初中学生对方程思想的理解
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摘要
方程思想是初中数学知识体系中最基本的数学思想之一，它在代数、几何的学习中都有着广泛的应用，掌握方程思想不仅能够使初中学生更好地理解和掌握初中数学的基本知识，而且对学生进一步学习高一级的数学知识，以及物理、化学等其它学科知识都将起到非常重要的作用。
本文通过对初中四个年级414名学生的数学测试和访谈，对以下4个方面进行了研究：初中学生对方程概念的理解；初中学生应用方程思想的主动意识；初中学生列方程的常用策略；影响初中学生正确列方程的主要因素。得出以下结论：
1．学生对方程概念的理解是比较片面和肤浅的，特别是学生对方程中等号作用的理解比较狭隘；
2．初中学生在低年级阶段，应用方程思想的主动意识有一定的局限性，特别是6年级的学生对算术方法有一定的依赖性，随着年级的升高应用方程的主动意识逐渐增强；
3．初中学生常用的列方程的策略主要有以下5种：
①公式策略；②归类策略；③综合策略；④分析策略；⑤图表策略。
4．影响初中学生正确列方程的主要因素有：
①心理因素：学生对掌握和应用方程方法的自信心不足，比较依赖惯用的算术方法；
②学生的阅读和表述水平：不能正确地将基本的数量关系转化成等量关系；
③问题中数量关系的复杂程度：不能深入地将实际问题中隐含的数量关系挖掘出来；
④生活经验与社会经验有限：不能准确理解题目中具有实际意义的专有术语；
⑤对问题类型的熟悉程度：不能自然地检索或构造出与问题情景的刺激其相匹配的解题策略。
最后，从本研究的结果出发，笔者对初中阶段方程思想的教学，提出了一些建议及有待继续探讨的若干问题。
The Equation Thinking is one of the most basic Mathematics Thoughts. It is widely used in learning of Algebra and Geometry. To master the Equation Thinking can not only help junior students to understand and master the basic math knowledge better. It also plays an important role in helping students learn further math knowledge as well as other subjects such as physics and chemistry.
Through math tests and interviews among 414 students from Grade6 to Grade9, the thesis studies the following 4 aspects: Junior students' understanding on the concept of the equation; Junior students' initiative on applying equation thinking; The common strategies used by junior students to make equation; The main influencing factors on students' making equations correctly. The conclusions are as follow:
1. Students' understanding on the concept of equation is unilateral and superficial, especially for the understanding on the function of the equal-sign in an equation.
2. The initiative on applying equation thinking is limited among the junior students in lower grades. And the Grade 6 students tend to use arithmetic methods. But the initiative will be developed gradually when students go to higher grades.
3. The common strategies widely used by junior students to make equations:①Formula Strategy②Classification Strategy③Integration Strategy④Analysis Strategy⑤Chart Strategy
4. The main influencing factors on students' making equations correctly
①The mental factors: Lack confidence of mastering and applying equation methods but rely on arithmetic methods;
②The level of reading and expressing: Cannot change the basic quantities relation into equality relation correctly;
③The complexity of the quantities relation in the problem: Cannot dig out the quantities relation deep hidden in the problem;
④The limit of life experience and social experience: Cannot understand the terms with practical meanings in the problem;
⑤Not familiar with the problem: Cannot naturally search or make up the solving strategic according to the problem situation.
At last, according to the results of the study, the author raises some advice and several issues on the equation teaching among junior students for further study.
本文通过对初中四个年级414名学生的数学测试和访谈，对以下4个方面进行了研究：初中学生对方程概念的理解；初中学生应用方程思想的主动意识；初中学生列方程的常用策略；影响初中学生正确列方程的主要因素。得出以下结论：
1．学生对方程概念的理解是比较片面和肤浅的，特别是学生对方程中等号作用的理解比较狭隘；
2．初中学生在低年级阶段，应用方程思想的主动意识有一定的局限性，特别是6年级的学生对算术方法有一定的依赖性，随着年级的升高应用方程的主动意识逐渐增强；
3．初中学生常用的列方程的策略主要有以下5种：
①公式策略；②归类策略；③综合策略；④分析策略；⑤图表策略。
4．影响初中学生正确列方程的主要因素有：
①心理因素：学生对掌握和应用方程方法的自信心不足，比较依赖惯用的算术方法；
②学生的阅读和表述水平：不能正确地将基本的数量关系转化成等量关系；
③问题中数量关系的复杂程度：不能深入地将实际问题中隐含的数量关系挖掘出来；
④生活经验与社会经验有限：不能准确理解题目中具有实际意义的专有术语；
⑤对问题类型的熟悉程度：不能自然地检索或构造出与问题情景的刺激其相匹配的解题策略。
最后，从本研究的结果出发，笔者对初中阶段方程思想的教学，提出了一些建议及有待继续探讨的若干问题。
The Equation Thinking is one of the most basic Mathematics Thoughts. It is widely used in learning of Algebra and Geometry. To master the Equation Thinking can not only help junior students to understand and master the basic math knowledge better. It also plays an important role in helping students learn further math knowledge as well as other subjects such as physics and chemistry.
Through math tests and interviews among 414 students from Grade6 to Grade9, the thesis studies the following 4 aspects: Junior students' understanding on the concept of the equation; Junior students' initiative on applying equation thinking; The common strategies used by junior students to make equation; The main influencing factors on students' making equations correctly. The conclusions are as follow:
1. Students' understanding on the concept of equation is unilateral and superficial, especially for the understanding on the function of the equal-sign in an equation.
2. The initiative on applying equation thinking is limited among the junior students in lower grades. And the Grade 6 students tend to use arithmetic methods. But the initiative will be developed gradually when students go to higher grades.
3. The common strategies widely used by junior students to make equations:①Formula Strategy②Classification Strategy③Integration Strategy④Analysis Strategy⑤Chart Strategy
4. The main influencing factors on students' making equations correctly
①The mental factors: Lack confidence of mastering and applying equation methods but rely on arithmetic methods;
②The level of reading and expressing: Cannot change the basic quantities relation into equality relation correctly;
③The complexity of the quantities relation in the problem: Cannot dig out the quantities relation deep hidden in the problem;
④The limit of life experience and social experience: Cannot understand the terms with practical meanings in the problem;
⑤Not familiar with the problem: Cannot naturally search or make up the solving strategic according to the problem situation.
At last, according to the results of the study, the author raises some advice and several issues on the equation teaching among junior students for further study.
引文
[1] 上海市教育委员会编．上海市中小学数学课程标准(试行稿)．上海：上海教育出版社，2004
[2] 布鲁纳教育过程．上海：上海人民出版社，1973
[3] 皮连生．教育心理学．上海：上海教育出版社，2007
[4] 李士锜．PME：数学教育心理．上海：华东师范大学出版社，2001
[5] 林永伟．叶立军．数学史与数学教育．浙江：浙江大学出版社，2004
[6] 陈向明．教师如何做质的研究．北京：教育科学出版社，2001
[7] 黄璋．应用题一题多解．上海：同济大学出版社出版，1985
[8] 何小亚．与新课程同行——数学学与教的心理学．华南理工大学出版社，2003
[9] B．S．布鲁梅，J．W．威尔逊．中学数学学习评价．上海：华东师范大学出版社，1989
[10] Carolyn Kieran．学校代数的学习和教学．数学教与学研究手册，422-481．上海：上海教育出版社，1999
[11] 史宁中，孔凡哲．方程思想及其课程教学设计．课程·教材·教法，2004年第9期
[12] 覃涛．培养中学生方程思想的意义和途径．华中师范大学，2005
[13] 李汇云．试谈数学中的方程思想．数学教学通讯，2001年第3期(总第136期)
[14] 张志存．例谈方程思想在解题中的应用．中国教育与教学．2006年10月，第4卷第8期
[15] 邱卫平．数学教学中的认知情景设定——关于观摩“曰历中的方程”一课的相关思考．深圳外国语学校校刊《阳光深外》，2005年第2期
[16] Mollie Mac Gregor, kaye Stacey. Cognitive Models Underlying students' Formulation of Simple linear Equations. Jounial for Research in Mathematics Education, 24: 217-232. 1993
[2] 布鲁纳教育过程．上海：上海人民出版社，1973
[3] 皮连生．教育心理学．上海：上海教育出版社，2007
[4] 李士锜．PME：数学教育心理．上海：华东师范大学出版社，2001
[5] 林永伟．叶立军．数学史与数学教育．浙江：浙江大学出版社，2004
[6] 陈向明．教师如何做质的研究．北京：教育科学出版社，2001
[7] 黄璋．应用题一题多解．上海：同济大学出版社出版，1985
[8] 何小亚．与新课程同行——数学学与教的心理学．华南理工大学出版社，2003
[9] B．S．布鲁梅，J．W．威尔逊．中学数学学习评价．上海：华东师范大学出版社，1989
[10] Carolyn Kieran．学校代数的学习和教学．数学教与学研究手册，422-481．上海：上海教育出版社，1999
[11] 史宁中，孔凡哲．方程思想及其课程教学设计．课程·教材·教法，2004年第9期
[12] 覃涛．培养中学生方程思想的意义和途径．华中师范大学，2005
[13] 李汇云．试谈数学中的方程思想．数学教学通讯，2001年第3期(总第136期)
[14] 张志存．例谈方程思想在解题中的应用．中国教育与教学．2006年10月，第4卷第8期
[15] 邱卫平．数学教学中的认知情景设定——关于观摩“曰历中的方程”一课的相关思考．深圳外国语学校校刊《阳光深外》，2005年第2期
[16] Mollie Mac Gregor, kaye Stacey. Cognitive Models Underlying students' Formulation of Simple linear Equations. Jounial for Research in Mathematics Education, 24: 217-232. 1993