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## К вопросу формирования принципов перевода первичных баллов ЕГЭ по математике в 100-балльную шкалу

The paper discusses in detail the scale of translation of primary points scored by school graduates in the unified state exam in mathematics, used from 2013 to the present time. Based on the analysis of the dynamics of these scales, a conclusion is made about the annual increase in the "average" 100-point result, as well as the presence of a significant increase in the final grade compared with the linear scale. Additionally, the authors describe the effect of reducing the value of primary points as they approach the maximum.

One merit of Trends in International Mathematics and Science Study (TIMSS) is that apart from a direct school students cognitive appraisal, it enables to collect information on teachers of these students, on their education, work experience and teaching practices. The first difference method was used to determine how teachers characteristics were associated with students achievements and to overcome restrictions of TIMSS correlation design. In addition, effects of teachers characteristics were evaluated by the conventional regressions method. The discovered associations differed across subject areas, and the first difference method results differed from the conventional correlation analysis results. For mathematics the first difference method revealed negative association of reproductive tasks and collaborative learning with achievements, and tasks aimed at comprehension and development of metasubject skills showed positive association. For natural science reproductive tasks showed, on the contrary, positive association, while tasks aimed at comprehension and development of metasubject skills either did not produce any effects, or they were negative. Also, for natural science, unlike mathematics, a teachers experience considerably influenced students achievements.

This article consider The project of the scientific and educational Center for integration of multimedia technologies in science, education and culture, as space-technological environment for the implementation of innovative scientific and educational projects of the 21st century, which should become the support for the master's programs, especially interdisciplinary; at the intersection of science, art and information technologies, and implementation of innovative scientific and commercial projects, which are to become a master's thesis.

The three already traditional volumes of the WDS Proceedings you are holding in the hands are composed of the contributions which have been presented during the 21st Annual Conference of Doctoral Students that was held in Prague, at Charles University, Faculty of Mathematics and Physics from May 29 to June 1, 2012. In this year, 100 student manuscripts were submitted to publishing and 88 were accepted after the review process.

The present study tested the possibility of operationalizing levels of knowledge acquisition based on Vygotskyђs theory of cognitive growth. An assessment tool (SAMMath) was developed to capture a hypothesized hierarchical structure of mathematical knowledge consisting of procedural, conceptual, and functional levels. In Study 1, SAM-Math was administered to 4th-grade students (N = 2,216). The results of Rasch analysis indicated that the test provided an operational definition for the construct of mathematical competence that included the three levels of mastery corresponding to the theoretically based hierarchy of knowledge. In Study 2, SAM-Math was administered to students in 4th, 6th, 8th, and 10th grades (N = 396) to examine developmental changes in the levels of mathematics knowledge. The results showed that the mastery of mathematical concepts presented in elementary school continued to deepen beyond elementary school, as evidenced by a significant growth in conceptual and functional levels of knowledge. The findings are discussed in terms of their implications for psychological theory, test design, and educational practice.

This article presents the results of a pilot study assessing the level of formation of a stochastic competence among teachers of mathematics. Besides, the indicators that reflect the competence of formation of stochastic students are identified and ranked in order of importance. Different instruments (questionnaires, tests, assignments) have been used to solve the problem under study.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.