The aim of this research was to investigate how I engaged in constructivist teaching (CT) when helping a group of low-performing Grade 7 students to develop new meanings of notation as they started to learn formal algebra. Data was collected over a period of one scholastic year, in which I explored the teacher-student dynamics during my mathematics lessons, where students learnt new representations and interpretations of notation with the help of the computer software Grid Algebra. Analysing video recordings of my lessons, I observed myself continuously changing my teaching purpose as I negotiated between the mathematics I intended to teach and the mathematics being constructed by my students. These shifts of focus and purpose were used to develop a conceptual framework called Mathematics-Negotiation-Learner (M-N-L). Besides serving as a CT model, the M-N-L framework was found useful to determine the extent to which I managed to engage in CT during the lessons and also to identify moments where I lost my sensitivity to students constructions of knowledge. The effectiveness of my CT was investigated by focusing on students learning, for which reason I developed the analytical framework called CAPS (Concept-Action-Picture-Symbol). The CAPS framework helped me to analyse how students developed notions about properties of operational notation, the structure and order of operations in numerical and algebraic expressions, and the relational property of the equals sign. Grid Algebra was found to be a useful tool in helping students to enrich their repertoire of representations and to develop new interpretations of notation through what I defined as informal- and formal-algebraic activities. All students managed to transfer these representations and interpretations of notation to pen-and-paper problems, where they successfully worked out traditionally set substitution-and-evaluation tasks.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-07-25T11:54:43Z No. of bitstreams: 1 Danilo Gustavo Bispo.pdf: 1616324 bytes, checksum: cbc43e66f323825f005da2c275dfa256 (MD5) / Made available in DSpace on 2018-07-25T11:54:43Z (GMT). No. of bitstreams: 1 Danilo Gustavo Bispo.pdf: 1616324 bytes, checksum: cbc43e66f323825f005da2c275dfa256 (MD5) Previous issue date: 2018-04-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The present work aims to expose the study of some aspects that permeated the emergence of Computing Theory of Alan Mathison Turing at the beginning of the 20th century. According to previous research, it is suggested that this occurred in part, due to a theoretical consequence of the development of mathematics. At the end of the nineteenth century there was an effort expended by some thinkers whose aim was to clarify the nature of truth in mathematics, a problem put in sharp evidence by the turbulent stage in the development of mathematical thinking between about 1870 and 1940 with the emergence of new types of geometry. This stage can also be interpreted as "the crisis of fundamentals" and visualized as the outcome of the development phase corresponding to the emergence of contemporary science. Within this context, the idea will now be to broaden the research to identify details of the projects that were not successful but that also had their contribution or influenced in some way the emergence of the theory that established key concepts for the digital computer model / O presente trabalho tem como objetivo expor o estudo de alguns aspectos que permearam o surgimento da Teoria da Computação de Alan Mathison Turing no início do século XX. De acordo com pesquisa, sugere-se que isso tenha ocorrido em parte, devido a uma consequência teórica do desenvolvimento da matemática. No final do século XIX houve um esforço despendido por parte de alguns pensadores cujo o intuito estava em clarificar a natureza da verdade em matemática, problema posto em aguda evidência pela turbulenta etapa no desenvolvimento do pensamento matemático ocorrido entre aproximadamente 1870 e 1940 com o surgimento de novos tipos de geometria. Esta etapa também pode ser interpretada como "a crise dos fundamentos" e visualizada como o desfecho da fase de desenvolvimento correspondente ao surgimento da ciência contemporânea. Dentro deste contexto, a ideia será agora ampliar a investigação procurando identificar detalhes dos projetos que não foram bem-sucedidos, mas que também tiveram sua contribuição ou influenciaram de algum modo o surgimento da teoria que estabeleceu conceitos chave para o modelo do computador digital

Nesse trabalho utiliza-se a transformada Z como método de resoluções de equações de diferenças, visando modelos matemáticos discretos, com o principal objetivo desenvolver um material didático, em língua portuguesa, tendo em vista que grande parte das referências encontradas estão em língua inglesa. Tem como etapas: definição da transformada Z unilateral, condição de existência, propriedades, transformada Z inversa, incluindo demonstrações, exemplos e exercícios propostos. O material proposto serve de base para o estudo da matemática discreta, direcionado a professores e alunos na disciplina de cálculo em curso de engenharia, no estudo de processamento de sinais e sistemas de controle de dados amostrados; aos professores de ensino básico como curiosidade e extensão dos da matemática discreta; e às demais áreas que exploram a matemática discreta tais como engenharia, economia e computação. / In this work the Z-transform is used to solve difference equations, aiming discrete mathematical models, with the main objective to develop a courseware in Portuguese, since most of the references are in English. The Z transformation approach has the following steps: definition of the unilateral Z-transform, existence condition, properties and inverse Z-transform, including demonstrations, examples and proposed exercises. The proposed material can be used to study discrete mathematics, by teachers and students in calculus classes for engineering courses, such as signal processing and sampled-data control systems; by High School teachers as a curiosity and further study of discrete math; and other related areas such as as Engineering, Economics and Computing.

Nesse trabalho utiliza-se a transformada Z como método de resoluções de equações de diferenças, visando modelos matemáticos discretos, com o principal objetivo desenvolver um material didático, em língua portuguesa, tendo em vista que grande parte das referências encontradas estão em língua inglesa. Tem como etapas: definição da transformada Z unilateral, condição de existência, propriedades, transformada Z inversa, incluindo demonstrações, exemplos e exercícios propostos. O material proposto serve de base para o estudo da matemática discreta, direcionado a professores e alunos na disciplina de cálculo em curso de engenharia, no estudo de processamento de sinais e sistemas de controle de dados amostrados; aos professores de ensino básico como curiosidade e extensão dos da matemática discreta; e às demais áreas que exploram a matemática discreta tais como engenharia, economia e computação. / In this work the Z-transform is used to solve difference equations, aiming discrete mathematical models, with the main objective to develop a courseware in Portuguese, since most of the references are in English. The Z transformation approach has the following steps: definition of the unilateral Z-transform, existence condition, properties and inverse Z-transform, including demonstrations, examples and proposed exercises. The proposed material can be used to study discrete mathematics, by teachers and students in calculus classes for engineering courses, such as signal processing and sampled-data control systems; by High School teachers as a curiosity and further study of discrete math; and other related areas such as as Engineering, Economics and Computing.

Orientadores: Edson Eiji Matsura, Jarbas Honorio de Miranda / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Agricola / Made available in DSpace on 2018-08-09T21:31:09Z (GMT). No. of bitstreams: 1 Souza_AndersonLuizde_D.pdf: 4529373 bytes, checksum: 402972ab3e735c73df6c25a04dbbd1e9 (MD5) Previous issue date: 2007 / Resumo: Conhecer a dinâmica da água e de solutos no solo é condição imprescindível para a produção agrícola sustentável. O sucesso destes modelos numéricos em descrever a dinâmica da água e de solutos no solo está intimamente ligado à precisão com que os parâmetros físicohídricos, envolvidos nestes processos são determinados. De uma maneira geral, a determinação dos parâmetros de transporte de íons no solo esta associado ao ajuste de modelos teóricos a dados associados a ensaios experimentais de deslocamento miscível. Tal procedimento além dos erros numéricos associados ao ajuste destes parâmetros, também é susceptível a erros experimentais. Sendo assim, mostra-se necessário a proposição de técnicas de modelagem que incorporem, de maneira apropriada as imprecisões intrínsecas ao estudo experimental da dinâmica da água e de soluto no solo. Neste sentido, este trabalho teve como objetivo avaliar o potencial do uso da lógica fuzzy, sobretudo de um modelo baseado em regras fuzzy (MBRF), na simulação numérica destes processos. A utilização da teoria dos conjuntos fuzzy é vantajosa à medida que permite que as incertezas envolvidas na dinâmica da água e de solutos no solo sejam absorvidas pela utilização de conjuntos de regras. A base de regras foi construída a partir de um conjunto de dados teóricos, denominado conjunto de treinamento, obtido a partir da simulação numérica das Equações Diferenciais Parciais que regem as dinâmicas da água no solo. O conjunto de treinamento foi obtido utilizando-se o modelo computacional Hydrus-2D. A construção do MBRF, utilizado para descrever a dinâmica da água no solo na direção vertical foi feita utilizando-se o ambiente computacional MATLAB 6.5. Paralelamente a modelagem da dinâmica da água, o estudo do transporte de potássio no solo foi feito considerando-se apenas o fluxo de massa. Os resultados obtidos pelo MBRF foram comparados com resultados obtidos no estudo experimental da dinâmica da água e de solutos em colunas verticais de solo equipadas com sondas de TDR e extratores de solução. Face aos resultados obtidos, pode-se concluir que, neste trabalho, muito embora os resultados apresentados pelo MBRF na simulação da dinâmica de água e de solutos no solo tenham sido apenas razoáveis, o uso da teoria de conjuntos fuzzy aplicada ao estudo destes processos, devido a sua simplicidade de formulação e interpretação, apresentou-se como uma importante ferramenta para auxiliar na tomada de decisão de processos que busquem o uso racional da água e do solo nas atividades agrícolas / Abstract: To know the water and solute dynamics is a indispensable condition for the sustainable agricultural production. The numeric models success in escribe those processes is intimately linked to the precision with the physical- hydric arameters involved in these processes are determineted. In a general way, the determination of the ions transport parameters are obtained by the adjustment of theoretical models to experimental data of miscible displacement studies. Such procedure, besides numerical mistakes, associated to the adjustment of these parameters, is also associated to experimental errors. In this way, the proposition of modeling techniques that incorporate, in an appropriate way, those intrinsic imprecisions are necessary. For this reason, this work¿s goal is evaluate the rule based fuzzy model¿s (RBFM) potential in the numeric simulation of these processes. The use of the fuzzy theory is advantageous once it allows to incorporate the uncertainties involved in the soil water and solutes dynamics using a set of rules. The base of rules was built using a set of theoretical data, denominated training set, obtained by the numeric simulation of the Equations that govern the water dynamics in the soil profile. The training set was obtained using the computational model Hydrus-2D. The construction of the RBFM, used to describe the vertical water dynamics, was done the computational software MATLAB 6.5. Parallel to the water dynamics modeling, a potassium transport study was made considering just the mass flow process. The results obtained with RBFM were compared with experimental data obtained in study of the water and solute dynamics in a vertical soil column, equipped with TDR probes and solution extractors. Face to the obtained results, it can be concluded that, in this work, the presented RBFM wasn¿t very efficient in describe water and solute dynamics the dynamics. In the other hands, apart of those results, the development of this modeling technique showed a great potential in the soils water and solute dynamics, reinforcing the importance of more investigations in this way / Doutorado / Agua e Solo / Doutor em Engenharia Agrícola

We investigate into the limitations of the sum-product algorithm for binary low density parity check (LDPC) codes having isolated short cycles. Independence assumption among messages passed, assumed reasonable in all configurations of graphs, fails the most in graphical structures with short cycles. This research work is a step forward towards understanding the effect of short cycles on error floors of the sum-product algorithm. We propose a modified sum-product algorithm by considering the statistical dependency of the messages passed in a cycle of length 4. We also formulate a modified algorithm in the log domain which eliminates the numerical instability and precision issues associated with the probability domain. Simulation results show a signal to noise ratio (SNR) improvement for the modified sum-product algorithm compared to the original algorithm. This suggests that dependency among messages improves the decisions and successfully mitigates the effects of length-4 cycles in the Tanner graph. The improvement is significant at high SNR region, suggesting a possible cause to the error floor effects on such graphs. Using density evolution techniques, we analysed the modified decoding algorithm. The threshold computed for the modified algorithm is higher than the threshold computed for the sum-product algorithm, validating the observed simulation results. We also prove that the conditional entropy of a codeword given the estimate obtained using the modified algorithm is lower compared to using the original sum-product algorithm.

We implement a package for the construction of a new class of Low Density Parity Check (LDPC) codes based on a new random high girth graph construction technique, and study the performance of the codes so constructed on both the Additive White Gaussian Noise (AWGN) channel as well as the Binary Erasure Channel (BEC). Our codes are “near regular”, meaning thereby that the the left degree of any node in the Tanner graph constructed varies by at most 1 from the average left degree and so also the right degree. The simulations for rate half codes indicate that the codes perform better than both the regular Progressive Edge Growth (PEG) codes which are constructed using a similar random technique, as well as the MacKay random codes. For high rates the ARG (Almost Regular high Girth) codes perform better than the PEG codes at low to medium SNR’s but the PEG codes seem to do better at high SNR’s. We have tried to track both near codewords as well as small weight codewords for these codes to examine the performance at high rates. For the binary erasure channel the performance of the ARG codes is better than that of the PEG codes. We have also proposed a modification of the sum-product decoding algorithm, where a quantity called the “node credibility” is used to appropriately process messages to check nodes. This technique substantially reduces the error rates at signal to noise ratios of 2.5dB and beyond for the codes experimented on. The average number of iterations to achieve this improved performance is practically the same as that for the traditional sum-product algorithm.