Search

Introdução: A pesquisa encontra o seu fundamento em factos científicos de artigos que discutem estratégias didáticas utilizadas no ensino e na aprendizagem do efeito fotoelétrico, tendo como base o uso das simulações computacionais. Objetivo: Desenvolver uma estratégia eficaz de utilização de simulações computacionais no ensino do tópico “efeito fotoelétrico” de modo a melhorar as aprendizagens dos alunos. Métodos: O estudo foi realizado numa Escola Secundária Pública localizada no distrito de Inharrime, a sul de Moçambique. Para a recolha de dados foi realizada uma pesquisa qualitativa e quantitativa, contando com a participação de uma amostra de quarenta (40) alunos do 12o ano do ensino médio, divididos em quatro turmas de dez alunos cada, sendo uma turma de controlo e as outras três de grupo experimental (onde foi aplicada a estratégia metodológica aqui proposta, baseada na realização das atividades laboratoriais seguidas da utilização de uma simulação computacional). Nas quatro turmas foi aplicado um pré-teste com o objetivo de avaliar o nível inicial de conhecimentos dos alunos, e um pós-teste para avaliar não só o contributo das atividades laboratoriais assim como da simulação computacional nas aprendizagens do tópico “efeito fotoelétrico”. Resultados: Os resultados deste estudo mostram que o uso de simulações computacionais associadas ou não com uma experiência laboratorial, contribui para uma aprendizagem mais eficaz sobre o tópico “efeito fotoelétrico”. Conclusão: De acordo com os resultados desta pesquisa, sugere-se a realização de uma experiência laboratorial aliada ao uso de simulações computacionais como uma alternativa para a prática pedagógica, uma vez que facilita a compreensão dos conteúdos por parte dos alunos, resultando em aprendizagens mais significativas.

This study introduces two new sequences: the incomplete Edouard and the incomplete Edouard–Lucas numbers. In addition, we establish some of the properties, identities, and recurrence relations of these sequences. The relations of these new sequences with balancing, Lucas-balancing, incomplete balancing, and incomplete Lucas-balancing numbers are explored, and some new identities are provided. Moreover, the functions for generating the incomplete Edouard and incomplete Edouard–Lucas numbers are stated.

In this study, we investigate some new properties of the sequence of bi-periodic Fibonacci numbers with arbitrary initial conditions, through an approach that combines the matrix aspect and the fundamental Fibonacci system. Indeed, by considering the properties of the eigenvalues of their related 2×2 matrix, we provide a new approach to studying the analytic representations of these numbers. Moreover, the similarity of the associated 2×2 matrix with a companion matrix, allows us to formulate the bi-periodic Fibonacci numbers in terms of a homogeneous linear recursive sequence of the Fibonacci type. Therefore, the combinatorial aspect and other analytic representations formulas of the Binet type for the bi-periodic Fibonacci numbers are achieved. The case of bi-periodic Lucas numbers is outlined, and special cases are exposed. Finally, some illustrative examples are given.

O presente trabalho incide sobre a importância da motivação dos alunos para a aprendizagem da geometria, centrando as atenções nos recentes manuais escolares de matemática, em adoção para o ano letivo de 2022/23, e na integração de referências ao quotidiano e vivências dos alunos que os possam interessar e levar a relacionar os conteúdos disciplinares com o seu dia-a-dia. A partir da análise de dois novos manuais, constatamos que apesar dessa integração, continuamos a deparar-nos com alguma falta de incentivo à curiosidade e motivação, baseada em situações de ligação e articulação com a realidade dos alunos.

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers.

In 2020, education found itself involved in a whirlwind of metamorphosis, of transformations that required responses to the emergencies dictated by the pandemic. In Portugal, many schools opted for Microsoft 365 as the platform of choice for providing adequate resources for teaching and learning processes, while also ensuring remote teaching in an integrated and inclusive way. In this context, we carried out an investigation with the objectives of knowing the opinion of teachers about the use of Microsoft 365 in their classes and identify their degree of satisfaction with its use. The methodology adopted had a descriptive and exploratory nature, following a mixed paradigm. A total of 101 primary and secondary school teachers from schools in Northern Portugal participated in the study. A questionnaire and an interview were used as data collection instruments. The results showed that the most respondents revealed a high level of satisfaction with the use of Microsoft 365, and that its use was accompanied by the employment of active methodologies. Moreover, despite the lack of initial or ongoing training of teachers in the use of this technology and the students’ lack of digital competence, Microsoft 365 proved to be an adequate response to the confinement and ensured students’ learning in a safe environment.

Technologies came to revolutionize the teaching of technical design; they brought resources that enabled 3D and 2D graphic simulation. The objective of this research is to understand how the implementation of the dynamic software GeoGebra on the topic of cylindrical sections can improve 3D spatial visualization in students because of the challenges in understanding the 3D representations and transporting them to the 2D sheet. This is a quasi-experimental case study in a mixed qualitative and quantitative approach which was conducted in a grade 12 class on the subject of descriptive geometry. The instruments used were a pre-test, post-test and Santa Barbara Solids Test. The students of the experimental group used GeoGebra and obtained better results in the post-test than the students in the control group, which suggests an improvement in learning; however, the Santa Barbara Solids Test results showed that the difficulties in visualizing views of the projected solids in a two-dimensional plane persisted. This study is also intended to test software in a classroom with students who have never experienced a 3D technology in their educational practices and direct future research in the complex content of descriptive geometry with the help of software.

A generalization of complex, dual, and hyperbolic numbers has recently been defined as hybrid numbers. In this study, using the Leonardo Pisano numbers and hybrid numbers we investigate Leonardo Pisano polynomials and hybrinomials. Furthermore, we also describe the basic algebraic properties and some identities of the Leonardo Pisano polynomials and hybrinomials.

Resumo Neste artigo descrevemos um livro do professor, usado na Província de Angola em 1962, para orientação do ensino dos alunos do 1.º ano do ensino primário rural dessa região. O livro analisado ilustra como era o ensino para os alunos que iniciavam a frequência escolar em zonas rurais e que incluía crianças indígenas. Este livro testemunha aspetos didáticos de vanguarda relativamente ao que se verificava em Portugal continental na época. Palavras-Chave: Ensino elementar; Livro de texto; Ãfrica; Portugal; Educação. The Teacher's book: "Didactics of the lessons of the 1ST year of rural primary education", in Angola in the 60s - an overview Abstract In this paper we describe a teacher's book, used in 1962 in the Province of Angola in order to guide the education of students in the 1st year of elementary school in this rural region. This book shows how the teaching for students who started school attendance in rural areas including indigenous children. In the book we can find didactic aspects of avant-garde relative to what it was in continental Portugal at that time. Keywords: Elementary education; Text book; Africa; Portugal; Education. Libro del profesor: "Didáctica de las Lecciones del primer año de enseñanza primaria rural" en Angola en los años 60: descripción general Resumen En este artículo se describe el libro de un profesor, que se utiliza en la provincia de Angola en 1962 para guiar la educación de los estudiantes en el primer año de la educación primaria rural en la región. El libro analizado ilustra cómo era la escuela para los estudiantes que iniciaron la asistencia escolar en las zonas rurales y que incluían los niños indígenas. Este libro testigo aspectos didácticos de vanguardia en relación a lo que ocurrió en Portugal continental en el momento. Palabras clave: La educación primaria; Libro de texto; Ãfrica; Portugal; Educación.

Creativity is an indispensable competence for the future of higher education students. Creative thinking skills play an important role in modern society. Creativity is acknowledged as a crucial aspect of the business, research and development or arts. This study presents the assessment results of an intervention using cooperative learning and the conventional teaching method in order to promote creative thinking in higher education. The design used was quasi-experimental pretest and posttest intervention using the Creative Intelligence test (CREA), with experimental and control groups of higher education students of the Communication and Multimedia course in a Linear Algebra class. The participants were 50 students from a Portuguese public university. Results showed that students who participated in the intervention scored significantly higher in the creativity test at the end of the intervention and indicated that creative thinking, and divergent thinking abilities in particular, can be enhanced through the kind of intervention that was proposed in the study.

This article aims to identify mathematical relations associated with a musical instrument called Marimba;, it is a wooden artefact and with iron trestles, built in three villages in the province of Malanje, in Angola. We did an analysis of how to build this instrument in order to identify some mathematical relationships underlying the construction. We used the ethnographic method of data collection, through direct observation, semi-structured interviews with craftsmen, a photographic collection of Marimbas, videos related to their construction and, tuning and we studied this process in three villages of this Angolan’s province. The selected villages are - the village "Caiongo" and the village "Eucalyptus", both located, both, in the region of Calandula and the village “Mufuma” in the Kiwaba Nzoji region. These villages were selected because they are the communes (term used for the third-level administrative units in Angola after the municipalities) to build more Marimba in the Malanje province. The analysis of the elements - gourds and keys - used in its construction made it possible to identify some mathematical relationships related to their area and volume. The relationship between the areas and volumes found show differences between the sizes of these Marimba elements from one region to another. This type of study, using tools like this and others, can be introduced in the classroom to motivate students for learning mathematical concepts, showing its application to the reality that they are very close and that, no doubt, can contribute to the promotion of a cultural identity that should not be forgotten from generation to generation.

In this article we analyze the research studies carried out by northeast Portuguese about the existing Ethnomathematics knowledge in traditional professions and artefacts. We refer to craft trades such as cooperage, tinsmithing, locksmithing and carpentry, and artefacts built by these craftsmen, such as barrels and vats, oil cans and other canned vessels, wrought-iron balconies, and yokes. We reflect on these studies and we disclose their potential in educational terms. We verified that it is possible to apply ethnomathematical knowledge and know-how in the context of the mathematics classroom in the 2nd and 3rd cycles of Portuguese education, and that this has its advantages. The mathematical contents are contextualized and the students discover uses of mathematics that they did not suspect, although being very close to them. In short, artisans feel their work valued and students become more aware of the culture of their region, in addition to learning math in an experimental and more engaging way.

Este trabalho apresenta como uma aluna do 5º ano de escolaridade de uma escola pública em Portugal resolveu o trabalho de férias de estatística usando o seu quotidiano como ponto de partida para a abordagem dos conteúdos. A metodologia é qualitativa, pois queríamos efetuar uma análise da atividade realizada do ponto de vista de ato pedagógico e registar aspetos do seu enfoque semiótico. A contribuição deste trabalho verifica-se no facto de que uma atividade de educação básica, sendo realizada num contexto de reconhecimento do aluno também pode facilitar essa aprendizagem dos conteúdos.

Este estudo tem como objetivo investigar aspetos da prática de ensino de um professor de Matemática, relativa a Sistemas de Equações no ano letivo de 1989/1990. Os dados foram recolhidos a partir das orientações curriculares oficiais, do manual adotado, do caderno diário de um aluno e da entrevista efetuada ao professor. Procedeu-se à análise de conteúdo e ao cruzamento dos dados. O estudo considerado mostra que as aulas do professor foram marcadas por um estilo de ensino expositivo, baseado na resolução de exercícios e no manual.

This work describes the process of conception and development of an educational game of mathematics called "Mathematical Trivial", by students of the eighth grade of schooling, in a public school in Portugal. The game allows to practice the most varied mathematical contents through small adaptations, in any year of elementary education. In this work, we investigated whether the design of an educational game promotes the learning of mathematical concepts and identified the implications of students' attitudes towards mathematics. The methodology used was a case study design. This is a mixed case study because there is a combination of tools and techniques associated with qualitative and quantitative studies. The data collection was done through naturalistic observation, the application of tasks and the application of a questionnaire survey. Statistical analysis of the data was performed using spss software. The results suggest that the use of the educational game, as a resource to support the classes, produces a significant improvement of the students 'attitudes towards mathematics, as well as, it contributes to develop students' motivation. In addition, the process of constructing the educational game points to an improvement in the learning of geometrical concepts inherent to the game, in a playful and dynamic way.

Creativity is a multifaceted term and in complex definition, due to the different definitions that have appeared over the years. However, in spite of this, its importance today is essential, especially in a quality educational context. To this end, the students of the 3rd cycle of basic education were given the opportunity to identify probable concepts referring to the meaning of creativity. An investigation study was developed during the school years 2013/2014 and 2014/2015. The methodology used was a case study design. The case study consisted of a case study, multiple, consisting of three cases, Case I, Case II and Case III, referring respectively to the three years of schooling (7th, 8th and 9th), involving five classes from two groups of public schools in the north of Portugal. The data collection instrument was a questionnaire. In the data treatment, was opted for the analysis of the content of the answers given by the students participating in this questionnaire survey and the formulation of analysis categories. The results allowed to conclude that the conceptions of creativity of the participants are different. In Case I, the conception of creativity of these students is associated with Imagination, while in the other two cases (Case II and III), creativity was characterized as being something new and different.

Muitas têm sido as sequências de números estudadas e que servem de motivação para o ensino de matemática. É disso exemplo a bem conhecida sequência de Fibonacci, não só pela sua ligação com várias formas da natureza, com a arte, com o belo, com a harmonia e o número de ouro, mas também pelos mais variados desafios que proporciona aos professores para ensinar conteúdos matemáticos usando as suas propriedades. Neste artigo dedicamos a nossa atenção a uma outra sequência de números - a sequência prateada - ainda pouco divulgada na literatura, apresentando, sucintamente, algumas das suas interessantes propriedades e mostrando que também ela pode ser útil no ensino de alguns conteúdos matemáticos.

Neste artigo damos especial ênfase às matemáticas envolvidas na almotolia – artefacto de latoaria – ao procurarmos contribuir para a valorização, divulgação e preservação de uma das mais antigas artes e ofícios tradicionais – o latoeiro – da região portuguesa de Trás-os-Montes e Alto Douro. Desenvolvemos um estudo sobre saberes (etno)matemáticos envolvidos nos processos de construção da almotolia, apoiados em entrevistas semiestruturadas a cinco latoeiros desta região e em pesquisa bibliográfica e de campo, com recurso à observação participante. Identificamos a utilização de geometria elementar. Com os resultados obtidos é possível criar tarefas matemáticas envolvendo os processos de contrução da almotolia para o ensino básico português (6 a 15 anos).

Este trabalho, em desenvolvimento pelos autores, dinamiza a construção e utilização de recursos educativos, de inspiração Etnomatemática, que incorporam elementos pertencentes à cultura ancestral, de alunos e professores, de duas regiões limítrofes da Península Ibérica – Galiza e Norte de Portugal. No acompanhamento de todo o processo de construção e utilização dos recursos pretendemos compreender como os mesmos podem influenciar os processos de ensino e de aprendizagem, esperando que se evidenciem como um meio capaz de induzir alterações nesses processos e desta forma contribuir para uma melhoria na aprendizagem da Matemática.

O presente trabalho tem como objetivo investigar aspetos do percurso escolar e da prática de ensino no primeiro ano da carreira docente de um professor de Matemática, relativa ao tema Sistemas de Equações, para um maior conhecimento do professor. O método usado foi o da análise documental, em particular do seu acervo pessoal. O estudo considerado mostra que o professor desde muito cedo escolheu a profissão docente e as aulas no ano letivo de 1999/2000 foram marcadas por um estilo de ensino expositivo, baseado na resolução de exercícios e que recorreu ao manual adotado e uma preocupação sumativa na avaliação.

In this paper, we introduce the bicomplex generalized tribonacci quaternions. Furthermore, Binet’s formula, generating functions, and the summation formula for this type of quaternion are given. Lastly, as an application, we present the determinant of a special matrix, and we show that the determinant is equal to the n th term of the bicomplex generalized tribonacci quaternions.

Given the diversity of meanings nowadays, creativity and mathematical creativity are difficult to define. However, it is crucial to do so, whether in society, in education or in mathematics education. Creativity is not valued as an integral part of the classroom context of mathematics or in Teacher practice, possibly due to the constant changes of Portuguese curricular. In this sense, it is necessary to promote the development of creativity as an integral part of the pedagogical practice of each teacher. The teacher’s role is fundamental to allow students opportunities to use the didactical resources, such as the mathematical tasks. In the 21st century, creativity and mathematical creativity have become important in Portuguese curricula through the implementation of the Students Profile in the end of compulsory schooling. This profile appeals to the development of creativity regarding of values, competences and guiding Principles, in order to mobilize schools and society for a better education. It is desirable that a young person, after leaving compulsory schooling, becomes a citizen capable of thinking critically, autonomously and creatively. Thus, in order to look for creative teaching approaches and strategies for the mathematical contents, it was decided to question students on the alluded concepts. This research took place in the academic year 2017/2018, with 14 participants aged between 15 and 21 years old (mean ages M = 17.36 and DP = 1.82) of two classes from one group of Portuguese state schools. The methodology was qualitative and interpretative, using a multiple case study design, one in a 10th grade class and another in a 12th. In data processing related to the analysis of the written documents, content analysis was used, and examples that represent each category were presented. The results achieved indicated, for these participants, that conceptions of creativity and mathematical creativity were not divergent in the two cases. In fact, the participants associated creativity with the category of Create something new and different and mathematical creativity with the category of Innovation.

No sentido de perceber se a criatividade quando aplicada nas práticas letivas de matemática poderá ou não marcar a diferença na formação dos educadores e dos professores e, consequentemente, na formação dos mais novos, preparámos uma ação de formação destinada a educadores de infância e professores do 1.º ciclo do ensino básico. A metodologia de trabalho e as tarefas criadas para essa ação foram pensadas de modo a fomentar a criatividade. Efetuámos quatro estudos de caso múltiplos (os grupos de trabalho nas sessões) e posteriormente, quatro estudos de caso individuais com duas educadoras e duas professoras. Neste artigo apresentamos os dois estudos de caso relativos às duas educadoras.

Prospective studies concerning liver disease in pediatric cystic fibrosis patients are scarce. The present study aimed to describe the prevalence and clinical expression of cystic fibrosis - related liver disease, in a cohort of 62 pediatric patients. Descriptive study, resulting from the prospective evaluation, between 1994 and 2009, of 62 pediatric patients (age