A) The need to study long-term mathematical questions

A basic characteristic of an algebrized mathematical activity and, in particular, of functional-algebraic modelling, is the requirement of long-term goals that can only be reached through systematic work extended in time. This was a trait of our instructional proposal: starting with a question that was not immediately solvable but required the construction and progressive enlargement of the models considered. This requirement appeared to be an important obstacle to the experimentation and can be explained, partially at least, as a consequence of the way the study of mathematics is interpreted in secondary schools, that is, in terms of the dominant cultural notion of study. It is in fact a conception clearly compatible with the limited, rigid and isolated character of mathematical organizations studied in such institutions (Bosch, Fonseca, Gascón 2004). It is also reflected in the way students carry out limited tasks and change the activity several times, even during the same session. On the contrary, our proposal is based on an activity that needs to integrate different mathematical objects in a functional way, working with a set of MOs that usually appear in an incomplete and isolated way. Thus, the type of study designed, with its long-term objectives, the non-definitive answers and the rising of new questions, go against the dominant epistemologic and didactic conception of the teaching and learning of mathematics.

B) The connection between numerical and functional language

To carry out the passage from an arithmetic solution to the construction of a functional-algebraic model, it is necessary to turn “numerical questions” into questions the answer of which is not a concrete number but a relation between variables. At the beginning of the workshop, the students had real difficulties with this, always trying to find “the number solution”, as they are used to doing. It was the teacher who had to clearly highlight the new didactic contract that was being established, using the “reality” of the situation to justify it. In any case, it seems that it is necessary to work more on the “question of the questions”, ie the kind of problem that is really approached, the kind of solution that is “receivable”, etc.

C) The role of WIRIS to facilitate functional-algebraic modeling

The symbolic calculator WIRIS CAS was used during the workshop to use mathematical techniques in a more fluent way and with less difficulty than their “paper and pencil” versions. WIRIS was thus supposed to help students carry out a richer exploratory activity. It was used to carry out a lot of trials and the exploration of different cases (different values of the parameters).

However, we did not succeed in making students question the techniques used (their scope, economy or efficiency). This is an essential point because any modeling activity requires the systematic interpretation of intermediate results and the questioning of the adequacy between model and system.

What did appear, with the help of WIRIS, was a certain degree of articulation between the algebraic work with formulas and the graphs obtained by considering any letter of the formula as independent variables.

Because it goes towards the didactic organization of current secondary schools, our proposal has highlighted various constraints that hinder the study of a long-term question and the use of functional-algebraic models to deal with it. A symbolic calculator like WIRIS CAS helps to overcome some of these constraints.