Atividades de investigação em Geometria: uma experiência no 2º ano de escolaridade

Dissertação apresentada à Escola Superior de Educação de Lisboa para obtenção de grau de mestre em Ciências da Educação, especialidade Educação Matemática na Educação Pré-Escolar e nos 1.º e 2.º Ciclos do Ensino Básico

Reasoning and proof in classroom (9th grade)

This paper is research oriented and pretends to contribute toward giving empirical evidence about how students develop their reasoning and how they achieved to a proof construction in school context. Its main theme is epistemology. It describes the way in which four students in 9th Grade explored a task related with the discovery of symmetry axes in various geometric figures. The proof constructed by students had essentially an explaining function and it was related with the symmetry axes of regular polygons. The teacher’s role in meaning negotiation of the proof and its need is described through illustrative episodes. The paper presents part of a study which purpose is to analyse the nature of mathematical proof in classroom, its role and the nature of the relationship between the construction of a proof and the social interactions. Assuming a social perspective, attention is focussed on the social construction of knowledge and on the structuring resources that shape mathematical experience. The study’s methodology has an interpretative nature. One outcome of the study discussed here is that students develop first a practical understanding with no awareness of the reasons founding mathematical statements and after a theoretical one leading them to a proof elaboration.

Geometric picture of generalized-CP and Higgs-family transformations in the two-Higgs-doublet model

In the two-Higgs-doublet model (THDM), generalized-CP transformations (phi(i) -> X-ij phi(*)(j) where X is unitary) and unitary Higgs-family transformations (phi(i) -> U-ij phi(j)) have recently been examined in a series of papers. In terms of gauge-invariant bilinear functions of the Higgs fields phi(i), the Higgs-family transformations and the generalized-CP transformations possess a simple geometric description. Namely, these transformations correspond in the space of scalar-field bilinears to proper and improper rotations, respectively. In this formalism, recent results relating generalized CP transformations with Higgs-family transformations have a clear geometric interpretation. We will review what is known regarding THDM symmetries, as well as derive new results concerning those symmetries, namely how they can be interpreted geometrically as applications of several CP transformations.

Vertex component analysis: a geometric-based approach to unmix hyperspectral data

Hyperspectral remote sensing exploits the electromagnetic scattering patterns of the different materials at specific wavelengths [2, 3]. Hyperspectral sensors have been developed to sample the scattered portion of the electromagnetic spectrum extending from the visible region through the near-infrared and mid-infrared, in hundreds of narrow contiguous bands [4, 5]. The number and variety of potential civilian and military applications of hyperspectral remote sensing is enormous [6, 7]. Very often, the resolution cell corresponding to a single pixel in an image contains several substances (endmembers) [4]. In this situation, the scattered energy is a mixing of the endmember spectra. A challenging task underlying many hyperspectral imagery applications is then decomposing a mixed pixel into a collection of reflectance spectra, called endmember signatures, and the corresponding abundance fractions [8–10]. Depending on the mixing scales at each pixel, the observed mixture is either linear or nonlinear [11, 12]. Linear mixing model holds approximately when the mixing scale is macroscopic [13] and there is negligible interaction among distinct endmembers [3, 14]. If, however, the mixing scale is microscopic (or intimate mixtures) [15, 16] and the incident solar radiation is scattered by the scene through multiple bounces involving several endmembers [17], the linear model is no longer accurate. Linear spectral unmixing has been intensively researched in the last years [9, 10, 12, 18–21]. It considers that a mixed pixel is a linear combination of endmember signatures weighted by the correspondent abundance fractions. Under this model, and assuming that the number of substances and their reflectance spectra are known, hyperspectral unmixing is a linear problem for which many solutions have been proposed (e.g., maximum likelihood estimation [8], spectral signature matching [22], spectral angle mapper [23], subspace projection methods [24,25], and constrained least squares [26]). In most cases, the number of substances and their reflectances are not known and, then, hyperspectral unmixing falls into the class of blind source separation problems [27]. Independent component analysis (ICA) has recently been proposed as a tool to blindly unmix hyperspectral data [28–31]. ICA is based on the assumption of mutually independent sources (abundance fractions), which is not the case of hyperspectral data, since the sum of abundance fractions is constant, implying statistical dependence among them. This dependence compromises ICA applicability to hyperspectral images as shown in Refs. [21, 32]. In fact, ICA finds the endmember signatures by multiplying the spectral vectors with an unmixing matrix, which minimizes the mutual information among sources. If sources are independent, ICA provides the correct unmixing, since the minimum of the mutual information is obtained only when sources are independent. This is no longer true for dependent abundance fractions. Nevertheless, some endmembers may be approximately unmixed. These aspects are addressed in Ref. [33]. Under the linear mixing model, the observations from a scene are in a simplex whose vertices correspond to the endmembers. Several approaches [34–36] have exploited this geometric feature of hyperspectral mixtures [35]. Minimum volume transform (MVT) algorithm [36] determines the simplex of minimum volume containing the data. The method presented in Ref. [37] is also of MVT type but, by introducing the notion of bundles, it takes into account the endmember variability usually present in hyperspectral mixtures. The MVT type approaches are complex from the computational point of view. Usually, these algorithms find in the first place the convex hull defined by the observed data and then fit a minimum volume simplex to it. For example, the gift wrapping algorithm [38] computes the convex hull of n data points in a d-dimensional space with a computational complexity of O(nbd=2cþ1), where bxc is the highest integer lower or equal than x and n is the number of samples. The complexity of the method presented in Ref. [37] is even higher, since the temperature of the simulated annealing algorithm used shall follow a log( ) law [39] to assure convergence (in probability) to the desired solution. Aiming at a lower computational complexity, some algorithms such as the pixel purity index (PPI) [35] and the N-FINDR [40] still find the minimum volume simplex containing the data cloud, but they assume the presence of at least one pure pixel of each endmember in the data. This is a strong requisite that may not hold in some data sets. In any case, these algorithms find the set of most pure pixels in the data. PPI algorithm uses the minimum noise fraction (MNF) [41] as a preprocessing step to reduce dimensionality and to improve the signal-to-noise ratio (SNR). The algorithm then projects every spectral vector onto skewers (large number of random vectors) [35, 42,43]. The points corresponding to extremes, for each skewer direction, are stored. A cumulative account records the number of times each pixel (i.e., a given spectral vector) is found to be an extreme. The pixels with the highest scores are the purest ones. N-FINDR algorithm [40] is based on the fact that in p spectral dimensions, the p-volume defined by a simplex formed by the purest pixels is larger than any other volume defined by any other combination of pixels. This algorithm finds the set of pixels defining the largest volume by inflating a simplex inside the data. ORA SIS [44, 45] is a hyperspectral framework developed by the U.S. Naval Research Laboratory consisting of several algorithms organized in six modules: exemplar selector, adaptative learner, demixer, knowledge base or spectral library, and spatial postrocessor. The first step consists in flat-fielding the spectra. Next, the exemplar selection module is used to select spectral vectors that best represent the smaller convex cone containing the data. The other pixels are rejected when the spectral angle distance (SAD) is less than a given thresh old. The procedure finds the basis for a subspace of a lower dimension using a modified Gram–Schmidt orthogonalizati on. The selected vectors are then projected onto this subspace and a simplex is found by an MV T pro cess. ORA SIS is oriented to real-time target detection from uncrewed air vehicles using hyperspectral data [46]. In this chapter we develop a new algorithm to unmix linear mixtures of endmember spectra. First, the algorithm determines the number of endmembers and the signal subspace using a newly developed concept [47, 48]. Second, the algorithm extracts the most pure pixels present in the data. Unlike other methods, this algorithm is completely automatic and unsupervised. To estimate the number of endmembers and the signal subspace in hyperspectral linear mixtures, the proposed scheme begins by estimating sign al and noise correlation matrices. The latter is based on multiple regression theory. The signal subspace is then identified by selectin g the set of signal eigenvalue s that best represents the data, in the least-square sense [48,49 ], we note, however, that VCA works with projected and with unprojected data. The extraction of the end members exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. As PPI and N-FIND R algorithms, VCA also assumes the presence of pure pixels in the data. The algorithm iteratively projects data on to a direction orthogonal to the subspace spanned by the endmembers already determined. The new end member signature corresponds to the extreme of the projection. The algorithm iterates until all end members are exhausted. VCA performs much better than PPI and better than or comparable to N-FI NDR; yet it has a computational complexity between on e and two orders of magnitude lower than N-FINDR. The chapter is structure d as follows. Section 19.2 describes the fundamentals of the proposed method. Section 19.3 and Section 19.4 evaluate the proposed algorithm using simulated and real data, respectively. Section 19.5 presents some concluding remarks.

A didactic sequence of elementary geometric optics Informed by history and philosophy of science

The concepts and instruments required for the teaching and learning of geometric optics are introduced in the didactic processwithout a proper didactic transposition. This claim is secured by the ample evidence of both wide- and deep-rooted alternative concepts on the topic. Didactic transposition is a theory that comes from a reflection on the teaching and learning process in mathematics but has been used in other disciplinary fields. It will be used in this work in order to clear up the main obstacles in the teachinglearning process of geometric optics. We proceed to argue that since Newton’s approach to optics, in his Book I of Opticks, is independent of the corpuscular or undulatory nature of light, it is the most suitable for a constructivist learning environment. However, Newton’s theory must be subject to a proper didactic transposition to help overcome the referred alternative concepts. Then is described our didactic transposition in order to create knowledge to be taught using a dialogical process between students’ previous knowledge, history of optics and the desired outcomes on geometrical optics in an elementary pre-service teacher training course. Finally, we use the scheme-facet structure of knowledge both to analyse and discuss our results as well as to illuminate shortcomings that must be addressed in our next stage of the inquiry.

Conceito de volume

Este estudo analisa o modo como se desenvolve a aprendizagem do conceito de volume nos alunos do 6º ano de escolaridade, no quadro de uma proposta pedagógica que dá ênfase a actividades que apelam à visualização e ao raciocínio espacial. O seu objectivo principal foi o de compreender as ideias que os alunos do 6.º ano têm sobre volume e perceber como se desenvolvem quando são envolvidos numa experiência de ensino, tendo por base uma cadeia de tarefas que apelam à visualização e ao raciocínio espacial. O estudo seguiu uma metodologia de investigação qualitativa baseada em estudos de caso. A proposta pedagógica foi desenvolvida em quatro aulas; três de noventa minutos e uma de quarenta e cinco minutos, durante os 2º e 3º períodos do ano lectivo de 2009/2010. A recolha de dados envolveu a realização de gravações áudio, em ambiente de sala de aula, dos alunos que constituíram os estudos de caso, registo de observações do desempenho dos alunos e os documentos produzidos por estes. Os resultados mostram que, ao longo da proposta pedagógica, os alunos adquiriram estratégias de contagem que lhes permitiram criar estruturas, para compreender a organização dos paralelepípedos e desenvolver o conceito de volume. iii ABSTRACT This study examines how 6th graders’ students develop the concept of volume in a learning experience context which emphasizes the visualization and spatial reasoning. The main objective of this study is to understand the ideas of 6th grade students about volume and see how they develop them when they are involved in a learning experience, based on a sequence of tasks that call for visualization and spatial reasoning. The study followed a qualitative research methodology based on case studies. The learning experience was developed in four classes, three ninety minutes period and one forty-five minutes, during the 2nd and 3rd terms of the school year 2009/2010. Data collection involved the use of audio recordings in the classroom environment, recording observations of student performance and the documents produced by them. The results showed that, over the learning experience, students acquired counting strategies that allowed them to create structures for understanding the organization of cubes arrays and develop the concept of volume.

Learning dependent sources using mixtures of Dirichlet: applications on hyperspectral unmixing

This paper is an elaboration of the DECA algorithm [1] to blindly unmix hyperspectral data. The underlying mixing model is linear, meaning that each pixel is a linear mixture of the endmembers signatures weighted by the correspondent abundance fractions. The proposed method, as DECA, is tailored to highly mixed mixtures in which the geometric based approaches fail to identify the simplex of minimum volume enclosing the observed spectral vectors. We resort then to a statitistical framework, where the abundance fractions are modeled as mixtures of Dirichlet densities, thus enforcing the constraints on abundance fractions imposed by the acquisition process, namely non-negativity and constant sum. With respect to DECA, we introduce two improvements: 1) the number of Dirichlet modes are inferred based on the minimum description length (MDL) principle; 2) The generalized expectation maximization (GEM) algorithm we adopt to infer the model parameters is improved by using alternating minimization and augmented Lagrangian methods to compute the mixing matrix. The effectiveness of the proposed algorithm is illustrated with simulated and read data.

A hybrid approach to multi-agent decision-making

In the aftermath of a large-scale disaster, agents' decisions derive from self-interested (e.g. survival), common-good (e.g. victims' rescue) and teamwork (e.g. fire extinction) motivations. However, current decision-theoretic models are either purely individual or purely collective and find it difficult to deal with motivational attitudes; on the other hand, mental-state based models find it difficult to deal with uncertainty. We propose a hybrid, CvI-JI, approach that combines: i) collective 'versus' individual (CvI) decisions, founded on the Markov decision process (MDP) quantitative evaluation of joint-actions, and ii)joint-intentions (JI) formulation of teamwork, founded on the belief-desire-intention (BDI) architecture of general mental-state based reasoning. The CvI-JI evaluation explores the performance's improvement

Estruturas tradicionais vs. termoargila em zonas sísmicas

Com a crescente divulgação no mercado português de métodos construtivos de alvenaria resistente tipo Termoargila, compara-se neste trabalho a sua rentabilidade económica, em relação à execução em betão armado, com paredes não estruturais de alvenaria. Estudam-se três tipologias de estruturas com geometria regular (1 piso, 2 pisos, 4 pisos), em zonas sísmicas A e D segundo o Regulamento de Segurança e Acções. A análise dos resultados permite verificar a eficiência dos métodos construtivos para cada tipologia de edifício, assim como os seus custos. Analisa-se se o motivo pelo qual em Portugal não é corrente a aplicação de soluções estruturais de alvenaria resistente tipo Termoargila, se unicamente económico ou se existe uma inércia dos intervenientes na construção, privilegiando os métodos construtivos tradicionais.

Developmental and evolutionary assumptions in a study about the impact of premature birth and low income on mother–infant interaction

In order to study the impact of premature birth and low income on mother–infant interaction, four Portuguese samples were gathered: full-term, middle-class (n=99); premature, middle-class (n=63); full-term, low income (n=22); and premature, low income (n=21). Infants were filmed in a free play situation with their mothers, and the results were scored using the CARE Index. By means of multinomial regression analysis, social economic status (SES) was found to be the best predictor of maternal sensitivity and infant cooperative behavior within a set of medical and social factors. Contrary to the expectations of the cumulative risk perspective, two factors of risk (premature birth together with low SES) were as negative for mother–infant interaction as low SES solely. In this study, as previous studies have shown, maternal sensitivity and infant cooperative behavior were highly correlated, as was maternal control with infant compliance. Our results further indicate that, when maternal lack of responsiveness is high, the infant displays passive behavior, whereas when the maternal lack of responsiveness is medium, the infant displays difficult behavior. Indeed, our findings suggest that, in these cases, the link between types of maternal and infant interactive behavior is more dependent on the degree of maternal lack of responsiveness than it is on birth status or SES. The results will be discussed under a developmental and evolutionary reasoning

Modalidades síncronas e assíncronas no ensino da radiologia do sistema nervoso: estilos e resultados de aprendizagem

Lidar com as problemáticas da saúde, implica um domínio dos processos cognitivos (raciocínio, resolução de problemas e tomada de decisão) e de desempenhos práticos, o que obriga a afectação de um conjunto de atitudes e comportamentos específicos. Este estudo, implementou e avaliou o impacto de experiências pedagógicas desenvolvidas com os estudantes da unidade curricular Radiologia do Sistema Nervoso (RSN) da Escola Superior de Tecnologia da Saúde de Lisboa (ESTeSL). Aplicaram-se metodologias de ensino mistas (presenciais e virtuais) utilizadas na leccionação teórica e prática no ano curricular 2008/2009. Para a avaliação do perfil de aprendizagem dos estudantes foi aplicado o método de Honey & Munford e para a avaliação e monitorização dos conhecimentos aplicaram-se check list baseadas nos conteúdos programáticos. A monitorização das ferramentas da plataforma moodle complementaram a restante informação. Verificou-se uma progressão de aprendizagem positiva para um grupo de estudantes maioritariamente do estilo reflexivo (média=10,6 estudantes). As conclusões apontaram para um impacto positivo quanto à aplicação das metodologias híbridas com maior índice de sucesso para a metodologia assíncrona. Verificou-se também mais flexibilidade no acesso aos conteúdos porém com algumas limitações tais como residência inicial por parte dos estudantes, maior carga de trabalho para os docentes, falta de terminais para acesso à plataforma e pouca experiência de todos os envolvidos no domínio e manipulação da plataforma. ABSTRACT - This study focused on the role of cognitive processes (reasoning, problem solving and decision making) and performance practice in the formation of attitudes and behaviours relating to health issues. It was conducted to evaluate the effects of pedagogical experiences on students who participated in the course in radiography in the Nervous System Imaging Unit (RSN) of the Lisbon Health School of Technology. Mixed (face-to-face and virtual) teaching methodologies were used in theory and practice sessions. Honey and Munford’s method was used to evaluate the learning profile of students. To monitor and evaluate students’ knowledge acquisition, check lists based on program topics were applied. Other information was supplied through the learning platform of Moodle. The student group with mostly a reflective learning style increased their knowledge. The asynchronous method was shown to produce a higher success rate and more flexibility in accessing content but also registered some limitations such as resistance by students, increased workload for teachers, lack of access to the platform and inexperience of all involved in handling the platform.

Evaluation of intermolecular interactions in thioxanthone derivatives: substituent effect on crystal diversity

A family of 9H-thioxanthen-9-one derivatives and two precursors, 2-[(4-bromophenyl) sulfanyl]-5-nitrobenzoic acid and 2-[(4-aminophenyl) sulfanyl]-5-nitrobenzoic acid, were synthesized and studied in order to assess the role of the different substituent groups in determining the supramolecular motifs. From our results we can conclude that Etter's rules are obeyed: whenever present the -COOH head to head strong hydrogen bonding dimer, R-2(2)(8) synthon, prevails as the dominant interaction. As for -NH2, the best donor when present also follows the expected hierarchy, an NH center dot center dot center dot O(COOH) was formed in the acid precursor (2) and an NH center dot center dot center dot O(C=O) in the thioxanthone (4). The main role played by weaker hydrogen bonds such as CH center dot center dot center dot O, and other intermolecular interactions, pi-pi and Br center dot center dot center dot O, as well as the geometric restraints of packing patterns shows the energetic interplay governing crystal packing. A common feature is the relation between the p-p stacking and the unit cell dimensions. A new synthon notation, R`, introduced in this paper, refers to the possibility of accounting for intra- and intermolecular interactions into recognizable and recurring aggregate patterns.

Synthesis of optimization of [11C] Pittsburgh compound B by the captive solvent method

Mestrado em Medicina Nuclear.

Avaliação de um sistema CAD na deteção de lesões em imagens mamográficas

Mestrado em Radiações Aplicadas às Tecnologias da Saúde - Área de especialização: Imagem Digital por Radiação X.

Fundamentos didatológicos para a construção de unidades curriculares integradas

Neste trabalho, com base na análise de diferentes formas de conceber e entender a integração didática na perspetiva curricular portuguesa e da consideração do trinómio didático - globalização da aprendizagem / motivação / ensino e aprendizagem integrados nos primeiros anos de escolaridade –, propomos, do ponto de vista científico - didático, uma reflexão sobre os princípios e fundamentos da sequenciação didática e da construção de unidades didáticas integradoras, com o objetivo de dar contributos significativos para que, desde a formação de professores, se possam criar bases teórico-práticas de definição de percursos de ensino e aprendizagem com potencial efetivo de integração nas dimensões epistemológica e curricular.