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Документ A Markov Chain Representation of the “5 E’s” Instructional Treatment(СумДПУ імені А. С. Макаренка, 2019) Voskoglou Michael Gr.; Воскоглой Майкл Гр.Formulation of the problem. The socio-constructive theories of learning have become very popular during the last decades for teaching mathematics. The “5 E’s” is an instructional model based on the principles of social constructivism that has recently become very popular, especially in school education, for teaching mathematics. Each of the 5 E's describes a phase of learning which begins with the letter "E" – Engage, Explore, Explain, Elaborate, Evaluate. Depending on the student reactions, there are forward or backward transitions between the three middle phases (explore, explain, elaborate) of the 5E’s model during the teaching process. The “5 E's” model allows students and teachers to experience common activities, to use and build on prior knowledge and experience and to assess their understanding of a concept continually. Materials and methods. Probabilistic methods of analysis are used. Results. The mathematical representation of the “5 E’s” model is attempted by applying an absorbing Markov chain on its phases. A Markov Chain (MC) is a stochastic process that moves in a sequence of steps (phases) through a set of states and has a one-step memory. A finite MC having as states Si the corresponding phases Ei, i = 1, 2,…, 5, of the “5 E’s” instructional model is introduced. A classroom application is also presented illustrating the usefulness of this representation in practice. The following application took place recently at the Graduate Technological Educational Institute of Western Greece for teaching the concept of the derivative to a group of fresher students of engineering. Conclusions. The Markov chain representation of the “5 E’s” model provides a useful tool for evaluating the student difficulties during the teaching process. This is very useful for reorganizing the instructor’s plans for teaching the same subject in future.Документ A Note on the Graphical Representation of the Derivatives(2017) Воскоглой Майкл Гр.; Voskoglou Michael Gr.In the article at hands an alternative definition of the concept of the derivative is presented, which makes no use of limits. This definition is based on an old idea of Descartes for calculating the slope of the tangent at a point of a curve and holds for all the algebraic functions. Caratheodory extended this definition to a general definition of the derivative in terms of the concept of continuity. However, although this definition has been used successfully by many German mathematicians, it is not widely known in the international literature, nor it is used in the school book texts. After presenting Caratheodory’s definition, the article closes by describing methods for calculating the derivative at a point of a function y = f(x) with the help of a suitably chosen table of values of f(x), and for designing of the graph of the derivative function f΄(x) given the graph, but not the formula, of f(x). These methods are based on the graphical representation of the derivative, which should be reclaimed better in general for teaching purposes.Документ Application of Fuzzy Relation Equations to Assessment of Analogical Problem Solving Skills(СумДПУ імені А. С. Макаренка, 2018) Воскоглой Майкл Гр.; Voskoglou Michael Gr.Analogical reasoning is a very important part of the human cognition in general for creativity and scientific discovery and in particular it is a very useful method for solving mathematical problems by retrieving from the memory similar problems solved in the past and adapting their solutions for use with the target problem. On the other hand, the student assessment is an essential task of Education, because, apart of being a social need and demand, it helps the instructors in designing their future plans for a more effective teaching procedure. However, frequently an instructor is not sure about the exact grade corresponding to each student’s performance. Therefore, the student assessment is characterized by a degree of vagueness and/or uncertainty. Consequently, fuzzy logic, due to its property of characterizing the ambiguous real life situations with multiple values, becomes a reach resource of assessment techniques to be applied in such vague cases. In this work fuzzy relation equations are used as a tool for evaluating student analogical problem solving skills. The fuzzy relation equations are obtained by the composition of binary fuzzy relations, which are fuzzy sets defined on the Cartesian product of two crisp sets. The compositions of binary fuzzy relations are conveniently performed in terms of the membership matrices of them. The same elements of the membership matrices are used in these compositions as would be used in the regular multiplication of matrices, but the product and sum operations are here replaced with the min and max operations respectively. The notion of fuzzy relation equations was first proposed by Sanchez in 1976 and later was further investigated by other researchers. A classroom application and other suitable examples are also presented in this article illustrating our results. Further, the present work is connected to our earlier research efforts on utilizing several other fuzzy Logic techniques as tools for the assessment of the student performance.Документ Current Problems and Future Perspectives of Mathematics Education(СумДПУ імені А. С. Макаренка, 2018) Воскоглой Майкл Гр.; Voskoglou Michael Gr.From the origin of mathematics as an autonomous science two extreme philosophies about its orientation have been tacitly emerged: Formalism, where emphasis is given to the axiomatic foundation of the mathematical content and intuitionism, which focuses on the connection of the mathematical existence of an entity with the possibility of constructing it, thus turning the attention to problem-solving processes. Although none of the existing schools of mathematical thought, including formalism and intuitionism, have finally succeeded to find a solid framework for mathematics, most of the recent advances of this science were obtained through their disputes about the absolute mathematical truth. In particular, during the 19th and the beginning of the 20th century, the paradoxes of the set theory was the reason of an intense “war” between formalism and intuitionism, which however was extended much deeper into the mathematical thought. All these disputes created serious problems yo the sensitive area of mathematics education, the most characteristic being probably the failure of the introduction of the “New Mathematics” to the school curricula that distressed students and teachers for many years. In the present work current problems of mathematics education are investigated, such as the role of computers in the process of teaching and learning mathematics, the negligence of the Euclidean Geometry in the school curricula, the excessive emphasis given sometimes by the teachers to mathematical modeling and applications with respect to the acquisition of the mathematical content by students, etc. The future perspectives of teaching and learning mathematics at school and out of it are also discussed. The article is formulated as follows: A short introduction is attempted in the first Section to the philosophy of mathematics .The main ideas of formalism and intuitionism and their effects on the development of mathematics education are exposed in the next two Sections. The fourth Section deals with the main issues that currently occupy the interest of those working in the area of mathematics education and the article closes with the general conclusions stated in the fifth Section that mainly concern the future perspectives of mathematics education.Документ Fuzzy Numbers as an Assessment Tool in the Apos/Ace Instructional Treatment for Mathematics(СумДПУ імені А. С. Макаренка, 2016) Воскоглой Майкл Гр.; Voskoglou Michael Gr.У статті використовується комбінація методів трикутних нечітких чисел (TFNs) та центру тяжіння (COG) як техніки дефазифікації для оцінки знань і навичок студентів універистету у процесі навчання математики у рамках APOS/ ACE.Документ Machine Learning Techniques for Teaching Mathematics(СумДПУ імені А. С. Макаренка, 2020) Воскоглой Майкл Гр.; Voskoglou Michael Gr.; Abdel-Badeeh M. Salem; Абдел-Баді М. СалемFormulation of the problem. Famous social thinkers of our times are speaking about a forthcoming new industrial revolution that will be characterized by the development of an advanced Internet of things and energy, and by the cyber-physical systems controlled through it. There is no doubt that our students should take full advantage of the potential that the new digital technologies can bring for improving their learning skills. Materials and methods. This treatise has a review character. The methods of analysis used are based on already reported researches. Results. The article focuses on the role that the artificial teaching and learning of mathematics could play for education in the forthcoming era of the new industrial revolution Starting with a brief review of the traditional learning theories and methods of teaching mathematics, the article continues by studying the use of computers and of applications of artificial intelligence in mathematics education. Conclusions. The advantages and disadvantages of artificial with respect to traditional learning are discussed as well as the perspectives for future research on the subject.Документ Management of Fuzzy Data in Education(СумДПУ імені А. С. Макаренка, 2019) Voskoglou Michael Gr.; Воскоглой Майкл Гр.Formulation of the problem. Some years ago the unique tool in hands of the scientists for handling the situations of uncertainty that frequently appear in problems of science, technology and of the everyday life, used to be the theory of Probability. However, nowadays the theory of Fuzzy Sets initiated by Zadeh in 1965 and its extensions and generalizations followed in the recent years have given a new dynamic to this field. Materials and methods. Mathematical methods of analysis are used. Results. In the present work a model is developed for handling the fuzzy data appearing in the field of Education. The model is based on the calculation of the possibilities of the profiles involved in the corresponding situations, which, according to the British economist Schackle and many other researchers, are more suitable than the fuzzy probabilities for studying the human behaviour. A classroom application to learning mathematics is also presented illustrating the importance of the model in practice. The general model is extended for studying the combined results of the evaluation of fuzzy data obtained from two (or more) different sources and an example is provided to emphasize the usefulness of this extension for real situations in education. Conclusions. The management and evaluation of the fuzzy data obtained by the operation mechanisms of large and complex systems is very important for real life and science applications. A developed model evaluates such kind of data in terms of the corresponding membership degrees and possibilities. The examples for the process of learning a subject-matter in the classroom and the example for a market's research illustrate the applicability and usefulness of the model to practical problems. The general character of the proposed model enables its application to a variety of other human and machine activities for a description of such kind of activities and this is one of main targets for future research.Документ Modes of Thinking in Problem Solving(СумДПУ імені А. С. Макаренка, 2020) Voskoglou Michael Gr.; Воскоглой Майкл Гр.Formulation of the problem: Problem Solving affects our daily lives in a direct or indirect way for centuries. Volumes of research have been written about it and attempts have been made by scientists to make it accessible to all in various degrees. The modes of thinking used in Problem Solving is, therefore, a very interesting and timely subject to study. The present paper discusses the main modes of thinking involved in PS, which are Critical Thinking (CrT), Computational Thinking (CT) and Statistical Thinking (ST). Emphasis is given to ST and in particular to Bayesian Reasoning due to its great importance for everyday life and science that only recently has been fully recognized. We start with a brief description of the use of CrT in PS. Next ST and the necessity of combining it with CrT in PS is discussed, while Bayesian reasoning is studied separately afterwards. Next the role of CT for solving complex technological problems is examined and the paper closes with the final conclusion. Materials and methods. The methods of analysis used are based on a synthesis of already reported researches and on suitable examples illustrating our results. Results. The article studies the role of Statistical Thinking in Problem Solving, where the problem is considered with its wide meaning (not mathematical problems only). Particular emphasis is given to Bayesian Reasoning, whose importance in everyday life and science applications has been only recently fully recognized. Critical and Computational Thinking, the other two main modes of thinking used in Problem Solving, are also discussed Conclusions. Problem Solving is a complex cognitive process that needs the combination of several modes of thinking in order to be successful. Those modes, apart from the simple spontaneous thinking, include Critical, Statistical and Computational Thinking.Документ Use of Fuzzy Numbers for Assessing the Acquisition of the van-Hiele Levels in Geometry(СумДПУ імені А. С. Макаренка, 2016) Воскоглой Майкл Гр.; Voskoglou Michael Gr.It is generally accepted that students face many difficulties in constructing proofs of theorems and solutions of problems of the Euclidean Geometry. The van Hiele theory of geometric reasoning suggests that students can progress through five levels of increasing structural complexity. It has been also proved by other researchers that these levels are continuous characterized by transitions between the successive levels. This means that from the teacher’s point of view there exists fuzziness about the degree of student acquisition of each level, which suggests that principles of Fuzzy Logic could be used for the evaluation of student geometric reasoning skills. Here a combination of Triangular Fuzzy Numbers (TFNs) and of the Centre of Gravity (COG) deffuzzification technique is applied for evaluating the acquisition of the van Hielie levels by students. It is further shown that the use of the Yager index instead of the COG technique leads to the same assessment conclusions. Other fizzy methods applied in earlier works are also discussed and an example on high school student acquisition of 3-dimensional geometrical concepts is presented illustrating our results.