1. Modeling- Use of Graphs. A Category in Calculus that Redefines Variation in
Situations of Modeling Movement.
Liliana Suárez Téllez, Francisco Cordero Osorio
lsuarez@cinvestav.mx , Cinvestav-IPN
TSG #16: Research and development in the teaching and learning of calculus.
Abstract
We are interested in favoring the development of a functional mathematics in the
educational system. Therefore, the importance and relevance of our research is in the
formulated theoretical-methodological aspects. In order to achieve our objective we
established the concept “Modeling-Use of graphs” 1 as part of the theoretical frame of
the research. Through this concept we are able to propose hypotheses about the nature
of the social construction of knowledge about Calculus associated to variation and
change.
The result of this research is the assumption of an epistemology to characterize
mathematical modeling through the use of graphs. Such epistemology of the social
construction of knowledge is conformed by two important aspects; the functioning, that
is to say, those circumstances related to its use and its modeling which are useful to
problem solving and to integrate theory, and its form, in which tasks are determined by
the function but at the same time by new forms and functions that can be generated
from them (Cordero & Flores, 2007).
This epistemological approach consists of the systematic study of the use of
mathematical knowledge in specific learning situations (Buendía & Cordero, 2005).
Oresme’s work, about the figuration of qualities, provides an explanation of the
transformation of the use of mathematical knowledge in our times to deal with
undergoing problems when working with change and variation situations (Oresme,
1968). The transformation characterized in this work, from the debate between
functioning and form of the use of geometric figures, highlights key elements to the
epistemological assumption about the use of graphs in situations of modeling movement
to redefine change and variation.
This socioepistemological approach provides a new status for modeling and for the use
of graphs, which orients them as generators of knowledge rather than plain abilities or
activities. In this way, we are able to propose a modeling category in Calculus that
redefines variation in situations of modeling movement. The socioepistemology of the
modeling- use of graphs can be used in situations to work with students and it is
integrated by sequences called Modeling Movement Situations (SMM). Our hypothesis
is that variation is redefined through the modeling- use of graphs proposal. Such
epistemological hypothesis can be analyzed with elements of Didactic Engineering.
We also have some evidence about the elements of function of figuration that come out
as results from a SMM when identifying the relationships between the graphs’ position
and speed that participants manage to make as arguments to explain the variation in a
situation of change (Suárez et al, 2005). We also have evidence of different ways of the
1
In this paper “Modeling- Use of Graphs” (“Modelación-Graficación”) refers to an epistemology to
characterize mathematical modeling through the use of graphs.

2. use of graphs as a result of the characterization of the meanings and procedures that
participants use to establish the relationships between the graphs’ position and the speed
in a situation of change in different stages such as in the situation design; in which the
form is established, the arguments are built and used.
Acknowledged
This research is funded under a grant from the CONACyT for the following proyect:
Construcción social del conocimiento matemático avanzado. Estudios sobre la
reproducibilidad y la obsolescencia de situaciones didácticas: de la investigación a la
realidad del aula (CLAVE: 41740-S).
References
- Buendia, G. & Cordero, F. (2005). Prediction and the Periodical Aspect as Generators
of Knowledge in a Social Practice Framework: A Socioepistemological Study.
Educational Studies in Mathematics, 58, 3, 299-333.
- Cordero, F. (2006). La modellazione e la rappresentazione grafica nella matematica
scolastica. La Matemática e la sua Didattica, 20, 1, 59-79.
-Cordero, F. y Flores, R. (2007). El uso de las gráficas en el discurso matemático
escolar. Un estudio socioepistemológico en el nivel básico a través de los libros de
texto. Revista Latinoamericana de Investigación en Matemática Educativa. 10, 1, 7-38.
- Oresme, N. (1968). Tractatus de configurationibus qualitatum et motumm. En Clagett,
M. Nicole Oresme and the Medieval Geometry of Qualities and Motions. Madison:
University of Wisconsin Press.
- Suárez L., Carrillo C, & López J. (2005). Diseño de gráficas a partir de actividades de
modelación. Acta Latinoamericana de Matemática Educativa, 18, 405-410.
http://www.matedu.cicata.ipn.mx/Publicaciones/alme%2018.pdf