陈省身·几何原本·欧拉示性数——从三角形内角和定理、高斯-邦尼公式到阿蒂亚-辛格指标定理

列昂纳德.欧拉(Leonhard Euler,1707-1783)是18世纪数学的中心人物.欧拉示性数是大量几何课题的源泉和出发点.本文从述评陈省身(1911-2004)求学和事业发展的历程及其相关言论的新视角,论述了欧拉对19世纪和20世纪数学的深刻影响及其数学与物理相结合的思想.数学的统一性反映了数学的本质.正如2002年国际数学家大会名誉主席陈省身指出的,"我们甚至可以预见纯数学与应用数学的统一".它揭示了未来数学发展的一个新的时代.     

Matlab语言在求初等数论函数值的应用

Matlab语言是计算机数学语言,在高等应用数学的各分支中应用广泛．文章利用Matlab语言编写程序求初等数论中Euler函数和数论函数的值．     

Bridging History of the Concept of Function with Learning of Mathematics: Students’ Meta-Discursive Rules,Concept Formation and Historical Awareness

In this paper we present a matrix-organised implementation of an experimental course in the history of the concept of a function. The course was implemented in a Danish high school. One of the aims was to bridge history of mathematics with the teaching and learning of mathematics. The course was designed using the theoretical frameworks of a multiple perspective approach to history, Sfard’s theory of thinking as communicating, and theories from mathematics education about concept image, concept definition and concept formation. It will be explained how history and extracts of original sources by Euler from 1748 and Dirichlet from 1837 were used to (1) reveal students’ meta-discursive rules in mathematics and make them objects of students’ reflections, (2) support students’ learning of the concept of a function, and (3) develop students’ historical awareness. The results show that it is possible to diagnose (some) of students’ meta-discursive rules, that some of the students acted according to meta-discursive rules that coincide with Euler’s from the 1700s, and that reading a part of a text by Dirichlet from 1837 created obstacles for the students that can be referenced to differences in meta-discursive rules. The experiment revealed that many of the students have a concept image that was in accordance with Euler’s rather than with our modern concept definition and that they have process oriented thinking about functions. The students’ historical awareness was developed through the course with respect to actors’ influence on the formation of mathematical concepts and the notions of internal and external driving forces in the historical development of mathematics.     

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

Problem 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.     

数学教育与教育数学

数学教育的现代发展,以及各国进行数学课程改革浪潮的涌现,提出了教育数学的研究课题。它似可与纯粹数学、应用数学并列为第三个研究领域。与“新数学”、“大众数学”相比,教育数学将能均衡好数学教育的“教育方面”与“数学方面”的关系和促进数学教育改革的健康发展。进行教育数学的研究,可先从数学现实、数学难点、数学新点等方面着手,然后再进行理论体系建设。     

Teacher self-assessment: A mechanism for facilitating professional growth

Self-assessment is a powerful technique for improving achievement. In this article we outline a theory of teacher change that links self-assessment by teachers to their professional growth. This theory provides avenues for peers and change agents to influence teacher practice. We apply the theory to change in mathematics teaching and report an explanatory case study in which use of the self-assessment tool, in combination with other elements, contributed to change in the instructional practice of a grade 8 mathematics teacher. Provision of a self-assessment tool contributed to teacher growth by: (1) influencing the teacher's definition of excellence in teaching and increasing his ability to recognize mastery experiences; (2) helping the teacher select improvement goals by providing him with clear standards of teaching, opportunities to find gaps between desired and actual practices, and a menu of options for action; (3) facilitating communication with the teacher's peer; and (4) increasing the influence of external change agents on teacher practice. The study argues that providing a self-assessment tool is a constructive strategy for improving the effectiveness of in-service provided it is bundled with other professional growth strategies: peer coaching, observation by external change agents, and focused input on teaching strategies     

小波分析的魅力

小波分析是继Fourier分析之后纯粹数学和应用数学完美结合的又一光辉典范。本从理论、应用和发展三个方面论述了小波分析所表现出的迷人风采和无穷魅力。     

The ��M�� in SMEC: a short history of the mathematics education presence

In this paper I examine the history of the integration of mathematics education into the Science Education Centre, which had been established by physicist, John de Laeter, within the School of Science and Engineering at Curtin University in Perth, Australia. De Laeter’s vision for science education was that teachers should have access to professional education that allowed them to extend their discipline and pedagogical knowledge using strategies that brought together theory and practice in ways that were meaningful for teachers. This model was expanded when mathematics education was also included, paving the way also for technology education. I present the history of this integration laying out the themes that are important for the continued educational effectiveness of the Science and Mathematics Education Centre (SMEC) and the role that mathematics education has played in this process. As the title suggests, this article focuses on the activities of the group of mathematics educators who have worked within the Science and Mathematics Education Centre of Curtin University since it was established 30 or so years ago and who have contributed to its reputation. The two streams operated then and now more-or-less independently in matters of student thesis topic choice but offered students opportunities for interaction that might not have been available if the “M” had not been incorporated into the Science Education Centre (SEC). This article’s focus is on the mathematics educators who contributed to the Centre’s success and reputation, highlighting the synergistic relationship between mathematics and science that helped to make SMEC a leading center for mathematics and science education.     

计算数学在高职计算机专业教学中的作用

计算数学从一种纯数学的计算方法研究渐渐形成了当今的科学与工程计算、计算机数值实验与模拟的方法、理论研究的一个复杂的系统工程。在高职计算机专业教学中要渗透计算数学的常用思想与方法,包括递推法、迭代法、切线逼近法、累加法等。     

Developmental Relations Among Motor and Cognitive Processes and Mathematics Skills

This study explored transactional associations among visuomotor integration, attention, fine motor coordination, and mathematics skills in a diverse sample of one hundred thirty‐five 5‐year‐olds (kindergarteners) and one hundred nineteen 6‐year‐olds (first graders) in the United States who were followed over the course of 2 school years. Associations were dynamic, with more reciprocal transactions occurring in kindergarten than in the later grades. Specifically, visuomotor integration and mathematics exhibited ongoing reciprocity in kindergarten and first grade, attention contributed to mathematics in kindergarten and first grade, mathematics contributed to attention across the kindergarten year only, and fine motor coordination contributed to mathematics indirectly, through visuomotor integration, across kindergarten and first grade. Implications of examining the hierarchical interrelations among processes underlying the development of children's mathematics skills are discussed.     

Kosambi’s vision of science A critical appreciation

Damodar Dharmananda Kosambi (1907–1966) was an eminent mathematician, scientist, innovative Marxist, a maverick scholar, and historian of contemporary India. Combining both versatility and a certain sense of virtuosity Kosambi made seminal contributions to a wide variety of disciplines spread across from pure and applied mathematics to archaeology and Indology. In the course of these writings he provided an incisive analysis of various developments in science and technology and novel perspectives onmany topics related to mathematics (especially in number theory and statistics), genetics and India’s past history, present concerns, and future growth.     

Euler数与集合的纯偶排列数

引入集合的纯偶排列数,给出了纯偶排列数的一些性质,用纯偶排列数得到了Euler数及正切数的一种简洁的表示形式,利用Akiyama-Tanigawa算法给出了Euler数表,并且给出Euler数几个同余式.     

拉格朗日中值定理的应用

总结了高等数学中拉格朗日中值定理五个方面的应用，并举例加以说明．     

木星-火星-航天器系统的零速度曲面和转移轨道的数值研究

针对木星-火星-航天器系统,根据欧拉-拉格朗日方程建立了航天器的动力学方程,并通过数值模拟,研究了系统的零速度曲面和Jacobi常数的几何结构的关系,设计了一条可用于多任务轨道的从火星至木星的转移轨道。     

Problems of general and technical education

During the twentieth century, the refounding of mathematics and its establishment as an intellectual entity of high interest [1], as well as the formulation of the theory of relativity and quantum theory, contributed significantly to the revision or discarding of many traditional scientific concepts, methods, and activities. The resulting upheaval has indicated that such a process was not a minor crisis in science, but rather a scientific revolution. The radical transformation of the relevant intellectual configurations, along with other social transformations, contributed to the acceptance, today, of a multi-approach mode — to understand the reality as well as to accept the continuous experimentation with new patterns of thought (paradigms).     

“七桥问题”及其对数学教育的启示

通过对欧拉《哥尼斯堡桥》一文的分析,揭示了欧拉解决“哥尼斯堡七桥问题”的过程中体现的重要数学思想、策略和方法,尤其是数学化思想。这些思想、方法和策略正是我国目前数学教育的薄弱环节,对数学创新型人才的培养和数学问题解决的教学具有重要的启示。     

调和级数案例教学

由一个数学问题引出了调和级数,并通过对调和级数的敛散性的研究,讲授了级数收敛的定义,子列收敛定理、正项级数收敛的判断准则、单调有界原理以及欧拉常数等高数知识,最后介绍了如何利用欧拉常数计算一些数列的极限值.     

MATHEMATICS AS AN IM/PURE KNOWLEDGE SYSTEM: SYMBIOSIS, (W)HOLISM AND SYNERGY IN MATHEMATICS EDUCATION

The problem of culturally decontextualised mathematics education faced by Nepali students, teachers and teacher educators has often been oriented by the view of the nature of mathematics as a body of pure knowledge, which gives rise to an exclusive emphasis on an ideology of singularity, epistemology of objectivism, language of universality and logic of certainty whilst developing curriculum, conceiving pedagogies and implementing assessment strategies in school mathematics education and mathematics teacher education programmes. With epistemic referents of dialectical logics and performative imagination, an alternative view of the nature of mathematics as an impure knowledge is discussed with its possible disempowering features, such as essentialism, hegemony and dualisms. Finally, an inclusive view of the nature of mathematics as im/pure knowledge system is articulated with the help of various forms of dialectics.     

模糊学视角中的大文学

模糊方法起源于数学,但是对于不确定性更强的文学是更有效的认识工具。大文学是个“正则模糊概念”,其核心部分是纯文学,其边缘部分是亚文学。如果动态地看问题,从文学向非文学领域伸展是“泛文学”,反向的是“潜文学”。“大文学理论”表现为对文学与对理论的双重超越。“前文学”时期的大文学是自在的综合智慧,实际上是大文化,“后文学”时期的大文学需要自觉的积极开发,向影视与网络、向市场经济的推进使文学边缘化了,但是使文学性中心化了。     

利用一些函数证明不等式

在数学学习过程中,不等式是十分重要的内容,而不等式的证明则是不等式知识的重要组成部分。而利用中值定理、泰勒公式、拉格朗日函数等函数证明不等式,可以拓宽证明不等式的不同思路,使得不等式有更好的应用,最终提高学生灵活运用数学知识的能力。