Reflective thinking in mathematics. About students' constructions of fractions

Sammanfattning: This thesis deals with students' constructions of mathematical knowledge in the case of fractions from a radical constructivist perspective. Fractions are the first more abstracted mathematics met by students. Even if they have a rich experience of sharing and partitioning things in their everyday life dealing with fractions often give rise to difficulties in school. The thesis consists of two parts. The first part constitutes a discourse on radical constructivism in the research field of mathematics education. The second part is an empirical study wherein we discuss earlier empirical investigations of students' achievements on fractions and our own empirical inquiry. Constructivism has become one of the major influences on contemporary research in mathematics education. In our discourse we give an account of its development in the light of three main changes in the philosophy of science and mathematics during the 20th century. Radical constructivism is a theory of active knowing. Knowledge does not depict or represent an experiencer-independent reality. It serves to organize the subject's experiential world. Our world is not an unchanging independent structure, but the result of distinctions that generate a physical and a social environment to which we adapt as best we can. Reality is a reality created by the subject. Jean Piaget, in his research programme the genetic epistemology, offers two accounts for the development of human understanding. His structuralist account is concerned with the organisation of human knowledge in a system of an overarching structure. His constructivist account focus on how these structures develop through a process of equilibration. Equilibration is a process of self-regulation leading to better structures from existing ones. Learning rational numbers remains a serious obstacle in students' mathematical development. When looking at the application of rational numbers to real-world problems from an educational point of view rational numbers appears in numerous forms. A full understanding of fractions seem to require exposure to numerous rational notions. Some Swedish and foreign empirical studies are reviewed. Former attempts to elaborate hiearchical models and learning levels seem to have failed. The aim of the empirical study was to investigate the schemes that the students develop when working with fractions. We strive to build models of students' mathematical knowledge in terms of coordinated schemes of actions and operations in a Piagetian sense. The method used is the constructivist teaching experiment. It has been a long term interaction between the researcher and two groups of students, A (n=24) and B (n=19), in the intermediate grades during four years. We have made classroom observations, written tests, teaching episodes and clinical interviews with six students from each group. The schemes have been analysed and interpreted in terms of part-whole and part-part relations, N-distraction and differentiation between spatial and logico-arithmetical structures. Students' development of mathematical knowledge is achieved in an interplay between mathematical and contextual senses of a problem. The task of the teacher is to organize activities in the classroom as part of a conscious mathematization of students' experiential reality.