六年級學(xué)生一元一次方程學(xué)習(xí)認(rèn)知困難分析
發(fā)布時(shí)間：2019-05-27 00:21
【摘要】：隨著國內(nèi)外對代數(shù)學(xué)研究的重視,關(guān)于方程的研究也逐漸進(jìn)入研究者的視野。早在18世紀(jì)末至19世紀(jì)初,人們就漸漸把代數(shù)理解為研究方程理論的科學(xué)。一元一次方程無疑是方程理論中最基礎(chǔ)、最根本的內(nèi)容。但是,目前關(guān)于一元一次方程的研究主要集中在一元一次方程在解決實(shí)際問題中的作用及在此過程中思維的轉(zhuǎn)變過程,以及學(xué)生在學(xué)習(xí)一元一次方程學(xué)習(xí)過程中容易出現(xiàn)的錯(cuò)誤進(jìn)行簡單的歸類,沒有對學(xué)生出現(xiàn)這些錯(cuò)誤背后的原因進(jìn)行深入的分析與探討。本文主要采取質(zhì)的研究方法,對學(xué)生在學(xué)習(xí)一元一次方程出現(xiàn)的認(rèn)知錯(cuò)誤進(jìn)行實(shí)證研究。筆者首先對學(xué)生在學(xué)習(xí)一元一次方程整章時(shí)的作業(yè)卷、測試卷、練習(xí)冊中出現(xiàn)的錯(cuò)誤進(jìn)行收集、整理、歸納;對不同類型的錯(cuò)誤進(jìn)行分類、精簡,最后選取最具代表性的案例作為本研究的依據(jù)。在對學(xué)生作業(yè)卷、測試卷、練習(xí)冊中出現(xiàn)的錯(cuò)誤進(jìn)行簡單的分類后,又在已有的對求解一元一次方程的研究的基礎(chǔ)上,形成了具有以下三個(gè)維度的分析框架:主要從算術(shù)中的“運(yùn)算定勢”、算術(shù)中的結(jié)構(gòu)化概念或策略理解不足、代數(shù)中的概念或策略理解不足三個(gè)方面對學(xué)生在學(xué)習(xí)一元一次方程中出現(xiàn)的認(rèn)知錯(cuò)誤及其原因進(jìn)行分析和探究。研究結(jié)果表明:(1)算術(shù)中的“運(yùn)算定勢”下的概念混淆或操作遺漏,主要有以下三個(gè)方面:算術(shù)中等號“=”的程序性理解使學(xué)生將方程的求解過程與代數(shù)式運(yùn)算混淆;算術(shù)中帶分?jǐn)?shù)的“并列”寫法在方程求解過程中被當(dāng)成整數(shù)部分與分?jǐn)?shù)部分的乘法;去分母時(shí)利用算術(shù)中的通分策略,忽略對整數(shù)部分的操作。(2)算術(shù)中結(jié)構(gòu)化概念或策略理解不足下的概念或操作錯(cuò)誤,主要有以下三個(gè)方面:對“+”“-”意義的理解仍然停留在“運(yùn)算符號”,不能將其理解為性質(zhì)符號或者表示相反意義的量,給學(xué)生理解“項(xiàng)”“移項(xiàng)”等概念或進(jìn)行“移項(xiàng)”操作造成了一定干擾;對等式性質(zhì)理解不足,影響學(xué)生對“移項(xiàng)”概念及操作的正確理解;乘法對加法的分配率理解不足使學(xué)生不能在求解方程過程中正確地進(jìn)行去括號操作。(3)代數(shù)中的概念或策略理解不足妨礙概念、原理的理解或策略的使用,主要有以下五個(gè)方面:不能正確理解一元一次方程的“三要素”(是方程、含有一個(gè)未知數(shù),未知數(shù)的次數(shù)為1)需要同時(shí)滿足,使學(xué)生不能正確理解或判斷一元一次方程;不能正確理解方程的同解原理,使學(xué)生不能正確理解方程的解的概念;不能正確理解參數(shù)的意義,致使學(xué)生不能接受含有字母的代數(shù)式作為答案;不能真正理解去括號法則,導(dǎo)致學(xué)生在去括號過程中出現(xiàn)各種各樣的錯(cuò)誤;在列方程解應(yīng)用題的過程中,容易關(guān)注方程的解而忽略問題的解,或當(dāng)應(yīng)用題中含有多個(gè)未知量時(shí),學(xué)生往往不能找出恰當(dāng)?shù)臉?biāo)準(zhǔn)量、不能找到未知量與未知量的關(guān)系,或者當(dāng)應(yīng)用題中含有多個(gè)復(fù)雜的等量關(guān)系時(shí),學(xué)生不容易找出正確的等量關(guān)系列方程。總之,本研究不僅對學(xué)生在學(xué)習(xí)一元一次方程時(shí)出現(xiàn)的錯(cuò)誤進(jìn)行簡單的分類、歸納,而且從認(rèn)知層面對學(xué)生出現(xiàn)的這些錯(cuò)誤進(jìn)行了細(xì)致、深入的分析和探究。不僅為了解學(xué)生在學(xué)習(xí)一元一次方程時(shí)出現(xiàn)的錯(cuò)誤、認(rèn)知水平提供了一個(gè)視角;也為教師在教授一元一次方程時(shí)采取相應(yīng)的有針對性教學(xué)策略提供了很好的依據(jù)。
[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.
【學(xué)位授予單位】：上海師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：G623.5
本文編號：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.
【學(xué)位授予單位】：上海師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：G623.5
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