Apollonius圆的一些新性质及其应用

通过讨论Apollonius圆的一些新性质,并利用它巧妙地证明了本原Pythagoras数组的参数表示形式。     

What difference does it make? Children's views of the differences between some quadrilaterals

This paper describes an attempt to explore aspects of the language used in children's written work in a secondary mathematics classroom as a means of assessing their mathematical understanding. It focuses on children's conceptualisations of polygons and their attempts to differentiate between them. It also examines the use of database and concordancing software in deepening our understanding of a potentially major source of students' (mis) conceptions: the published materials they engage with. It describes their usefulness in providing a systematic and efficient method of analysis of materials to explore patterns within them. The method is also used to identify possible sources of students' misconceptions about the properties of polygons based on their likely exposure to potentially misleading examples in published materials. This revised version was published online in July 2006 with corrections to the Cover Date.     

Moebius变换的一个新特征

H．Haruki和T．M．Rassias，利用Apollonius四边形，给出了一个在复平面上亚纯且在复平面的一个非空区域内单叶解析的函数是Moebius变换的充要条件。我们采用了一种纯几何的证明方法,说明的这些解析性限制是不必要的。     

Providing a Foundation for Deductive Reasoning: Students' Interpretations when Using Dynamic Geometry Software and Their Evolving Mathematical Explanations

A key issue for mathematics education is howchildren can be supported in shifting from `because it looks right' or`because it works in these cases' to convincing arguments which work ingeneral. In geometry, forms of software usually known as dynamicgeometry environments may be useful as they can enable students tointeract with geometrical theory. Yet the meanings that students gain ofdeductive reasoning through experience with such software is likely to beshaped, not only by the tasks they tackle and their interactions with theirteacher and with other students, but also by features of the softwareenvironment. In order to try to illuminate this latter phenomenon, and todetermine the longer-term influence of using such software, this paperreports on data from a longitudinal study of 12-year-old students'interpretations of geometrical objects and relationships when using dynamicgeometry software. The focus of the paper is the progressivemathematisation of the student's sense of the software, examining theirinterpretations and using the explanations that students give of thegeometrical properties of various quadrilaterals that they construct as oneindicator of this. The research suggests that the students' explanations canevolve from imprecise, `everyday' expressions, through reasoning that isovertly mediated by the software environment, to mathematicalexplanations of the geometric situation that transcend the particular toolbeing used. This latter stage, it is suggested, should help to provide afoundation on which to build further notions of deductive reasoning inmathematics.This revised version was published online in September 2005 with corrections to the Cover Date.