探索法解题例说

在解决数学难题时,一般很难一眼就看出准确的解题思路.这时,如果我们在认真审题,充分理解题意的基础上,按照一定的方向,通过试探,摸索规律,就有可能寻找到解决问题的途径,这样的解题思想方法称为探索法.     

例说解题反思

反思是对知识的回忆再现,是对方法的总结归纳,是对题目实质的再挖掘．反思可以提高数学的解题质量,培养解题能力,使我们更深入地钻研知识,拓宽思路和视野．著名数学家费赖登塔尔指出：反思是数学活动的核心和动力．以下就解题反思举例说明．     

复数中配偶法解题例说

在复数问题中,对已知关系式两端同加或同乘以某复数,配成zz/-或成对研究z,z/-以及含z、z/-的有关函数式,我们称这种方法为配偶法．这一方法渗透了整体思想,解题思路异常简明．     

例说“极端性”原理在数学解题中的运用

“极端性”原理是解决数学问题的一个重要方法，从极端情形(最大值、最小值、极端有利、极端不利、边界情形、极端位置等)入手分析，往往能发现解决问题的突破口．此法不仅在解竞赛问题中用途广泛．事实上，在平时的解题过程中，为了寻求更清晰的解题思路，更简洁的运算方法，我们也会不经意地去“走极端”，本文例举说明．     

例析图形在解题中的作用

图形是数学解题的一个组成部分，平面几何和立体几何能借助图形形象地反映问题的条件与结论之间的内在联系，启发解题思路；代数中的许多问题可通过构造图形，揭示问题的隐含条件，发现简洁明了而富有创意的解题方法；试题中的选择题、填空题借助图形可以简化解题过程，检验解题结果；数学教学中通过优美图形的展示和简洁解法的讲授可以培养学生解题的创新能力．     

平面几何解题思路探析

探寻平面几何解题的一般性思路,并应用该思路解决具体实例问题,并提出学生的认知结构是解题的基础,注意解题思维过程的整体推进,以及根据问题难度做出相应调整等对解题思路的一些思考.拓展平面几何解题的相关研究,为学生几何解题探寻一些方法,同时给平面几何解题教学带来些许启示.     

例说利用对称优化解题

一道优秀的数学题能体现数学知识、信息与思想方法的合理搭配与有机结合 ,成为数学对象及其关系在一定逻辑形式下组成的一个关系结构 ,在教学过程中 ,适时、适度地引导学生去弄清问题的关系结构 ,挖掘数学问题中关系结构的和谐性与对称美 ,能简化运算 ,优化解题思路 .是实现“发展学生智力 ,培养学生能力”的重要手段 .1　熟悉常见的对称关系　抓住问题中连接数学元素之间某些对应关系(如相等、互逆、互否、同解等 )的对称性 ,通过互逆关系合理变更问题的结构 ,使问题的解决明朗化 .例 1　若函数 ｙ =ｆ(ｘ) 的反函数为 ｇ(ｘ) ,且ｆ(ａｂ) …     

例说建立递推关系式解题

在解某些数学问题时,若能根据问题的实质和特征,建立递推关系式,就会使问题很巧妙地得到解决.下面举例说明,供读者参考.     

例说用自乘法解题

所谓“自乘法”,就是根据题目的结构特征,进行自乘运算解题的一种方法,运用这种方法解题,往往能化繁为简,变难为易,得到简捷合理的解题途径,兹举例说明．     

The Kissing Triangles: The Aesthetics of Mathematical Discovery

Papert's (1978) appeal to reconsider the power and possibilities of the aesthetic in mathematics learning is often ignored in mathematics education research. This paper begins with the premise, put forth by Dewey (1934), that the aesthetic structures many dimensions of inquiry and experience. In the same way that using particular paintings, musical compositions, or even everyday experiences has been instrumental to attempts by philosophers to understand the aesthetic dimensions of meaning and experience in artistic domains, I propose that analysing a particular encounter with mathematics may help reveal the nature and role of the often nebulous responses of elegance, beauty, and `fit' to which mathematicians lay claim in their mathematical activity. To achieve this, I draw on and adapt the defining features of the aesthetic character of experience set forth by the aesthetician Beardsley (1982). This, in turn, sheds light on the role thataesthetics can play in mathematical inquiry and experience, and provides initial categories and conjectures that can be used to investigate the potential roles of aesthetics in mathematics learning contexts.This revised version was published online in September 2005 with corrections to the Cover Date.     

The riddle of aesthetic principles

The problem of aesthetic principles and that of the nature of aesthetic reasons get confronted. If aesthetic reasons play an important role in our aesthetic evaluations and judgments, then both some general aesthetic principles and rules could support them (aesthetic generalism) or again their nature may be particularistic (aesthetic particularism). A recent argument in support of aesthetic generalism as proposed by Oliver Conolly and Bashshar Haydar is presented and criticized for its misapprehension of particularism. Their position of irreversible aesthetic generalism is questioned. Aesthetic particularism is restated by the help of proposals by Jonathan Dancy’s version of moral particularism.     

关于高等代数题解的五种方法

介绍处理高等代数习题的标准形式化方法、公理化方法、模型转换方法、形式统一性方法和形态联想方法,并给出实例加以说明.     

Exploring the relationship between metacognitive and collaborative talk during group mathematical problem-solving – what do we mean by collaborative metacognition?

The purpose of this study was to enhance our understanding of the relationship between collaborative talk and metacognitive talk during group mathematical problem-solving. Research suggests that collaborative talk may mediate the use of metacognitive talk, which in turn is associated with improved learning outcomes. However, our understanding of the role of group work on the individual use of metacognition during problem-solving has been limited because research has focused on either the individual or the group as a collective. Here, primary students (aged nine to 10) were video-recorded in a naturalistic classroom setting during group mathematical problem-solving sessions. Student talk was coded for metacognitive, cognitive and social content, and also for collaborative content. Compared with cognitive talk, we found that metacognitive talk was more likely to meet the criteria to be considered collaborative, with a higher probability of being both preceded by and followed by collaborative talk. Our results suggest that collaborative metacognition arises from combined individual and group processes.     

A theory,implementation and applications of human-like understanding

Operation logic is a formal logic with well-defined formulas as semantic language clauses and with modus ponens rules as a method of reasoning. Operation logic can be implemented on any database management system (as the so-called OLS) having a universal general knowledge database and enabling understanding of data stored in the database. Semantic language clauses have necessary and sufficient properties for being able to describe any process in the world. Semantic language is the deepest level of any natural language, the level of data storing, understanding and reasoning. OLS can be a tool for studying implementation possibilities of human-like consciousness, for building artificial experts and artificial encyclopedias and for constructing semantic mathematical theories of anthropoecosystems (which is such an exact theory that qualitative information can be used with meaning completely defined by the user). In the paper the theory (and complete information enabling implementation) is presented for human-like understanding, topic-focus division of clauses, for human-like problem solving (program synthesis and verification) and for semantic mathematical analyses. Many examples are presented.     

Investigating secondary students beliefs about mathematical problem-solving

Many studies over the past 30 years have highlighted the important role of students’ beliefs for successful problem-solving in mathematics. Given the recent emphasis afforded to problem-solving on the reformed Irish secondary school mathematics curriculum, the main aim of this study was to identify Irish students’ (n = 975) beliefs about the field. A quantitative measure of these beliefs was attained through the use of the Indiana Mathematical Belief Scale, an existing 30-item (five-scale) self-report questionnaire. A statistical analysis of the data revealed that students who were further through their secondary education had a stronger belief that not all problems could be solved by applying routine procedures. In contrast, the same students held less positive beliefs than their younger counterparts that they could solve time-consuming problems and that conceptual understanding was important. The analysis also indicated that gender had a significant impact on three of the five belief scales.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

Students' attitude towards computer algebra systems (CAS) and their choice of using CAS in problem-solving

We explore students choice of using computer algebra systems (CAS) in problem-solving relative to their self-reported attitude towards learning mathematics with CAS. Our research design is a case study of nine Norwegian upper-secondary mathematics students with a wide range of attitude towards CAS. Our findings on routine problems indicate that (1) students use CAS whenever students perceive the problem as time-consuming regardless of their attitude towards CAS, and (2) students attitude affects their use of CAS whenever students perceive the problem as non-time-consuming. Norway, among other countries, has implemented CAS as an essential digital resource towards learning mathematics in upper-secondary school. Our discussion focuses on the implications of our findings have on local mathematics educators and national policy-makers.     

基于算法的《任意项级数审敛法》教学与反思

基于算法的《任意项级数审敛法》教学设计与实践顺应新课程改革趋势,符合教育学、心理学规律,培养了学生解决问题的意识和能力,值得推广.     

High school prospective teachers’ problem-solving reasoning that involves the coordinated use of digital technologies

The aim of this study was to characterize and discuss ways of reasoning that prospective high school mathematics teachers develop and exhibit in a problem-solving scenario that involves the coordinated use of digital technologies. A conceptual framework that includes Virtual Learning Spaces (VLS) and Resources, Activities, Support and Evaluation (RASE) essentials is used to introduce and support a problem-solving approach to structure learners’ problem-solving activities that encouraged them to share ideas, discuss and extend mathematical discussions beyond formal settings. Main results indicated that prospective high school teachers relied on a set of tool affordances (dragging objects, looking and exploring object’s loci, using sliders, quantifying and visualizing mathematical relations, etc.) to formulate, explore and identify properties or relations to share, discuss and support mathematical conjectures. In this context, the participants recognized and valued the importance of using several tools to both dynamically represent and explore mathematical tasks and to share and constantly refine their mathematical ideas and problem-solving approaches.