A secondary school student's understanding of the concept of function - A case study.

This article is concerned with an average student's understanding of the notion of function. It presents an analysis of a dialogue held with Kasia, a sixteen-years-old pupil of a general academic secondary school. The conversation concentrates on a non-standard problem – a functional equation. The theoretical framework used here is the PROCEPT THEORY formulated by Gray and Tall (1993, 1994). The article diagnoses Kasia's procept of function revealed during the dialogue. Her difficulties with understanding this notion can be described as based on a very limited procept of function and a misinterpretation of the symbols used in the functional equation. Three kinds of sources of the difficulties are identified: firstly, the intrinsic ambiguities in the mathematical notation, secondly, the restricted context in which some symbols occur in teaching and a limited choice of mathematical tasks at school, and, thirdly, Kasia's idiosyncratic interpretation of mathematical tasks. Moreover, the paper describes several important changes in understanding of symbols and of the concept of function that took place in the course of the dialogue. This article shows that we can use functional equations as both a research and a didactic tool in mathematics education. [ABSTRACT FROM AUTHOR]