7-9年级学生数学符号语言的理解与表示
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摘要
自从《全日制义务教育数学课程标准(讨论稿)》将“符号感”与“空间观念”、“推理能力”一样作为数学学习内容的核心概念提出以来,对学生数学符号使用的研究就成为一个比较活跃的领域。然而,多数研究集中在数学符号的特点和作用以及对“符号感”或者“符号意识”的内涵、外延分析和培养途径等方面,这些研究往往立足于思辨或者经验的简单总结,缺乏系统地对学生在数学学习中使用数学符号的真实情况的剖析。为此,本文将数学符号放在数学、心理学、语言学以及教育学等多学科的视角下,对7-9年级学生数学符号语言的理解与表示进行了较为系统的研究。研究的意义在于:丰富了数学教育教学的理论和实践研究;从数学符号语言理解与表示的角度为解决学困生问题寻找新途径;为教师的符号教学设计及反思提供实践依据和理论支撑。
研究从数学符号语言的理解和表示两个方面入手,主要围绕着学生的符号理解与表示在哪些方面存在不同表现和典型错误?这些不同表现的本质是什么?依据这些不同表现及其本质,教师应该怎样开展符号教学?等问题展开。
针对这些问题,本文对7-9年级的496位学生进行了问卷调查,并针对问卷中出现的具体情况进行了访谈,以此了解学生完成问卷过程中的真实想法;还对12位学生进行了基于任务的访谈,进一步探究学生数学符号语言理解和表示的过程。对问卷和访谈收集来的数据运用统计方法作了进一步的分析。
本文的研究基础一方面来自于对数学符号语言的基本认识,另一方面来自于哲学、教育学、心理学、语言学等学科的成熟理论,具体说来,主要运用了“表示——联系”的理解理论、“认识论的三角形”模型等理论分析框架。
本文获得的主要研究结论是:
(1)对于同样的数学符号语言,学生的理解结果存在一些不同,理解过程也存在一定的差异。这些不同和差异主要体现在理解的广度、符号之间关系的理解、字母意义的理解以及理解策略等四个方面。
(2)对于同样的数学意义,学生的符号表示结果存在一些不同,表示过程也存在一定的差异。这些不同和差异主要体现在符号表示的多样性、系统性、准确性以及策略性等四个方面。
(3)随着年级的增高,学生在符号理解和表示的多个方面并没有获得实质性的提高,其它数学知识的增长并不预示着在数学符号语言理解和表示方面一定得到了提高。
(4)学生在符号理解和表示上的不同表现与他们所具有的学业水平是比较一致的。总的说来,学优生在很多方面都明显的优于中等生和学困生。学生在数学符号语言的理解和表示上所存在的问题也是制约学生数学学习发展的一个重要因素。
(5)在数学符号语言理解和表示的过程中,学生面对的数学符号语言不仅仅是简约的、精确的、通用的,而且还是复杂的、多样的、自足的。
(6)导致学生在符号理解和表示上出现不同表现的原因是多方面的,本质上可以将它们归于两个方面:语言知识和符号意识。具体说来就是语言知识掌握的多与少、系统与零碎的不同,与其它知识联系紧密与疏松的不同,语言知识运用的方法有与无、灵活与呆板的不同;符号意识的强与弱、优与劣的不同。它们既单独对学生的符号理解和表示产生影响,也结合在一起产生作用。
(7)教师通常能够将相关的符号教学内容纳入日常的教学活动之中,但是,还是“顺带”进行,总体来说,符号教学还是一种无目的、无意识的行为。具体的教学过程中也存在一些诸如“死记硬背”、“重形式、轻内容”等不良行为。为此,提出了一些教学建议:符号教学应该成为一种有意识的、自觉的行为;符号教学应该对数学符号语言的学科特点和教育特点都有所关注;符号教学应该兼顾“形式”与“内容”;符号教学应该充分关注学生的行为。
本文力图的创新之处在于:其一、较为系统地获得了7-9年级学生数学符号语言理解和表示基本情况;其二、将数学符号视为一种语言,从语言的角度来发现问题、分析问题、解决问题;其三、关注数学符号语言的教学,依据学生数学符号语言的理解与表示中的问题以及符号教学中的不足提出相应的教学建议。
关于学生的数学符号语言运用的研究是一个较为广阔的领域,本文只是对其中的一些部分有所涉及,对学生心理层面的剖析深入程度不够,对所提出的符号教学建议是否具有实践价值也还没有进行系统研究,这些都将是我们进一步研究的方向。
Since "symbol sense" is looked on as a core concept of mathematics learning as "space idea" and " reason capability " in the standard of mathematics curriculum for compulsory education, studying the students' the use of mathematical symbols gradually becomes a more active field. These studies are focused on the characteristics and functions of mathematical symbols, focused on the meaning and training's approaches of "symbol sense". These studies are based on theory or experience and lack the systematic analysis to the real situation of the students using mathematical symbols. In this paper, the use of mathematical symbols being placed on the scope of language, the studies is to the real situation of the students using the language of mathematical symbols.
The article will study the understanding and representation of mathematical symbols. The studies envelop several problems: Which differences the students have in the understanding and representation of mathematical symbols? What these differences' essence is? Based on these differences and essence, how to teach the language of mathematical symbols?
To solve these problems, 496 students are surveyed though the questionnaires. In order to understand the students' ideas in the process of understanding and representation of mathematical symbols interview, some students are interviewed. 12 students are concerned with the task-based interviews to further explore the process of understanding and representation. To these data from the questionnaires and the interviews, the article uses the methods of statistic to analyze.
The basic of theories that the article is according as are from the cognition to the language of mathematical symbols, also are from the philosophy, pedagogy, psychology and linguistics. The article exerts basically the theory of representing and connecting about understand and the model of the epistemological triangle.
In the article, the studies' main conclusions are:
i) To the same language of mathematical symbols, the understanding results and process of students are some differences. These differences are reflected in four aspects: the understanding extent, the understanding to the relationship between the symbols, the understanding to the letters' meaning, and the understanding strategy.
ii) To the same mathematical meaning, the symbolic representation's results and process are some differences. These differences are reflected in four aspects: multiformity, system, veracity and strategy.
iii) With the grade increasing, the students don't obtain the substantial improvement in the understanding and representation of mathematical symbols. In other words, the growth of the other mathematical knowledge does not indicate the growth of understanding and representation of mathematical symbols.
iv) The students' level of mathematical achievement is accord with their situation in the understanding and representation of mathematical symbols. In general, the higher students are obviously better than the poorer students in understanding and representation of mathematical symbols. In other words, the problems existing the students' understanding and representation of mathematical symbols is also constrained the development of students in the mathematics.
v) In the process of the understanding and representation, the mathematical symbols not only are concise, accurate and universal, but also complicated, faint and self-sufficiency.
vi) The reasons to induce the differences in the understanding and representation of mathematical symbols are from multi-aspects. But, they can be concluded two aspects in substance: the knowledge of language and symbol sense. In the embody, the differences in the knowledge of language are between the more and the little knowledge of language which the students hold, are between the systematic knowledge of language and the fragmentary knowledge of language, are between the tightness and the loosen which the knowledge of language is contact with the other knowledge, are between the flexible ways and the inflexible ways to handle knowledge of language. The differences in the symbol sense are the exact sense and the wrong sense, are the strong sense and the weak sense. The affection to the understanding and representation of mathematical symbols is both produced because of uniting the knowledge of language with the symbol sense and produced because of the singleness.
vii) The teachers are usually able to think over the symbolic teaching in the mathematical teaching. But, as a whole, the symbolic teaching is on the way in the other mathematical teachings. The symbolic teaching is still a kind of aimless and unconscious action. There are also some problems in the process of symbolic teaching as to learn by rote. For this reason, some suggestions are put forward: the symbolic teaching should be self-conscious and intentional; the symbolic teaching should pay attention to the characteristics of mathematical symbols in the education; the symbolic teaching not only should take into account the symbolic form, but also should consider the symbolic content; the symbolic teaching also should adequately consider the students' behaviors.
This article has three innovative aspects. First, the article obtains systemically the students' situation in the understanding and representation of mathematical symbols. Second, the article considers the mathematical symbols as a language and analyzes the problems and the reasons from the language of view. Third, the article pays attention to the teaching of mathematical symbols. According to the problems and shortages of students' understanding and representation to mathematical symbols, the article puts forward some suggestion to the teaching of mathematical symbols.
The study is a broad field about the language of mathematical symbols in the middle school. This article only involves some parts in the field. The article is not more sufficient in the psychological level's analysis. The article also doesn't study the actual effect of the teaching suggestions. Hence, the article can continue the study according to the above-mentioned shortage.
研究从数学符号语言的理解和表示两个方面入手,主要围绕着学生的符号理解与表示在哪些方面存在不同表现和典型错误?这些不同表现的本质是什么?依据这些不同表现及其本质,教师应该怎样开展符号教学?等问题展开。
针对这些问题,本文对7-9年级的496位学生进行了问卷调查,并针对问卷中出现的具体情况进行了访谈,以此了解学生完成问卷过程中的真实想法;还对12位学生进行了基于任务的访谈,进一步探究学生数学符号语言理解和表示的过程。对问卷和访谈收集来的数据运用统计方法作了进一步的分析。
本文的研究基础一方面来自于对数学符号语言的基本认识,另一方面来自于哲学、教育学、心理学、语言学等学科的成熟理论,具体说来,主要运用了“表示——联系”的理解理论、“认识论的三角形”模型等理论分析框架。
本文获得的主要研究结论是:
(1)对于同样的数学符号语言,学生的理解结果存在一些不同,理解过程也存在一定的差异。这些不同和差异主要体现在理解的广度、符号之间关系的理解、字母意义的理解以及理解策略等四个方面。
(2)对于同样的数学意义,学生的符号表示结果存在一些不同,表示过程也存在一定的差异。这些不同和差异主要体现在符号表示的多样性、系统性、准确性以及策略性等四个方面。
(3)随着年级的增高,学生在符号理解和表示的多个方面并没有获得实质性的提高,其它数学知识的增长并不预示着在数学符号语言理解和表示方面一定得到了提高。
(4)学生在符号理解和表示上的不同表现与他们所具有的学业水平是比较一致的。总的说来,学优生在很多方面都明显的优于中等生和学困生。学生在数学符号语言的理解和表示上所存在的问题也是制约学生数学学习发展的一个重要因素。
(5)在数学符号语言理解和表示的过程中,学生面对的数学符号语言不仅仅是简约的、精确的、通用的,而且还是复杂的、多样的、自足的。
(6)导致学生在符号理解和表示上出现不同表现的原因是多方面的,本质上可以将它们归于两个方面:语言知识和符号意识。具体说来就是语言知识掌握的多与少、系统与零碎的不同,与其它知识联系紧密与疏松的不同,语言知识运用的方法有与无、灵活与呆板的不同;符号意识的强与弱、优与劣的不同。它们既单独对学生的符号理解和表示产生影响,也结合在一起产生作用。
(7)教师通常能够将相关的符号教学内容纳入日常的教学活动之中,但是,还是“顺带”进行,总体来说,符号教学还是一种无目的、无意识的行为。具体的教学过程中也存在一些诸如“死记硬背”、“重形式、轻内容”等不良行为。为此,提出了一些教学建议:符号教学应该成为一种有意识的、自觉的行为;符号教学应该对数学符号语言的学科特点和教育特点都有所关注;符号教学应该兼顾“形式”与“内容”;符号教学应该充分关注学生的行为。
本文力图的创新之处在于:其一、较为系统地获得了7-9年级学生数学符号语言理解和表示基本情况;其二、将数学符号视为一种语言,从语言的角度来发现问题、分析问题、解决问题;其三、关注数学符号语言的教学,依据学生数学符号语言的理解与表示中的问题以及符号教学中的不足提出相应的教学建议。
关于学生的数学符号语言运用的研究是一个较为广阔的领域,本文只是对其中的一些部分有所涉及,对学生心理层面的剖析深入程度不够,对所提出的符号教学建议是否具有实践价值也还没有进行系统研究,这些都将是我们进一步研究的方向。
Since "symbol sense" is looked on as a core concept of mathematics learning as "space idea" and " reason capability " in the standard of mathematics curriculum for compulsory education, studying the students' the use of mathematical symbols gradually becomes a more active field. These studies are focused on the characteristics and functions of mathematical symbols, focused on the meaning and training's approaches of "symbol sense". These studies are based on theory or experience and lack the systematic analysis to the real situation of the students using mathematical symbols. In this paper, the use of mathematical symbols being placed on the scope of language, the studies is to the real situation of the students using the language of mathematical symbols.
The article will study the understanding and representation of mathematical symbols. The studies envelop several problems: Which differences the students have in the understanding and representation of mathematical symbols? What these differences' essence is? Based on these differences and essence, how to teach the language of mathematical symbols?
To solve these problems, 496 students are surveyed though the questionnaires. In order to understand the students' ideas in the process of understanding and representation of mathematical symbols interview, some students are interviewed. 12 students are concerned with the task-based interviews to further explore the process of understanding and representation. To these data from the questionnaires and the interviews, the article uses the methods of statistic to analyze.
The basic of theories that the article is according as are from the cognition to the language of mathematical symbols, also are from the philosophy, pedagogy, psychology and linguistics. The article exerts basically the theory of representing and connecting about understand and the model of the epistemological triangle.
In the article, the studies' main conclusions are:
i) To the same language of mathematical symbols, the understanding results and process of students are some differences. These differences are reflected in four aspects: the understanding extent, the understanding to the relationship between the symbols, the understanding to the letters' meaning, and the understanding strategy.
ii) To the same mathematical meaning, the symbolic representation's results and process are some differences. These differences are reflected in four aspects: multiformity, system, veracity and strategy.
iii) With the grade increasing, the students don't obtain the substantial improvement in the understanding and representation of mathematical symbols. In other words, the growth of the other mathematical knowledge does not indicate the growth of understanding and representation of mathematical symbols.
iv) The students' level of mathematical achievement is accord with their situation in the understanding and representation of mathematical symbols. In general, the higher students are obviously better than the poorer students in understanding and representation of mathematical symbols. In other words, the problems existing the students' understanding and representation of mathematical symbols is also constrained the development of students in the mathematics.
v) In the process of the understanding and representation, the mathematical symbols not only are concise, accurate and universal, but also complicated, faint and self-sufficiency.
vi) The reasons to induce the differences in the understanding and representation of mathematical symbols are from multi-aspects. But, they can be concluded two aspects in substance: the knowledge of language and symbol sense. In the embody, the differences in the knowledge of language are between the more and the little knowledge of language which the students hold, are between the systematic knowledge of language and the fragmentary knowledge of language, are between the tightness and the loosen which the knowledge of language is contact with the other knowledge, are between the flexible ways and the inflexible ways to handle knowledge of language. The differences in the symbol sense are the exact sense and the wrong sense, are the strong sense and the weak sense. The affection to the understanding and representation of mathematical symbols is both produced because of uniting the knowledge of language with the symbol sense and produced because of the singleness.
vii) The teachers are usually able to think over the symbolic teaching in the mathematical teaching. But, as a whole, the symbolic teaching is on the way in the other mathematical teachings. The symbolic teaching is still a kind of aimless and unconscious action. There are also some problems in the process of symbolic teaching as to learn by rote. For this reason, some suggestions are put forward: the symbolic teaching should be self-conscious and intentional; the symbolic teaching should pay attention to the characteristics of mathematical symbols in the education; the symbolic teaching not only should take into account the symbolic form, but also should consider the symbolic content; the symbolic teaching also should adequately consider the students' behaviors.
This article has three innovative aspects. First, the article obtains systemically the students' situation in the understanding and representation of mathematical symbols. Second, the article considers the mathematical symbols as a language and analyzes the problems and the reasons from the language of view. Third, the article pays attention to the teaching of mathematical symbols. According to the problems and shortages of students' understanding and representation to mathematical symbols, the article puts forward some suggestion to the teaching of mathematical symbols.
The study is a broad field about the language of mathematical symbols in the middle school. This article only involves some parts in the field. The article is not more sufficient in the psychological level's analysis. The article also doesn't study the actual effect of the teaching suggestions. Hence, the article can continue the study according to the above-mentioned shortage.
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