A teoria dos grafos no ensino.

Abstract

Muitas vezes, para resolver uma determinada situação problemática temos tendência a fazer um esquema, ou um modelo, que nos facilite na organização dos dados e na estruturação das ideias e do pensamento. Com base nesses modelos, conseguimos visualizar melhor qual é a solução para o nosso problema ou, então, definir uma estratégia para a sua resolução. A resolução de problemas deverá ser encarada como uma metodologia, através da qual são desenvolvidos diversos conteúdos e não como um conteúdo por si só. Assim, enquanto professores, cabe‐nos o papel de preparar tarefas, partindo da resolução de problemas do quotidiano dos alunos, de modo a que estes se sintam motivados e integrados nas próprias actividades. Esta motivação e integração poderá ser alcançada através da modelação de realidades vivenciadas pelos alunos, os quais devem ter um papel central na construção dos conhecimentos. Uma das áreas que nos permite obter com relativa facilidade uma simbiose entre a resolução de problemas e a modelação é a Teoria dos Grafos pois, em muitas situações, o tipo de modelos utilizados são grafos, que não são mais do que esquemas onde se utilizam pontos ligados por linhas conforme a relação que é estabelecida no problema. Neste trabalho, podemos encontrar alguma informação relativamente à origem dos grafos, alguns conceitos gerais sobre grafos bem como alguns exemplos de suas aplicações. Por fim, e com o intuito de evidenciar as suas potencialidades inerentes e as da sua exploração na sala de aula, partindo de situações possivelmente consideradas pelos alunos como não estando relacionadas com a Matemática, apresentamos um conjunto de tarefas que constituem uma sugestão para a abordagem e desenvolvimento de conteúdos constantes na disciplina de Matemática Aplicada às Ciências Sociais. Graph theory goes back to the XVIII century, when Euler introduced its basic ideas to solve the famous problem of the KÄonigsberg's bridges. However, in the last few decades, graph theory has been established, by its own right, as an important mathematical tool in a wide variety of areas of knowledge, such as operational research, engineering, genetics, sociology, geography, ecology, numerical analysis, parallel computation, telecommunica- tions and chemistry. Besides, it is usual to say that a considerable number of problems in a wide variety of sciences can be modeled by a graph and solved using graph theory. For example, it is possible to calculate the di®erent combinations of °ights between two cities, to determinate if it is possible or not to walk in every street of a city without walking in a street twice and to know the number of colours we need to colour a map. Until the 90s of the last century, graph theory was taught just in university. With the rise of new subjects in high school, namely, Mathematics Applied to Social Sciences, graph theory gained a place in the o±cial curriculum; therefore,the non-university students are exposed to an introduction to this theory. In this context, some relevant questions arise in the spirit of people interested in peda- gogic and scienti¯c problems. Is graph theory introduced and analysed with the adequate mathematical rigour? Can it be that the attempt of dealing with younger students, comes at the expense of scienti¯c quality? The main goal of this work is, on one hand, to show a personal perspective about the ap- proach that is done to graph theory with non-university students, and on the other hand, to put the question of why graph theory doesn't appear in the Mathematics'curriculum (level A). Is graph theory just relevant for students interested in social sciences? The thesis starts, after an intuitive approach, with a compilation of the fundamental re- sults, some elementary and others with more complexity, of graph theory. Concepts that are thought to be essential are selected in order for the high school students to build a coherent and consistent body of knowledge. Making such selections will show, from a personal perspective, how the approach of graph theory should be, forming a basis for a critical analysis of the current implementation of graph theory in high school. In the second part of the thesis, the third chapter, we summarise some applications of graph theory in some areas of science. For each application, we show how graph theoryGraph theory goes back to the XVIII century, when Euler introduced its basic ideas to solve the famous problem of the KÄonigsberg's bridges. However, in the last few decades, graph theory has been established, by its own right, as an important mathematical tool in a wide variety of areas of knowledge, such as operational research, engineering, genetics, sociology, geography, ecology, numerical analysis, parallel computation, telecommunica- tions and chemistry. Besides, it is usual to say that a considerable number of problems in a wide variety of sciences can be modeled by a graph and solved using graph theory. For example, it is possible to calculate the di®erent combinations of °ights between two cities, to determinate if it is possible or not to walk in every street of a city without walking in a street twice and to know the number of colours we need to colour a map. Until the 90s of the last century, graph theory was taught just in university. With the rise of new subjects in high school, namely, Mathematics Applied to Social Sciences, graph theory gained a place in the o±cial curriculum; therefore,the non-university students are exposed to an introduction to this theory. In this context, some relevant questions arise in the spirit of people interested in peda- gogic and scienti¯c problems. Is graph theory introduced and analysed with the adequate mathematical rigour? Can it be that the attempt of dealing with younger students, comes at the expense of scienti¯c quality? The main goal of this work is, on one hand, to show a personal perspective about the ap- proach that is done to graph theory with non-university students, and on the other hand, to put the question of why graph theory doesn't appear in the Mathematics'curriculum (level A). Is graph theory just relevant for students interested in social sciences? The thesis starts, after an intuitive approach, with a compilation of the fundamental re- sults, some elementary and others with more complexity, of graph theory. Concepts that are thought to be essential are selected in order for the high school students to build a coherent and consistent body of knowledge. Making such selections will show, from a personal perspective, how the approach of graph theory should be, forming a basis for a critical analysis of the current implementation of graph theory in high school. In the second part of the thesis, the third chapter, we summarise some applications of graph theory in some areas of science. For each application, we show how graph theory is used, in the ¯rst stage, to model the problem, and in the latter stage, to solve the problem. The problems that are mentioned consist of determining the shortest path in a weighted graph, the problem of minimisation, problems that need the use of trees, and the problem of colouring graphs and maps. In chapter number four, we give a re°ection on how to approach graph theory to non- university students, going through basic school and high school. In the basic school we identify some contents that already belong to the curriculum, and also, propose some activities adequate for the age of the student at each stage (1st, 2nd, 3rd cycle), that, in many cases, are being taught in a spontaneous way, without the conscience that there exists a theory that supports the activities. For the high school, we summarise some proposals of activities for the students of Mathematics Applied to Social Sciences and we question the formal absence of graph theory in the Mathematics'curriculum (level A).