高中学生对导数概念的理解研究
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摘要
数学理解已经成为继“问题解决”之后国际数学教育界关心的又一中心话题,是数学教育研究与实践的焦点。导数在世界各地都已成为高中课程的教学内容,大多数国家都将其作为高中选修课程,这也是世界性的方向。
本文通过问卷调查和访谈等方法,研究以下几个问题:(1)高三数学“导数”学完后,学生是如何理解导数概念的?(2)学生导数概念理解中常见问题有哪些?(3)学生导数概念学习困难的主要原因。
本研究结果表明:学生的导数概念表象主要是曲线切线的斜率,物理运动方程中的瞬时速度。很少有学生的导数概念表象是标准的瞬时变化率。学生对于导数的理解是机械的静态的,不能从动态的无限趋近的角度去理解。不能在不同的情境中对导数赋予正确的意义。学生的协变推理能力整体来看水平不高。高中学生对函数概念理解的常见问题有:
1.概念意象的片面或错误;
2.对变化率、平均变化率、导数三个概念没有清晰的认识,不清楚三者之间的关系;
3.对导数的理解受到情境的影响显著;
4.对导数定义式的内涵没有清晰的理解;
5.对导数概念的理解停留在静态上和常量数学阶段。
Mathematical understanding has become another central topic the mathematical educators concerned about after "problem solving" in the world, and it’s also become the concern that international education research and practice is focused on. As a field of study and practice direction, mathematical understanding of learning and teaching has recently attracted public attention in the field of education ,and also has achieved much results. Derivatives has become the courses of high school around the world.
This paper studies the following questions through questionnaire survey and interview:
(1) After“derivative”study in high school, how about the students’understanding of the derivative concept?
(2) What kind of questions they will meet about derivative concept?
(3)What’s the main cause on learning the derivative concept? These findings indicated that: student's derivative concept representation is mainly the curve tangent slope and the instantaneous speed in physics equation of motion. The students’understanding of the derivative is just static, not dynamic. They cannot entrust with the correct significance in the different situation to the derivative. Overall, the students' covariational reasoning faculty level is not high.
The high school students’questions about understanding of the function concept are following:
1.The concept image wrong;
2. They have not clear understanding about the rate of change, average rate of change and derivative;
3. The influence on the understanding receives the situation to the derivative are remarkable;
4. They have no clear understanding on derivative definition-like connotation;
5. They just stay on the stage about the derivative concept's understanding in static and the constant mathematics.
本文通过问卷调查和访谈等方法,研究以下几个问题:(1)高三数学“导数”学完后,学生是如何理解导数概念的?(2)学生导数概念理解中常见问题有哪些?(3)学生导数概念学习困难的主要原因。
本研究结果表明:学生的导数概念表象主要是曲线切线的斜率,物理运动方程中的瞬时速度。很少有学生的导数概念表象是标准的瞬时变化率。学生对于导数的理解是机械的静态的,不能从动态的无限趋近的角度去理解。不能在不同的情境中对导数赋予正确的意义。学生的协变推理能力整体来看水平不高。高中学生对函数概念理解的常见问题有:
1.概念意象的片面或错误;
2.对变化率、平均变化率、导数三个概念没有清晰的认识,不清楚三者之间的关系;
3.对导数的理解受到情境的影响显著;
4.对导数定义式的内涵没有清晰的理解;
5.对导数概念的理解停留在静态上和常量数学阶段。
Mathematical understanding has become another central topic the mathematical educators concerned about after "problem solving" in the world, and it’s also become the concern that international education research and practice is focused on. As a field of study and practice direction, mathematical understanding of learning and teaching has recently attracted public attention in the field of education ,and also has achieved much results. Derivatives has become the courses of high school around the world.
This paper studies the following questions through questionnaire survey and interview:
(1) After“derivative”study in high school, how about the students’understanding of the derivative concept?
(2) What kind of questions they will meet about derivative concept?
(3)What’s the main cause on learning the derivative concept? These findings indicated that: student's derivative concept representation is mainly the curve tangent slope and the instantaneous speed in physics equation of motion. The students’understanding of the derivative is just static, not dynamic. They cannot entrust with the correct significance in the different situation to the derivative. Overall, the students' covariational reasoning faculty level is not high.
The high school students’questions about understanding of the function concept are following:
1.The concept image wrong;
2. They have not clear understanding about the rate of change, average rate of change and derivative;
3. The influence on the understanding receives the situation to the derivative are remarkable;
4. They have no clear understanding on derivative definition-like connotation;
5. They just stay on the stage about the derivative concept's understanding in static and the constant mathematics.
引文
[1]陈宁.找准在中学数学作为选修课的导数教学的切入点[J].中学数学教学参考,2005,(5):9-10.
[2]唐艳.基于APOS理论的数学概念教学设计[J].上海中学数学,2005,(12):22-24.
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[4]张文辉,王光明.数学认知理解的研究综述[J].曲阜师范大学学报,2005,31(1):120-123.
[5]马复.试论数学理解的两种类型—从R.斯根普的工作谈起[J].数学教育学报.2002,11(1):21-23.
[6]王爱珍.数学理解及理解障碍的探究[J].广东教育学院学报,2004,24(2):27-30.
[7]陈琼,翁凯庆.试论数学学习中的数学理解学习[J].数学教育报,2003,12(1):6-11.
[8]王秀明,王家铧,李忠海.寓“理解”与数学概念教学[J].数学教育学报,2005,14(2):26-28.
[9]李善良.关于数学概念意象的研究[J]. 数学教育学报,2004,13(2):13-15.
[10]李善良.数学概念学习研究综述[J]. 数学教育学报,2002,11(3):6-11.
[11]张耀. 数学概念教学研究综述[J].运城学院学报,2005,23(2):39-41.
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[13]马复.设计合理的数学教学[M].北京:高等教育出版社,2003.
[14]施良方.学习论[M]. 北京:人民教育出版社,2000.
[15]谢竺.浅谈导数概念的教学[J].乌鲁木齐成人教育学院学报,2006,14(2).
[16]李莉,谢琳.关于学习极限概念认知障碍的研究与分析[J]. 数学教育学报,2006,15(1):42-44.
[17]李延林,汪香志.普通高中课程标准实验教科书数学(选修1-1) [M].北京:北京师范大学出版社,2006.
[18]钱珮玲. 普通高中课程标准实验教科书数学(选修2-2)[M].北京: 人民教育出版社,2005.
[19] 濮 安 山 , 史 宁 中 . 从 APOS 理 论 看 高 中 生 对 函 数 概 念 的 理 解 [J]. 数 学 教 育 学报,2007,16(2):48-50.
[20]黄燕玲,喻平.对数学理解的再认识[J].数学教育学报,2002,11(3):40-43.
[21]张云标.谈谈导数的教学[J].数学教学研究,2006,(3):12-13.
[22]吴涛.关于导数教学的建议[J].中学数学月刊,2004,(2):13-14.
[23] 田德宇.浅谈导数概念的教学[J].中国教育导刊,2005,(1),55-56.
[24] 李善良.数学概念学习中的错误分析[J]. 数学教育学报,2002,11(3):6-11.
[25]乔连生.APOS:一种建构主义的数学学习理论[J].全球教育展望,2001,(3):16-18.
[26] Baron,E.& Avital,S1 986.Rate of change over interval as a property of functions:all algebraic and numerical approach . International Journal of Mathematics Education in Science&Technology,17(1):7 1-77.
[27] Carlson,M.2002.Applying covariation reasoning while modeling dynamic events:a framework and a study.Journal for Research in Mathematics Education,33,5:351-377.
[28] Confrey,J.& Smith,E.1994.Exponential functions,rates of change,and themultiplicative unit.Educational Studies in Mathematics,26:135-164.
[29] Orton,A.1984.Understanding Rate of Change.Mathematics in School,1984(11).
[30] Davis,R.B.& Vinner,S.1986.The notion of limit:some seemingly unavoidable misconception stages.Journal of Mathematical Behavior,5:281-303.
[2]唐艳.基于APOS理论的数学概念教学设计[J].上海中学数学,2005,(12):22-24.
[3]中华人民共和国教育部.普通高中数学课程标准(实验)[M].北京:人民教育出版社,2003.
[4]张文辉,王光明.数学认知理解的研究综述[J].曲阜师范大学学报,2005,31(1):120-123.
[5]马复.试论数学理解的两种类型—从R.斯根普的工作谈起[J].数学教育学报.2002,11(1):21-23.
[6]王爱珍.数学理解及理解障碍的探究[J].广东教育学院学报,2004,24(2):27-30.
[7]陈琼,翁凯庆.试论数学学习中的数学理解学习[J].数学教育报,2003,12(1):6-11.
[8]王秀明,王家铧,李忠海.寓“理解”与数学概念教学[J].数学教育学报,2005,14(2):26-28.
[9]李善良.关于数学概念意象的研究[J]. 数学教育学报,2004,13(2):13-15.
[10]李善良.数学概念学习研究综述[J]. 数学教育学报,2002,11(3):6-11.
[11]张耀. 数学概念教学研究综述[J].运城学院学报,2005,23(2):39-41.
[12]王林全,刘美伦,张安庆.高中数学新课程实验与探索(下册) [M].北京:高等教育出版社,2004.
[13]马复.设计合理的数学教学[M].北京:高等教育出版社,2003.
[14]施良方.学习论[M]. 北京:人民教育出版社,2000.
[15]谢竺.浅谈导数概念的教学[J].乌鲁木齐成人教育学院学报,2006,14(2).
[16]李莉,谢琳.关于学习极限概念认知障碍的研究与分析[J]. 数学教育学报,2006,15(1):42-44.
[17]李延林,汪香志.普通高中课程标准实验教科书数学(选修1-1) [M].北京:北京师范大学出版社,2006.
[18]钱珮玲. 普通高中课程标准实验教科书数学(选修2-2)[M].北京: 人民教育出版社,2005.
[19] 濮 安 山 , 史 宁 中 . 从 APOS 理 论 看 高 中 生 对 函 数 概 念 的 理 解 [J]. 数 学 教 育 学报,2007,16(2):48-50.
[20]黄燕玲,喻平.对数学理解的再认识[J].数学教育学报,2002,11(3):40-43.
[21]张云标.谈谈导数的教学[J].数学教学研究,2006,(3):12-13.
[22]吴涛.关于导数教学的建议[J].中学数学月刊,2004,(2):13-14.
[23] 田德宇.浅谈导数概念的教学[J].中国教育导刊,2005,(1),55-56.
[24] 李善良.数学概念学习中的错误分析[J]. 数学教育学报,2002,11(3):6-11.
[25]乔连生.APOS:一种建构主义的数学学习理论[J].全球教育展望,2001,(3):16-18.
[26] Baron,E.& Avital,S1 986.Rate of change over interval as a property of functions:all algebraic and numerical approach . International Journal of Mathematics Education in Science&Technology,17(1):7 1-77.
[27] Carlson,M.2002.Applying covariation reasoning while modeling dynamic events:a framework and a study.Journal for Research in Mathematics Education,33,5:351-377.
[28] Confrey,J.& Smith,E.1994.Exponential functions,rates of change,and themultiplicative unit.Educational Studies in Mathematics,26:135-164.
[29] Orton,A.1984.Understanding Rate of Change.Mathematics in School,1984(11).
[30] Davis,R.B.& Vinner,S.1986.The notion of limit:some seemingly unavoidable misconception stages.Journal of Mathematical Behavior,5:281-303.