# EPISTEMOLOGY OF THE VARIATION

It is a fact that every situation and every phenomenon are not permanently static. This stance on things makes sure people develops variational thinking. However, when students want to describe situations or phenomena mathematically, it is very hard for them to do it, even this is a natural way of thinking.

In this paper it is carried out an epistemological study on the concept of variation for the objetive of providing the identifiers that describe and identify the variation concept, and then in the future designing learning activities for engineering students.

This work is founded on the theory of Mathematics in the Context of Sciences, which reflects on the link that should exist between mathematics and the sciences requiring it, with the purpose of considering the learning and teaching processes as a system in which the five phases are involved: curricular (developed since 1984), didactic (initiated since 1987), epistemological (dealt with since 1988), professorate (defined in 1990) and cognitive (studied since 1982).

The research incidents in epistemological phase, where there have been several investigations that verify that most of mathematics included in engineering courses arises in the context of specific problems in different areas of knowledge, and over time lose their context, offering, in consequence, a pure mathematics taught in classes that makes no sense to students will not be mathematician (Camarena, 1990, 2000, 2001). So, the mathematics required in engineering schools has arisen from the context of areas of knowledge where needed, and, as time passes, textbooks present it decontextualized from its origin, as a finished knowledge, which possesses mathematical formalism and a structure making it too abstract for students; consequently, the students can not describe situations or phenomena mathematically.

While searching, it was founded that the development of skills in mathematical modeling for contextualized events allows the development of variational mathematical thinking. So the identifiers are the next. Recognizing the varying being the first identifier in the concept of variation in mathematical modeling. Determining the elements making vary that one which is our center of analysis; that is, determining the dependency of elements on each other, being the second identifier. Identifying and associating variables with the elements of variation, as well as the usual nomenclature for variables, this is the third identifier. Determining the manner in which one variable changes in function of another is the fourth identifier. To know and understand the concept of function, is the fifth identifier. The sixth identifier in the concept of variation is prediction based on established data. The seventh identifier in the concept of variation are the mathematical modeling identifiers; these are identify variables and constants of the event, it includes the identification of what changes and what remains constant, implicitly or explicitly; identify mathematical concepts implicitly or explicitly; validating the mathematical relation modeling the event. Finally, this allows to conclude that the development of skills at modeling mathematically contextualized events enables the development of mathematical variational thinking.