Thought and Variational Language in the Introduction to Analysis

Abstract

We present a pedagogical approach based on research in Mathematics Education. With this, we attempt to construct a base of meanings for mathematical analysis processes and concepts, especially that taught at a university level. We begin with activities for the construction, among students, of a universe of graphic forms, which is in turn expanded and structured; and then we continue with the development of the notion of prediction of the phenomena of flows supported by the Newton binomial. The combination of both tasks, which we sustain in this hypothesis, fosters the development of thought and variational language. This approach has been put into operation with humanities and physical sciences and engineering students with promising results. Naturally, the subject order of the contents of the preparatory course and the analysis course have been modified noticeably, because we now put the notions of the curve and analyticalness in the center of the design of didactic situations. This approach has made it possible to use the Taylor Series as the principal support, the mathematical objective to predict the future state of that which flows in a variational situation extracted from the knowledge of reference in order to discipline the student.

In recent times we have witnessed the appearance in the bosom of mathematics educators, academic university sectors that deal with the study of the so-called advanced thought processes in math subjects in higher education. The subjects tackled are after basic algebra and they usually have subject matter that goes for analysis onwards. This amazing growth has been possible, in our opinion, thanks to two principal factors; the first is due to the growing interest of professional mathematicians in the affairs of teaching and learning and the second is the result of the stability and maturity that has been reached in research communities that are organized around academic groups with their own paradigms. The name in Spanish .Matem?ica Educativa., gives our discipline a geographic and conceptual location; the term Mathematics Education has been used in the Anglo-Saxon world, while on continental Europe it has been called Did?tica de las Matem?icas, Didactique des Math?atiques, Didaktik der Mathematik. Now, it is accepted as a functional premise that our discipline studies the process of the constitution, transmission and acquisition of various different mathematical contents in a school situation. It is not reduced to a search for a .good way. to teach a certain previously fixed notion, but one that allows us to assume the organization of an activity as an object of study, for example, whose declared intention is the learning of a degree of knowledge even if the goal is not reached. The purpose of research in our field is to positively affect the teaching system; improve teaching methods and contents and to propose the conditions for the stable functioning of the teaching situation so that math is not only treated as a subject, but we want to understand how and why it is learned and why knowledge is structured for teaching purposes.