六年级学生一元一次方程学习认知困难分析
发布时间：2019-05-27 00:21
【摘要】：随着国内外对代数学研究的重视,关于方程的研究也逐渐进入研究者的视野。早在18世纪末至19世纪初,人们就渐渐把代数理解为研究方程理论的科学。一元一次方程无疑是方程理论中最基础、最根本的内容。但是,目前关于一元一次方程的研究主要集中在一元一次方程在解决实际问题中的作用及在此过程中思维的转变过程,以及学生在学习一元一次方程学习过程中容易出现的错误进行简单的归类,没有对学生出现这些错误背后的原因进行深入的分析与探讨。本文主要采取质的研究方法,对学生在学习一元一次方程出现的认知错误进行实证研究。笔者首先对学生在学习一元一次方程整章时的作业卷、测试卷、练习册中出现的错误进行收集、整理、归纳;对不同类型的错误进行分类、精简,最后选取最具代表性的案例作为本研究的依据。在对学生作业卷、测试卷、练习册中出现的错误进行简单的分类后,又在已有的对求解一元一次方程的研究的基础上,形成了具有以下三个维度的分析框架:主要从算术中的“运算定势”、算术中的结构化概念或策略理解不足、代数中的概念或策略理解不足三个方面对学生在学习一元一次方程中出现的认知错误及其原因进行分析和探究。研究结果表明:(1)算术中的“运算定势”下的概念混淆或操作遗漏,主要有以下三个方面:算术中等号“=”的程序性理解使学生将方程的求解过程与代数式运算混淆;算术中带分数的“并列”写法在方程求解过程中被当成整数部分与分数部分的乘法;去分母时利用算术中的通分策略,忽略对整数部分的操作。(2)算术中结构化概念或策略理解不足下的概念或操作错误,主要有以下三个方面:对“+”“-”意义的理解仍然停留在“运算符号”,不能将其理解为性质符号或者表示相反意义的量,给学生理解“项”“移项”等概念或进行“移项”操作造成了一定干扰;对等式性质理解不足,影响学生对“移项”概念及操作的正确理解;乘法对加法的分配率理解不足使学生不能在求解方程过程中正确地进行去括号操作。(3)代数中的概念或策略理解不足妨碍概念、原理的理解或策略的使用,主要有以下五个方面:不能正确理解一元一次方程的“三要素”(是方程、含有一个未知数,未知数的次数为1)需要同时满足,使学生不能正确理解或判断一元一次方程;不能正确理解方程的同解原理,使学生不能正确理解方程的解的概念;不能正确理解参数的意义,致使学生不能接受含有字母的代数式作为答案;不能真正理解去括号法则,导致学生在去括号过程中出现各种各样的错误;在列方程解应用题的过程中,容易关注方程的解而忽略问题的解,或当应用题中含有多个未知量时,学生往往不能找出恰当的标准量、不能找到未知量与未知量的关系,或者当应用题中含有多个复杂的等量关系时,学生不容易找出正确的等量关系列方程。总之,本研究不仅对学生在学习一元一次方程时出现的错误进行简单的分类、归纳,而且从认知层面对学生出现的这些错误进行了细致、深入的分析和探究。不仅为了解学生在学习一元一次方程时出现的错误、认知水平提供了一个视角;也为教师在教授一元一次方程时采取相应的有针对性教学策略提供了很好的依据。
[Abstract]:With the emphasis of the research on the generation of mathematics at home and abroad, the research on the equation has gradually entered the field of the researchers. As early as the end of the 18th century to the beginning of the 19th century, people gradually understood the algebra as the science of the study of the theory of the equation. The one-dimensional unitary equation is no doubt the most basic and fundamental content in the theory of the equation. However, the present study on the one-dimensional unitary equation is mainly focused on the function of the one-dimensional unitary equation in solving the practical problems and the transformation process of thinking in this process, and the simple classification of the error that the students can easily occur during the learning process of the one-dimensional unitary equation, There is no further analysis and discussion of the reasons behind these errors. This paper mainly takes a qualitative research method to carry out an empirical study on the cognitive errors of the students in the learning of the one-dimensional unitary equation. In this paper, we first collect, sort, and sum up the errors in the work volume, test volume and exercise book when the students are studying the whole chapter of the one-dimensional unitary equation, classify and streamline the errors of different types, and select the most representative cases as the basis for this study. After a simple classification of the errors in the student's job volume, the test volume and the exercise book, the analysis framework with the following three dimensions is formed on the basis of the existing research on solving the one-dimensional equation, mainly from the "operational fixed potential" in the arithmetic, The structural concept or strategy understanding in the arithmetic is not enough, the concept or the strategy in the algebra is not enough to understand the cognitive errors and the causes of the students in learning the one-dimensional unitary equation. The results show that: (1) The concept confusion or operation omission in the "operational fixed potential" of arithmetic mainly includes the following three aspects: the procedural understanding of the arithmetic middle "=" makes the students confuse the solving process of the equation with the algebraic operation; The "juxtaposition"-writing method with a score in arithmetic is used as a multiplication of an integer part and a fractional part in the solving process of the equation; when the denominator is used, the operation of the integer part is ignored by using the general sub-strategy in the arithmetic. (2) The concept or operation error under the understanding of the structural concept or the strategy in the arithmetic mainly includes the following three aspects: the concept of the "It is not to be understood as a property symbol or an amount that represents the opposite sense, which is to be understood by the student."-"The sense of meaning is still on" operation symbol "Move Item" term "Move Item" or the like or the "+" operation is caused to have certain interference; and the understanding of the equation property is insufficient, The understanding of the students' understanding of the concept and operation of the "Move Item" is affected, and the insufficient understanding of the ratio of the multiplication to the addition makes the students unable to perform the bracket operation correctly in the process of solving the equation. (3) The concept or strategy in the algebra is not an obstacle to the understanding of the concept and the principle, or the use of the strategy, mainly including the following five aspects: the "three elements" of the one-dimensional equation cannot be correctly understood (it is an equation, it contains an unknown number, the number of unknowns is 1) needs to be met at the same time, In order to make the students unable to correctly understand or judge the one-dimensional equation, we can't correctly understand the principle of the same solution of the equation, so that the students can't understand the concept of the solution of the equation correctly; the significance of the parameters cannot be correctly understood, so that the students can't accept the algebraic expression containing the letter as the answer; in that course of solving the problem of the application of the equation, it is easy to pay attention to the solution of the equation and ignore the solution of the problem, Students are often unable to find the proper amount of the standard, do not find the relationship between the amount of equivalence and the amount of the equivalent, or when the application problem contains a plurality of complex equal relationships, the students can't easily find the correct equivalent relationship equation. In conclusion, this study not only makes a simple classification and induction to the mistakes that the students have made in the learning of the one-dimensional equation, but also makes a detailed and in-depth analysis and exploration of the errors of the students from the cognitive level. It not only provides a visual angle for understanding the errors and the cognitive level of the students in the learning of the one-dimensional unitary equation, but also provides a good basis for teachers to take corresponding targeted teaching strategies in the teaching of the one-dimensional unitary equation.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：G623.5
本文编号：2485755
[Abstract]:With the emphasis of the research on the generation of mathematics at home and abroad, the research on the equation has gradually entered the field of the researchers. As early as the end of the 18th century to the beginning of the 19th century, people gradually understood the algebra as the science of the study of the theory of the equation. The one-dimensional unitary equation is no doubt the most basic and fundamental content in the theory of the equation. However, the present study on the one-dimensional unitary equation is mainly focused on the function of the one-dimensional unitary equation in solving the practical problems and the transformation process of thinking in this process, and the simple classification of the error that the students can easily occur during the learning process of the one-dimensional unitary equation, There is no further analysis and discussion of the reasons behind these errors. This paper mainly takes a qualitative research method to carry out an empirical study on the cognitive errors of the students in the learning of the one-dimensional unitary equation. In this paper, we first collect, sort, and sum up the errors in the work volume, test volume and exercise book when the students are studying the whole chapter of the one-dimensional unitary equation, classify and streamline the errors of different types, and select the most representative cases as the basis for this study. After a simple classification of the errors in the student's job volume, the test volume and the exercise book, the analysis framework with the following three dimensions is formed on the basis of the existing research on solving the one-dimensional equation, mainly from the "operational fixed potential" in the arithmetic, The structural concept or strategy understanding in the arithmetic is not enough, the concept or the strategy in the algebra is not enough to understand the cognitive errors and the causes of the students in learning the one-dimensional unitary equation. The results show that: (1) The concept confusion or operation omission in the "operational fixed potential" of arithmetic mainly includes the following three aspects: the procedural understanding of the arithmetic middle "=" makes the students confuse the solving process of the equation with the algebraic operation; The "juxtaposition"-writing method with a score in arithmetic is used as a multiplication of an integer part and a fractional part in the solving process of the equation; when the denominator is used, the operation of the integer part is ignored by using the general sub-strategy in the arithmetic. (2) The concept or operation error under the understanding of the structural concept or the strategy in the arithmetic mainly includes the following three aspects: the concept of the "It is not to be understood as a property symbol or an amount that represents the opposite sense, which is to be understood by the student."-"The sense of meaning is still on" operation symbol "Move Item" term "Move Item" or the like or the "+" operation is caused to have certain interference; and the understanding of the equation property is insufficient, The understanding of the students' understanding of the concept and operation of the "Move Item" is affected, and the insufficient understanding of the ratio of the multiplication to the addition makes the students unable to perform the bracket operation correctly in the process of solving the equation. (3) The concept or strategy in the algebra is not an obstacle to the understanding of the concept and the principle, or the use of the strategy, mainly including the following five aspects: the "three elements" of the one-dimensional equation cannot be correctly understood (it is an equation, it contains an unknown number, the number of unknowns is 1) needs to be met at the same time, In order to make the students unable to correctly understand or judge the one-dimensional equation, we can't correctly understand the principle of the same solution of the equation, so that the students can't understand the concept of the solution of the equation correctly; the significance of the parameters cannot be correctly understood, so that the students can't accept the algebraic expression containing the letter as the answer; in that course of solving the problem of the application of the equation, it is easy to pay attention to the solution of the equation and ignore the solution of the problem, Students are often unable to find the proper amount of the standard, do not find the relationship between the amount of equivalence and the amount of the equivalent, or when the application problem contains a plurality of complex equal relationships, the students can't easily find the correct equivalent relationship equation. In conclusion, this study not only makes a simple classification and induction to the mistakes that the students have made in the learning of the one-dimensional equation, but also makes a detailed and in-depth analysis and exploration of the errors of the students from the cognitive level. It not only provides a visual angle for understanding the errors and the cognitive level of the students in the learning of the one-dimensional unitary equation, but also provides a good basis for teachers to take corresponding targeted teaching strategies in the teaching of the one-dimensional unitary equation.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：G623.5
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