What we know and what we do not know about Turán numbers

The numbers which are traditionally named in honor of Paul Turán were introduced by him as a generalization of a problem he solved in 1941. The general problem of Turán having anextremely simple formulation but beingextremely hard to solve, has become one of the most fascinatingextremal problems in combinatorics. We describe the present situation and list conjectures which are not so hopeless.     

Sur la conception des objets et des méthodes mathématiques dans les textes philosophiques de d'Alembert

This paper is devoted to the conception of mathematical objects and methods according to d'Alembert. We first recall his vision of the place of mathematics in the knowledge of nature, then the internal hierarchy of the various fields of this science, based on their degree of abstraction from sensations (41 and 2). Then we come to the ideas of definitions, primitive ideas, simple ideas, and their generation as well as their generalization (43 and 4). Then, having looked at what he means by quantities, numbers, quantities, as well as his conception of the objects and rules of algebra as abstract ideas by generalization (45), we approach the question of the reality of mathematical objects with the example of the irrational (46). The following paragraphs of the text are devoted to the difficulties encountered in various fields and the way d'Alembert tries to solve them: algebra and negative quantities (47); principles of geometry (48); the notion of limit as the basis of infinitesimal calculus (49). His reflections, even if unfinished, were not without posterity (410).     

A locus for transnational exchanges: European mathematical journals for students and teachers, 1860s–1914

While there were a few mathematical journals aimed at teachers and students as early as the 1840s, it was only in the late 19th century that they became more numerous in Europe. This article is based on the analysis of a corpus of European mathematical journals published between the 1860s and World War I, selected in the first place because they were aimed at high school teachers and high school or/and first two years university students, which are often referred to as “intermediate journals”. All these journals had focused on the teaching of mathematics and, as such, they were shaped by the educational context of the country in which they were published. However, leafing through theses journals, one is struck by the fact that the mathematics they published was in fact highly commensurable, and can see that they were the locus of transnational exchanges on mathematical knowledge. This article shows that several aspects of “internationalisation” were in fact at stake in mathematical journals for students: making knowledge from elsewhere available and of publicizing to the whole world the mathematics produced in one country; making people from different countries collaborate. Finally, it focuses on the effects of transnational exchanges between journals for teachers and students: what was the mathematical knowledge that was circulated through them, and in what respect was it different from that published in other mathematical journals?     

Euler and the structure of mathematics

When Euler began his mathematical career, analysis was not an autonomous and self-founding field of mathematics but only a method for solving geometrical problems. During Euler's activity, analysis emerged as a new field of mathematics whose methods differed from those of geometry. Moreover, Euler changed the architecture of this science and regarded analysis as lying at the heart of mathematics, while geometry and mechanics were considered as a sort of applied analysis. The new structure of mathematical sciences was successful, but this success soon showed the inadequacy of the Eulerian foundations of analysis.     

The reality of Mathematics

The power of mathematics is discussed as a way of expressing reasoning, aesthetics and insight in symbolic non-verbal communication. The human culture of discovering mathematical ways of thinking in the enterprise of exploring the understanding of the nature and the evolution of our world through hypotheses, theories and experimental affirmation of the scientific notion of algorithmic and non-algorithmic ‘computation’, is examined and commended upon.     

关于模糊数贴近度问题的研究

利用模糊集和区间数的贴近度理论,讨论了模糊数的贴近度问题.通过区间数与模糊数之间的关系,根据区间数贴近度的一般表示形式,给出了构造一些模糊数贴近度的具体计算公式的方法;并通过实例说明了所得到的贴近度公式的有效性和实用性,解决了常用贴近度公式所不能解决的问题.     

Developing the roots of modelling conceptions: ‘mathematical modelling is the life of the world’

A study conducted with 25 Year 6 primary school students investigated the potential for a short classroom intervention to begin the development of a Modelling conception of mathematics on the way to developing a sense of mathematics as a way of thinking about life. The study documents the developmental roots of the cognitive activity, actions and conceptions of both modelling and mathematics that these beginners to modelling displayed. Understanding the conceptions of mathematics that students might hold or be developing and how these can be influenced in early schooling are essential ingredients in any plans for introducing modelling seriously into primary school classrooms. The majority of the students (22/25) were identified as displaying a developing conception of modelling as a way of problem handling. The three other students displayed the developmental roots of a way of understanding the world conception of modelling. These three students also displayed a Modelling conception of mathematics with one showing indications of developing towards a Life conception of mathematics.     

浅析对数学史教育的认识与实施

数学史教育在大学数学教育中具有重要的意义.有助于学生理解数学知识的本质,培养学生的思维能力.有必要开设数学史课程或在专业课教学中渗透数学史知识;有适合学生的专门数学史教材;进行数学史教育要联系学生的数学知识,要深入浅出;要注意数学精神的宣传.     

What we can learn from analyzing the teacher’s role in collective argumentation

In this paper, I use analyses of collective argumentation in a variety of classroom settings, from elementary school to a university-level differential equations class to illustrate various roles the teacher plays. These include initiating the negotiation of classroom norms that foster argumentation as the core of students’ mathematical activity, providing support for students as they interact with each other to develop arguments, and supplying argumentative supports (data, warrants, and backing) that are either omitted or left implicit. We gain two important insights from these analyses. First, an emphasis on argumentation can be used productively to provide openings in mathematical discussions for new mathematical concepts and tools to emerge. Second, the analyses demonstrate that teachers need to have both an in-depth understanding of students’ mathematical conceptual development and a sophisticated understanding of the mathematical concepts that underlie the instructional activities being used.     

The origins of mathematics education research in the UK: a tribute to Brian Griffiths

The meeting of the British Society for Research into Learning Mathematics held at the University of Southampton on 21st June 2008 was dedicated to the memory of Brian Griffiths, who had died earlier that month. At the meeting it was suggested that I write a paper tracing the development of research in mathematics education in the UK up to the writing of Mathematics: Society and Curricula, a book that Brian and I co-authored, published in 1974. This paper is dedicated to Brian's memory: I hope that it will be considered a fitting tribute to a great friend and colleague, and to an outstanding mathematician and mathematics educator.     

Small-group searches for mathematical proofs and individual reconstructions of mathematical concepts

This is a study of mathematics students working in small groups. Our research methodology allows us to examine how individual ideas develop in a social context. The research perspective used in this study is based on a co-constructive view of learning. Groups of three or four undergraduate mathematics majors, with prior experience writing mathematical proofs together, were asked to prove three statements. Computer software, such as Geometers Sketchpad, was available. Group work sessions were videotaped. Later, individuals viewed segments of the group video and were asked to reflect on group activities. Students in some groups did not share a common conception of proof, which seemed to hamper their collaboration. We observed interactions that fit with the co-constructive theory, with bidirectional interactions that shaped both group and individual conceptions of the tasks. These changes in understanding may result from parallel and successive internalization and externalization of ideas by individuals in a social context.     

The valuation of life contingencies: A symmetrical triangular fuzzy approximation

This paper extends the framework for the valuation of life insurance policies and annuities by Andrés-Sánchez and González-Vila (2012, 2014) in two ways. First we allow various uncertain magnitudes to be estimated by means of fuzzy numbers. This applies not only to interest rates but also to the amounts to be paid out by the insurance company. Second, the use of symmetrical triangular fuzzy numbers allows us to obtain expressions for the pricing of life contingencies and their variability that are closely linked to standard financial and actuarial mathematics. Moreover, they are relatively straightforward to compute and understand from a standard actuarial point of view.     

小议数学发展的哲学问题及其在数学与计算机科学关系中的表现

本文从哲学角度讨论数学与实践的联系 ,并以数学与计算机科学的关系为例说明这种联系 .