Sobre o ensino superior da matemática : a geometria e os professores do 1.º ciclo : "novos desafios, velhas deficiências"

Existe, na sociedade actual, um consenso generalizado a respeito da importância da formação (científica e pedagógica) dos professores de Matemática. No entanto este sentimento não tem sido, em Portugal, devidamente monotorizado nem no caso da formação científica nem no caso dos professores do 1.º ciclo. Apresentamos, neste artigo, um estudo que i) analisa os desafios que, na área da Matemática e de acordo com as directivas oficiais, se colocam aos professores do 1.º ciclo; ii) regista a formação matemática oferecida, em Portugal, a estes profissionais e iii) avalia as competências matemáticas de um grupo de professores e futuros professores quando nos centramos no caso dos conceitos geométricos elementares. Defendemos, como resultado da investigação, a necessidade urgente de se olhar para o conteúdo matemático como verdadeiro núcleo fundamental de qualquer programa de formação de professores do 1.º ciclo.Different vectors of the society worldwide agree on the importance of the training (both scientifically and pedagogically) given to Mathematics teachers. However this belief has not been, in Portugal, followed by research on the implementation of correspondent means neither in the case of the scientific training of the teachers nor in the case of primary school teachers. In this article we are presenting i) an analysis of the actual official directives posed to these professionals; ii) some facts about the actual mathematical training offered by the national Institutions (both Universities and Polytechnics) and finally iii) the evaluation of mathematical competencies of a group formed by teachers and future teachers, in the case of elementary geometrical concepts. As a result we defend that the scientific training offered actually to these professionals must be urgently reviewed in order to become the nucleus of any training program to primary school teachers.Fundação para a Ciência e a Tecnologia (FCT)

Aventuras numéricas no cálculo do e

Faz-se uma análise dos erros de arredondamento no cálculo de aproximações do número de Neper com a expressão (1+1/n)^n.Fundação para a Ciência e a Tecnologi

Study on the fundamental concept of “measure” and its history

This paper exhibits a small part of a PhD degree project on the theme: “The fundamental concept of Measure: epistemological and pedagogical aspects related with the first six years of schooling”. It aims at presenting the analyzis of the fundamental concept of measure at the so called elementary levels of schooling, in Portugal. We intend to: 1) distinguish elementary concepts and fundamental concepts; 2) explain why do we look at the mathematical concept of measurement as not only an elementary concept but, and above all, a fundamental concept in mathematics; 3) foresee implications of this distinction for teaching at the elementary level.CIEC - Centro de Investigação em Estudos da Criança, UM (UI 317 da FCT

Topics of physics in virtual experimentation

The use of information and communication technologies in education for different school levels is becoming an usual practice. Concepts that many times are presented to the students from a static point of view are more easily apprehended when adequate technologies are used to give them greater dynamism. Bearing in mind the real difficulties in the implementation of an experimental teaching of Physics, it is possible nowadays to present to the students activities inserted in a context of new technologies that encourage the experimentation and the sense of discovery, two essential teaching aspects of the Experimental Sciences. The project “Topics of Physics in Virtual Experimentation” intends to promote the virtual experimentation in the classroom, through the exploration by the students of computational applications and the development of small investigation projects. All the materials will be available in the project site http:// fisicaexpvirtual.com.Fundação para a Ciência e a Tecnologia (FCT

A historical approach of the fundamental concept of measurement: Measuring “Time”, in Portuguese textbooks for 5th and 6th grades

In this article we present part of a PhD project on “The fundamental concept of Measure: epistemological and pedagogical aspects related to the first six years of schooling”. It aims at analyzing the historical approach of the fundamental concept of Measure as presented in Portuguese mathematics textbooks for the 2nd cycle of Basic Education. In particular it intends: 1) To explain why do we look at the mathematical concept of Measurement as not only elementary but, above all, a fundamental concept in mathematics; 2) to identify a historical approach, in textbooks, inasmuch as it contributes to promotion and enrichment of learning the concept; 3) to analyse activities (problems and exercises) suggested and requiring Measurement. Our specification of the concept of Measurement will be, in this article, “Time”

A história dos programas de matemática para a formação dos professores do 1º Ciclo do Ensino Básico em Portugal: o conceito fundamental de medida

Estamos convictas de que a preservação das memórias é, tal como o próprio Heródoto (séc. V a. C.) defendia, a melhor forma para se compreender o presente e para se preparar o futuro. É nosso propósito compreender a evolução dos programas de formação dos professores do 1.º Ciclo do Ensino Básico, em Portugal, nomeadamente no que respeita ao conhecimento matemático e perceber que dimensão é atribuída ao conceito fundamental de Medida. Neste artigo reportar-nos-emos, em particular, a uma fonte primária histórica sobre a uniformização dos pesos e medidas: o Mappa do Systema decimal em Nomenclatura Portugueza (1812-1814)

The fundamental concept of measure in elementary mathematics textbooks

In this article we present the work in progress at a PhD degree in Estudos da Criança, Área de Conhecimento de Matemática Elementar, Universidade do Minho (Portugal). The focus on this paper is the analysis of the fundamental concept of “measure” as seen in Portuguese textbooks directed to the first six years of schooling. The concept of “measure” plays an important role in the construction of mathematical knowledge, in particular at the elementary stages of education. Our main aims are: (1) to define a set of criteria based on both scientific and methodological aspects (related to the concept of measurement) to be applied on the analysis of mathematics textbooks; (2) to identify criteria for developing didactical material for coherently teaching the concept of measure during the first six years of schooling; (3) to apply our results to programs for both initial training for teachers and “recycling” teaching careers

Francisco Gomes Teixeira: o conceito de reta tangente no Curso de Analyse Infinitesimal

Francisco Gomes Teixeira destacou - se como matemático mas também como pedagogo. O seu “ Curso ” foi utilizado como livro de texto durante vários anos nas Universidades do Porto e de Coimbra e é um exemplo na forma como deve ser composto e revisto um livro de texto. Neste artigo analisaremos a parte relativa ao tratamento dado por Gomes Teixeira ao conceito de reta tangente, explicitando as lições pedagógicas que se podem retirar desse tratamento.FEDER através do “Programa Operacional Factores de Competitividade – COMPETE” e por fundos Portugueses através da FCT FEDER através do “Programa Operacional Factores de Competitividade – COMPETE” e por fundos Portugueses através da FCT –“Fundação para a Ciência e Tecnologia”, incluída no Projeto PEst – C/MAT/UI0013/201

Reflection upon a "Method for Studying Maths", by José Monteiro da Rocha (1734-1819)

This was a workshop designed to acquaint the participants with a broad spectrum of mathematical ideas and viewpoints presented in a manuscript written, in the 18th century, by the Portuguese priest (and very influential scholar during the reform of the University of Coimbra) José Monteiro da Rocha, as an introduction to a mathematics course. Based on both Monteiro da Rocha’s life and on his own comments on the teaching and learning of mathematics in Portugal, we have strong reasons to date this unpublished manuscript around 1760. At that time, the study of mathematics as well as the study of other areas, was at such a low level that some scholars were quite concerned with the learning content as well as the methods of learning; as a consequence we can find some important suggestions for dealing with these problems. An example of this is Luís António de Verney’s “Verdadeiro Método de Estudar” (“True Method of Studying”), published in 1750 which acquired quite a large circulation and, even nowadays, is used as a source of research. Less known than Verney’s work, and other similar texts produced all over Europe at that time, is the above cited manuscript by Monteiro da Rocha. In this work he echoed his concern with the teaching and learning of mathematics, emphasysing the lack of teachers of mathematics while, in his own words, “waking up his nationals from the lethargic state [in which they were living]”. In particular he suggested “writing in vulgar language” (i.e. Portuguese instead of Latin) for spreading mathematical knowledge among Portuguese citizens and he stimulated the national pride by reporting on the example of Portugal’s “glorious maritime discoveries”. In the present workshop: i) We presented a brief summary on the life and works of José Monteiro da Rocha, referring, in particular, to his activity in designing and writing the rules for the big reform of the University of Coimbra (1772) where, for the first time, a Faculty of Mathematics was created in Portugal. We also referred to his activity as a professor of the same university and his relationships with the academics of Real Academia das Ciências de Lisboa; we will also report on the international spreading of some of his works on Astronomy. ii) We referred to the ideas/models presented in other textbooks of Monteiro da Rocha’s times. iii) We proposed the analysis of some specific parts of the above cited introduction, by reading them from an English version of the original manuscript distributed to the participants. iv) We lead a follow-up discussion/reflection on - The actuality of da Rocha’s ideas, for example on his understanding of what is mathematical knowledge; - The development/history of the models for presenting/introducing mathematics textbooks; - The historical-didactical facts that we can learn/apply, at the present time where there is an acknowledged decrease of interest in studying mathematics.Fundação para a Ciência e Tecnologi (FCT

The teaching of the concept of tangent line using original sources

FEDER Funds through "Programa Operacional Factores de Competitividade COMPETE" and by Portuguese Funds through FCT- "Fundação para a Ciência e a Tecnologia", within the project PEst-C/MAT/UI0013/201