This research presents a study of the teaching and learning of the notion of affine function, aiming to understand how this mathematical content has been worked with high school students in São Paulo, and to verify whether or not there is flexibility in the development of problems on the level of both mobilisable and available knowledge.
The used framework for the results analysis and discussion is the one of Robert, A (1997) on the three levels of student’s expected knowledge. In addition, we have considered Douady’s (Douady, R. 1984,1992) change of frames, Duval’s theory of semiotic representations (Duval, R. 1993,1995, 2003), and Chevallard’s anthropological approach (Chevallard, Y. 1992, 1996, 1999).
When analyzing this question, we have found out that there are few works addressing such problem, and that, in both, high school and undergraduation levels, the student faces difficulties in working with the notion of affine function, because s/he often does not carry the needed competencies and abilities to perform tasks involving the content. These difficulties can be associated with the lack of mobilization of mathematical knowledge acquired at the elementary school, and end up damaging the student’s scholar and professional performance. This leads to a non interesting situation, difficult to be surpassed, since the lacking of such knowledge tends to increase during his/her school life.
On the other hand, teacher seems to have few work choices, since they do not have opportunities to work this aspect of the question and, perhaps, that is why the students are not capable of performing tasks involving flexibility in different knowledge levels.
The institutional analysis has been performed by means of textbooks, high school National Standards and the curricular proposal for São Paulo State. The analysis of the personal relations developed by the students has been obtained from the study of the results from the SARESP test. The study considered the distinctions in terms of representation register required for the construction of the analysis frame, which is an instrument that made possible to present the articulation possibilities among different forms of knowledge associated to the notion of affine functions and the symbolic representations that support it.
In this way, one can verify that the proposal for the articulation of different forms of knowledge of affine function and its respective representations, even if they are not treated using such terminology, are considered as essential, as shown in the institutional analysis, based on the study of the curricular proposal of the São Paulo State, in the National Standards and in high school textbooks. However, there still is a huge distance between what is institutionally proposed and what the students were able to perform in the questions related to affine functions in SARESP test, which treated the articulation among graphic representation of linear function with the notion of directly proportional quantities; the identification of the function by means of the formula f(x)=ax+b, simultaneous linear equations with two unknown, the identification of a straight line in the Cartesian plane, using the function algebraic law; the linear function formation law by means of the formula f(x)= ax+b; the identification of the linear coefficient as a point that belongs to the ordinate axis; and simultaneous first degree equations with two unknown.
