数学符号意义及其获得能力培养的研究
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摘要
为什么随着年级的增加,许多学生感觉数学越来越难学、越来越枯燥,普遍出现“听而不懂”、“懂而不会”、“会而不对”问题?对小学和初中数学教材中的数学概念、数学符号、数学图表、数学公式、数学定理、数学关键词进行分类统计的结果表明,小学生平均每学期需要学习42个新符号,而初中生每学期需要学习120个新符号,几乎是小学生学习量的3倍。对小学、初中、高中三个阶段学生的问卷调查表明,学生的数学符号意义获得能力普遍较低,38%的学生不认识学过的数学符号,45%的学生只能说出数学符号的一个意义,只有17%的学生能够想到二个或二个以上的意义,而且三个学段学生的数符号意义获得能力无显著差异。这些数据表明,随着年级增加,数学符号的数量急剧增加,形式越来越简洁,意义越来越复杂,学生的数学符号意义获得能力却仍处在低水平,没有得到相应提升,是导致学生数学学习困难的根本原因。为此,本课题提出了研究假设：培养和提高学生的数学符号意义获得能力是解决上述问题的有效方法。
首先,概括阐述了符号学的基本方法和基本原理,作为本研究的理论基础。符号学理论认为,任何事物的存在状态和变化规律既受内部组成要素的影响,也受外部环境因素的影响,始终处在由内部要素和外部因素组成的关系结构中；符号是包含符号形式(记号)和符号意义(记号表象)的统一体,不能脱离记号谈论符号意义,也不能脱离符号意义谈论记号；符号都不是孤立存在的,它本身是一个结构,又处于更大的符号结构中；研究符号意义需要全面构建相互关联的包括要素结构、联结结构和意义结构三个层次的符号结构。
其次,应用符号学理论分析教学活动中的符号现象,探讨符号学理论和方法的教学意蕴,对传统的“符号”、“知识”、“学习”、“教学”进行新的诠释。符号本质上是一种能够刺激人的感官,使人产生意义联想的客观存在形式,是一种可以替代认识对象的“感官刺激物”。教学活动中可以刺激学生产生意义联想,帮助学生理解教学内容的实物、模型、手势、视频、教材等一切东西都可看作符号,视作教学资源。知识是由知识外部表征(记号结构)与知识内部表征(认知结构)组成的统一体,本质上是一种符号结构。人的任何想法都可以通过符号以“直观”的方式直接地或通过符号结构以“意会”方式间接地传递给他人。个体知识的外部表征构成了与现实世界相对应的个体的“记号世界”,个体知识的内部表征构成了与“记号世界”相对应的个体的“经验世界”。由记号结构和认知结构构成的符号结构,代表了个体的所有知识和经验,代表了个体适应和改造现实世界的综合能力。人类的某一感官不可能同时感知整个客观事物,只能感知它的部分属性。感知到的属性被感知者赋予意义后就建立了一个刺激物(记号)与意义(感觉表象)的联结,成为自然符号。当感觉表象被感性思维加工成与客观事物对应的知觉表象(感性经验)时,与感觉表象对应的符号就联结成自然符号结构,并与客观事物建立了对应关系。当感觉表象被理性思维加工成客观世界中不存在的知觉形象(概念)时,人类就需要创造人工符号来表征它,并使建立在概念基础上的理性经验与人工符号结构形成对应关系。因此,学习知识的过程本质上是建构符号结构的过程,具体包括客观事物的经验化、经验的符号化、符号的经验化三个相互转换过程。知识的教学就是教师帮助学生建构符号结构的过程。
再次,应用符号学理论和方法重新界定了数学符号、数学符号意义、数学符号意义获得能力的内涵,分析了影响数学符号意义获得能力培养的主要因素和困难,并结合数学概念教学、数学命题教学和数学问题教学进行了案例研究。在教学活动中,数学符号是一切承载数学信息的符号,主要包括数学自然符号、数学模型符号、数学语音符号、数学文字符号、数学专业符号、数学图表符号、数学行为符号七大类。数学符号意义是指在数学符号刺激下被激活的整个数学符号结构,主要包括数学符号的语符意义、基本意义、转换意义、隐性意义、美学意义、个性化意义、操作意义七种意义,它可通过联想到的所有数学符号的记号的数量来测量。数学符号意义获得能力是指在数学符号刺激下建构包含这该数学符号的数学符号结构的能力,主要包括数学符号的形式感性能力、意义联想能力、意义转换能力、意义整合能力和记号操作能力五大能力。影响数学符号意义获得能力培养的因素主要是数学教师的数学符号观和教学资源观、数学教学观和教学方法观。在数学教学实践,数学教师应转变观念,依据《数学课程》的“三维”教学目标要求,科学选择、安排、呈现数学符号资源,灵活应用符号结构分析方法,传授学生建构数学符号意义结构的基本方法和思维模式,探讨数学符号的多元表征,全面建构数学符号意义结构,并使之内化为学生自己的认知结构,提升学生的数学素养,促进学生的全面发展。
最后,概括了本研究的基本逻辑：(1)无法获得数学符号丰富的数学意义是学生害怕、讨厌数学,感觉数学难学的主要原因；(2)教师忽视数学符号教学是导致学生数学符号意义获得能力较低的主要原因；(3)教师片面的数学符号观和知识观是导致教师忽视数学符号教学的主要原因；(4)数学符号结构中蕴含了数学知识的所有信息,需要学习者去感知、发现、领悟和建构；(5)获得数学符号结构中的数学信息需要学生具备较高的数学符号意义获得能力；(6)培养数学符号意义获得能力的核心是超越数学符号“是什么”的传统思维,努力思考它“意味着什么”；(7)培养学生的数学符号意义获得能力需要教师转变片面的符号观、知识观、学习观和教学观。本研究的最终结论是：培养和提高学生的数学符号意义获得能力是解决“数学难学”、“数学枯燥”,“听而不懂”、“懂而不会”“会而不对”等教学难题的一种有效的、可行的、具有操作性的途径和方法。
Why do students feel more difficult and boring to learn mathematics in the higher grade, not understand what teachers teach repeatedly, not do what they have understood, and do wrong with what they can do? A statistics of symbols of the mathematics teaching materials show that the quantity of new symbols the junior students need to learn is3times the elementary students. A questionnaires survey to the students' mathematical Symbol-meaning-obtain-ability found that38%have symbol recognition difficulty,45%only get one meaning,17%can get two or more meaning, and all stages students have no obvious difference. It shows that the numbers of mathematical symbols with concise form and complex meaning increase sharply but the students' mathematical symbol meaning obtaining ability is still in low level is the root cause of mathematical learning difficulty. For this, the research puts forward a hypothesis:to cultivate students' mathematical symbol meaning obtain ability is the effective method to solve the above problem.
First of all, this paper expounds the basic methods and basic principle of semiotics. The existence state and change law of things is influenced by both internal elements and external environment factors. Each symbol is a union of symbolic form (sign) and exists in symbol structure. To get the symbol meaning need study the element structure, coupling structure and meaning structure three levels of symbol structure.
Secondly, this paper discusses the semiotics' teaching implication and the new interpretation of traditional "symbol","knowledge","learning" and "teaching". Symbols is a kind of objective existence form to stimulate people's senses in essence which include both language symbols and other known object to cause meaning association. All things that can stimulate students producing meaning association, help students understand the content of the teaching in teaching activities such as model, gestures, video, teaching material and so on should be regarded as symbols, as teaching resources. Knowledge is the union of knowledge external (sign) and knowledge internal representation (experience) and it should not only as experience, and should be regarded as symbol structure. Any human thoughts can transfer to others through the symbol directly or through the symbol structure indirectly. The external representation of Individual knowledge constitutes the individual "mark world"(mark structure) with the real world correspondingly, and the internal representation of the individual constitutes "cognitive world"(cognitive structure) with "mark world" correspondingly. Symbol structure with mark structure and cognitive structure represent the individual all knowledge and the comprehensive ability to adapt to and remake the real world. The human sensory can only sense the part attributes of the whole things. The stimulus (sign) sensing attribute and meaning (feel representation) are united together to become a natural symbol. When feeling representation constitute perceptual representation (perceptual experience) by perceptual thinking process, the sign with sensory representation correspondingly coupling into natural symbol structure and Establish corresponding relation with objective things. When feeling representation constitute a perceptual image (concept) not existing in the world by rational thinking process, human beings need create artificial symbols to represent the image and establish corresponding relation between the rational experience based on the concept and artificial symbol structure. Therefore, the process of learning knowledge is essentially the process to construct symbol structure which includes three mutual conversion processes:objective things become experience; symbol become experience; experience become symbol. Knowledge teaching is the process that teacher help students construct symbol structure.
thirdly, based on the semiotics the study define connotation of the mathematical symbol, mathematical symbol meaning, mathematical symbol meaning obtain ability, analyzes the factors and difficulties in cultivating the ability to obtain mathematical symbol meaning, and make case studies in the mathematics concept teaching, mathematical proposition teaching and mathematical problem-solving teaching. In the semiotics perspective, mathematical symbol is symbol bearing mathematical information, mainly including mathematics natural symbol, mathematics model symbol, mathematics phonetic symbol, mathematics text symbols, mathematics professional symbol, mathematics chart symbol, mathematics behavior symbol. Mathematical symbol meaning is the whole mathematical symbol structure activated by the mathematical symbols, mainly including the mathematical symbol's form significance, basic meaning, conversion significance, implicit meaning and aesthetic significance, personalized significance, operational meaning seven kind of significance, and it can be measure with the number of marks in the mathematical symbol structure. Mathematical symbol meaning obtaining ability refers to the capacity constructing mathematical symbol structure with the mathematical symbol stimulus, mainly including the mathematical symbol form perceptual ability, meaning association ability, meaning transformation ability, meaning integrated ability and mark operation ability. The factors to Influence the cultivation of mathematical symbol significance are mainly mathematics teachers' mathematical symbol view, teaching resource view, mathematics teaching view and teaching method view. In the mathematics teaching, teachers should change teaching view, select, arrange and present mathematical symbol resources scientifically, apply symbol structure analysis method flexibly, teach students the basic method and thinking mode to construct mathematical symbol meaning structure, discusses the multiple representation of mathematical symbol, construct the mathematical symbol meaning structure comprehensively, and make it become students'cognitive structure in order to promote students' mathematical quality, to promote the all-round development of students on the basis of the request of mathematics curriculum.
Finally, the paper summarize the basic logic of this study:(1) not getting mathematical symbol rich mathematical meaning is the main cause making students be afraid of, and hate mathematics, feeling difficult to learn mathematics;(2) the teachers'ignoring mathematical symbol teaching is the main reason to lead to students' mathematical symbol meaning obtain ability lower;(3) the teacher'one-sided mathematical symbol view and knowledge view is the main reason causing teachers ignore mathematical symbol teaching;(4) mathematical symbol structure contains all information of the mathematical knowledge which need the learner' perception, discovery, digestion and construction;(5) getting the information of mathematical symbol structure need students with higher mathematical symbol meaning obtain ability;(6) the core to training mathematical symbol meaning obtain ability is to think "what it means" more than "what it is";(7) training students'mathematical symbol meaning obtain ability need teachers to change their one-sided symbolic view, knowledge view, learning view and teaching view. The final conclusion is that cultivating and improving the students' mathematical symbol meaning obtain ability is the effective, feasible, operational ways and methods to solve some problems in teaching.
首先,概括阐述了符号学的基本方法和基本原理,作为本研究的理论基础。符号学理论认为,任何事物的存在状态和变化规律既受内部组成要素的影响,也受外部环境因素的影响,始终处在由内部要素和外部因素组成的关系结构中；符号是包含符号形式(记号)和符号意义(记号表象)的统一体,不能脱离记号谈论符号意义,也不能脱离符号意义谈论记号；符号都不是孤立存在的,它本身是一个结构,又处于更大的符号结构中；研究符号意义需要全面构建相互关联的包括要素结构、联结结构和意义结构三个层次的符号结构。
其次,应用符号学理论分析教学活动中的符号现象,探讨符号学理论和方法的教学意蕴,对传统的“符号”、“知识”、“学习”、“教学”进行新的诠释。符号本质上是一种能够刺激人的感官,使人产生意义联想的客观存在形式,是一种可以替代认识对象的“感官刺激物”。教学活动中可以刺激学生产生意义联想,帮助学生理解教学内容的实物、模型、手势、视频、教材等一切东西都可看作符号,视作教学资源。知识是由知识外部表征(记号结构)与知识内部表征(认知结构)组成的统一体,本质上是一种符号结构。人的任何想法都可以通过符号以“直观”的方式直接地或通过符号结构以“意会”方式间接地传递给他人。个体知识的外部表征构成了与现实世界相对应的个体的“记号世界”,个体知识的内部表征构成了与“记号世界”相对应的个体的“经验世界”。由记号结构和认知结构构成的符号结构,代表了个体的所有知识和经验,代表了个体适应和改造现实世界的综合能力。人类的某一感官不可能同时感知整个客观事物,只能感知它的部分属性。感知到的属性被感知者赋予意义后就建立了一个刺激物(记号)与意义(感觉表象)的联结,成为自然符号。当感觉表象被感性思维加工成与客观事物对应的知觉表象(感性经验)时,与感觉表象对应的符号就联结成自然符号结构,并与客观事物建立了对应关系。当感觉表象被理性思维加工成客观世界中不存在的知觉形象(概念)时,人类就需要创造人工符号来表征它,并使建立在概念基础上的理性经验与人工符号结构形成对应关系。因此,学习知识的过程本质上是建构符号结构的过程,具体包括客观事物的经验化、经验的符号化、符号的经验化三个相互转换过程。知识的教学就是教师帮助学生建构符号结构的过程。
再次,应用符号学理论和方法重新界定了数学符号、数学符号意义、数学符号意义获得能力的内涵,分析了影响数学符号意义获得能力培养的主要因素和困难,并结合数学概念教学、数学命题教学和数学问题教学进行了案例研究。在教学活动中,数学符号是一切承载数学信息的符号,主要包括数学自然符号、数学模型符号、数学语音符号、数学文字符号、数学专业符号、数学图表符号、数学行为符号七大类。数学符号意义是指在数学符号刺激下被激活的整个数学符号结构,主要包括数学符号的语符意义、基本意义、转换意义、隐性意义、美学意义、个性化意义、操作意义七种意义,它可通过联想到的所有数学符号的记号的数量来测量。数学符号意义获得能力是指在数学符号刺激下建构包含这该数学符号的数学符号结构的能力,主要包括数学符号的形式感性能力、意义联想能力、意义转换能力、意义整合能力和记号操作能力五大能力。影响数学符号意义获得能力培养的因素主要是数学教师的数学符号观和教学资源观、数学教学观和教学方法观。在数学教学实践,数学教师应转变观念,依据《数学课程》的“三维”教学目标要求,科学选择、安排、呈现数学符号资源,灵活应用符号结构分析方法,传授学生建构数学符号意义结构的基本方法和思维模式,探讨数学符号的多元表征,全面建构数学符号意义结构,并使之内化为学生自己的认知结构,提升学生的数学素养,促进学生的全面发展。
最后,概括了本研究的基本逻辑：(1)无法获得数学符号丰富的数学意义是学生害怕、讨厌数学,感觉数学难学的主要原因；(2)教师忽视数学符号教学是导致学生数学符号意义获得能力较低的主要原因；(3)教师片面的数学符号观和知识观是导致教师忽视数学符号教学的主要原因；(4)数学符号结构中蕴含了数学知识的所有信息,需要学习者去感知、发现、领悟和建构；(5)获得数学符号结构中的数学信息需要学生具备较高的数学符号意义获得能力；(6)培养数学符号意义获得能力的核心是超越数学符号“是什么”的传统思维,努力思考它“意味着什么”；(7)培养学生的数学符号意义获得能力需要教师转变片面的符号观、知识观、学习观和教学观。本研究的最终结论是：培养和提高学生的数学符号意义获得能力是解决“数学难学”、“数学枯燥”,“听而不懂”、“懂而不会”“会而不对”等教学难题的一种有效的、可行的、具有操作性的途径和方法。
Why do students feel more difficult and boring to learn mathematics in the higher grade, not understand what teachers teach repeatedly, not do what they have understood, and do wrong with what they can do? A statistics of symbols of the mathematics teaching materials show that the quantity of new symbols the junior students need to learn is3times the elementary students. A questionnaires survey to the students' mathematical Symbol-meaning-obtain-ability found that38%have symbol recognition difficulty,45%only get one meaning,17%can get two or more meaning, and all stages students have no obvious difference. It shows that the numbers of mathematical symbols with concise form and complex meaning increase sharply but the students' mathematical symbol meaning obtaining ability is still in low level is the root cause of mathematical learning difficulty. For this, the research puts forward a hypothesis:to cultivate students' mathematical symbol meaning obtain ability is the effective method to solve the above problem.
First of all, this paper expounds the basic methods and basic principle of semiotics. The existence state and change law of things is influenced by both internal elements and external environment factors. Each symbol is a union of symbolic form (sign) and exists in symbol structure. To get the symbol meaning need study the element structure, coupling structure and meaning structure three levels of symbol structure.
Secondly, this paper discusses the semiotics' teaching implication and the new interpretation of traditional "symbol","knowledge","learning" and "teaching". Symbols is a kind of objective existence form to stimulate people's senses in essence which include both language symbols and other known object to cause meaning association. All things that can stimulate students producing meaning association, help students understand the content of the teaching in teaching activities such as model, gestures, video, teaching material and so on should be regarded as symbols, as teaching resources. Knowledge is the union of knowledge external (sign) and knowledge internal representation (experience) and it should not only as experience, and should be regarded as symbol structure. Any human thoughts can transfer to others through the symbol directly or through the symbol structure indirectly. The external representation of Individual knowledge constitutes the individual "mark world"(mark structure) with the real world correspondingly, and the internal representation of the individual constitutes "cognitive world"(cognitive structure) with "mark world" correspondingly. Symbol structure with mark structure and cognitive structure represent the individual all knowledge and the comprehensive ability to adapt to and remake the real world. The human sensory can only sense the part attributes of the whole things. The stimulus (sign) sensing attribute and meaning (feel representation) are united together to become a natural symbol. When feeling representation constitute perceptual representation (perceptual experience) by perceptual thinking process, the sign with sensory representation correspondingly coupling into natural symbol structure and Establish corresponding relation with objective things. When feeling representation constitute a perceptual image (concept) not existing in the world by rational thinking process, human beings need create artificial symbols to represent the image and establish corresponding relation between the rational experience based on the concept and artificial symbol structure. Therefore, the process of learning knowledge is essentially the process to construct symbol structure which includes three mutual conversion processes:objective things become experience; symbol become experience; experience become symbol. Knowledge teaching is the process that teacher help students construct symbol structure.
thirdly, based on the semiotics the study define connotation of the mathematical symbol, mathematical symbol meaning, mathematical symbol meaning obtain ability, analyzes the factors and difficulties in cultivating the ability to obtain mathematical symbol meaning, and make case studies in the mathematics concept teaching, mathematical proposition teaching and mathematical problem-solving teaching. In the semiotics perspective, mathematical symbol is symbol bearing mathematical information, mainly including mathematics natural symbol, mathematics model symbol, mathematics phonetic symbol, mathematics text symbols, mathematics professional symbol, mathematics chart symbol, mathematics behavior symbol. Mathematical symbol meaning is the whole mathematical symbol structure activated by the mathematical symbols, mainly including the mathematical symbol's form significance, basic meaning, conversion significance, implicit meaning and aesthetic significance, personalized significance, operational meaning seven kind of significance, and it can be measure with the number of marks in the mathematical symbol structure. Mathematical symbol meaning obtaining ability refers to the capacity constructing mathematical symbol structure with the mathematical symbol stimulus, mainly including the mathematical symbol form perceptual ability, meaning association ability, meaning transformation ability, meaning integrated ability and mark operation ability. The factors to Influence the cultivation of mathematical symbol significance are mainly mathematics teachers' mathematical symbol view, teaching resource view, mathematics teaching view and teaching method view. In the mathematics teaching, teachers should change teaching view, select, arrange and present mathematical symbol resources scientifically, apply symbol structure analysis method flexibly, teach students the basic method and thinking mode to construct mathematical symbol meaning structure, discusses the multiple representation of mathematical symbol, construct the mathematical symbol meaning structure comprehensively, and make it become students'cognitive structure in order to promote students' mathematical quality, to promote the all-round development of students on the basis of the request of mathematics curriculum.
Finally, the paper summarize the basic logic of this study:(1) not getting mathematical symbol rich mathematical meaning is the main cause making students be afraid of, and hate mathematics, feeling difficult to learn mathematics;(2) the teachers'ignoring mathematical symbol teaching is the main reason to lead to students' mathematical symbol meaning obtain ability lower;(3) the teacher'one-sided mathematical symbol view and knowledge view is the main reason causing teachers ignore mathematical symbol teaching;(4) mathematical symbol structure contains all information of the mathematical knowledge which need the learner' perception, discovery, digestion and construction;(5) getting the information of mathematical symbol structure need students with higher mathematical symbol meaning obtain ability;(6) the core to training mathematical symbol meaning obtain ability is to think "what it means" more than "what it is";(7) training students'mathematical symbol meaning obtain ability need teachers to change their one-sided symbolic view, knowledge view, learning view and teaching view. The final conclusion is that cultivating and improving the students' mathematical symbol meaning obtain ability is the effective, feasible, operational ways and methods to solve some problems in teaching.
引文
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③李雅云,李宝庆,小学生数学语言学习的调查研究[J],教学与管理,2011,(29)：20-22.
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