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    Basic Education and Fundamental Ideas – Clear Combination of Mathematical Structures
    (СумДПУ імені А. С. Макаренка, 2020) Шмельцер Неллі; Schmelzer Nelli; Кляйне Міхаель; Kleine Michael
    Formulation of the problem. At present, the focus on competence is an important part of discussions about mathematics lessons. In such discussions, particular attention is given to basic mathematical education. In this article, we substantiate the importance of mathematical competence in mathematical work and introduce mathematical work as a modeling cycle. The focus is on the processes of transforming reality into mathematics. In particular, transformational processes contribute to a better mathematical understanding of students and thus contribute to improving the quality of teaching of mathematics. Materials and methods. In order to achieve our goals, we use in this article an empirical methods and general methods of scientific cognition: benchmarking to clarify different views on a problem and determining the direction of research, systematization and generalization to formulate conclusions and recommendations, summarize the author's pedagogical experience and observations. Results. In Chapter 1.2, we describe the process of mathematical work at different stages, using the example of a typical problem. The above example is intended to clearly disclose the processes of thinking and work according to the theoretical model proposed in Chapter 1.1. It should also not be assumed that in a general situation, mathematical work can be comprehensively described by the example that we are studying. However, a competency-oriented teaching methodology is used to help students develop new strategies and heuristics to work with mathematics as a science. In order for students to develop their mathematical competence, mental models are called, which we call fundamental ideas. The construction of such cognitive structures is called the formation of fundamental ideas. This process is characterized by fixing the meaning of the new terms in terms of known factual connections, constructing mental objects that describe the term, and applying this objects to new contexts. Training involves both extending and changing existing foundational ideas as well as building new ideas. Accordingly, in Chapter 2.1 we use an example of probability to illustrate how various aspects of probability can be understood in terms of such a fundamental concept and how the development of fundamental ideas can occur. Significant in this article is a new approach that focuses on competence with a modeling cycle and a basic conception of foundational ideas. Conclusions. The approach developed emphasizes the importance of considering mathematical work as a process and linking the individual levels of foundational ideas to a basic concept. The use of the proposed structure enables teachers to more effectively identify, interpret, and appropriately remove misunderstood students' basic mathematical ideas.