Conceitos e Definição

A matemática possui em sua fundamentação dois entes de extrema importância – Conceito e Afirmação. Qualquer estudo sério de matemática, isto é, que tem por objetivo um entendimento mais aprofundado, deve ser fundamentado sobre esses dois elementos. Os livros didáticos e a exposição dos conteúdos pelo professor, em geral, têm uma metodologia composta pela apresentação de cada conceito por meio de uma indrodução formal ou através de uma motivação (um problema, um questionamento, uma história, etc.). Após a apresentação dos conceitos, são retiradas deles, afirmações que podem ser provadas ou admitidas como verdades. Dessa forma, o ensino de matemática tem sua composição principal baseada em um conjunto de conceitos e afirmações. Atualmente esses dois elementos são banalizados pelos estudantes e algumas vezes pelo próprio professor, ou seja, não são enfatizado com a sua devida importância; é mais fácil fazer um treinamento prático, estabelecendo um algoritmo para resolução de problemas e exercícios, que “perder tempo“ e despender esforços para uma compreensão da teoria. Neste post, faremos uma discussão sobre Conceito e, consequentemente, sobre Definição, elementos importantes para uma boa compreensão da composição de uma estrutura matemática. E, no próximo post, falaremos sobre Afirmação e sua implicações no ensino e aprendizagem da matemática.

Conceito

Conceito, em matemática, pode ser entendido como um conjunto de informações, que se tem de um dado objeto, que nos torna capaz de identificar esse objeto e distingui-lo de outros de naturezas diferentes.

Quando possuímos (conhecemos) o conceito de um determinado objeto (conteúdo), nem sempre podemos explica-lo. Por exemplo, muitos estudantes sabem identificar números reais, porém não sabem dizer formalmente (sem dar margens a dúvidas e dupla interpretações) o que é um número real.

Assim, podemos entender o conceito de duas maneiras: quando conhecemos o objeto, isto é, sabemos identifica-lo, distingui-lo, mas não sabemos explicar de forma precisa o que o objeto é; uma outra forma, é quando a gente conhece o objeto e consegue explica-lo formalmente. No primeiro caso, dizemos que possuímos um conceito intuitivo do objeto e, no segundo caso, conhecemos uma definição desse objeto.

Definição

Uma definição é uma formalização de um conceito. Definir é dar uma denominação a um grupo de objetos que possui uma ou mais característica em comum. Uma definição é composta de três elementos: O que está sendo definido (o novo nome que está sendo dado ao objeto), o objeto da definição (quem está sendo renomeado) e as características/propriedades do objeto (quais especialidades que esse objeto possui).

Exemplo 1:

Definição: “Número primo é todo número natural, diferente de 1, que têm como divisores apenas o 1 e ele próprio”.

O que está sendo definido: Número Primo;

Objeto da Definição: Número Natural;

Características do Objeto: Ter apenas os divisores 1 e ele próprio.

Objeto sem definição (primitivo)

Se observarmos bem sobre o que já foi dito, podemos perceber que uma definição não cria um novo objeto, apenas renomeia um objeto já existente. Dessa forma, toda definição precisa estar fundamentada em um objeto que já exista.  E esse objeto que fundamenta uma definição pode também estar definido e sua definição fundamentada sobre outro objeto que, por sua vez, também pode estar definido, e assim por diante. Para entendermos melhor observe o Exemplo 2.

Exemplo 2:

Considere a seguinte Definição:

Progressão Aritmética é uma sequência de números reais, onde cada número, a partir do segundo, é a soma do anterior com um mesmo número real, denominado razão”.

O que está sendo definido: progressão aritmética;

Objeto da definição de progressão aritmética: sequência de números reais;

Propriedade do objeto: cada elemento, a partir do segundo, é a soma do anterior com um mesmo número real.

Assim, Progressão Aritmética é uma sequência de Números Reais. E o que é uma sequência de Números Reais?

Podemos definir sequência de números reais por:

Uma Sequência de Números Reais é qualquer função definida dos Números Naturais nos Reais”.

O que está sendo definido:  sequencia de números reais;

Objeto da definição de seqüência:  função;

Característica do objeto: associa cada número natural a um real.

Logo, uma Progressão Aritmética é uma Sequência de Números Reais que, por sua vez, é uma Função. Novamente poderíamos perguntar – o que é uma função?

A definição de função necessitaria de um novo objeto, que precisaria estar, também, definido como outro objeto, e assim por diante. Isso acontecerá até o momento em que não teremos mais nenhum objeto para fundamentar uma definição. Neste caso, o objeto é denominado primitivo.

É importante que, antes de fundamentar uma definição sobre um objeto primitivo, se possua uma idéia de significado desse objeto, ou seja, se tenha um conceito intuitivo dele. Porém, nem sempre será necessário começar toda teoria através de um objeto primitivo, pois dependendo do estágio em que se encontra o nível do indivíduo, é inviável. No exemplo citado anteriormente, se o público já estudou funções em cursos anteriores, podemos partir do princípio que o conceito de função é intuitivo, e todas as outras definições podem ser fundamentadas a partir desse conceito. Uma outra situação que pode ocorrer é quando a definição requer uma compreensão aquém do individuo; por exemplo, em estudos básicos e mesmo em alguns mais avançados, usamos frequentemente os números reais sem defini-los, admitindo como sendo de conhecimento público. Isso ocorrem pela dificuldade em se definir números reais. Porém é preciso tomar cuidado ao admitir um objeto como intuitivo sem conceituá-lo, ou seja, sem apresentar alguma situação que provoque uma compreensão do objeto, de forma que o individuo consiga identifica-lo e distingui-lo de outros de naturezas diferentes; senão, corremos o risco de que todos os novos conceitos, ou definições, fiquem fundamentados sobre um objeto sem significado para o individuo.

Na matemática estamos admitindo, o tempo todo, objetos como sendo intuitivos, mas devemos tomar cuidado – o que é intuitivo para uns pode não ser para outros. Por exemplo, números inteiros, equações, polinômios, etc. são considerados intuitivos para a maioria dos estudantes do ensino médio, porém, não o são para estudantes do ensino fundamental. Assim, ao observar cada objeto procure analisa-lo com o objetivo de saber se ele é intuitivo para você, ou seja, se você consegue identificá-lo e distingui-lo de outros de naturezas distintas. Se existir alguma dúvida quanto isso, é importante procurar exemplos ou situações que sejam capazes de produzir uma idéia intuitiva dele.

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