GRAPH ISOMORPHISMS AND MATRIX SIMILARITY: SWITCHING BETWEEN REPRESENTATIONS

A proof whether two graphs (possibly oriented graphs or multigraphs, etc.) are isomorphic or not can be derived by various methods. Some of them are reasonable for small numbers of vertices and/or edges, but not for larger numbers. Switching from iconic representation to a matrix representation transforms the problem of Graph Theory into a problem in Linear Algebra. The support provided by a Computer Algebra System is analyzed, in particular with regard to the building of new mathematical knowledge through a transition from graphical to algebraic representation. Moreover two important issues are discussed: a. the need for more than one representation; b. the direction of the switch between representations, which is non standard, from graphical to algebraic

From conic intersections to toric intersections: The case of the isoptic curves of an ellipse

Starting from the study of the orthoptic curves of parabolas and ellipses, we generalize to the case of isoptic curves for any angle, i.e. the geometric locus of points from which a parabola or an ellipse are viewed under a given angle. This leads to the investigation of spiric curves and to the construction of these curves as an actual intersection of a self-intersecting torus with a plane. The usage of a Computer Algebra System facilitated this investigation

ChatGPT may excel in States Medical Licensing Examination but falters in basic Linear Algebra

The emergence of ChatGPT has been rapid, and although it has demonstrated positive impacts in certain domains, its influence is not universally advantageous. Our analysis focuses on ChatGPT's capabilities in Mathematics Education, particularly in teaching basic Linear Algebra. While there are instances where ChatGPT delivers accurate and well-motivated answers, it is crucial to recognize numerous cases where it makes significant mathematical errors and fails in logical inference. These occurrences raise concerns regarding the system's genuine understanding of mathematics, as it appears to rely more on visual patterns rather than true comprehension. Additionally, the suitability of ChatGPT as a teacher for students also warrants consideration

Dynamic construction of a family of octic curves as geometric loci

We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations, of the given curves. Finally, we highlight the involved mathematical or computational issues arising when reflecting on the outputs obtained in each case

Instrumented Techniques and Reflective Thinking in Analytic Geometry

In a previous study that explored epistemological perspectives on solving problems with Computer Algebra Systems (CAS) we concluded that awareness of the special ways that the software utilizes symbols in algebraic manipulations and in implicit plotting should be encouraged (Zehavi, 2004). Such awareness is required for, and encouraged by treating geometry analytically with a symbolic-graphical system. In this paper we compare a traditional solution of a problem in analytic geometry with CAS-based solutions to the same problem. The discussion will focus on the role of reflective thinking, namely selection of techniques, monitoring of the solution process, insight, and conceptualization, play in the creation of instrumented techniques (Guin & Trouche, 1999). Teachers, who experienced learning activities from a resource e-book for teaching analytic geometry with CAS, contributed to the design of tasks and to the analysis of instrumented techniques

BBC'22

The International Conference BBC'22 aims to provide an opportunity for all academic and non-academics to share their personal experiences and projects, presenting their contributions and getting feedback from other attendees.info:eu-repo/semantics/publishedVersio

Literature Review: Problem Based Learning Berbasis Isu Sosiosaintifik dalam Menjawab Tantangan Keterampilan Abad 21 Pembelajaran IPA

Abstract:. Science learning at this time must of course lead to the challenges of 21st century skills. Initially teacher-centred learning has now shifted to learner-centred learning. At this time, with the demands of the times, learning developments must be varied in order to build a construct of understanding and encouragement so that active learning occurs. One of them is problem-based learning found in everyday life which often intersects with students' daily lives. The aim of this research is to find out how Problem Based Learning based on socioscientific issues can answer the challenges of 21st century skills in science learning. The research method applied in writing this article is a study of literature originating from various books that are likely to be published from 2018-2023. The results obtained in the research show that there are many advantages of Problem Based Learning when it is integrated with scientific issues in the surrounding environment, this shows the activeness of learning when there is a problem that must be solved. In science learning, the application of Problem Based Learning found 3 dimensions, namely attitudes, processes and products. From this dimension, various foundations emerge in the form of outcomes that answer the 4 skills of the 21st century, namely Critical Thinking, Creative Thinking, Collaboration, and Communication.Abstrak: Pada pembelajaran IPA pada saat ini tentu harus mengarah pada tantangan keterampilan abad 21. Semula pembelajaran yang berpusat pada guru sekarang beralih pada pembelajaran yang berpusat pada peserta didik. Pada saat ini dengan tuntutan zaman maka perkembangan pembelajaran harus bervariasi agar dapat membangun suatu konstruk pemahaman dan dorongan agar terjadi pembelajaran yang aktif. Salah satunya dengan pembelajaran berbasis masalah yang ditemukan dalam kehidupan sehari-hari yang sering bersingungan dengan keseharian peserta didik. Tujuan penelitian ini adalah untuk mengetahui bagaimana pembelajaran Problem Based Learning berbasis isu sosiosaintifik ini dalam menjawab tantangan keterampilan abad 21 pada pembelajaran IPA. Metode penelitian yang diterapkan pada penulisan artikel ini adalah studi literatur yang berasal dari berbagai buku dengan rentan terbit dari 2018-2023. Hasil yang diperoleh dalam penelitian menunjukkan banyak kelebihan pembelajaran Problem Based Learning ini ketika di integrasikan dengan isu-isu ilmiah yang ada di lingkungan sekitar, hal ini menunjukkan keaktifan pembelajaran ketika terdapat ada masalah yang harus dipecahkan. Pada pembelajaran IPA pada penerapan Problem Based Learning menemukan 3 dimensi yaitu sikap, proses, dan produk. Dari dimensi ini memunculkan berbagai pondasi berupa outcome yang menjawab 4 skill keterampilan abad 21 yaitu Critical Thinking, Creative Thinking, Collaboration, dan Communication

On isoptics and isochordal-viewed curves

In this paper, some results involving isoptic curves and constant $\phi$-width curves are given for any closed curve. The non-convex case, as well as non-simple shapes with or without cusps are considered. Relating the construction of isoptics to the construction given in Holditch’s theorem, a kind of curves is defined: the isochordal-viewed curves. The explicit expression of these curves is given together with some examples. Integral formulae on the area of their isoptics are obtained and a Barbier-type theorem is derived. Finally, a characterization for isochordal-viewed hedgehogs and curves of constant $\phi$-width is given in terms of an angle function

The usage of digital technologies in the university training of future bachelors (having been based on the data of mathematical subjects)

This article demonstrates that mathematics in the system of higher education has outgrown the status of the general education subject and should become an integral part of the professional training of future bachelors, including economists, on the basis of intersubject connection with special subjects. Such aspects as the importance of improving the scientific and methodological support of mathematical training of students by means of digital technologies are revealed. It is specified that in order to implement the task of qualified training of students learning econometrics and economic and mathematical modeling, it is necessary to use digital technologies in two directions: for the organization of electronic educational space and in the process of solving applied problems at the junction of the branches of economics and mathematics. The advantages of using e-learning courses in the educational process are presented (such as providing individualization of the educational process in accordance with the needs, characteristics and capabilities of students; improving the quality and efficiency of the educational process; ensuring systematic monitoring of the educational quality). The unified structures of “Econometrics”, “Economic and mathematical modeling” based on the Moodle platform are the following ones. The article presentsthe results of the pedagogical experiment on the attitude of students to the use of e-learning course (ELC) in the educational process of Borys Grinchenko Kyiv University and Alfred Nobel University (Dnipro city). We found that the following metrics need improvement: availability of time-appropriate mathematical materials; individual approach in training; students’ self-expression and the development of their creativity in the e-learning process. The following opportunities are brought to light the possibilities of digital technologies for the construction and research of econometric models (based on the problem of dependence of the level of the Ukrainian population employment). Various stages of building and testing of the econometric model are characterized: identification of variables, specification of the model, parameterization and verification of the statistical significance of the obtained results