【作者】 胡典顺；

【导师】 郭元祥；

【作者基本信息】 华中师范大学 ， 课程与教学论， 2009， 博士

【摘要】 本文以数学哲学和数学教育哲学的前沿成果为背景,力图从一个新的视角——数学意义——探求数学价值,反思数学教育中存在的突出问题,探讨数学教育回归数学意义的基本路径,并分析这些基本路径的教学意蕴。人的存在总是牵涉到意义,人是一种意义的存在,有意义的东西才能使人快乐。数学意义的问题自古希腊以来就备受关注。柏拉图指出,忘却正方形的对角线与其边之比是不可公度的这个事实,那他就不配有人的名字。他把对数学重要性的无知形容为像“猪一般”。随着新一轮数学课程改革的进行,数学意义的问题是近年来的一个热门话题。尽管有不少人对此进行了大量探索和研究,但从总体上来讲,大部分研究只是从一个方面,在一定范围内对数学意义进行了研究,研究不系统,不深入,不全面。本文的中心论题是“数学意义”。围绕“数学意义”这一中心论题,本文着重探讨了以下几个核心问题:数学的本质是什么?数学为什么是一个意义的领域?其意义体现在哪里?数学教育中,数学意义缺失了吗?为什么会造成这样的状况?其深层原因是什么?反映了怎样的数学哲学根源?数学教育中,回归数学意义的具体途径有哪些?这些途径包含着怎样的数学哲学观?对数学意义认识不清的根源,在于没有从意义哲学、文化哲学、教育哲学,特别是数学哲学和数学教育哲学的角度对此进行深入剖析。事实上,如果回避数学本质的问题,有关数学意义的争议就不会有结果,就不会提出一种更有说服力、更令人满意的有关数学之意义和本质的哲学解释。本文中,通过对数学的传统认识、现代隐喻以及数学知识的特征的讨论。笔者认为,数学是一个复杂的多元体,数学是不断发展的,任何从数学的某些特征对数学进行的描述都是不完善的。它们要么过于狭窄,要么过于宽泛。数学是动态的、可误的、经验的与拟经验的,数学是一种文化。数学寻求理解遍及我们周围的物质世界,以及我们思想中的各种模式。数学是关于模式而不只是关于数或形的科学。知识具有三个不可分割的组成部分:符号表征、逻辑形式和意义。在本文中,笔者提出“数学是一个意义的领域”这一观点,并将“数学意义”解释为“数学对物理世界、客观知识世界以及精神世界的一种指向”。由于人们对数学意义的认识从数学应用的角度论述得多,从人的发展的角度论述得少。即以科学的立场的研究较多,以教育的立场的研究较少。所以,本文中,笔者从科学的立场和教育的立场两个维度对“数学意义”进行了讨论。并认为,从科学的立场来看,数学是时代的特征,数学是美妙的乐章,数学是科学的皇后、科学的仆人、科学的伙伴。从教育的立场来看,数学是具备公民资格的前提,数学是现代人的基本素质,数学培养人的优秀品质,数学教人思维,数学提升审美能力,数学促进人的终身发展等等。调查研究表明,数学教育中数学意义缺失现象普遍存在。无论是对数学本质的理解,还是对数学的目的、价值,以及对数学教学的认识都存在偏差。在不少学生心目中,数学是一种符号游戏,是杀死脑细胞的工具,难打败的憎恨者,从而直接导致数学学习的异化。数学学习是一个强制的过程,是一个痛苦的过程,它使学习者与知识和成绩、学习者之间互相对立等等。其深层原因在于人们对数学以及数学教学的认识存在一些误区。在数学教学中人们往往认为:数学等于升学工具,学数学就是做题,数学是绝对真理,数学等于逻辑,数学教学应该强调公理体系,数学教学应该很抽象等等。本文中,结合数学哲学和数学教育哲学研究的大量成果对这些错误认识进行了深入的剖析。笔者认为,数学教育中的种种困惑与迷茫,都与数学意义的失落密切相关。要走出被异化的数学教育,必须回归数学意义。在文中,笔者提出了一种新的数学教育观——走向意义的数学教育。并认为,走向意义的数学教育应该成为新的教育方向,新的教育追求。那么,数学教育如何回归数学意义呢?在论述数学的对话性、文化性和经验性,并反思数学教育中存在的种种误区的基础上,作者提出从“数学对话”、“数学文化”、“数学经验”等方面回归数学意义。从数学对话来看,笔者认为,意义因对话而获得,意义因对话而丰富。没有对话,就没有交流。没有交流,就没有真正的教育。数学对话有重要的教育价值。无论是从数学哲学和数学发展来看,还是从数学知识来看,数学都具有对话性。从而,在数学对话教学中,要让学生感受数学对话,重视数学交流,体现过程价值。从数学文化来看,数学一直是文明和文化的重要组成部分。数学教育要考虑两个层次:具体的知识层次和无形的文化层次。在文中,从数学知识教育的特点,数学知识教育的缺失,以及数学知识教育的根源,对数学知识教育进行了深刻的反思。笔者认为,由知识教育到文化教育是数学教育的必然选择。从数学经验来看,数学具有经验性,数学经验具有重要的价值,所有的学习都涉及到原有经验的迁移。从而,在数学教育中,应该回归生活世界,重视数学经验,关注数学表征。更多还原

【Abstract】 This paper attempts to explore the value of mathematics from a new perspective,based on the update findings of mathematical philosophy and mathematical education philosophy study,with reflection on the prominent problems in mathematics education and analysis of the implications of the fundamental way back to the meaning of mathematics.The existence of human being is always related to the meaning,since human being is one kind of the existence of meaning and only the one full of meaning may bring about happiness.The meaning of mathematics has drawn much attention since Ancient Greece times.Plato indicated that if one forgets the fact that the ratio of diagonal lines to sides in square is non-commensurable,he deserves to be deprived of his name.He thought the one who ignores the importance of mathematics is as stupid as goose.With the promotion of the new round of mathematical curriculum reform,the meaning of mathematics meaning has become a hot topic in recent years.Although many people have carried out a large number of explorations and researches on this topic,most of them undertake the research only from one aspect in general.In a certain scope,those researches they carried out are more in a scientific position,but lack of the one in an educational position,and thereby those are not systematic, in-depth,and comprehensive.This paper focuses on the meaning of mathematics,with emphasis on the central subjects as follows:what is the essence of mathematics? Why does mathematics belong to the field of meaning? How to embody the meaning? Is mathematics education lacking in the meaning? What’s the reason? What is the following concrete ways to deal with it and what are the philosophy perspectives in those ways?I believe that the root of failing to recognize the meaning is the ignorance of analysis from the perspective of meaning philosophy,cultural philosophy and educational philosophy,especially from the mathematical philosophy and mathematical education philosophy perspective.In fact,if we avoid taking the nature of mathematics into consideration,the controversy over mathematical meaning will never come to an end.Therefore,it will be impossible to propose a more convincing and satisfactory philosophical interpretation of the meaning and the nature of mathematics.In this paper,with the exploration on the nature of mathematics,the author considers mathematics as a kind of science and culture which embodying the characteristics of dynamic,misleading,experiential or to-be -experiential.Mathematics seeks to interpret the meaning of the physical world around us thoroughly and understand different models in our minds.It is a kind of science concerning models instead of a science only concerning numbers or figures.Based on these,I introduce -"Mathematics is a field related to the meaning"-this point of view,and make an in-depth discussion on the meaning of mathematics both in a scientific and educational position.Knowledge consists of three indivisible parts:symbolic feature,logic form and meaning.In this paper,the author proposes the idea that mathematics belongs to the field of meaning and explains that the meaning of mathematics relates mathematics to the physical,objective knowledge and spiritual world.Since there are many comments on the meaning of mathematics from the view of application and there are few discussions from the perspective of the development of human being which is concentrating on the standpoint of science rather than education,this paper will discuss on the meaning of mathematics from both the standpoint of scienco and education.From the perspective of science,mathematics is like a piece of music,the Queen of science,the servant of science,the company of science.From the perspective of education,mathematics is a prerequisite to become citizenship,which can equip one with excellent quality,the ability of thinking and aesthetic ability and so on.Investigations and studies show that the absence of mathematical meaning is common in mathematical education which directly leads to the alienation of mathematical teaching.Meanwhile there are some deviations in the understanding of mathematics teaching,both in the essence and the purpose and value of mathematics.Many students regard mathematics as a kind of symbol play to kill brain cells and invincible enemy.Learning mathematics is a process of compulsion and depression, which makes the learner standing the opposition towards the knowledge,the outcome and other learners.The deep reason behind this is that people have some misunderstanding of mathematics and mathematics teaching.It is considered that mathematics is as a tool for entering schools in mathematics teaching,the purpose of learning mathematics is to solve the problems,mathematics is an absolute truth,mathematical is equal to logic,mathematical teaching should emphasize on axiomatic system,mathematical teaching should be very abstract and so on.In this paper,I conduct an in-depth analysis on these ideas through the combination of the forefront results of researches on mathematical philosophy and the philosophy of mathematical education.I believe that all kinds of perplexities and confusions in the mathematical education are closely related to the absence of mathematical meaning.In the paper,I propose a new perspective of mathematical education——the meaning of mathematical education.And I believe,a mathematical education moving towards the meaning should be regarded as the new orientation and the pursuit of education.Then,how does the mathematical education return to mathematical meaning? Based on discussions on the three characteristics of mathematics-dialogue,culture and experience,together with reflection on all kinds of misconceptions in mathematical education,I put forward an idea that we can return to mathematical meaning from the "mathematical dialogue",the "mathematical culture" and the "mathematical experience" aspect.From the aspect of mathematics dialogue,the author suggests that the meaning can be obtained and becnme rich from the dialogue.Without dialogue,there is no communication and no real education.Those dialogues have great educational value and mathematics embodies the nature of dialogue,not only from the aspect of mathematical philosophy and development but also from the aspect of the mathematical knowledge.Therefore,it should make mathematical dialogues with students in mathematical teaching,with great attention paid to the mathematical communication and the value of process.From the aspect of mathematics culture,mathematics is an essential part of civilization and culture.In mathematics education,it should be considered from two aspects:concrete knowledge aspect and invisible culture aspect.There is deep introspection of mathematical knowledge education from the analysis of the characteristics,the losses,and the root of mathematical knowledge education. The author indicates that it is inevitable choice from knowledge education to culture education.From the aspect of mathematics experience,mathematics has a great empirical value.All learning about mathematics involve into the transference of experience.So it should return to the real life world with great importance paid to the mathematical experience and characterization in mathematics teaching.更多还原