arXiv subjects
Publications and source records attributed to Aslanbek Naziev.
The paper examines the construction of a course in mathematical analysis at a pedagogical university, aimed at developing the ability of future mathematics teachers to detect and solve problems related to finding proofs. Key words: teaching mathematics, teacher training, discovery of proofs, quantifiers, mathematical analysis
We study the (in)dependence of additivity and homogeneity conditions in the definition of linear mappings between vector spaces over the same scalar field. Unlike other works on the subject, dealing with particular fields like real or complex numbers, or with particular mappings like continuous or measurable, we consider the general case. This enables us to obtain complete picture. Namely, for the prime field, and only in this case, the conditions are dependent (additivity implies homogeneity). For the non-prime field they are independent: neither of conditions implies the other. Thus, the problem posed in the previous version of the paper, is solved here.
The exposition of the theory of structure species in Bourbaki's tractate takes only a few pages but still is quite difficult. However, in the exercises, Bourbaki outlines another approach that is based on the notion of structure type rather than the notion of echelon construction scheme. Herewith, relying on the permissible limits common sense, one can proceed without complicating consideration the concept of the balanced string of signs, and then get quite affordable and at the same time sufficiently rigorous exposition. That's what we are doing by following the path proposed in this doctoral thesis. To make the presentation as accessible as possible, we have included in it a large number of examples. The work ends with the definition of kind structure. The author hopes to continue work, including in her consideration of the three main species of the general structures combined structures and structures with morphisms.
The purpose of this paper is to emphasize the role of language in the process of teaching and learning mathematics. We will begin with the definition of mathematics given by Cassiodorus (in its essential features repeated in Kolmogorov's famous article "Mathematics" from the "Great Soviet Encyclopedia" and explored in the author's book "Humanitarian oriented mathematics teaching in general school" (Ryazan, Ryazan Institute of Education Development, 2000, 112 p.). This definition illuminates the exceptional role of language in the building, discovering, understanding, teaching, and learning of mathematics. We show that this role is predetermined by the abstract character of mathematics. The abstract object cannot be shown; it can only be described with the help of language. This gives rise to the problem of identifying abstract entities on the grounds of their descriptions. We accentuate the way the language of set theory helps finding the solution to this problem. We illustrate this through the examples from elementary geometry and elementary algebra. And we especially underline the importance of this solution when working with events in the probability theory. After that, we are going to show how proof and language help to form and develop imagery thinking and, thereby, illuminate the possibility of the left brain hemisphere to participate in the formation of imagery thinking. Keywords: mathematics, teaching, and learning, language, imagery thinking, functional brain asymmetry