Generalization of patterns: the tension between algebraic thinking and algebraic notation

This study explores the attempts of a group of preservice elementary school teachers to generalize a repeating visual number pattern. We discuss students' emergent algebraic thinking and the variety of ways in which they generalize and symbolize their generalizations. Our results indicate that students' ability to express generality verbally was not accompanied by, and did not depend on, algebraic notation. However, participants often perceived their complete and accurate solutions that did not involve algebraic symbolism as inadequate.This revised version was published online in September 2005 with corrections to the Cover Date.     

Capturing congruence with a turtle

This note compares turtle geometry and Euclidean geometry with respect to their treatment of similarity and difference of plane figures. It is observed that while the Euclidean notion of congruence faithfully captures a common perception of sameness, the turtle expression of this idea is too weak. To deal with this insufficiency we add a new turtle operation, FLIP, which turns the turtle upside down. This brings the turtle's power to express invariance of shape up to Euclid's.The problem and its solution are viewed briefly from the perspectives of mathematics, computer science and education. The mathematical discussion compares the turtle group and the Euclidean group. The computational discussion focuses on the issue of expressive power of a language and how it may be enhanced. The educational discussion suggests a classroom implementation of the above ideas.     

From visual reasoning to logical necessity through argumentative design

Our main goal in this study is to exemplify that a meticulous design can lead pre-service teachers to engage in productive unguided peer argumentation. By productivity, we mean here a shift from reasoning based on intuitions to reasoning moved by logical necessity. As a subsidiary goal, we aimed at identifying the kinds of reasoning processes (visual, inquiry-based, and deductive) pre-service teacher's students adopt, and how these reasoning processes are interwoven in peer-unguided argumentation. We report on a case study in which one dyad participating in a pre-service teachers program solved a mathematical task. We relied on three principles to design an activity: (a) creating a situation of conflict, (b) creating a collaborative situation, and (c) providing a device for checking hypotheses/conjectures. We show how the design afforded productive argumentation. We show that the design of the task entailed argumentation which first relied on intuition, then intertwined the activities of conjecturing and checking conjectures by means of various hypotheses-testing devices (measurement, manipulations, and dynamic change of figures with Dynamic Geometry software), leading to a conflict between conjectures and the outcome of the manipulation of DG software. Peer argumentation then shifted to abductive and deductive considerations towards the solution of the mathematical task. These beneficial outcomes resulted from collaborative rather than adversarial interactions as the students tried to accommodate their divergent views through the co-elaboration of new explanations.     

在中国"入世"背景下看中西文化趋同与差异

近年来,中国与西方国家在文化、贸易等领域的交往日益增多.在此过程中,中西方各自不同的价值观念、民族心理、语言习惯、民俗风情、生活方式等相互渗透、融合,同时又不可避免地发生碰撞与冲突,从而出现了中西文化诸多的趋同与差异现象,本文在中国"入世"背景下对这些现象进行了深入的分析.     

The role of contradiction and uncertainty in promoting the need to prove in Dynamic Geometry environments

In many geometrical problems, students can feel that the universalityof a conjectured attribute of a figure is validated by their action in adynamic geometry environment. In contrast, students generally do not feelthat deductive explanations strengthen their conviction that a geometricalfigure has a given attribute. In order to cope with students' convictionbased on empirical experience only and to create a need for deductiveexplanations, we developed a collection of innovative activities intended tocause surprise and uncertainty. In this paper we describe two activities, thatled students to contradictions between conjectures and findings. We analyzethe conjectures, working methods, and explanations given by the studentswhen faced with the contradictions that arose.     

"自下而上"式大学生价值观教育方法研究

15岁到25岁是人价值观探索、形成和发展的时期,一旦形成稳定的价值观就很难再改变.就整个学校教育而言,高校是青年学生价值观教育的关键时期,也是最后的机会.如今高校价值观教育主要采用老师为主导的至上而下的教学模式,这种方法最大的缺陷是抛开了学生自身的成长经历和学生面临的现实问题.本研究通过访谈和问卷调查了当前大学生普遍关注的156个问题以及在不同价值判断下所反映的观点和态度.本文以访谈案例的真实生活经验为模板,以价值观调研结果为参考,编写了两难故事.以讨论课形式探索两难故事的价值困境,提出了自下而上的价值观教育模式.     

近红外光谱分析快速检测技术在丝棉混纺织物成分分析中的应用研究

以1260个丝/棉混纺织品作为研究对象,对样品颜色及其成分分布进行分析。按样品丝含量的不同进行分段细分,利用偏最小二乘法分别建立丝/棉混纺织品中丝含量的定量校正模型,提高了校正集和验证集的准确率。为验证模型的准确性和实用性,对600个未参加建模的样品进行预测,并与经典方法的结果进行对比分析,结果表明利用近红外法与经典方法所得结果不存在显著差异,说明该技术在丝棉成分分析领域有着广阔的应用前景。