Summary:
The concept of the limit is fundamental but at the same time difficult to be managed by high school students who encounter this concept for the first time in the case of functions. As it comes of research data, the high school teachers of mathematics agree that students confront difficulties in understanding the notion of the limit. Students approximate the concept most of the time as a procedure. In this paper we suggest a didactical suggestion for teaching the concept of the limit in a school environment. To be more specific, the teaching was on students of the first and second high school classes and its purpose was to make students to have a first contact with infinite procedures and the sense of the limit in the specific case that the sequence tends to zero.
The structure of the paper was designed in a way that the reader will be able to obtain a timeless image of the concepts of the limit and the infinities as these were shaped through centuries. For this reason, except for the introduction, in chapter two (2) we refer to historical data and the timeless evolution of these concepts. In chapter three (3) we highlight the role that the analytic programs and school play in the teaching of the limit. In chapter four (4) we highlight the finds of the international research and bibliography that refer to the teaching of mathematics and especially to the concept of the limit. The next chapters are dedicated to the way that the procedure was planned and carried out as also to its results.
The procedure was planned and carried out in two phases, in the first phase the students confronted the problem of calculating the acreage of the circle via a work sheet and the use of proper software, with the purpose of highlighting the necessity of infinite procedures and their introduction in them. In the second phase, students confronted the problem of dichotomy, which aimed at the intuitive discovery of the limit and then in the output of the results through the formal definition.
Twenty-two (22) students participated in the procedure. The students were in the first and second class of a high school in the southern suburbs of Athens. Eighteen (18) of them were in the second class and four (4) in the first and the procedure lasted four didactic hours. Students worked in pairs and their findings were announced to the whole team, so discussions were evoked. The role of the teacher was clarifying and organizing and his interventions aimed only to the normal conduct of the procedure, which was videotaped and recorded.
The results came of not only by analyzing the work sheets but also by the careful observation of the videotaped and recorded data.
Keywords:
Infinite procedures, sequence, intuition,definition.
