## Thought and Variational Language in the Introduction
to Analysis
*Ricardo Cantoral* & *Rosa - Maria Farfan*
`rcantor@cinvestav.mx` & `rfarfan@cinvestav.mx`
Center of Research and Advanced Studies of the National
Polytechnic Institute
Cinvestav IPN
**Mexico**
### 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.
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