arXiv subjects
Publications and source records attributed to Ioannis Rizos.
Let $a,b$ be positive, relatively prime, integers. We prove, using induction, that for every $d > ab-a-b$ there exist $x,y\in\mathbb{Z}_{\geq 0}$, such that $d=ax+by$. As a byproduct, we obtain a constructive recursive algorithm for identifying appropriate $x,y$ as above.
Let $a,b$ be positive, relatively prime, integers. Our goal is to characterize, in an elementary way, all positive integers $c$ that can be expressed as a linear combination of $a,b$ with non-negative integer coefficients and discuss the teaching perspectives of our methods.
In this paper we first study the isoperimetric problem in the case of integer triangles, as well as Alcuin's sequence and how it relates to the number of different integer triangles with a given perimeter. We then present and compare two different approaches to the above problem. The first approach is due to a university professor (a working mathematician), who uses a well-known technique of calculating the number of elements of finite sets, while the second one is due to a high school student with a special interest in mathematics and is purely number-theoretic. Furthermore, some related instructor perspectives are discussed.
In this paper we discuss, from a historical and philosophical point of view, a variation of the meaning of the five postulates in Euclidean Geometry and we make a short reference to D. Hilberts formalism. We examine, throughout the ages, the question what is Geometry by studying the segmentation and the unification of various geometries, been introduced by F. Kleins Erlangen Program. We concisely present three out of nine Cayley-Klein geometries and then we conclude with some didactic perspectives.
Geometry is essentially a global language, which is fully understood in different times, countries and cultures. The proof of a geometric theorem (e.g. the Pythagorean Theorem) or a geometric construction (e.g. the construction of an equilateral triangle with straightedge and compass) is recognized in more or less the same way by everyone. This fact, in combination with other parameters, brings out a human face for geometric and mathematical achievements in general, thus allowing people and civilizations to come together.
Around the globe Covid-19 pandemic has influenced not only the education, but also our everyday life, among other aspects. In Greece, distance learning started to get in use widely in tertiary education, since the first national lockdown was announced and reshaped education in many ways. In the University of Thessaly, in the Department of Mathematics, undergraduate students opt a lot of different courses to attend and due to the Covid-19 crisis all of them are taught via web platforms. Some of the most significant theoretical subjects are "Calculus" (with applications in Science and Mechanics), "Physics" (Classical Mechanics) and "Philosophy of Science". In addition, some other applied subjects are "Programming Languages" and "Digital Technologies in Mathematics Education" and the impact the above have, generally in education and society itself. In this paper, we describe the different ways students have reacted regarding learning Sciences in a distance learning environment. We split our case study in two parts. The first one is about the way students experience e-learning and the second one is about their suggestions for the next day. Integrating e-questionnaires and interviews and taking into account parameters like economic factors and the permanent residence issue, we asked the students about their preferences among face-to-face learning, distance learning and a blended model. Remarks about the academic life and the possible ways of taking the extra step, after the Covid-19 crisis ceases to exist, are made.
From antiquity the conceptual perception of space changed painfully and at a relatively slow pace. It went through mythological descriptions, religious beliefs, metaphysical worldviews and cosmological models with a mechanistic structure, until it reached the Newtonian perception of the infinite and homogeneous three-dimensional continuum, which dominates to this day as a "common perception" even in education. This perception of space, which today seems self-evident, is the culmination of a long historical development, which we will try to outline. Our aim is to briefly present the conceptual course of models related to natural space from antiquity to the 20th century, focusing on the theories and models that can be mathematized. At the same time we try to highlight one of the most important starting points of geometric concepts, which is the need for mathematical modeling of physical space, as in modern education there is no such view.
One of the demands of modern teaching is the interdisciplinary approach of cognitive subjects and in particular that of the natural sciences, with simultaneous engagement of teachers and students. At the same time, teaching scenarios seem to be gaining ground in the methodology of teaching and learning Mathematics in school. In this paper we analyze the concepts of the teaching scenario and interdisciplinarity, and we briefly present the basic results of a teaching experiment using teaching scenario, focused on the teaching of Minkowskian Metric in two dimensions.