World Of Mathematics Research Articles

What is the problem in problem-based learning in higher education mathematics

ABSTRACTProblem and Project-Based Learning (PBL) emphasise collaborate work on problems relevant to society and emphases the relation between theory and practice. PBL fits engineering students as preparation for their future professions but what about mathematics? Mathematics is not just applied mathematics, but it is also a body of abstract knowledge where the application in society is not always obvious. Does mathematics, including pure mathematics, fit into a PBL curriculum? This paper argues that it does for two reasons: (1) PBL resembles the working methods of research mathematicians. (2) The concept of society includes the society of researchers to whom theoretical mathematics is relevant. The paper describes two cases of university PBL projects in mathematics; one in pure mathematics and the other in applied mathematics. The paper also discusses that future engineers need to understand the world of mathematics as well as how engineers fit into a process of fundamental-research-turned-into-applied-science.

Words or numbers: The essence of language teachers' un-appreciation of mathematics

As proved by varied researches, English language teachers, with their burning passion in the realm of words, have constantly updated themselves with educational trends and timely methods and techniques with regard to teaching this discipline so that they can be efficient and effective in the learning process. Armed with pedagogical skills in language teaching, language teachers, as observed, apparently do not dwell themselves so much on the world of numbers, or simply, world of Mathematics, which in their roles as educators, plays a vital role with regard to measurement of student learning. Hence, this paper considered this inquiry as an attempt to understand language teacher’s unappreciation of Mathematics at the University of Perpetual Help System Laguna. Research questions occurred as to what unappreciation of mathematics meant to the participant and as to how they experienced this phenomenon. The study, employing psychological phenomenology, analyzed the participants’ verbatim responses which resulted in six themes (Inevitable Unappreciated Discipline, Adjusting for Oneself, Mental Burden Experience, Complicating Life, Avoiding Expansion of Intellectual Horizon, Realizing Reality) categorized into textural (what) and structural (how) descriptions of the phenomenon. The research study concludes with the participants having clear images of Mathematics as evidenced and detailed that they truly dislike the subject as it creates much difficulty and puzzlement in their minds. The varied scenarios depicting the participants’ unpleasant experiences with Mathematics clearly affected and heightened their unapppreciation of this area of study.

Ronald S. Calinger. Leonhard Euler: Mathematical Genius in the Enlightenment. xvii + 669 pp., illus., apps., bibl., index. Princeton, N.J./Oxford: Princeton University Press, 2016. $55, £40.95 (cloth). Leonhard Euler. Correspondence. Edited by Franz Lemmermeyer and Martin Mattmüller. 2 parts. (Opera Omnia, 4A: Commercium Epistolicum, 4.) xiii + 1,248 pp., illus., figs., bibl., index. Basel: Springer, 2015. $458, £148.50 (cloth).

Ronald S. Calinger. <i>Leonhard Euler: Mathematical Genius in the Enlightenment</i>. xvii + 669 pp., illus., apps., bibl., index. Princeton, N.J./Oxford: Princeton University Press, 2016. $55, £40.95 (cloth). Leonhard Euler. <i>Correspondence</i>. Edited by Franz Lemmermeyer and Martin Mattmüller. 2 parts. (Opera Omnia, 4A: Commercium Epistolicum, 4.) xiii + 1,248 pp., illus., figs., bibl., index. Basel: Springer, 2015. $458, £148.50 (cloth).

The Neuroscience of Moral Judgment and Aquinas on Moral Expertise

The Neuroscience of Moral Judgment and Aquinas on Moral Expertise Stephen Napier CONTEMPORARY MORAL psychology is discovering some unflattering aspects of our moral intuitions. Quite simply, they are unreliable, or, at least, we have every reason to think that our intuitions are not attuned to morally relevant features of our environment. Because moral intuitions are the starting point for a person's moral judgments and reasoning, faulty intuitions can affect someone's entire system of moral beliefs. What is more, the agent whose beliefs result from these faulty intuitions would not know otherwise. As E. F. Schumacher reminded us,1 just as by the eye one can see everything except the eye through which one sees, so the moral eye can see everything except the eye through which it sees. The same is true for whole systems of moral beliefs according to which everything looks okay "from the inside" or from the perspective of the agent, but, viewed sideways on, a moral belief system may be radically disordered. This paper attempts to put contemporary cognitive science of moral judgment into conversation with Aquinas's moral epistemology. Aquinas has provided us with a cognitive therapy for our moral intuitions, one that stands as a viable remedy for the problem of faulty intuitions. My goal here is to explain why and how his moral epistemology addresses the problem of faulty intuitions. A more ambitious goal would be to argue that his [End Page 31] solution is the best solution to our epistemic predicament. Though in fact I think this is the case, I cannot argue for it here. The first section aims to highlight the problem contemporary moral psychology is discovering about our moral intuitions. The second section outlines and explains Aquinas's solution to this problem. I explain this in two stages. First, I show that Aquinas's solution to the problem of faulty intuitions is only partly the acquisition of the requisite virtues. Full rehabilitation of our moral intellects requires realizing the gifts of the Holy Spirit—in particular, understanding, knowledge, wisdom, and counsel. In the second stage I explain why the virtues and gifts are solutions to our problem. To understand why they are necessary we must understand how they operate on our moral intellects. Here I advert to recent work done on other areas of expertise with an eye toward finding family resemblances between expertise in chess and other domains, and moral expertise. We can appreciate better why the virtues and gifts are necessary if we have some grasp of how they are supposed to work, and we can see this on analogy with other forms of expertise. I. The Problem The following nomenclature will provide the framework for the remainder of our discussion. The term "intuition" in philosophy has a rather diverse meaning. Following Elijah Chudnoff2 and William Tolhurst,3 I understand intuition to mean something like an intellectual seeming analogous to a perceptual seeming. To use Aquinas's terminology, an intuition is something like an apprehension according to which "the intellect merely has a likeness of a thing existing outside the soul, as a sense has a likeness when it receives the species of a sensible thing. But when the intellect begins to judge about the thing it has apprehended, then its judgment is something proper [End Page 32] to itself."4 An intuition may be taken as an apprehension of something qua kind or species. Intuitions are intentional states in that they are about the world—"world" understood to include the perceptual environment, mathematical world, or the world of values and goods. They have a mind-to-world direction of fit, which means that they are representations of how the world actually is, or appears to be. But intuitions, as understood here, are not beliefs. One can have an intuition—a seeming—that, for example, the Muller-Lyer Illusion indicates two lines of different lengths. Click for larger view View full resolution Upon being informed that this is an illusion and/or measuring the lines, any subsequent viewing of the Muller-Lyer lines will involve a seeming that they are different lengths, but one would not believe them to be different lengths...

Physical Physics – Getting students Active in Learning Materials Science

ABSTRACTPhysics forms the core of any Materials Science Programme at undergraduate level. Knowing the properties of materials is fundamental to developing and designing new materials and new applications for known materials.“Physical Physics” is a physics education approach which is an innovative and promising instruction model that integrates physical activity with mechanics and material properties. It aims to significantly enhance the learning experience and to illustrate how physics works, while allowing students to be active participants and take ownership of the learning process. It has been successfully piloted with undergraduate students studying mechanics on a Games Development Programme. It is a structured guided learning approach which provides a scaffold for learners to develop their problem solving skills.The objective of having applied physics on a programme is to introduce students to the mathematical world. Today students view the world through smart devices. By incorporating student recorded videos into the laboratory experience the student can visualise the mathematical world. Sitting in a classroom learning about material properties does not easily facilitate an understanding of mathematical equations as mapping to a physical reality. In order to get the students motivated and immersed in the real mathematical and physical world, an approach which makes them think about the cause and effect of actions is used. Incorporating physical action with physics enables students to assimilate knowledge and adopt an action problem solving approach to the physics concept. This is an integrated approach that requires synthesis of information from various sources in order to accomplish the task. As a transferable skill, this will ensure that the material scientists will be visionary in their approach to real life problems.

Gaspare Berti’s Legacy

Documentation regarding the practical mathematicians in the early modern age is as rare as it is precious. In fact, where it exists, it permits us to document the culture of mathematics at a time of strong interchanges between the ‘artisan epistemology’ and erudite scientific culture. This paper will present a complete edition of the post-mortem inventory of the Roman mathematician Gaspare Berti (1601–1643), which was discovered among the notary papers of the Roman Court of Auditor Camerae. This document is of great interest, both generally and in particular. On the one hand, it sheds light on a figure who has remained unknown for centuries, except for his pioneering work on the vacuum in the early 17th century. On the other hand, thanks to an exceptional wealth of details, through the inventory we are given a deeper look from within at the ‘trading zone’ between practical and theoretical mathematics in the particular context of Baroque Rome. This almost photographic documentation contextualizes the lively world of practical mathematics, allowing comparison with the ‘big narrative’ of its alleged decline after Galileo’s condemnation.

Analysis of real world problems in mathematics textbooks of early twentieth and twenty-first century Turkish education: political and social reflections

This paper aims to show that historical eras from the perspective of socio-political approaches can be linked to the context of mathematical real world problems in conformity with the eras they were used. Three mathematics high school textbooks were selected from the Hamidian era [1876–1909], Early Republican era [1923–28] and the Contemporary era [2002–]. A content analysis was carried out by taking the political events into consideration and linked them with the context of the problems. The problem contexts were examined under four general categories, military, gender, religion, and work force. Findings of the study indicated that mathematics could be related to socio-political aspects of history.

THE MAYER VIETORIS SEQUENCE CALCULATING DE RHAM COHOMOLOGY

De Rham cohomology it is very obvious that it relies heavily on both topology as well as analysis. We can say it creates a natural bridge between the two. To understand and be able to explain what exactly de Rham cohomology is to the world of mathematics we need to know de Rham groups. This is the reasons to calculate the de Rham cohomology of a manifold. This is usually quite difficult to do directly. We work with manifold. Manifold is a generalization of curves and surfaces to arbitrary dimension. A topological space M is called a manifold of dimension k if : · M is a topological Hausdorff space . · M has a countable topological base. · For all m∈M there is an open neighborhood U⊂M such that U is homeomorphic to an open subset V of ℝk. There are many different kinds of manifolds like topological manifolds, ℂk - manifolds, analytic manifolds, and complex manifolds, we concerned in smooth manifolds. A smooth manifold can described as a topological space that is locally like the Euclidian space of a dimension known. An important definion is homeomorphism. Let X, Y be topological spaces, and let f: X⟶Y e a bijection. If both f and the inverse function f−1: X⟶Y are continuous, then f is called a homeomorphism We introduce one of the useful tools for this calculating, the Mayer – Vietoris sequence. Another tool is the homotopy axiom. In this material I try to explain the Mayer – Vietoris sequence and give same examples. A short exact sequence of cochain complexes gives rise to a long exact sequence in cohomology, called the Mayer - Vietories sequence. Cohomology of the circle (S^1), cohomology of the spheres (S^2). Homeomorphism between vector spaces and an open cover of a manifold. we define de Rham cohomology and compute a few examples.

The Mathematics of Love

On the face of it, affairs of the human heart are far removed from the world of mathematics. On the one hand, the mathematicians amongst us might see the lack of precision in the former as reason enough to dismiss it from the proper domain of mathematical analysis. On the other hand, the romantics amongst us might see the cold, calculating nature of mathematics as an inappropriate tool for the study of love. The Mathematics of Love goes some way toward dispelling both these lines of thought and shows how mathematics can, indeed, shed light on love, sex, and romance. The book is not written for a mathematical audience; it’s written for a lay audience and presupposes very little by way of mathematical training. The few technical details presented are introduced with such care and patience that even those averse to mathematics should not be put off—such people may even warm to mathematics once some of what it can do is better appreciated. Indeed, this is a large part of Hannah Fry’s mission in this book: “my great hope is that a little bit of insight into the mathematics of love might just inspire you to have a little bit more love for mathematics” (p. xii). The fact that the book is not written for a mathematical audience does not mean, however, that there is no interest here for mathematicians. The book is engaging, humorous, and full of nice examples of applications of various branches of mathematics to human relationships. It would make a good supplementary text for teaching purposes at both high school and undergraduate level. I would imagine that one could build an interesting, if somewhat idiosyncratic, undergraduate course around this book, if the

Where is the bigger picture in the teaching and learning of mathematics?

This article presents an interpretive analysis of three different mathematics teaching cases to establish where the bigger picture should lie in the teaching and learning of mathematics. We use pre-existing data collected through pre-observation and post-observation interviews and passive classroom observation undertaken by the third author in two different Grade 11 classes taught by two different teachers at one high school. Another set of data was collected through participant observation of the second author’s Year 2 University class. We analyse the presence or absence of the bigger picture, especially, in the teachers’ questioning strategies and their approach to content, guided by Tall’s framework of three worlds of mathematics, namely the ‘conceptual-embodied’ world, the ‘proceptual-symbolic’ world and the ‘axiomatic-formal’ world. Within this broad framework we acknowledge Pirie and Kieren’s notion of folding back towards the attainment of an axiomatic-formal world. We argue that the teaching and learning of mathematics should remain anchored in the bigger picture and, in that way, mathematics is meaningful, accessible, expandable and transferable.

Exploring the convergence of sequences in the embodied world using GeoGebra

This study addresses the embodied approach of convergence of numerical sequences using the GeoGebra software. We discuss activities that were applied in regular calculus classes, as a part of a research which used a qualitative methodology and aimed to identify contributions of the development of activities based on the embodiment of concepts, with the use of software, to the understanding of the convergence of numerical sequences and to the transition of mathematical thinking from elementary to advanced. Such activities had an exploratory nature and were constructed based on the theoretical frameworks of Advanced Mathematical Thinking and the Three Worlds of Mathematics. For the data collection were used the recordings of the computer screens, audio recordings of the discussion between students, student records in the responses of the activities and field notes of the researchers. The data were analyzed considering relations between the embodiment of the concept of convergence and proceptualization processes and axiomatization and also considering the transition from elementary mathematical thinking to advanced. The results indicate that the activities enabled the formation of the mental image that the convergence of sequences occurs when the sequence terms are approaching a certain value, and thus propitiated the embodiment of the concept of convergence and established cognitive roots for formal definition of convergence by the limit and further theoretical development. They also indicate that relations between different representations were established, which contributed to abstraction of the concept of convergence of sequences and thus for the transition between elementary and advanced mathematical thinking.

Rings of small rank over a Dedekind domain and their ideals

In 2001, M. Bhargava stunned the mathematical world by extending Gauss's 200-year-old group law on integral binary quadratic forms, now familiar as the ideal class group of a quadratic ring, to yield group laws on a vast assortment of analogous objects. His method yields parametrizations of rings of degree up to 5 over the integers, as well as aspects of their ideal structure, and can be employed to yield statistical information about such rings and the associated number fields. In this paper, we extend a selection of Bhargava's most striking parametrizations to cases where the base ring is not Z but an arbitrary Dedekind domain R. We find that, once the ideal classes of R are properly included, we readily get bijections parametrizing quadratic, cubic, and quartic rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss composition for which Bhargava is famous. We expect that our results will shed light on the analytic distribution of extensions of degree up to 4 of a fixed number field and their ideal structure.

About Ennio De Giorgi

In this paper, after briefly presenting Andrea Parlangeli’s book Uno spirito puro—Ennio De Giorgi, I recall, through my memories as a student, the figure of Ennio De Giorgi in the 1960s, soon after his arrival in Pisa, showing the extraordinary spirit of collaboration and incitement to research he managed to bring about. Finally, I describe the birth of Gamma-convergence, one of the most celebrated theories De Giorgi created, which had a great diffusion in the mathematical world.

Pre-service teachers understanding of continuity of functions in the context of assessment items

Continuous and non-continuous functions as school topics are dealt with in grades 10 to 12 at South African high schools as prescribed by the Curriculum and Assessment Policy Statement. This article reports on the use of a combined framework for mathematical thinking that incorporated the ideas of the Action-Process-Object-Schema (APOS) and Three Worlds of Mathematics (TWM) theories developed in a previous article (Brijlall and Maharaj 2011). The present study is qualitative in that it reports on those pre-service teachers' mental constructions of the concept of continuity of single-valued functions, obtained from an analysis of their responses to examination questions. The four pre-service teachers in the study specialised in the teaching of mathematics for the Further Education and Training (FET) high school curriculum at an education faculty in a South African university. One of the rationales for assessment was to obtain informed feedback on the effect of instructional treatments. The pre-service teachers' responses in their final examination were analysed using a modified framework for reflective abstraction (MFRA). The current article is follow-up to a two-tiered concurrent approach which engaged pre-service teacher with an instructional design worksheet and collaborations in order to develop mathematical understandings of the concept of continuity. In the study it was found that the mathematics pre-service teachers worked in a symbolic world of the MFRA. There was no clear demarcation between the embodied and symbolic worlds of the MFRA. The pre-service teachers either used either one or the other, or moved flexibly between them when offering their explanations.

Out of Bounds

Writing reliable code means understanding bounds. Only a finite amount of memory is available for computation, only a finite amount of time exists to do it, and every object we store and modify must be finite. Resources are similarly bounded. Stacks are bounded, queues are bounded, file system capacity is bounded, and even numbers are bounded. This makes the world of computer science very different from the world of mathematics, but too few people take this into account when they write code.

신체적-정신적 장애를 극복하고 학문적 기여를 한 수학자들과 특수수학교육 환경

인류의 진보를 이끈 많은 수학자 중에는 장애를 극복하고 커다란 업적을 이룬 장애인 수학자들이 적지 않다. 그리고 이들의 수학자로서의 성공은 장애와 수학을 연결하는 좋은 모델이 될 수 있다. 따라서 본 논문에서는 국내 외에서 신체적 또는 정신적 장애를 극복하고 수학적 발전에 기여한 니콜라스 선더슨, 오일러, 루이스 캐롤, 솔로몬 레프셰츠, 루이스 앙투안, 가스통 줄리아, 레프 폰트랴긴, 아브라함 네메스, 존 내쉬, 버나드 모린, 아나톨리 뷔투쉬킨, 로렌스 바젯, 노베르토 살리나, 시어도어 카진스키, 리처드 보처즈, 디미트리 카네브스키, 황윤성, 엠마뉴엘 지록, 김인강, 재커리 배틀(한국이름: 이정남), 프라티쉬 다타 등과 같은 수학자들의 사례를 소개하고, 특수수학교육 환경에 대하여 논한다. There are lots of disabled mathematicians who overcame their disabilities and made great achievement to the world of mathematics. In this article, we introduce disabled mathematicians who overcome their disabilities and contributes to the development of mathematics: Nicholas Saunderson, Leonhard Euler, Lewis Carroll, Solomon Lefschetz, Louis Antoine, Gaston Maurice Julia, Lev Semenovich Pontryagin, Abraham Nemeth, John Nash, Bernard Morin, Anatoli G. Vitushkin, Lawrence W. Baggett, Norberto Salinas, Theodore John Kaczynski, Richard E. Borcherds, Dimitri Kanevsky, Hwang Yun-seong, Emmanuel Giroux, Kim In-kang, Zachary J. Battles, and Pratish Datta. As well, we classify mathematics education environments and the role education played in helping these mathematicians overcome their disabilities and other obstacles. Then, we discuss educational environmental changes in the 21st century for special mathematics education.

In Memoriam: Slavik Jablan 1952–2015

After a long and brave battle with a serious illness, our dear friend and colleague Slavik Jablan passed away on 26 February 2015. [...]

Bernulf Kanitscheider.Natur und Zahl: Die Mathematisierbarkeit der Welt[Nature and Number: The Mathematizability of the World]. Berlin: Springer Verlag, 2013. ISBN: 978-3-642-37707-5 (hbk); 978-3-642-37708-2 (e-book). Pp. vii + 385

The title of Bernulf Kanitscheider's Natur und Zahl has an even better ring in English: Nature and Number. The book is a wide-ranging discussion of the mathematical nature of the physical world. It takes the form of a sort of perambulation through history, beginning with the ancient Pythagorean philosophers, with stops along the way to note analogies or parallels in contemporary science, especially cosmology and astrophysics, Kanitscheider's area of specialization, in order to illustrate the viewpoints under discussion. Unfortunately, this procedure results in a rambling and unsystematic exploration of the issues and eventually breaks down in the later chapters, where Kanitscheider tends to desert the historical approach in favor of a more thematic discussion. The book presents itself as an exploration of the reason for the astonishing applicability (Anwendbarkeit) of mathematics in physics, a question neglected, according to Kanitscheider, in Germanic philosophy in contrast to anglophone philosophy (pp. 12–13). In fact, however, Kanitscheider treats this issue only tangentially in the book. The actual question with which he struggles is the compatibility of mathematics with naturalism. Kanitscheider is a committed naturalist, and he is acutely aware of the problems for his metaphysical worldview posed by mathematical truth (which he wants to affirm) and the underlying ontology of mathematical platonism (which he wants to avoid). Thus, the book's center of gravity is its longest chapter ‘Naturalism in the world of mathematics’.1

The Age of STEM: educational policy and practice across the world in science, technology, engineering and mathematics

The Age of STEM: educational policy and practice across the world in science, technology, engineering and mathematics

Worldly beings becoming human beings: Differentiations and articulations within our different ways of being

These excellent papers by Pablo Rojas and Mariagrazia Grantella, both in their own very different ways, begin to bring into view aspects of our social psychological functioning that Descartes’ mechanical–mathematical world view has occluded, i.e., made rationally invisible to us. They both emphasize the degree to which we have our being within already flowing, intra-mingling, strands of both physical and social activities that influence us more than we can influence them. Rojas’ interest is in our coming to feel so “at home,” so to speak, in moving around on a piano keyboard, that we can come to relate to it as we relate to our own vocal tracts in singing—skills that we can develop (but not easily) by rigorous training. Grantella too, in turning to Vico’s notion that the early people’s “were almost entirely body, and practically not at all reflection,” makes a similar point: we need to replace our rationalistic interest in abstract entities with an interest in origins and processes, and to focus on our human ways of being and of living our lives. My only point of criticism of these two excellent paper is that I think that they still start too late in the day.