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义务教育阶段学生数学符号意识发展水平研究

【作者】 朱立明

【导师】 马云鹏;

【作者基本信息】 东北师范大学, 课程与教学论, 2018, 博士

【摘要】 数学符号的教育意义在于它超越了所要表达的具有象征性的事物,使数学从冗长繁琐的文字表达束缚下释放出来,数学在真正意义上成为传达思想的“语言”。在义务教育阶段,儿童的思维逐渐开始由形象思维向抽象思维转变,学生的这一思维转换的过程恰恰是数学符号形成与发展的一个缩影,这个阶段所接触的数学符号及其意义,成为理解数学本质的标志,以及以后学习数学和其他知识的重要基础,是学生能够更好地进行数学学习的前提保障,只有掌握最基础、最简单的数学符号,才能更好地进行数学运算与推理、思考与表达。1990年,希尔伯特在其《数学问题》中说:“算术符号是文字化的图形,而几何图形则是图形化的公式;没有一个数学家能缺少这些图形化的公式,正如数学演算中他们不能不使用加、脱括号的操作或者其他分析符号一样。”20世纪以来,希尔伯特的形式主义数学哲学风行世界,数学形式化成为当今数学的主流。时至今日,一切严谨的数学,都是通过形式化的数学符号加以表示的。古代数学大多用自然语言来书写,后来渐渐地使用字母、图形和公式。这是因为数学符号不会像自然语言那样带有明显的歧义,它是提升数学确定性的重要手段,为科学思维从普遍性、深刻性的角度把握数学对象的本质提供保障。在数学学习中,数学符号给人以冰冷的魅力之感,我们更应该感受符号形式化背后的火热的内涵。符号意识写入我国《义务教育数学课程标准》(2001年标准提出符号感),数学符号意识的培养的研究相对滞后。由于教育理念从以知识为本到以人为本的转变,发展学生的符号意识已成为中小学数学课程的目标,是我国义务教育阶段的一种数学核心素养,体现数学学科的逻辑性与学生的认知水平。从已有研究来看,关于义务教育阶段学生数学符号意识发展水平的研究并不多见,在数学教育中,依然缺乏关于数学符号意识的较为系统的研究。因此,本研究从学生数学符号意识发展的视角,探求义务教育阶段学生数学符号意识的内涵、分析层次及其发展水平,为培养学生数学符号意识提供理论上的借鉴。本研究以数学思想与数学核心素养作为切入点,以量化研究方法为主,辅以质化研究方法进行讨论分析,通过厘清数学符号意识内涵,构建数学符号意识的分析层次,最终划分学生数学符号意识的发展水平。首先,通过对数学符号意识的关键词进行频率统计,经过两轮专家咨询,得到数学符号意识的分析层次。其次,在该分析层次及相关理论的基础之上,根据题项设计的基本原理与选择的基本原则,编制义务教育阶段学生数学符号意识的测评问卷,对义务教育阶段1至9年级学生数学符号意识进行测评。对测评结果进行统计分析,并对个别学生进行访谈。通过研究得出以下主要结论:第一,义务教育阶段学生数学符号意识可以从四个层次进行分析。这四个层次分别是数学符号的感知与识别、数学符号的理解与运算、数学符号的联想与推理、数学符号的抽象与表达。四个分析层次是由低到高不断提升的,数学符号的感知与识别是数学符号意识的最低层次,数学符号的抽象与表达是数学符号意识的最高层次。第二,义务教育阶段学生数学符号意识存在三个节点,即二年级与三年级之间、四年级与五年级之间、七年级与八年级之间,这三个节点将义务教育分成四个集合,说明义务教育阶段学生数学符号意识发展过程中存在着一个“关键年龄”,这个“关键年龄”就是我们在培养学生数学符号意识过程中特别需要注意的关键所在,是义务教育阶段学生数学符号意识发展水平划分的依据。第三,义务教育阶段学生数学符号意识可以划分为四个发展水平,学生数学符号意识的发展是从简单到复杂,从低级到高级,从具象到抽象,呈现层次性发展的,随着年级的增长学生数学符号意识发展水平逐渐升高。一年级和二年级的特征是数学符号意识具有可感性与具体化,称之为经验观察水平,三年级和四年级的特征是数学符号意识具有深刻性与特殊化,称之为本质内化水平,五年级、六年级和七年级的特征是数学符号意识具有逻辑性与形式化,称之为理性辩证水平,八年级和九年级的特征是数学符号意识具有整体性与模式化,称之为结构普适水平,这与《义务教育数学课程标准》中学段的划分是有区别的。从这个意义上来看,虽然小学与初中的数学教育在形式上是分开的,课程标准是分段的,但是在学生数学符号意识的发展上具有一定的连贯性。

【Abstract】 The education significance of mathematical symbols is that it transcends symbolic things that it expresses,which releases maths from the shackles of cumbersome expression of words and language,and thus mathematics in the real sense,has become the “Language” of conveying thoughts.At the stage of compulsory education,the children’ thinking begins to gradually shift from image thinking to abstract thinking,and the transformation process of thinking is the epitome of the formation and development of mathematical symbols.The mathematical symbols and its meaning students learn at this stage will become the sign of Understanding of mathematics essence and the important foundation of mathematics study later,it is the premise of learning mathematics better.Only through mastering the most basic and simplest mathematical symbols can they reason,think and express better.1990,Hilbert said in his mathematical problems: “the math symbols are text graphics,and the geometry is a graphical formula;Any mathematician can not lack these graphical formulas,just as they have to apply the operation of adding or removing brackets or other analytical symbols in the mathematics operation.” Since the 20 th century,Hilbert formalism mathematical philosophy is popular around the world,and mathematical formalization has become the mainstream in mathematics research now.Today,all the rigorous mathematics,are represented by the formal mathematical symbols.Most of the ancient mathematics were written in natural languages,then gradually letters,graphics and equations were used.This is because the mathematical symbols are unlike natural languages with no obvious ambiguity,and it is an important way to improve mathematical certainty,providing guarantee to grasp the essence of math from the perspective of the universality and profundity for scientific thinking.In mathematics learning,therefore,mathematical symbols impress people with the sense of icy charm,we should feel more about hot connotation behind formalized symbols.Symbol sense was written into our mathematics curriculum standard in compulsory education in 2011,so the research on the cultivation of mathematical symbol sense is lagging behind.Because education philosophy from knowledge-based to people-oriented and developing the students’ consciousness of symbols has become a primary and secondary school mathematics curriculum goals.It is the core mathematical competencies during stage of compulsory education in China,and it also reflects the logic of mathematics and students’ cognitive level.On former researches,there are not many researches about the level of development ofstudents’ mathematical symbol sense within the entire stage of compulsory education.In mathematics education,systematic empirical research on the mathematical symbol sense is still insufficient.Therefore,from the perspective of development of the mathematical symbol sense,this research explores the connotation of students’ mathematical symbol sense during the stage of compulsory education and analyzes hierarchy construction and its development level,in order to provide the theory reference for developing students’ mathematical symbol sense.With the mathematical thought and the core of mathematical conpetencies as an entry point,this research is discussed mainly with quantitative method,supplemented by qualitative method,the research is analysed by clarifying the meaning of mathematical symbol sense,constructing the analysis level of the mathematical symbol sense,and finally dividing the development level of the students’ mathematics symbol sense.First of all,through the mathematical symbol sense on keywords frequency statistics,after two experts consultation,the analysis of the levels of student’ mathematical symbol sense comes out.Next,on the basis of the analysis hierarchy and the related theory,according to the basic principle of item design and basic principles of selecting,assessment questionnaire of consciousness of mathematical symbols in stage of compulsory education is prepared,to evaluate students’ consciousness of mathematical symbols for the stage of compulsory education from grade 1 to 9.Finally,the researcher makes statistical analysis of the students’ answers,and interview some individual students.The following conclusions are made from this study:Firstly,students’ mathematical symbol sense in the stage of compulsory education can be analyzed from four levels,which is the perception and recognition of mathematical symbols,understanding and operation of mathematical symbols,association and reasoning of the mathematical symbols,and the abstract and expression of mathematical symbols.These four levels of analysis are improved from low to high,and mathematical symbols of perception and recognition is the lowest level of the consciousness,while the abstract and expression of mathematical symbols is the highest level.Secondly,students’ mathematical symbol sense in compulsory education exists three demarcation,namely grade two and grade three,grade four and grade five,grade seven and grade eight.Compulsory education is divided into four set by the demarcation,which explains that there exits “a key age” in the development of mathematical symbol sense in compulsory education.The “age” is what we need to pay special attention to in the process of developing students’ symbol sense,and it is also the division basis of developing level of students’ mathematical symbol sense in compulsory education.Thirdly,students’ mathematical symbol sense in compulsory education can be divided into four developing levels,the development of which has distinct gradation from simplicity to complexity,from low level to high level,from the concrete to the abstract,and the symbolsense develops as the growth of their grades.Symbol sense in grade 1 and grade 2 is characterized by perception and and embodiment,thus we call this experience observation level.Symbol sense in grade 3 and grade 4 is characterized by profundity and specialization,thus we can this nature internalization level.Symbol sense in grade 4,grade 5 and grade 6 is characterized by logicality and formalization,thus we call it rational dialectical level.Symbol sense in grade 8 and grade 9 is characterized by integrity and pattern,thus we call this structure universal level,which is quite different from the division of Curriculum Standard of High school Mathematics.In this sense,although teaching form in primary school and junior high school mathematics is separated,and curriculum standard is segmented,there is a certain continuity in the development of students’ mathematical symbol sense.

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  • 【网络出版投稿人】 东北师范大学
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