenomenon. The fringes disappearance phenomenon in the self-mixing interferometry occurs whenever the external round-trip phase at free-running state is modulated by either external modulation such as external cavity length changes or internal modulation when the laser injection current is modulated with a high back-scattered light power. This phenomenon has been observed by many authors for harmonic motion or vibration application and more recently in the case of the absolute distance measurement scheme when the laser injection current is modulated in the triangle waveform. This phenomenon is highly dependent on the feedback parameter C and it is described in detail based on the coupled cavity model. The primary cause for fringes disappearance is demonstrated to be the expansion of the excess phase equation stable solutions range with the increment of the parameter C, thus reducing the number of stable solutions for a given phase stimulus. This new approach in the modelling of the fringe disappearance phenomenon allows determination of the C values for which a pair of fringes are expected to disappear and as a consequence correlates the number of missing fringes to the value of C. This approach is validated both by a behavioural model of the laser under optical feedback and by a series of measurements in the SMI absolute distance configuration
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Yepremyan, Astrik. "Of Proofs, Mathematicians, and Computers." Scholarship @ Claremont, 2015. http://scholarship.claremont.edu/scripps_theses/723.
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As computers become a more prevalent commodity in mathematical research and mathematical proof, the question of whether or not a computer assisted proof can be considered a mathematical proof has become an ongoing topic of discussion in the mathematics community. The use of the computer in mathematical research leads to several implications about mathematics in the present day including the notion that mathematical proof can be based on empirical evidence, and that some mathematical conclusions can be achieved a posteriori instead of a priori, as most mathematicians have done before. While some mathematicians are open to the idea of a computer-assisted proof, others are skeptical and would feel more comfortable if presented with a more traditional proof, as it is more surveyable. A surveyable proof enables mathematicians to see the validity of a proof, which is paramount for mathematical growth, and offer critique. In my thesis, I will present the role that the mathematical proof plays within the mathematical community, and thereby conclude that because of the dynamics of the mathematical community and the constant activity of proving, the risks that are associated with a mistake that stems from a computer-assisted proof can be caught by the scrupulous activity of peer review in the mathematics community. Eventually, as the following generations of mathematicians become more trained in using computers and in computer programming, they will be able to better use computers in producing evidence, and in turn, other mathematicians will be able to both understand and trust the resultant proof. Therefore, it remains that whether or not a proof was achieved by a priori or a posteriori, the validity of a proof will be determined by the correct logic behind it, as well as its ability to convince the members of the mathematical community—not on whether the result was reached a priori with a traditional proof, or a posteriori with a computer-assisted proof.
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Efimova, Hagsröm Inga. "Matematiskt resonemang på högstadiet : En studie av vilka strategier högstadieelever väljer vid matematiska resonemangsföringar." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-9266.
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Arbetets syfte är att undersöka hur högstadieelever för matematiskt resonemang. De frågeställningar som studien inriktas på är vilka lösningsstrategier elever väljer då de resonerar matematiskt såväl som vad det finns för skillnader och likheter mellan de yngre elevernas lösningar och de äldre elevernas lösningar. Undersökningen genomfördes i två klasser, den ena i årskurs 8 och den andra i årskurs 9, på en grundskola. Eleverna fick lösa uppgifter, vilka uppmanade dem att föra matematiskt resonemang, individuellt. Resultatet av studien visar att majoriteten av undersökta elever har valt att resonera deduktivt. Jämförelsen av elevers lösningar i två årskurser visar att årskurs 9 elevers resonemangsföring präglas av större förtrogenhet med den algebraiska demonstrationen. Resultatet visar även att elever med högre kunskaper om algebra oftare visar benägenheter till att vidaregeneralisera de givna påståendena.The purpose of this study is to examine secondary school students’ strategies of reasoning. The study inquires into which strategies students choose when reasoning mathematically as well as differences and similarities between the younger students’ solutions and the older students’ solutions. The study was conducted in two classes, in years 8 and 9 respectively, at a secondary school. The students were asked to solve tasks, which encouraged them to reason mathematically, on individual basis. The study revealed that the majority of students had chosen to reason deductively. The comparison of students’ presented answers in two years showed that the ninth-graders’ solutions are characterized of greater skill when it comes to algebraic demonstrations. The results of the study also reveal that students with stronger algebraic abilities attempt more often to generalize the given mathematical statements further.
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Souba, Matthew. "From the Outside Looking In: Can mathematical certainty be secured without being mathematically certain that it has been?" The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1574777956439624.
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Paulino, Zita da Conceição Russo. "A demonstração matemática com recurso a um ambiente de geometria dinâmica: um estudo de caso com alunos de 10º ano de escolaridade." Master's thesis, Universidade de Évora, 2012. http://hdl.handle.net/10174/11572.
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Esta investigação analisa como um ambiente de geometria dinâmica, o Geogebra, influencia o processo de demonstração matemática desenvolvido por alunos de 10.º ano.
Constituindo uma dificuldade para muitos alunos, a demonstração impõe-se como um processo matemático essencial, realçado nas atuais orientações curriculares. Os ambientes de geometria dinâmica, com tarefas adequadas, podem apoiar os alunos desde a formulação de conjeturas à sua demonstração.
Optou-se por uma abordagem interpretativa e qualitativa, concretizada através de um estudo de caso de uma turma, sujeita a uma intervenção didática focada na resolução de tarefas que requeriam demonstração.
Concluiu-se que a demonstração pode estar viva na aula de Matemática, e ser vista como o culminar do processo investigativo que é facilitado pelo Geogebra. O alcance deste ultrapassa o estímulo à formulação e testes de conjeturas, auxiliando também a realização das próprias demonstrações, dado que as construções produzidas podem constituir, em muitos casos, a base das demonstrações; ### Abstract:
This research analyzes how a dynamic geometry environment, Geogebra, influences the process of mathematical demonstration developed by students of the 10th grade.
Representing a difficulty for many students, the demonstration is a key mathematical process, highlighted in the current curriculum guidelines. With appropriate tasks, dynamic geometry environments can support students from the formulation of conjectures to their demonstration.
A qualitative and interpretative approach was chosen, implemented through a case study of a class undergoing a didactic intervention focused on solving tasks requiring demonstration.
The conclusion is that the demonstration can be alive in math class and it is the high point of the investigative process enabled by Geogebra, whose scope goes beyond stimulating the formulation of conjectures and their test. Geogebra also contributes to perform the demonstrations themselves, as the constructions produced often constitute the basis of the demonstrations.
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Peske, Wendy Ann. "A topological approach to nonlinear analysis." CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2779.
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A topological approach to nonlinear analysis allows for strikingly beautiful proofs and simplified calculations. This topological approach employs many of the ideas of continuous topology, including convergence, compactness, metrization, complete metric spaces, uniform spaces and function spaces. This thesis illustrates using the topological approach in proving the Cauchy-Peano Existence theorem. The topological proof utilizes the ideas of complete metric spaces, Ascoli-Arzela theorem, topological properties in Euclidean n-space and normed linear spaces, and the extension of Brouwer's fixed point theorem to Schauder's fixed point theorem, and Picard's theorem.
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Terrematte, Patrick Cesar Alves. "A integra??o do tutorial interativo TryLogic via IMS Learning Tools Interoperability: construindo uma infraestrutura para o ensino de L?gica atrav?s de estrat?gias de demonstra??o e refuta??o." Universidade Federal do Rio Grande do Norte, 2013. http://repositorio.ufrn.br:8080/jspui/handle/123456789/18685.
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Previous issue date: 2013-06-03Logic courses represent a pedagogical challenge and the recorded number of cases of failures and of discontinuity in them is often high. Amont other difficulties, students face a cognitive overload to understand logical concepts in a relevant way. On that track, computational tools for learning are resources that help both in alleviating the cognitive overload scenarios and in allowing for the practical experimenting with theoretical concepts. The present study proposes an interactive tutorial, namely the TryLogic, aimed at teaching to solve logical conjectures either by proofs or refutations. The tool was developed from the architecture of the tool TryOcaml, through support of the communication of the web interface ProofWeb in accessing the proof assistant Coq. The goals of TryLogic are: (1) presenting a set of lessons for applying heuristic strategies in solving problems set in Propositional Logic; (2) stepwise organizing the exposition of concepts related to Natural Deduction and to Propositional Semantics in sequential steps; (3) providing interactive tasks to the students. The present study also aims at: presenting our implementation of a formal system for refutation; describing the integration of our infrastructure with the Virtual Learning Environment Moodle through the IMS Learning Tools Interoperability specification; presenting the Conjecture Generator that works for the tasks involving proving and refuting; and, finally to evaluate the learning experience of Logic students through the application of the conjecture solving task associated to the use of the TryLogic
A disciplina de L?gica representa um desa o tanto para docentes como para discentes, o que em muitos casos resulta em reprova??es e desist?ncias. Dentre as dificuldades enfrentadas pelos alunos est? a sobrecarga da capacidade cognitiva para compreender os conceitos l?gicos de forma relevante. Neste sentido, as ferramentas computacionais de aprendizagem s?o recursos que auxiliam a redu??o de cen?rios de sobrecarga cognitiva, como tamb?m permitem a experi?ncia pr?tica de conceitos te?ricos. O presente trabalho prop?e uma tutorial interativo chamado TryLogic, visando ao ensino da tarefa de Demonstra??o ou Refuta??o (DxR) de conjecturas l?gicas. Trata-se de uma ferramenta desenvolvida a partir da arquitetura do TryOcaml atrav?s do suporte de comunica??o da interface web ProofWeb para acessar o assistente de demonstra??o de teoremas Coq. Os objetivos do TryLogic s?o: (1) Apresentar um conjunto de li??es para aplicar estrat?gias heur?sticas de an?lise de problemas em L?gica Proposicional; (2) Organizar em passo-a-passo a exposi ??o dos conte?dos de Dedu??o Natural e Sem?ntica Proposicional de forma sequencial; e (3) Fornecer aos alunos tarefas interativas. O presente trabalho prop?e tamb?m apresentar a nossa implementa??o de um sistema formal de refuta??o; descrever a integra??o de nossa infraestrutura com o Ambiente Virtual de Aprendizagem Moodle atrav?s da especi ca??o IMS Learning Tools Interoperability ; apresentar o Gerador de Conjecturas de tarefas de Demonstra??o e Refuta??o e, por m, avaliar a experi?ncia da aprendizagem de alunos de L?gica atrav?s da aplica??o da tarefa de DxR em associa??o ? utiliza??o do TryLogic
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Dumbravă, Ştefania-Gabriela. "Formalisation en Coq de Bases de Données Relationnelles et Déductives -et Mécanisation de Datalog." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLS525/document.
Full textAbstract:
Cette thèse présente une formalisation en Coq des langages et des algorithmes fondamentaux portant sur les bases de données. Ainsi, ce fourni des spécifications formelles issues des deux approches différentes pour la définition des modèles de données: une basée sur l’algèbre et l'autre basée sur la logique.A ce titre, une première contribution de cette thèse est le développement d'une bibliothèque Coq pour le modèle relationnel. Cette bibliothèque contient les modélisations de l’algèbre relationnelle et des requêtes conjonctives. Il contient aussi une mécanisation des contraintes d'intégrité et de leurs procédures d'inférence. Nous modélisons deux types de contraintes: les dépendances, qui sont parmi les plus courantes: les dépendances fonctionnelles et les dépendances multivaluées, ainsi que leurs axiomatisations correspondantes. Nous prouvons formellement la correction de leurs algorithmes d'inférence et, pour le cas de dépendances fonctionnelles, aussi la complétude.Ces types de dépendances sont des instances de contraintes plus générales : les dépendances génératrices d'égalité (equality generating dependencies, EGD) et, respectivement, les dépendances génératrices de tuples (tuple generating dependencies, TGD), qui appartiennent a une classe encore plus large des dépendances générales (general dependencies). Nous modélisons ces dernières et leur procédure d'inférence, i.e, "the chase", pour lequel nous établissons la correction. Enfin, on prouve formellement les théorèmes principaux des bases de données, c'est-à-dire, les équivalences algébriques, la théorème de l' homomorphisme et la minimisation des requêtes conjonctives.Une deuxième contribution consiste dans le développement d'une bibliothèque Coq/ssreflect pour la programmation logique, restreinte au cas du Datalog. Dans le cadre de ce travail, nous donnons la première mécanisations d'un moteur Datalog standard et de son extension avec la négation. La bibliothèque comprend une formalisation de leur sémantique en theorie des modelés ainsi que de leur sémantique par point fixe, implémentée par une procédure d'évaluation stratifiée. La bibliothèque est complétée par les preuves de correction, de terminaison et de complétude correspondantes. Cette plateforme ouvre la voie a la certification d' applications centrées donnéesThis thesis presents a formalization of fundamental database theories and algorithms. This furthers the maturing state of the art in formal specification development in the database field, with contributions stemming from two foundational approches to database models: relational and logic based.As such, a first contribution is a Coq library for the relational model. This contains a mechanization of integrity constraints and of their inference procedures. We model two of the most common dependencies, namely functional and multivalued, together with their corresponding axiomatizations. We prove soundness of their inference algorithms and, for the case of functional ones, also completeness. These types of dependencies are instances of equality and, respectively, tuple generating dependencies, which fall under the yet wider class of general dependencies. We model these and their inference procedure,i.e, the chase, for which we establish soundness.A second contribution consists of a Coq/Ssreflect library for logic programming in the Datalog setting. As part of this work, we give (one of the) first mechanizations of the standard Datalog language and of its extension with negation. The library includes a formalization of their model theoretical semantics and of their fixpoint semantics, implemented through bottom-up and, respectively, through stratified evaluation procedures. This is complete with the corresponding soundness, termination and completeness proofs. In this context, we also construct a preliminary framework for dealing with stratified programs. This work paves the way towards the certification of data-centric applications