ESEV - DCEN - Artigo em revista científica, indexada ao WoS/Scopus
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- An overview on the history of actuarial calculus in Portugal until the late 19th centuryPublication . martins, ana patríciaPension funds, namely survivors' pension funds were established in Portugal in the late 18th century, in mutual benefit societies. The first Portuguese life assurance companies were created in the first half of the 19th century but their activity was not very extensive. Only in the 20th century did both institutions become actuarially-based. In this paper, we provide an overview of the development of actuarial calculus in Portugal until the late 19th century, in order to contextualise the scientific progress of those institutions, mainly the former. This paper lays the groundwork for further research on the history of actuarial calculus in Portugal.
- Conceções de Professores do Ensino Básico sobre a Prova Matemática: influência da experiência profissionalPublication . Viseu, Floriano; Menezes, Luís; Fernandes, José António; Gomes, Alexandra; Martins, Paula MendesProof is an activity that plays a key role in the construction of mathematical knowledge, which is why it acquires relevance in mathematics programs. Admitting that teachers' conceptions about proof influence how it is handled in the classroom, we tried to investigate the conceptions of Portuguese mathematics teachers from the 3rd cycle of basic education (grade 7 to 9) on different aspects of mathematical proof and the relationship that the teachers’ experience has in these conceptions. Adopting a mixed methodological approach, we collected data through a questionnaire answered by 72 teachers, and an interview with two teachers with different professional experience. The results show that teachers, especially those with less experience, consider that mathematical proof has a distinct nature from other disciplines, it is an essential activity for the construction of mathematical knowledge and its function is to verify and explain the truth of a statement by developing reasoning and mathematical communication. With respect to student participation in the activity of proving, teachers with more teaching experience are the ones that most highlight it, which allows students to realize the nature of this activity. In curriculum terms, teachers with less experience are the ones that most agree with the presence of proof in early grades, while emphasizing that this activity only makes sense in some program topics.
- Conceptions and practices of Iberian mathematics teachers about humourPublication . Floriano, Viseu; Menezes, Luís; Amante, Susana; Flores, PabloThis study focuses on the conceptions and practices of Iberian (Portuguese and Spanish) mathematics teachers regarding instructional humour. Specifically, the study aims to answer the following questions: (1) How do Iberian mathematics teachers view humour and how do they appreciate a sense of humour? (2) What educational value do Iberian mathematics teachers ascribe to humour in the teaching and learning process of this subject? (3) How do Iberian mathematics teachers use humour in mathematics teaching? and (4) What differences, regarding humour and its use in the educational context, are found among Iberian mathematics teachers, as determined by their professional experience? A mixed methodology was adopted for this study, with a greater quantitative emphasis, combining quantitative analysis with descriptive and inferential statistics. Iberian mathematics teachers (N=1087) from all educational levels participated in the study. The data was collected through an online questionnaire, organised according to three dimensions of analysis: (1) Humour and sense of humour; (2) Educational value of humour in the teaching and learning of mathematics; and (3) Use of humour in the teaching of mathematics. The results reveal that Iberian teachers who teach mathematics recognise the meaning of humour, feel they have a sense of humour and appreciate it in others, find reasons to use it in mathematics teaching and have seen it being used or use it in their classes to create a good learning environment and make students think. The results show differences over the teachers’ careers, especially in their use of humour and the purpose they make of it. However, the magnitude of the effect suggests the need to consider other variables in addition to the teachers’ professional experience, such as the level of training, the type of training and the cultural context in which the school is located.
- Currículo do pré-secundário em Timor-Leste e sua articulação com o secundário geralPublication . Capelo, Ana; Cabrita, IsabelProblemas herdados nas últimas décadas conduziram Timor-Leste a reestruturar os currículos do ensino pré-secundário – EPS – e do secundário geral – ESG. Atendendo à importância de uma adequada articulação vertical ao longo dos ciclos de escolaridade e diante do cumprimento de uma primeira edição completa do EPS, importa estudar se tal currículo está alinhado com o do ESG, em particular no que respeita às Ciências e Tecnologia. Este artigo centra-se na análise do currículo do EPS e na avaliação da sua articulação com o do ESG. Não obstante existam décalages entre o currículo prescrito, o implementado e o apropriado, é relevante haver sintonia no nível dos currículos para uma educação e formação de boa qualidade, fundamentais à consecução dos Objetivos de Desenvolvimento do Milénio.
- Ensino e aprendizagem da Matemática no contexto da implementação de um novo programa: os pais contam?Publication . Menezes, Luís; Viseu, Floriano; Conceição, SandraThis study was conducted in the context of the implementation, in Portugal, of a new program of mathematics for primary education, which introduced profound changes over the previous one. In the process of mathematics learning is widely recognized the importance of parents/Guardians (G). Its influence is dependent on the knowledge they have, and on their involvement with the school. It is in this context that this research, which adopts a mixed methodology, stands, trying to understand what parents/G know and think about mathematics program, and what support they provide to their children in the learning of this discipline. Data analysis, resulting from the application of a questionnaire and an interview (with parents/G of two schools, one urban and another one rural), shows that they realize a set of changes in learning of their children, particularly in mental calculation, problem-solving and mathematical reasoning. Parents/G have a favourable opinion about the mathematics program. In terms of parental support in mathematics, parents/G show to be attentive to their children and consider their action essential.
- Formulations of the inclusion–exclusion principle from Legendre to Poincaré, with emphasis on Daniel Augusto da SilvaPublication . martins, ana patrícia; Sousa, TeresaThe inclusion–exclusion principle is a simple, intuitive, and extremely versatile result. It is one of the most useful methods for counting and it can be used in different areas of mathematics. In the eighteenth century, the first uses of this result that appear in the literature are related to the study of problems of games of chance. However, the first formulations of this principle appear, independently by several authors, only in the nineteenth century. In this article, we study the formulations obtained by Adrien-Marie Legendre, Daniel Augusto da Silva, James Joseph Sylvester, and Henri Poincaré. We highlight the contribution of the Portuguese mathematician Daniel Augusto da Silva, since his formulation can be applied to different problems of number theory, whenever collections of numbers satisfying certain properties are involved, and this is the reason why his formulation stands out compared with all the others.
- Perspectivas de Professores de Matemática sobre o Humor e o seu Valor EducacionalPublication . Menezes, Luís; Fernandes, José António; Viseu, Floriano; Ribeiro, António Augusto Gaspar; Flores, PabloAt present, humour has an increasing acceptance and appreciation, because we recognize the positive contributions to the well-being of people in contexts as diverse as health, business, and entertainment. What happens in Education and, in particular, in Mathematics teaching? To answer this question, we designed this quantitative study with a sample of 1088 Iberian mathematics teachers (Portuguese and Spanish), that teach from the early years through to higher education. Through the use of a questionnaire, we seek to find out if Portuguese and Spanish teachers who teach Mathematics, throughout the various levels of education, appreciate humor, what perspective they have of it and its educational value, and if they use it in teaching the discipline. The study also sought to determine if the teaching level in which teachers teach mathematics has an influence on the perception and use of humor for instructional purposes. The study also investigates if there are any significant differences between Portuguese and Spanish teachers in relation to these aspects. The results show that the overwhelming majority of teachers, at all levels of teaching, recognize the meaning of humor, consider that they have a sense of humor, justify its use in mathematics teaching and have already seen it being used or used it in their classes, with the goal of creating a good learning environment and making students think. The study reveals differences among teachers at different levels of teaching, indicating that there is a greater predisposition of teachers in the initial years, in comparison to other years, to value and use humor in mathematics teaching. The study also reveals differences between Portuguese and Spanish teachers, with Portuguese presenting a higher average agreement in most items.
- Semiotic Representations in The Learning of Rational Numbers by 2nd Grade Portuguese StudentsPublication . Viseu, Floriano; Pires, Ana Luísa; Menezes, Luís; Costa, Ana MariaThe use of different registers to represent mathematical concepts enhances understanding. For example, rational numbers can assume pictorial, symbolic and natural language representations and this kind of change improves learning. Based on these assumptions, a teaching experiment for the learning of rational numbers by 2nd grade students was conducted, so as to allow for an understanding of how semiotic representations contribute to the learning of rational numbers, particularly with concern to unit fractions. Using a qualitative methodology and a content analysis of the students' written productions, the study shows a greater use of the pictorial representation register compared to the other types. Students' main difficulties in learning rational numbers are related to the pictorial representation of unit fractions and to an understanding of the concept of fraction itself. Some of these difficulties result from errors such as the misrepresentation of unit fractions in the case of the pictorial register, the association of the concept "half" with multiple unit fractions, the absence of the fraction bar when it comes to the symbolic register, the use of everyday terms to represent fractions when students rely on the natural language register, and the misrepresentation of rational numbers when the graphic register is used.
- Spectra and Randic Spectra of Caterpillar Graphs and Applications to the EnergyPublication . Andrade, Enide; Gomes, Helena Margarida dos Santos Vasconcelos; Robbiano, MaríaLet H be an undirected simple graph with vertices v_1,...,v_k and G_1,...,G_k be a sequence formed with k disjoint graphs G_i,i=1,...,k. The H-generalized composition (or H-join) of this sequence is denoted by H[G_1,...,G_k]. In this work, we characterize the caterpillar graphs as a H-generalized composition and we study their spectra and Randi\'c spectra, respectively. As an application, we obtain an improved and tight upper bound for the Energy and the Randi\'c energy of these interesting trees.
- The role performed by the teacher’s question in the learning of quadratic function in an exploratory mathemPublication . Silva, Anaís Veloso; Viseu, Floriano; Menezes, LuísCommunication plays a key role in teaching and learning processes. Questions are a communicational act greatly used by teachers to structure their discourse, establish dynamics, and foster interaction between the different participants in the classroom. In view of these potentialities of questions in the classroom context, we have developed a teaching experiment with the aim of understand the role of the teacher's question in the learning of topics on functions. Considering the nature of this aim, a methodology of qualitative and interpretative nature was used. The data collection was based on the students’ written productions and on the audio and video recordings of a mathematics class of a Grade 10 educational group (in northern Portugal). Data analysis is based on content analysis techniques, crossing collected data and categories emerging from the literature. The study revealed that the teacher’s questions alternated between confirmation, focalization, and inquiry, with inquiry prevailing. Questions aimed at testing the student’s knowledge gave both the teacher and actual student important information. Questions that focused the student’s attention on a particular detail enabled the students to organize their reasoning and structure their answer. Questions that required the students to explain or justify their thoughts were those that proved to most contribute to the development of the student’s reasoning process.