An Empirical Study in Growth of Discrete and Continuously Defined Functions via the Combined Action of a Socratic Interview and a Computer Generated Tool

Abstract

Starting with an intuitive understanding of the idea of one-to-one correspondence via counting
collections, we develop the notion of functional relationship in discrete and continuous settings. This
concept image build up enables us to pursue the study of growth and rate of growth in the context of
sequences and simple functions given by analytic expressions as well as the study under which
conditions growth and its rate coincide, triggering the appearance of the number e as an added bonus.
Under the constructivist perspective which focuses in individual thinking and the continuing act of
creating learning opportunities, we shall use a carefully planned semi-structured Socratic interview in
conjunction with a computer-generated tool. The dialog tests the solidity of those ideas and arguments,
which students bring and that are based in prior educative experiences, when challenged to dig out and
verbalize their own beliefs and questioning techniques try to ease the difficulties most students have with
concept translations to and from verbal representations (some more considerable than others) which are
the key to understanding and communicating effectively. The tool, apart from easing the burden of
calculations, promotes higher levels of engagement encouraging students‟ participation, provides them
with numerical data and visual realizations which, paradoxically as it may sound, facilitate abstraction by
linking the processes of discovery, understanding and conceptualization.