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| 1 | Approach to education from the viewpoint of the author’s theory of modeling is considered in the paper. The attitude to undertaking additional construction and analysis of the drawing for enrichment of a geometric model (external and internal) has allowed to form compact and efficient set of recommendations for solution of geometric problems «on construction». Study of geometry, founded on this approach, forms knowledge and skills in the field of mathematical modeling. | 1462 | ||||
| 2 | Adequacy of a Model // Tomsk State Pedagogical University Bulletin. 2006. Issue 3 (54). P. 11-15 The author’s approach to definition of the notion «a model» is considered, in which we refuse from the requirement of analogy between a prototype and its image. Only structure of a model as a pair «interface component – prototyping object» is fixed. The level of likeness between a prototype and its image is estimated by a special notion «adequacy of a model». The notion «adequacy» is discussed in this paper. | 1505 | ||||
| 3 | The mathematical phenomenon includes the mathematical concept, the system of axioms, theorem, method, algorithm, etc. The purpose of this research is determination of all the options of perception to the mathematical phenomenon relevant for the education system and identifying some of the learning features for each of these options. A set of three postulates is chosen (the postulate on the priority result of educational and mathematical activity, the postulate of the didactic relevance of the components of activity, the postulate of the priority component of educational and mathematical activity). It is shown that if these postulates are implemented from the didactic point of view, then only two variants of the relation to mathematical phenomena are applicable: 1) the mathematical phenomenon as the subject of activity (in particular, as information to be memorized); 2) mathematical phenomenon as a tool of activity. The study of the mathematical phenomenon always begins in a situation when this phenomenon acts as an object of activity. However, to form a view of this phenomenon, the teacher must create conditions, the learning environment in which the learner naturally desires to consider this phenomenon as an instrument of activity. The examples of reflection of these variants of relationship in theory and practice of teaching mathematics are given. It is pointed out that for the students to take a mathematical phenomenon as a versatile phenomenon associated with other mathematical and non-mathematical phenomena, it is necessary that both variants of the relation to the mathematical phenomenon should be represented in the educational process. Keywords: methods of teaching mathematics, learning theory | 1528 | ||||
| 4 | Teaching mathematical activity is almost completely reduced to learning how to manage information processing. The author identifies three levels of mathematical information processing: the level of typical algorithms (for example, calculating the value of an arithmetic operation), the level of typical strategies (for example, solving equations) and the level of methodology (for example, the choice of a mathematical language for solving a task). Work at the level of typical algorithms and the simplest strategies of activity can be carried out without a deep understanding of the phenomena involved. But work at the level of strategies, and even more so, at the level of methodology, already requires understanding. We propose to interpret the term “understanding” as a system of associations with a mathematical phenomenon, i.e., concept, theorem, strategy, decision, etc. With a formal-constructive interpretation of the model, the considered interpretation of understanding includes the “creation of meanings” and other components of understanding. To control the completeness of the system of associations, it is proposed to apply the classification of associations with a phenomenon: 1) as an object or product of modeling activity (prototype, image and an interface, i.e. a system of information exchange between a prototype and an image); 2) as with a tool or operations; 3) as the management system and its components (with the motive of activity, with typical goals, with typical strategies and, in particular, algorithms, with systems for evaluating the adequacy of the model). The conditions for the success of the formation of priority associations among students are indicated: 1) adjustment of educational and methodological support; 2) adjustment of control and measuring materials; 3) purposeful formation of basic associations and, in particular, learning to translate information into different mathematical languages; 4) multi-aspect positioning of the studied phenomena relative to the information already learned by the student. Keywords: understanding, model, mathematical phenomenon, association with a mathematical phenomenon, classification of associations | 889 | ||||
| 5 | Mathematical activity is the exchange of information and its processing. Typically the focus is on learning to process information. For example, we highlight information processing at the level of: 1) standard algorithms; 2) standard activity strategies; 3) methodology. But working with information begins with its presentation, which is characterized by the language used, the pace and style of transmitting information, etc. But the influence of the level of fixation of the order of presentation of information units has not been studied enough. The purpose is to study different models of presentation of information units and their impact on the process of learning mathematics. The authors developed different models of presentation and primary perception of information units and built an axiomatic theory that helped to systematize and study the influence of different approaches to the presentation of information on teaching mathematics. The article presents a new approach to studying the influence of the order of information presentation on mathematics learning. This has both theoretical and practical significance, since the results of the study can be used to develop more effective methods of teaching mathematics, as well as to improve educational programs. During the study, three options for the presentation of information units and their primary perception were identified: sequential presentation of information units (oral speech), one-time presentation of all units of information with a fixed priority order of their analysis (text message), one-time presentation of all units of information with arbitrary order of their analysis (table, drawing). Each option has a different impact on the effectiveness of teaching mathematics. The authors came to the conclusion that it is important to take into account the order in which information is presented when developing educational programs and methods for teaching mathematics. This allows you to increase the efficiency of the educational process. Keywords: teaching mathematics, information, presentation of information, axiomatic theory | 593 | ||||