Teaching Undergraduate Mathematics on the Internet

This is Part 2 of a two-part study of how APOS theory may be used to provide cognitive explanations of how students and mathematicians might think about the concept of infinity. We discuss infinite processes, describe how the mental mechanisms of interiorization and encapsulation can be used to conceive of an infinite process as a completed totality, explain the relationship between infinite processes and the objects that may result from them, and apply our analyses to certain mathematical issues related to infinity.     

现代数学基础理论中的一个不可解危机

从新构建的基础理论学的角度,讨论同时存在于现有集合论中两种性质上截然不同的“无穷集合”概念——潜无穷式的与实无穷式的无穷集合．指出现有数学中通常所使用的“无穷”概念极端的模糊不清,一方面混淆了原来的“有穷”概念,另一方面使人们无法理解与解析数学中与“无穷”相关的许多数量形式的性质,因此导致许多错误的数学行为．数学基础理论中这个致命缺陷是导致第二次与第三次数学危机的最根本原因．因此,只有对传统的“无穷”概念进行修正,构建新的“有穷-无穷”理论体系及相关的数量体系才是彻底解决第二次与第三次数学危机的惟一途径．     

APOS理论视角下无穷概念认知分析

无穷概念一直是学生学习的一个困难.APOS理论是一种建构主义的数学学习理论,包含操作、过程、对象和图式4个阶段.内化和凝聚是APOS理论的两个重要的心理机制.教师对无理数的认识要从历史视角下关注数的发展,从发生教学法角度上进行无理数教学设计,让学生从概念发展的过程阶段转化到对象阶段.     

Bolzano‘s Approach to the Paradoxes of Infinity: Implications for Teaching

In this paper we analyze excerpts of Paradoxes of the Infinite, the posthumous work of Bernard Bolzano (1781–1848), in order to show that Georg Cantor‘s (1845–1918) approach to the problem of defining actual mathematical infinity is not the most natural. In fact, Bolzano‘s approach to the paradoxes of infinity is more intuitive, while remaining internally coherent. Bolzano‘s approach, however, had limitations. We discuss implications for teaching, which include a better understanding of the responses of students to situations involving actual mathematical infinity, for it is possible to draw a kind of parallel between these responses and Bolzano‘s reasoning.     

Tacit Models and Infinity

The paper analyses several examples of tacit influences exerted by mental models on the interpretation of various mathematical concepts in the domain of actual infinity. The influences of the respective tacit models, being generally uncontrolled consciously, may lead to erroneous interpretations, to contradictions and paradoxes. The paper deals especially with the unconscious effect of the figural-pictorial models of statements related to the infinite sets of geometrical points (on a segment, a square, or a cube) related to the concepts of function and derivative and to the spatial interpretation of time and motion in Zeno's paradoxes. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interactions between defining,explaining and classifying: the case of increasing and decreasing sequences

This paper describes a study in which we investigated relationships between defining mathematical concepts — increasing and decreasing infinite sequences — explaining their meanings and classifying consistently with formal definitions. We explored the effect of defining, explaining or studying a definition on subsequent classification, and the effect of classifying on subsequent explaining and defining. We report that (1) student-generated definitions and explanations were highly variable in content and quality; (2) explicitly considering the meaning of the concept facilitated subsequent classification, and giving a personal definition or explanation had a greater effect than studying a given definition; (3) classifying before defining or explaining resulted in significantly poorer definitions and explanations. We discuss the implications of these results for the teaching of abstract pure mathematics, relating our discussion to existing work on the concept image/concept definition distinction and on working with examples.     

集合论悖论与辩证逻辑

从辩证逻辑视角出发，对无穷集合论的基本思想、认识论背景、悖论表现形式、矛盾消除方案及悖论实质作了系统的评述。指出无穷集合既是最大的集合(即实无穷集合成完全的集合)，又不是最大的集合，而是潜无穷集合或不完全的集合。其辩证本质即在于此。     

The lure of internationalization: paradoxical discourses of transnational student mobility,linguistic diversity and cross-cultural exchange

This paper scrutinizes a set of paradoxes arising from a mismatch between contemporary discourses that praise and promote mobility in and internationalization of higher education, and the everyday effects of mobility and internationalization on university teaching and learning practice. We begin with a general characterization of the discourse of mobility and internationalization in a European context and then turn to Denmark as a specific case in which we unfold and discuss three paradoxes in turn: internationalization and linguistic pluralism, internationalization and intercultural understanding and, finally, internationalization and competitiveness. We then link our deconstruction of the three paradoxes to a critique of the concept of “parallel language policy,” widely promoted in the Nordic context, and show how it potentially undermines the ideals of internationalization.     

Young Peoples' Ideas of Infinity

This paper considers views of infinity of young people prior to instruction in the methods mathematicians use in dealing with infinity. To avoid overlap with other papers in this special issue on infinity, reference to limit notions and Cantorian views of infinity are kept to the minimum. A partially historical account of studies examining young peoples' ideas of infinity is presented. Methodological problems in accessing such ideas is a sub-theme of this paper. The four main sections deal with: potential pitfalls for research in this area and the work of Piaget; issues concerning the contradictory nature of infinity and infinity as a process and as a object; infinite numbers; contexts and tasks. This revised version was published online in July 2006 with corrections to the Cover Date.     

数学概念的理解问题

对数学概念的理解问题是数学学习理论中的重要理论问题．就刻划数学概念理解的重要概念——数学概念的心理表征和数学概念理解的认知基础进行分析，从而对数学概念的理解问题有更深入的认识。     

Zooming in on infinitesimal 1–.9.. in a post-triumvirate era

The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis “...” in the real formula \(\hbox{.999\ldots = 1}\). Infinitesimal-enriched number systems accommodate quantities in the half-open interval [0,1) whose extended decimal expansion starts with an unlimited number of repeated digits 9. Do such quantities pose a challenge to the unital evaluation of the symbol “.999...”? We present some non-standard thoughts on the ambiguity of the ellipsis in the context of the cognitive concept of generic limit of B. Cornu and D. Tall. We analyze the vigorous debates among mathematicians concerning the idea of infinitesimals.     

从“工作流”到“学习流”：E-learning系统设计的新视角

目前大多数E-learning支持系统基于内容对象设计,由此造成的学习者控制、网络迷失和认知超载等问题已成为制约E—learning发展的瓶颈。本文借鉴“工作流”的理念,结合学习活动的流程化特性,提出了“学习流”的概念,并建立了学习流的数学模型——ATC模型,在此基础上分析了学习流的构成要素和控制策略设计,试图为E—learning系统的设计和开发提供新的思路和视角。最后简要介绍了本研究开发的基于“学习流”理念的管理系统功能架构,该系统已于2006年成功地应用于一项由教育部主持的面向全国的教师培训项目。研究结果表明,学习者对于“学习流”管理系统支持下的培训有着很高的评价和认同度;采用基于“学习流”的方式组织和管理学习活动,可以大大提高系统的易用性和学习活动的组织效率,避免学习者网络迷航,从而为实施有效教学、促进有意义的学习创造了条件。     

The Boundary Between Finite and Infinite States Through the Concept of Limits of Sequences

In this article, attempts were made to examine students’ thinking about the concepts of infinity and their ideas about transiting from finite to infinite states through the concept of limits of sequences. The participants included 78 senior high-school students ranging in age between 17 and 19 years old. The data were collected through a questionnaire and an interview with all of the subjects. The findings showed that the students’ understanding of infinity is related to finite situations and many students consider infinite processes as a generalized form of finite processes. In the present study, the most common mistakes committed by students were related to consideration of infinity as a number and application of known finite results to infinite states.     

关于极限论对Berkeley悖论之解释方法的分析与研究

该文是文献[5]的续篇,由于文献[1]之2．5论证指出：当代极限论没有给Berkeley悖论留下任何有关0/0一类悖论生成余地或隙缝．（文献[1]P26）,文献[5]在确认潜无限（poi）与实无限（aci）在ZFC框架内是无中介矛盾对立面（p,]p）前提下,论证结论是当代极限论所留给Berkeleyn悖论有关0/0一类悖论的,远不止一个隙缝,而是一个大窟窿．但文献[1]之2．5与2．6还有诸多针U对文献[2]之6．7．3的质疑,本文旨在质疑文献[1]对文献[2]的每一条质疑,结论是文献[1]针对文献[2]之任何一条相关的质疑都是没有根据的．并在逻辑数学层面上对文献[1]中所谓“双相无限”概念略作评论．     

Nonlinearity, conservation law and shocks

We present in two parts, a mathematical theory of conservation laws using the language of physics. In Part I we explain the concept of a special type of nonlinearity which appears in an important class of evolutionary processes governed by hyperbolic partial differential equations. For simplicity, we develop the theory using a simple model equation. We show that it is possible to extend the concept of solutions with discontinuities with the help of a conservation form of the equation.     

Strategy Selection and Metacognition

The concept of metacognition is one of the most important developments in the contemporary study of cognition, especially with regard to problem solving and the transfer of cognitive skills. Its study has followed the experimental paradigm with researchers looking for universal principles; metacognition has only a small role to play in looking for explanations of individual differences in cognition. In this paper we are attempting a theoretical analysis of a number of interrelated issues with regard to their importance for metacognition in the light of some current empirical work. It considers mainly the role of these processes in strategy selection, especially in light of the impasse‐based theories of problem solving, and explores the relationship of individual differences to metacognition.     

Preservice education of math teachers using paradoxes

This is a report on a naturalistic study of the role mathematical paradoxes can play in the preservice education of high school mathematics teachers. The study examined the potential of paradoxes as a vehicle for: (a) sharpening student-teachers' mathematical concepts; (b) raising their pedagogical awareness of the constructive role of fallacious reasoning in the development of mathematical knowledge. Course material development and data collection procedures are described. Results obtained in parts of the study through written responses and class-videotapes are analyzed and discussed. The findings indicate that the model of dealing with paradoxes as applied in this study has relevance to such aspects of mathematics education as cognitive conflicts, motivation, misconceptions and constructive learning.This is a second report on this study. The first report (Movshovitz-Hadar, 1988) was a brief one, and focused on the problem, the procedures and findings in a general way. The present report is focused in details on one task. The authors intend to present an across tasks analysis in a third report to be presented at AERA 90.     

Ascertaining teachers’ perceptions of working with adolescents diagnosed with attention‐deficit/hyperactivity disorder

There are few studies that have directly examined teachers’ perceptions of children with ADHD, particularly children in the adolescent age range. The purpose of this study was to examine the perceptions of general‐education ninth through twelfth grade teachers regarding working with adolescent students with ADHD. Sorted responses from a sample of 100 general‐education high school teachers were analysed and then represented visually using a concept‐mapping technique. The final concept map suggests that teachers perceive adolescents with ADHD along two dichotomies of thematic clusters on an orthogonal axis. Clusters at the top area of the concept map suggest perceptions related to high confidence and willingness, while clusters at the bottom suggest perceptions related to uneasiness and frustration. Clusters along the left side of the concept map suggest perceptions related to behaviour issues, while clusters along the right side suggest perceptions related to classroom teaching issues. Central to the vertical and horizontal dichotomies is a solitary central dimension, which highlights training as an important component of teachers’ perceptions of adolescents with ADHD.     

Generating multiple solutions for a problem: A comparison of the responses of U.S. and Japanese students

A task involving simple mathematics, yet complex in its call for the generation of multiple solution methods, was administered to about 150 U.S. students, most of whom were in fourth grade. Written responses were examined for correctness, evidence of strategy use and mode of explanation. Results for the U.S. sample were also compared to those obtained from about 200 Japanese fourth-grade students. Students in both countries (a) produced multiple solutions and explanations of their solutions, (b) exhibited almost identical patterns and frequency of strategy use across response occasions, and (c) used the same kinds of explanations, with a majority of the responses involving solution explanations that combined both visual and verbal/symbolic features. Nevertheless, Japanese students tended to produce explanations involving more sophisticated mathematical ideas (multiplication rather than addition) and formalisms (mathematical expressions rather than verbal explanations) than did U.S. students.     

统一无穷理论是消解悖论的典型案例

从数理辩证逻辑的观点看“统一无穷理论”,可以消解涉及无穷小的微积分悖论和涉及无穷大的集合论悖论,并能引出与有穷实数相互谐调一致的,涉及大小实无穷常数与最大最小潜无穷常数间的加、减、乘、除、乘方、开方的非标准数计算,由此讨论了“统一无穷理论”存在的合理性.