Didactic Contract; Conceptions of Teaching and Learning; Theory of Didactic Situations; Fractions; Difficulties.
This research aimed to analyze the Didactic Contract and the teaching and learning concepts of
Mathematics as influencing factors in the performance of fractions of students from the 5th year
of Elementary School in two municipal schools in Jaboatão dos Guararapes - PE. We use as
foundations the Theory of Didactic Situations, especially the Didactic Contract, since this
phenomenon is circumscribed in the didactic relationship in terms of the components of the
didactic triangle (teacher, students and knowledge). From this context, we also support studies
on teaching and learning concepts of Mathematics in Câmara dos Santos, among others that we
consider pertinent. In general, our research involved a theoretical propositional and an empirical
part. Thus, we organized our investigation into three correlated moments. In the first, we build
the theoretical bases for the sketch of the assignment of types of Didactic Contract inspired by
the learning models proposed by Charnay, based on the concept of the aforementioned
phenomenon. In the second, a semi-structured interview with three teachers of those years and,
in the third, the monitoring of classes reserved for teaching fractions. Each teacher explained,
in the interview, that they work within different conceptions of teaching and learning. The first
was associated with the socio-constructivist, the second with both the socio-constructivist and
the baldista, and the third, the ladder and the baldista. During stages two and three, it was
possible to observe, as results, discrepancies in the conceptions of teaching and learning
between the teaching practice of the teachers and their answers to the semi-structured interview.
Starting from the hypothesis of the Didactic Contract and the conceptions of teaching and
learning as complementary theoretical constructs that allowed us to see different things, it was
possible to better understand the origin of the difficulties encountered by students in the early
years of elementary school to deal adequately with fractions. The analysis of the study data
gave us indications of the constitutive elements of this phenomenon, among them: the teachers'
expectations in the students' prior knowledge about fractions and in their deepening; the
occurrence of the Topaze and Pigmaleão effects; and the establishment of rules such as that the
numerator of a fraction must be smaller than the denominator, of the idea of fraction as division
and two independent numbers, which denoted the Didactic Contract and teaching conceptions
as influencing factors in the performance of fractions . We also found some conceptual
weaknesses of the teachers.