Students' interpretations of some concepts of mathematical analysis

The research presented in this thesis examines the reasons lying behind the difficulties displayed by young learners with Real Analysis, as the subject is usually developed in first year English university courses and in Greece in the final year of the Lyceums for those to follow mathematics or sciences at universities.We make a cognitive study of the ways in which young mathematicians make conceptual formations of the basic ideas of Analysis. These formations were investigated through clinical interviews following pupils' answers to questions papers that we designed; these were of unconventional type relative to their prescribed examination papers.Both investigatory procedures were aimed to elicit the learners' thoughts on the limit concept, the real line and the notion of function, i.e. the foundation stones of Real Analysis. The answers, a representative sample of which is included in the thesis, were studied in a `microscopic' way, in order to enable us to draw a satisfactory portrait of pupils'predominant cognitive images. Of this portrait , the most significant characteristic we identified is pupils' conceptualization of the continuum, which was nearer to that of Cauchy than to that of Weierstrass (apparent especially among the English pupils).Arguments are given to justify our choice of this `subjective' style of research rather than a statistical approach. In this connexion the thesis contains various essays on relevant topics, for what we felt was appropriate both to set the subject in a general context and in a framework suitable for discussing our own work.First there is an essay of a philosophical nature concentrating on mathematical intuition, then one dealing with previous research in the field. Finally we present a historical account of the development of Calculus up to its Arithmetization in Analysis. These essays give a coherent framework for discussing the initial naive steps which any pupil is bound to resort to, to some degree, when he (she) faces the subject for the first time. Our investigation found such naivety and our historical account helps to give a sense of how similar thoughts to theirs were originally present in the minds of well known mathematicians, and in fact played a large part in shaping the development of Analysis.