Números complexos: uma proposta de mudança metodológica para uma aprendizagem significativa no ensino médio

This work presents a proposal of a methodological change to the teaching and learning of the complex numbers in the Secondary education. It is based on the inquiries and difficulties of students detected in the classrooms about the teaching of complex numbers and a questioning of the context of the mathematics teaching - that is the reason of the inquiry of this dissertation. In the searching for an efficient learning and placing the work as a research, it is presented a historical reflection of the evolution of the concept of complex numbers pointing out their more relevant focuses, such as: symbolic, numeric, geometrical and algebraic ones. Then, it shows the description of the ways of the research based on the methodology of the didactic engineering. This one is developed from the utilization of its four stages, where in the preliminary analysis stage, two data surveys are presented: the first one is concerning with the way of presenting the contents of the complex numbers in math textbooks, and the second one is concerning to the interview carried out with High school teachers who work with complex numbers in the practice of their professions. At first, in the analysis stage, it is presented the prepared and organized material to be used in the following stage. In the experimentation one, it is presented the carrying out process that was made with the second year High school students in the Centro Federal de Educação tecnológica do Rio Grande do Norte CEFET-RN. At the end, it presents, in the subsequent and validation stages, the revelation of the obtained results from the observations made in classrooms in the carrying out of the didactic sequence, the students talking and the data collection

Uma análise dos questionamentos dos alunos nas aulas de números complexos

This work presents a contribution for the studies reffering to the use of the History of Mathematics focusing on the improvement of the Teaching and Learning Process. It considers that the History of Matematics, as a way of giving meaning to the discipline and improve the quality of the Teaching and Learning Process. This research focuses on the questions of the students, classified in three categories of whys: the chronological, the logical and the pedagogical ones. Therefore, it is investigated the teaching of the Complex Numbers, from the questions of the students of the Centro Federal de Educação Tecnológica do Rio Grande do Norte (Educational Institution of Professional and Technology Education from Rio Grande do Norte). The work has the following goals: To classify and to analyse the questions of the students about the Complex Numbers in the classes of second grade of the High School, and to collate with the pointed categories used by Jones; To disccus what are the possible guidings that teachers of Mathematics can give to these questions; To present the resources needed to give support to the teacher in all things involving the History of Mathematics. Finally, to present a bibliographic research, trying to reveal supporting material to the teacher, with contents that articulate the Teaching of Mathematics with the History of Mathematics. It was found that the questionings of the pupils reffers more to the pedagogical whys, and the didatic books little contemplate other aspects of the history and little say about the sprouting and the evolution of methods of calculations used by us as well

O uso pedagógico de uma seqüência didática para a aquisição de algumas idéias relacionadas ao conceito de números complexos

The aim of the present work is to contribute to the teaching-learning process in Mathematics through an alternative which tries to motivate the student so that he/she will learn the basic concepts of Complex Numbers and realize that they are not pointless. Therefore, this work s general objective is to construct a didactic sequence which contains structured activities that intends to build up, in each student s thought, the concept of Complex Numbers. The didactic sequence is initially based on a review of the main historical aspects which begot the construction of those numbers. Based on these aspects, and the theories of Richard Skemp, was elaborated a sequence of structured activities linked with Maths history, having the solution of quadratic equations as a main starting point. This should make learning more accessible, because this concept permeates the students previous work and, thus, they should be more familiar with it. The methodological intervention began with the application of that sequence of activities with grade students in public schools who did not yet know the concept of Complex Numbers. It was performed in three phases: a draft study, a draft study II and the final study. Each phase was applied in a different institution, where the classes were randomly divided into groups and each group would discuss and write down the concepts they had developed about Complex Numbers. We also use of another instrument of analysis which consisted of a recorded interview of a semi-structured type, trying to find out the ways the students thought in order to construct their own concepts, i.e. the solutions of the previous activity. Their ideas about Complex Numbers were categorized according to their similarities and then analyzed. The results of the analysis show that the concepts constructed by the students were pertinent and that they complemented each other this supports the conclusion that the use of structured activities is an efficient alternative for the teaching of mathematics

Os Polímeros como Sistemas Complexos

Neste trabalho, através de simulações computacionais, identificamos os fenômenos físicos associados ao crescimento e a dinâmica de polímeros como sistemas complexos exibindo comportamentos não linearidades, caos, criticalidade auto-organizada, entre outros. No primeiro capítulo, iniciamos com uma breve introdução onde descrevemos alguns conceitos básicos importantes ao entendimento do nosso trabalho. O capítulo 2 consiste na descrição do nosso estudo da distribuição de segmentos num polímero ramificado. Baseado em cálculos semelhantes aos usados em cadeias poliméricas lineares, utilizamos o modelo de crescimento para polímeros ramificados (Branched Polymer Growth Model - BPGM) proposto por Lucena et al., e analisamos a distribuição de probabilidade dos monômeros num polímero ramificado em 2 dimensões, até então desconhecida. No capítulo seguinte estudamos a classe de universalidade dos polímeros ramificados gerados pelo BPGM. Utilizando simulações computacionais em 3 dimensões do modelo proposto por Lucena et al., calculamos algumas dimensões críticas (dimensões fractal, mínima e química) para tentar elucidar a questão da classe de universalidade. Ainda neste Capítulo, descrevemos um novo modelo para a simulação de polímeros ramificados que foi por nós desenvolvido de modo a poupar esforço computacional. Em seguida, no capítulo 4 estudamos o comportamento caótico do crescimento de polímeros gerados pelo BPGM. Partimos de polímeros criticamente organizados e utilizamos uma técnica muito semelhante aquela usada em transições de fase em Modelos de Ising para estudar propagação de danos chamada de Distância de Hamming. Vimos que a distância de Hamming para o caso dos polímeros ramificados se comporta como uma lei de potência, indicando um caráter não-extensivo na dinâmica de crescimento. No Capítulo 5 analisamos o movimento molecular de cadeias poliméricas na presença de obstáculos e de gradientes de potenciais. Usamos um modelo generalizado de reptação para estudar a difusão de polímeros lineares em meios desordenados. Investigamos a evolução temporal destas cadeias em redes quadradas e medimos os tempos característicos de transporte t. Finalizamos esta dissertação com um capítulo contendo a conclusão geral denoss o trabalho (Capítulo 6), mais dois apêndices (Apêndices A e B) contendo a fenomenologia básica para alguns conceitos que utilizaremos ao longo desta tese (Fractais e Percolação respectivamente) e um terceiro e ´ultimo apêndice (Apêndice C) contendo uma descrição de um programa de computador para simular o crescimentos de polímeros ramificados em uma rede quadrada

Estudo de sistemas complexos com interações de longo alcance : percolação, redes e tráfego

In this thesis we investigate physical problems which present a high degree of complexity using tools and models of Statistical Mechanics. We give a special attention to systems with long-range interactions, such as one-dimensional long-range bondpercolation, complex networks without metric and vehicular traffic. The flux in linear chain (percolation) with bond between first neighbor only happens if pc = 1, but when we consider long-range interactions , the situation is completely different, i.e., the transitions between the percolating phase and non-percolating phase happens for pc < 1. This kind of transition happens even when the system is diluted ( dilution of sites ). Some of these effects are investigated in this work, for example, the extensivity of the system, the relation between critical properties and the dilution, etc. In particular we show that the dilution does not change the universality of the system. In another work, we analyze the implications of using a power law quality distribution for vertices in the growth dynamics of a network studied by Bianconi and Barabási. It incorporates in the preferential attachment the different ability (fitness) of the nodes to compete for links. Finally, we study the vehicular traffic on road networks when it is submitted to an increasing flux of cars. In this way, we develop two models which enable the analysis of the total flux on each road as well as the flux leaving the system and the behavior of the total number of congested roads

Estudo de propriedades críticas em sistemas complexos: método de procura automática em sistemas tipo Ising e percolação de longo alcance

The new technique for automatic search of the order parameters and critical properties is applied to several well-know physical systems, testing the efficiency of such a procedure, in order to apply it for complex systems in general. The automatic-search method is combined with Monte Carlo simulations, which makes use of a given dynamical rule for the time evolution of the system. In the problems inves¬tigated, the Metropolis and Glauber dynamics produced essentially equivalent results. We present a brief introduction to critical phenomena and phase transitions. We describe the automatic-search method and discuss some previous works, where the method has been applied successfully. We apply the method for the ferromagnetic fsing model, computing the critical fron¬tiers and the magnetization exponent (3 for several geometric lattices. We also apply the method for the site-diluted ferromagnetic Ising model on a square lattice, computing its critical frontier, as well as the magnetization exponent f3 and the susceptibility exponent 7. We verify that the universality class of the system remains unchanged when the site dilution is introduced. We study the problem of long-range bond percolation in a diluted linear chain and discuss the non-extensivity questions inherent to long-range-interaction systems. Finally we present our conclusions and possible extensions of this work

Representações espectrais de sistemas complexos: aplicações à síntese de superfícies brownianas fracionárias anisotrópicas, filtragem de sinais e identificação de correlações

In this thesis, we study the application of spectral representations to the solution of problems in seismic exploration, the synthesis of fractal surfaces and the identification of correlations between one-dimensional signals. We apply a new approach, called Wavelet Coherency, to the study of stratigraphic correlation in well log signals, as an attempt to identify layers from the same geological formation, showing that the representation in wavelet space, with introduction of scale domain, can facilitate the process of comparing patterns in geophysical signals. We have introduced a new model for the generation of anisotropic fractional brownian surfaces based on curvelet transform, a new multiscale tool which can be seen as a generalization of the wavelet transform to include the direction component in multidimensional spaces. We have tested our model with a modified version of the Directional Average Method (DAM) to evaluate the anisotropy of fractional brownian surfaces. We also used the directional behavior of the curvelets to attack an important problem in seismic exploration: the atenuation of the ground roll, present in seismograms as a result of surface Rayleigh waves. The techniques employed are effective, leading to sparse representation of the signals, and, consequently, to good resolutions

Práticas institucionais/discursivas acerca dos cuidados com os bebês prematuros e/ou de baixo peso: o programa canguru

The Kangaroo Program was implemented in Brazil in 2000 through the Unified Health System (Sistema Único de Saúde SUS) sustained with a humanized rethoric of health care assistance. This program adopts the skin-to-skin contact contributing to the mother-infant bond, breastfeeding and promoting security in mother s care. The users of SUS are encouraged to live in the maternity ward to follow the baby health improvement. However, it was verified in previous observations that mothers participation in the Kangaroo Program has been done through an imposed practice. Therefore, this study intended to understand the texts that permeate the kangaroo practice. This research was developed through two studies: 1) an historic exploration of motherhood concept and an analysis of how the motherhood is presented in the official document that orients the program; 2) an analysis of institutional dynamic of Kangaroo Program, emphasizing the study about the health workers everyday practice, the mothers view about their life in the maternity wards, and the attendance practice. It is highlighted that the relation between this two studies allowed the comprehension abouthow the official discourses can influence the health workers behaviors and how their viewpoint and position can shape the everyday work in a public health program. This research, supported by Institutional Ethnography, considers that people s practices and experiences are socially organized and shaped by broad social forces. The discourse method was used in the documental analysis and in the analysis of qualitative data from empiric research. The research showed that the kangaroo program has been an excellent way to save resources and to improve some baby s biologic and psychological aspects. However, this program has failed to consider the social, economic and cultural complexity of mothers and the structural limitation of the health care system. The official document uses the economic and medical approach, following the hegemonic biomedical model and the life style of the people that don t use the public health system. Consequently, the program has not been successful because it is planned without people participation. On the other hand, it was verified that although some professionals are committed with their work, the mainly does not consider mothers participation as an active process, using the institutional power as a social control to keep mothers uninformed about the possibility to leave the maternity wards. As a result, the research also showed that mothers perceive the program as mandatory and not as option that can improve pleasure moments. It is, therefore, necessary to consider the complex social determinants of health that can increase mothers participation in the Kangaroo Program. Bringing these issues into debate can be a reflective exercise on citizenship and governance, allowing spaces for the improvement of public health programs

Complexos metálicos de cobalto, níquel e cobre com a Pirazina-2-carboxamida e 4- hidrazida ácida piridincarboxílica: síntese e caracterização

This work involved the synthesis, characterization and proposing the molecular structure of coordination compounds involving ligands pyrazine-2-carboxamide (PZA) and 4- hydrazide acidic pyridine carboxylic (INH) and metals of the first transition series (M = Co2+, Ni2+ and Cu2+). For the characterization of the compounds used were analytical techniques such as infrared absorption spectroscopy average (FT-IR) molar conductivity measurements, CHN elemental analysis, EDTA Complexometric, measurement of melting point, X-ray diffraction by powder method, Thermogravimetry (TG) and Differential Thermal Analysis (DTA) and Simultaneous Differential Scanning Calorimetry (DSC). The absorption spectra in the infrared region suggested that the ligand coordination to the metal center occurs through the carbonyl oxygen atom and nitrogen alpha pyrazine ring to those complexes formed with PZA. For INH complexes with metal-ligand coordination is through the carbonyl oxygen and nitrogen of the terminal hydrazide grouping. The conductivity measurements of the complexes in aqueous solution they suggest to all behavior of the type 1:2 electrolytes, and conduct of non-electrolytes in acetonitrile. The results obtained by CHN elemental analysis and EDTA Complexometric allowed to infer the stoichiometry of the compounds synthesized. For all of the complexes obtained was possible to record the melting points, neither of which melted near the melting temperature of the free ligands. The X-ray diffraction showed that the complexes of pyrazinamide exhibited diffraction lines, suggesting that these compounds are crystalline, while compounds of isoniazid, with the exception of cobalt, exhibited diffraction lines, indicating that they are crystalline. The results from the TG-DTA and DSC allowed information regarding the dehydration and thermal decomposition of these complexes

Síntese, caracterização e estudo fotofísico de complexos de 4,4- diaminoestilbeno-2,2-dissulfonato com íons lantanídeos

This is a work involving fundamental studies of chemistry where the synthesis and structural characterization, as well as a possible future application of these new compounds as luminescent sensors or sunscreen agents, complexes with 4,4 diaminostilbene-2,2-disulfonic (DSD) and trivalent lanthanide ions La3+, Nd3+, Eu3+, Gd3+ and Yb3+, were synthesized in the ratio of 3 mmol: 1 mmol (DSD: lanthanides). The complexes obtained with these ions were present in powder form and were characterized by complexometric titration with EDTA CHN Elemental analysis, molecular absorption spectroscopy in the ultraviolet region, the absorption spectroscopy in the infrared, thermal analysis (TG / DTG), Nuclear Magnetic Resonance - NMR 1H and Luminescence Spectroscopy. The complexometric titration and CHN analysis, confirmed the TG / DTG which suggest that these complexes have the following general chemical formulas: [La2(C14H12S2O6N2)2(H2O)2Cl2].7H2O,[Nd2(C14H12S2O6N2)2(H2O)2Cl2].6H2O,[Eu2(C14 H12S2O6N2)2(H2O)2Cl2].7H2O,[Gd2(C14H12S2O6N2)2(H2O)2Cl2].4H2O e [Yb2(C14H12S2O6N2)2(H2O)2].6H2O. The disappearance of the bands in the infrared spectrum at 2921 cm-1 and 2623 cm-1 and the displacement of the bands in the spectra of the amine complex indicate that the lanthanide ion is coordinated to the oxygen atoms and the sulfonate groups of the nitrogens amines, suggesting the formation of the dimer. The disappearance of the signal and the displacement signal SO3H amines in the 1H NMR spectrum of this complex are also indicative coordination and dimer formation. The Thermogravimetry indicates that the DSD is thermally stable in the range of 40º to 385°C and their complexes with lanthanide ions exhibit weight loss between 4 and 5 stages. The Uv-visible spectra indicated that the DSD and complexes exhibit cis isomers. The analysis of luminescence indicates that the complexes do not exhibit emission in the region of the lanthanides but an intense emission part of the binder. This is related to the triplet states of the ligand, which are in the lowest energy state emitting lanthanide ions, and also the formation of the dimer that suppress the luminescence of ion Eu3+. The formation of dimer was also confirmed by calculating the europium complex structure using the model Hamiltonian PM6 and Sparkle

Obtenção de dispersões de complexos polieletrolíticos à base de quitosana e poli(ácido metacrílico) e análise de adsorção de albumina bovina sérica

Dispersions composed of polyelectrolyte complexes based on chitosan and poly(methacrylic acid), PMAA, were obtained by the dropping method and template polymerization. The effect of molecular weight of PMAA and ionic strength on the formation of chitosan/poly(methacrylic acid), CS/PMAA, complexes was evaluated using the dropping method. The increase in molecular weight of PMAA inhibited the formation of insoluble complexes, while the increase in ionic strength first favored the formation of the complex followed by inhibiting it at higher concentrations. The polyelectrolyte complexation was strongly dependent on macromolecular dimensions, both in terms of molecular weight and of coil expansion/contraction driven by polyelectrolyte effect. The resultant particles from dropping method and template polymerization were characterized as having regions with different charge densities: chitosan predominating in the core and poly(methacrylic acid) at the surface, the particles being negatively charged, as a consequence. Albumin was adsorbed on templatepolymerized CS/PMAA complexes (after crosslinking with glutardialdehyde) and pH was controlled in order to obtain two conditions: (i) adsorption of positively charged albumin, and (ii) adsorption of albumin at its isoelectric point. Adsorption isotherms and zeta potential measurements showed that albumin adsorption was controlled by hydrogen bonding/van der Waals interactions and that brushlike structures may enhance adsorption of albumin on these particles

Estudo de Fractalidade e Evolução Dinâmica de Sistemas Complexos

In this work, the study of some complex systems is done with use of two distinct procedures. In the first part, we have studied the usage of Wavelet transform on analysis and characterization of (multi)fractal time series. We have test the reliability of Wavelet Transform Modulus Maxima method (WTMM) in respect to the multifractal formalism, trough the calculation of the singularity spectrum of time series whose fractality is well known a priori. Next, we have use the Wavelet Transform Modulus Maxima method to study the fractality of lungs crackles sounds, a biological time series. Since the crackles sounds are due to the opening of a pulmonary airway bronchi, bronchioles and alveoli which was initially closed, we can get information on the phenomenon of the airway opening cascade of the whole lung. Once this phenomenon is associated with the pulmonar tree architecture, which displays fractal geometry, the analysis and fractal characterization of this noise may provide us with important parameters for comparison between healthy lungs and those affected by disorders that affect the geometry of the tree lung, such as the obstructive and parenchymal degenerative diseases, which occurs, for example, in pulmonary emphysema. In the second part, we study a site percolation model for square lattices, where the percolating cluster grows governed by a control rule, corresponding to a method of automatic search. In this model of percolation, which have characteristics of self-organized criticality, the method does not use the automated search on Leaths algorithm. It uses the following control rule: pt+1 = pt + k(Rc − Rt), where p is the probability of percolation, k is a kinetic parameter where 0 < k < 1 and R is the fraction of percolating finite square lattices with side L, LxL. This rule provides a time series corresponding to the dynamical evolution of the system, in particular the likelihood of percolation p. We proceed an analysis of scaling of the signal obtained in this way. The model used here enables the study of the automatic search method used for site percolation in square lattices, evaluating the dynamics of their parameters when the system goes to the critical point. It shows that the scaling of , the time elapsed until the system reaches the critical point, and tcor, the time required for the system loses its correlations, are both inversely proportional to k, the kinetic parameter of the control rule. We verify yet that the system has two different time scales after: one in which the system shows noise of type 1 f , indicating to be strongly correlated. Another in which it shows white noise, indicating that the correlation is lost. For large intervals of time the dynamics of the system shows ergodicity

Crescimento de prematuros de extremo baixo peso nos primeiros dois anos de vida

OBJETIVO: Analisar o padrão de crescimento de prematuros de extremo baixo peso (EBP) até 24 meses de idade corrigida, a influência da displasia broncopulmonar (DBP) e os fatores de risco para falha de crescimento. MÉTODOS: Coorte de prematuros <1.000g de gestação única, nascidos e acompanhados em um centro terciário. O crescimento foi avaliado por meio de escores-z para peso, comprimento e perímetro cefálico ao nascimento, com 40 semanas, aos 3, 6, 12, 18 e 24 meses de idade corrigida. Dentre 81 sobreviventes, 70 foram estudados e estratificados em dois grupos: DBP (n=41) e sem DBP (n=29). Foi realizada análise bivariada com teste t ou Mann-Whitney, qui-quadrado ou Exato de Fisher, e análise multivariada com regressão logística. RESULTADOS: em ambos os grupos, o escore-z de peso diminuiu significantemente entre o nascimento e 40 semanas. Houve um pico de incremento nos escores-z de peso, comprimento e perímetro cefálico entre 40 semanas e três meses. No grupo sem DBP, os escores-z atingiram a faixa normal a partir dos seis meses e assim permaneceram até 24 meses de idade corrigida. Crianças com DBP tiveram menores escores-z de peso e perímetro cefálico no primeiro ano, mas equipararam-se às sem DBP no segundo ano de vida. A regressão logística mostrou que catch-down no escore-z de peso com 40 semanas foi fator de risco para falha de crescimento. CONCLUSÕES: Prematuros EBP apresentam catch-up precoce do crescimento nos primeiros dois anos. Crianças com DBP têm pior crescimento ponderal. A restrição do crescimento pós-natal prediz a falha de crescimento nos primeiros anos.

Efeitos da fisioterapia respiratória convencional versus aumento do fluxo expiratório na saturação de O2, freqüência cardíaca e freqüência respiratória, em prematuros no período pós-extubação

INTRODUÇÃO: Recém-nascidos (RN) prematuros apresentam elevada morbidade respiratória e necessidade de ventilação mecânica, assim, a fisioterapia respiratória é parte integrante da assistência neonatal. Objetivo: Comparar os efeitos da fisioterapia respiratória convencional (FRC) versus aumento do fluxo expiratório (AFE), na saturação de O2 (SpO2), freqüência cardíaca (FC) e na freqüência respiratória (FR) em prematuros no período pós-extubação. Método: Ensaio clínico randomizado realizado na UTI Neonatal do Hospital das Clínicas de BotucatuUNESP, comparando duas técnicas fisioterapêuticas, aplicadas em recém-nascidos prematuros, nas primeiras 48 horas pós-extubação. Para a análise estatística foram utilizados o teste t Student, Mann-Whitney, Qui-quadrado e o teste exato de Fisher, com nível de significância em 5%. Resultados: Os dois grupos de estudo: Grupo FRC (n= 20) e grupo AFE (n= 20), não diferiram quanto à idade gestacional (média de 28 semanas) e peso de nascimento (média de 1100 gramas). em ambos os grupos a síndrome do desconforto respiratório (SDR) foi o principal diagnóstico. A mediana da idade no início da fisioterapia foi de sete dias no grupo AFE e 11 dias na FRC. Ambas as técnicas produziram aumento significativo da SpO2 aos 10 e 30 minutos, sem alterações na FR. A FC aumentou significativamente após a FRC e não se alterou após o AFE. Conclusão: Os resultados sugerem que o AFE é menos estressante que a FRC e pode ser aplicado em prematuros no período pós-extubação. Nestes recém-nascidos o AFE parece ser seguro e benéfico a curto prazo.