Jaime Carvalho Silva

University of Coimbra | UC · Department of Mathematics

Desde su fundación que la UNESCO tuvo muchas intervenciones importantes en la educación matemática. Intentamos mostrar cómo hay cinco iniciativas y documentos que son extremamente significativas en nuestros tiempos y porqué. Estos documentos e iniciativas deben ser aprovechados para inspirar la práctica de la enseñanza de la Matemática en nuestras...

Disponível em: https://hdl.handle.net/10316/115228 . Neste trabalho apresentamos a estratégia utilizada para ultrapassar a dificuldade colocada pelo absentismo (devidamente justificado) dos estudantes numa unidade curricular de didática especifica da Matemática. Em face da dificuldade em monitorar as situações didáticas e adidáticas necessárias ao...

Geometry has been taught for millennia and we have rather detailed references of the geometry taught in Scribal Schools in Babylon, 4000 years ago. Today, with the computer revolution and Artificial Intelligence available, nobody seems to be proposing we stop teaching Geometry. But which Geometry should we teach, why and how? Which are the driving...

Que conhecimento tem havido em Portugal sobre a Matemática e os seus maiores cultores? Neste trabalho analisamos os conteúdos de História da Matemática em duas enciclopédias portuguesas do século XIX: E. M. Campagne – Dicionário Universal de Educação e Ensino. Trasladado a portuguez por Camillo Castello Branco (1873); Diccionario popular historico,...

As novas Aprendizagens Essenciais (AE) para o Ensino Secundário valorizam con-sideravelmente o uso da tecnologia, em múltiplas dimensões. Nas referênciasé até citado um interessante livro em formato GeoGebra de Jorge Cássio. Nas Apren-dizagens Essenciais para o 10º ano de escolaridade da disciplina de Matemática A podemos ler: A abordagem explorató...

Na nossa época ninguém pode ignorar os dados numéricos ou gráficos com que somos constantemente confrontados. Por exemplo, na coluna do Provedor do Leitor do jornal "Público" do dia 14 de outubro de 2023 o tema em discussão é "Sondagens, fichas técnicas e fiabilidade". O problema levantado por um leitor a que o Provedor tenta responder, tem a ver c...

This study describes an experiment conducted to explore the effectiveness of using Moodle forums as a support for face-to-face classes and as a tool to promote student engagement and enhance success rates in a mathematics course at the higher education level. Three groups were selected: two groups used collaborative work and MOODLE forums for shari...

This chapter focuses on the relations between mathematics and other disciplines through the lens of school curricula. The debates between teaching mathematics as a 'service' discipline and per se is old. In mandatory curricula, it is generally mentioned that mathematics is useful and serves valuable purposes in everyday life, other disciplines and...

This chapter contains the biographical portraits of the 54 members of the Central/Executive Committee of ICMI who passed away in the first 100 years of ICMI.Each of these portraits consists of a section of general biographical information and a section containing contributions to education and dissemination of mathematical culture. In the same vein...

How can STEM education be enriched by means of digital technology? This key question was addressed by the “Research On STEM Education in the Digital Age” conference (ROSEDA). It took place in Porto from February 23 to 25, 2023, and focused on teaching and learning of science, technology, engineering, and mathematics with digital technology. During...

Problem solving is a central aspect of teaching Mathematics. The solving of real and concrete problems must be a central concern of mathematics teaching, as José Sebastião e Silva (1914-1972) said more than 50 years ago: "The mathematics teacher must be, first of all, a teacher of mathematisation, that is, it must get the student to reduce concrete...

La presente revisión integrativa tiene como objetivo conocer los principales factores que influyen en el rendimiento de los estudiantes portugueses, singapurenses, holandeses, españoles y brasileños en Alfabetización Matemática en las ediciones del PISA, de 2000 a 2012, de acuerdo con la literatura publicada. De los estudios surgieron cinco factore...

We consider the generalized Goursat–Darboux problem for a third-order linear PDE with real coefficients. Our purpose is to find necessary conditions for the problem to be well-posed in the Gevrey classes Γ s with s > 1. It is proved that there exists some critical index s 0 such that if the Goursat–Darboux problem is well-posed in Γ s for s > s 0 ,...

The origins, rationale and development of the course ‘Mathematics Applied to the Social Sciences’ (MACS), created in 2001, is described and some ideas around the teaching of mathematics in secondary schools for students other than the future scientists and engineers are discussed.

We consider the generalized Goursat-Darboux problem for a third order linear PDE with real constant coefficients. Our purpose is to find necessary conditions for the problem to be well-posed in the Gevrey classes. Since this problem can be reduced to the Cauchy problem using permutations of independent variables, we solve it for a ODE with complex...

We consider the generalized Goursat-Darboux problem for a third order linear PDE with real constant coefficients. Our purpose is to find necessary conditions for the problem to be well-posed in the Gevrey classes. Since this problem can be reduced to the Cauchy problem using permutations of independent variables, we solve it for a ODE with complex...

The present integrative review has the goal of knowing the main factors influencing the results of Portuguese and Brazilian students in the editions of PISA, from 2000 to 2012, according to the published literature. Five factors and ten sub-factors have been identified. The more determinant factors in the considered studies were the educational sys...

Resumo – Neste trabalho apresentamos o projeto ReM@t – Recuperar a Matemática a Distância cujo objetivo principal é apoiar, através da conceção de um curso aberto, livre, gratuito e permanentemente acessível pela internet, os muitos estudantes que sentem dificuldades e constrangimentos na aprendizagem da Matemática, no Ensino Secundário, no Ensino...

This paper presents the " ReM@t-Recovering Mathematic at a distance " project. It includes the implementation of a free open course, designed to help students to improve their mathematical skills, namely those that they are expected to master when in higher education. The lack of mathematical knowledge is often mentioned as a problem for students t...

In this paper we present briefly some ideas behind the use of technology in schools, as well as some difficulties. We present some reasons why graphing calculators are still important for mathematics teaching, even with the entrance in the school scene of much more sophisticated technologies. Some difficulties around national examinations are summa...

Preliminary version of a paper presented at the meeting “O INSUCESSO EM MATEMÁTICA: CONTRIBUTOS DA INVESTIGAÇÃO” (Lisboa, 23-24 Abril 2004) and never published.

Atas do XXXI Encontro Nacional de Professores de Matemática, ProfMat2015, realizado na Escola Secundária Gabriel Pereira, em Évora, entre 26 e 28 de Março de 2015.

Com este projeto de investigação pretende-se estudar as implicações do contexto cultural nas aprendizagens matemáticas. Em particular, procura-se analisar como é que as crianças se vão apropriando do novo conceito de rede multiplicativa e o mobilizam num ambiente cultural aberto de resolução de problemas, explorando um elemento económico e cultural...

Bajo los auspicios reformadores del Marqués de Pombal, los nuevos estatutos de la Universidad de Coimbra, ratificados por el rey José I en junio de 1772, representaron una importante revisión de los principales estudios en Portugal. Hacia un largo tiempo que los Estatutos de la Universidad de Coimbra no fueron revisados, y la revisión de los Estatu...

The use of technology in schools has been one of the most debated topics around mathematics education. In some countries there is a huge investment, in others there is a down-scaling. Malaysia decided in 2013 to put its 10 million students to use Google laptops and Google apps, while Australia in the same year decided it would not continue funding...

O presente texto tem como objetivo delinear uma reflexão sobre algumas questões relativas à chamada Educação Sombra (vulgo “explicações”) na Contemporaneidade. Nesta reflexão damos visibilidade ao impacto contemporâneo das explicações nos resultados escolares em Portugal e no resto do mundo; ao papel dos explicadores; ao fator de discriminação que...

A presente comunicação resultou de uma investigação realizada no campo das aprendizagens algébricas contextualizadas em que a folha de cálculo surgiu como suporte tecnológico na resolução de problemas. Para além do desenvolvimento de outras competências pôde-se concluir que o uso continuado desta ferramenta promove a aplicação de estratégias multip...

School mathematics reflects the wider aspect of mathematics as a cultural activity. From the philosophical point of view, mathematics must be seen as a human activity both done within individual cultures and also standing outside any particular one. From the interdisciplinary point of view, students find their understanding both of mathematics and...

People have studied, learned and used mathematics for over four thousand years. Decisions on what is to be taught in schools, and how, are ultimately political, influenced by a number of factors including the experience of teachers, expectations of parents and employers, and the social context of debates about the curriculum. The ICMI study is posi...

Differential and Integral Calculus courses are nowadays, in Portuguese and worldwide universities, experiencing the challenge of integrating new technologies, namely graphing calculators and computers equipped with specific software like CAS (Computer Algebraic Systems); this happens on top of a traditional low success rate, thus heating more the d...

La Internet es una herramienta completamente nueva en el mundo del profesor. Lo que ofrece es intrínsecamente diferente de todo lo que havia antes. Tenemos que estar muy atentos para poder determinar a tiempo cuales son los aspectos de la Internet que son importantes para la escuela, en particular para la formación del profesorado. Una de las maner...

A Análise Matemática é ainda, nas nossas Universidades, tradicionalmente ensinada na forma de “papel e lápis”. No entanto, o conteúdo da disciplina compreende conceitos reconhecidamente subtis, cuja abordagem se inicia no Ensino Secundário e se estende depois por diversos semestres – durante a formação universitária inicial dos nossos alunos – das...

Neste texto sao apresentadas algumas propostas de marcos significativos cujo objectivo é provar, sem margem para dúvidas, que a actividade matemática na Universidade de Coimbra no período que medeia entre a Reforma Pombalina de 1772 e a chegada a Portugal de António Aniceto Monteiro em 1936 após o doutoramento em França foi significativa, no contex...

We give an overview of the content of the mathematics course, the type and quality of the professors, the textbooks used and the methods of teaching at the Faculty of Mathematics of the University of Coimbra during the 19th century. We see that progressively the level of mathematics studies was growing attaining an interesting degree of quality in...

Desde o século XVIII que o sistema educativo português tem sido objecto de discussões frequentes e de bastante controvérsia. Contudo, nem sempre essas discussões têm em conta a realidade concreta do sistema, sendo por vezes evidente um confrangedor desajustamento entre as soluções propostas e o que se passa na realidade nas escolas. Portugal não te...

Sobre os Novos Programas de Matemática A, B e MACS que entraram em vigor em setembro de 2014.

Reconhecer a importância da literacia quantitativa (ou numeracia ou literacia matemática) resume-se apenas em querermos ensinar matemática no nosso tempo virados para o futuro. Queremos?

“A Magia dos Números” (http://nautilus.fis.uc.pt/mn) é um programa informático constituído por diversos módulos independentes (neste momento 14), relacionados com números, curiosidades numéricas e conceitos matemáticos. Está integrado no portal científico "Mocho" (http://www.mocho.pt). Todos os módulos têm como objectivo principal incentivar o gost...

People have studied, learned and used mathematics for over four thousand years. Decisions on what is to be taught in schools, and how, are ultimately political, influenced by a number of factors including the experience of teachers, expectations of parents and employers, and the social context of debates about the curriculum. The ICMI study is posi...

Motivated essentially by several recent studies of the Goursat-Darboux problems for third-order partial differential equations, asymptotic representations are obtained here for some multiple hypergeometric functions (a type of generalized Lauricella functions) when one of the variables tends to infinity and the others tend to zero. Several examples...

of the Round Table According of the suggestion of the Round Table Committee of 3EMC, this RT will be devoted to problems of philosophy of mathematics. Specifically, we will dis-cuss the following questions: 1. How do mathematicians conceive of what is mathematics (a retrospective summary of changes of various attitudes between 1900 and 2000). Highl...

Exames nacionais do Ensino Secundário em Portugal

A unica iniciativa em Portugal que tem vindo consis-tentemente a popularizar a Matemática e a servir de alguma forma como identificação de talentos em Matemática tem sido a iniciativa da SPM denominada "Olimpíadas Portuguesas de Matemática". Recentemente tive oportunidade de participar no encontro "Educational treatment of young talent in mathemati...

Neste livro apresentam-se os textos referentes às comunicações proferidas no Ciclo de Conferências que se realizou no âmbito da chamada Revisão Curricular do Ensino Secundário que decorreu em 1997/1998.

In this paper we establish a comparison of the 25 contraction conditions considered in a previous paper of Rhoades [16], thus completing the comparison first made in a systematic way in that paper.

It is proved that if a generalized Goursat-Darboux problem is C∞-well posed then the operator cannot contain derivatives with respect to one of the variables.

In many countries, the teaching of mathematics has always taken a classical lecture form, for the majority of teachers, in a more obvious way at the secondary school and the university levels. There have been very successful experiences, major changes in the official syllabus have taken place almost everywhere, materials were produced in big quanti...

ent accepted it and it is now a compulsory course for future school teachers in the second semester of their fourth year of studies. We have been the two responsible for this course over the years and we would like to discuss briefly how it was conducted. We have 2 or 3 hours a week of theory and demonstrations and 3 hours a week of practical work...

Como é do conhecimento de muitos professores, o Departamento do Ensino Secundário encomendou a uma Equipa Técnica, formada pelos colegas Graziela Fonseca, Arsélio Martins e Jaime Carvalho e Silva, "uma proposta de Ajustamento do Programa do Ensino Secundário que integrasse os ensinamentos colhidos com o período de experimentação e generalização de...

One of the concepts related to the exponential function that is less well understood is the concept of "exponential growth". A lot of people in everyday life speak about it since Thomas Malthus (1766-1834) used it to describe population growth, but very few understand it; they want to say that the growth is very fast, but they confuse growth with t...

Uma abordagem crítica dos novos programas (de 1991), e alguns exemplos concretos de utilização na sala de aula. Publicado na revista Educação & Matemática https://em.apm.pt/index.php/em/article/view/375

A proposta de Novos Programas de Matemática para o Ensino Secundário parece, na minha opinião, padecer de grandes defeitos que aconselham uma grande reformulação. Publicado na revista Educação & Matemática https://em.apm.pt/index.php/em/article/view/273

We show that when one line approaches a vertical line and another line approaches a horizontal line, the product of the slopes can approach any real number or +oo or — oo or nothing, thus clarifying preceding statements of Subramaniam and Das. We stress that the slope of a vertical line does not exist and explain why all these discussions are very...

A situação atual (1985) do ensino universitário não pode ser considerada muito brilhante. Um dos meios existentes para o melhoramento dos métodos e técnicas pedagógicas é o da discussão e troca de experiências através de revistas mais ou menos especializadas.

We find a necessary and sufficient condition for a Goursat problem for a third order partial differential operator with constant coefficients of the form C2(Dx, Dy) Dt + C3(Dx, Dy) to be C∞-well posed, showing at the same time that a necessary and sufficient condition of Hasegawa cannot be extended. The result can be generalised to operators of hig...

We find a necessary and sufficient condition for a Goursat problem for a third order partial differential operator with constant coefficients of the form \[ C 2 ( D x , D y ) D t + C 3 ( D x , D y ) {C_2}({D_x},{D_y}){D_t} + {C_3}({D_x},{D_y}) \] to be C ∞ {C^\infty } -well posed, showing at the same time that a necessary and sufficient condition o...

This paper presents part of a study (Costa, 2005) intending to create, explore and refine a theoretical model for visual-spatial thinking that includes three visual-spatial thinking modes along with the thinking processes associated to them. This paper will focus on the final theoretical model. Many researchers have emphasized the value of the visu...

We consider the generalized Goursat-Darboux problem for a large class of third order operators. We will prove that this problem is C ∞ well posed if and only if the operator is decomposed as a sum of certain third order hyperbolic operators and a first order operator.