World Of Mathematics Research Articles

닮은 도형에서 넓이비의 이중성: 과정-개념과 과정화-개념화

Tall expressed the three worlds of mathematics using a diagram, and discussed the role of the ‘blending forms’ combining ‘embodied forms’ and ‘symbolic forms’ in the development of individual thinking. According to Tall’s viewpoint, ‘procept’ and ‘proceptual’ are distinguished. The ‘blending form’ proposed by Tall can be reinterpreted in regards to ‘procept’ and ‘proceptual’. Based on the two perspectives of ‘procept’ presented in textbooks through didactical transformation and ‘proceptual’ that the students are actually learning, this study investigated the relationship between similarity ratio and area ratio in the 2015 revised mathematics textbooks. Consequently, most textbooks exhibited ‘procept’ in the exploring task and the generalizing process of area ratios between similar figures. However, the ‘symbolic form’ described in the textbook proved to be insufficient to provide the students with opportunity to explore ‘proceptual’. Particularly, it was discussed Klein’s similarity transformation as a means to supplement the ‘embodied forms’. As a result of this study, it was established that the didactical transformation considering both ‘procept’ and ‘proceptual’ was required. The results of this study could be extended to offer pedagogical implications for school mathematics.

Conexões entre grafos e matrizes na modelagem de problemas matemáticos

A teoria dos grafos é muito importante no mundo matemático como uma excelente forma de conexão com o mundo real. Utilizando-se a teoria de grafos dirigidos é possível transformar muitos dos problemas cotidianos em problemas matemáticos, de forma a fazer um estudo exato em cada caso. Neste trabalho são exploradas as matrizes relacionadas aos diversos tipos de grafos, como a matriz de vértices, a qual é associada a um grafo dirigido, e a matriz de adjacência. Mais do que isso, são construídas matrizes de conexões de vários passos, de forma a criar múltiplas conexões entre os vértices de um grafo dirigido. Em seguida, colocaremos diversas aplicações destes resultados na forma de exemplos.

The Sixth Seminar Nasional Pendidikan Matematika Universitas Ahmad Dahlan 2018

The Preface of the Seminar Nasional Pendidikan Matematika (SENDIKMAD) 2018Puguh Wahyu PrasetyoEditor in Chief of SENDIKMAD’s 2018 Publication, Universitas Ahmad Dahlan Kampus IV UAD, Jl. Ringroad Selatan, Kragilan, Tamanan, Banguntapan, Bantul, Daerah Istimewa Yogyakarta 55191E-mail: puguh.prasetyo@pmat.uad.ac.idPrefaceThe Sixth Seminar Nasional Pendidikan Matematika Ahmad Dahlan is a biennial event of Department of Mathematics Education of Universitas Ahmad Dahlan. The objectives are to improve mathematics teaching and to expand mathematics contributions to the society. The main topics of the conference are divided into five categories namely Analysis, Statistics, Algebra, Applied Mathematics, and Mathematics Education.The keynote presentations are provided especially to show the contribution of Mathematician and Mathematics Educators in the world of mathematics and mathematics education towards research and knowledge sharing where our conference theme for this year is Developing literation skills and High Order Thinking Skills by Innovative Mathematics Learning in Industry Era 4.0. The main event is the talk of the Minister for the Ministry of Education and Culture of the Republic of Indonesia, Professor Dr. Muhadjir Effendy, M.A.P as the first keynote speaker. We have two another keynote speakers coming from Universitas Muhammadiyah Malang, Professor Dr. Yus Mochamad Cholily and Universitas Gadjah Mada, Dr. Nanang Susyanto, M.Sc.We also have a speaker in workshop session coming from Universitas Ahmad Dahlan, Dr. Rully Charitas Indra Prahmana, S.Si., M.Pd. SENDIKMAD 2018 was an overwhelming success, attracting the delegates, speakers and sponsors from many countries and provided great intellectual and social interaction for the participants. Without their support, the conference would not have been successfully organized. I trust that all the participants found their involvement in the Conference both valuable and rewarding. Our wish is that all participants would enjoy this conference, contribute effectively toward it and take back with you knowledge, experiences, contacts and happy memories of this conference and especially with this beautiful kingdom of Yogyakarta.Dr. Puguh Wahyu Prasetyo, S.Si., M.Sc

How Students’ Mathematical Ideas Emerged through Flow of Lesson in Classroom Using Lesson Study and Open Approach

This study was aimed to analyze how students’ mathematical ideas emerged through a flow of lesson in mathematics classroom using Lesson Study and Open Approach. A context of the study was the Open Approach as a teaching approach and the Lesson Study as a way to improve the quality of teaching (Inprasitha, 2015a). The flow of lesson, likewise, is considered to be a tool for accessing to students’ ideas when the students are involved in problem solving and connection between students’ real world and mathematical real world (Inprasitha, 2017b). Ethnographic study was employed as a research methodology in this qualitative study by using a participative research design to form a lesson study team for collaborative designing of lesson plans guided by Thai version of Japanese mathematics textbooks. Research results were shown that the students’ mathematical ideas emerged, through the flow of lesson, when the students’ ideas of a problem solving were mathematized as follows: 1) the mathematical ideas were extended through representations of real world, 2) the mathematical ideas were extended and generalized through semi-concrete aids, and 3) the mathematical ideas were generalized through representations of mathematical world. The processes of mathematization (Isoda & Katagiri, 2012), furthermore, were accomplished when the students’ mathematical ideas become “how to” or tools for learning for the next lessons.

Primary school students’ mathematics motivation and anxieties

Mathematics is necessary to enable children to gain the knowledge and skills required for daily life, it teaches them how to solve a problem, enables them to gain ways of thinking and prepares them for the future. The prejudice developed towards mathematics affects the students’ perceptions related to the mentioned course. The students’ low motivation and high anxiety concerning mathematics are among the most important problems encountered. This research was conducted with students in the third and fourth grades in primary schools in Istanbul province to examine their mathematics motivation and anxieties. It was concluded that students’ mathematics motivation was at a medium level and their mathematics anxieties were very low and there were no differences according to gender. Mathematics motivations among fourth grade students were higher than that of third grade students. It can be recommended that classroom teachers should prepare the environment in which they can introduce the entertaining world of mathematics to students. Keywords: Primary school student, mathematics, mathematics motivation and mathematics anxiety

Riemannian Curvature of a Sliced Contact Metric Manifold

Contact geometry become a more important issue in the mathematical world with the works which had done in the 19th century. Many mathematicians have made studies on contact manifolds, almost contact manifolds, almost contact metric manifolds and contact metric manifolds. Many different studies have been done and papers have been published on Sasaki manifolds, Kähler manifolds, the other manifold types and submanifolds of them. In our previous studies we get the characterization of indefinite Sasakian manifolds. In order to get the characterization of indefinite Sasakian manifolds, firstly we defined sliced contact metric manifolds and then we examined the features of them. As a result we obtain a sliced almost contact metric manifold which is a wider class of almost contact metric manifolds. Thus, we constructed a sliced which is a contact metric manifold on an almost contact metric manifold where the manifold is not a contact metric manifold. Sliced almost contact metric manifolds generalized the almost contact metric manifolds. Then, we study on the sliced Sasakian manifolds and the submanifolds of them. Moreover we analyzed some important properties of the manifold theory on sliced almost contact metric manifolds.In this paper we calculated the -sectional curvature and the Riemannian curvature tensor of the sliced almost contact metric manifolds. Hence we think that all these studies will accelerate the studies on the contact manifolds and their submanifolds.

Interplay between mathematical journals on various scales 1850–1950

This paper is meant as an introduction to a special issue on circulating mathematics via journals, from ca 1850 to 1950. It intends to study the practices, actors, places and institutions involved in mathematical circulation. Journals constitute one of the main carriers of mathematical knowledge and hence the main vehicle for its circulation. The focus of the paper is on ‘mathematical’ (understood very broadly) journals and the various forms of interplay between them. In order to understand and analyse the processes of circulation that are an important feature of mathematical production, the interactions that take place between journals are considered across different scales, from local practices and representations to more global appraisals. We shall use the notion of circulation space in order to break down the representations of the mathematical world in terms of the centres and peripheries suggested by the geographical area, with its massive publishing centres at different times – Paris, London, Berlin, Leipzig, New York, etc. Besides the geographical dimensions, the focus is on connections between journals, polarisations, competition, rivalries, etc.

Adventures in shape and space – and time

As a youth entering the sixth form to study Mathematics, Further Mathematics and Physics I enjoyed the riches of the school's mathematics library and in particular three books which appealed to me, A mathematician's apology [1], A book of curves [2] and On growth and form [3].Hardy's book [1] is one that an impressionable, young mathematician should not read unguided. It left me with the impression that the proper pursuit of mathematics was as a pure subject, of no use or application, to be studied for its own sake; to my regret, I held to this view for several years before finally being able to shake it off through teaching Newtonian mechanics. Looking across mathematics teaching today I seem to observe great interest in geometry, number and algebra ‘curiosities’ that are rooted entirely in mathematics. This in itself is no bad thing, since it clearly draws us and our students into the fascinating world of mathematics. But what of the applications of mathematics? Might they be equally fascinating? Surely we do not want to lure our students into Hardy's trap?

Rapid evaluation of vignetting-free high-performance systems

System assessment for design often involves averages, such as the standard deviation of the wavefront error, which are estimated by ray tracing through a sample of points within the entrance pupil. General-purpose sampling and weighting schemes are presented, and it is also shown that optical design can benefit from tailored versions of these schemes. It turns out that the type of Gaussian quadrature that has long been recognized for efficiency in this domain requires about 40% to 50% more ray tracing to attain comparable accuracy to generic versions of the new schemes. Even greater efficiency gains can be won, however, by tailoring such sampling schemes to the optical context where azimuthal variation in the wavefront is generally weaker than the radial variation. These new schemes are special cases of what is known in the mathematical world as cubature. Our initial results also led to the consideration of simpler sampling configurations that approximate the newfound cubature schemes. We report on the practical application of a selection of such schemes and make observations that aid in the discovery of cubature schemes relevant to optical design of systems with circular pupils.

Integration of Discreteness and Continuity in Forming Mathematical World View Аmong Students

Introduction. When studying mathematics the principle of a wholeness of contents, of integration of its separate components is not always followed. The problem is in how to give students not just the sum of knowledge of separate elements of mathematics, but some holistic integrated system of ideas of the world of mathematics. The purpose of the article is to consider the way of forming an integral mathematical world view. Materials and Methods. To solve the problem, the article draws on philosophical views on the scientific picture of the world as a special form of systematization and integration of knowledge, as well as a trinitarian methodology and historical analysis. The trinitarian methodology has been increasingly used in the postnon-classical worldview. This methodology presupposes the presence of the third element in addition to two binary oppositions, which is necessary to solve the problem of binary contradictions, their integration into a single whole, as a measure of their compromise, as an ar bitrator, as a condition for their existence. Results. Based on the trinitarian methodology, the article shows that the unity of discreteness and continuity, the possibility of their integration into a single whole can be provided with the help of a fractality as the third component. The fractality is just as fundamental structural property of a matter as discreteness and continuity. Thus, it is shown that fractal geometry is not just a new branch of mathematics, it is one of the most important components of the mathematics’ world view of. By studying this section, it is possible when teaching mathematics to students the integration of continuity and discreteness, to develop a holistic intergrated mathematical world view in students. Discussion and Conclusions. For practical work in high school and university, it is also important that the study of fractal geometry contributes to the solution of the main tasks set in the Concept of Development of Mathematical Education in Russia. This is primarily to improve students’ motivation to study mathematics, developing cognitive activity among them, bringing together the learning and research, and solving the problem of the aesthetic orientation of education. Fractal geometry is also a means of integration in the teaching of mathematics and information technology. Therefore, there are all grounds for introducing schoolchildren and students to it.

GeoGebra: A Technological Soft Ware for Teaching and Learning of Calculus in Nigerian Schools

Technology has grown rapidly in today’s world and so the teaching and learning of mathematics have become more and more necessary. A new technological software- GeoGebra is to make the teaching and learning of the subject more practical, easy and simple to understand. Calculus is the mathematics of change. Calculus is an important branch of mathematics which many students find difficult and complex to comprehend. GeoGebra as a dynamic software, easy-to-use which promotes learning and enhance understanding of calculus concepts. It is a free new technological tool, a new means of utilizing technology with visual aids to assist students to interact with mathematical concepts independently or in teams, in the classroom or at home. This study demonstrated practical ways of constructing calculus concepts. Showed the simulations of tracing and identifying the solutions of solving problems in Calculus. The study suggested that Mathematics Teachers should be encouraged to use GeoGebra and other technological tools to improve the quality of teaching Mathematics. Also, GeoGebra should be introduced to Mathematics educators in Nigeria to enable students to explore the world of mathematics in a wider spectrum and think more creatively.

Is a straight line the shortest path?

Is the shortest path from A to B the straight line between them? Your first response might be to think it's obviously so. But in fact you know that it's not quite that straightforward. Your sat-nav knows it's not that straightforward. It asks whether you would like it to find the shortest route or the fastest route, because finding the best path depends on knowing what exactly you mean by ‘long’. Likewise, if you're on a walk in the mountains, there's a good chance you'd rather follow the path around the head of the valley, rather than heading down the steep slope and up the other side.The same sorts of considerations apply in mathematical worlds. I use the mountainside image because it is my preferred way of thinking of a Riemannian metric. Pick an abstract surface S. A Riemannian metric on S gives a well-behaved distance function. By force of habit I tend to picture S as sitting somehow within the physical world. Probably, I'm looking at it from the outside. But if I change viewpoint, so that I am walking around on S, I can picture how the topography affects the idea of the ‘shortest path’.

Algebraic Reasoning in Early Grade: Promoting through Lesson Study and Open Approach

The objective of this study is to analyze what characteristics are of early grade students’ algebraic reasoning in context of open approach and lesson study. Ethnographic study was employed to conduct in this qualitative study. The study was carried out in one mathematics classroom which is a case study of this study and it is the case of classroom which has been using open approach and lesson study since 2006. The 3 teachers are as a member of school lesson study team participated the study as informant and so were 10 students from the class. The data were collected through 9 consecutive lessons by observation with audio-video tape recording, interview, students’ written works and daily field notes. The lessons were designed by carrying into 4 steps of open approach: posing problem, students’ self-learning, whole class discussion and comparison, and summing-up by connecting students’ emergent mathematical ideas. All activities were guided by Thai version of 1st grade Japan mathematics textbook. The results showed that characteristics of first grade students’ algebraic reasoning are as follows: 1) using algebraic expressions to represent addition situation and posing situation to represent given expressions, 2) constructing and using a tool to find problem results more easily, 3) extending solutions to another domain of number, 4) using various representations to justify their ways of thinking, and 5) reasoning about relations among numbers. The algebraic reasoning occurred under the condition that teachers and students had connected among 3 worlds oriented to Inprasitha’s approach: real world, semi-concrete world, and mathematics world.

A Research Approximation to Generalized Riemann Derivatives by Integral Operator Families

Approximation theory has very important applications of polynomial approximation in various areas of functional analysis, Harmonic analysis, Fourier analysis, application mathematic, operator theory in the field generalized derivatives and numerical solutions of differential and integral equations, etc. Integral operators is very important in Harmonic and Fourier analysis. The study of approximation theory is a well-established area of research which deals with the problem of approximating a function <i>f</i> by means of a sequence <i>L_n</i> of positive linear operators. Generalized derivatives (Riemann, Peano and Taylor derivative) are more general than ordinary derivative. Approximation theory is very important for mathematical world. Nowadays, many mathematicians are working in this field.

The First Ahmad Dahlan International Conference on Mathematics and Mathematics Education

Rully Charitas Indra PrahmanaEditor in Chief of AD-INTERCOMME’s Publication, Universitas Ahmad Dahlan, Jl. Pramuka Kav. 5 Pandeyan, Umbulharjo, Yogyakarta, Indonesia.E-mail: rully.indra@mpmat.uad.ac.idPrefaceThe first Ahmad Dahlan International Conference on Mathematics and Mathematics Education (AD-INTERCOMME) is a collaborative event of Department of Mathematics and Department of Mathematics Education (Undergraduate and Postgraduate) Universitas Ahmad Dahlan. The objectives are to improve mathematics teaching and to expand mathematics contributions to the society. This international conference is a biannual conference, started in 2017. The main topics of the conference are divided into five categories namely Analysis, Statistics, Algebra, Applied Mathematics, and Mathematics Education.The keynote presentations are provided especially to show the contribution of Mathematician and Mathematics Educator in the world of mathematics and mathematics education towards research and knowledge sharing where our conference theme for this year is ‘Current Issues in Mathematics and Mathematics Education’. We have four keynote speakers coming from Goethe-University Frankfurt, Germany, Prof. Dr. Mathias Ludwig and Dr. Xenia-Rosemarie Reit, South East Asian Ministers of Education Organization (SEAMEO) QITEP in Mathematics, Assoc. Prof. Allan Leslie White, and Universitas Ahmad Dahlan, Indonesia, Assoc. Prof. Dr. Suparman. We also have two keynote speakers in workshop session coming from Universitas Gadjah Mada, Dr. Indah Emilia Wijayanti, and Universitas Ahmad Dahlan, Dr. Rully Charitas Indra Prahmana.

Géometrie Et Arts-Graphiques Pour Les Pprentissages Fondamentaux A L’ecole Elementaire

Geometric activities are a great way for children, especially those in elementary school, to enter into the mathematical world. To handle, discover, invent, recognize, and identify progressively geometric forms, the child is required to play in space and time with multiple tools and material in order to create new and representative forms. The results obtained have high performance and it helps the teacher to achieve some objectives in all domain of learning, and in particular in the acquisition of transversal skills such as writing and reading. Our suggestions set out ways of improving the objectives set in relation to the undeniable importance of the use of geometric shapes in the practice of graphic art in elementary schools.

Fostering Transit between Real World and Mathematical World: Some Phases on the Modelling Cycle

This study shows how, in the initial training of mathematics teachers, it is possible to promote processes of abstraction and mathematisation through modelling a real situation with the support of auxiliary material to mediate understanding. By adapting elements of the theoretical and methodological framework called Abstraction in Context (AiC), participants’ discussions while building a mathematical model—in a nested epistemic actions—are analysed. Two specific points are discussed in this paper. The first aims to identify how different types of knowledge emerge when an individual is faced with a modelling task. The second is regarding the use of auxiliary material as a means of metaphorising a situation. It was evidenced how the material favours the construction of a mathematical model through the simplification and idealisation that it brings. The meaning constructed for the model is supported in recognising a decreasing behaviour as a part of a whole.

On the Uniform Computational Content of Computability Theory

We demonstrate that the Weihrauch lattice can be used to classify the uniform computational content of computability-theoretic properties as well as the computational content of theorems in one common setting. The properties that we study include diagonal non-computability, hyperimmunity, complete consistent extensions of Peano arithmetic, 1-genericity, Martin-Löf randomness, and cohesiveness. The theorems that we include in our case study are the low basis theorem of Jockusch and Soare, the Kleene-Post theorem, and Friedberg’s jump inversion theorem. It turns out that all the aforementioned properties and many theorems in computability theory, including all theorems that claim the existence of some Turing degree, have very little uniform computational content: they are located outside of the upper cone of binary choice (also known as LLPO); we call problems with this property indiscriminative. Since practically all theorems from classical analysis whose computational content has been classified are discriminative, our observation could yield an explanation for why theorems and results in computability theory typically have very few direct consequences in other disciplines such as analysis. A notable exception in our case study is the low basis theorem which is discriminative. This is perhaps why it is considered to be one of the most applicable theorems in computability theory. In some cases a bridge between the indiscriminative world and the discriminative world of classical mathematics can be established via a suitable residual operation and we demonstrate this in the case of the cohesiveness problem and the problem of consistent complete extensions of Peano arithmetic. Both turn out to be the quotient of two discriminative problems.

Число как формообразующий принцип музыкальной композиции

Цель работы. Исследование посвящено некоторым аспектам взаимодействия музыки и математики, а именно – композиторским замыслам, основу которых составляют фундаментальные мировые константы (такие как число Пи и число Эйлера) либо статистические данные, воплощенные авторами некоторым образом в звуковой ткани их произведений. Помимо диалога музыкантов с математикой, в статье приводится пример обратного диалога (математические исследования Леонарда Эйлера в области музыки) и рассматривается возникающий в связи с этим вопрос о диссонансе в музыке. Методология состоит в использовании компаративного, аналитического, историко-логического методов. Научная новизна работы заключается в расширении бытующих представлений о сопутствующей роли математической составляющей в процессе композиторского творчества до уровня, когда она становится формообразующей первоосновой. Выводы. Объект математического мира, становясь отправной идейной точкой музыкальной композиции и внедряясь в структуру музыкальной ткани, трансформирует ее сообразно со своими потребностями. Трансплантируясь в звуковую сферу, этот объект порождает череду уже всецело музыкальных мыслей автора, однако его символическая первооснова продолжает напоминать о себе на протяжении всего музыкального развития.Ключевые слова: число Пи, число Эйлера, музыкальная композиция, музыкальное формообразование, семантика диссонанса.

The symbolic world of mathematics

In understanding upper secondary school students’ interpretations of information in symbolic representations of a distance-time-relation, little attention has been paid to the analysis of the condition of the conceptual development related to utterances. Understanding this better can help improve the teaching of attribute and information in symbolic representations of different phenomena. Two theoretical perspectives have been used to conduct the analysis: Tall and Vinner's theoretical perspectives on learning and Gray’s &amp; Talls’s theory of three mathematical worlds. The findings provide evidence that a detailed analyse of student’s utterances show difference in quality related to student’s interpretations of a distance-time relation. The qualities were related to student’s concept images of functions and derivatives.