Ewolucja znaczenia słowa w matematyce jako problem dydaktyczny

The research problem of this investigation is what are the sources of misunderstandings between the teacher and a student that spontaneously occur during a mathematical discourse. Several episodes of mathematics classes are here analyzed, in which the meaning of a word or expression was different for the teacher and a student. They are analyzed with respect to four kinds of cognitive obstacles identified previously: Different understanding of the context of a situation or a problem. The teacher while designing a problem or a context situation, which aims at bringing a concept closer to the student, is able to extract the concept out of the context, knows what is important and what is not. A projection concept ą context situation is functioning in her thinking. A student solving the problem designed by the teacher will try to exploit all his school and out-of-school knowledge. Associating experience may be linked with different mathematical concepts or be detached from mathematics. The students' mathematical knowledge may prove to be different from that expected by the teacher, and the out-of-mathematics knowledge may cause some aspects of the situation, for the teacher unimportant, to dominate. Focusing on own goals. The teacher knows what she intends to achieve when proposing to solve a certain problem. It can be getting a skill or discovering some property or connections, important for the science of mathematics. Those objectives often determine the desired way of handling the problem, deviations of it being considered faulty. The student is focused on finding the solution and doing it with least effort. So she/he will apply approaches that she/he knows and thinks to be most efficient. Focusing on different pieces of information. Understanding of an utterance is possible because of all words being kept in memory, so that the meaning of each can be located in a context. The sense of the whole utterance depends on which part of it has been highlighted. Different meanings assigned to the same key word. Building once own mathematics is a long-lasting process, in the course of which some words may change their meanings. Different meanings assigned to the same words stimulate enacting different procedures and operations attached to the given concept. As a result, the same word evokes different properties, connections, and relations. Example A mathematics class, 2nd year of junior secondary school (17-year-old students). The topic: Transforming algebraic expressions with the use of formulas. Teacher: Represent the given expression in the form of an algebraic sum: $$ (x + 3)^2 + (2x + 5)(2x - 5) $$ Teacher 01: Who would come to do this example? Student 01: (sitting) But there is a sum here already, though. T02: Who? Come, come. S02: But there is nothing to do here. T03: Why? Nothing can be transformed? S03: Well, one could, but a sum is there already. T04: So, if one could then calculate. S04: But... T05: Please come to the blackboard. S05: (reluctantly): $(x + 3)^2 + (2x + 5)(2x - 5) = (x^2 + 6x + 9) + (4x^2 - 25)$. T06: Speed up, remove those brackets. S06: $x^2 + 6x + 9 + 4x^2 - 25$. T07: What next? S07: underlines similar terms. T08: How is it called? Redu... S08: Reducing similar terms. $5x^2 + 6x - 16$. T09: Thank you, sit down please. S09: (returning to his chair) Ha, and it is a difference coming out, not a sum... Different meanings assigned to the same key word For the teacher, an algebraic sum should be composed of monomials with the plus signs between them. The student associated with the word sum its original meaning: the adding operation, usually coded with +. On the blackboard he saw numbers and letters: symbols used in algebra. To him, an "algebraic sum" referred to a situation where the plus sign occurred in-between algebraic symbols. Different understanding of the context of a situation or a problem In previous lessons students practiced translation of a verbal formula to the symbolic one and inversely. The student may have associated the present task with the former one where calling a sum the expression $(x+3)^2+(2x+5)(2x-5)$ would be approved. Hence his reaction But there is a sum here already, though. Focusing on different pieces of information To the teacher, importance of the task was in Represent the given expression. Representing (transforming) an expression is meant as a process, in which the student should show her/his ability to use the $(a + b)^2$ formula, to reduce similar terms, and to get the "simplest form". To the student, the most important was the word sum. In the given situation no action was needed as the expression was a sum already. Focusing on own objectives The event shows that the student's action was forced by the teacher. The former for long could not catch the teacher's intention. She/he worked under the pressure of the teacher's expectations, yet her/his own understanding of the objective did not fit her/his action. The teacher on her part expected executing an algorithm, showing skill, as well as theoretical knowledge. In further analysis the authors show that the dominant reason of a misunderstanding is the difference of meanings assigned to words by the teacher and the student. There are two causes of this phenomenon: the teacher uses words, common to her, to which the student is not accustomed, in the course of learning meanings of mathematical words evolve. The research has made the authors aware of a cultural gap, which exists in the mathematics classroom. The teacher's mathematical culture in incomparable with that of a student