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## Proceedings of the PME and Yandex Russian conference: Technology and Psychology for Mathematics Education

Proceedings of the Moscow regional conference of The International Group for the Psychology of Mathematics Education (PME) and Yandex are presented.

The competence of mathematical modelling is well conceptualized, thought a much-debated question is how to develop it in schools. International achievement test PISA consider mathematics achievements from the modelling perspective (formulating, employing and interpreting), and provides us with comprehensive data to analyze school factors in math results from the comparative perspective. The goal of our study was to estimate the effect of teaching practices on students’ achievements in different PISA mathematical processes while controlling prior achievements (TIMSS).

Processing of mathematical operations and solving numerical tasks implicate a distributed set of brain regions. These regions include the superior and inferior parietal lobules that underlie numerical processing such as size judgments, and additional prefrontal regions that are needed for formal mathematical operations such as addition, subtraction and multiplication [Arsalidou, Taylor, 2011]. Critically, little is known about the connectivity between these regions and the association between math performance and the anatomical structure of white matter tracts. The present study investigates connectivity and white matter tracks associated with networks related to math performance: arcuate fasciculus (AF) and superior longitudinal fasciculus (SLF). Participants performed a computerized task with mathematical operations (addition, subtraction, multiplication, and division) with three levels of difficulty; accuracy and reaction time were recorded. Diffusion tensor imagining (DTI) recordings provided indices on fractional anisotropy (FA) — a measure of the direction of white matter tracks in the brain. The relation between FA and math performance scores is reported.

The current study has three components: (a) a functional magnetic resonance imaging (fMRI) meta-analyses of past literature on mathematical operations; (b) a behav- ioral study to validate a math protocol with parametric changes in the difficulty of math problems that use addition, subtraction, multiplication and division; and (c) an fMRI study that examines the brain correlates of mathematical operations (addition, subtraction, multiplication and division) in relation to subjective effort.

This project investigated for the first time the Right-Left-Right hypothesis using func- tional brain indices related to solving addition, subtraction, multiplication and divi- sion problems. The classic ideas of hemispheric dominance (i.e., visual-spatial abilities in the right hemisphere, verbal ones in the left hemisphere) cannot explain the study’s findings as the material provided were all numerical. We adopt a hypothesis derived from cognitive development to predict that hemispheric involvement stems from an interaction between an individual’s mental-attentional capacity and the mental demand of the task [Pascual-Leone, 1987; Arsalidou, Pascual-Leone, Johnson, 2010]. To test this hypothesis, we adopted a parametric design with several levels of difficulty (easy, within the individual’s competence level and above the individual’s competence level). Our results provide new insights on the brain correlates of mathematical problem solving as a function of operation and difficulty.

The competence of mathematical modelling is well conceptualized, thought a much-debated question is how to develop it in schools. International achievement test PISA consider mathematics achievements from the modelling perspective (formulating, employing and interpreting), and provides us with comprehensive data to analyze school factors in math results from the comparative perspective. The goal of our study was to estimate the effect of teaching practices on students’ achievements in different PISA mathematical processes while controlling prior achievements (TIMSS).

Traditions of mathematical education in Russia on both school and university level, research done by Russian scientists and its impact on the development of mathematics is considered by many a unique and valuable part of the world cultural heritage. In the present paper, we describe the development of mathematical education in Russian universities after 1955 — a period that proved to be most fruitful.

It is widely known that Soviet school of exact sciences, was among the strongest in the world, particularly in terms of physics and mathematics. Why? This is the question we would like to address in this paper by collecting and summarizing different viewpoints on this issue expressed by prominent mathematicians. Many of them witnessed the most fruitful period, the “golden years” of Soviet science and played a major role in the subsequent development of Soviet/Russian mathematics. There is little controversy in the explanations provided by different people; the only essential differences are in the emphases. Thus the list of factors may be regarded as precisely determined. This paper simply aims at communicating them to a non-mathematical community interested in issues of science and education.

The significance of the education in the field of philosophy of mathematics as the part of both philosohpy and mathematics at the universities is the subject of the article.

Over the past century, educational psychologists and researchers have posited many theories to explain how individuals learn, i.e. how they acquire, organize and deploy knowledge and skills. The 20th century can be considered the century of psychology on learning and related fields of interest (such as motivation, cognition, metacognition etc.) and it is fascinating to see the various mainstreams of learning, remembered and forgotten over the 20th century and note that basic assumptions of early theories survived several paradigm shifts of psychology and epistemology. Beyond folk psychology and its naïve theories of learning, psychological learning theories can be grouped into some basic categories, such as behaviorist learning theories, connectionist learning theories, cognitive learning theories, constructivist learning theories, and social learning theories.

Learning theories are not limited to psychology and related fields of interest but rather we can find the topic of learning in various disciplines, such as philosophy and epistemology, education, information science, biology, and – as a result of the emergence of computer technologies – especially also in the field of computer sciences and artificial intelligence. As a consequence, machine learning struck a chord in the 1980s and became an important field of the learning sciences in general. As the learning sciences became more specialized and complex, the various fields of interest were widely spread and separated from each other; as a consequence, even presently, there is no comprehensive overview of the sciences of learning or the central theoretical concepts and vocabulary on which researchers rely.

The *Encyclopedia of the Sciences of Learning* provides an up-to-date, broad and authoritative coverage of the specific terms mostly used in the sciences of learning and its related fields, including relevant areas of instruction, pedagogy, cognitive sciences, and especially machine learning and knowledge engineering. This modern compendium will be an indispensable source of information for scientists, educators, engineers, and technical staff active in all fields of learning. More specifically, the Encyclopedia provides fast access to the most relevant theoretical terms provides up-to-date, broad and authoritative coverage of the most important theories within the various fields of the learning sciences and adjacent sciences and communication technologies; supplies clear and precise explanations of the theoretical terms, cross-references to related entries and up-to-date references to important research and publications. The *Encyclopedia* also contains biographical entries of individuals who have substantially contributed to the sciences of learning; the entries are written by a distinguished panel of researchers in the various fields of the learning sciences.

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.