Numerous
research works report about difficulties encountered by students with the
notion of function. Il appears that a large part of them are due to obstacles
of both epistemological and didactical nature (Chauvat 1999) [1]
as will be shown in what follows.

Epistemological considerations

Historically, the notion of function took a
long time to fully develop. First, the notion of curve appeared in
Greek
mathematics. Their notion of curve, mainly as a geometric locus, served
as a
tool for solving geometric problems until Descartes (17th century).
Descartes starts considering a curve as a set of points characterized
by a
distinctive property, called symptom, which can be expressed by means
of an
algebraic relation. A symptom emphasizes an algebraic relationship
between the
involved quantities, but conceals the functional relationship of
dependency
between them. With Leibniz, Newton and Bernoulli, the notion of
function
becomes a mathematical tool, and later, with Euler in the 18th century,
it becomes an object of study. New types of problems, such as
searching for minimums or maximums or study of tangents to curves, gave
rise to
the development of differential and integral calculus and led to the
notion of
numerical function where two conceptions coexist: function=analytical
expression and function=curve. As Sierpinska (1992) [2]
points out, they are both obstacles to the general notion of function.
Indeed,
the conception function=curve (i.e., its graphical representation) does
not
allow to perceive a dynamic process the function models: one quantity
varies in
relation with variations of an other one. The perception of this
relational dynamics
is eclipsed by the statics of a finished drawing.

*Didactical considerations*

The
teaching of functions in the French schools seems to reinforce this obstacle
(Chauvat 1999) [1]. In fact, students first encounter curves and functions through
the example of linear function linked to the study of proportional relationship
between quantities (Grades 7 and 8). The general notion of function is taught
later (Grade 10) and is based on the notion and use of graphical
representations, which leads to assimilation by the students of a function with
its curve.

*Theoretical frameworks*

First,
anthropological theory of didactics (Chevallard, 1992) [3] is applied
to an analysis of both the textbook used in the experimental Grade
10 class and of Alnuset, aiming at identifying praxeologies (types of
tasks,
techniques that can be used to solve these types of tasks, and
technologies
providing a justification of the techniques) related to the notions of
function, equation and inequation existing in these two institutions.

Second, our educational goal
is to help students conceptualise mathematical notions. We refer here
to
Vergnaud’s approach to conceptualisation, which situated knowledge
within an
action rather than within a theory (Vergnaud, 1996) [4]. He
distinguishes between operational form of a knowledge, which is used in
action of problem solving, and predicative form of a knowledge, oral or
written, which shows only a part of the former form.

[1] Chauvat, G. (1999), Courbes et fonctions au collège, “Petit x”
51, 23-44.

[2] Sierpinska, A. (1992), On understanding the notion of function. In The
concept of function: Aspects of epistemology and pedagogy, MAA Notes n°25,
25-58.

[3] Chevallard,
Y. (1992). Concepts fondamentaux de la didactique : perspectives apportées par
une approche anthropologique, *Recherches en didactique des mathématiques *12(1),
73-112.

[4] Vergnaud,
G. (1996). Au fond de l’action, la conceptualisation, in J.M. Barbier (Ed.): *Savoirs
théoriques et savoirs d’action *(pp. 275-292), Paris: PUF.