Mathematical Concepts and Proofs from Nicole Oresme

This paper examines the mathematical work of the French bishop, Nicole Oresme (c. 1323–1382), and his contributions towards the development of the concept of graphing functions and approaches to investigating infinite series. The historical importance and pedagogical value of his work will be considered in the context of an undergraduate course on the history of calculus.     

History of the Infinitely Small and the Infinitely Large in Calculus

The infinitely small and the infinitely large are essential in calculus. They have appeared throughout its history in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels – as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the 17^th through the 20^th centuries. We will also present `didactic observations' at relevant places in the historical account. This revised version was published online in July 2006 with corrections to the Cover Date.     

数学分析的教改札记

以信息社会为特征的新世纪开始便提出扫“数学盲”的历史任务,脱盲的标志则是要懂得微积分基础,微分与积分是数学分析的主体结构.21世纪的大学数学教育应该搞好数学分析课程改革与建设,开展教学研究:(1)数学分析教改与教材建设必须融于大学与中小学数学教育改革的整体之中.(2)注意问题之间的密切相关性与系统性,从唯有源头活水来出发,挖掘问题的内在联系,培养学生掌握看问题的角度.(3)注意各门学科之间的交叉关系,将高等代数、实变函数、泛函分析、点集拓扑等相关内容与数学分析进行整合.     

Using the onto-semiotic approach to identify and analyze mathematical meaning when transiting between different coordinate systems in a multivariate context

The main objective of this paper is to apply the onto-semiotic approach to analyze the mathematical concept of different coordinate systems, as well as some situations and university students’ actions related to these coordinate systems. The identification of objects that emerge from the mathematical activity and a first intent to describe an epistemic network that relates to this activity were carried out. Multivariate calculus students’ responses to questions involving single and multivariate functions in polar, cylindrical, and spherical coordinates were used to classify semiotic functions that relate the different mathematical objects.     

流形概念的起源与发展

流形概念是20世纪数学的最具代表性的基本观念之一。文章分析与探讨流形概念的起源与发展。特别地，研究了流形概念从产生到其概念的精确的数学描述这一历史过程。     

Teaching functions in a graphic milieu: What forms of knowledge enable students to conjecture and prove?

In many countries, the first concepts of calculus (such as functions)are taught by looking at examples, noticing their properties and generalizing from them in some implicit ways. Students have no means to discuss the general truth of a statement, or to examine the validity of a theorem, relative to the mathematical field. This knowledge is nonetheless required by teachers at the university level. The question is, therefore, if it is possible to organize activities for beginning calculus students, which would nevertheless lead them to working on statements and validity of theorems. This paper presents a teaching approach related to the concept of function, which aimed at leading students working within a graphic milieu to producing, discussing and testing the validity of mathematical statements and theorems. The intention of the approach was to use the procedural aspect of the graphs to provide a favorable milieu for linking the intuitive and the formal knowledge(such as required at the university for establishing proofs). The approach was experimented with a group of students. After the experiment, the students indeed became able to think of functions as objects and to engage with questions of validity of mathematical statements.This revised version was published online in October 2005 with corrections to the Cover Date.     

计算机在微积分教学中的作用例谈

计算机应该在微积分教学改革中发挥作用。而且,我们更需要从实例的角度对它的作用进行探讨,以便举一反三。文章结合微积分的教学,从计算机的使用目的、方式与层次等三方面阐述了计算机在教学中的作用:构建基于计算机的实验探究环境,实现微积分知识的多重表示与联系,以达到理解微积分的目的。     

级数收敛的概念评注

从级数、函数列的收敛理论出发,建立数项级数和函数项级数的收敛理论,即数项级数的收敛归结为它的部分和数列收敛是数学分析(高等数学)教学中很重要的一个环节;本文就是从级数收敛的定义出发来分析和探讨级数收敛的概念的.     

The appropriation of ‘enlightenment’ in modern Korea and Japan: Competing ideas of the enlightenment and the loss of the individual subject

Abstract

In recent decades in Korea, many significant changes in political, social and cultural dimensions have been held by the citizen’s initiative, where the revitalization of citizenship and strong civic unity have played a role. Yet, in regard to the characteristic of Korean citizenship, it seems that the aspect of individual subject has not been fully matured or issued; that is, there is a dissymmetry between the strong civic unity and a weak individual subject. This paper attempts to explore a possible historical account of why this has been the case by examining the historical development of the concept of enlightenment in modern Korea and Japan. ‘Enlightenment’, as a modern concept in Korea, was imported via Japan in the period from the late nineteenth century to the early twentieth century as in many other new concepts such as ‘democracy’ or ‘nation’. However, by comparison to the Western idea of the Enlightenment, its modern concept, Korean or Japanese, developed a different meaning in each own context, while lacking its original meaning essential to the creation of the ‘modern individual subject’ as a ‘citizen’. Hence, in modern Korea and Japan, the word ‘enlightenment’ is regarded as a historical concept with no contemporary relevance.     

从历史寓言走向文化寓言——新世纪内蒙古草原电影创作的审美转型

20世纪90年代以塞夫、麦丽丝为代表的蒙古族导演用他们气势宏大的历史题材系列影片蜚声中国影坛。进入21世纪以来,更多的蒙古族导演则将镜头对准草原民族当下的生存状态和现实境遇,即由对草原民族的历史寓言讲述转变为对草原民族文化寓言的书写。而这种审美转型主要体现在三个方面,即：由对历史事件的恢弘展现转向了对当下现实状态的深刻描绘、由对英雄祖先的讴歌赞美转向了对普通牧民生活与命运的关怀、由对草原民族传统艺术及民俗风情的奇观展示转向了对民族性格与文化内涵的深刻揭示。     

A cognitive analysis of Cauchy’s conceptions of function,continuity, limit and infinitesimal,with implications for teaching the calculus

In this paper, we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It highlights the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern set-theoretic formulation of mathematical analysis. This offers a re-evaluation of the relationship between the natural geometry and algebra of elementary calculus that continues to be used in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of non-standard analysis using infinitesimal concepts. It suggests that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students, and to make an effort to understand the rationale of differing theoretical viewpoints.     

Encrypted objects and decryption processes: problem-solving with functions in a learning environment based on cryptography

This paper introduces an applied problem-solving task, set in the context of cryptography and embedded in a network of computer-based tools. This designed learning environment engaged students in a series of collaborative problem-solving activities intended to introduce the topic of functions through a set of linked representations. In a classroom-based study, students were asked to imagine themselves as cryptanalysts, and to collaborate with the other members of their small group on a series of increasingly difficult problem-solving tasks over several sessions. These tasks involved decrypting text messages that had been encrypted using polynomial functions as substitution ciphers. Drawing on the distinction between viewing functions as processes and as objects, the paper presents a detailed analysis of two groups’ developing fluency with regard to these tasks, and of the aspects of the function concept underlying their problem-solving approaches. Results of this study indicated that different levels of expertise with regard to the task environment reflected and required different aspects of functions, and thus represented distinct opportunities to engage those different aspects of the function concept.     

浅谈学生极限思维的培养

在微积分课程中，极限作为一个极为重要的基本概念，贯穿于该学科始终。微积分中其他一些重要概念，如导数、微分、积分、级数等都用极限来定义，理解和掌握极限对微积分课程的学习至关重要。因此，培养学生的极限思维，对于学生准确把握该课程的概念与体系，运用极限的思维方法解决相关问题，都有非常重要的作用。     

数学建模融于微积分教学的探索与实践

当前大学微积分的教学改革，应强调对数学原理和背景的介绍，强调微积分的数学建模方法和计算功能。据此提出了以“启发应用意识，提高应用能力”为宗旨的一整套教学改革方案，并进行了教学改革的实践。结果证明：这一改革方案的实施能卓有成效地提高学生学习积极性、提高学生数学知识应用能力和综合素质。     

Revealing educationally critical aspects of rate

Rate (of change) is an important but complicated mathematical concept describing a ratio comparing two different numeric, measurable quantities. Research referring to students’ difficulties with this concept spans more than 20 years. It suggests that problems experienced by some calculus students are likely a result of pre-existing limited or incorrect conceptions of rate. This study investigated 20 Australian Year 10 students’ understanding of rate as revealed by phenomenographic analysis of interviews. Eight conceptions of rate emerged, leading to the identification of four educationally critical aspects of the concept which address gaps in students’ thinking. In addition, the employment of phenomenography, to reveal conceptions of rate, is described in detail.     

取消农业税后乡镇政权体制改革问题研究述评

随着农业税的取消,乡镇政权建设正面临着一场历史性的巨大变革;乡镇政权体制改革已成为目前社会各界讨论的热点问题。社会各界对这一问题的研究成果,可从乡镇建制的去留、乡镇规模的确定、乡镇职能的转变、乡镇人员的精减及分流安置、乡镇政权体制改革的模式选择及相关改革措施的配套等六个方面加以述评。     

拉格朗日中值定理证明及其应用

拉格朗日中值定理是微分学突出的成果,在微积分中占有非常重要的地位,且它是微分学的基础定理之一,是沟通函数与导数之间的桥梁,在理论及其应用上都有极其重要的意义。通过对定理的再认识,对拉格朗日中值定理的应用做了一定研究,主要探讨了拉格朗日中值定理在求极限、证明不等式、证明函数单调性等方面的应用。     

浅谈幂级数的和函数的求法及其应用

幂级数的和函数的求法是高等数学中的重点、难点内容.本文运用几个初等函数的展开式及相关知识,对幂级数的和函数的求法进行了讨论,给出了求幂级数的和函数的一般步骤,并介绍了利用幂级数求常数项级数的和的步骤.     

浅议高等数学在经济定量分析中的应用

随着经济的发展,高等数学和经济的关系越来越密切,本文主要从微积分、无穷等比级数、数学归纳法三个方面阐述了高等数学在经济定量分析中的重要应用。