Os fractais articulados à teoria do caos: uma possibilidade de ensino na educação básica
Data
2023-10-09
Autores
Ervolino, Lucas de Lima 
Orientadores
Kluth, Verilda Speridião
Tipo
Dissertação de mestrado
Título da Revista
ISSN da Revista
Título de Volume
Resumo
A presente dissertação tem o objetivo de apresentar uma possibilidade de ensino dos fractais articulados à Teoria do Caos na Educação Básica. Desse modo, buscou-se por meio da hermenêutica filosófica, estabelecer uma dialógica entre pergunta e resposta no que se refere à “Como introduzir os fractais na Educação Básica tendo como inspiração a Teoria do Caos?”. Assim, o processo investigativo teve início na História da Matemática e da Ciência, de modo a compreender o processo de construção e concepção dos fractais. Para ir além, fez-se necessário investigar o caos da perspectiva filosófica e matemática, possibilitando distinguir o caos do senso comum que, por sua vez, pode remeter à desordem - do caos matemático que se fundamenta na sensibilidade às condições iniciais. Desse modo, os Sistemas Dinâmicos - caracterizados por mudanças ocorridas em alguns sistemas com o decorrer do tempo - designam-se como a rota pela qual o caos transita e se expressa, visto que estes possibilitam descrever os fenômenos intrínsecos da natureza e da vida que, em sua grande maioria, são fenômenos não lineares e não determinísticos. No entanto, introduzir os fractais na Educação Básica tendo como inspiração a Teoria do Caos, consiste em identificar o elemento que conecta o caos aos fractais: o atrator estranho. As órbitas desses atratores no espaço de fase desvelam estruturas fractais intrincadas e se apresentam como um caminho viável para desenvolver estes dois conceitos da Matemática contemporânea no contexto da sala de aula. Para possibilitar a inserção das supracitadas temáticas, esta dissertação propõe, ao final, uma sequência didática orientada para o ensino dos fractais articulados ao caos, tendo como referencial a Educação Básica.
The aim of this dissertation is to present the possibility of teaching fractals in conjunction with Chaos Theory in Basic Education. Thus, through philosophical hermeneutics, we sought to establish a dialog between question and answer with regard to "How to introduce fractals in Basic Education with Chaos Theory as inspiration?". This investigative process began in the History of Mathematics and Science, in order to understand the process of construction and conception of fractals. Furthermore, it was necessary to investigate chaos from a philosophical and mathematical perspective, making it possible to distinguish between common sense chaos - which refers to disorder - and mathematical chaos, which is based on sensitivity to initial conditions. Dynamic Systems - characterized by changes that occur in some systems over time - are designated as the route through which chaos transits and expresses itself, since they make it possible to describe the intrinsic phenomena of nature and life, the vast majority of which are non-linear and non-deterministic phenomena. However, introducing fractals in Basic Education, inspired by Chaos Theory, consists of understanding the element that makes it possible to establish this connection between chaos and fractals: the strange attractor. The orbits of these attractors in phase space reveal ornamental fractal structures and are a viable way of developing these two concepts of modern mathematics in the classroom. In order to enable the insertion of these themes, this dissertation proposes a didactic sequence oriented towards the teaching of fractals articulated with chaos, using Basic Education as a reference.
The aim of this dissertation is to present the possibility of teaching fractals in conjunction with Chaos Theory in Basic Education. Thus, through philosophical hermeneutics, we sought to establish a dialog between question and answer with regard to "How to introduce fractals in Basic Education with Chaos Theory as inspiration?". This investigative process began in the History of Mathematics and Science, in order to understand the process of construction and conception of fractals. Furthermore, it was necessary to investigate chaos from a philosophical and mathematical perspective, making it possible to distinguish between common sense chaos - which refers to disorder - and mathematical chaos, which is based on sensitivity to initial conditions. Dynamic Systems - characterized by changes that occur in some systems over time - are designated as the route through which chaos transits and expresses itself, since they make it possible to describe the intrinsic phenomena of nature and life, the vast majority of which are non-linear and non-deterministic phenomena. However, introducing fractals in Basic Education, inspired by Chaos Theory, consists of understanding the element that makes it possible to establish this connection between chaos and fractals: the strange attractor. The orbits of these attractors in phase space reveal ornamental fractal structures and are a viable way of developing these two concepts of modern mathematics in the classroom. In order to enable the insertion of these themes, this dissertation proposes a didactic sequence oriented towards the teaching of fractals articulated with chaos, using Basic Education as a reference.