Różne sposoby rozumienia pojęcia parametru przez uczniów gimnazjum w toku badań długoterminowych w trakcie zintegrowanej nauki matematyki i informatyki
Treść / Zawartość
The object of interest in traditional mathematics teaching are first of all contents connected with concepts like triangles, numbers, vectors etc. and their properties. A teacher focuses pupils' attention mainly on these concepts. Letters (variables) are simply used (Konior, 2002) and treated as if they were already known by pupils and represented the part of alphabet earlier known to them. It is clear that the assumption that pupils will use mathematical alphabet symbols in a conscious manner is wrong as long as nobody has familiarized them with this alphabet. It is necessary to ask a question how algebra looks in pupils' the eyes. According to Turnau (1990) it is not possible to specify a purpose of executable operation without understanding particular letters in a task clearly, and it is not possible to plan this operation rationally. Understanding and proper interpretation of a letter is not an easy task for pupils. A context decides about its meaning in the algebraic text, phase of reasoning, subjective interpretation, as well as customs concerning the kind of used letters (Turnau, 1990). Proper interpretation of a letter in a given algebraic structure is a task often overgrowing pupils. Feeling and intuitive recognition turns out to be more popular strategy than interpreting of a letter in way of certain reasoning. Pupils know concepts like variable, unknown, or parameter, however, they do not always use them consciously, and therefore they are not understood by them entirely. I specify different meanings of letters on the base of literature (Turnau, 1990; Konior, 2002; Giessen van de, 2001; Drijvers, 2003) and I present them on the drawing below (fig. 14): I have decided to devote a particular attention to the letter standing for parameter. After the review of literature I state, that there has been in Poland no research concerning understanding by pupils of the concept of parameter . However there are such researches abroad. Firs was led by Carel van de Giessen (2001) who described classroom experiences of 16-17 years old pupils connected with visualization of different meanings of the concept of parameter during solving tasks with the use of Graphic Calculus program. The second appeared in Holland, at the University of Utrecht, where Paulus Drijvers (2003) described research on the understanding of the concept of parameter using information technology. In the experiment of the mentioned author there were four meanings of the concept of parameter classified: placeholder, generalizer, changing quantity and unknown. In my research I have investigated different kinds of understanding of the concept of parameter by students at the age of thirteen during their three-year learning at school. Research of Drijvers differs with mine because it did not have a continuous character and it was not planned on several years, while my observation is led in a course of three-year work with pupils in secondary school (13-16 years old children). In my research I implement a didactical conception supported by information technology, concerning the understanding of the concept of parameter, performing also its verification in school conditions. Because of the fact, that my research is continuous, information techniques and interface problems have an in uence on solving mathematical tasks by pupils only in initial period of my research. Besides, a Dutch group carrying the research did not consist of teachers of the investigated groups of students. Contrary to that, I do teach mathematics and informatics, and my group is also younger. Pupils, during realization of my conception, in comparison with those from Holland, solve many more tasks from different sections of mathematics, which contain parameter. After some time, my research tasks with parameters did not seem as new type of tasks to the pupils. Investigated problem: If and in what degree computer programs can be helpful in the process of forming the concept of parameter? I have carried three-year preliminary research in the oldest class, and proper research in the youngest one, which also lasted three years. The described groups have not been selected by no means. Besides, mathematics lessons are led by me in both classes both in the traditional manner, and by means of software. My research is consistent with the idea of natural experiment. I have also used designed research method in my investigations, leaned observation and analysis of documents. Due to the use of Camtasia Studio program, I have registered solving of tasks by some pupils, that gives capability of observing the movie of their work. As a research tool also serves me TI InterActiveą, which includes texts and objects created in this program by pupils, which they recorded on the disk of their computer. This investigative tool is created by pupils independently in the course of solving a task. Presently I am in the course of proper research in third class of secondary school. During preliminary research, I created lists of tasks related to the concept of parameter, concerning different sections of mathematics, which are solved by pupils in the first, second and third class of secondary school. Each list includes also the task with parameter, which pupils solve during the posttest, which finishes the realization of some section of mathematics. Preliminary research served also for the verification of these lists. The analysis of solutions of problems from lessons and from the posttest supplies information of pupils' progress in the understanding of different meanings of the concept of parameter in different mathematical contexts. For me, as a teacher-researcher it is a situation, which gives me full image of effects of the process concerning forming the concept of parameter. I have conducted pretest at the beginning of the first secondary class, which allowed me to find out, which meaning of the concept of parameter pupils understand. The analysis of solutions of this test by Karolina, a female pupil of class I b, in which I carried my proper research, has supplied information, that the meanings of parameter she was aware of are placeholder and generalizer. In this article I present analysis of solutions of problems from four posttests during the first class of secondary school, which supplies information of her progress in the understanding of different meanings of the concept of parameter in contexts of: geometry, algebra, equations and functions.