Conjunto dos números irracionais: a trajetória de um conteúdo não incorporado às práticas escolares

AUTOR(ES)
DATA DE PUBLICAÇÃO

2008

RESUMO

The main objective of this work is to investigate the difficulties that appeared along the history for the development of the mathematical content irrational numbers and which are the approaches present in the text books. The subject irrational numbers is considered important in the basic education of Mathematics and it comes for the students, in the text books, as an obstacle to its full understanding. One of the aspects that can justify such situation is the complexity that the subject shows. However, the irrational number can be worked in a historical-epistemological process, by doing a study of how the transformation of scientific object to an object of teaching in a praxeological organization has been processing. That organization is the final result of a mathematical activity that presents two inseparable aspects: the mathematical practice, that consists of tasks and techniques, and the speech based on that practice that is constituted by technologies and theories. Our analyses point that factors exist which interfere in the process of teaching-learning of irrational numbers related with the praxeological organization of that content in the collections of the text books of the 70s, 90s and 2000. The proof of the irrationality with traditional Euclidian approach served as parameter to evaluate the degree of difficulty and to analyze the type of tasks, techniques and the theoretical-technological speech for the demonstration of the irrational number. The organization points that the most difficulty is in the axiomatic system that should satisfy to two conditions: to be solid, it means, the postulates cannot contradict each other for themselves or for their consequences; to be complete and enough, in the sense of having conditions to prove true or false all propositions formulated in the context of the theory in subject. The proof of the irrationality in a modern Dedekind approach analyzed by the type of tasks, techniques and for the theoretical-technological speech enlarges the numeric domain, joining to the rational numbers a new category of irrational numbers that fill out the gaps of the numeric straight line. To build techniques to modify and to enlarge the concept of irrationality of other numbers is an approach that explores numbers in the form a+b√2, with rational a and b, and that contributes to overcome the idea that there are few irrational numbers

ASSUNTO(S)

estudo histórico-epistemológico organization praxeological text book analysis matematica (elementar) -- livros didaticos números irracionais organização praxeológica study historical-epistemological matematica -- historia numeros irracionais matematica análise de livro didático irrational numbers

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