Crenças e dificuldades de futuros professores de matemática no domínio dos números racionais

Abstract

This thesis work was dedicated to a survey about beliefs and difficulties of a group of first-year students from a degree course in Mathematics of the University Center in Belo Horizonte (MG) concerning fractions. The main objectives of the survey were to identify these students` remembrances with this content in basic school and the possible origins of those beliefs and difficulties. In the case of beliefs, the research was grounded on the idea of domain-specific beliefs as introduced by Törner (2000). For this researcher, different fields of Mathematics have different features and therefore have specificbeliefs that could not be explained by global beliefs about Mathematics. In this thesis work, the specific domain considered was the one of rational numbers .The difficulties were treated as hindrances of mathematical actions when someone is engaged in a situation that involves handling with specific content and evidenced through emotional expression (fear, apprehension, blockage) in the direction proposed by Sanmartí (2009). The data was produced from the students answers to questionnaires and interviews and field notes. The treatment of the data relative to the beliefs was based on Clements (2000) model. The main feature of this model consists on a cyclic and constant study of protocols. That model allowedthe researcher to find four dimensions of students beliefs about fractions: a) fractions as a process, a number to be solved, as division (action); b) fraction as (not) the answer of a problem; c) fraction teaching and learning; d) negative self-efficacy. The students difficulties in relation to fractions were identified mainly through emotions of fear, apprehension, blockage, described by the students during the interviews. It was also sought to identify the origins of these beliefs through the students remembrances from their experiences of fraction teaching and learning in basic schooling. The participant students reports showed the littleattention of teachers to treating fractions in basic school, being summarized as traditional, very basicteaching, with routine activities of pizza fractioning, reducing fractions to least common multiple and giving priority to integers. The research results, obtained from the participantstudents` remembrances and later from three case studies, have strongly indicated that the origins of students beliefs and difficulties in the domain of rational numbers, particularly fractions, come from the students basic schooling, precisely the way of teaching they were submitted to. The understanding constructed in this research was that, once the beliefs were originated at a certain point in the pupils school life, it begins a vicious circle involving unfavorable relations among teaching and learning, beliefs and difficulties. To this respect this research takes into a reflection about the perverse role of school: it is supposed that the school function is to provide conditions for teaching and learning that favor meaningful comprehension of mathematical content, particularly fractions. However, what the results of this thesis pointed out, for the case of fractions, was that school itself seems to be the responsible for the neglect and limitation of the teaching of this content. An important pedagogical implication coming from this survey consists on the suggestion of incorporation of thematic about the role of affection in teaching and learning in the teachers professional-development courses.