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- Esquemas de demonstração para proposições de Álgebra Linear com valor lógico verdadePublication . Beites, P. D.; Branco, Maria Luísa; Costa, CecíliaO presente trabalho tem por objetivo analisar esquemas de demonstração em produções de alunos numa unidade curricular de álgebra linear do ensino superior. Concretamente, estudamos as representações de três alunos, e suas transformações, em tarefas que envolvem proposições de álgebra linear com valor lógico verdade, procurando aceder ao seu raciocínio matemático. As conclusões do estudo indicam que o nível de desempenho em matemática de um aluno pode não traduzir o nível de desempenho desse aluno na elaboração de demonstrações. Sugerem também uma possível ligação entre os conteúdos, e a experiência, associados ao primeiro contacto dos alunos com a demonstração e as representações por eles utilizadas nos esquemas de demonstração. As conclusões apontam ainda para os significados atribuídos pelos alunos a esquemas de demonstração não dedutivos. Em sentido lato, os resultados mostram a necessidade de intervenções na sala de aula, com base em investigação, de modo a gerar soluções para os desafios do ensino-aprendizagem da demonstração no ensino superior.
- Arithmetic for closed ballsPublication . Beites, P. D.; Nicolás, A. P.; Vitoria, JoseInspired by circular complex interval arithmetic, an arithmetic for closed balls in Rn is pursued. In this sense, the properties of certain operations on closed balls in Rn, some of which related either to the Hadamard product of vectors or to the 2-fold vector cross product when n ∈ {3, 7}, are studied. In particular, known results for operations on closed balls in C, which can be identified with R2, are extended to closed balls in Rn.
- A Note on Standard Composition Algebras of Types II and IIIPublication . Beites, P. D.; Nicolás, AlejandroSome identities satis ed by certain standard composition algebras, of types II and III, are studied and become candidates for the characterization of the mentioned types.[...]
- Desterritorialização didática de erros relacionados com (x+y)^2Publication . Peel, Misleine; Beites, P. D.; Simões, F.Será que (x+y)2=x2+y2 representa um erro? A resposta, sim ou não, depende do universo matemático em que se esteja a trabalhar. Com efeito, é um erro se, por exemplo, x e y forem números complexos arbitrários e “+” denotar a adição usual de números complexos. [...]
- Vector cross product differential and difference equations in R-3 and in R-7Publication . Beites, P. D.; Nicolás, A. P.; Saraiva, Paulo; Vitoria, JoseThrough a matrix approach of the 2-fold vector cross product in R^3 and in R^7, some vector cross product di erential and di erence equations are studied. Either the classical theory or convenient Drazin inverses, of elements belonging to the class of index 1 matrices, are applied.
- Que símbolos são estes?Publication . Bento, Sandra; Beites, P. D.; Simões, FátimaNo domínio de conteúdo Primitivas e Cálculo Integral, do novo Programa de Matemática do Ensino Secundário, surge uma lista de descritores norteados pelos objetivos gerais: Definir a noção de primitiva; Abordar intuitivamente a noção de integral definido; Resolver problemas. [...]
- Using Learning Analytics to evaluate the quality of multiple-choice questions: a perspective with Classical Test Theory and Item Response TheoryPublication . Azevedo, José Manuel; Oliveira, Ema; Beites, P. D.Purpose The purpose of this paper is to find appropriate forms of analysis of multiple-choice questions (MCQ) to obtain an assessment method, as fair as possible, for the students. The authors intend to ascertain if it is possible to control the quality of the MCQ contained in a bank of questions, implemented in Moodle, presenting some evidence with Item Response Theory (IRT) and Classical Test Theory (CTT). The used techniques can be considered a type of Descriptive Learning Analytics since they allow the measurement, collection, analysis and reporting of data generated from students’ assessment. Design/methodology/approach A representative data set of students’ grades from tests, randomly generated with a bank of questions implemented in Moodle, was used for analysis. The data were extracted from a Moodle database using MySQL with an ODBC connector, and collected in MS ExcelTM worksheets, and appropriate macros programmed with VBA were used. The analysis with the CTT was done through appropriate MS ExcelTM formulas, and the analysis with the IRT was approached with an MS ExcelTM add-in. Findings The Difficulty and Discrimination Indexes were calculated for all the questions having enough answers. It was found that the majority of the questions presented values for these indexes, which leads to a conclusion that they have quality. The analysis also showed that the bank of questions presents some internal consistency and, consequently, some reliability. Groups of questions with similar features were obtained, which is very important for the teacher to develop tests as fair as possible. Originality/value The main contribution and originality that can be found in this research is the definition of groups of questions with similar features, regarding their difficulty and discrimination properties. These groups allow the identification of difficulty levels in the questions on the bank of questions, thus allowing teachers to build tests, randomly generated with Moodle, that include questions with several difficulty levels in the tests, as it should be done. As far as the authors’ knowledge, there are no similar results in the literature.
- On skew-symmetric matrices related to the vector cross product in R^7Publication . Beites, P. D.; Nicolás, Alejandro; Vitoria, JoseA study of real skew-symmetric matrices of orders 7 and 8, de ned through the vector cross product in R7, is presented. More concretely, results on matrix properties, eigenvalues, (generalized) inverses and rotation matrices are established.
- Generalized Derivations of Multiplicative n-Ary Hom-Omega Color AlgebrasPublication . Beites, P. D.; Kaygorodov, I.; Popov, Y.We generalize the results of Leger and Luks, Zhang R. and Zhang Y.; Chen, Ma, Ni, Niu, Zhou and Fan; Kaygorodov and Popov about generalized derivations of color n-ary algebras to the case of n-ary Hom- Omega color algebras. Particularly, we prove some properties of generalized derivations of multiplicative n-ary Hom- Omega color algebras. Moreover, we prove that the quasiderivation algebra of any multiplicative n-ary Hom-Omega color algebra can be embedded into the derivation algebra of a larger multiplicative n-ary Hom- Omega color algebra.
- (Não) São números complexosPublication . Beites, P. D.No âmbito dos números complexos, reflete-se sobre duas modificações que se destacam na passagem do anterior para o atual Programa de Matemática A do Ensino Secundário. _A luz do novo currículo e da investigação em Didática da Matemática, apresentam-se algumas propostas de tarefas com números complexos para a sala de aula.