论数学证明的教育价值

数学证明是指根据某个或某些真实命题和概念去断定另一命题的真实性的推理过程.数学证明的教育价值体现在:数学证明是理解数学知识特别是公式(定理)不可缺少的基本方法,是开发大脑的有效途径,可以激发许多人的学习兴趣,有利于培养中国国民的理性精神.     

论数学证明的教育价值

数学证明是指根据某个或某些真实命题和概念去断定另一命题的真实性的推理过程.数学证明的教育价值体现在:数学证明是理解数学知识特别是公式(定理)不可缺少的基本方法,是开发大脑的有效途径,可以激发许多人的学习兴趣,有利于培养中国国民的理性精神.     

数学证明教学策略探索

证明可以分为实用性证明和理性的证明两类.学生学习数学证明前已有实用性证明观念,教师在数学证明入门教学中常常忽视学生的实用性证明观念.数学证明入门教学的有效策略是:实现由实用性证明到理性证明的过渡,并应有较为合理的证明教学观.     

数学证明教育价值研究文献述评

数学证明的教育价值是数学证明教育研究中要探讨的首要问题,人们数学证明教育观的转变主要依赖于对数学证明做出的教育价值判断.对于数学证明的教育价值问题,以下几方面有待研究:数学证明有哪些重要的教育价值,实证性的量化研究,影响数学证明教育价值实现的因素.     

证明的作用和挑战对数学教学的影响

上个世纪,人们重新审视数学的本质,出现了很多的关于证明的观点,如:"证明是数学实践的反映","证明是促进数学理解的一个极其重要的工具","证明可以用来交流数学理解"等.最近,证明在数学和数学教育中的作用受到了质疑,甚至有人预测,证明将消失.这种质疑来自数学内部、数学教育、社会价值和经验科学的挑战,对数学教学产生了重要影响.     

G.H.哈代的数学观及其当代数学教育意义--兼论我国中小学《国家数学课程标准》的修改与完善

辨证地理解G .H .哈代的数学观 ,对当代数学教育和修订我国的《国家数学课程标准》极具启发意义。关于数学对象 ,哈代持极端的实在论观点 ,它有利于理解数学“主观的”客观实在性 ;关于数学本质 ,哈代持极端的完美主义 ,它有利于理解“为数学而数学”的理性追求和理性批判精神 ;关于数学证明 ,哈代持“内部的”和“外部的”两种证明的观点 ,它有利于理解数学的“整体观”和“文化功能”。     

主客观相统一原则下民事诉讼证明方法的选择

民事诉讼证明方法的选择应坚持主客观相统一的原则,在主观上应有利于实现诉讼证明主体所追求的价值目标——形式正义和实质正义,在客观上应当遵循方法的特征和规律,并符合民事诉讼证明活动的特点。具体地说应坚持:多学科证明方法的交叉和融合;抽象证明方法论和具体证明方法的并用;理论理性之证明方法和实践理性之证明方法的协同使用;形式合理性之证明方法和内容合理性及可接受性之证明方法的同等关注;逻辑证明方法和经验证明方法的并用和互补;程序性证明方法和说服性证明方法的并行使用;形式正义和实质正义不同视角下证明方法的区别运用;自向证明方法和他向证明方法的协力运用。     

数学证明教学要教什么

数学证明在数学教学中占据着重要的作用．在数学教学中,数学证明要教些什么呢？本文从让学生顺利地从实用性证明过渡到理性证明,要淡化形式、注重实质,要善于揭示过程、培养推理能力以及把握好合情推理和演绎推理的关系四个方面进行了阐述．     

谈谈中学数学证明

数学证明首先在几何学领域里开始,公元前3世纪产生的欧几里德《几何原本》在两千多年的时间里一直是数学证明的范例.罗巴切夫斯基几何学的产生,使人们对数学证明的认识大大加深,并随之产生了现代公理体系.数学证明在本质上是一种方法论.学生学习欧氏几何、经过这种论证方法的训练,其作用不限于几何学、甚至不限于数学,对于学生学习其他学科,对于学生未来走向社会都是很有益处的,这便是其教育价值所在.     

思想道德教育价值评价的合理性

探讨思想道德教育价值评价的合理性 ,必须从两方面入手 ,一方面是要弄清思想道德教育价值评价合理性的基础 ;另一方面是要证明思想道德教育价值评价的合理性。本文认为思想道德教育价值评价的合理性是相对合理性 ,是合情理性。为了把握这种合理性 ,必须遵循方向性原则、全面性原则、客观性原则、教育性原则和主体性原则     

Cauchy微分中值定理证法思路探析

为了培养学生的数学思维,提高学生的创新能力,从多角度和多方位对Cauchy微分中值定理的证明方法进行了探讨,归纳出了利用罗尔定理、同增量性、单调性、行列式、定积分、复合函数等证明Cauchy微分中值定理的方法.利用分析法分析了构造适当辅助函数证明的思路.     

Proof schemes and learning strategies of above-average mathematics students

What patterns can be observed among the mathematical arguments above-average students find convincing and the strategies these students use to learn new mathematical concepts? To investigate this question, we gave task-based interviews to eleven female students who had performed well in their college-level mathematics courses, but who differed in the number of proof-oriented courses each had taken. One interview was designed to elicit expressions of what students find convincing. These expressions were categorized according to the proof schemes defined by Harel and Sowder (1998). A second interview was designed to elicit expressions of what strategies students use to learn a mathematical concept from its definition, and these expressions were classified according to the learning strategies described by Dahlberg and Housman (1997). A qualitative analysis of the data uncovered the existence of a variety of phenomena, including the following: All of the students successfully generated examples when asked to do so, but they differed in whether they generated examples without prompting and whether they successfully generated examples when it was necessary to disprove conjectures. All but one of the students exhibited two or more proof schemes, with one student exhibiting four different proof schemes. The students who were most convinced by external factors were unsuccessful in generating examples, using examples, and reformulating concepts. The only student who found an examples-based argument convincing generated examples far more than the other students. The students who wrote and were convinced by deductive arguments were successful in reformulating concepts and using examples, and they were the same set of students who did not generate examples spontaneously but did successfully generate examples when asked to do so or when it was necessary to disprove a conjecture.     

公式蕴含的证法

本论文介绍了公式蕴含的几种证法:真值表法、等价演算法、直接证法、间接证法等,灵活应用公式蕴含的证明方法,有利于逻辑推理的顺利进行。     

高等代数中反证法的三点注记

针对高等代数中许多结论、定理的证明有时虽然可以用构造法、数学归纳法等其他方法证明,但证明过程较复杂;有时结果虽然是数值却无法用求解的方法来求解,提出了用反证法来证明或求解的思想,从而达到了化复杂为简明、化难为易的效果.     

The Notion of Proof in the Context of Elementary School Mathematics

Despite increased appreciation of the role of proof in students’ mathematical experiences across all grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major features of any given argument – foundation, formulation, representation, and social dimension – so that the argument could count as proof at the elementary school level. My examination is situated in an episode from a third-grade class, which presents a student’s argument that could potentially count as proof. In order to examine the extent to which this argument could count as proof (given its four major elements), I develop and use a theoretical framework that is comprised of two principles for conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, at once, honest to mathematics as a discipline and honoring of students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that students’ experiences with proof in school have coherence. The two principles offer the basis for certain judgments about whether the particular argument in the episode could count as proof. Also, they support more broadly ideas for a possible conceptualization of the notion of proof in the elementary grades.     

利用二次求导确定函数单调性法证明一些不等式

利用二次求导确定函数单调性的方法,证明了高等数学中一些常见的不等式.     

代数学中基本定理证明的探讨

代数学基本定理的经典证明用到较多的代数知识,且难以理解,文章探讨用数学分析的方法予以证明。该证明从复变多项式无非零最小模引入,并在此基础上简单证明了代数学的基本定理。     

Deficit or Difference? the Role of Students' Epistemologies of Mathematics in their Interactions with Proof

The ability to handle proof is the focus of a number of well-documented complaints regarding students' difficulties in encountering degree-level mathematics. However, in addition to observing that proof is currently marginalised in the UK pre-university mathematics curriculum with a consequent skills deficit for the new undergraduate mathematics student, we need to look more closely at the nature of the gap between expert practice and the student experience in order to gain a full explanation. The paper presents a discussion of first-year undergraduate students' personal epistemologies of mathematics and mathematics learning with illustrative examples from 12 student interviews. Their perceptions of the mathematics community of practice and their own position in it with respect to its values, assumptions and norms support the view that undergraduate interactions with proof are more completely understood as a function of institutional practices which foreground particular epistemological frameworks while obscuring others. It is argued that enabling students to access the academic proof procedure in the transition from pre-university to undergraduate mathematics is a question of fostering an epistemic fluency which allows them to recognise and engage in the process of creating and validating mathematical knowledge.     

Analysing proof-related competences in Estonian,Finnish and Swedish mathematics curricula—towards a framework of developmental proof

Many countries are revising their mathematics curriculum in order to elevate the role of proof and argumentation at all school levels and for all student groups. Yet, we have very little research on how proof-related competences are aimed to be developed in the mathematics curricula of different countries in Grades 1 to 12. This article contributes to filling this gap by analysing and comparing three countries’ curricula from the perspective of developmental proof. For this purpose, we created an analytical frame of proof-related competences that could be connected to the development of students’ understanding and skills concerning argumentation and mathematical proof. The analysis reveals three quite different trajectories with specific characteristics, shortcomings and strengths.